This paper is the supplement to the section 2 of the paper ” Floating bundles and their applications ” LaTeXMLCite .
Below we study some properties of category , connected with cobordism rings of FBSP .
In particular , we shall show that it is the tensor category .
In LaTeXMLCite the series LaTeXMLMath where LaTeXMLMath was defined .
Recall that it corresponds to the direct limit LaTeXMLMath of the maps LaTeXMLMath where LaTeXMLMath is the canonical FBSP over LaTeXMLMath ( LaTeXMLMath ) .
In LaTeXMLCite some properties of LaTeXMLMath were studied .
In particular , it was shown that LaTeXMLEquation where LaTeXMLMath is the counit of the Hopf algebra LaTeXMLMath and LaTeXMLMath is the formal group of geometric cobordisms .
Let LaTeXMLMath be the map LaTeXMLEquation .
The commutativity of the following diagram LaTeXMLEquation allows us to define on the algebra LaTeXMLMath the structure of LaTeXMLMath -module such that LaTeXMLMath acts as the multiplication by LaTeXMLMath Let us denote this LaTeXMLMath -module by LaTeXMLMath Let us consider LaTeXMLMath as a Hopf algebra .
Recall that LaTeXMLMath LaTeXMLMath is the module coalgebra over LaTeXMLMath i. e. LaTeXMLMath is the homomorphism of coalgebras .
Proof .
The proof follows from the following commutative diagram ( LaTeXMLMath ) : LaTeXMLEquation .
Let us consider the next commutative diagram ( LaTeXMLMath ) : LaTeXMLEquation where LaTeXMLMath is the FBSP over LaTeXMLMath induced by the map LaTeXMLMath ( the definition of LaTeXMLMath was given in LaTeXMLCite ) .
Clearly that the bundle LaTeXMLMath ( with fiber LaTeXMLMath ) is ( “ external “ ) Segre ’ s product of the canonical FBSP over LaTeXMLMath and LaTeXMLMath By definition , put LaTeXMLEquation .
LaTeXMLEquation We have the homomorphism of LaTeXMLMath -modules LaTeXMLEquation defined by the fiber map LaTeXMLMath ( recall that LaTeXMLMath is the comultiplication in the Hopf algebra LaTeXMLMath ) .
Clearly that the restriction LaTeXMLMath coincides with LaTeXMLMath Let LaTeXMLMath be a FBSP over a finite LaTeXMLMath -complex LaTeXMLMath with fiber LaTeXMLMath Recall ( LaTeXMLCite ) that if LaTeXMLMath and LaTeXMLMath are sufficiently large then there exist a classifying map LaTeXMLMath and the corresponding fiber map LaTeXMLEquation which are unique up to homotopy and up to fiber homotopy respectively .
Let LaTeXMLMath be Segre ’ s product of LaTeXMLMath with the trivial FBSP LaTeXMLMath Let us pass to the direct limit LaTeXMLEquation where LaTeXMLMath LaTeXMLMath as LaTeXMLMath The stable equivalence class of FBSP ( see LaTeXMLCite ) LaTeXMLMath may be unique restored by the direct limit LaTeXMLMath We have also a classifying map LaTeXMLMath and the corresponding fiber map LaTeXMLEquation .
Let us define the category LaTeXMLMath by the following way .
LaTeXMLMath is the class of direct limits LaTeXMLMath of FBSP over finite LaTeXMLMath -complexes LaTeXMLMath ( in other words , the class of stable equivalence classes of FBSP ) ; LaTeXMLMath is the set of fiber maps LaTeXMLEquation such that its restrictions to any fiber LaTeXMLMath are isomorphisms .
Applying the functor of unitary cobordisms LaTeXMLMath to an object LaTeXMLMath we get the LaTeXMLMath -module LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is a classifying map for LaTeXMLMath It is clear that LaTeXMLMath where LaTeXMLMath is the homomorphism , induced by an embedding of a point LaTeXMLMath In other words , for any object in the category LaTeXMLMath there exists the canonical morphism LaTeXMLMath Hence there exist the initial object LaTeXMLMath and the final object LaTeXMLMath in the category LaTeXMLMath Let ’ s consider a pair LaTeXMLMath where LaTeXMLMath Let ’ s define their “ tensor product “ as the object LaTeXMLMath ( recall that LaTeXMLMath is the comultiplication in the Hopf algebra LaTeXMLMath ) .
The category LaTeXMLMath is the tensor category with the just defined tensor product and the unit LaTeXMLMath Proof .
The proof is trivial .
For example , the associativity axiom follows from the identity LaTeXMLMath which follows from the next commutative diagram ( LaTeXMLMath ) : LaTeXMLEquation where LaTeXMLMath is external Segre ’ s product of the canonical FBSP over LaTeXMLMath and LaTeXMLMath ( it is the bundle over LaTeXMLMath with fiber LaTeXMLMath ) .
LaTeXMLMath Note that there exist the canonical homomorphisms LaTeXMLMath : LaTeXMLEquation such that LaTeXMLMath
We give an interpretation of the q-Catalan numbers in frameworks of the random matrix theory and weighted partitions of the set of integers .
Key words : random matrices , Catalan numbers , non-commutative random variables AMS subject classification Primary : 15A52 Secondary : 05A18 In a joint discussion LaTeXMLCite , Christian Mazza asked , what one can obtain when regarding the weighted pairings of LaTeXMLMath points under the condition that they are non-crossing ?
In the present note we give one possible answer to this question .
Let us consider a set of LaTeXMLMath -dimensional random matrices LaTeXMLMath determined on the same probability space .
We assume that these matrices are real symmetric and LaTeXMLEquation where LaTeXMLMath is the family of Gaussian random variables with zero mathematical expectation and the covariance matrix LaTeXMLEquation .
Relations ( 1 ) mean that the probability distribution of the matrix LaTeXMLMath with given LaTeXMLMath is equal to that of GOE LaTeXMLCite .
Certainly we assume LaTeXMLMath to be real , even , and positively defined function .
In this note , our main subject is the product LaTeXMLEquation and we will be related mainly with the simplest case when LaTeXMLEquation .
If LaTeXMLMath , we obtain the product of real gaussian random variables .
In this case we can use the integration by parts formula for a Gaussian random vector LaTeXMLMath with zero mean value : LaTeXMLEquation where LaTeXMLMath is a nonranom function .
Then the mathematical expectation LaTeXMLEquation is given by the sum of weighted partitions of the set LaTeXMLMath of LaTeXMLMath labelled points into LaTeXMLMath pairs .
Here LaTeXMLMath denotes one particular partition , LaTeXMLMath stands for the set of all possible partitions , and LaTeXMLMath means that given a partition LaTeXMLMath , the product is taken over pairs that form LaTeXMLMath and whose elements are denoted by LaTeXMLMath .
Certainly , if LaTeXMLMath , then LaTeXMLEquation .
It is interesting to study ( 4 ) in the limit LaTeXMLMath , namely , the asymptotic behaviour of the value LaTeXMLEquation in the dependence of the parameter LaTeXMLMath .
It is proved that there exists critical value LaTeXMLMath such that ( 5 ) is positive for LaTeXMLMath and negative for LaTeXMLMath LaTeXMLCite .
Now let us turn to the case of non-commutative random variables , namely , to the limit of LaTeXMLMath .
Theorem 1.1 The average value of LaTeXMLMath LaTeXMLEquation converges as LaTeXMLMath in average LaTeXMLEquation the limit is given by equality LaTeXMLEquation where LaTeXMLMath are determined by recurrent relations LaTeXMLEquation with initial conditions LaTeXMLMath .
The numbers LaTeXMLMath , where LaTeXMLMath , are known as the q-Catalan numbers LaTeXMLCite .
To discuss this result , let us note that the numbers LaTeXMLMath are represented by the sum over all possible partitions LaTeXMLMath of the points LaTeXMLMath to the set of LaTeXMLMath non-crossing pairs ; each partition is weighted by powers of LaTeXMLMath ( cf .
( 4 ) ) LaTeXMLEquation such that LaTeXMLEquation .
Each non-crossing partition into pairs LaTeXMLMath can be identified with one of the half-plane rooted trees LaTeXMLMath of LaTeXMLMath edges LaTeXMLCite .
Thus LaTeXMLMath represents the total number of the elements LaTeXMLMath .
Regarding definition ( 8 ) , one can easily deduce that LaTeXMLMath satisfy the following recurrent relations LaTeXMLEquation with the conditions LaTeXMLMath .
Indeed , following the reasoning by Wigner LaTeXMLCite , let us consider the subsum of ( 8 ) over those partitions , where the first point LaTeXMLMath is paired with the last point LaTeXMLMath ; we denote by LaTeXMLMath the corresponding weighted sum .
Then it is easy to observe that LaTeXMLEquation .
To complete the derivation of ( 9 ) , it remains to consider the sum over partitions where LaTeXMLMath is paired with LaTeXMLMath .
Relations ( 8 ) and ( 9 ) answer the question of C. Mazza LaTeXMLCite and theorem 1.1 establishes relations of ( 8 ) with the product of random matrices of infinite dimensions .
In this connection , let us make one more remark about relations of our results with the non-commutative probability theory LaTeXMLCite .
In frameworks of this approach , the set of random matrices LaTeXMLMath with LaTeXMLMath represents in the limit LaTeXMLMath a family of free random variables LaTeXMLMath with respect to the mathematical expectation LaTeXMLMath ( 6 ) .
Free random variables represent a non-commutative analogue of jointly independent scalar random variables .
In particular , according to the rules adopted to compute the moments of free random variables ( see e.g .
LaTeXMLCite ) , we recover the ( ordinary ) Catalan numbers LaTeXMLEquation known since the pioneering work by Wigner on the eigenvalue distribuion of large random matrices LaTeXMLCite .
In LaTeXMLCite one can find a detailed analysis of the multiplicative functions over the non-crossing partitions , where , in particular , several generalizations of the Catalan numbers appear .
However , the q-Catalan numbers are not present in LaTeXMLCite .
In this context , conditions ( 1 ) can be regarded as a starting point to define the non-commutative analogs LaTeXMLMath of the correlated scalar ( gaussian ) random variables .
The standard rules to compute the average value LaTeXMLMath could be added there by conditions LaTeXMLEquation .
Returning to the generalization of the Catalan numbers ( 8 ) , let us note that as well as for the scalar case ( 5 ) , there should be a critical value LaTeXMLMath in the sense that the limit LaTeXMLMath of ( 5 ) exhibits different behaviour as LaTeXMLMath in dependence whether LaTeXMLMath or LaTeXMLMath .
This can be explained by the observation that the trees of LaTeXMLMath that have a vertex of large degree are relatively rare ( see e.g .
LaTeXMLCite ) .
However , the weight ( 8 ) with LaTeXMLMath ascribes to such trees the probability greater than , say , to binary trees .
This makes the subject of weighted non-crossing pairings reach and interesting .
Proof of Theorem 1.1 .
Our goal is to derive recurrent relation ( 9 ) .
We rewrite ( 6 ) in the form LaTeXMLEquation and compute the mathematical expectation with the help of ( 3 ) and then ( 1 ) : LaTeXMLEquation .
LaTeXMLEquation In this relations we mean that for LaTeXMLMath and LaTeXMLMath the expression in curly brackets takes the forms LaTeXMLMath and LaTeXMLMath , respectively .
Then we obtain relation LaTeXMLEquation .
LaTeXMLEquation To complete the proof of Theorem 1.1 , it remains to prove the following three items : 1 ) the moments of the normalized traces factorize LaTeXMLEquation in the limit LaTeXMLMath ; 2 ) the odd moments are zero LaTeXMLEquation and the average is invariant with respect to simultaneous shifts of all values of the subscripts LaTeXMLEquation and finally 3 ) the value of LaTeXMLEquation remains bounded as LaTeXMLMath .
With these items in mind , we can easily derive from ( 10 ) that ( 7 ) takes place and LaTeXMLMath satisfy ( 9 ) .
Relations ( 12 ) and ( 13 ) trivially follow from the definitions .
Regarding ( 14 ) , we observe that it is equal to LaTeXMLEquation where the number of factors LaTeXMLMath is LaTeXMLMath .
Now the question of its behavior as LaTeXMLMath is reduced to the problem of computing the expected value of ( 6 ) where LaTeXMLMath factors are subjected to certain permutation .
Thus , ( 14 ) can be estimated in terms of LaTeXMLMath that is sufficient for us .
Let us describe the scheme of the proof of ( 11 ) based on the following standard procedure ( see e.g .
LaTeXMLCite ) .
Let us denote LaTeXMLMath .
Then we can write relations LaTeXMLEquation .
LaTeXMLEquation To compute the last mathematical expectation , we use again the identity ( 3 ) with LaTeXMLMath .
Repeating the same computations as above , we derive with the help of ( 1 ) equality LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This somewhat cumbersome relation has rather simple structure .
Indeed , using identity LaTeXMLEquation we observe that LaTeXMLMath is expressed as the sum of terms of the following forms : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Also we see that in ( 15 ) the total number of factors LaTeXMLMath is decreased by 2 .
Thus , we conclude that repeating this procedure with respect to all terms LaTeXMLMath in ( 15 ) , we obtain the finite number ( depending on LaTeXMLMath ) of terms that involve products of LaTeXMLMath and LaTeXMLMath .
The only expressions that has no factor LaTeXMLMath are LaTeXMLEquation .
This averaged product can be treated as before with the help of ( 3 ) .
We obtain relations that express LaTeXMLMath in terms of the sum of LaTeXMLMath and terms that have factors LaTeXMLMath .
Taking into account that LaTeXMLMath is finite , we arrive at ( 11 ) .
Theorem 1.1 is proved .
LaTeXMLMath
We show how to obtain the mixture connection in an infinite dimensional information manifold and prove that it is dual to the exponential connection with respect to the Fisher information .
We also define the LaTeXMLMath -connections and prove that they are convex mixtures of the extremal LaTeXMLMath -connections .
The search for a fully-fledged infinite dimensional version of Information Geometry is an ongoing enterprise .
The original ideas date back to the work of Amari LaTeXMLCite , Efron LaTeXMLCite and most notably Dawid LaTeXMLCite , in which it was suggested using the natural exponential structure of the space of probability measures to obtain a manifold structure .
The first mathematically rigorous construction of such a manifold is , however , due to Pistone and Sempi LaTeXMLCite , using the theory of Orlicz spaces as their most important analytical tool .
In their seminal paper , the notion of exponential convergence is presented and an atlas consisting of maps with values in Orlicz spaces is proposed to cover the set of all probability measures equivalent to a given one .
They then prove that the induced topology obtained from this atlas is equivalent to the one with exponential convergence and use this fact to prove that the atlas satisfies the conditions necessary in order to give rise to a LaTeXMLMath Banach manifold .
The Banach spaces used are the so called exponential Orlicz spaces , denoted by LaTeXMLMath .
We review their construction in section 3 , with the difference that we use the Banach space LaTeXMLMath , the completion of the bounded random variables in the norm of LaTeXMLMath .
We also present a direct proof that the construction yields a LaTeXMLMath Banach manifold without using the concept of exponential convergence .
Having defined the information manifold , the next step in the programme is to define the analogues of a metric and dual connections , ideas that play a leading role in parametric information geometry .
A proposal for exponential and mixture , as well as for the intermediate LaTeXMLMath -connection , has been advocated by Gibilisco and Pistone LaTeXMLCite .
However , we argue that their elegant definition does not properly generalise the original ideas from the parametric case .
Their connections each act on a different vector bundle instead of all acting on the tangent bundle as in the finite dimensional case .
The duality observed between them does not involve any metric , while in parametric information geometry dual connections with respect to one metric can fail to be dual with respect to an arbitrarily different metric .
We present in section 4 our proposal for the infinite dimensional exponential and mixture connections , together with the appropriate concept of duality , as well as the generalised metric that makes them dual to each other .
In section 5 we also show that these definitions reduce to the familiar ones for finite dimensional submanifolds and that exponential and mixture families are geodesic for the exponential and mixture connections respectively .
We then move to the subject of LaTeXMLMath -connections in section 6 , where we again rearrange the definitions of LaTeXMLCite in order to have them all acting on the same bundle and with the desired relation between them , the exponential and the mixture connections still holding .
We present here the aspects of the theory of Orlicz spaces that will be relevant for the construction of the information manifold .
Similarly oriented short introductions to the subject can be found in LaTeXMLCite .
For more comprehensive accounts the reader is refered to the monographs LaTeXMLCite and LaTeXMLCite .
Recall first that a Young function is a convex function LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Note that , in this generality , LaTeXMLMath can vanish on an interval around the origin ( as opposed to vanishing iff LaTeXMLMath ) and it can also happen that LaTeXMLMath , for LaTeXMLMath , although it must be continuous where it is finite ( due to convexity ) .
In the absence of these annoyances , most of the theorems have stronger conclusions .
This will be the case for the following three Young functions used in information geometry : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation ( in the sequel , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath will always refer to these particular functions , with other symbols being used to denote generic Young functions ) .
Any Young function LaTeXMLMath ( including those with a jump to infinity ) admits an integral representation LaTeXMLEquation where LaTeXMLMath is nondecreasing , left continuous , LaTeXMLMath and LaTeXMLMath for LaTeXMLMath if LaTeXMLMath .
We define the complementary ( conjugate ) function to LaTeXMLMath as the Young function LaTeXMLMath given by LaTeXMLEquation where LaTeXMLMath is the generalised inverse of LaTeXMLMath , that is LaTeXMLEquation .
One can verify that LaTeXMLMath and LaTeXMLMath are a complementary pair .
Two Young functions LaTeXMLMath and LaTeXMLMath are said to be equivalent if there exist real numbers LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
For example , the functions LaTeXMLMath and LaTeXMLMath are equivalent .
There are several classifications of Young functions according to their growth properties .
The only one we are going to need for the construction of the information manifold is the so called LaTeXMLMath -class .
A Young function LaTeXMLMath satisfies the LaTeXMLMath -condition if LaTeXMLEquation for some constant LaTeXMLMath .
Examples of functions in this class are LaTeXMLMath and the function LaTeXMLMath .
Now let LaTeXMLMath be a measure space .
The theory of Orlicz spaces can be developed using a general measure LaTeXMLMath .
However , in several important theorems , to get necessary and sufficient conditions , instead of just sufficient ones , one needs to impose a couple of technical restrictions on the measure .
In this paper , we are going to assume without further mention that all our measures have the finite subset property and are diffuse on a set of positive measure LaTeXMLCite .
The reader must be aware that some of the results we are going to state do not hold if these conditions are not assumed and is refered to LaTeXMLCite for the full version of the theorems when unrestricted measures are considered .
The finite subset condition only excludes pathological cases like LaTeXMLMath if LaTeXMLMath and LaTeXMLMath otherwise .
It is satisfied , for instance , by all LaTeXMLMath -finite measures .
We also mention that the Lebesgue measure on the Borel LaTeXMLMath -algebra of LaTeXMLMath is diffuse on a set of positive measure , as are many other measures likely to appear in applications of information geometry .
The Orlicz class associated with a Young function LaTeXMLMath is defined as LaTeXMLEquation .
It is a convex set .
However , it is a vector space if and only if the function LaTeXMLMath satisfies the LaTeXMLMath -condition .
The Orlicz space associated with a Young function LaTeXMLMath is defined as LaTeXMLEquation .
It is easy to prove that this is a vector space and that it coincides with LaTeXMLMath iff LaTeXMLMath satisfies the LaTeXMLMath -condition .
Moreover , if we identify functions which differ only on sets of measure zero , then LaTeXMLMath is a Banach space when furnished with the Luxembourg norm LaTeXMLEquation or with the equivalent Orlicz norm LaTeXMLEquation where LaTeXMLMath is the complementary Young function to LaTeXMLMath .
If two Young functions are equivalent , the Banach spaces associated with them coincide as sets and have equivalent norms .
For example , LaTeXMLMath .
A key ingredient in the analysis of Orlicz spaces is the generalised Hölder inequality .
If LaTeXMLMath and LaTeXMLMath are complementary Young functions , LaTeXMLMath , LaTeXMLMath , then LaTeXMLEquation .
It follows that LaTeXMLMath for any pair of complementary Young functions , the inclusion being strict in general .
Suppose now that the measure space is finite .
Then it is clear that LaTeXMLMath .
Let LaTeXMLMath denote the closure of LaTeXMLMath in the LaTeXMLMath -norm and define also LaTeXMLEquation .
In general , we have that LaTeXMLMath .
In the next lemma , we collect for later use the results for the case of a continuous Young function vanishing only at the origin .
We need the following definition first .
We say that LaTeXMLMath has an absolutely continuous norm if LaTeXMLMath for each sequence of measurable sets LaTeXMLMath .
In terms of the Orlicz norm , this is equivalent to the statement that for every LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLEquation provided LaTeXMLMath and LaTeXMLMath .
Suppose that LaTeXMLMath and let LaTeXMLMath be a complementary pair of Young functions , LaTeXMLMath continuous , LaTeXMLMath iff LaTeXMLMath .
Then : LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath iff f has an absolutely continuous norm .
Furthermore , LaTeXMLMath is separable iff LaTeXMLMath is separable .
If , moreover , LaTeXMLMath satisfies the LaTeXMLMath -condition , then LaTeXMLMath .
As consequences of this lemma , we obtain LaTeXMLMath and LaTeXMLMath .
Consider the set LaTeXMLMath of all the LaTeXMLMath -almost everywhere strictly positive probability densities relative to the measure LaTeXMLMath , that is , LaTeXMLEquation .
For each point LaTeXMLMath , let LaTeXMLMath be the exponential Orlicz space over the measure space LaTeXMLMath .
The measure LaTeXMLMath inherits all the good properties assumed for LaTeXMLMath ( finite subset property and diffusiveness ) in addition to being finite , so that all the statements from the last section hold for LaTeXMLMath .
Instead of using the whole of LaTeXMLMath as the model Banach space for the manifold to be constructed , we restrict ourselves to LaTeXMLMath and take its closed subspace of LaTeXMLMath -centred random variables LaTeXMLEquation as the coordinate Banach space .
For definiteness , we choose to work with the Orlicz norm LaTeXMLMath , although everything could be done with the equivalent Luxemburg norm LaTeXMLMath , and use the notation LaTeXMLMath when it is necessary to specify the base point LaTeXMLMath .
In probabilistic terms , the set LaTeXMLMath has the characterisation given in the following lemma , whose proof is a simple adaptation of the one given in LaTeXMLCite for the case of LaTeXMLMath .
LaTeXMLMath coincides with the set of random variables for which the moment generating function is finite for all LaTeXMLMath .
Proof : If LaTeXMLMath , then LaTeXMLEquation which implies LaTeXMLEquation .
Conversely , if LaTeXMLMath for all LaTeXMLMath , then both LaTeXMLMath and LaTeXMLMath are finite , so LaTeXMLMath for all LaTeXMLMath , which means that LaTeXMLMath .
In particular , the moment generating functional LaTeXMLMath ( otherwise known as the partition function ) is finite on the whole of LaTeXMLMath .
Let LaTeXMLMath be the open unit ball of LaTeXMLMath and consider the map LaTeXMLEquation .
LaTeXMLEquation Denote by LaTeXMLMath the image of LaTeXMLMath under LaTeXMLMath .
We verify that LaTeXMLMath is a bijection from LaTeXMLMath to LaTeXMLMath , since LaTeXMLEquation implies that LaTeXMLMath is a constant random variable .
But since LaTeXMLMath , we must have LaTeXMLMath .
Then let LaTeXMLMath be the inverse of LaTeXMLMath on LaTeXMLMath .
One can check that LaTeXMLEquation .
LaTeXMLEquation and also that , for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation Now suppose that LaTeXMLMath for some LaTeXMLMath .
Then we can write it as LaTeXMLEquation for some LaTeXMLMath .
Using the formula just obtained , we find LaTeXMLEquation .
Since LaTeXMLMath , we have that LaTeXMLEquation .
Consider an open ball of radius LaTeXMLMath around LaTeXMLMath in the topology of LaTeXMLMath , that is , consider the set LaTeXMLEquation and let LaTeXMLMath be small enough so that LaTeXMLMath .
Then the image in LaTeXMLMath of each point LaTeXMLMath under LaTeXMLMath is LaTeXMLEquation .
We claim that LaTeXMLMath .
Indeed , applying LaTeXMLMath to it we find LaTeXMLEquation so LaTeXMLEquation where LaTeXMLMath and we use the notation LaTeXMLMath for the LaTeXMLMath -norm .
It follows from the growth properties of LaTeXMLMath that there exists LaTeXMLMath such that LaTeXMLMath .
Moreover , it was found in LaTeXMLCite that LaTeXMLMath , so there exists a constant LaTeXMLMath such that LaTeXMLMath .
Therefore , the previous inequality becomes LaTeXMLEquation .
Thus , is we choose LaTeXMLEquation we will have that LaTeXMLEquation which proves the claim .
What we just have proved is that LaTeXMLMath consists entirely of interior points in the topology of LaTeXMLMath , so LaTeXMLMath is open in LaTeXMLMath .
We then have that the collection LaTeXMLMath satisfies the three axioms for being a LaTeXMLMath –atlas for LaTeXMLMath ( see LaTeXMLCite ) .
Moreover , since all the spaces LaTeXMLMath are toplinear isomorphic , we can say that LaTeXMLMath is a LaTeXMLMath –manifold modelled on LaTeXMLMath .
As usual , the tangent space at each point LaTeXMLMath can be abstractly identified with LaTeXMLMath .
A concrete realisation has been given in LaTeXMLCite , namely each curve through LaTeXMLMath is tangent to a one-dimensional exponential model LaTeXMLMath , so we take LaTeXMLMath as the tangent vector representing the equivalence class of such a curve .
Since we are using LaTeXMLMath instead of LaTeXMLMath to construct the manifold , we need the following corresponding definition for the maximal exponential model at each LaTeXMLMath : LaTeXMLEquation .
One can verify that LaTeXMLMath is the connected component of LaTeXMLMath containing LaTeXMLMath .
In the parametric version of information geometry , Amari and Nagaoka have introduced the concept of dual connections with respect to a Riemannian metric .
For finite dimensional manifolds , any continuous assignment of a positive definite symmetric bilinear form to each tangent space determines a Riemannian metric .
In infinite dimensions , we need to impose that the tangent space is self-dual and that the bilinear form is continuous .
Since our tangent spaces LaTeXMLMath are not even reflexive , let alone self-dual , we abandon the idea of having a Riemannian structure on LaTeXMLMath and propose a weaker version of duality , the duality with respect to a continuous scalar product .
When restricted to finite dimensional submanifolds , the scalar product becomes a Riemannian metric and the original definition of duality is recovered .
Let LaTeXMLMath be a continuous positive definite symmetric bilinear form assigned continuously to each LaTeXMLMath .
A pair of connection LaTeXMLMath are said to be dual with respect to LaTeXMLMath if LaTeXMLEquation for all LaTeXMLMath and all smooth curves LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath denote the parallel transports associated with LaTeXMLMath and LaTeXMLMath , respectively .
Equivalently , LaTeXMLMath are dual with respect to LaTeXMLMath if LaTeXMLEquation for all LaTeXMLMath and all smooth vector fields LaTeXMLMath and LaTeXMLMath .
We stress that this is not the kind of duality obtained when a conection LaTeXMLMath on a bundle LaTeXMLMath is used to construct another connection LaTeXMLMath on the dual bundle LaTeXMLMath as defined , for instance , in LaTeXMLCite .
The latter is a construction that does not involve any metric or scalar product and the two connections act on different bundles , while Amari ’ s duality is a duality with respect to a specific scalar product ( or metric , in the finite dimensional case ) and the dual connections act on the same bundle , the tangent bundle .
The infinite dimensional generalisation of the Fisher information is given by LaTeXMLEquation .
This is clearly bilinear , symmetric and positive definite .
Also , since LaTeXMLMath , the generalised Hölder inequality gives LaTeXMLEquation which implies the continuity of LaTeXMLMath .
The use of exponential Orlicz space to model the manifold induces naturally a globally flat affine connection on the tangent bundle LaTeXMLMath , called the exponential connection and denoted by LaTeXMLMath .
It is defined on each connected component of the manifold LaTeXMLMath , which is equivalent to saying that its parallel transport is defined between points connected by an exponential model LaTeXMLCite .
If LaTeXMLMath and LaTeXMLMath are two such points , then LaTeXMLMath and the exponential parallel transport is given by LaTeXMLEquation .
LaTeXMLEquation It is well defined , since LaTeXMLMath and LaTeXMLMath are subsets of the same set LaTeXMLMath , so the exponential parallel transport just subtracts a constant from LaTeXMLMath to make it centred around the right point .
We now want to define the dual connection to LaTeXMLMath with respect to the Fisher information .
We begin by proving the following lemma .
Let LaTeXMLMath and LaTeXMLMath be two points in the same connected component of LaTeXMLMath .
Then LaTeXMLMath , for all LaTeXMLMath .
Proof : From the hypothesis , LaTeXMLMath has absolutely continuous norm in LaTeXMLMath , so for for every LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath implies LaTeXMLEquation .
But since LaTeXMLMath , as they are the completion of the same set LaTeXMLMath under equivalent norms ( recall that LaTeXMLMath and LaTeXMLMath are supposed to be connected by an exponential model ) , we have that LaTeXMLMath , in the sense that they are the same set furnished with equivalent norms .
We then use ( LaTeXMLRef ) to conclude that LaTeXMLEquation where we used the facts that LaTeXMLMath iff LaTeXMLMath LaTeXMLCite and that there exists a constant LaTeXMLMath such that LaTeXMLMath .
Since LaTeXMLMath was arbitrary , this proves that LaTeXMLMath has absolutely continuous norm in LaTeXMLMath .
The lemma then follows from lemma LaTeXMLRef and the fact that LaTeXMLMath is centred around LaTeXMLMath .
We can then define the mixture connection on LaTeXMLMath , as LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath in the same connected component of LaTeXMLMath .
We notice that it is also globally flat and prove the following result .
The connections LaTeXMLMath and LaTeXMLMath are dual with respect to the Fisher information .
Proof : We have that LaTeXMLEquation where , to go from the second to the third line above , we used that LaTeXMLMath is centred around LaTeXMLMath .
We begin this section recalling that the covariant derivative for the exponential connection has been computed in LaTeXMLCite and found to be LaTeXMLEquation where LaTeXMLMath is a differentiable vector field , LaTeXMLMath is a tangent vector at LaTeXMLMath , LaTeXMLMath denotes the expected value with respect to the measure LaTeXMLMath and LaTeXMLMath denotes the directional derivative in LaTeXMLMath of LaTeXMLMath composed with some patch LaTeXMLMath as a map between Banach spaces .
We first notice that this gives the usual covariant derivative for the exponential connection in parametric information geometry LaTeXMLCite .
For if LaTeXMLMath is a coordinate system in a finite dimensional submanifold of LaTeXMLMath we can put LaTeXMLMath and LaTeXMLMath as the LaTeXMLMath -representation ( see LaTeXMLCite ) of the vector field LaTeXMLMath , then ( LaTeXMLRef ) reduces to LaTeXMLEquation which is the classical finite dimensional result .
Accordingly , parametric exponential models are flat submanifolds of LaTeXMLMath .
We can also verify that one-dimensional exponential models of the form LaTeXMLEquation are geodesics for LaTeXMLMath , since if LaTeXMLMath is the vector field tangent to LaTeXMLMath at each point LaTeXMLMath LaTeXMLCite , then ( LaTeXMLRef ) gives LaTeXMLEquation .
As we emphasised in the previous section , the definition given in LaTeXMLCite for the mixture connection differs from ours ( due to the different concepts of duality employed ) , so we have to compute its covariant derivative according to the definition given here , at least to have the notation right .
Let LaTeXMLMath be a smooth curve such that LaTeXMLMath and LaTeXMLMath and let LaTeXMLMath be a differentiable vector field .
Then LaTeXMLEquation where LaTeXMLMath .
Proof : LaTeXMLEquation .
Again this reduces to the parametric result for the case of submanifolds of LaTeXMLMath .
Put LaTeXMLMath and LaTeXMLMath in ( LaTeXMLRef ) to obtain LaTeXMLEquation which is the classical finite dimensional result .
The mixture connection owes its name to the fact that in the parametric version of information geometry a convex mixture of two densities describes a geodesic with respect to LaTeXMLMath .
To verify the same statement in the non-parametric case , we first need to check that a convex mixture of two points in a connected component of LaTeXMLMath remains in the same connected component .
If LaTeXMLMath and LaTeXMLMath are two points in LaTeXMLMath for some LaTeXMLMath , then LaTeXMLEquation belongs to LaTeXMLMath for all LaTeXMLMath .
Proof : We begin by writing LaTeXMLEquation for some LaTeXMLMath .
To simplify the notation , let us define LaTeXMLEquation .
We want to show that , if we write LaTeXMLEquation then LaTeXMLMath is an element of LaTeXMLMath , so that LaTeXMLEquation and LaTeXMLEquation .
All we need to prove is that both LaTeXMLMath and LaTeXMLMath are finite for all LaTeXMLMath .
We have that LaTeXMLEquation which implies LaTeXMLEquation .
Thus LaTeXMLEquation since both LaTeXMLMath and LaTeXMLMath are in LaTeXMLMath .
As for the other integral , observe that LaTeXMLEquation .
Therefore LaTeXMLEquation since LaTeXMLMath .
We can now verify that a family of the form LaTeXMLEquation is a geodesic for LaTeXMLMath .
Let LaTeXMLMath be the vector field tangent to LaTeXMLMath at each point LaTeXMLMath , then ( LaTeXMLRef ) gives LaTeXMLEquation .
In this section , we define the infinite dimensional analogue of the LaTeXMLMath -connections introduced in the parametric case independently by Chentsov LaTeXMLCite and Amari LaTeXMLCite .
We use the same technique proposed by Gibilisco and Pistone LaTeXMLCite , namely exploring the geometry of spheres in the Lebesgue spaces LaTeXMLMath , but modified in such a way that the resulting connections all act on the tangent bundle LaTeXMLMath .
We begin with Amari ’ s LaTeXMLMath -embeddings LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Observe that LaTeXMLEquation so LaTeXMLMath , the sphere of radius LaTeXMLMath in LaTeXMLMath ( we warn the reader that throughout this paper , the LaTeXMLMath in LaTeXMLMath refers to the fact that this is a sphere of radius LaTeXMLMath , while the fact that it is a subset of LaTeXMLMath is judiciously omitted from the notation ) .
According to Gibilisco and Pistone LaTeXMLCite , the tangent space to LaTeXMLMath at a point LaTeXMLMath is LaTeXMLEquation where LaTeXMLMath .
In our case , LaTeXMLEquation so that LaTeXMLEquation .
Therefore , the tangent space to LaTeXMLMath at LaTeXMLMath is LaTeXMLEquation .
We now look for a concrete realisation of the push-forward of the map LaTeXMLMath when the tangent space LaTeXMLMath is identified with LaTeXMLMath as in the previous sections .
Since LaTeXMLEquation the push-forward can be formally implemented as LaTeXMLEquation .
LaTeXMLEquation For this to be well defined , we need to check that LaTeXMLMath is an element of LaTeXMLMath .
Indeed , since LaTeXMLMath for all LaTeXMLMath , we have that LaTeXMLEquation so LaTeXMLMath .
Moreover LaTeXMLEquation which verifies that LaTeXMLMath .
The sphere LaTeXMLMath inherits a natural connection obtained by projecting the trivial connection on LaTeXMLMath ( the one where parallel transport is just the identity map ) onto its tangent space at each point .
For each LaTeXMLMath , a canonical projection from the tangent space LaTeXMLMath onto the tangent space LaTeXMLMath can be uniquely defined , since the spaces LaTeXMLMath are uniformly convex LaTeXMLCite , and is given by LaTeXMLEquation .
LaTeXMLEquation When LaTeXMLMath and LaTeXMLMath , the formula above gives LaTeXMLEquation .
LaTeXMLEquation We are now ready to define the LaTeXMLMath -connections .
In what follows , LaTeXMLMath is used to denote the trivial connection on LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath be a smooth curve such that LaTeXMLMath and LaTeXMLMath and let LaTeXMLMath be a differentiable vector field .
The LaTeXMLMath -connection on LaTeXMLMath is given by LaTeXMLEquation .
A formula like ( LaTeXMLRef ) deserves a more wordy explanation .
We take the vector field LaTeXMLMath and push it forward along the curve LaTeXMLMath to obtain LaTeXMLMath .
Then we take its covariant derivative with respect to the trival connection LaTeXMLMath in the direction of LaTeXMLMath , the push-forward of the tangent vector LaTeXMLMath .
The result is a vector in LaTeXMLMath , so we use the canonical projection LaTeXMLMath to obtain a vector in LaTeXMLMath .
Finally , we pull it back to LaTeXMLMath using LaTeXMLMath .
The next theorem shows that the relation between the exponential , the mixture and the LaTeXMLMath -connections just defined is the same as in the parametric case .
Its proof resembles the calculation in the last pages of LaTeXMLCite , except that all our connections act on the same bundle , whereas in LaTeXMLCite each one is defined on its own bundle-connection pair .
The exponential , mixture and LaTeXMLMath -connections on LaTeXMLMath satisfy LaTeXMLEquation .
Proof : Let LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath as in definition LaTeXMLRef .
Before explicitly computing the derivatives in ( LaTeXMLRef ) , observe that since LaTeXMLMath for each LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In particular , for LaTeXMLMath , we get LaTeXMLEquation .
We can now look more closely at ( LaTeXMLRef ) .
It reads LaTeXMLEquation .
At this point we make use of ( LaTeXMLRef ) in the integrand above to obtain LaTeXMLEquation .
A direct application of ( LaTeXMLRef ) gives the following .
The connections LaTeXMLMath and LaTeXMLMath are dual with respect to the Fisher information LaTeXMLMath .
In this paper , we give decompositions of the moonshine module LaTeXMLMath with respect to subVOAs associated to extremal Type II codes over LaTeXMLMath for an integer LaTeXMLMath .
Those subVOAs are isomorphic to the tensor product of 24 copies of the charge conjugation orbifold VOA .
Using such decompositions , we obtain some elements of type LaTeXMLMath ( LaTeXMLMath odd ) and LaTeXMLMath ( LaTeXMLMath even ) of the Monster simple group Aut LaTeXMLMath .
The notation of a vertex operator algebra ( VOA ) is introduced in LaTeXMLCite .
One of the most interesting examples of a VOA is the moonshine module LaTeXMLMath constructed in LaTeXMLCite .
The automorphism group of LaTeXMLMath is the Monster , the largest sporadic finite simple group .
The moonshine module LaTeXMLMath has many subalgebras having good symmetry ( LaTeXMLCite ) .
The decompositions of LaTeXMLMath with respect to those subalgebras are computed in LaTeXMLCite .
In particular , LaTeXMLMath contains a subVOA isomorphic to the tensor product of LaTeXMLMath copies of LaTeXMLMath , called a Virasoro frame in LaTeXMLCite , and the decomposition of LaTeXMLMath is computed .
Since LaTeXMLMath is isomorphic to LaTeXMLMath algebra at LaTeXMLMath , we should study the first member of the unitary series of LaTeXMLMath algebras .
For details of LaTeXMLMath algebras , see LaTeXMLCite and references given there .
In particular , LaTeXMLMath algebra at LaTeXMLMath is realized as the fixed-point subspace LaTeXMLMath of the lattice VOA LaTeXMLMath corresponding to the rank one lattice LaTeXMLMath with LaTeXMLMath with respect to the LaTeXMLMath automorphism of the lattice .
Its fusion rules have a nice symmetry according to LaTeXMLCite and for each embedding of LaTeXMLMath into LaTeXMLMath , we get a LaTeXMLMath element of the Monster LaTeXMLCite .
In this paper , we consider more general case LaTeXMLMath with LaTeXMLMath for an integer LaTeXMLMath .
Since the Leech lattice LaTeXMLMath contains many elements of squared length LaTeXMLMath and LaTeXMLMath contains LaTeXMLMath as a subVOA , LaTeXMLMath contains many copies of LaTeXMLMath .
We consider a set of LaTeXMLMath mutually orthogonal pairs of opposite vectors of LaTeXMLMath with squared length LaTeXMLMath .
We call such a set a LaTeXMLMath -frame of LaTeXMLMath .
The existence of LaTeXMLMath -frame of LaTeXMLMath is shown in LaTeXMLCite .
By using a LaTeXMLMath -frame , we show that LaTeXMLMath contains a subVOA isomorphic to the tensor product of LaTeXMLMath copies of LaTeXMLMath .
Then we give the decomposition of LaTeXMLMath as a LaTeXMLMath -module .
By using the fact that there exists the natural bijection between equivalence classes of LaTeXMLMath -frames and extremal Type II codes of length LaTeXMLMath over LaTeXMLMath , the decompositions of LaTeXMLMath are described in terms of such codes .
By the fusion rules of LaTeXMLMath , we obtain automorphisms of LaTeXMLMath with respect to the decompositions .
More precisely , we have LaTeXMLMath elements ( LaTeXMLMath is odd ) and LaTeXMLMath elements ( LaTeXMLMath is even ) of the Monster simple group .
Moreover , we give new expressions of McKay-Thompson series for LaTeXMLMath elements and obtain formulas of modular functions .
Professor Ching Hung Lam studies the subject independently .
Throughout the paper , we will work over the field LaTeXMLMath of complex numbers unless otherwise stated .
We denote the set of integers by LaTeXMLMath and the ring of the integers modulo LaTeXMLMath by LaTeXMLMath .
Acknowledgements .
The author wishes to thank Professor Atsushi Matsuo , my research supervisor , for his advice and warm encouragement .
He also thanks Professor Masaaki Kitazume and Professor Ching Hung Lam for helpful advice .
In this section , we recall or give some definitions and facts necessary in this paper .
In this section , we give the charge conjugation orbifold VOA LaTeXMLMath and its irreducible modules .
The description of LaTeXMLMath in this paper is slightly different from those given in LaTeXMLCite because it is useful to show the main theorem .
Let LaTeXMLMath be an even lattice of rank one with LaTeXMLMath .
Set LaTeXMLMath and we regard LaTeXMLMath as a subgroup of LaTeXMLMath .
We extend the form LaTeXMLMath to a LaTeXMLMath -bilinear form on LaTeXMLMath .
Let LaTeXMLMath be the dual lattice of LaTeXMLMath .
Let LaTeXMLMath be the trivial extension of LaTeXMLMath by the order LaTeXMLMath cyclic group LaTeXMLMath .
Form the induced LaTeXMLMath -module LaTeXMLMath , where LaTeXMLMath denotes the group algebra and LaTeXMLMath acts on LaTeXMLMath as multiplication by LaTeXMLMath .
We choose a section LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
Set LaTeXMLMath .
For LaTeXMLMath , we set LaTeXMLMath , where LaTeXMLMath is regarded as a subset of LaTeXMLMath and LaTeXMLMath is the subspace of LaTeXMLMath spanned by LaTeXMLMath .
It is well known that LaTeXMLMath has a VOA structure and LaTeXMLMath has a LaTeXMLMath -module structure for each LaTeXMLMath ( cf .
LaTeXMLCite ) .
Note that all the irreducible LaTeXMLMath -modules are given by the set LaTeXMLMath ( cf .
LaTeXMLCite ) .
For convenience , we denote LaTeXMLMath by LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath : LaTeXMLMath be a group homomorphism such that LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath be the automorphism of LaTeXMLMath given by LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the unique commutative algebra automorphism of LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a representative of a coset in LaTeXMLMath .
We will extend LaTeXMLMath on LaTeXMLMath as a LaTeXMLMath -module isomorphism .
Let LaTeXMLMath : LaTeXMLMath be the LaTeXMLMath -module isomorphism such that LaTeXMLMath .
When LaTeXMLMath , the linear map LaTeXMLMath , LaTeXMLMath is an isomorphism of LaTeXMLMath -modules .
Then we have the automorphism LaTeXMLMath of LaTeXMLMath as a LaTeXMLMath -module such that LaTeXMLMath .
For each LaTeXMLMath -stable subspace LaTeXMLMath of LaTeXMLMath , let LaTeXMLMath denote the LaTeXMLMath eigenspaces of LaTeXMLMath with respect to LaTeXMLMath respectively .
Note that LaTeXMLMath is a subVOA of LaTeXMLMath .
Let LaTeXMLMath be the subgroup of LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be irreducible LaTeXMLMath -modules over LaTeXMLMath such that LaTeXMLMath acts by LaTeXMLMath and LaTeXMLMath acts by LaTeXMLMath respectively .
Note that LaTeXMLMath acts by LaTeXMLMath on LaTeXMLMath .
Set LaTeXMLMath .
Then it has an irreducible LaTeXMLMath -twisted LaTeXMLMath module structure .
Moreover , LaTeXMLMath and LaTeXMLMath give all the inequivalent irreducible LaTeXMLMath -twisted LaTeXMLMath -modules ( cf .
LaTeXMLCite ) .
Let LaTeXMLMath be the unique commutative algebra automorphism of LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
We consider the linear automorphism of LaTeXMLMath given by LaTeXMLMath , where LaTeXMLMath is the identity operator of LaTeXMLMath .
By abuse of notation , we denote both automorphisms of LaTeXMLMath and LaTeXMLMath by LaTeXMLMath .
Since LaTeXMLMath on LaTeXMLMath , LaTeXMLMath is an automorphism as a LaTeXMLMath -module .
We denote the LaTeXMLMath eigenspaces of LaTeXMLMath with respect to LaTeXMLMath by LaTeXMLMath respectively .
Then LaTeXMLMath becomes an irreducible LaTeXMLMath -module .
Note that LaTeXMLMath and its irreducible modules defined above are isomorphic to those given in LaTeXMLCite .
By Theorem LaTeXMLMath of LaTeXMLCite , the set LaTeXMLEquation gives all inequivalent irreducible LaTeXMLMath -modules .
Let us review the moonshine module LaTeXMLMath from LaTeXMLCite for what we need in this paper .
Let LaTeXMLMath be the Leech lattice with the positive-definite LaTeXMLMath -bilinear form LaTeXMLMath .
It is a unique positive-definite even unimodular lattice of rank LaTeXMLMath without roots .
Let LaTeXMLMath be the central extension of LaTeXMLMath by the cyclic group LaTeXMLMath : LaTeXMLEquation with the commutator map LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath be the automorphism of LaTeXMLMath given by LaTeXMLMath for LaTeXMLMath .
We denote the center of LaTeXMLMath by LaTeXMLMath .
Set LaTeXMLMath .
Then LaTeXMLMath is a normal subgroup of LaTeXMLMath ; consider the group LaTeXMLMath .
Note that the center of LaTeXMLMath is LaTeXMLMath .
By Theorem LaTeXMLMath of LaTeXMLCite , we have the following proposition .
LaTeXMLMath LaTeXMLCite LaTeXMLMath Set LaTeXMLMath and let LaTeXMLMath be a character of LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath be a maximal abelian subgroup of LaTeXMLMath and let LaTeXMLMath : LaTeXMLMath be a character with LaTeXMLMath .
Then LaTeXMLMath is the unique irreducible LaTeXMLMath -module on which LaTeXMLMath acts by LaTeXMLMath , where LaTeXMLMath is the one-dimensional LaTeXMLMath -module corresponding to LaTeXMLMath .
Moreover , LaTeXMLMath , where LaTeXMLMath ranges over the characters of LaTeXMLMath whose restriction to LaTeXMLMath is LaTeXMLMath , and LaTeXMLMath .
Set the induced LaTeXMLMath -module LaTeXMLMath , where LaTeXMLMath acts on LaTeXMLMath as multiplication by LaTeXMLMath .
We extend LaTeXMLMath to LaTeXMLMath linearly .
Since LaTeXMLMath fixes LaTeXMLMath , we view the automorphism LaTeXMLMath as an automorphism of LaTeXMLMath .
Set LaTeXMLMath .
We regard LaTeXMLMath as a subgroup of LaTeXMLMath .
We extend LaTeXMLMath to a LaTeXMLMath -bilinear form on LaTeXMLMath .
Let LaTeXMLMath be the irreducible LaTeXMLMath -module given in Proposition LaTeXMLRef .
Set LaTeXMLMath and LaTeXMLMath .
For convenience , we also denote LaTeXMLMath by LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the unique commutative algebra automorphism of LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the unique commutative algebra automorphism of LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath is an automorphism of LaTeXMLMath , where LaTeXMLMath is the identity operator on LaTeXMLMath .
Thus we have LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath -fixed-point subspace of LaTeXMLMath and LaTeXMLMath is the LaTeXMLMath -fixed-point subspace of LaTeXMLMath .
In this subsection , we give some terminology on a code over LaTeXMLMath ( cf .
LaTeXMLCite ) .
A ( linear ) code LaTeXMLMath of length LaTeXMLMath over LaTeXMLMath is a LaTeXMLMath -submodule of LaTeXMLMath .
We denote the image of LaTeXMLMath with respect to the canonical map LaTeXMLMath by LaTeXMLMath .
We fix an ordered basis of LaTeXMLMath and denote LaTeXMLMath -th element of this basis by LaTeXMLMath , where LaTeXMLMath only appears in the LaTeXMLMath -th position .
An element of LaTeXMLMath is called a codeword .
The Euclidean weight Ewt LaTeXMLMath on LaTeXMLMath is given by LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
We define the inner product of LaTeXMLMath and LaTeXMLMath in LaTeXMLMath by LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
The dual code LaTeXMLMath of C is defined as LaTeXMLMath .
LaTeXMLMath is self-orthogonal if LaTeXMLMath and LaTeXMLMath is self-dual if LaTeXMLMath .
We define a Type II code over LaTeXMLMath to be a self-dual code with all codewords having Euclidean weight divisible by LaTeXMLMath .
It is well known that LaTeXMLMath , where LaTeXMLMath is the symmetric group .
LaTeXMLMath acts on LaTeXMLMath by the permutation of the coordinate positions and the change of the signs of some positions of LaTeXMLMath .
Two codes LaTeXMLMath and LaTeXMLMath over LaTeXMLMath are called equivalent if they both have length LaTeXMLMath and if there exists LaTeXMLMath such that LaTeXMLMath .
A LaTeXMLMath -frame of a lattice of rank LaTeXMLMath is a set of LaTeXMLMath mutually orthogonal pairs of opposite vectors of squared length LaTeXMLMath .
We denote the group of isometries of a lattice LaTeXMLMath by LaTeXMLMath .
Since LaTeXMLMath preserves the inner product , LaTeXMLMath acts on the set of LaTeXMLMath -frames of LaTeXMLMath .
We say that LaTeXMLMath -frames LaTeXMLMath , LaTeXMLMath of LaTeXMLMath are equivalent if there exists LaTeXMLMath such that LaTeXMLMath .
We define an extremal Type II code of length LaTeXMLMath over LaTeXMLMath to be a Type II code with minimum Euclidean weight LaTeXMLMath .
LaTeXMLCite For any positive integer LaTeXMLMath , there exists an extremal Type II code of length LaTeXMLMath over LaTeXMLMath .
Using the fact that the Leech lattice LaTeXMLMath is the unique positive definite unimodular lattice in dimension LaTeXMLMath without roots , it is easy to see that for LaTeXMLMath , equivalence classes of LaTeXMLMath -frames of LaTeXMLMath are the same as equivalence classes of extremal Type II codes of length LaTeXMLMath over LaTeXMLMath .
More precisely , for a LaTeXMLMath -frame LaTeXMLMath of LaTeXMLMath , we have an extremal Type II code LaTeXMLMath , where LaTeXMLMath is the sublattice of LaTeXMLMath generated by LaTeXMLMath and LaTeXMLMath is the dual lattice of LaTeXMLMath , and for an extremal Type II code LaTeXMLMath of length LaTeXMLMath over LaTeXMLMath , the lattice constructed by the generalized Construction A with LaTeXMLMath is the Leech lattice and contains a LaTeXMLMath -frame .
By direct calculation , we have the following proposition .
Let LaTeXMLMath be a LaTeXMLMath -frame of the Leech lattice LaTeXMLMath and let LaTeXMLMath be the sublattice of LaTeXMLMath generated by LaTeXMLMath .
Let LaTeXMLMath be a code over LaTeXMLMath and let LaTeXMLMath be a binary code .
Set LaTeXMLMath over LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath is an elementary abelian LaTeXMLMath -group with order LaTeXMLMath .
If LaTeXMLMath is odd , then LaTeXMLMath is a Type II code .
Let LaTeXMLMath be a LaTeXMLMath -frame of LaTeXMLMath and LaTeXMLMath be an extremal Type II code corresponding to LaTeXMLMath .
Let LaTeXMLMath be the sublattice of LaTeXMLMath generated by LaTeXMLMath .
Then it is easy to see that LaTeXMLMath , where we regard the both codes as binary codes .
In this section , for an integer LaTeXMLMath , we give the decomposition of LaTeXMLMath as a LaTeXMLMath -module associated with an extremal Type II code over LaTeXMLMath .
It is easy to see the embedding of LaTeXMLMath into LaTeXMLMath .
But the LaTeXMLMath tensor product of the involution of LaTeXMLMath given in LaTeXMLCite is not the same involution LaTeXMLMath of LaTeXMLMath .
By using the definition of LaTeXMLMath given in Section 1.1 , we will clarify the problem .
For convenience , we use the following notation .
For LaTeXMLMath , we set LaTeXMLEquation .
Note that LaTeXMLMath as a LaTeXMLMath -module for LaTeXMLMath , where LaTeXMLMath is an operator which changes signs of some positions .
In particular , for LaTeXMLMath , we set LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLMath be an element of LaTeXMLMath .
LaTeXMLMath is the fixed-point subspace of LaTeXMLMath with respect to LaTeXMLMath .
LaTeXMLMath is the LaTeXMLMath eigenspace of LaTeXMLMath with respect to LaTeXMLMath .
The following is our main theorem .
For LaTeXMLMath , let LaTeXMLMath be an extremal Type II code of length LaTeXMLMath over LaTeXMLMath and set LaTeXMLMath , which is binary code .
Set LaTeXMLMath over LaTeXMLMath .
Then there exists an embedding of LaTeXMLMath into LaTeXMLMath as a subVOA such that LaTeXMLMath decomposes into LaTeXMLMath -modules as the following : LaTeXMLEquation .
LaTeXMLEquation This decomposition is uniquely determined by the extremal Type II code of length LaTeXMLMath over LaTeXMLMath , up to the action of LaTeXMLMath .
The rest of this section is devoted to the proof of the theorem .
In this subsection , we will give the key lemma for Theorem LaTeXMLRef .
It is an extension of Theorem D.6 of LaTeXMLCite .
More precisely , we consider the case of a dual lattice .
We will use the lemma to identify LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be an even integral lattice and let LaTeXMLMath be the lattice VOA associated to LaTeXMLMath .
By LaTeXMLCite , LaTeXMLMath is a LaTeXMLMath -module .
We choose a section LaTeXMLMath , LaTeXMLMath such that the LaTeXMLMath -cocycle with respect to it is LaTeXMLMath -bilinear .
A lift of LaTeXMLMath of LaTeXMLMath is an automorphism LaTeXMLMath of LaTeXMLMath as a LaTeXMLMath -module such that for all LaTeXMLMath , there is a scalar LaTeXMLMath so that LaTeXMLMath .
Set LaTeXMLMath and set LaTeXMLMath .
For LaTeXMLMath , we set the automorphism LaTeXMLMath of LaTeXMLMath as a LaTeXMLMath -module by LaTeXMLMath : LaTeXMLMath , and we set LaTeXMLMath .
By Theorem D.6 of LaTeXMLCite , we have the following proposition .
For lifts LaTeXMLMath and LaTeXMLMath of LaTeXMLMath of LaTeXMLMath , there exists an element LaTeXMLMath such that LaTeXMLMath .
Moreover , we consider the case of LaTeXMLMath .
Let LaTeXMLMath be lifts of LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be a set of representatives of LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath are conjugate by an element of LaTeXMLMath whose restriction on LaTeXMLMath is the identity map if and only if LaTeXMLMath for LaTeXMLMath .
In order to simplify the proof , we assume LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath be a set of representatives of LaTeXMLMath containing LaTeXMLMath .
First , we assume LaTeXMLMath for LaTeXMLMath .
By a conjugation of LaTeXMLMath , we assume LaTeXMLMath for LaTeXMLMath .
Namely , if LaTeXMLMath for LaTeXMLMath , we set LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
It is easy to see that LaTeXMLMath for LaTeXMLMath .
Note that LaTeXMLMath , and for LaTeXMLMath , LaTeXMLMath is generated by LaTeXMLMath as a LaTeXMLMath -module .
Since LaTeXMLMath is LaTeXMLMath -module automorphism and LaTeXMLMath -cocycle is a LaTeXMLMath -bilinear map , we have LaTeXMLMath for LaTeXMLMath by the direct calculation .
Next , we assume LaTeXMLMath for LaTeXMLMath such that LaTeXMLMath is the identity map .
For LaTeXMLMath , we set LaTeXMLMath , where LaTeXMLMath .
Then we have LaTeXMLMath for LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath on LaTeXMLMath , we have LaTeXMLMath for LaTeXMLMath .
∎ In this subsection , we give the decomposition of LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath -frame of LaTeXMLMath corresponding to the code LaTeXMLMath and let LaTeXMLMath be the sublattice of LaTeXMLMath generated by LaTeXMLMath .
Since LaTeXMLMath is an even lattice of rank one with LaTeXMLMath , we have LaTeXMLMath .
From LaTeXMLCite , it follows that LaTeXMLMath is decomposed as LaTeXMLEquation as a LaTeXMLMath -module .
It is well known that LaTeXMLEquation as a LaTeXMLMath -module , where LaTeXMLMath and LaTeXMLMath is a codeword of LaTeXMLMath .
Now , we have two involutions of LaTeXMLMath as a LaTeXMLMath -module .
One is LaTeXMLMath given in Section LaTeXMLRef and the other one is LaTeXMLMath .
Note that LaTeXMLMath is the automorphism of LaTeXMLMath as a LaTeXMLMath -module given in Section LaTeXMLRef .
We will apply Proposition LaTeXMLRef and Lemma LaTeXMLRef to LaTeXMLMath , and identify LaTeXMLMath and LaTeXMLMath .
Since we consider LaTeXMLMath in this section , we use the notation LaTeXMLMath instead of LaTeXMLMath .
By LaTeXMLCite , we can choose a LaTeXMLMath -bilinear LaTeXMLMath -cocycle LaTeXMLMath such that LaTeXMLMath .
By Proposition LaTeXMLRef there exists LaTeXMLMath such that LaTeXMLMath on LaTeXMLMath .
Therefore we identify LaTeXMLMath with LaTeXMLMath on LaTeXMLMath , and we have the inclusion of subVOAs LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the real forms of LaTeXMLMath and LaTeXMLMath as constructed in LaTeXMLCite .
By LaTeXMLCite , those have the positive-definite symmetric bilinear forms .
By the above inclusion , we have the embedding LaTeXMLMath .
In this embedding , the positive-definite symmetric bilinear form of the subVOA is the restriction of that of LaTeXMLMath .
Next , in order to apply to lemma LaTeXMLRef , we have to check the hypothesis of the lemma .
Let LaTeXMLMath be the element of LaTeXMLMath corresponding to LaTeXMLMath .
For LaTeXMLMath , we have LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are linear on LaTeXMLMath , we can choose LaTeXMLMath and the set LaTeXMLMath of representatives of LaTeXMLMath such that LaTeXMLMath on LaTeXMLMath .
Therefore we can use lemma LaTeXMLRef , and we have LaTeXMLMath and LaTeXMLMath are conjugate by LaTeXMLMath .
So , we identify LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
We consider the LaTeXMLMath -fixed-point subspace of LaTeXMLMath .
We obtain the following lemma ( cf .
LaTeXMLCite ) .
Let LaTeXMLMath be a codeword of LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is LaTeXMLMath -invariant , and LaTeXMLMath is the LaTeXMLMath -fixed-point subspace .
If LaTeXMLMath , then LaTeXMLMath is LaTeXMLMath -invariant , and the LaTeXMLMath -fixed-point subspace LaTeXMLMath is isomorphic to LaTeXMLMath as a LaTeXMLMath -module .
By Lemma LaTeXMLRef , ( LaTeXMLRef ) and ( LaTeXMLRef ) , we obtain Theorem LaTeXMLRef ( i ) .
In this subsection , we give the decomposition of LaTeXMLMath .
Set LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is a LaTeXMLMath -module , LaTeXMLMath is a LaTeXMLMath -module .
Note that LaTeXMLMath acts on LaTeXMLMath as multiplication by LaTeXMLMath .
Let LaTeXMLMath be a maximal abelian subgroup of LaTeXMLMath such that LaTeXMLMath .
Note that LaTeXMLMath .
By Proposition LaTeXMLRef , LaTeXMLMath decomposes into the direct sum of all the irreducible LaTeXMLMath -modules on which LaTeXMLMath acts by LaTeXMLMath .
Therefore LaTeXMLMath decomposes into the direct sum of irreducible LaTeXMLMath -modules on which LaTeXMLMath acts by LaTeXMLMath .
By Proposition LaTeXMLRef ( LaTeXMLMath ) , we have LaTeXMLMath .
Then any multiplicities of the irreducible LaTeXMLMath -module is LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is a LaTeXMLMath -module .
Thus we obtain LaTeXMLEquation where Irr LaTeXMLMath is the set of the characters for LaTeXMLMath and LaTeXMLMath is the one-dimensional module LaTeXMLMath with a character LaTeXMLMath .
By using the canonical map LaTeXMLMath : LaTeXMLMath , we view LaTeXMLMath as a LaTeXMLMath -module .
Note that the kernel of LaTeXMLMath is LaTeXMLMath .
Therefore , as a LaTeXMLMath -module , LaTeXMLMath is decomposed LaTeXMLEquation where LaTeXMLMath is the multiplicity .
By ( LaTeXMLRef ) , we have LaTeXMLMath .
By ( LaTeXMLRef ) , LaTeXMLMath has a decomposition , LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath -twisted LaTeXMLMath -module corresponding to the LaTeXMLMath -module LaTeXMLMath .
As in Section 2.3 , we fix the ordered basis of LaTeXMLMath as LaTeXMLMath .
Let LaTeXMLMath be a sublattice of LaTeXMLMath and set LaTeXMLMath .
For LaTeXMLMath , we set the character of LaTeXMLMath LaTeXMLMath .
For a character LaTeXMLMath such that LaTeXMLMath , we set LaTeXMLMath , where LaTeXMLMath .
For LaTeXMLMath , we set LaTeXMLMath .
Then we have LaTeXMLEquation .
Now , let us determine the multiplicities LaTeXMLMath .
Let LaTeXMLMath be an element of LaTeXMLMath such that LaTeXMLMath .
Set LaTeXMLMath .
Then LaTeXMLMath if and only if there exists LaTeXMLMath such that LaTeXMLMath , namely , LaTeXMLMath for any LaTeXMLMath .
Since LaTeXMLMath for LaTeXMLMath , we have LaTeXMLMath .
Therefore LaTeXMLMath for any LaTeXMLMath if and only if LaTeXMLMath for any LaTeXMLMath Since LaTeXMLMath , LaTeXMLMath if and only if LaTeXMLMath .
By ( LaTeXMLRef ) , we have an isomorphism LaTeXMLEquation .
Next , we fix a character LaTeXMLMath and consider the space LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
Thus , we get an isomorphism LaTeXMLMath .
Note that we have LaTeXMLMath , because LaTeXMLMath acts by LaTeXMLMath on LaTeXMLMath and LaTeXMLMath acts by LaTeXMLMath on LaTeXMLMath .
Therefore , for LaTeXMLMath with LaTeXMLMath , we have an isomorphism of LaTeXMLMath -modules LaTeXMLEquation .
By LaTeXMLMath and LaTeXMLMath , we get Theorem LaTeXMLRef ( ii ) .
In this section , we give the character of LaTeXMLMath and give some automorphisms of LaTeXMLMath .
We recall the character of a VOA .
Let LaTeXMLMath be a VOA and the character of LaTeXMLMath is given by LaTeXMLMath .
In order to give the character of LaTeXMLMath , we consider the symmetrized weight enumerator of a code over LaTeXMLMath .
The symmetrized weight enumerator of a code LaTeXMLMath over LaTeXMLMath is defined as : LaTeXMLEquation where LaTeXMLMath denotes the number of LaTeXMLMath such that LaTeXMLMath .
Note that the symmetrized weight enumerators of equivalence codes are same .
For LaTeXMLMath , we set LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Note that for LaTeXMLMath , we have LaTeXMLMath /2 , and LaTeXMLMath .
By Theorem LaTeXMLRef , we obtain the following corollary .
Let LaTeXMLMath be an extremal Type II code of length LaTeXMLMath over LaTeXMLMath .
Then we have LaTeXMLEquation .
It is easy to see that LaTeXMLMath , where LaTeXMLMath is the theta function associated with LaTeXMLMath .
Since LaTeXMLMath , we obtain LaTeXMLEquation .
This equation is given in Remark LaTeXMLMath of LaTeXMLCite .
Let LaTeXMLMath be the set of inequivalent irreducible LaTeXMLMath -modules .
If LaTeXMLMath is odd then we define the map LaTeXMLMath setting by LaTeXMLEquation and if LaTeXMLMath is even then we defined the map LaTeXMLMath setting by LaTeXMLEquation .
The fusion algebra of LaTeXMLMath is the vector space LaTeXMLMath equipped products LaTeXMLMath given by fusion rules , where we regard LaTeXMLMath as a formal element .
The definition of fusion rules is given in LaTeXMLCite .
An automorphism of the fusion algebra LaTeXMLMath is a linear automorphism LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
By the fusion rules of LaTeXMLMath determined in LaTeXMLCite , we have the following proposition ( cf .
LaTeXMLCite ) .
The linear map of the fusion algebra of LaTeXMLMath , LaTeXMLMath , is an automorphism of the fusion algebra of LaTeXMLMath .
Suppose we are given a decomposition LaTeXMLEquation as a LaTeXMLMath -module , where LaTeXMLMath is the multiplicity .
For each LaTeXMLMath , we define a linear automorphism LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation for LaTeXMLMath .
By Proposition LaTeXMLRef , we have the following proposition ( cf .
LaTeXMLCite ) .
LaTeXMLMath is a VOA automorphism of LaTeXMLMath .
By the definition of LaTeXMLMath , LaTeXMLMath fixes any element of LaTeXMLMath .
In particular , LaTeXMLMath fixes the Virasoro element and the vacuum vector .
Note that the fusion rules of tensor products of modules are the tensor products of the fusion rules of those modules .
Let LaTeXMLMath .
Let LaTeXMLMath : LaTeXMLMath be a map such that LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath be elements of LaTeXMLMath .
By the definition of fusion rules , we have LaTeXMLMath for LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is the vertex operator of LaTeXMLMath and LaTeXMLMath is fusion rules for LaTeXMLMath .
Therefore we have LaTeXMLMath .
∎ By Theorem LaTeXMLRef , we have the following proposition .
Suppose LaTeXMLMath is odd .
In decompositions of LaTeXMLMath given by Theorem LaTeXMLRef , LaTeXMLMath is a LaTeXMLMath element of the Monster .
In fact , LaTeXMLMath , where LaTeXMLMath is a non-split extension of LaTeXMLMath LaTeXMLMath largest simple Conway group LaTeXMLMath by the extra-special group LaTeXMLMath .
By Theorem LaTeXMLRef and Proposition LaTeXMLRef , LaTeXMLMath acts by LaTeXMLMath on LaTeXMLMath and acts by LaTeXMLMath on LaTeXMLMath .
By LaTeXMLCite , the centralizer of LaTeXMLMath in the Monster simple group is LaTeXMLMath .
Since LaTeXMLMath commutes with LaTeXMLMath , we have LaTeXMLMath .
Since LaTeXMLMath preserves the space LaTeXMLMath for any LaTeXMLMath , we have LaTeXMLMath .
It is well known that there are the four types of Monster elements LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath contained in LaTeXMLMath .
Since LaTeXMLMath is an order LaTeXMLMath element , LaTeXMLMath is a LaTeXMLMath element of the Monster.∎ Suppose LaTeXMLMath is even .
In decompositions of LaTeXMLMath given by Theorem LaTeXMLRef , LaTeXMLMath is a LaTeXMLMath element of the Monster .
More precisely , LaTeXMLMath acts by LaTeXMLMath on LaTeXMLMath and acts by LaTeXMLMath on LaTeXMLMath , namely , LaTeXMLMath given in LaTeXMLMath of LaTeXMLCite .
In this section , we assume that LaTeXMLMath is odd .
Then we give the McKay-Thompson series for 4A elements LaTeXMLMath given in Section 3.2 .
The expressions of it are different from LaTeXMLCite , and we obtain formulas of modular functions .
We recall the McKay-Thompson series .
Let LaTeXMLMath be an element of the Monster simple group and let LaTeXMLMath be the moonshine module .
Then the McKay-Thompson series for LaTeXMLMath is given by LaTeXMLMath , where LaTeXMLMath is the trace of the action of LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath : LaTeXMLMath be the composite map of the projection map LaTeXMLMath with respect to LaTeXMLMath -th elements and the canonical map LaTeXMLMath .
Note that for LaTeXMLMath , the automorphism LaTeXMLMath acts by the multiplication LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be the automorphism of LaTeXMLMath given in Section 3.2 .
In the decomposition given in Theorem 2.1 , we have LaTeXMLEquation where LaTeXMLMath is given in Section 3.1 .
LaTeXMLEquation .
By direct calculation , we have ( i ) .
Since LaTeXMLMath contains the all LaTeXMLMath element , the numbers of elements of LaTeXMLMath whose LaTeXMLMath -th coordinate is LaTeXMLMath , and whose LaTeXMLMath -th coordinate is LaTeXMLMath are equal .
By the definitions of LaTeXMLMath , we have LaTeXMLMath respectively .
Therefore we have ( ii ) .
∎ Let LaTeXMLMath be an extremal Type II code over LaTeXMLMath .
The McKay-Thompson series for the LaTeXMLMath element LaTeXMLMath is given by LaTeXMLEquation .
In LaTeXMLCite , all the McKay-Thompson series are computed .
So we have the following formulas of modular functions .
LaTeXMLEquation where the Dedekind LaTeXMLMath -function LaTeXMLMath .
Let LaTeXMLMath be a bounded domain in LaTeXMLMath whose boundary has a Minkowski dimension LaTeXMLMath .
Suppose that LaTeXMLMath , LaTeXMLMath an infinite discrete subset of LaTeXMLMath , is a frame of exponentials for LaTeXMLMath , with frame constants LaTeXMLMath , LaTeXMLMath .
Then if LaTeXMLEquation where LaTeXMLMath depends only on the ambient dimension LaTeXMLMath and LaTeXMLMath denotes the Minkowski content , then every cube of sidelength LaTeXMLMath contains at least one element of LaTeXMLMath .
We give examples that illustrate the extent to which our estimates are sharp .
Let LaTeXMLMath be a domain of finite Lebesgue measure in LaTeXMLMath .
Suppose that LaTeXMLMath has a frame of exponentials of the form LaTeXMLMath , LaTeXMLMath , a discrete infinite subset of LaTeXMLMath , with frame constants LaTeXMLMath , LaTeXMLMath , in the sense that LaTeXMLEquation where LaTeXMLMath , and LaTeXMLMath denotes the standard Fourier transform .
In this paper we will work with frames rather than exponential basis because LaTeXMLMath of every bounded domain has frames , whereas exponential basis are generally hard to come by .
( See LaTeXMLCite ) .
The following quantities were introduced by Beurling .
See LaTeXMLCite .
LaTeXMLEquation where LaTeXMLMath is a cube of sidelength LaTeXMLMath centered at LaTeXMLMath , and let LaTeXMLEquation .
It follows from results proved by Landau ( LaTeXMLCite , see also LaTeXMLCite ) that if LaTeXMLMath is a bounded domain then LaTeXMLEquation .
If the set LaTeXMLMath is actually an orthogonal basis for LaTeXMLMath then the inequality LaTeXMLMath is actually an equality for both LaTeXMLMath and LaTeXMLMath .
These results show that , asymptotically , a sufficiently large cube centered at any point contains the number of elements of LaTeXMLMath at least equal to its volume multiplied by the Lebesgue measure of LaTeXMLMath .
In this paper we will show that if the Minkowski dimension , LaTeXMLMath , of the boundary LaTeXMLMath is smaller than the ambient dimension LaTeXMLMath , then there exists LaTeXMLEquation where LaTeXMLMath only depends on LaTeXMLMath and LaTeXMLMath denotes the LaTeXMLMath -dimensional upper Minkowski content of LaTeXMLMath , such that a cube of sidelength LaTeXMLMath centered at any point contains at least one element of LaTeXMLMath .
Note that if LaTeXMLMath is , say , piecewise smooth , then LaTeXMLMath and LaTeXMLMath .
A note on notation .
The letter LaTeXMLMath below shall denote a generic constant which may change from line to line .
We shall give more precise information about the constants when appropriate .
Our main result is the following .
Let LaTeXMLMath denote a domain in LaTeXMLMath with finite non-zero Lebesgue measure whose boundary LaTeXMLMath has Minkowski dimension LaTeXMLMath in the sense that LaTeXMLEquation .
Then there exists LaTeXMLMath depending only on LaTeXMLMath , such that if LaTeXMLEquation then LaTeXMLEquation for every LaTeXMLMath , and any set LaTeXMLMath such that LaTeXMLMath is an exponential frame for LaTeXMLMath , with frame constants LaTeXMLMath , LaTeXMLMath where LaTeXMLMath denotes the cube of sidelength LaTeXMLMath centered at LaTeXMLMath .
In other words , our result shows , at least if LaTeXMLMath , that if LaTeXMLMath has a fixed volume , then the maximum gap between the elements of LaTeXMLMath is no larger than a fixed constant times the the Minkowski content of the boundary .
Moreover , the rate of increase depends on the Minkowski dimension of LaTeXMLMath .
This idea is illustrated by the following simple example .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
We can take LaTeXMLMath .
It is not hard to see that the largest cube that does not intersect LaTeXMLMath has sidelength LaTeXMLMath .
The measure of LaTeXMLMath is LaTeXMLMath .
It follows that LaTeXMLEquation so LaTeXMLMath grows linearly with LaTeXMLMath .
We now spice up the above example to illustrate the fractal phenomenon .
Let LaTeXMLMath be a domain constructed by taking a square LaTeXMLMath and replacing the upper and lower segments by identical fractal curves of Minkowski dimension LaTeXMLMath .
It is not hard to see that LaTeXMLMath may be taken to be LaTeXMLMath .
( See LaTeXMLCite ) .
We now blow up the domain by the factor of LaTeXMLMath ( i.e we apply the matrix LaTeXMLMath , where LaTeXMLMath is the identity matrix ) .
Let LaTeXMLMath denote the image of LaTeXMLMath under that mapping .
The set LaTeXMLMath must now be taken to be LaTeXMLMath , which tells us that LaTeXMLMath in Theorem 1 should be LaTeXMLMath .
On the other hand , LaTeXMLMath , and LaTeXMLMath , so Theorem 1 gives us LaTeXMLMath .
The following example shows that if the Lebesgue measure LaTeXMLMath the conclusion of Theorem 1 no longer holds .
Let LaTeXMLMath denote the Cantor type set consisting of numbers that do not have LaTeXMLMath or LaTeXMLMath in their base LaTeXMLMath expansion .
Let LaTeXMLMath denote the unique probability measure supported on LaTeXMLMath ( see LaTeXMLCite ) given by the equation LaTeXMLEquation .
One can check that LaTeXMLEquation .
If LaTeXMLMath is the set of non-negative integers whose base LaTeXMLMath expansion does not contain LaTeXMLMath or LaTeXMLMath , then LaTeXMLMath is an orthonormal basis of LaTeXMLMath .
( See LaTeXMLCite ) .
In particular this shows that the conclusion of Theorem 1 fails miserably in this case .
In this example we shall see that there exist families of domains with piecewise smooth boundaries such that the volume of each domain is LaTeXMLMath , the length of the boundary tends to infinity , but LaTeXMLMath , in the sense of Theorem 1 , may always be taken to be LaTeXMLMath , for any LaTeXMLMath .
Let LaTeXMLMath denote the unit square in LaTeXMLMath where the upper and lower edges are replaced by a sawtooth function with LaTeXMLMath teeth where the height of each tooth is LaTeXMLMath .
The length LaTeXMLMath goes to infinity as LaTeXMLMath .
The set LaTeXMLMath for each LaTeXMLMath is LaTeXMLMath , so LaTeXMLMath , in the sense of Theorem 1 , may always be taken to be LaTeXMLMath , for any LaTeXMLMath .
This says that the inequality LaTeXMLMath does not sharply describe the behavior of LaTeXMLMath in this case .
However , the proof of Theorem 1 ( see the discussion at the end of the proof of Theorem 1 below ) shows that in some cases LaTeXMLMath may be taken to be LaTeXMLMath , where LaTeXMLMath depends only on LaTeXMLMath .
We shall see that the example given in this paragraph falls into that category .
In all the previous examples we used frames which were actually orthogonal bases .
However , this phenomenon persists in the cases when orthogonal exponential basis do not exist and we have to make do with frames .
Let LaTeXMLMath denote the disc of radius LaTeXMLMath in LaTeXMLMath centered at the origin .
It was shown in LaTeXMLCite that LaTeXMLMath is frame for LaTeXMLMath with constants LaTeXMLMath .
Note that we do not have orthogonal basis becuase , in particular , that would imply that LaTeXMLMath .
It is well known that LaTeXMLMath does not have orthogonal basis of exponentials .
See LaTeXMLCite .
It is clear that LaTeXMLMath , in the sense of Theorem 1 must be taken to be greater than LaTeXMLMath , which is exactly what Theorem 1 predicts .
The key estimate ( see Lemma 9 below ) involved in the proof of Theorem 1 is LaTeXMLEquation for any LaTeXMLMath , where LaTeXMLMath depends only on the dimension and on the frame constant LaTeXMLMath .
This estimate is similar to the estimate that comes up in the theory of irregularities of distributions , ( see LaTeXMLCite , p.110 ) , namely that for any domain LaTeXMLMath whose boundary is a piecewise LaTeXMLMath curve LaTeXMLMath LaTeXMLEquation .
In fact , our proof of the estimate LaTeXMLMath given in Lemma 9 below uses an idea from the proof of the estimate LaTeXMLMath given by Brandolini , Colzani , and Travaglini in LaTeXMLCite .
The proof of Theorem 1 is based on the following sequence of lemmae .
For any LaTeXMLMath define LaTeXMLEquation and let LaTeXMLMath denote the standard Fourier transform LaTeXMLEquation .
Let LaTeXMLMath , and let LaTeXMLMath denote the characteristic function of LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
The proof is straightforward .
Let LaTeXMLMath be as above .
Then LaTeXMLEquation and LaTeXMLEquation with LaTeXMLMath , where LaTeXMLMath depends only on LaTeXMLMath .
We note again that even though the estimate LaTeXMLMath is best possible over all LaTeXMLMath ’ s , for special choices of LaTeXMLMath , the estimate is frequently much better .
( See Example 5 above ) .
To prove LaTeXMLMath note that the left hand side equals LaTeXMLMath .
The proof of LaTeXMLMath is similar .
The key lemma is the following .
( See LaTeXMLCite for a similar argument ) .
Let LaTeXMLMath be as above and let LaTeXMLMath be such that LaTeXMLMath is a frame of LaTeXMLMath with frame constants LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath , and LaTeXMLMath as in Lemma 8 .
To prove Lemma 9 chose LaTeXMLMath boxes LaTeXMLMath and LaTeXMLMath vectors LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
Clearly this can be done in any dimension LaTeXMLMath , for a sufficiently large LaTeXMLMath .
Now , by triangle inequality LaTeXMLEquation .
By Lemma 7 , the frame property , and Lemma 8 we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
On the other hand , by LaTeXMLMath , Lemma 7 , the frame property , and Lemma 8 we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since LaTeXMLMath is a frame for LaTeXMLMath if and only if LaTeXMLMath is also a frame for LaTeXMLMath ( with the same frame constants ) for any LaTeXMLMath , and our estimates do not depend on the choice of LaTeXMLMath , it is sufficient to consider the case LaTeXMLMath .
By the frame property and Lemma 7 we get LaTeXMLEquation .
Using Lemma 9 we see that if LaTeXMLMath , LaTeXMLEquation .
So by LaTeXMLMath and LaTeXMLMath LaTeXMLEquation which proves that if LaTeXMLMath , then LaTeXMLEquation .
Moreover , the above proof shows that LaTeXMLMath where LaTeXMLMath depends only on LaTeXMLMath .
In the proof above the key estimate is LaTeXMLMath .
While this is the best possible estimate uniform in LaTeXMLMath , in the proof we are have a wide choice of LaTeXMLMath ’ s as long as LaTeXMLMath and the estimates LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are satisfied .
This observation can be used to handle the family of examples given by Example 5 above .
For convenience we take LaTeXMLMath .
We can now take all LaTeXMLMath ’ s in the proof of Theorem 1 of the form LaTeXMLMath and for this choice of LaTeXMLMath ’ s it is easy to check that LaTeXMLMath , where LaTeXMLMath is a uniform constant , since the ” teeth ” of LaTeXMLMath ’ s point in the LaTeXMLMath -direction .
Since LaTeXMLMath is uniformly bounded above and below , the lack of sharpness of Theorem 1 exposed in Example 5 is resolved for this family of examples .
References
We show , using a Knapp-type homogeneity argument , that the LaTeXMLMath restriction theorem implies a growth condition on the hypersurface in question .
We further use this result to show that the optimal LaTeXMLMath restriction theorem implies the sharp isotropic decay rate for the Fourier transform of the Lebesgue measure carried by compact convex finite hypersurfaces .
Let LaTeXMLMath be a smooth compact finite type hypersurface .
Let LaTeXMLMath , where LaTeXMLMath denotes the Lebesgue measure on LaTeXMLMath .
Let LaTeXMLMath , where LaTeXMLMath denotes the standard Fourier transform of LaTeXMLMath .
It is well known ( see LaTeXMLCite , LaTeXMLCite ) that if LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath for LaTeXMLMath .
A natural question to ask is , does the boundedness of LaTeXMLMath , LaTeXMLMath , imply that LaTeXMLMath ?
In this paper we will show that this is indeed the case if LaTeXMLMath is a smooth convex finite type hypersurface in the sense that the order of contact with every tangent line is finite .
( See LaTeXMLCite ) .
Under a more general condition , called the finite polyhedral type assumption , ( see Definition 1 below ) , we will show that the LaTeXMLMath restriction theorem with LaTeXMLMath implies that LaTeXMLMath , where LaTeXMLMath , where LaTeXMLMath denotes the tangent hyperplane to LaTeXMLMath at LaTeXMLMath .
Our plan is as follows .
We will first use a variant of the Knapp homogeneity argument to show that if LaTeXMLMath satisfies the finite polyhedral type condition and LaTeXMLMath , with LaTeXMLMath , then LaTeXMLMath for each LaTeXMLMath .
If the surface is , in addition , convex and finite type , then the result due to Bruna , Nagel , and Wainger ( see LaTeXMLCite ) implies that LaTeXMLMath .
If the surface is not convex finite type , then we do not , in general , know how to conclude that LaTeXMLMath implies that LaTeXMLMath .
A gap remains .
Let LaTeXMLMath be a smooth compact hypersurface in LaTeXMLMath .
Let LaTeXMLMath denote the projection of LaTeXMLMath onto LaTeXMLMath .
We say that LaTeXMLMath is of finite polyhedral type if there exists a family of polyhedra LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath do not depend on LaTeXMLMath , and the number of vertices of LaTeXMLMath is bounded above independent of LaTeXMLMath , where LaTeXMLMath denotes the characteristic function of LaTeXMLMath .
The motivation for the definition of finite polyhedral type is the standard homogeneity argument due to Knapp .
In order to prove the sharpness of the LaTeXMLMath restriction theorem for hypersurfaces with non-vanishing Gaussian curvature , Knapp approximated such a surface with a box with side-lengths LaTeXMLMath , LaTeXMLMath small .
He then took LaTeXMLMath to be the inverse Fourier transform of the characteristic function of that box .
It is not hard to see that LaTeXMLMath .
On the other hand , using the fact that the Fourier transform of the box in question is LaTeXMLMath , it is not hard to see that LaTeXMLMath .
It follows that LaTeXMLMath , which is the known positive result due to Stein and Tomas .
The crucial part of this calculation is the approximation of the surface with a box with appropriate dimensions .
Definition 1 and Theorem 2 are generalizations of this phenomenon .
It should also be noted that it is not hard to see that the hypersurface LaTeXMLMath does not satisfy the finite polyhedral type condition .
Thus , it makes sense to think of the finite polyhedral type condition as a generalization of convexity .
Let LaTeXMLMath be of finite polyhedral type .
Consider the estimates LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation for each LaTeXMLMath .
Then LaTeXMLMath implies LaTeXMLMath and LaTeXMLMath implies LaTeXMLMath .
Further , LaTeXMLMath implies LaTeXMLMath if LaTeXMLMath is in addition convex and finite type .
The fact that LaTeXMLMath implies LaTeXMLMath is essentially the Stein-Tomas restriction theorem .
( See LaTeXMLCite , LaTeXMLCite ) .
The fact that LaTeXMLMath implies LaTeXMLMath in the case of convex finite type hypersurfaces is due to Bruna , Nagel , and Wainger .
( See LaTeXMLCite ) .
So it remains to prove that LaTeXMLMath implies LaTeXMLMath , and that convex finite type hypersurfaces are of finite polyhedral type .
( See Theorems 3 and 4 below ) .
Let LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath is mixed homogeneous in the sense that there exist integers LaTeXMLMath , LaTeXMLMath , such that LaTeXMLMath , LaTeXMLMath if LaTeXMLMath , LaTeXMLMath is the remainder in the sense that LaTeXMLMath and LaTeXMLMath is a constant .
Then LaTeXMLMath is of finite polyhedral type .
Theorem 3 implies that convex finite type hypersurfaces are of finite polyhedral type via the following representation result due to Schulz .
( See LaTeXMLCite .
See also LaTeXMLCite ) .
Let LaTeXMLMath be a convex finite type function such that LaTeXMLMath and LaTeXMLMath .
Then , after perhaps applying a rotation , we can write LaTeXMLMath , where LaTeXMLMath is a convex polynomial , mixed homogeneous in the sense of Theorem 3 , and LaTeXMLMath is the remainder in the sense of Theorem 3 .
Locally , LaTeXMLMath is a graph of a smooth function LaTeXMLMath , such that LaTeXMLMath , and LaTeXMLMath .
If we consider a sufficiently small piece of our hypersurface , LaTeXMLMath , where LaTeXMLMath is a compact set in LaTeXMLMath containing the origin , and , without loss of generality , LaTeXMLMath .
Since LaTeXMLMath , it suffices to show that LaTeXMLMath .
Let LaTeXMLMath be a function such that LaTeXMLMath is the characteristic function of the set LaTeXMLMath , where LaTeXMLMath is the polyhedron containing the set LaTeXMLMath given by the definition of finite polyhedral type .
Let ’ s assume for a moment that LaTeXMLMath .
Since the restriction theorem holds , we must have LaTeXMLMath which implies that LaTeXMLMath .
Since LaTeXMLMath , it follows that LaTeXMLMath .
By the definition of LaTeXMLMath it follows that LaTeXMLMath .
This completes the proof provided that we can show that LaTeXMLMath .
More generally , we will show that if LaTeXMLMath is a polyhedron in LaTeXMLMath , then LaTeXMLMath , where LaTeXMLMath depends on the dimension and the number of vertices of LaTeXMLMath and LaTeXMLMath denotes the volume of LaTeXMLMath .
We give the argument in two dimensions , the argument in higher dimensions being similar .
Break up LaTeXMLMath as a union of disjoint ( up to the boundary ) triangles LaTeXMLMath , LaTeXMLMath .
Since LaTeXMLMath , it suffices to carry out the argument for LaTeXMLMath , where LaTeXMLMath is assumed to be a triangle .
Since translations don ’ t contribute anything in this context , we may assume that one of the vertices of the triangle is at the origin .
Break up this triangle , if necessary , into two right triangles .
Refine the original decomposition so that it consists of right triangles .
Rotate the right triangle so that it is in the first quadrant and one of the sides is on the LaTeXMLMath -axis .
We now apply a linear transformation mapping this triangle ( denoted by LaTeXMLMath ) into the triangle with the endpoints LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
It is easy to check by an explicit computation that the Fourier transform of the characteristic function of this triangle has the LaTeXMLMath norm ( crudely ) bounded by LaTeXMLMath .
Let LaTeXMLMath denote the linear transformation taking the triangle LaTeXMLMath to the unit triangle above .
We see that LaTeXMLEquation so LaTeXMLEquation .
Since LaTeXMLMath , we see that LaTeXMLMath , where the LaTeXMLMath ’ s are the triangles from the ( refined ) original decomposition .
Adding up the estimates we get LaTeXMLEquation .
In higher dimensions the proof is virtually identical with triangles replaced by LaTeXMLMath dimensional simplices , i.e the convex hull of LaTeXMLMath points that are not contained in any LaTeXMLMath dimensional plane .
Since we have assumed that the number of vertices of LaTeXMLMath is bounded above , it follows that LaTeXMLMath , as desired .
As before , it is enough to consider the set LaTeXMLMath .
It will be clear from the proof below that if we shrink the support sufficiently , then LaTeXMLMath due to our assumptions on the remainder term LaTeXMLMath .
Let LaTeXMLMath .
Our plan is as follows .
We first prove that LaTeXMLMath .
Then , we will find a polyhedron of suitable area that contains the set LaTeXMLMath .
We shall obtain the polyhedra for all values of LaTeXMLMath by homogeneity .
Going into polar coordinates , LaTeXMLMath , LaTeXMLMath , we see that LaTeXMLMath .
This proves that LaTeXMLMath .
We now find a box LaTeXMLMath with sides parallel to the coordinate axes , such that LaTeXMLMath , and LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath be a mixed homogeneous function of degree LaTeXMLMath defined by the condition LaTeXMLMath , where LaTeXMLMath denotes the boundary of LaTeXMLMath .
Let LaTeXMLMath be the polyhedron such that the boundary LaTeXMLMath .
It is not hard to see that LaTeXMLMath .
Moreover , LaTeXMLMath by the calculation made in the previous paragraph .
This completes the proof of Theorem 3 .
Let LaTeXMLMath be a proper , smooth and geometrically connected curve over a global field LaTeXMLMath .
In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of LaTeXMLMath to the order of the Brauer group of a proper regular model of LaTeXMLMath .
We thereby partially answer a question of Grothendieck .
Let LaTeXMLMath be a global field , i.e .
LaTeXMLMath is a finite extension of LaTeXMLMath ( the “ number field case ” ) or is finitely generated and of transcendence degree 1 over a finite field ( the “ function field case ” ) .
In the number field case we let LaTeXMLMath denote a nonempty open subscheme of the spectrum of the ring of integers of LaTeXMLMath , and when LaTeXMLMath is a function field in one variable with finite field of constants LaTeXMLMath , we let LaTeXMLMath denote a nonempty open subscheme of the unique smooth complete curve over LaTeXMLMath whose function field is LaTeXMLMath .
We will write LaTeXMLMath for the set of primes of LaTeXMLMath not corresponding to a point of LaTeXMLMath .
Further , we will write LaTeXMLMath for the separable algebraic closure of LaTeXMLMath and LaTeXMLMath for the Galois group of LaTeXMLMath over LaTeXMLMath .
The completion of LaTeXMLMath at a prime LaTeXMLMath will be denoted by LaTeXMLMath .
Assume now that a connected , regular , 2-dimensional scheme LaTeXMLMath is given together with a proper morphism LaTeXMLMath whose generic fiber LaTeXMLMath is a smooth geometrically connected curve over LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) denote the index ( resp .
period ) of LaTeXMLMath .
These integers may be defined as the least positive degree of a divisor on LaTeXMLMath and the least positive degree of a divisor class in LaTeXMLMath , respectively , where LaTeXMLMath .
For each prime LaTeXMLMath of LaTeXMLMath , we will write LaTeXMLMath and LaTeXMLMath for the analogous quantities associated to the curve LaTeXMLMath .
It is known that there are only finitely many primes LaTeXMLMath for which LaTeXMLMath .
Further , Lichtenbaum [ 21 ] has shown that LaTeXMLMath equals either LaTeXMLMath or LaTeXMLMath for each prime LaTeXMLMath .
We will write LaTeXMLMath for the number of primes LaTeXMLMath for which LaTeXMLMath .
These primes were called “ deficient ” in [ 35 ] , and we sometimes refer to LaTeXMLMath as the number of deficient primes of LaTeXMLMath .
Now let LaTeXMLMath be the Jacobian variety of LaTeXMLMath .
It has long been known ( see [ 46 ] , §3 ) that there exist close connections between the Brauer group LaTeXMLMath of LaTeXMLMath and the Tate-Shafarevich group LaTeXMLMath of LaTeXMLMath .
These connections were explored at length by Grothendieck in his paper [ 15 ] , in a more general setting than the one considered here .
The following theorem can be extracted from [ 15 ] , p. 121 .
Theorem ( Grothendieck ) .
In the function field case , suppose that LaTeXMLMath for all primes LaTeXMLMath of LaTeXMLMath .
Then there exist a finite group LaTeXMLMath of order LaTeXMLMath , a finite group LaTeXMLMath of order dividing LaTeXMLMath , and an exact sequence LaTeXMLEquation .
Regarding this result , Grothendieck ( op .
cit. , p. 122 ) asked for the exact order of LaTeXMLMath .
This problem was solved by Milne in [ 28 ] ( see also [ 29 ] , III.9.6 ) , who used Cassels-Tate duality to compute the order of LaTeXMLMath .
In order to state Milne ’ s result , which covers both the function field and number field cases , we need the following definition .
Let LaTeXMLEquation .
Thus LaTeXMLMath is the group of elements in the Brauer group of LaTeXMLMath becoming trivial on LaTeXMLMath for all LaTeXMLMath .
Then one has the Theorem ( Milne ) .
Assume that LaTeXMLMath for all primes LaTeXMLMath of LaTeXMLMath and that LaTeXMLMath contains no nonzero infinitely divisible elements .
Then the period of LaTeXMLMath equals its index , i.e .
LaTeXMLMath , and there exist finite groups LaTeXMLMath and LaTeXMLMath of order LaTeXMLMath and an exact sequence LaTeXMLEquation .
In particular , if one of LaTeXMLMath or LaTeXMLMath is finite , then so is the other , and their orders are related by LaTeXMLEquation .
Grothendieck ( loc .
cit . )
went on to pose the problem of making explicit the relations between LaTeXMLMath and LaTeXMLMath when the integers LaTeXMLMath are no longer assumed to be equal to one .
In this paper we generalize the methods developed by Milne in [ 28 ] to prove the above theorem and obtain the following stronger result , which may be viewed as a partial solution to Grothendieck ’ s problem .
Assume that the integers LaTeXMLMath are relatively prime in pairs ( i.e. , LaTeXMLMath for all LaTeXMLMath ) and that LaTeXMLMath contains no nonzero infinitely divisible elements .
Then there is an exact sequence LaTeXMLEquation in which LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are finite groups of orders LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
Here LaTeXMLMath is the number of deficient primes of LaTeXMLMath defined previously .
In particular , if one of LaTeXMLMath or LaTeXMLMath is finite , then so is the other , and their orders are related by LaTeXMLEquation .
Some immediate corollaries are Suppose that one of LaTeXMLMath or LaTeXMLMath is finite .
Assume also that LaTeXMLMath for all LaTeXMLMath ( which holds for instance if LaTeXMLMath has genus 1 ) , and that these integers are relatively prime in pairs .
Then LaTeXMLMath , and LaTeXMLEquation .
In particular if LaTeXMLMath for all LaTeXMLMath , then LaTeXMLEquation .
Note that the last formula in the statement of Corollary 1 is precisely the formula of Milne stated before .
Assume that one of LaTeXMLMath or LaTeXMLMath is finite and that LaTeXMLMath for all LaTeXMLMath ( the latter holds for instance if LaTeXMLMath has genus 2 ) .
Then LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are as in the statement of the theorem .
In the function field case our main result , when combined with a formula of Gordon [ 10 , p. LaTeXMLMath ] , should imply the expected equivalence of the conjectures of Artin and Tate for LaTeXMLMath [ 46 , Conj .
C ] and Birch and Swinnerton-Dyer for LaTeXMLMath [ 46 , Conj .
B ] ( this is Tate ’ s “ elementary ” conjecture ( d ) of [ 46 ] ) , at least under the additional assumption that the structural morphism LaTeXMLMath is cohomologically flat in dimension 0 .
We expect to address this issue in a separate publication .
I am grateful to the people of Fondecyt for their financial support and their patience .
I also thank the library staff at IMPA ( Rio de Janeiro , Brasil ) for their bibliographical assistance .
We keep the notations introduced in the previous section .
Thus in particular LaTeXMLMath is a regular connected scheme of dimension 2 equipped with a proper morphism LaTeXMLMath whose generic fiber LaTeXMLMath is a smooth geometrically connected curve over LaTeXMLMath .
Remark .
The LaTeXMLMath -scheme LaTeXMLMath is a proper regular model of its generic fiber .
Conversely , if we start with a ( geometrically connected ) proper and smooth curve LaTeXMLMath over LaTeXMLMath , then there is a closed immersion LaTeXMLMath for some LaTeXMLMath and we can obtain a LaTeXMLMath -scheme LaTeXMLMath as above , with generic fiber LaTeXMLMath , by applying Lipman ’ s desingularization process [ 2 ] , [ 22 ] to the schematic image of LaTeXMLMath in LaTeXMLMath .
The Picard scheme of LaTeXMLMath , LaTeXMLMath , is a smooth group scheme over LaTeXMLMath whose identity component , LaTeXMLMath , is an abelian variety , the Jacobian variety of LaTeXMLMath .
Henceforth , we will write LaTeXMLMath for LaTeXMLMath and ( in accordance with previous notations ) LaTeXMLMath for LaTeXMLMath .
The following holds .
If LaTeXMLMath is any field containing LaTeXMLMath such that LaTeXMLMath is nonempty , then LaTeXMLMath ( for the basic facts on the relative Picard functor , see [ 4 , §8.1 ] or [ 12 ] ) .
Further , there is an exact sequence of LaTeXMLMath -modules LaTeXMLEquation where LaTeXMLMath is the degree map on LaTeXMLMath .
In particular LaTeXMLMath may be identified with LaTeXMLMath , the subgroup of LaTeXMLMath consisting of divisor classes of degree zero ( in this paper we shall regard the elements of LaTeXMLMath mainly as classes of divisors .
See below ) .
Now we observe that LaTeXMLMath .
Further , there is an exact sequence ( deduced from the preceding one by taking LaTeXMLMath -invariants ) LaTeXMLEquation where LaTeXMLMath is the period of LaTeXMLMath as defined previously .
For any regular connected scheme LaTeXMLMath , LaTeXMLMath will denote the field of rational functions on LaTeXMLMath .
There is an exact sequence LaTeXMLEquation which induces an exact sequence LaTeXMLEquation .
Note that LaTeXMLMath is a finite abelian group since it is finitely generated ( by the Mordell-Weil theorem ) and isomorphic to a subgroup of the torsion group LaTeXMLMath .
Similarly , LaTeXMLMath is finite and there is an exact sequence LaTeXMLEquation .
We also have an exact sequence LaTeXMLEquation where LaTeXMLMath is the index of LaTeXMLMath as defined previously .
We note that LaTeXMLMath may also be defined as the greatest common divisor of the degrees of the fields LaTeXMLMath over LaTeXMLMath such that LaTeXMLMath .
The period of LaTeXMLMath divides its index , i.e .
LaTeXMLMath , and LaTeXMLEquation .
Proof .
The first assertion of the lemma follows from the definitions .
Regarding the second , an application of the snake lemma to the commutative diagram LaTeXMLEquation yields , using ( 1 ) and ( 2 ) above , an exact sequence LaTeXMLEquation .
The lemma is now immediate .
∎ It is clear from the definitions that if LaTeXMLMath is any field containing LaTeXMLMath , then the index ( resp .
period ) of LaTeXMLMath divides the index ( resp .
period ) of LaTeXMLMath .
Thus for any prime LaTeXMLMath of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are , respectively , the index and period of LaTeXMLMath .
Further , since LaTeXMLMath for all but finitely many primes LaTeXMLMath ( see for example [ 18 , p. LaTeXMLMath , Remark 1.6 ] ) , we conclude that there are only finitely many primes LaTeXMLMath for which LaTeXMLMath .
The analogous statement with LaTeXMLMath replaced by LaTeXMLMath holds true as well , since LaTeXMLMath for each LaTeXMLMath .
We now write LaTeXMLMath ( resp .
LaTeXMLMath ) for the least common multiple of the integers LaTeXMLMath ( resp .
LaTeXMLMath ) .
Clearly , LaTeXMLMath and LaTeXMLMath .
Further , since LaTeXMLMath for each LaTeXMLMath as mentioned earlier , we have LaTeXMLEquation .
We now consider the map LaTeXMLEquation .
We have LaTeXMLEquation and LaTeXMLEquation .
Proof .
That the image of LaTeXMLMath is exactly LaTeXMLMath follows easily from the fact that LaTeXMLMath .
The rest of the lemma is clear.∎ Next we consider the map LaTeXMLEquation given by LaTeXMLMath .
Since the kernel of LaTeXMLMath is LaTeXMLMath , the following lemma is clear We have LaTeXMLEquation and LaTeXMLEquation .
For each prime LaTeXMLMath of LaTeXMLMath we will write LaTeXMLMath for the Galois group of LaTeXMLMath over LaTeXMLMath .
From the cohomology sequence associated to the exact sequence of LaTeXMLMath -modules LaTeXMLMath we get an exact sequence LaTeXMLEquation .
Now by work of Lichtenbaum [ 21 ] and Milne [ 26 ] ( see also [ 28 , Remark I.3.7 ] ) , there exists a perfect pairing LaTeXMLEquation where LaTeXMLMath denotes LaTeXMLMath if LaTeXMLMath is archimedean and LaTeXMLMath otherwise .
Relative to this pairing , the annihilator of the image of LaTeXMLMath in LaTeXMLMath under the canonical map LaTeXMLMath is exactly the image of LaTeXMLMath in LaTeXMLMath under the map in ( 3 ) .
Consequently the following holds We have LaTeXMLEquation .
We now combine the previous lemma with an analogue of Lemma 1.1 to obtain We have LaTeXMLEquation .
Remark .
The archimedean case of Lemma 1.5 was originally established by Witt in 1935 [ 50 ] .
For an interesting review of this and other related classical results in terms of étale cohomology , see [ 38 , §20.1 ] .
We now derive a slight variant of the snake lemma ( Proposition 1.6 below ) .
It is one of the basic ingredients of the proof of the Main Theorem .
Consider the following exact commutative diagram in the category of abelian groups LaTeXMLEquation ( we have labeled only those maps which are relevant to our purposes ) .
We have an induced exact commutative diagram LaTeXMLEquation where LaTeXMLMath is the composition of LaTeXMLMath with the canonical map LaTeXMLMath .
An application of the snake lemma to the above diagram using the fact that LaTeXMLMath yields To any exact commutative diagram of the form ( 4 ) there is associated an exact sequence LaTeXMLEquation where LaTeXMLMath is as defined above .
The foolowing result supplements Proposition 1.6 .
With the above notations , there is an exact sequence LaTeXMLEquation .
Proof .
This is nothing more than the kernel-cokernel sequence [ .. , I.0.24 ] for the pair of maps LaTeXMLMath .
The maps in the exact sequence of the lemma are the natural ones , e.g .
LaTeXMLMath is the composite LaTeXMLMath .∎ All cohomology groups below will be either Galois cohomology groups or étale cohomology groups .
We will view LaTeXMLMath as a subgroup of LaTeXMLMath in the standard way , i.e. , by identifying it with the decomposition group of some fixed prime of LaTeXMLMath lying above LaTeXMLMath .
For each LaTeXMLMath , LaTeXMLMath will denote the usual invariant map of local class field theory .
The LaTeXMLMath -torsion subgroup of an abelian group LaTeXMLMath will be denoted by LaTeXMLMath .
We begin by recalling a fundamental exact sequence .
Since LaTeXMLMath for all LaTeXMLMath [ 27 , Ex .
2.23 ( b ) , p. LaTeXMLMath 110 ] , the Hochschild-Serre spectral sequence LaTeXMLEquation yields ( see [ 7 , XV.5.11 ] ) an exact sequence LaTeXMLEquation where the zero at the right-hand end comes from the fact that LaTeXMLMath [ 29 , I.4.21 ] .
( We have used here the well-known facts that LaTeXMLMath and that the Brauer group of a regular scheme of dimension LaTeXMLMath agrees with the cohomological Brauer group of the scheme . )
Similarly , for each prime LaTeXMLMath of LaTeXMLMath there is an exact sequence LaTeXMLEquation .
For each prime LaTeXMLMath of LaTeXMLMath , we have LaTeXMLEquation .
Proof .
By Lemma 1.5 , LaTeXMLMath is a subgroup of LaTeXMLMath of order LaTeXMLMath .
On the other hand the invariant map LaTeXMLMath induces an isomorphism LaTeXMLMath , whence the lemma follows.∎ We now consider the commutative diagram LaTeXMLEquation where the direct sums extend over all primes of LaTeXMLMath , the vertical maps are the natural ones and LaTeXMLMath .
This diagram is of the form considered at the end of Section 2 , and we may therefore apply to it Proposition 1.6 above .
Before doing so , however , we call upon ( a ) There is an exact sequence LaTeXMLEquation ( b ) There is an exact sequence LaTeXMLEquation where LaTeXMLMath is the set of primes of LaTeXMLMath not corresponding to a point of LaTeXMLMath .
Proof .
Assertion ( a ) is one of the major theorems of class field theory .
See [ 47 , §11 ] .
Assertion ( b ) is proved in [ 28 , Lemma 2.6 ] .∎ We now apply Proposition 1.6 to the diagram above using the preceding lemma .
We get an exact sequence LaTeXMLEquation where LaTeXMLMath is induced by LaTeXMLMath and LaTeXMLEquation .
Now by Lemma 1.1 , the order of LaTeXMLMath equals LaTeXMLMath .
Regarding the kernel and cokernel of LaTeXMLMath , the following holds .
We have LaTeXMLEquation and the map LaTeXMLMath induces an isomorphism LaTeXMLEquation .
Proof .
By combining Lemmas 1.7 , 2.1 and 2.2 we obtain an exact sequence LaTeXMLEquation .
Now for each LaTeXMLMath the invariant map LaTeXMLMath induces an isomorphism LaTeXMLMath , and it follows that the kernel and cokernel of the middle map in the above exact sequence may be identified with the kernel and cokernel of the map LaTeXMLMath considered in Section 2 .
The proposition now follows from Lemma 1.2.∎ We summarize the results obtained so far .
There is an exact sequence LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are finite groups of orders LaTeXMLEquation .
LaTeXMLEquation We now prove Assume that the integers LaTeXMLMath are relatively prime in pairs .
Then LaTeXMLEquation .
Proof .
There is an exact commutative diagram LaTeXMLEquation in which the vertical maps are the natural ones and the nontrivial horizontal maps are induced by the maps LaTeXMLMath and LaTeXMLMath coming from ( 5 ) and ( 6 ) .
Arguing as in the proof of Lemma 2.1 ( using Lemma 1.4 in place of Lemma 1.5 there ) , we conclude that for each LaTeXMLMath the image of LaTeXMLMath in LaTeXMLMath equals LaTeXMLMath .
It follows that LaTeXMLMath injects into LaTeXMLMath , where LaTeXMLEquation is induced by LaTeXMLMath .
Now arguing as in the proof of Proposition 2.3 , we see that the order of LaTeXMLMath equals LaTeXMLMath .
By hypothesis this number is 1 , whence the theorem follows.∎ We now recall from the Introduction the integer LaTeXMLMath , which was defined to be the number of primes LaTeXMLMath of LaTeXMLMath for which LaTeXMLMath .
As noted just before the statement of Lemma 1.2 , LaTeXMLMath according as LaTeXMLMath or LaTeXMLMath .
Now we observe that LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
Consequently if the integers LaTeXMLMath are relatively prime in pairs , then LaTeXMLMath .
Thus Corollary 2.4 and Theorem 2.5 together imply Assume that the integers LaTeXMLMath are relatively prime in pairs .
Then there is an exact sequence LaTeXMLEquation in which LaTeXMLMath and LaTeXMLMath are finite groups of orders LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
We turn now to the problem of relating LaTeXMLMath to LaTeXMLMath .
There is an exact commutative diagram LaTeXMLEquation in which LaTeXMLMath is the diagonal map of Section 2 and the rows come from the cohomology sequences of LaTeXMLEquation over LaTeXMLMath and over LaTeXMLMath .
Applying the snake lemma to the above diagram yields the exact sequence LaTeXMLEquation .
Let LaTeXMLMath .
Then , by Lemma 1.3 , LaTeXMLEquation .
Further , the order of LaTeXMLMath is LaTeXMLMath .
Consequently the following holds Suppose that the integers LaTeXMLMath are relatively primes in pairs .
Then there is an exact sequence LaTeXMLEquation in which LaTeXMLMath is a finite group of order LaTeXMLEquation .
In what follows we will view LaTeXMLMath as a subgroup of LaTeXMLMath by identifying it with its image in LaTeXMLMath under the map in Proposition 2.7 .
Putting together Corollary 2.6 and Proposition 2.7 , we obtain Assume that the integers LaTeXMLMath are relatively prime in pairs .
Then there is an exact sequence LaTeXMLEquation in which LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are finite groups of orders LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath .
It remains only to compute LaTeXMLMath , the image of LaTeXMLMath in LaTeXMLMath under the map in Corollary 2.8 , or equivalently , the image of the composite map LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are the maps in Corollary 2.6 and Proposition 2.7 , respectively .
We will show that LaTeXMLMath has the order stated in the Introduction by generalizing [ 28 , Lemma 2.11 ] .
We begin by recalling from [ 28 , Remark 2.9 ] the explicit description of the map LaTeXMLMath .
We write LaTeXMLMath for the canonical map LaTeXMLMath .
Represent LaTeXMLMath by a cocycle LaTeXMLMath , and let LaTeXMLMath be such that LaTeXMLMath .
Then LaTeXMLMath and , because LaTeXMLMath , it can be pulled back to an element LaTeXMLMath ( here LaTeXMLMath ) .
On the other hand , LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be such that LaTeXMLMath .
Then LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLMath be the class of LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath is the canonical map LaTeXMLMath induced by the identity map on LaTeXMLMath .
We note that , for any divisor LaTeXMLMath on LaTeXMLMath such that neither LaTeXMLMath nor LaTeXMLMath has a zero or a pole in the support of LaTeXMLMath , LaTeXMLEquation ( see for example [ 21 , §4 ] ) .
Here LaTeXMLMath is the value at LaTeXMLMath of the cup-product pairing LaTeXMLEquation which is induced by the evaluation pairing LaTeXMLMath , and similarly for LaTeXMLMath .
Since LaTeXMLMath with LaTeXMLMath , we see that LaTeXMLMath is represented by LaTeXMLMath .
Now choose a divisor LaTeXMLMath on LaTeXMLMath of degree LaTeXMLMath such that neither LaTeXMLMath nor LaTeXMLMath has a zero or a pole in the support of LaTeXMLMath ( this is always possible ; see [ 19 , App .
2 ] ) .
Then LaTeXMLMath is represented by LaTeXMLMath .
We now recall from [ 29 , Remark I.6.12 ] the definition of a pairing LaTeXMLEquation which annihilates only the divisible part of LaTeXMLMath .
Let LaTeXMLMath be represented by LaTeXMLMath , and let LaTeXMLMath with LaTeXMLMath .
Write LaTeXMLEquation .
LaTeXMLEquation Then LaTeXMLMath in LaTeXMLMath with LaTeXMLMath .
Moreover , LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be a secod element of LaTeXMLMath and define LaTeXMLMath as for LaTeXMLMath .
Set LaTeXMLEquation where LaTeXMLMath denotes the cup-product pairing induced by the evaluation pairing mentioned earlier .
This definition is independent of the choices made , and LaTeXMLMath and LaTeXMLMath can be chosen so that LaTeXMLMath and LaTeXMLMath are defined .
Moreover , LaTeXMLMath .
Now let LaTeXMLMath be the natural isomorphism which is induced by multiplication by LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be a generator of LaTeXMLMath and let LaTeXMLMath .
Then the composite LaTeXMLEquation is LaTeXMLMath .
Proof .
Let LaTeXMLMath and define LaTeXMLMath and LaTeXMLMath as above .
Then LaTeXMLMath where LaTeXMLMath is represented by LaTeXMLMath for some divisor LaTeXMLMath of degree LaTeXMLMath on LaTeXMLMath .
On the other hand , we can choose LaTeXMLMath to be represented by LaTeXMLMath where LaTeXMLMath , LaTeXMLMath any closed point on LaTeXMLMath .
Define LaTeXMLEquation .
Then LaTeXMLMath satisfies LaTeXMLEquation .
Further , LaTeXMLMath , which means that we may choose LaTeXMLMath .
Thus LaTeXMLMath , where LaTeXMLMath is represented by LaTeXMLMath .
Consequently LaTeXMLEquation .
LaTeXMLEquation as claimed.∎ Assume that LaTeXMLMath contains no nonzero infinitely divisible elements .
Then LaTeXMLEquation .
Proof .
Under our hypothesis the pairing ( 9 ) is nondegenerate , and the proposition shows that LaTeXMLMath is isomorphic to the dual of LaTeXMLMath .
The corollary now follows from ( 8 ) .∎ The Main Theorem of the Introduction may now be obtained by combining Corollaries 2.8 and 2.10 .
As shown in the proof of Theorem 2.5 , the order of LaTeXMLMath divides LaTeXMLMath .
We believe that it is always equal to LaTeXMLMath .
This is equivalent to the following plausible statement .
Consider the nondegenerate pairing LaTeXMLEquation which is the sum of the local pairings LaTeXMLEquation induced by Lichtenbaum duality ( see the discussion preceding the statement of Lemma 1.4 ) .
Then , relative to this pairing , the image of the diagonal map LaTeXMLEquation is the exact annihilator of the image of the diagonal map LaTeXMLEquation .
Assuming ( for simplicity ) that LaTeXMLMath is finite , the above conjectural statement implies in addition that the exact sequence LaTeXMLEquation ( which is the continuation of the exact sequence ( 7 ) .
Here LaTeXMLMath and LaTeXMLMath are the natural maps LaTeXMLMath and LaTeXMLMath ) splits into two short exact sequences LaTeXMLEquation and LaTeXMLEquation the second of which is the dual of LaTeXMLEquation .
Thus if the above conjecture is true , then we can give a complete answer to Grothendieck ’ s question stated in the Introduction .
One of the connections between Number Theory and Mathematical Physics that emerged in recent years is Arithmetical Quantum Chaos .
On the physical side we have the problem of how the notion of chaos in classical dynamical systems can be transfered to quantum mechanical systems .
Here Gutzwiller ’ s trace formula is a useful quantitative tool which connects the lengths of closed orbits in the classical picture with the energy eigenvalues in the quantum mechanical picture .
Unfortunately it is an asymptotic relation without rigorous error estimates .
But in the special case of motion on a surface of constant negative curvature generated by a discontinuous group , Gutzwiller ’ s trace formula is Selberg ’ s trace formula and is therefore exact .
Furthermore , for particular choices of groups connections with well understood number theoretical objects can be exploited which are not available for general groups .
Numerical experiments led to surprising results : For a generic group the eigenvalue statistics of the Laplacian on the surface seem to be in accordance with the Gaussian Orthogonal Ensemble ( see Bohigas , Giannoni , Schmit LaTeXMLCite ) .
But for arithmetic groups it is closer to Poisson distribution ( LaTeXMLCite , Aurich , Steiner LaTeXMLCite ; see also LaTeXMLCite ) .
It seems that the high degeneracy of the length spectrum is responsible for the Poissonian behaviour of the eigenvalues .
Note that a similar phenomenon was observed by Hejhal and Selberg for quaternion groups ( LaTeXMLCite , Theorems 17.1 and 18.8 ) .
They used the high degeneracy of the length spectrum to prove exceptionally large lower bounds for integral means of the remainder term in Weyl ’ s asymptotic law for the eigenvalues .
Luo and Sarnak LaTeXMLCite used the same phenomenon in their study of the “ number variance ” ( the mean square for the remainder term in short intervals ) for general arithmetic ( not necessarily congruence ) groups .
They proved that for arithmetic groups , the length spectrum without multiplicities has at most linear growth and conjectured that this property characterizes arithmetic groups , a fact later proved by Schmutz LaTeXMLCite in the noncompact case .
In an attempt to understand the Poissonian behaviour for the arithmetic group LaTeXMLMath , Bogomolny , Leyvraz and Schmit LaTeXMLCite calculated the two point correlation function for the eigenvalues .
Their arguments are not mathematically rigorous and are given in two steps : First Selberg ’ s trace formula along with heuristic arguments is used to reduce the pair correlation function of the eigenvalues to that of the lengths of closed orbits .
Second a heuristic version of the Hardy-Littlewood method is used to express the latter correlation function as an infinite product with easily calculable factors .
Making the first step in LaTeXMLCite mathematically rigorous is extremely hard without a new methodological tool .
Selberg ’ s trace formula seems too weak for this purpose .
In the present paper it will be shown how the second step can be made rigorous with an approach different from LaTeXMLCite .
There is another way of looking at the result of this paper .
Since the structure of the Selberg trace formula is the same as that of Weil ’ s explicit formula in prime number theory ( see LaTeXMLCite ) one can look upon the lengths of closed geodesics as the logarithms of some sort of generalized primes .
Thus the theorem below is an analogue of the Hardy-Littlewood twin prime conjecture .
In order to state the main result , let LaTeXMLMath be the complex upper half plane and LaTeXMLMath .
This group acts discontinuously on LaTeXMLMath by Möbius transformations .
Let LaTeXMLMath be the Riemann surface generated by LaTeXMLMath .
The closed geodesics on LaTeXMLMath are in one-to-one correspondence to conjugacy classes of primitive hyperbolic elements in LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath , let LaTeXMLMath be the number of closed geodesics on LaTeXMLMath which correspond to primitive conjugacy classes with trace LaTeXMLMath .
Set LaTeXMLMath .
Then the function LaTeXMLMath has mean value LaTeXMLMath .
Set LaTeXMLMath .
For LaTeXMLMath , the limit LaTeXMLEquation exists .
Its value is given by LaTeXMLEquation where for a prime LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Furthermore , for LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This theorem was already stated in LaTeXMLCite and was made plausible on heuristic grounds .
In the present paper , a different approach is used which exploits the connection of LaTeXMLMath with a sum of class numbers of primitive binary quadratic forms ( see ( LaTeXMLRef ) below ) .
The main step is to show that LaTeXMLMath is almost periodic and to compute its Fourier coefficients .
Then the theorem follows from Parseval ’ s equation .
To this end a method is used that has already been applied in LaTeXMLCite and LaTeXMLCite .
It might be of interest to note that almost periodic functions — albeit on the real line — already found applications to trace formulas for integrable geodesic flows ( see LaTeXMLCite for an overview ) .
The present paper is organized as follows : In section 2 the function LaTeXMLMath is reduced to class numbers .
Section 3 contains a short review of almost periodic arithmetical functions .
Furthermore , the function LaTeXMLMath is shown to be almost periodic by approximating it with a carefully choosen periodic function .
This will considerably simplify the computation of the product representation of LaTeXMLMath .
In section 4 this Euler product is derived and its local factors are computed in section 5 .
Acknowledgement .
I would like to express my sincere gratitude to Prof. Zeév Rudnick for bringing this problem to my attention .
I gained much from conversations with him and from the stimulating atmosphere which he and his co-organizers created at the DMV seminar “ The Riemann Zeta Function and Random Matrix Theory ” .
In order to state the connection with class numbers we need some notation .
Proofs of the number theoretic facts in this section can be found for example in LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
The letter LaTeXMLMath will always stand for a positive non-square discriminant ( i.e .
LaTeXMLMath ) .
A primitive binary quadratic form is a polynomial LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Its discriminant is LaTeXMLMath .
Two such forms LaTeXMLMath , LaTeXMLMath , are called equivalent if there is a matrix LaTeXMLMath with LaTeXMLMath .
A fundamental theorem in number theory now states that the number LaTeXMLMath of equivalence classes of primitive binary quadratic forms with discriminant LaTeXMLMath is finite .
This number is one of the important quantities in number theory since it appears in a surprising variety of situations .
The Pellian equation LaTeXMLMath has infinitely many solutions in integers LaTeXMLMath .
The solution LaTeXMLMath with LaTeXMLMath and LaTeXMLMath minimal is called the fundamental solution since all the other solutions can be generated from it by a simple law .
Set LaTeXMLMath .
The trace of LaTeXMLMath is defined as LaTeXMLMath .
The one-to-one correspondence between primitive conjugacy classes in LaTeXMLMath and closed geodesics on LaTeXMLMath is as follows : Every primitive hyperbolic LaTeXMLMath has two real fixed points ( one of them can be LaTeXMLMath ) .
The orthogonal circle in LaTeXMLMath which ends in these fixed points induces a closed geodesic on LaTeXMLMath of length LaTeXMLMath where LaTeXMLMath .
The possible values for LaTeXMLMath and their multiplicities are described in The lengths of closed geodesics on LaTeXMLMath are the numbers LaTeXMLMath with multiplicities LaTeXMLMath .
From this proposition and the preceding description it follows that LaTeXMLEquation .
Quantitative results about class numbers are often derived using Dirichlet ’ s class number formula .
First we must define Jacobi ’ s character and Dirichlet L-series .
Define LaTeXMLMath to be completely multiplicative with LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath is a Dirichlet character modulo LaTeXMLMath ( the quadratic reciprocity law is used to prove the LaTeXMLMath -periodicity ) .
For LaTeXMLMath , the series LaTeXMLEquation is uniformly convergent and defines a holomorphic function .
For a positive non-square discriminant LaTeXMLMath , we have LaTeXMLEquation .
In the next section this formula will be used to prove the almost periodicity of LaTeXMLMath .
In principle this could be done by writing LaTeXMLEquation .
Note that the sum on the right hand side is a periodic function of LaTeXMLMath and can be used to approximate LaTeXMLMath .
But this procedure has three severe drawbacks : First the approximation is not particularly good .
Therefore we will use a smoothed version of the series representation of LaTeXMLMath instead .
Second we must approximate a sum of values LaTeXMLMath with the condition LaTeXMLMath .
Breaking up this condition into easier summations , as must be done to process it further , would make things much more difficult .
Third we want to compute the Fourier coefficients of LaTeXMLMath and show that they are multiplicative .
This would be near to impossible with the above approach .
Instead an approximating periodic function is used which already incorporates some sort of multiplicativity .
From the multiplicativity of LaTeXMLMath the Euler product LaTeXMLEquation follows .
Thus it seems more reasonable to use a partial product of this representation than a partial sum of the series as approximating function .
The standard reference for almost periodic arithmetical functions is LaTeXMLCite .
Here the necessary material will be reviewed briefly .
Let LaTeXMLMath .
For LaTeXMLMath , define the seminorm LaTeXMLEquation .
LaTeXMLMath is called LaTeXMLMath -limit periodic if for every LaTeXMLMath there is a periodic function LaTeXMLMath with LaTeXMLMath .
The set LaTeXMLMath of all LaTeXMLMath -limit periodic functions becomes a Banach space with norm LaTeXMLMath if functions LaTeXMLMath with LaTeXMLMath are identified .
If LaTeXMLMath , we have LaTeXMLMath as sets ( but they are endowed with different norms ! ) .
There is the more general notion of LaTeXMLMath -almost periodic function which will not be used in this paper ( they are defined as above but with arbitrary trigonometric polynomials for LaTeXMLMath instead of periodic functions ) .
For all LaTeXMLMath , the mean value LaTeXMLEquation exists .
The space LaTeXMLMath is a Hilbert space with inner product LaTeXMLEquation .
For LaTeXMLMath , define LaTeXMLMath , LaTeXMLMath .
For all LaTeXMLMath , the Fourier coefficients LaTeXMLMath , LaTeXMLMath , exist .
For LaTeXMLMath , we have LaTeXMLMath ( this comes from the fact that LaTeXMLMath can be approximated by linear combinations of functions LaTeXMLMath with LaTeXMLMath , and that LaTeXMLMath equals LaTeXMLMath if LaTeXMLMath and LaTeXMLMath otherwise ; for almost periodic functions it is no longer true ) .
In LaTeXMLMath , we have the canonical orthonormal base LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
Limit periodic ( and almost periodic ) functions have a couple of nice properties .
They can be added , multiplied and plugged into continuous functions and , under certain conditions , the result is again a limit periodic ( almost periodic ) function .
They have mean values and limit distributions .
Here we will use Parseval ’ s equation .
In Corollary LaTeXMLRef we will prove that LaTeXMLMath for all LaTeXMLMath .
As a side result in section 5 , we get LaTeXMLMath .
Thus for LaTeXMLMath , we have LaTeXMLMath ( where LaTeXMLMath ) , and Parseval ’ s equation gives LaTeXMLEquation .
LaTeXMLEquation The last equation follows easily since LaTeXMLMath is real valued .
If we can compute LaTeXMLMath and show that it is multiplicative our main theorem follows .
The approximation of LaTeXMLMath by periodic functions is done in two steps .
First we bring Dirichlet L-series into the picture .
For LaTeXMLMath , LaTeXMLMath , define LaTeXMLEquation where we must remember that LaTeXMLMath will always run through non-square positive discriminants .
For LaTeXMLMath , we have LaTeXMLMath .
For LaTeXMLMath fixed , the powers LaTeXMLMath , LaTeXMLMath , of the fundamental unit give all solutions LaTeXMLMath of the Pellian equation LaTeXMLMath with LaTeXMLMath by way of the rule LaTeXMLEquation .
For every such solution we have LaTeXMLMath .
Thus Proposition LaTeXMLRef gives LaTeXMLEquation .
Since LaTeXMLEquation it follows from ( LaTeXMLRef ) that LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath ( this can be seen , e.g .
from Proposition LaTeXMLRef and the estimates LaTeXMLMath and LaTeXMLMath ; the latter follows easily by partial summation from the orthogonality relation for characters ) , it follows that for LaTeXMLMath with LaTeXMLMath , LaTeXMLEquation .
Thus LaTeXMLEquation for all LaTeXMLMath ; here LaTeXMLMath denotes the divisor function and LaTeXMLMath was used .
This implies LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation which proves the lemma .
∎ In the crucial second step LaTeXMLMath is approximated by a sum of partial products of the Euler product of LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath , define LaTeXMLEquation .
Then LaTeXMLMath , where LaTeXMLEquation and LaTeXMLEquation .
Fix LaTeXMLMath with LaTeXMLMath and choose LaTeXMLMath with LaTeXMLMath .
The next lemma shows that LaTeXMLMath is negligible in the LaTeXMLMath -norm as LaTeXMLMath .
For LaTeXMLMath , we have LaTeXMLEquation .
Hölder ’ s inequality gives LaTeXMLEquation .
For LaTeXMLMath , this gives LaTeXMLEquation .
The second sum on the right hand side is LaTeXMLEquation as LaTeXMLMath ( see LaTeXMLCite ) .
This means that the values LaTeXMLMath are constant in the mean when ordered according to the sizes of their fundamental units .
Thus the lemma follows .
∎ In order to estimate LaTeXMLMath we must compare LaTeXMLMath with a partial product of its Euler product .
This is done by comparing both terms with a smoothed version of the Dirichlet series for LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation here the conditions on LaTeXMLMath and LaTeXMLMath are as in LaTeXMLMath .
The third term can be estimated easily .
For LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
Since LaTeXMLMath for LaTeXMLMath , we see that for LaTeXMLMath the inner sum in LaTeXMLMath is LaTeXMLEquation .
LaTeXMLEquation Hölder ’ s inequality now gives LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath was used .
The last sum is LaTeXMLEquation ( see LaTeXMLCite or LaTeXMLCite ) , and the lemma follows .
∎ So far we did not use the oscillation of the Jacobi character .
For the estimation of LaTeXMLMath we must take it into account .
For LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation where LaTeXMLMath is the squarefree kernel of LaTeXMLMath and LaTeXMLMath .
See LaTeXMLCite , estimate ( 2.7 ) .
∎ For LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation where LaTeXMLMath .
Split the first sum in LaTeXMLMath into two sums depending on whether LaTeXMLMath or LaTeXMLMath .
Thus LaTeXMLMath .
A trivial estimate gives LaTeXMLEquation and thus LaTeXMLEquation since LaTeXMLMath .
Hölder ’ s inequality gives LaTeXMLEquation .
LaTeXMLEquation Thus for LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation Applying Lemma LaTeXMLRef to the innermost sum gives the estimate LaTeXMLEquation .
LaTeXMLEquation which together with ( LaTeXMLRef ) proves the lemma .
∎ In order to estimate LaTeXMLMath we must show that the error LaTeXMLEquation which comes from smoothing the Dirichlet series expansion of LaTeXMLMath , is small for large LaTeXMLMath .
This is done by representing LaTeXMLMath as an integral over a vertical line in the critical strip LaTeXMLMath and using information about the location of the non-trivial zeros of LaTeXMLMath .
The Dirichlet series for LaTeXMLMath is absolutely and uniformly convergent on the line LaTeXMLMath .
Sterling ’ s formula gives LaTeXMLEquation for LaTeXMLMath , LaTeXMLMath , with some constant LaTeXMLMath .
Using Mellin ’ s formula LaTeXMLEquation we get LaTeXMLEquation .
On the other hand , for LaTeXMLMath , we have LaTeXMLEquation ( this is easily seen by partial summation ) .
Thus the line of integration in ( LaTeXMLRef ) may be moved to the line LaTeXMLMath with some LaTeXMLMath .
Taking into account the pole of the integrand at LaTeXMLMath , and using the residue theorem we get for ( LaTeXMLRef ) the expression LaTeXMLEquation and thus LaTeXMLEquation .
We must now considerably reduce the exponent LaTeXMLMath of LaTeXMLMath in ( LaTeXMLRef ) .
In order to see the principle let us first assume the Generalized Riemann Hypothesis which says that for all Dirichlet characters LaTeXMLMath modulo LaTeXMLMath , the L-series LaTeXMLMath has only zeros with real part LaTeXMLMath in the critical strip LaTeXMLMath .
From this the Generalized Lindelöf Hypothesis follows and in particular for all LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
From ( LaTeXMLRef ) it follows that for LaTeXMLMath , LaTeXMLEquation .
This shows that LaTeXMLEquation and for LaTeXMLMath and LaTeXMLMath , we get LaTeXMLEquation .
Here it is important that the exponent of LaTeXMLMath is negative .
Taking LaTeXMLMath to be a power of LaTeXMLMath with small exponent therefore lets the contribution of LaTeXMLMath vanish as LaTeXMLMath .
The next lemma gives an estimate which for our purposes is as good as ( LaTeXMLRef ) and can be proved unconditionally .
There are LaTeXMLMath such that for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
Choose LaTeXMLMath with LaTeXMLMath .
Choose LaTeXMLMath .
Define the rectangle LaTeXMLEquation .
If LaTeXMLMath and LaTeXMLMath has no zeros in LaTeXMLMath then a standard argument ( see , for example , Titchmarsh LaTeXMLCite , Theorem 14.2 ) shows that for LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation with some constant LaTeXMLMath depending on LaTeXMLMath .
Together with ( LaTeXMLRef ) and ( LaTeXMLRef ) this gives LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Next we must show that LaTeXMLMath can not have a zero in LaTeXMLMath too often .
From zero density estimates it follows that LaTeXMLEquation ( see LaTeXMLCite , Lemma 4.11 or LaTeXMLCite , estimate ( 2.6 ) ) .
A trivial estimation of ( LaTeXMLRef ) gives LaTeXMLEquation ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) and Hölder ’ s inequality give LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation which proves the lemma .
∎ Now the results are collected .
For LaTeXMLMath , we have LaTeXMLEquation .
For LaTeXMLMath , choose LaTeXMLMath .
Then Lemmas LaTeXMLRef , LaTeXMLRef and LaTeXMLRef show that LaTeXMLEquation .
LaTeXMLEquation Since the series LaTeXMLMath converges , we have for LaTeXMLMath fixed LaTeXMLEquation .
Together with Lemma LaTeXMLRef this proves the proposition .
∎ In this section we exploit the particular construction of LaTeXMLMath by writing it as a product of functions each depending only on a single prime .
For LaTeXMLMath a prime , LaTeXMLMath and LaTeXMLMath , set LaTeXMLMath if LaTeXMLMath and , in case LaTeXMLMath , if LaTeXMLMath is a discriminant ( for LaTeXMLMath this is automatically fulfilled ) .
Set LaTeXMLMath otherwise .
Define LaTeXMLEquation .
For LaTeXMLMath , we have LaTeXMLMath .
In the definition of LaTeXMLMath , write LaTeXMLMath with LaTeXMLMath .
There is a discriminant LaTeXMLMath with LaTeXMLMath iff LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath for LaTeXMLMath , the last condition is equivalent to LaTeXMLMath .
If these conditions are fulfilled , we have LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , for LaTeXMLMath .
Thus LaTeXMLMath for LaTeXMLMath .
This proves the lemma .
∎ For LaTeXMLMath , we have LaTeXMLMath .
In particular , LaTeXMLMath with respect to the LaTeXMLMath -norm .
For LaTeXMLMath with LaTeXMLMath fixed and LaTeXMLMath with LaTeXMLMath it follows from Lemma LaTeXMLRef and Proposition LaTeXMLRef that LaTeXMLEquation here LaTeXMLEquation as LaTeXMLMath since LaTeXMLMath .
Furthermore , LaTeXMLEquation as LaTeXMLMath , since LaTeXMLEquation .
Thus LaTeXMLMath .
For LaTeXMLMath arbitrary and LaTeXMLMath , we have LaTeXMLMath by Hölder ’ s inequality .
Thus LaTeXMLMath for all LaTeXMLMath .
Since the LaTeXMLMath -th summand of LaTeXMLMath is LaTeXMLMath -periodic ( LaTeXMLMath -periodic in case LaTeXMLMath ) and the series representing LaTeXMLMath is uniformly convergent , the function LaTeXMLMath is uniformly limit periodic , i.e .
LaTeXMLMath ; here LaTeXMLMath is the set of all functions which can be approximated to an arbitrary accuracy by periodic functions with respect to the supremum norm .
Since LaTeXMLMath is closed under multiplication it follows from Lemma LaTeXMLRef that LaTeXMLMath for all LaTeXMLMath .
This gives LaTeXMLMath for all LaTeXMLMath .
∎ Next the Fourier coefficients of LaTeXMLMath are computed in terms of the Fourier coefficients of the LaTeXMLMath .
In particular , this will show their multiplicativity .
For all primes LaTeXMLMath , we have LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , choose LaTeXMLMath for all LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLEquation .
From Corollary LaTeXMLRef it follows that for arbitrary LaTeXMLMath there is some LaTeXMLMath with LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
For all LaTeXMLMath it follows from ( LaTeXMLRef ) that there is some LaTeXMLMath and coefficients LaTeXMLMath , LaTeXMLMath , such that LaTeXMLEquation where LaTeXMLMath denotes the supremum norm and LaTeXMLMath .
From Lemma LaTeXMLRef it follows that LaTeXMLEquation .
Thus LaTeXMLEquation .
For LaTeXMLMath we have LaTeXMLMath .
Furthermore , LaTeXMLMath iff LaTeXMLMath for LaTeXMLMath .
Therefore the orthogonality relation for the exponential function gives LaTeXMLEquation .
Similarly , LaTeXMLMath for LaTeXMLMath and thus LaTeXMLEquation .
This gives LaTeXMLEquation .
In the next section we will compute LaTeXMLMath and thereby show that LaTeXMLMath .
This gives ( a ) and LaTeXMLEquation .
Since LaTeXMLMath is arbitrary , ( b ) follows .
∎ From Corollary LaTeXMLRef it follows that ( LaTeXMLRef ) holds .
Here the series on the right hand side is absolutely convergent ( plug in LaTeXMLMath ) .
Thus Lemma LaTeXMLRef gives LaTeXMLEquation where for LaTeXMLMath prime and LaTeXMLMath , we define LaTeXMLEquation .
The last step is to calculate LaTeXMLMath .
In particular , this will show that LaTeXMLMath which is left over from the proof of Lemma LaTeXMLRef .
For LaTeXMLMath prime , LaTeXMLMath , define LaTeXMLEquation .
Then LaTeXMLMath , where the series is uniformly convergent .
The calculation will only be done for LaTeXMLMath .
The case LaTeXMLMath is similar but somewhat more elaborate .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Case 1 : LaTeXMLMath .
Since LaTeXMLMath is LaTeXMLMath -periodic , we have LaTeXMLMath .
Case 2 : LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation The cardinality of the first set is LaTeXMLMath and that of the second is LaTeXMLMath ( see for example the proof of Lemma 3.3 in LaTeXMLCite ) .
Thus LaTeXMLMath .
Case 3 : LaTeXMLMath , LaTeXMLMath .
Define LaTeXMLEquation .
Then LaTeXMLEquation .
LaTeXMLEquation Case 4 : LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation .
Setting LaTeXMLMath gives LaTeXMLEquation where LaTeXMLMath denotes the Legendre symbol .
Case 4.1 : LaTeXMLMath .
We have LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath if LaTeXMLMath and LaTeXMLMath otherwise .
The last sum can be reduced to the Gaussian sum associated to the Legendre character which can be computed explicitely ( see for example LaTeXMLCite , Chapter 2 ) .
This gives for the above quantity the value LaTeXMLEquation .
Therefore LaTeXMLEquation .
Case 4.2 : LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Now we can calculate the Fourier coefficients LaTeXMLEquation .
Case 1 : LaTeXMLMath .
Then LaTeXMLEquation .
Case 2 : LaTeXMLMath .
Since LaTeXMLMath , we get LaTeXMLEquation .
LaTeXMLEquation Case 3 : LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Finally LaTeXMLMath can be computed .
Case 1 : LaTeXMLMath .
Then LaTeXMLEquation .
The innermost sum is LaTeXMLMath for LaTeXMLMath and LaTeXMLMath otherwise .
Since LaTeXMLEquation the value for LaTeXMLMath follows as given in the theorem .
Case 2 : LaTeXMLMath .
Then for LaTeXMLMath , we have LaTeXMLEquation .
Thus LaTeXMLEquation .
A short calculation gives the value in the theorem .
∎
TAUP 2668-2001 Relativistic Mechanics of Continuous Media S. Sklarz Department of Chemical Physics , Weizmann Institute of Science , Rehovot 76100 , Israel shlomo.sklarz @ weizmann.ac.il L. P. Horwitz Raymond and Beverly Sackler Faculty of Exact Science , School of physics , Tel Aviv University , Ramat Aviv 69978 , Israel In this work we study the relativistic mechanics of continuous media on a fundamental level using a manifestly covariant proper time procedure .
We formulate equations of motion and continuity ( and constitutive equations ) that are the starting point for any calculations regarding continuous media .
In the force free limit , the standard relativistic equations are regained , so that these equations can be regarded as a generalization of the standard procedure .
In the case of an inviscid fluid we derive an analogue of the Bernoulli equation .
For irrotational flow we prove that the velocity field can be derived from a potential .
If in addition , the fluid is incompressible , the potential must obey the d ’ Alembert equation , and thus the problem is reduced to solving the d ’ Alembert equation with specific boundary conditions ( in both space and time ) .
The solutions indicate the existence of light velocity sound waves in an incompressible fluid ( a result known in previous literature LaTeXMLCite ) .
Relaxing the constraints and allowing the fluid to become linearly compressible one can derive a wave equation , from which the sound velocity can again be computed .
For a stationary background flow , it has been demonstrated that the sound velocity attains its correct values for the incompressible and nonrelativistic limits .
Finally viscosity is introduced , bulk and shear viscosity constants are defined , and we formulate equations for the motion of a viscous fluid .
We wish to formulate manifestly covariant equations for the motion of a continuum on a fundamental level , and to demonstrate their use in an interesting problem .
In attempting to reach a consistent theory one meets a few difficulties : The general motion of a continuum may include accelerations , making the use of standard special relativistic dynamics problematic .
The usual procedure for dealing with an accelerating particle , would be to go from one inertial frame to the next , as the particle accelerates , always keeping it instantaneously in an inertial frame LaTeXMLCite .
In the case of a continuum however , this procedure is impossible , for each of the infinitesimal volume elements accelerates , in general , at different rates and directions , and so only one specific portion of the volume can be kept at rest at any particular time .
Secondly the above procedure may not be manifestly covariant , as an accelerating particle momentarily at rest in some frame , is obviously not equivalent to an identical particle moving at constant speed .
The accelerating particle , for instance , will be radiating if it is charged .
Our claim is that keeping a particle momentarily at rest can , at best , yield an approximation to the real motion , but is by no means a consistent covariant treatment of the system .
Another severe difficulty in writing down relativistic equations of hydrodynamics lies in the relativistic property of simultaneity .
In order to define an extended body in mathematical terms , a certain configuration of the body is chosen arbitrarily and is referred to as the reference configuration .
In the nonrelativistic theory , each particle , or infinitesimal volume element of the body , can then be labeled throughout the evolution by its position in that reference configuration .
The difficulty in constructing a relativistic theory of hydrodynamics lies in the fact that each inertial system will have its own notion of time and simultaneity , and they will not be able to agree on any set of particles to be considered as existing simultaneously at any given time LaTeXMLMath .
It therefore becomes impossible to define a consistent reference configuration .
In 1941 Stueckelberg LaTeXMLCite postulated that events ( not particles ) are to be considered as the fundamental dynamical objects of the theory of motion .
The dynamics of an event is governed by 8 independent parameters consisting of the space-time coordinates LaTeXMLMath take the values 1 , 2 , 3 , related to Euclidean space , whereas Greek indices LaTeXMLMath take the values 0 , 1 , 2 , 3 and correspond to the four dimensions of Minkowski space-time LaTeXMLMath and energy momenta LaTeXMLMath associated with each event , and parameterized by an invariant time .
Based on this , Horwitz and Piron in 1973 proposed that there exists a universal time LaTeXMLMath , by means of which dynamical interactions are correlated LaTeXMLCite .
This universal time may be identified with the Robertson-Walker time of the expansion of the universe , the time read on a freely falling ideal clock .
The evolution of a system of LaTeXMLMath events through LaTeXMLMath is generated by an invariant function LaTeXMLMath on this LaTeXMLMath dimensional phase space , by means of the covariant Hamilton equations LaTeXMLCite : LaTeXMLEquation .
The collection of events along each world line corresponds to a particle , and hence the evolution of the state of the LaTeXMLMath -event system describes , a posteriori , the history in space and time of an LaTeXMLMath -particle system .
In the case of a free particle , for instance , we choose for the Hamiltonian : LaTeXMLEquation where LaTeXMLMath is a positive parameter , which is a given intrinsic property of an event , having the dimension of mass and emerging as the Galilean target mass in the nonrelativistic limit LaTeXMLMath .
This results in LaTeXMLEquation so that we have for the physical velocity ( we take LaTeXMLMath in what follows unless otherwise specified ) , LaTeXMLEquation in total agreement with the usual Einstein kinematics .
As LaTeXMLMath are each taken to be independent variables of the motion , the particles are not necessarily on mass shell , and the particle mass squared : LaTeXMLMath , is not contrained to be constant ( except for a free particle ) , but is rather to be determined as a solution of the dynamical equations .
Note that ( we use the metric LaTeXMLMath ) LaTeXMLEquation and is unity only for the “ mass-shell ” value LaTeXMLMath , Hence LaTeXMLMath corresponds to the proper time of an ideal free clock on its mass shell .
On mass shell , the time obeys the relation : LaTeXMLEquation so that LaTeXMLMath is precisely the time interval measured in the laboratory between two signals emitted by a source , traveling with velocity LaTeXMLMath , with interval LaTeXMLMath , according to the Lorenz transformation .
If the emitter is not on shell there is a factor LaTeXMLMath .
The red shift imposed by general relativity , LaTeXMLMath is another example of the effect of dynamics , in this case , the acceleration due to gravity ( contained in LaTeXMLMath ) , on the observed time interval .
The observed time interval is therefore influenced by forces LaTeXMLCite that may move the energy momentum off shell .
This theory has since been applied to fundamental questions such as the Newton-Wigner position operator and the Landau-Peierls relation LaTeXMLCite , and to dynamical problems such as two body bound states LaTeXMLCite and scattering and electromagnetic interactions in classical mechanics LaTeXMLCite , quantum mechanics , and quantum field theory LaTeXMLCite .
In order to formulate equations of motion for continuous media , we therefore turn to the above mentioned procedure of Stueckelberg , Horwitz and Piron LaTeXMLCite , and all following discussions will be conducted in the framework of this theory .
As a consequence of the existence of a universal time which parameterizes all systems , it becomes possible to define a certain configuration of events in space-time LaTeXMLMath , at some definite initial universal time LaTeXMLMath , and then follow the evolution of the system through LaTeXMLMath .
The universality of LaTeXMLMath eliminates any ambiguity as to the identification of events and their distribution in space-time .
Labeling the various events by their initial configuration LaTeXMLMath , one can thus express their respective positions at any later time LaTeXMLMath : LaTeXMLEquation .
The velocity and acceleration of an event labeled LaTeXMLMath at time LaTeXMLMath is thus : LaTeXMLEquation .
LaTeXMLEquation The Eulerian velocity field describes the velocities of events passing a certain point in space-time , as a function of the universal time : LaTeXMLEquation .
We now wish to express the acceleration in terms of the Eulerian velocity field : LaTeXMLEquation .
LaTeXMLEquation In space-time , due to mass-energy equivalence , there is no a priori conservation of mass of individual particles ; rather we shall be dealing with a “ conservation of events ” .
Let LaTeXMLMath , be the density of events ( this quantity is a scalar since the four volume element is invariant ) at the space-time point LaTeXMLMath at time LaTeXMLMath multiplied by LaTeXMLMath , the intrinsic , given mass dimension property attributed to each event .
Consider now a small volume of fluid .
The flux of events through a closed surface enclosing the volume must cause a decrease in the density of events within the volume .
It is in place to stress at this point that the surface mentioned is a three dimensional surface , the direction of which is defined by a four vector normal to that surface in four dimensional space-time .
Hence , for example , we define LaTeXMLMath as a volume of three dimensional space , its direction being the time direction .
Likewise , the volume over which the integration is carried out is obviously the four dimensional space-time .
The flux of events past an infinitesimal surface LaTeXMLMath is : LaTeXMLMath , where LaTeXMLMath depending as to whether LaTeXMLMath is space-like ( LaTeXMLMath ) or time-like ( LaTeXMLMath ) .
The reason for introducing this additional sign is that for LaTeXMLMath and LaTeXMLMath both pointing out of the bounded volume , in a time-like direction , the vector product LaTeXMLMath is negative whereas the flux of events must be positive LaTeXMLCite .
Therefore : LaTeXMLEquation or , using the four-dimensional Gauss law , LaTeXMLCite LaTeXMLEquation we have , LaTeXMLEquation and so taking the volume to be infinitely small : LaTeXMLEquation which is the equation of continuity for the flow field .
Note that in the case where the density of events LaTeXMLMath is constant , the above equation reduces to : LaTeXMLEquation .
In a continuum we focus on an infinitesimal volume element and describe the forces acting on it .
In general one talks about two kinds of forces : Forces which are proportional to the density of mass , charge or number of events , within the volume .
The force will therefore be LaTeXMLMath , where LaTeXMLMath is the density of events .
Surface contact forces , which are exerted by the continuum of events surrounding the volume under consideration , and are transferred through its surface .
The number of events neighboring a small portion of the surface , LaTeXMLMath , in a locally homogenous fluid , are obviously proportional to its size , hence we can assume the force to be proportional to the surface it is acting upon .
In mathematical terms we assert that the force LaTeXMLMath on an element of surface LaTeXMLMath tends to zero as LaTeXMLMath tends to zero , but that the ratio LaTeXMLMath tends to a definite limit .
This assumption enables us to define a stress four tensor , the components of which describe the force in the LaTeXMLMath direction , exerted on a surface in the LaTeXMLMath direction : LaTeXMLEquation .
The surface force on an elementary volume unit , in the LaTeXMLMath direction is therefore ( by the usual argument ) : LaTeXMLEquation .
Summing up the forces acting on an infinitesimal volume element and equating to the density of acceleration within the volume , we get the equations of motion : LaTeXMLEquation .
By introducing a manifestly covariant Boltzmann equation , Horwitz , Shashoua and Schieve LaTeXMLCite were able to write down equations for conserved quantities in a statistical mechanical framework .
It is interesting to note that the equations of continuity and motion derived above from a hydrodynamical point of view are equivalent to those presented by these authors for the conservation of some constant quantity and for the conservation of momentum respectively .
In order to make these equations of motion operational it is necessary to formulate constitutive equations for the fluid .
These will define the way by which the stress forces acting at a certain point in the fluid LaTeXMLMath , are governed by its physical state ( and/or history ) .
These laws however are determined by physical properties of the material ; we shall make some simplifying assumptions about the nature of the fluid , comparable to those which lead to the Navier-Stokes equations in the non-relativistic case .
In the following work we shall formulate a potential theory for inviscid , irrotational , incompressible fluids .
Then , relaxing the constraints , we shall allow the fluid to become linearly compressible , an assumption which will lead to acoustic modes within the flow .
Finally we shall introduce equations for the motion of viscous flow .
In the current section we shall show how Eq .
( LaTeXMLRef ) and ( LaTeXMLRef ) transform into the standard relativistic equations of hydrodynamics , which in turn give the nonrelativistic limit by taking LaTeXMLMath .
We begin with a few preliminary remarks which will assist us in what follows .
Note the following connection between the relativistic force term LaTeXMLMath and its nonrelativistic counterpart LaTeXMLMath : LaTeXMLEquation .
Taking the space components we can write LaTeXMLEquation .
A rearrangement of terms gives the physically measureable force LaTeXMLMath in terms of the relativistic force , LaTeXMLEquation .
Another useful expression which connects derivatives of the velocity field LaTeXMLMath with its nonrelativistic physically observable counterpart LaTeXMLMath , will be used later and can be proved as follows ( we denote by LaTeXMLMath a derivative with respect to any variable ) : LaTeXMLEquation .
We remind ourselves too , that LaTeXMLMath , and that in the nonrelativistic ( on-shell ) limit LaTeXMLMath .
To put our equations into correspondence with standard results , we must note that any interaction with an apparatus or some macroscopical object corresponds to an interaction with a worldline , an integration over LaTeXMLMath is therefore necessary .
In performing this procedure to obtain the on shell limit , we assume that correlations in LaTeXMLMath survive only over a short interval , thus a term of the form : LaTeXMLMath can be factored to obtain LaTeXMLMath in a LaTeXMLMath averaged sense .
LaTeXMLCite .
LaTeXMLMath and LaTeXMLMath have support only arround the zero mode then LaTeXMLEquation .
LaTeXMLEquation Applying this procedure to equations ( LaTeXMLRef ) and ( LaTeXMLRef ) causes the LaTeXMLMath derivative terms in both equations to drop due to the assumption that both LaTeXMLMath and LaTeXMLMath vanish pointwise as LaTeXMLMath .
From Eq .
( LaTeXMLRef ) we then get : LaTeXMLEquation .
LaTeXMLEquation where all quantities are from now to be understood as being LaTeXMLMath averaged .
Written in the form ( LaTeXMLRef ) , Eq .
( LaTeXMLRef ) can be interpreted as expressing the conservation of energy density ( or the continuity of energy flow ) , whereas the second form ( LaTeXMLRef ) states that the conservation of mass is to be corrected by a relativistic term .
This can be seen by noting that the first bracketed term in ( LaTeXMLRef ) , is just the regular nonrelativistic equation of continuity which gives the change in time of the mass of a small volume of three dimensional space .
In nonrelativistic hydrodynamics this term should obviously be zero , for the mass is conserved .
This however is not the case for a relativistic fluid , as an increase in the energy also has to be taken into account in the conservation law .
This is precisely the meaning of the second term .
It expresses the change of energy of the small volume of fluid moving with the particle .
This change of energy causes a corresponding non vanishing change of mass which must balance it out in such a way that the total be zero .
It should be noticed that the derivatives of LaTeXMLMath which appear in the second term of ( LaTeXMLRef ) are in the standard relativistic limit , of the order LaTeXMLMath , and so are small compared to the first term .
Therefore when taking the nonrelativistic limit this second term can be neglected , and one is left with the standard equation for the conservation of mass : LaTeXMLEquation .
We shall now treat the space and time components of Eq .
( LaTeXMLRef ) separately .
We consider a fluid free of any external forces and therefore take LaTeXMLMath .
For the sake of simplicity of notation we assume here that LaTeXMLMath .
A physical justification for this simplification will be forthcoming in the next section , but the generality of our following discussion is not based on or restricted by it .
Taking the time component we have : LaTeXMLEquation or , LaTeXMLEquation .
This gives the time change of energy moving with the fluid .
Inserting the standard relativistic limit for LaTeXMLMath , and neglecting terms of order LaTeXMLMath we are left with LaTeXMLEquation where we made use in the third line of the equation of continuity ( LaTeXMLRef ) .
Eq ( LaTeXMLRef ) is evidently an equation for the change in nonrelativistic kinetic energy .
The first term on the right hand side can be interpreted as the change of kinetic energy within a small volume of space whereas the second term is the flux of energy leaving that volume .
Turning to the space components , and using relation ( LaTeXMLRef ) we get : LaTeXMLEquation .
Inserting Eq .
( LaTeXMLRef ) and rearranging terms yields LaTeXMLEquation .
By remembering relation ( LaTeXMLRef ) one can notice at once that the left hand side of Eq .
( LaTeXMLRef ) is simply the physical force acting on an infinitesimal volume of fluid , whereas the right hand side gives the rate of change of the physical velocity moving with the fluid .
In fact it can clearly be seen that in the nonrelativistic limit LaTeXMLMath Eq .
( LaTeXMLRef ) simply becomes the non-relativistic Euler ’ s equation of hydrodynamics .
It is worth noting too , that if we accept a correspondance between the density of events LaTeXMLMath and the sum of the internal energy and pressure LaTeXMLMath ( also called the heat function ) , then Eq .
( LaTeXMLRef ) is precisely equivalent to the relativistic hydrodynamics equation derived by Weinberg LaTeXMLCite and Landau LaTeXMLCite in the forceless events framework of standard relativity for which LaTeXMLMath .
Under certain circumstances some problems in fluid dynamics can be solved in an approximation in which viscous forces are neglected , and the fluid is assumed to be incapable of sustaining shear forces .
If the fluid is assumed also to be spatially and temporally isotropic , we can write down the stress tensor as follows : LaTeXMLEquation where LaTeXMLMath is the metric tensor for flat Minkowski space and LaTeXMLMath is a Lorentz scalar .
The interpretation of the stress tensor components in this form , is as follows : The three space-space components relate to the force exerted within the fluid per unit surface per unit time , whereas the time-time component describes the forces acting in the time direction per unit volume of the fluid ( essentially mass-changing forces ) .
Substituting Eq .
( LaTeXMLRef ) into the equations of motion , LaTeXMLEquation this leads to LaTeXMLEquation .
Eq .
( LaTeXMLRef ) may be written in different form by introducing two new variables .
We define LaTeXMLMath , so that LaTeXMLMath .
Note that for incompressible fluid LaTeXMLMath is constant and we have simply LaTeXMLMath .
Assuming too that the body force can be derived from a potential function LaTeXMLMath , we can write : LaTeXMLEquation .
An important kinematic result which shall be used later , is known as the Circulation Theorem .
Consider a closed circuit LaTeXMLMath linking a continuous line of fluid events .
In general the space-time configuration of the loop LaTeXMLMath depends on LaTeXMLMath .
We denote this symbolically by LaTeXMLMath .
The kinematic Theorem to be proven is : LaTeXMLEquation where LaTeXMLMath is the space-time acceleration field of the fluid .
The proof can be obtained by transformation to the material coordinates LaTeXMLMath as follows : LaTeXMLEquation .
So then : LaTeXMLEquation .
The first integral on the right hand side of Eq .
( LaTeXMLRef ) , can be shown to vanish : LaTeXMLEquation since the path is closed .
The second can be transformed back to the spatial coordinates : LaTeXMLEquation proving the theorem .
Substituting for LaTeXMLMath in Eq .
( LaTeXMLRef ) , from the equation of motion ( LaTeXMLRef ) gives : LaTeXMLEquation where it has been assumed in the last step that the functions are single valued .
This leads generally to : LaTeXMLEquation .
If the motion of the fluid was generated from rest , so that for some initial time the circulation was zero , we get : LaTeXMLEquation for all times .
A necessary and sufficient condition , for Eq .
( LaTeXMLRef ) , with continuous LaTeXMLMath , is that LaTeXMLMath be expressible as a gradient of a potential function .
We write LaTeXMLEquation where LaTeXMLMath is defined as the velocity potential .
It has been shown above ( Eq .
( LaTeXMLRef ) that for an incompressible fluid LaTeXMLEquation so combining the two , we find that the velocity potential , LaTeXMLMath , satisfies the d ’ Alembert equation : LaTeXMLEquation providing an enormous simplification in the theory of inviscid irrotational flow .
Using the four dimensional Stokes theorem LaTeXMLCite we can convert the line integral in Eq .
( LaTeXMLRef ) into a spatial integral : LaTeXMLEquation .
If this is to be true for arbitrary surface LaTeXMLMath , the integrand must vanish , giving LaTeXMLEquation .
Using this we can evaluate the following expression : LaTeXMLEquation .
Inserting Eq .
( LaTeXMLRef ) into the equations of motion ( LaTeXMLRef ) , and rearranging terms , we obtain : LaTeXMLEquation .
Inserting Eq ( LaTeXMLRef ) , into ( LaTeXMLRef ) and rearranging terms , leads to further simplification : LaTeXMLEquation which means that : LaTeXMLEquation where LaTeXMLMath is some arbitrary function of LaTeXMLMath .
This can be considered a relativistic equivalent to the Bernoulli equations .
We return now to Eq .
( LaTeXMLRef ) and assert that under the assumptions made , our problem reduces to solution of the d ’ Alembert equation , given specific boundary conditions .
These require that the component of the velocity field normal to the boundary vanish everywhere on its surface .
Thus the motion of the fluid will be governed solely by the space-time geometry of the problem .
The solution can then be inserted back into the equations of motion ( LaTeXMLRef ) , determining the distribution of forces and stresses within the fluid .
The general solution to the d ’ Alembert equation ( LaTeXMLRef ) , can be written as an integral over all possible plane waves : LaTeXMLEquation where LaTeXMLMath is restricted to the shell LaTeXMLEquation and the LaTeXMLMath which determine the shape of the wave packet are to be determined by boundary and initial conditions .
It must be noted though , that there also always exists another trivial solution , because any solution to the Laplace equation with an additional linear time term , will also satisfy the d ’ Alembert equation .
The space part , LaTeXMLMath , by definition , vanishes under the Laplacian , while the time component being linear , vanishes when differentiated twice .
This trivial solution LaTeXMLEquation gives rise to a physical velocity field : LaTeXMLEquation which is clearly to be interpreted as the nonrelativistic flow of an incompressible fluid .
The total solution then should be written as a linear combination LaTeXMLEquation .
As for the physical motion of the fluid we now have : LaTeXMLEquation .
If we assume , as is reasonable for low energies , that the nonrelativistic part of the flow is far greater than the relativistic correction , LaTeXMLMath , then an expansion in a Taylor series gives : LaTeXMLEquation .
The first term is just the nonrelativistic , LaTeXMLMath independent flow of Eq .
( LaTeXMLRef ) , whereas the other two are of a smaller order and describe time dependent wave-like fluctuations .
It seems , then , that the covariant form for the potential flow gives rise to a physical velocity which can be divided into two parts LaTeXMLMath , an underlying nonrelativistic velocity field and an additional relativistic correction with a wave-like nature , due to disturbances in the background flow .
These disturbances can be envisioned as ripples riding above the usual nonrelativistic flow , at a speed which will be shown to be the speed of light .
The velocity of the disturbances can be calculated in various ways .
Writing LaTeXMLMath in Eq .
( LaTeXMLRef ) , and picking out a specific direction LaTeXMLMath , we can perform the integration over the magnitude LaTeXMLMath , getting : LaTeXMLEquation .
But from relation ( LaTeXMLRef ) , LaTeXMLMath , so the phase velocity LaTeXMLMath , of the wave front is : LaTeXMLEquation .
If the solution given above ( LaTeXMLRef ) , describes a wave packet , the group velocity of this packet can be determined through the stationary phase method .
The highest contribution to the integral will come from those LaTeXMLMath ’ s for which the phase is stationary : LaTeXMLEquation .
LaTeXMLEquation so , denoting by LaTeXMLMath , the group velocity of the wave packet , we get LaTeXMLEquation where in the last step we used relation ( LaTeXMLRef ) .
It is in point to stress that it is not the fluid which flows with the speed of light , rather it is the disturbances propagating in the flow field which travel with the above calculated group and phase velocities .
In previous work on the subject , there has been much debate as to the actual definition of incompressibility of relativistic fluids .
The most straight forward definition in the framework of standard relativity , would be to demand that an element of the fluid shall retain the same proper volume throughout its motion , or in other words that the expansion of an elementary world-tube of liquid shall be zero .
The difficulty lies in the fact that this definition yields infinite sound wave velocities .
This would violate the most basic assumptions of standard relativity which forbids any material particle or information to be propagated at a speed higher than light .
A disturbance such as a sound wave does carry energy , momentum and information and should therefore not break the light speed barrier .
For this precise reason Synge LaTeXMLCite introduced two more alternate definitions , the second of which actually defines an incompressible fluid as such that sound waves propagate within it at light velocity .
In comparison , the equations we have written down and the simple assertion that incompressibility is to be defined on events in four dimensional space-time , lead smoothly and naturally to light speed sound waves in agreement with Synge ’ s criterion .
Moreover it can be seen that as LaTeXMLMath , the sound waves speed , tends to infinity too , as would be expected of an incommpresible fluid in the nonrelativistic limit .
We wish to widen the scope of our study to encompass the behavior of a compressible fluid , and we shall show that this additional freedom introduces acoustic modes with sound velocities other than that of light .
We begin by assuming that the density and pressure vary only slightly from some constant values .
LaTeXMLEquation .
Generally the pressure is a function of the density so one can expand the pressure in a Taylor series : LaTeXMLEquation where higher derivatives in LaTeXMLMath have been neglected and we have in the last stage denoted the constant LaTeXMLMath .
Comparing equations ( LaTeXMLRef ) and ( LaTeXMLRef ) we get a linear relationship between the variations of density and pressure , LaTeXMLEquation .
We wish also to assume the three space components of the velocity field to be , in some sense , small .
The time component on the other hand must by definition be of the order of LaTeXMLMath and can therefore not be considered small .
This reasoning leads us to divide the velocity field too into two parts , LaTeXMLEquation where LaTeXMLMath is a constant LaTeXMLMath varies only slowly in spacetime relative to LaTeXMLMath ; this possibility will be discussed later on .
, pure , time-like vector and LaTeXMLMath is a small perturbation the nature of which we wish to determine .
It has been shown earlier that the motion of the fluid , neglecting unimportant body forces , is governed by the following two equations : LaTeXMLEquation .
LaTeXMLEquation We assume firstly , that all changes in LaTeXMLMath in equations ( LaTeXMLRef ) and ( LaTeXMLRef ) , are negligably small compared with the spacetime changes , and we can therefore omit the LaTeXMLMath derivatives from both equations .
This can be explained too on grounds of integrating the above equations over LaTeXMLMath and asserting that all physical quantities vanish as LaTeXMLMath or by taking the zero component of the frequencies in LaTeXMLMath .
All three arguments are essentially equivalent and are based on the fact that any apparatus in the laboratory is not capable of resolving fast changes in LaTeXMLMath , but can rather only measure averages over large periods of the universal time .
The resulting equations LaTeXMLEquation .
LaTeXMLEquation can now be linearized by inserting relations ( LaTeXMLRef ) and ( LaTeXMLRef ) and keeping only terms of the first order ( LaTeXMLMath , LaTeXMLMath and LaTeXMLMath being of the first order ) .
Higher order terms are neglected and so in this approximation the two equations reduce to LaTeXMLEquation .
LaTeXMLEquation where by remembering relation ( LaTeXMLRef ) , we have eliminated LaTeXMLMath from equation ( LaTeXMLRef ) .
The importance of the background vector LaTeXMLMath , now becomes apparent .
It ensures that the velocity field remains time-like and that the large time-like component does not get neglected in the linearization process .
By taking space derivatives of Eq .
( LaTeXMLRef ) it can be shown that LaTeXMLEquation implying that if the flow was initially irrotational everywhere , then it will remain so throughout the entire evolution and that the velocity field can be derived from a potential : LaTeXMLEquation .
Multiplying Eq .
( LaTeXMLRef ) by LaTeXMLMath one obtains LaTeXMLEquation where we have interchanged the raising and lowering of the indices on the left hand side .
The equations can now be solved for gradients of LaTeXMLMath by inserting Eq .
( LaTeXMLRef ) into ( LaTeXMLRef ) : LaTeXMLEquation .
Using relation ( LaTeXMLRef ) to replace the velocity field by the derivative of its potential LaTeXMLMath , and rearranging terms , we get LaTeXMLEquation which can alternately be written as LaTeXMLEquation or LaTeXMLEquation where we denote the tensor LaTeXMLMath and LaTeXMLMath .
Before resuming with an interpretation of the result ( LaTeXMLRef ) , we wish to express the variation in density of events LaTeXMLMath , in terms of the potential LaTeXMLMath .
Inserting relation ( LaTeXMLRef ) into Eq .
( LaTeXMLRef ) and rearranging terms we get LaTeXMLEquation implying that the term in parenthesis is constant .
By definition though , any constant part of LaTeXMLMath is to be included in LaTeXMLMath , so the additive constant must be identically zero and we are left with the following relationship LaTeXMLEquation .
We now return to Eq .
( LaTeXMLRef ) , and assume for simplicity that the background fluid is stationary i.e .
LaTeXMLMath is pure time-like and constant .
We then get LaTeXMLEquation or , on separating time and space derivatives , LaTeXMLEquation .
This is a wave equation for which the modified sound velocity is LaTeXMLEquation and we can investigate the various limits as follows .
When we take the fluid to its noncompressible limit , or mathematically LaTeXMLMath , we get for the sound velocity LaTeXMLMath .
This agrees with our conclusions of the previous section , namely that a noncompressible fluid gives rise to light speed sound waves .
On the other hand if we take the fluid to be ‘ sufficiently ’ compressible , or in other words LaTeXMLMath , the velocity becomes linear in LaTeXMLMath , LaTeXMLMath .
This is feasible for in the nonrelativistic limit when LaTeXMLMath , we then get the usual nonrelativistic result for compressible flow namely , LaTeXMLMath .
These results are summarized in figure LaTeXMLRef .
Until now we have dealt solely with “ ideal fluids ” which do not sustain shear stresses during motion .
The justification for our previous equations was that for many materials the shear stresses occurring during motion are small compared with the pressure .
In practice though , all liquids and gases are in fact able to sustain shear forces , and we now wish to take these into account in a consistent covariant manner .
In order to obtain equations describing the motion of a viscous fluid , we have to include some additional terms in the constitutive equations of the fluid .
We therefore write the stress tensor LaTeXMLMath in the form LaTeXMLEquation .
The second term is the extra stress tensor or viscosity stress tensor resulting from frictional forces between different layers of fluid .
We can , with some restrictive assumptions , establish a general form for the tensor LaTeXMLMath .
If we assume that our fluid does not “ remember ” its past history or initial configuration but rather that its motion is governed solely by the immediately preceding state , then LaTeXMLMath , must depend only on the velocity field of the fluid .
Processes of internal friction occur in a fluid only when different fluid particles move with different velocities , so that there is a relative motion between various parts of the fluid .
On the other hand if the distances between adjacent parts of the fluid are kept constant during the motion and assuming that forces between events are a function of the distance separating them LaTeXMLMath , then there will be no friction .
In other words a frictional dissipation could arise only in those regions of the fluid continuum undergoing distortion , excluding places at which the fluid moves uniformly as a rigid body .
Hence LaTeXMLMath must depend on derivatives of the velocity field rather than the velocity field itself .
When the velocity gradients are small we can , to some approximation , suppose LaTeXMLMath to be a linear function of the derivatives LaTeXMLMath omitting higher orders of powers and derivatives .
There can be no terms independent of LaTeXMLMath , since LaTeXMLMath must vanish for constant LaTeXMLMath .
A rigid rotation of the fluid in space time must also be excluded from affecting the viscous force , and we shall show that this implies that only the symmetric combination of derivatives LaTeXMLMath , can be contained in LaTeXMLMath .
We must note here that by rigid rotation in spacetime , we mean the group of transformations LaTeXMLMath , which keep the covariant distance LaTeXMLMath constant .
This is precisely the group of pure Lorentz transformations , which fulfill the requirement LaTeXMLEquation .
We can show quite generally that for an infinitesimal transformation LaTeXMLMath , the infinitesimal displacement tensor LaTeXMLMath , must be antisymmetric , as follows : LaTeXMLEquation where in the third line we have neglected terms of second order in LaTeXMLMath .
Comparison of Eq .
( LaTeXMLRef ) with ( LaTeXMLRef ) yields LaTeXMLEquation proving the antisymmetry of LaTeXMLMath .
Consider now a small region in the flow around a point LaTeXMLMath , at which the fluid can locally be considered to be rotating rigidly ( figure LaTeXMLRef ) .
Our claim is that at LaTeXMLMath there are no frictional forces within the fluid and therefore the the viscosity stress tensor must vanish at that point .
Let LaTeXMLMath represent the angular velocity of the fluid rotating in the LaTeXMLMath plane .
The velocity field round LaTeXMLMath will then by definition be LaTeXMLEquation where LaTeXMLMath here and in the following discussion is the displacement from the center of rotation LaTeXMLMath .
Now consider a fluid particle at some place LaTeXMLMath being rotated by an infinitesimal amount to a new position LaTeXMLMath .
There are two relations connecting LaTeXMLMath and LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation is an infinitesimal Lorenz transformation .
Inserting Eq .
( LaTeXMLRef ) into ( LaTeXMLRef ) , and comparing with Eq .
( LaTeXMLRef ) results in LaTeXMLEquation .
Using relation ( LaTeXMLRef ) we can get a form for LaTeXMLMath in terms of LaTeXMLMath LaTeXMLEquation proving that LaTeXMLMath too , is antisymmetric .
Taking derivatives of relation ( LaTeXMLRef ) while considering the fact that LaTeXMLMath is antisymmetric shows that the combination LaTeXMLMath vanishes for a rigidly rotating fluid whereas the antisymmetric combination LaTeXMLMath , does not .
As the viscosity stress tensor LaTeXMLMath must vanish for rigidly rotating motion , we deduce that it must contain just the symmetric combination of derivatives .
The most general tensor of rank two satisfying all the above conditions is LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are independent of the velocity .
It is convenient , however , to replace LaTeXMLMath and LaTeXMLMath by other constants and write the equation in another form which lends itself more readily to interpretation LaTeXMLEquation .
We shall call the constants LaTeXMLMath and LaTeXMLMath the shear viscosity and bulk viscosity respectively .
The terminology derives from considering the two following basic flows ( see figure LaTeXMLRef ) : For a “ pure ” shear flow LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are mutually orthogonal unit vectors so that the direction of the velocity gradient LaTeXMLMath is perpendicular to the direction LaTeXMLMath of the velocity , it can be shown that only the first term in LaTeXMLMath survives , leaving LaTeXMLMath .
So the coefficient LaTeXMLMath alone governs the resistance of the fluid to shear distortions .
For a “ pure ” radial flow LaTeXMLMath in which the material expands in a radial direction , the the first term in Eq .
( LaTeXMLRef ) vanishes leaving LaTeXMLMath , which shows that LaTeXMLMath is proportional to the resistance of the fluid to expansion of bulk .
This can be understood also by noticing that the expression in parentheses in Eq .
( LaTeXMLRef ) has the property of vanishing on contraction with respect to LaTeXMLMath and LaTeXMLMath .
The equations of motion can now be obtained by adding the expressions LaTeXMLMath to the left hand side of the nonviscous flow equation ( LaTeXMLRef ) .
Thus we have LaTeXMLEquation .
This is the most general form of the equations of motion of a viscous fluid .
If we assume , however , that the viscosity coefficients do not change noticeably throughout the fluid , then they may be regarded as constant and can therefore be taken outside the gradient operators .
We then have LaTeXMLEquation .
Further simplification can be obtained by assuming the bulk viscosity LaTeXMLMath small compared to the shear viscosity LaTeXMLMath .
If the fluid is incompressible then LaTeXMLMath and the third term on the left hand side of Eq .
( LaTeXMLRef ) vanishes giving LaTeXMLEquation .
On solving a specific dynamical problem one must also write down the boundary conditions for the equations of motion of a viscous fluid .
We assume that there are friction forces acting between the boundary surface and the fluid such that the layer of fluid immediately adjacent to the boundary is brought to complete rest .
Accordingly , the boundary conditions on the equations of motion require that the fluid velocity should vanish at fixed solid surfaces : LaTeXMLEquation .
We emphasize that for a viscous fluid both the normal and tangential velocity components must vanish , in contradistinction to ideal fluids for which it is required only that the normal component vanish .
In a general case with boundaries moving in LaTeXMLMath , the velocity of the fluid at the boundary should be equal to the velocity of the moving surface .
We have studied a continuum flow of events in space-time parameterized by LaTeXMLMath , the invariant , universal , historical time .
In the process of evolution , as LaTeXMLMath changes uniformly , the events move in space-time , generating a dense continuum of world lines which constitute a physical flow of particles .
The velocity field is in general a function of both space-time and LaTeXMLMath and is governed by equations of motion and continuity .
The solution of the equations for the velocity field provides , in principal , a solution to the physical problem , but is in general rather complicated .
We considered an inviscid fluid and derived an analogue of the Bernoulli equation .
In the special case of irrotational flow we proved that the velocity field can be derived by a potential .
If in addition , the fluid is incompressible , it has been shown that the potential must obey the d ’ Alembert equation , and thus the problem is reduced to solving the d ’ Alembert equation with specific boundary conditions .
The solutions consist of a background flow ( The nonrelativistic solution ) over which wave like ripples propagate with group and phase velocities equal to that of light .
This is to be compared with Synge LaTeXMLCite , who actually defines noncompressibility in such a way as to achieve this result .
Relaxing the constraints and allowing the fluid to become linearly compressible one can derive a wave equation , from which the sound velocity can again be computed .
For a stationary background flow , it has been demonstrated that the sound velocity , attains its correct values for the incompressible and nonrelativistic limits .
It is in principal possible , under some restrictions , to compute the sound velocities also for a non stationary and non uniform background flow .
Precise computations and predictions in this direction are yet to be studied .
Finally viscosity was introduced and the use of some general arguments of symmetry and isotropy enabled the formulation of equations for the motion of a viscous fluid .
Only introductory comments were made about this topic and the equations derived are yet to be applied to specific problems .
It must be emphasized that the velocity and density ( and any other ) fields that we mention , describe quantities related to the flow of events at a particular time LaTeXMLMath .
In order to derive a physically observable quantity related to particles , an averaging must be performed over LaTeXMLMath .
The method we use is that of integrating over LaTeXMLMath with the density of events as a weight function .
An exact transcription from event properties to world-line , or particle , properties , however , is not always possible due to nonlinearities of these quantities .
Alternatively the observed quantities can be extracted from the zero frequency ( in LaTeXMLMath ) components of the fields .
LaTeXMLMath as a function of the inverse compressibility LaTeXMLMath , and energy LaTeXMLMath respectively .
Infinitesimal rotation of fluid .
Two basic distortions : left : Radial flow .
right Shear flow .
Keywords : Riemann zeta function , Riemann zeros , Dirichlet series , Hadamard factorization , meromorphic functions , Mellin transform .
MSC2000 : 11Mxx , 30B40 , 30B50 .
Fonctions Zêta pour les zéros de Riemann Keywords : Riemann zeta function , Riemann zeros , Dirichlet series , Hadamard factorization , meromorphic functions , Mellin transform .
MSC2000 : 11Mxx , 30B40 , 30B50 .
This work proposes to investigate certain meromorphic functions defined by Dirichlet series over the nontrivial zeros LaTeXMLMath of the Riemann zeta function LaTeXMLMath , and to thoroughly compile their explicit analytical features .
If the Riemann zeros are listed in pairs as usual , LaTeXMLEquation then the Dirichlet series to be mainly studied read as LaTeXMLEquation extended to meromorphic functions of LaTeXMLMath , and parametrized by LaTeXMLMath — with emphasis on two cases , LaTeXMLMath and especially LaTeXMLMath .
Their analysis will simultaneously yield some results for the variant series LaTeXMLEquation .
Those Zeta functions are “ secondary ” : arising from the nontrivial zeros of a classic zeta function ( here , LaTeXMLMath ) ; and “ generalized ” : they admit an auxiliary shift parameter just like the Hurwitz zeta function ( LaTeXMLMath ) .
Such “ LaTeXMLMath -Zeta ” functions have occasionally appeared in the literature , but mostly through particular cases or under very specific aspects .
On the other hand , their abundance of general explicit properties seems to have been largely ignored , although it can be revealed by quite elementary means ( as we will do ) .
And with regard to the Riemann zeros , reputed to be highly elusive quantities , those properties constitute additional explicit information : this is enough to motivate a more comprehensive treatment ( and bibliography ) of the subject .
The present work just aims to do that , in a self-contained and very concrete way , as a kind of “ All you ever wanted to know about LaTeXMLMath -Zeta functions… ” handbook ( without prejudice to the usefulness of any single result by itself ) .
We begin by developing the background and our motivations in greater detail .
First , if a Selberg zeta function is used in place of LaTeXMLMath from the start ( assuming the simplest setting of a compact hyperbolic surface LaTeXMLMath here ) , then the LaTeXMLMath become the eigenvalues of the ( positive ) Laplacian on LaTeXMLMath , and the Zeta functions ( LaTeXMLRef ) turn into bona fide spectral ( Minakshisundaram–Pleijel ) zeta functions , for which numerous explicit results have indeed been displayed ( with the help of Selberg trace formulae : cf .
LaTeXMLCite for LaTeXMLMath , LaTeXMLCite for LaTeXMLMath , LaTeXMLCite for LaTeXMLMath ) .
Some transposition of those results to the Riemann case can then be expected , in view of the formal analogies between the trace formulae for Selberg zeta functions on the one hand , and the Weil “ explicit formula ” for LaTeXMLMath on the other hand LaTeXMLCite .
Indeed , a few symmetric functions over the Riemann zeros that resemble spectral functions have been well described , mainly LaTeXMLMath ( LaTeXMLCite , LaTeXMLCite chap.II ) .
Zeta functions like ( LaTeXMLRef ) have also been considered , but almost solely to establish their meromorphic continuation to the whole LaTeXMLMath -plane — apart from the earliest occurrence we found : a mention by Guinand LaTeXMLCite ( see also LaTeXMLCite ) of the series LaTeXMLMath ( LaTeXMLMath ) on one side of a functional relation ( eq .
( LaTeXMLRef ) below ) arising as an instance of a generalized Poisson summation formula .
Later , Delsarte introduced that function again ( as LaTeXMLMath in LaTeXMLCite ) to describe its poles qualitatively , displaying ( only ) its principal polar part at LaTeXMLMath , as LaTeXMLMath ; Kurokawa LaTeXMLCite made the same study at LaTeXMLMath , not only for LaTeXMLMath but also for Dedekind zeta functions and Selberg zeta functions for PSL LaTeXMLMath [ or congruence subgroups ] ( then , Zeta functions like ( LaTeXMLRef ) occur within the parabolic components ) ; and Matiyasevich LaTeXMLCite discussed the special values LaTeXMLMath .
Extensions in the style of the Lerch zeta function have also been studied ( LaTeXMLCite , LaTeXMLCite chap.VI ) .
Independently , Deninger LaTeXMLCite and Schröter–Soulé LaTeXMLCite considered a different Hurwitz-type family ( we keep their factor LaTeXMLMath just to avoid multiple notations ) , LaTeXMLEquation mainly to evaluate LaTeXMLMath ( as eq .
( LaTeXMLRef ) below ) ; earlier , Matsuoka LaTeXMLCite , then Lehmer LaTeXMLCite had focused upon the sums LaTeXMLEquation .
Here we easily recover LaTeXMLMath from the other Zeta function ( LaTeXMLRef ) ( but not the reverse ) , as LaTeXMLEquation .
The present work proposes a broader , and unified , description for all those LaTeXMLMath -Zeta functions , with a wealth of explicit results comparable to usual spectral zeta functions LaTeXMLCite .
Tools for the purpose could also be borrowed from spectral theory ( trace formulae , etc .
) , but the objects under scrutiny are more singular here ( the Zeta functions for the Riemann zeros manifest double poles , vs simple poles in the Selberg case ) ; this then imposes so many adaptations upon the classic procedures that a self-contained treatment of the Riemann case alone is actually simpler .
Even then , we can not get maximally explicit outputs for all cases at once ( e.g. , Weil ’ s “ explicit formula ” diverges for LaTeXMLMath ) , and our analysis has to develop gradually .
So , we begin ( Sec.2 ) by setting up a minimal abstract framework , sufficient to handle LaTeXMLMath ( with a permanent distinction between properties in the half-planes LaTeXMLMath and LaTeXMLMath respectively ) .
We next obtain a first batch of explicit results for the case LaTeXMLMath ( Sec.3 ) , then for general values of LaTeXMLMath ( Sec.4 ) .
Specializing to the case LaTeXMLMath in Sec.5 , we reach an almost explicit meromorphic continuation formula for LaTeXMLMath into the half-plane LaTeXMLMath , which immediately implies many more properties of that function , and is generalizable to LaTeXMLMath -series and other number-theoretical zeta functions .
In Sec.6 we exploit the latter results to sharpen the descriptions of both Hurwitz-type functions LaTeXMLMath and LaTeXMLMath .
Sec.7 provides a summary of the results ; essentially , a Table collects the main formulae for LaTeXMLMath and LaTeXMLMath , also referring to the main text like an index .
( Which text can in some sense be viewed , and simply used , as a set of “ notes ” for this Table ! )
Finally , Appendices A and B treat some subsidiary issues : a meromorphic continuation method for the Mellin transforms of Sec.2 , and certain numerical aspects .
For convenience , we recall the needed elementary results and notations LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation [ LaTeXMLMath is also a standard symbol for quantized energy levels , a concept often invoked purely rhetorically about the Riemann zeros ; let us then insist that our ( present ) work is totally decoupled from such interpretations , whereas it sees the Euler numbers as truly essential ! ]
Concerning the Riemann zeta function LaTeXMLMath LaTeXMLCite , we will need its special values , LaTeXMLEquation .
LaTeXMLEquation and its functional equation in the form LaTeXMLEquation where LaTeXMLMath is an entire function , which is even under the symmetry LaTeXMLMath ( this expresses the functional equation ) , and only keeps the nontrivial zeros of LaTeXMLMath .
In order to analyze LaTeXMLMath , we will be forced to invoke a particular Dirichlet LaTeXMLMath -series as well ( associated with the Dirichlet character LaTeXMLMath LaTeXMLCite ) LaTeXMLEquation .
LaTeXMLMath extends to an entire function , having the special values LaTeXMLEquation .
LaTeXMLEquation and a functional equation expressible as LaTeXMLEquation where the function LaTeXMLMath is entire and only keeps the nontrivial zeros of LaTeXMLMath .
Throughout this work , a numerical sequence LaTeXMLMath ( systematically labeled by positive integers LaTeXMLMath ) will be called admissible of order LaTeXMLMath if : ( i ) LaTeXMLMath ; ( or : complex LaTeXMLMath with LaTeXMLMath sufficiently bounded LaTeXMLCite , to provide an unconditionally valid framework “ in case of need ” ) ; ( ii ) LaTeXMLMath , making the following Weierstrass product converge , LaTeXMLEquation it then defines an entire “ Delta ” function LaTeXMLMath ( we omit the argument LaTeXMLMath except when an ambiguity may result ) ; ( iii ) for LaTeXMLMath , LaTeXMLMath , and it admits a complete uniform asymptotic expansion in some sector LaTeXMLMath , of a form governed by some strictly decreasing sequence of real exponents LaTeXMLMath , as LaTeXMLEquation .
LaTeXMLEquation ( “ generalized Stirling expansion ” , by extension from the case LaTeXMLMath LaTeXMLCite ) ; such a uniform expansion is repeatedly differentiable in LaTeXMLMath .
Then , the Dirichlet series LaTeXMLEquation defines the Zeta function of LaTeXMLMath , holomorphic in that half-plane .
The point LaTeXMLMath has to lie in the latter by assumption ( ii ) , which imposes LaTeXMLMath ; then LaTeXMLMath can moreover be expanded term by term in eq .
( LaTeXMLRef ) to yield the Taylor series LaTeXMLEquation .
Motivations : the idea here is to assume certain properties for an entire function LaTeXMLMath of order LaTeXMLMath , so as to generate a function LaTeXMLMath meromorphic in all of LaTeXMLMath with poles at the LaTeXMLMath , their maximum order LaTeXMLMath being 2 here ( as dictated by the specific form of eq .
( LaTeXMLRef ) ) .
Such LaTeXMLMath are very special instances of zeta-regularized infinite products , for which much more general frameworks exist ( e.g. , LaTeXMLCite ) .
However , singularities and essential complications definitely increase each time LaTeXMLMath or the integer part LaTeXMLMath can become larger ( chiefly , the formalism leaves LaTeXMLMath meaningful integration constants undetermined ) .
Efficiency then commands to minimize both latter parameters ( subject to LaTeXMLMath and LaTeXMLMath ) ; specially , LaTeXMLMath is simpler to handle than any LaTeXMLMath .
In this respect , spectral zeta functions frequently have LaTeXMLMath ( e.g. , for Laplacians on compact Riemannian manifolds , LaTeXMLMath [ dimension ] ) but only simple poles , hence the simplifying assumption LaTeXMLMath is suitable for them LaTeXMLCite ; by contrast , the present functions LaTeXMLMath will accept the lower value LaTeXMLMath but will need LaTeXMLMath .
Another difference is that for eigenvalue spectra , a “ partition function ” LaTeXMLMath is a natural starting point ; in the Riemann case , that type of function ( LaTeXMLMath ) exhibits a more remote and contrived structure LaTeXMLCite , while Delta-type functions are easily accessible ( by simple alterations of eq .
( LaTeXMLRef ) ) .
Thus , Riemann zeros and standard eigenvalue spectra have several mutually singular features , making their unified handling rather cumbersome .
The bounds LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath , with LaTeXMLMath , imply that the following Mellin transform of LaTeXMLMath , LaTeXMLEquation converges to a holomorphic function of LaTeXMLMath in the stated vertical strip .
Then , by standard arguments ( see App.A , and LaTeXMLCite ) , LaTeXMLMath extends to a meromorphic function on either side of that strip : LaTeXMLMath by virtue of the expansion ( LaTeXMLRef ) for LaTeXMLMath , LaTeXMLMath extends to all LaTeXMLMath , with LaTeXMLEquation .
LaTeXMLMath just because LaTeXMLMath has a Taylor series at LaTeXMLMath , LaTeXMLMath extends to all LaTeXMLMath , with LaTeXMLEquation .
Now , for LaTeXMLMath , the integral ( LaTeXMLRef ) can also be done term by term after inserting the expansion ( LaTeXMLRef ) , giving LaTeXMLEquation .
The meromorphic extension of LaTeXMLMath then entails that of LaTeXMLMath to the whole LaTeXMLMath -plane as well , i.e. , to the half-planes LaTeXMLMath by LaTeXMLMath , resp .
LaTeXMLMath by LaTeXMLMath independently , so that : LaTeXMLMath any non-integer pole LaTeXMLMath of LaTeXMLMath generates at most a double pole for LaTeXMLMath , with LaTeXMLEquation .
LaTeXMLMath any negative integer pole LaTeXMLMath of LaTeXMLMath generates at most a simple pole for LaTeXMLMath , through partial cancellation with the zeros of LaTeXMLMath , with LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation ( we call eqs .
( LaTeXMLRef ) “ trace identities ” by extension from spectral theory , specially when LaTeXMLMath , in which case explicit finite values for the LaTeXMLMath result ) ; LaTeXMLMath any pole at LaTeXMLMath of LaTeXMLMath gets fully cancelled by the double zero of LaTeXMLMath , giving LaTeXMLEquation .
LaTeXMLMath each positive integer pole of LaTeXMLMath ( LaTeXMLMath , from eq .
( LaTeXMLRef ) ) gets cancelled by a zero of LaTeXMLMath , giving LaTeXMLEquation but this last output just duplicates with the previous Taylor formula ( LaTeXMLRef ) .
Ultimately , all the poles of LaTeXMLMath lie in a single decreasing sequence LaTeXMLMath , and have maximum order LaTeXMLMath under the specific assumption ( LaTeXMLRef ) .
We now apply the previous framework to the Riemann zeros upon just slight changes with regard to the usual factorization of LaTeXMLMath around LaTeXMLMath .
The most convenient starting point is the entire function ( LaTeXMLRef ) , which keeps precisely the non-trivial zeros LaTeXMLMath of LaTeXMLMath , and has the familiar Hadamard product representation LaTeXMLCite LaTeXMLEquation .
Zeros are henceforth grouped in pairs as LaTeXMLMath according to eq .
( LaTeXMLRef ) .
Their corresponding counting function , LaTeXMLMath , follows a well-known estimate LaTeXMLMath , as LaTeXMLCite LaTeXMLEquation .
Thus , the sequence of zeros itself could be admissible of order 1 at best ; fortunately , a transformed sequence LaTeXMLMath and its Zeta function will immediately arise : LaTeXMLEquation ( the latter series will converge for LaTeXMLMath , again by the estimate ( LaTeXMLRef ) ) .
Indeed , once the zeros have been reordered in pairs , it first follows that LaTeXMLMath ( convergent sums ) , and then , LaTeXMLMath ; the parity property LaTeXMLMath thereupon imposes LaTeXMLMath , hence LaTeXMLEquation a classic result ( LaTeXMLCite , ch.12 ; LaTeXMLCite , Sec.3.8 ) .
All in all , the product formulae and functional equation boil down to LaTeXMLEquation .
LaTeXMLEquation i.e. , LaTeXMLMath has been rewritten as an infinite product LaTeXMLMath which is manifestly even ( under LaTeXMLMath ) , and will qualify as a Delta function of order LaTeXMLMath in the variable LaTeXMLMath , a much simpler situation than LaTeXMLMath ( naively suggested by eq .
( LaTeXMLRef ) ) .
We now derive the ensuing properties of LaTeXMLMath ( as compiled in Sec.7 , Table 1 ) .
As basic initial result , the sequence LaTeXMLMath is admissible of order LaTeXMLMath : it obviously fulfills assumptions ( i ) – ( ii ) ; the function LaTeXMLMath is entire of order 1 in LaTeXMLMath , hence LaTeXMLMath in LaTeXMLMath ; finally , a large- LaTeXMLMath expansion of the form ( LaTeXMLRef ) for LaTeXMLMath is easily obtained as follows .
In eq .
( LaTeXMLRef ) , for LaTeXMLMath , LaTeXMLMath can be replaced by its complete Stirling expansion and LaTeXMLMath can be deleted , giving LaTeXMLEquation whereupon LaTeXMLMath is to be substituted by the relevant solution branch of LaTeXMLMath , namely LaTeXMLEquation .
The resulting LaTeXMLMath -expansion then has all the required properties , with the exponents LaTeXMLMath — giving the order LaTeXMLMath — and leading coefficients LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This expansion can be computed to any power LaTeXMLMath in principle ; still , a reduced general formula for LaTeXMLMath looks inaccessible this way ; by contrast , all the LaTeXMLMath with LaTeXMLMath arise from the single substitution of eq .
( LaTeXMLRef ) into the prefactor LaTeXMLMath of LaTeXMLMath in eq .
( LaTeXMLRef ) , giving LaTeXMLEquation .
Now , by Sec.2.2 LaTeXMLMath – LaTeXMLMath ) , a large- LaTeXMLMath expansion for LaTeXMLMath translates into explicit properties of LaTeXMLMath for LaTeXMLMath .
Here , LaTeXMLMath gets a double pole at each half -integer LaTeXMLMath , with principal polar term LaTeXMLMath , and is regular elsewhere ; the leading pole LaTeXMLMath has LaTeXMLEquation .
At LaTeXMLMath , eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) deliver two explicit values , LaTeXMLEquation the latter makes the Stirling constant ( the value LaTeXMLMath , or regularized determinant ) also explicitly known for this sequence LaTeXMLMath .
Further quantities , tied to the yet unspecified general coefficients LaTeXMLMath , will also acquire explicit closed forms : the polar terms of order LaTeXMLMath and the finite values LaTeXMLMath for all LaTeXMLMath , see Table 1 ; but we will need a more indirect approach ( Sec.6.1 ) .
Now by Sec.2.2 LaTeXMLMath ) , LaTeXMLMath is holomorphic in the half-plane LaTeXMLMath , and the values LaTeXMLMath for LaTeXMLMath lie in the Taylor series of LaTeXMLMath at LaTeXMLMath , which can be specified here through eq .
( LaTeXMLRef ) .
We first expand LaTeXMLMath in powers of LaTeXMLMath , where LaTeXMLMath .
Eq .
( LaTeXMLRef ) directly implies LaTeXMLEquation then , from eq .
( LaTeXMLRef ) , LaTeXMLMath and LaTeXMLMath ( expressing the functional equation ) respectively yield the two Taylor series LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation The LaTeXMLMath in the last line are cumulants for the Stieltjes constants LaTeXMLMath of eq .
( LaTeXMLRef ) LaTeXMLCite ( cf .
also the LaTeXMLMath in ref .
LaTeXMLCite , Sec .
4 ) , i.e. , LaTeXMLEquation .
LaTeXMLEquation The identification of the three series ( LaTeXMLRef ) – ( LaTeXMLRef ) at each order LaTeXMLMath now yields a countable sequence of 3-term identities : the first one just restores the result LaTeXMLMath as in eq .
( LaTeXMLRef ) ; then , the subsequent ones likewise express the higher LaTeXMLMath in two ways , LaTeXMLEquation .
That short argument subsumes several earlier results .
The rightmost and center expressions in eq .
( LaTeXMLRef ) amount to formulae for LaTeXMLMath by Matsuoka LaTeXMLCite and Lehmer ( LaTeXMLCite , eq .
( 12 ) ) respectively ; the implied relations between the derivatives LaTeXMLMath and the Stieltjes constants LaTeXMLMath were also discussed in LaTeXMLCite , together with Euler–Maclaurin formulae for the LaTeXMLMath which parallel the specification of the LaTeXMLMath in eq .
( LaTeXMLRef ) .
As for the values LaTeXMLMath themselves , they are given by eq .
( LaTeXMLRef ) now using LaTeXMLMath as expansion variable , i.e. , LaTeXMLEquation .
But alternatively , LaTeXMLMath , and vice-versa : as shown below , LaTeXMLEquation ( LaTeXMLMath ) .
The LaTeXMLMath being already known from eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , LaTeXMLMath then reduces to an explicit affine combination ( over the rationals ) of LaTeXMLMath , LaTeXMLMath , and either LaTeXMLMath or LaTeXMLMath for LaTeXMLMath ( see Table 1 ) , as stated earlier by Matiyasevich LaTeXMLCite .
( More recently , LaTeXMLMath and LaTeXMLMath also got revived in studies of the distribution of primes LaTeXMLCite ) .
Since ref .
LaTeXMLCite uses eq .
( LaTeXMLRef ) for LaTeXMLMath without mentioning any proof , we sketch one .
First , if LaTeXMLMath , the expansion of the identity LaTeXMLMath in powers of LaTeXMLMath yields LaTeXMLMath for LaTeXMLMath .
By recursion , this must invert in the form LaTeXMLMath ; then , LaTeXMLMath has to be the singular part in the Laurent series of LaTeXMLMath around LaTeXMLMath , resulting in LaTeXMLMath .
Now , summing both sets of identities over the Riemann zeros LaTeXMLMath yields the stated decompositions ( LaTeXMLRef ) .
( We stress that their finite character is specific to the LaTeXMLMath as opposed to all other values LaTeXMLMath , LaTeXMLMath . )
Note added in proof .
— For completeness , we quote two other sets of identities for the sums LaTeXMLMath LaTeXMLCite : LaTeXMLEquation ( a countable sequence of “ sum rules ” , easy but unreported , which allow to eliminate any finite subset of odd values ) ; and the connection to Li ’ s coefficients ( cf .
[ 39 ] , thm 2 ) , LaTeXMLMath [ which allow to recast the Riemann Hypothesis as LaTeXMLMath ] : LaTeXMLEquation .
We begin to discuss the Hurwitz-like generalizations of the preceding case obtained by shifting the squared parameters LaTeXMLMath .
The obviously allowed translations ( LaTeXMLMath in the real case ) preserve the notion of admissible sequences ( with their values of LaTeXMLMath ) ; that validates the earlier definition ( LaTeXMLRef ) , as LaTeXMLMath .
The corresponding transformation of Delta functions ( as Hadamard products of order LaTeXMLMath ) only involves an explicit constant denominator to preserve their specific normalization LaTeXMLMath , as LaTeXMLEquation .
Alternatively , we might have opted to normalize Delta functions as zeta-regularized infinite products , i.e. , LaTeXMLEquation which are fully translation-covariant , but at the same time less explicit .
The overall benefit of this normalization is then dubious within the restricted scope of this work ; but here , it explains a dichotomy between algebraic and transcendental properties of Zeta functions , which roughly follows our overall division between LaTeXMLMath and LaTeXMLMath properties , but not quite .
Covariance implies that if we translate LaTeXMLMath , the expansion of LaTeXMLMath around the invariant point LaTeXMLMath can be recomputed to any order by straight substitution , yielding explicit polynomials in LaTeXMLMath as coefficients .
When LaTeXMLMath as here , then LaTeXMLMath LaTeXMLCite , hence the previous statement holds for the expansion ( LaTeXMLRef ) minus its term of order LaTeXMLMath ; i.e. , all the shifted coefficients LaTeXMLMath will be polynomial excepting LaTeXMLMath .
For LaTeXMLMath specifically , eqs .
( LaTeXMLRef – LaTeXMLRef ) ( at LaTeXMLMath ) imply that LaTeXMLEquation .
LaTeXMLEquation For functions like LaTeXMLMath , the consequences are that their polar parts and “ trace identities ” will depend polynomially on LaTeXMLMath ; furthermore , a single ( fixed- LaTeXMLMath ) large- LaTeXMLMath expansion , such as eq .
( LaTeXMLRef ) for LaTeXMLMath , suffices to express those LaTeXMLMath -dependences in full .
Precisely here , by eq .
( LaTeXMLRef ) : LaTeXMLMath keeps its rightmost ( LaTeXMLMath ) full polar part constant ( and given by eq .
( LaTeXMLRef ) ) , as well as its value at 0 ( LaTeXMLMath ) ; all its other poles ( of order 2 , except at LaTeXMLMath ) keep fixed locations .
( As a by-product , any difference function LaTeXMLMath is holomorphic for LaTeXMLMath ) .
We can specify such polynomial formulae in the half-plane LaTeXMLMath still further , but only later by a different path ( Sec.6 ) .
By contrast , all formulae for LaTeXMLMath in the half-plane LaTeXMLMath refer to Taylor coefficients of LaTeXMLMath around LaTeXMLMath finite , which evolve transcendentally with LaTeXMLMath : here they will only express in terms of LaTeXMLMath ( or LaTeXMLMath , from eq .
( LaTeXMLRef ) ) and its derivatives at LaTeXMLMath .
The first of those coefficients , LaTeXMLMath itself ( LaTeXMLMath ) , actually yields a special value lying at LaTeXMLMath , by eq .
( LaTeXMLRef ) : LaTeXMLEquation ( a result which fully matches eq .
( LaTeXMLRef ) below for the Hurwitz-type function LaTeXMLMath LaTeXMLCite ) .
Then , the Taylor coefficients of order LaTeXMLMath ( identical for LaTeXMLMath and LaTeXMLMath ) yield LaTeXMLEquation .
LaTeXMLEquation ( but we are in lack of more reduced closed forms for general LaTeXMLMath ) .
In summary , the polar parts of LaTeXMLMath and the special values LaTeXMLMath ( “ trace identities ” ) have polynomial expressions in LaTeXMLMath ; whereas LaTeXMLMath ) plus LaTeXMLMath at LaTeXMLMath are also special values , but only computable transcendentally .
( This conclusion is moreover fully obeyed for typical spectral zeta functions . )
An interesting option is now to shift the parameter LaTeXMLMath from its initial value LaTeXMLMath in LaTeXMLMath , to the most symmetrical value LaTeXMLMath .
By eq .
( LaTeXMLRef ) , the Hadamard product ( LaTeXMLRef ) becomes based at LaTeXMLMath as LaTeXMLEquation .
LaTeXMLEquation ( but very little is known about LaTeXMLMath LaTeXMLCite and we can not make this constant factor any more explicit , contrary to the special case LaTeXMLMath where that factor was LaTeXMLMath ) .
The factorized representation ( LaTeXMLRef ) then transforms to LaTeXMLEquation ( This Delta function is closest to the determinant of Riemann zeros used by Berry–Keating for other purposes LaTeXMLCite . )
We accordingly switch to the Zeta function of the sequence LaTeXMLMath LaTeXMLCite , or in short , LaTeXMLEquation .
Numerically , this new function looks almost indistinguishable from LaTeXMLMath ( see App.B ) .
( Also , by Sec.4 , LaTeXMLMath extends holomorphically to LaTeXMLMath . )
By contrast , the meromorphic continuation of LaTeXMLMath will prove to be distinctly simpler than that of LaTeXMLMath , thanks to specially explicit representation formulae in the half-plane LaTeXMLMath .
To obtain these , we now switch to a more powerful , special to LaTeXMLMath , approach ( whereas the earlier considerations would still describe LaTeXMLMath , but just to the same extent as LaTeXMLMath ) .
The factor LaTeXMLMath in eq .
( LaTeXMLRef ) has the structure of a spectral determinant built over the “ spectrum ” of trivial zeros in the variable LaTeXMLMath , namely LaTeXMLMath ( LaTeXMLMath is not exactly the zeta-regularized determinant , but again this will not matter here ) .
That spectral interpretation can be extended to the factor LaTeXMLMath , by treating the pole LaTeXMLMath ( of LaTeXMLMath ) as a “ ghost eigenvalue ” of multiplicity ( LaTeXMLMath ) .
A major role of the spectrum of trivial zeros is to make LaTeXMLMath asymptotically cancel LaTeXMLMath to all orders when LaTeXMLMath in LaTeXMLMath , given that LaTeXMLMath decreases exponentially there .
We therefore expect the spectral zeta function of the trivial zeros ( of LaTeXMLMath ) to play an important role ; this “ shadow zeta function of LaTeXMLMath ” ( for short ) involves both LaTeXMLMath itself and the partner function LaTeXMLMath , in the combination LaTeXMLEquation ( LaTeXMLMath in terms of the Hurwitz zeta function ) .
LaTeXMLMath has a single simple pole at LaTeXMLMath , of residue LaTeXMLMath , and admits the special values LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Remark : literally , the framework of Sec.2 excludes the sequence of trivial zeros ( of linear growth , and order LaTeXMLMath ) , but the truly relevant function here will be LaTeXMLMath , as Zeta function of the modified sequence LaTeXMLMath , which is admissible of order LaTeXMLMath again .
We start from a slight variant of the representation ( LaTeXMLRef ) for LaTeXMLMath , obtained through an integration by parts upon the Mellin formula ( LaTeXMLRef ) ( where LaTeXMLMath , by eq .
( LaTeXMLRef ) ) : LaTeXMLEquation .
We next introduce a ( regularized ) resolvent trace for the spectrum of trivial zeros , LaTeXMLEquation which has a simple pole of residue LaTeXMLMath at each trivial zero of LaTeXMLMath ; a corresponding function for the pole ( “ ghost ” ) at LaTeXMLMath is LaTeXMLEquation .
Then , upon insertion of the factorization formula ( LaTeXMLRef ) , eq .
( LaTeXMLRef ) yields LaTeXMLEquation .
Now a crucial feature of the case LaTeXMLMath is that this integral is also a Mellin transform with respect to the argument appearing in the factorized form of LaTeXMLMath ( namely the variable LaTeXMLMath , in eq .
( LaTeXMLRef ) ) .
As a consequence , the contribution to LaTeXMLMath from LaTeXMLMath ( and also LaTeXMLMath ) can be neatly extracted and evaluated , in closed and interpretable form .
Because the factor in brackets in eq .
( LaTeXMLRef ) is LaTeXMLMath at LaTeXMLMath ( due to the functional equation ) , LaTeXMLMath can be split as LaTeXMLEquation .
LaTeXMLEquation ( this splitting preserves the convergence strip LaTeXMLMath for both resulting integrals ) .
LaTeXMLMath can be split still further , once its integration path has been rotated by a small angle : either LaTeXMLMath or LaTeXMLMath , in order to bypass the poles of LaTeXMLMath and of LaTeXMLMath at LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation Now , LaTeXMLMath can be straightforwardly transformed into Hankel contour integrals and then computed in closed form ( by the residue calculus ) , giving LaTeXMLEquation both of which are explicit functions , meromorphic in the whole plane ; chiefly , LaTeXMLMath brings in the shadow zeta function ( LaTeXMLRef ) .
Then , again upon back-and-forth integrations by parts , LaTeXMLMath continue to LaTeXMLEquation and ( cf .
eq .
( LaTeXMLRef ) ) these integrals admit meromorphic extensions to the whole plane , with LaTeXMLEquation ( the difference LaTeXMLMath is entire ) .
All in all , we finally get two complex conjugate Mellin representations for LaTeXMLMath : LaTeXMLEquation and one real principal-value integral representation given by their half-sum , LaTeXMLEquation ( each of the above converges in the full half-plane LaTeXMLMath ) .
Another real form can be obtained with a regular integrand , directly from eq .
( LaTeXMLRef ) : LaTeXMLEquation however this last integral only converges in the strip LaTeXMLMath .
Remarks : - as analytical extension formulae , eqs .
( LaTeXMLRef – LaTeXMLRef ) are precise counterparts of the functional equation for LaTeXMLMath ; they also stand as more explicit forms of Guinand ’ s functional relation for LaTeXMLMath LaTeXMLCite , as discussed below ( eq .
( LaTeXMLRef ) ) .
- a similar formula exists for the function LaTeXMLMath of eq .
( LaTeXMLRef ) ( LaTeXMLCite , middle of p.149 ) , only requiring LaTeXMLMath ( which precisely avoids the problem raised above by the pole of LaTeXMLMath ) ; in comparison , the present results correspond to the fixed value LaTeXMLMath , since LaTeXMLMath by eq .
( LaTeXMLRef ) ; As we will elaborate next , analytical properties of LaTeXMLMath in the half-plane LaTeXMLMath are made totally straightforward by the Mellin formulae ( LaTeXMLRef – LaTeXMLRef ) ( while LaTeXMLMath is holomorphic in the half-plane LaTeXMLMath , where its defining series ( LaTeXMLRef ) converges ) .
Detailed results are also recollected in fully reduced form in Sec.7 , Table 1 .
A few leading properties of LaTeXMLMath in the half-plane LaTeXMLMath emerge more easily as special cases from Sec.4 ( although they can be drawn from eq .
( LaTeXMLRef ) as well ) : - LaTeXMLMath is a double pole , with the same full polar part as for LaTeXMLMath , eq .
( LaTeXMLRef ) ; - by specializing eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) , LaTeXMLEquation .
Otherwise , a Mellin representation like ( LaTeXMLRef ) gives a better global view of LaTeXMLMath over the whole half-plane LaTeXMLMath .
Indeed , its non-elementary part ( the integral ( LaTeXMLRef ) ) becomes regular there , hence can be ignored both for the polar analysis and ( thanks to the LaTeXMLMath factor ) for the “ trace identities ” at integer LaTeXMLMath : all such information lies then in the first term alone , accessible by mere inspection .
We thus obtain that : - LaTeXMLMath only has simple poles at the negative half -integers LaTeXMLMath , with residues LaTeXMLEquation ( hence , only the leading pole LaTeXMLMath stays double ) ; - at the negative integers , the “ trace identities ” read as LaTeXMLEquation ( both formulae ( LaTeXMLRef ) , ( LaTeXMLRef ) were fully reduced using eq .
( LaTeXMLRef ) ) .
- a ( LaTeXMLMath ) asymptotic formula follows for LaTeXMLMath , from the term-by-term substitution of the Euler product for LaTeXMLMath into the integrand of eq .
( LaTeXMLRef ) , giving LaTeXMLEquation where as usual , LaTeXMLMath if LaTeXMLMath for some prime LaTeXMLMath , else 0 .
( An asymptotic formula for LaTeXMLMath itself then follows from eq .
( LaTeXMLRef ) and the functional equations for LaTeXMLMath . )
Remark : in our notations , Guinand ’ s functional relation for LaTeXMLMath LaTeXMLCite reads as LaTeXMLEquation where LaTeXMLMath — subject to the Riemann Hypothesis LaTeXMLCite — clearly specifies a ( real-valued ) resummation of the divergent series in eq .
( LaTeXMLRef ) ( the asymptotic series for LaTeXMLMath ) .
Eq .
( LaTeXMLRef ) was only asserted for LaTeXMLMath , with no clue as to the analytic structure of either LaTeXMLMath or LaTeXMLMath elsewhere .
The present formulae ( LaTeXMLRef – LaTeXMLRef ) are thus resummed versions of eq .
( LaTeXMLRef ) , with a definitely more explicit content .
As stated before , LaTeXMLMath is regular in the half-plane LaTeXMLMath , where analytical results are identities directly obtainable by expanding the logarithm of the functional relation ( LaTeXMLRef ) in Taylor series around LaTeXMLMath .
Here we will extract those results from the Mellin representation ( LaTeXMLRef ) , invoking the meromorphic properties of its integral term in the whole plane as given by eq .
( LaTeXMLRef ) .
- for half-integer LaTeXMLMath : the residues of the two summands in ( LaTeXMLRef ) have to cancel given that LaTeXMLMath is analytic in the half-plane ; this imposes LaTeXMLEquation .
LaTeXMLEquation which simply amounts to LaTeXMLMath ( itself a consequence of the functional equation LaTeXMLMath ) ; that result further reduces , using eq .
( LaTeXMLRef ) , to the identity LaTeXMLEquation .
The case LaTeXMLMath is singular , but LaTeXMLMath directly yields LaTeXMLEquation - for integer LaTeXMLMath , the pole of the integral is cancelled by the zero of LaTeXMLMath , and the following explicit relation results , LaTeXMLEquation which can also be further reduced with the help of eq .
( LaTeXMLRef ) , see Table 1 .
Unfortunately , we hardly know anything else about the values LaTeXMLMath ( cf .
LaTeXMLCite for LaTeXMLMath ) .
To supplement the relation ( LaTeXMLRef ) with LaTeXMLMath for LaTeXMLMath odd , we can only refer to other formulae for LaTeXMLMath ( compiled in LaTeXMLCite ) , and to Euler–Maclaurin formulae for LaTeXMLMath ( valid at LaTeXMLMath ) with related numerical data LaTeXMLCite .
So , even at LaTeXMLMath , the transcendental values LaTeXMLMath ( eq .
( LaTeXMLRef ) ) and LaTeXMLMath currently remain more elusive than at the ( exceptional ) point LaTeXMLMath ( Sec.3.3 ) .
Furthermore , we found no reference at all to those values ( i.e. , LaTeXMLMath , LaTeXMLMath ) in the literature .
The results of Secs.5.2–4 for LaTeXMLMath are similar to those yielded by the “ sectorial ” trace formula for the analogous spectral zeta function LaTeXMLMath over a compact hyperbolic surface LaTeXMLMath LaTeXMLCite .
The present formulae for the Riemann case nevertheless show several distinctive features .
- as announced end of Sec.2.1 , the sequence LaTeXMLMath and the analogous spectrum of the Laplacian on LaTeXMLMath have mutually singular features : the former has the parameter values LaTeXMLMath ( LaTeXMLMath has its leading pole double , at LaTeXMLMath ) , whereas the latter more precisely has LaTeXMLMath ( LaTeXMLMath has all its poles simple , starting at LaTeXMLMath ) , hence this spectral analogy for the Riemann zeros holds only partially ; - in the continuation formulae ( LaTeXMLRef – LaTeXMLRef ) , LaTeXMLMath itself reenters as an additive component of the shadow zeta function LaTeXMLMath .
This is an altogether different incarnation of LaTeXMLMath from its initial , multiplicative involvement , which remains in the integral term and indirectly through the zeros , in the left-hand side LaTeXMLMath .
It is curious to find two such copies of LaTeXMLMath to coexist in one formula , especially with the additive LaTeXMLMath represented in its critical strip ; - however , those formulae relative to LaTeXMLMath are not fully closed as they also invoke the other Dirichlet series LaTeXMLMath ( as second additive component in the shadow zeta function LaTeXMLMath ) .
The question then arises whether LaTeXMLMath and other zeta functions can be handled on the same footing as LaTeXMLMath ( as in LaTeXMLCite ) , so we now outline a possible extension of eqs .
( LaTeXMLRef – LaTeXMLRef ) .
We assume that LaTeXMLMath is a Dirichlet zeta or LaTeXMLMath -series , having : - a single pole , at LaTeXMLMath and of order LaTeXMLMath ( typically , LaTeXMLMath or 1 ) ; - the asymptotic property LaTeXMLMath for all LaTeXMLMath ; - a functional equation of the form LaTeXMLEquation - LaTeXMLMath is an entire function of order LaTeXMLMath in the variable LaTeXMLMath , and - LaTeXMLMath is an entire function with all of its zeros lying in the half-plane LaTeXMLMath .
Then the Zeta functions LaTeXMLMath ( for the zeros of LaTeXMLMath ) and LaTeXMLMath ( for the zeros of LaTeXMLMath ) are related by this formula corresponding to eq .
( LaTeXMLRef ) ( we omit the others ) , LaTeXMLEquation .
Apart from LaTeXMLMath itself , with eq .
( LaTeXMLRef ) , the next independent example is LaTeXMLMath .
Its functional equation ( LaTeXMLRef ) has the form ( LaTeXMLRef ) with LaTeXMLMath ( no pole ) and LaTeXMLMath , LaTeXMLMath ; its spectrum of trivial zeros ( for LaTeXMLMath ) is LaTeXMLMath , giving as shadow zeta function LaTeXMLEquation .
Under LaTeXMLMath , all Mellin representations ( LaTeXMLRef – LaTeXMLRef ) coalesce into the single regular form LaTeXMLEquation and all consequences previously drawn for LaTeXMLMath have analogs for LaTeXMLMath .
Various such integral representations will naturally add , whenever the initial zeta functions combine nicely under multiplication .
For instance , eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) add up to : LaTeXMLEquation .
Here , the shadow zeta function purely invokes LaTeXMLMath ; on the other hand , under the integral sign we now find LaTeXMLMath so that the new multiplicative zeta function is LaTeXMLMath , also recognized as LaTeXMLMath times LaTeXMLMath , the zeta function of the ring of Gaussian integers LaTeXMLMath LaTeXMLCite .
Hence eq .
( LaTeXMLRef ) becomes LaTeXMLEquation thus , to isolate LaTeXMLMath , here in the additive position , we again had to allow a different zeta function elsewhere , this time LaTeXMLMath in the multiplicative position .
Likewise , by subtracting eq .
( LaTeXMLRef ) from ( LaTeXMLRef ) instead , we could get the shadow zeta function to be LaTeXMLMath ; then the counterpart of eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) is a fully explicit identity , LaTeXMLEquation whereas each of the two left-hand-side terms separately needs LaTeXMLMath ( or LaTeXMLMath for LaTeXMLMath ) .
The purpose of this Section is twofold .
First , we analyze the Zeta functions LaTeXMLMath more explicitly over the half-plane LaTeXMLMath than in Sec.4 , by exploiting the latest special properties of the function LaTeXMLMath ( with new results even for the case LaTeXMLMath ) .
Then , by the same approach , we ( briefly ) discuss the other Hurwitz-type Zeta functions LaTeXMLMath and LaTeXMLMath , defined through eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) respectively .
To describe the Hurwitz-type function LaTeXMLMath better , we now systematically expand it in terms of LaTeXMLMath , as LaTeXMLEquation .
Such an expansion can be formulated around any reference point LaTeXMLMath , but it will be specially useful for LaTeXMLMath as above .
For instance , coupled with eq .
( LaTeXMLRef ) ( say ) , it can express the meromorphic continuation of the general LaTeXMLMath to LaTeXMLMath , while we lack an analog of eq .
( LaTeXMLRef ) itself for any LaTeXMLMath .
For the polar structure of LaTeXMLMath at LaTeXMLMath , the series ( LaTeXMLRef ) reduces to LaTeXMLEquation then , importing the polar structure of LaTeXMLMath from eq .
( LaTeXMLRef ) , we get LaTeXMLEquation just by brute-force polar expansion of the right-hand side of eq .
( LaTeXMLRef ) .
Here , the polar part of order 2 at every LaTeXMLMath is clearly induced by the only such part of LaTeXMLMath ( at LaTeXMLMath ) , through the term with LaTeXMLMath in eq .
( LaTeXMLRef ) ; whereas the residue LaTeXMLMath is built from all residues of LaTeXMLMath at poles with LaTeXMLMath , as LaTeXMLEquation ( the residues LaTeXMLMath of LaTeXMLMath at LaTeXMLMath are known from eq .
( LaTeXMLRef ) ) .
Remark : for LaTeXMLMath , the full polar part ( LaTeXMLRef ) at LaTeXMLMath , independent of LaTeXMLMath , is recovered .
When LaTeXMLMath , the series ( LaTeXMLRef ) also terminates , as LaTeXMLEquation so that explicit “ trace identities ” for general LaTeXMLMath derive from those for LaTeXMLMath ( eq .
( LaTeXMLRef ) ) .
( For LaTeXMLMath , this result simplifies further , see Table 1 . )
Remark : in view of eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , the latter two results now imply a general- LaTeXMLMath formula for the coefficients LaTeXMLMath in the large- LaTeXMLMath expansion ( LaTeXMLRef ) of LaTeXMLMath .
( Hitherto we had such a formula just at LaTeXMLMath , not even at LaTeXMLMath , and knew only the other coefficients LaTeXMLMath for any LaTeXMLMath and LaTeXMLMath , by eqs .
( LaTeXMLRef – LaTeXMLRef ) . )
Our initial emphasis on the special case LaTeXMLMath might now seem misplaced : why didn ’ t we operate at once from LaTeXMLMath ?
In the first place , we saw the case LaTeXMLMath arise more readily from the standard product representation of LaTeXMLMath .
But mainly , the case LaTeXMLMath also enjoys certain special properties , this time with the values LaTeXMLMath and LaTeXMLMath ( Sec.3.3 ) , and since these evolve from transcendental functions of LaTeXMLMath ( Sec.4 ) , their expansions ( LaTeXMLRef ) around LaTeXMLMath are now infinite .
So , each case LaTeXMLMath and LaTeXMLMath has its own exceptional features , the former in the half-plane LaTeXMLMath , and the latter for LaTeXMLMath .
The function LaTeXMLMath as defined by eq .
( LaTeXMLRef ) is a priori more singular than LaTeXMLMath ( the sequence LaTeXMLMath itself has LaTeXMLMath and LaTeXMLMath , which would require a formalism more elaborate than in Sec.2 ) .
Fortunately , LaTeXMLMath can also be analyzed directly through its expansion around LaTeXMLMath , by analogy with eq .
( LaTeXMLRef ) ( see also LaTeXMLCite ) : LaTeXMLEquation .
This formula generates a pole for LaTeXMLMath now at every half-integer LaTeXMLMath , according to : LaTeXMLEquation .
Differences with eq .
( LaTeXMLRef ) arise due to the factor LaTeXMLMath vanishing whenever LaTeXMLMath .
Only the polar part at LaTeXMLMath remains the same as for LaTeXMLMath ( of order LaTeXMLMath and independent of LaTeXMLMath , given by eq .
( LaTeXMLRef ) ) ; all other poles LaTeXMLMath of LaTeXMLMath are now simple , of residues LaTeXMLEquation ( again , LaTeXMLMath is the residue given by eq .
( LaTeXMLRef ) ) .
In addition , at LaTeXMLMath the LaTeXMLMath -expansion of eq .
( LaTeXMLRef ) captures the finite part too : LaTeXMLEquation .
As for the function LaTeXMLMath , if we express it by eq .
( LaTeXMLRef ) in terms of LaTeXMLMath , then we find this combination to be less singular overall : by mere substitution of eq .
( LaTeXMLRef ) , LaTeXMLMath shows a simple pole at LaTeXMLMath , of residue LaTeXMLMath LaTeXMLCite , and all other possible poles at LaTeXMLMath cancel out , resulting in the holomorphy of LaTeXMLMath for all LaTeXMLMath with the computable finite values ( “ trace identities ” ) LaTeXMLEquation ( An alternative evaluation follows from Deninger ’ s continuation formula for LaTeXMLMath ( LaTeXMLCite , middle of p.149 ) , as LaTeXMLMath , whose agreement with eq .
( LaTeXMLRef ) can be verified . )
As for special values : first , LaTeXMLMath is expressible as well , in terms of LaTeXMLMath LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation ( the equivalence of the two forms follows from eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) for LaTeXMLMath , i.e. , LaTeXMLMath ) .
Now , the exponentiated left-hand side of eq .
( LaTeXMLRef ) precisely defines the zeta-regularized product LaTeXMLMath built upon the sequence LaTeXMLMath of Riemann zeros , while the right-hand side mainly involves LaTeXMLMath of eq .
( LaTeXMLRef ) .
So , eq .
( LaTeXMLRef ) is converting a zeta-regularized product ( LaTeXMLMath ) to Hadamard product form .
As an aside , we now verify that such a conversion formula is entirely fixed by universal rules for ( complex ) admissible sequences , specialized here to LaTeXMLMath ( as in LaTeXMLCite ) and LaTeXMLMath — since the Zeta functions LaTeXMLMath have just a simple pole at LaTeXMLMath .
Those rules yield these two prescriptions : LaTeXMLMath , and the large- LaTeXMLMath expansion of LaTeXMLMath shall only retain canonical ( or standard ) terms , namely : LaTeXMLMath for LaTeXMLMath .
Those conditions together fix LaTeXMLMath uniquely , and here , eqs .
( LaTeXMLRef – LaTeXMLRef ) for LaTeXMLMath as input precisely lead to eq .
( LaTeXMLRef ) as output .
Likewise , the special values LaTeXMLMath are expressible in terms of [ the higher Laurent coefficients of ] LaTeXMLMath , e.g. , by applying residue calculus to Deninger ’ s continuation formula ( LaTeXMLCite , p.149 ) .
Thus , a fair degree of structural parallelism finally shows up between the two Hurwitz-like families LaTeXMLMath and LaTeXMLMath LaTeXMLCite .
In way of conclusion , Table 1 collates the analytical results found for the two LaTeXMLMath -Zeta functions LaTeXMLMath and LaTeXMLMath .
Furthermore , corresponding results for the general LaTeXMLMath were derived in Sec.4 ( for LaTeXMLMath ) and 6.1 ( for LaTeXMLMath ) , and partly extended to the functions LaTeXMLMath and LaTeXMLMath in Sec.6.2 .
The other novel results we have developed here concern LaTeXMLMath in the half-plane LaTeXMLMath : the analytical continuation formulae ( LaTeXMLRef – LaTeXMLRef ) , and the LaTeXMLMath asymptotic formula ( LaTeXMLRef ) as corollary .
We also came across two elementary ( but unfamiliar to us ) formulae concerning LaTeXMLMath itself : eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) .
Appendix B gives information on some numerical aspects based on our use of the 100,000 first Riemann zeros ( made freely available on the Web by A.M. Odlyzko LaTeXMLCite , to whom we express our gratitude ) .
We also wish to thank C. Deninger , J.P. Keating , P. Lebœuf , V. Maillot , C. Soulé , and the Referee , for helpful references and comments .
We briefly recall the meromorphic continuation argument for a Mellin transform like eq .
( LaTeXMLRef ) , LaTeXMLMath , assuming the function LaTeXMLMath to be regular on LaTeXMLMath ( for simplicity ) , with LaTeXMLEquation as in eq .
( LaTeXMLRef ) ( asymptotic estimates are repeatedly differentiable ) ; and crucially , LaTeXMLMath .
Sequential directed integrations by parts can be used ( see LaTeXMLCite for details ) .
Step 1 : LaTeXMLMath converges for LaTeXMLMath , and in that strip , LaTeXMLEquation .
If LaTeXMLMath , this suffices : the new integral actually converges for LaTeXMLMath ( thanks to LaTeXMLMath for LaTeXMLMath ) , hence LaTeXMLMath is manifestly meromorphic in that wider strip , with a simple pole at LaTeXMLMath of residue LaTeXMLEquation furthermore , in the complementary strip LaTeXMLMath , backward integration by parts now yields LaTeXMLEquation .
Then the whole argument can be restarted from here , to extend LaTeXMLMath further ( across LaTeXMLMath ) , and so on : the case LaTeXMLMath thus gets settled .
Step 2 : if LaTeXMLMath , one more integration by parts upon eq .
( LaTeXMLRef ) yields LaTeXMLEquation .
All previous arguments then carry over , but the pole is now double , with LaTeXMLEquation and residue LaTeXMLMath ( from the residue calculus ) ; integrations by parts ( backwards , and split ) reduce the latter to LaTeXMLEquation .
The last two formulae thus generate eq .
( LaTeXMLRef ) for the leading double pole , and so on for LaTeXMLMath .
( More generally , if the factor of LaTeXMLMath in the expansion ( LaTeXMLRef ) is a polynomial of degree LaTeXMLMath in LaTeXMLMath , then LaTeXMLMath becomes a pole of order LaTeXMLMath for LaTeXMLMath LaTeXMLCite . )
As for meromorphic continuation in the other direction ( across LaTeXMLMath ) , it works likewise if LaTeXMLMath admits a LaTeXMLMath expansion : the previous arguments apply upon exchanging the bounds 0 and LaTeXMLMath under LaTeXMLMath .
E.g. , in the regular case , LaTeXMLMath expands in an entire series at LaTeXMLMath , and step 1 suffices ( as in the main text , where LaTeXMLMath ) .
We complete our analytical study by describing some very heuristic numerical work with LaTeXMLMath and LaTeXMLMath , mostly in the range LaTeXMLMath , and focusing on the simpler case of LaTeXMLMath .
( The same ideas apply for any generalized Zeta function LaTeXMLMath and for complex LaTeXMLMath , but the formulae get more involved . )
Here , the Riemann Hypothesis is de facto implied throughout ( there being no numerical counter-example ) .
Numerically , LaTeXMLMath looks almost indistinguishable from LaTeXMLMath for LaTeXMLMath , because LaTeXMLMath ( an empirical fact ; already , LaTeXMLMath ) .
If we expand LaTeXMLMath in terms of LaTeXMLMath according to eq .
( LaTeXMLRef ) , and make the roughest approximations , we get that LaTeXMLMath : i.e. , the very first correction term is only of relative size LaTeXMLMath .
As other related numerical observations : LaTeXMLEquation .
LaTeXMLEquation ( not only is LaTeXMLMath small , but moreover , LaTeXMLMath by eq .
( LaTeXMLRef ) and LaTeXMLMath by eq .
( LaTeXMLRef ) , hence LaTeXMLMath reflects how little the function LaTeXMLMath deviates from the parabolic shape LaTeXMLMath over the interval LaTeXMLMath ) .
We now focus on the numerical evaluation of LaTeXMLMath itself .
The defining series ( LaTeXMLRef ) converges more and more poorly as LaTeXMLMath ( with divergence setting in at LaTeXMLMath ) .
We then replace a far tail of that series ( LaTeXMLMath ) by an integral according to the integrated-density estimate ( LaTeXMLRef ) , and formally obtain a kind of Euler–Maclaurin formula , LaTeXMLEquation .
LaTeXMLEquation ( similar formulae can be written for LaTeXMLMath , LaTeXMLMath , finite parts at LaTeXMLMath , etc . ) .
The approximate remainder term LaTeXMLMath balances the dominant trend of the partial sums LaTeXMLMath .
It thereby accelerates the convergence of the partial sums in eq .
( LaTeXMLRef ) for LaTeXMLMath , while for LaTeXMLMath it counters their divergent trend , so that LaTeXMLMath converges ( as LaTeXMLMath ) when LaTeXMLMath ( LaTeXMLCite , p.116 , last line ) .
The next obstruction to convergence arises at LaTeXMLMath but is of another type : LaTeXMLMath displays erratic fluctuations in LaTeXMLMath ( roughly of the order LaTeXMLMath , according to LaTeXMLCite , eq .
( 2.5.7 ) ) , and those numerically blow up indeed ( as LaTeXMLMath ) when LaTeXMLMath .
Further convergence now requires to perform a damping of those fluctuations ( as argued previously for “ chaotic ” spectra LaTeXMLCite ) .
Here , a Cesaro averaging ( defined by LaTeXMLMath ) appears to work well initially ( results can be verified at LaTeXMLMath ) , but not very far down : already at LaTeXMLMath , the fluctuations of LaTeXMLMath itself retain a standard deviation LaTeXMLMath up to LaTeXMLMath .
So , instead of pursuing ever more severe ( and unproven , after all ) numerical regularizations as LaTeXMLMath decreases below LaTeXMLMath , we advocate the switch to the continuation formulae ( LaTeXMLRef – LaTeXMLRef ) for numerical work as well .
Thus , we first tested eq .
( LaTeXMLRef ) against eq .
( LaTeXMLRef ) for LaTeXMLMath , then used it to evaluate LaTeXMLMath , plus eq .
( LaTeXMLRef ) with LaTeXMLMath ( 3 terms sufficed ) to obtain LaTeXMLMath .
Table 2 gives a summary of the numerical results we obtained .
( We found no earlier analogs , except for the other special sums LaTeXMLMath in LaTeXMLCite , LaTeXMLCite . )
Let LaTeXMLMath be a compact group and let LaTeXMLMath be the Fourier algebra of LaTeXMLMath .
This is a commutative Banach algebra .
As a Banach space , LaTeXMLMath is defined as the predual of LaTeXMLMath , the von Neumann algebra of LaTeXMLMath .
Now weak amenability of LaTeXMLMath in the sense of LaTeXMLCite has been studied by Johnson LaTeXMLCite , and we strengthen one of his results with the following result .
Theorem .
Let LaTeXMLMath be a compact connected Lie group .
Then LaTeXMLMath is weakly amenable if and only if LaTeXMLMath is abelian .
Proof ( i ) Let LaTeXMLMath be abelian .
Then the Pontryagin dual LaTeXMLMath is a discrete abelian group .
Then we have LaTeXMLMath and so we have LaTeXMLEquation .
Now LaTeXMLMath is amenable .
Therefore LaTeXMLMath is weakly amenable .
( ii ) Let LaTeXMLMath be a non-abelian compact connected Lie group .
For Lie groups our reference is Bourbaki LaTeXMLCite especially Lie IX.5 and Lie IX.31 .
The Lie algebra LaTeXMLMath decomposes as LaTeXMLEquation where LaTeXMLMath is the centre of LaTeXMLMath and LaTeXMLMath is the derived algebra of LaTeXMLMath .
The derived algebra LaTeXMLMath is a semisimple Lie algebra .
If LaTeXMLMath is non-abelian then LaTeXMLMath is non-abelian and LaTeXMLMath .
Hence the root system LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is a maximal torus of LaTeXMLMath .
But LaTeXMLMath .
Hence there exists a root LaTeXMLMath .
Let LaTeXMLMath be the centralizer of the kernel of LaTeXMLMath .
Let LaTeXMLMath be the derived group ( commutator subgroup ) of LaTeXMLMath .
Then LaTeXMLMath is a connected compact semisimple Lie group of rank 1 .
Let LaTeXMLMath be an irreducible representation of LaTeXMLMath of degree LaTeXMLMath : LaTeXMLEquation .
Consider LaTeXMLMath .
This is a maximal torus of LaTeXMLMath .
There are only 2 possibilities for LaTeXMLMath .
( i ) LaTeXMLMath .
Consider the restriction LaTeXMLMath and the fact that LaTeXMLMath is a circle LaTeXMLMath .
Now LaTeXMLEquation with each LaTeXMLMath irreducible .
But LaTeXMLMath where LaTeXMLMath .
Consequently we have LaTeXMLMath .
( ii ) LaTeXMLMath .
A similar argument gives LaTeXMLMath for the characters occurring in LaTeXMLMath .
We now apply the remark in Johnson LaTeXMLCite ( p.373 ) to infer that LaTeXMLMath is not weakly amenable .
Corollary .
If LaTeXMLMath is any compact connected Lie group except LaTeXMLMath then LaTeXMLMath is not weakly amenable .
Mathematics Department , University of Manchester , Manchester M13 9PL , UK .
For a general crossed product LaTeXMLMath , of an algebra LaTeXMLMath by a Hopf algebra LaTeXMLMath , we obtain complexes simpler than the canonical ones , giving the Hochschild homology and cohomology of LaTeXMLMath .
These complexes are equipped with natural filtrations .
The spectral sequences associated to them is a natural generalization of the one obtained in LaTeXMLCite by the direct method .
We also get that if the LaTeXMLMath -cocycle LaTeXMLMath takes its values in a separable subalgebra of LaTeXMLMath , then the Hochschild ( co ) homology of LaTeXMLMath with coefficients in LaTeXMLMath is the ( co ) homology of LaTeXMLMath with coefficients in a ( co ) chain complex .
Let LaTeXMLMath be a group , LaTeXMLMath a strongly LaTeXMLMath -graded algebra and LaTeXMLMath an LaTeXMLMath -bimodule .
In LaTeXMLCite was shown that there is a convergent spectral sequence LaTeXMLEquation where LaTeXMLMath denotes the identity of LaTeXMLMath .
In LaTeXMLCite was shown that this result remains valid for LaTeXMLMath -Galois extensions ( in his paper the author deals with both the homology and the cohomology of these algebras ) .
An important particular type of LaTeXMLMath -Galois extensions are the crossed products with convolution invertible cocycle LaTeXMLMath , of an algebra LaTeXMLMath by a Hopf algebra LaTeXMLMath ( for the definition see Section one ) .
The purpose of our paper is to construct complexes simpler than the canonical ones , given the Hochschild ( co ) homology of LaTeXMLMath with coefficients in an arbitrary LaTeXMLMath -bimodule .
These complexes are equipped with canonical filtrations .
We show that the spectral sequences associated to them coincide with the ones obtained using a natural generalization of the direct method introduced in LaTeXMLCite , and with the ones constructed in LaTeXMLCite ( when these are specialize to crossed products ) .
In the case of group extensions these results were proved in LaTeXMLCite and LaTeXMLCite .
This paper is organized as follows : in Section 1 a resolution LaTeXMLMath of a crossed product LaTeXMLMath is given .
To accomplish this construction we do not use the fact that the cocycle is convolution invertible .
Moreover , we give a recursive construction of morphisms LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is the normalized Hochschild resolution , such that LaTeXMLMath and we show that LaTeXMLMath is homotopically equivalent to the identity map .
Consequently our resolution is a direct sum of the normalized Hochschild resolution .
We also recursively construct an homotopy LaTeXMLMath .
Both , the canonical normalized resolution and LaTeXMLMath are equipped with natural filtrations , which are preserved by the maps LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
In Section 2 , for an LaTeXMLMath -bimodule LaTeXMLMath , we get complexes LaTeXMLMath and LaTeXMLMath , giving the Hochschild homology and cohomology of LaTeXMLMath with coefficients in LaTeXMLMath respectively .
The filtration of LaTeXMLMath induces filtrations on LaTeXMLMath and LaTeXMLMath .
So , we obtain converging spectral sequences LaTeXMLMath and LaTeXMLMath .
Using the results of Section 1 , we get that these spectral sequences are the ones associated to suitable filtrations of the Hochschild normalized chain and cochain complexes LaTeXMLMath and LaTeXMLMath .
This allows us to give very simple proofs of the main results of LaTeXMLCite and LaTeXMLCite .
In Section 3 , we show that , if the cocycle is convolution invertible , then the complexes LaTeXMLMath and LaTeXMLMath are isomorphic to simpler complexes LaTeXMLMath and LaTeXMLMath respectively .
Then , we compute the term LaTeXMLMath and LaTeXMLMath of the spectral sequences obtained in Section 2 .
Moreover , using the above mentioned filtrations , we prove that if the LaTeXMLMath -cocycle LaTeXMLMath takes its values in a separable subalgebra of LaTeXMLMath , then the Hochschild ( co ) homology of LaTeXMLMath with coefficients in LaTeXMLMath is the ( co ) homology of LaTeXMLMath with coefficients in a ( co ) chain complex .
Finally , as an application we obtain some results about the LaTeXMLMath and LaTeXMLMath functors and an upper bound for the global dimension of LaTeXMLMath ( for group crossed products this bound was obtained in LaTeXMLCite ) .
In addition to the direct method developed in LaTeXMLCite , there are another two classical methods to obtain spectral sequences converging to LaTeXMLMath and with LaTeXMLMath -term LaTeXMLMath .
Namely the Cartan-Leray and the Grothendieck spectral sequences of a crossed product .
In Section 4 , we recall these constructions and we prove that these spectral sequences are isomorphic to the one obtained in Section 2 .
This generalizes the main results of LaTeXMLCite .
In a first appendix we give a method to construct ( under suitable hypothesis ) a projective resolution of the LaTeXMLMath -algebra LaTeXMLMath as LaTeXMLMath -bimodule , simpler than the canonical one of Hochschild .
This method , which can be considered as a variant of the perturbation lemma , is used to prove the main result of Section 1 .
The boundary maps of the resolution LaTeXMLMath are recursively defined in Section 1 .
In a second appendix we give closed formulas for these maps .
Let LaTeXMLMath be a LaTeXMLMath -algebra and LaTeXMLMath a Hopf algebra .
We will use the Sweedler notation LaTeXMLMath , with the summation understood and superindices instead of subindices .
Recall some definitions of LaTeXMLCite and LaTeXMLCite .
A weak action of LaTeXMLMath on LaTeXMLMath is a bilinear map LaTeXMLMath from LaTeXMLMath to LaTeXMLMath such that , for LaTeXMLMath , LaTeXMLMath 1 ) LaTeXMLMath , 2 ) LaTeXMLMath , 3 ) LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -algebra and LaTeXMLMath a Hopf algebra with a weak action on LaTeXMLMath .
Given a LaTeXMLMath -linear map LaTeXMLMath , let LaTeXMLMath be the LaTeXMLMath -algebra ( in general non associative and without LaTeXMLMath ) with underlying vector space LaTeXMLMath and multiplication map LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath .
The element LaTeXMLMath of LaTeXMLMath will usually be written LaTeXMLMath to remind us LaTeXMLMath is weakly acting on LaTeXMLMath .
The algebra LaTeXMLMath is called a crossed product if it is associative with LaTeXMLMath as identity element .
It is easy to check that this happens if and only if LaTeXMLMath and the weak action satisfy the following conditions : i ) ( Normality of LaTeXMLMath ) for all LaTeXMLMath , we have LaTeXMLMath , ii ) ( Cocycle condition ) for all LaTeXMLMath , we have LaTeXMLEquation iii ) ( Twisted module condition ) for all LaTeXMLMath , LaTeXMLMath we have LaTeXMLEquation .
In this section we obtain a resolution LaTeXMLMath of a crossed product LaTeXMLMath as an LaTeXMLMath -bimodule , which is simpler than the canonical one of Hochschild .
To begin , we fix some notations : 1 ) For each LaTeXMLMath -algebra LaTeXMLMath , we put LaTeXMLMath .
Moreover , given LaTeXMLMath we also let LaTeXMLMath denote the class of LaTeXMLMath in LaTeXMLMath .
2 ) We write LaTeXMLMath , LaTeXMLMath ( LaTeXMLMath times ) and LaTeXMLMath , for each natural number LaTeXMLMath .
3 ) Given LaTeXMLMath and LaTeXMLMath , we write LaTeXMLMath .
4 ) Given LaTeXMLMath and LaTeXMLMath , we write LaTeXMLMath and LaTeXMLMath .
5 ) Given LaTeXMLMath , we let LaTeXMLMath denote the comultiplication of LaTeXMLMath in LaTeXMLMath .
So , LaTeXMLMath .
6 ) Given LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we write LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath ( LaTeXMLMath ) and LaTeXMLMath ( LaTeXMLMath ) .
The groups LaTeXMLMath are LaTeXMLMath -bimodules in an obvious way and the groups LaTeXMLMath are LaTeXMLMath -bimodules via the left canonical action and the right action LaTeXMLEquation where LaTeXMLMath .
Let us consider the diagram of LaTeXMLMath -bimodules and LaTeXMLMath -bimodule maps LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are defined by : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
We have left LaTeXMLMath -module maps LaTeXMLMath and LaTeXMLMath , given by LaTeXMLMath for LaTeXMLMath .
Clearly LaTeXMLMath is a complex and LaTeXMLMath is a contracting homotopy of LaTeXMLEquation .
So , we are in the situation considered in Appendix A .
We define LaTeXMLMath -bimodule maps LaTeXMLMath ( LaTeXMLMath and LaTeXMLMath ) recursively , by : LaTeXMLEquation for LaTeXMLMath .
There is a relative projective resolution LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath is the multiplication map and LaTeXMLMath .
Let LaTeXMLMath be the map LaTeXMLMath .
The complex of LaTeXMLMath -bimodules LaTeXMLEquation is contractible as a complex of left LaTeXMLMath -modules .
A chain contracting homotopy LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) is given by LaTeXMLMath .
Hence , the theorem follows from Corollary A.2 of Appendix A∎ Let LaTeXMLMath and LaTeXMLMath be the maps recursively defined by LaTeXMLEquation .
We will prove , in Corollary A.2 , that the family LaTeXMLMath , LaTeXMLMath , defined by LaTeXMLMath and LaTeXMLEquation is a contracting homotopy of the resolution LaTeXMLMath introduced in Theorem 1.1.1 .
Let LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
We have : 1 ) LaTeXMLMath is the map given by LaTeXMLEquation .
LaTeXMLEquation 2 ) For each LaTeXMLMath , there are maps LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) , whose image is included in the LaTeXMLMath -submodule of LaTeXMLMath generated by all the elementary tensors LaTeXMLMath with LaTeXMLMath coordinates in the image of LaTeXMLMath , such that for LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath if LaTeXMLMath .
The computation of LaTeXMLMath can be obtained easily by induction on LaTeXMLMath , using that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
The assertion for LaTeXMLMath , with LaTeXMLMath , follows easily by induction on LaTeXMLMath and LaTeXMLMath , using the recursive definition of LaTeXMLMath ∎ In Appendix B we will give more precise formulas for the maps LaTeXMLMath completing the computation of the LaTeXMLMath ’ s .
Let LaTeXMLMath be the normalized Hochschild resolution of LaTeXMLMath .
As it is well known , the complex LaTeXMLEquation is contractible as a complex of left LaTeXMLMath -modules , with contracting homotopy LaTeXMLMath .
Let LaTeXMLMath be the contracting homotopy of LaTeXMLMath introduced in Remark 1.1.2 .
Let LaTeXMLMath and LaTeXMLMath be the morphisms of LaTeXMLMath -bimodule complexes , recursively defined by LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath is homotopically equivalent to the identity map .
An homotopy LaTeXMLMath is recursively defined by LaTeXMLMath and LaTeXMLMath , for LaTeXMLMath .
We prove both assertions by induction .
Let LaTeXMLMath and LaTeXMLMath .
Assuming that LaTeXMLMath , we get that on LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLMath on LaTeXMLMath .
Next , we prove that LaTeXMLMath .
It is clear that LaTeXMLMath .
Assume that LaTeXMLMath .
Since LaTeXMLMath , we have that , on LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation So , to finish the proof it suffices to check that LaTeXMLMath , which follows easily from the definition of LaTeXMLMath ∎ Let LaTeXMLMath and let LaTeXMLMath be the sub-bimodule of LaTeXMLMath generated by the tensors LaTeXMLMath such that at least LaTeXMLMath of the LaTeXMLMath ’ s belong to LaTeXMLMath .
The normalized Hochschild resolution LaTeXMLMath and the resolution LaTeXMLMath are filtered by LaTeXMLMath and LaTeXMLMath , respectively The maps LaTeXMLMath , LaTeXMLMath and LaTeXMLMath preserve filtrations .
Let LaTeXMLMath .
We claim that a ) LaTeXMLMath for all LaTeXMLMath , b ) LaTeXMLMath for all LaTeXMLMath , c ) LaTeXMLMath for all LaTeXMLMath , d ) LaTeXMLMath .
In fact a ) , b ) and c ) follow immediately from the definition of LaTeXMLMath .
Suppose d ) is valid for LaTeXMLMath .
Let LaTeXMLMath .
Using a ) and b ) , we get that for LaTeXMLMath , LaTeXMLEquation .
Since LaTeXMLMath , to prove d ) for LaTeXMLMath we only must check that LaTeXMLMath .
If LaTeXMLMath , then using a ) and b ) , we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and if LaTeXMLMath , then LaTeXMLMath , which together a ) and c ) , implies that LaTeXMLEquation .
From d ) follows immediately that LaTeXMLMath preserves filtrations .
Next , assuming that LaTeXMLMath preserve filtrations , we prove that LaTeXMLMath does it .
Let LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLEquation it suffices to see that LaTeXMLMath for LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Next , we prove that LaTeXMLMath preserves filtrations .
Assume that LaTeXMLMath does it .
Let LaTeXMLMath .
It is evident that LaTeXMLMath .
Since LaTeXMLMath , from d ) we get LaTeXMLEquation .
It remains to check that LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLMath .
Hence , if LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and if LaTeXMLMath , then LaTeXMLMath , and so LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath an LaTeXMLMath -bimodule .
In this section we use Theorem 1.1.1 in order to construct complexes LaTeXMLMath and LaTeXMLMath , simpler than the canonical ones , giving the Hochschild homology and cohomology of LaTeXMLMath with coefficients in LaTeXMLMath respectively .
These complexes have natural filtrations that allow us to obtain spectral sequences converging to LaTeXMLMath and LaTeXMLMath respectively .
Let LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ) be the morphisms defined by : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
The Hochschild homology of LaTeXMLMath with coefficients in LaTeXMLMath is the homology of the chain complex LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
It follows from the fact that LaTeXMLMath .
An isomorphism is provided by the maps LaTeXMLMath , defined by LaTeXMLMath ∎ Let LaTeXMLMath .
Clearly LaTeXMLMath is a filtration of LaTeXMLMath .
Using this fact we obtain : There is a convergent spectral sequence LaTeXMLEquation where LaTeXMLMath is considered as an LaTeXMLMath -bimodule via LaTeXMLMath .
The normalized Hochschild complex LaTeXMLMath has a filtration LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath -submodule of LaTeXMLMath generated by the tensors LaTeXMLMath such that at least LaTeXMLMath of the LaTeXMLMath ’ s belong to LaTeXMLMath .
The spectral sequence associate to this filtration is called the homological Hochschild-Serre spectral sequence .
Since , for each extension of groups LaTeXMLMath with LaTeXMLMath a normal subgroup , it is hold that LaTeXMLMath is a crossed product of LaTeXMLMath on LaTeXMLMath , the following theorem ( joint with Corollary 3.1.3 below ) gives , as a particular case , the homological version of the main results of LaTeXMLCite .
The homological Hochschild-Serre spectral sequence is isomorphic to the one obtained in Corollary 2.1.2.1 .
It is an easy consequence of Propositions 1.2.1 and 1.2.2 .
Let LaTeXMLMath be the LaTeXMLMath -submodule of LaTeXMLMath spanned by the set of all elements LaTeXMLMath ( LaTeXMLMath ) .
It is easy to see that LaTeXMLMath is a coideal in LaTeXMLMath .
Let LaTeXMLMath be the quotient coalgebra LaTeXMLMath .
Given LaTeXMLMath , we let LaTeXMLMath denote the class of LaTeXMLMath in LaTeXMLMath .
Given a subcoalgebra LaTeXMLMath of LaTeXMLMath and a right LaTeXMLMath -comodule LaTeXMLMath , we put LaTeXMLMath .
It is well known that if LaTeXMLMath decomposes as a direct sum of subcoalgebras LaTeXMLMath ( LaTeXMLMath ) , then LaTeXMLMath .
Now , let us assume that LaTeXMLMath is a Hopf bimodule .
That is , LaTeXMLMath is an LaTeXMLMath -bimodule and a right LaTeXMLMath -comodule , and the coaction LaTeXMLMath verifies : LaTeXMLEquation .
For each LaTeXMLMath , LaTeXMLMath is an LaTeXMLMath -comodule via LaTeXMLEquation .
Moreover , the map LaTeXMLMath is a map of complexes .
This fact implies that if LaTeXMLMath is a subcoalgebra of LaTeXMLMath , then LaTeXMLMath .
We consider the subcomplex LaTeXMLMath of LaTeXMLMath , with modules LaTeXMLMath , and we let LaTeXMLMath denote its homology .
By the above discussion , if LaTeXMLMath decomposes as a direct sum of subcoalgebras LaTeXMLMath ( LaTeXMLMath ) , then LaTeXMLMath .
Consequently LaTeXMLMath .
Finally , the filtration of LaTeXMLMath induces a filtration on LaTeXMLMath .
Hence we have a convergent spectral sequence LaTeXMLEquation where LaTeXMLMath is an LaTeXMLMath -bimodule via LaTeXMLMath .
Let us assume that LaTeXMLMath , LaTeXMLMath is cocommutative , LaTeXMLMath is commutative , LaTeXMLMath is symmetric as an LaTeXMLMath -bimodule and the cocycle LaTeXMLMath takes its values in LaTeXMLMath .
In LaTeXMLCite was obtained a decomposition of the canonical Hochschild complex LaTeXMLMath .
It is easy to check that the maps LaTeXMLMath and LaTeXMLMath are compatible with this decomposition .
Since LaTeXMLMath for all LaTeXMLMath , we obtain a decomposition of LaTeXMLMath , and then a decomposition of LaTeXMLMath .
Let LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath ) be the morphisms defined by : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
The Hochschild cohomology of LaTeXMLMath with coefficients in LaTeXMLMath is the homology of LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
It follows from the fact that LaTeXMLMath .
An isomorphism is provided by the maps LaTeXMLMath , defined by LaTeXMLMath ∎ Let LaTeXMLMath .
Clearly LaTeXMLMath is a filtration of LaTeXMLMath .
Using this fact we obtain : There is a convergent spectral sequence LaTeXMLEquation where LaTeXMLMath is considered as an LaTeXMLMath -bimodule via LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath -submodule of LaTeXMLMath consisting of maps LaTeXMLMath , for which LaTeXMLMath whenever LaTeXMLMath of the LaTeXMLMath ’ s belong to LaTeXMLMath .
The normalized Hochschild complex LaTeXMLMath is filtered by LaTeXMLMath .
The spectral sequence associated to this filtration is called the cohomological Hochschild-Serre spectral sequence .
The following theorem ( joint with Corollary 3.2.3 below ) gives , as a particular case , of the main results of LaTeXMLCite .
The cohomological Hochschild-Serre spectral sequence is isomorphic to the one obtained in Corollary 2.2.2.1 .
It is an easy consequence of Propositions 1.2.1 and 1.2.2 .
Assume that LaTeXMLMath , LaTeXMLMath is cocommutative , LaTeXMLMath is commutative , LaTeXMLMath is symmetric as an LaTeXMLMath -bimodule and the cocycle LaTeXMLMath takes its values in LaTeXMLMath .
Then , the Hochschild cohomology LaTeXMLMath has a decomposition similar to the one obtained in 2.1.4 for the Hochschild homology .
Let LaTeXMLMath and LaTeXMLMath an LaTeXMLMath -bimodule .
Assume that the cocycle LaTeXMLMath is invertible .
Then , the map LaTeXMLMath is convolution invertible and its inverse is the map LaTeXMLMath .
Under this hypothesis , we prove that the complexes LaTeXMLMath and LaTeXMLMath of Section 2 are isomorphic to simpler complexes .
These complexes have natural filtrations , which give the spectral sequences obtained in LaTeXMLCite .
Using these facts and a theorem of Gerstenhaber and Schack , we prove that if the LaTeXMLMath -cocycle LaTeXMLMath takes its values in a separable subalgebra of LaTeXMLMath , then the Hochschild ( co ) homology of LaTeXMLMath with coefficients in LaTeXMLMath is the ( co ) homology of LaTeXMLMath with coefficients in a ( co ) chain complex .
Finally , as an application we obtain some results about the LaTeXMLMath and LaTeXMLMath functors and an upper bound for the global dimension of LaTeXMLMath .
Let LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ) be the morphisms defined by : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the complex LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
The map LaTeXMLMath , given by LaTeXMLEquation is an isomorphism of complexes .
Consequently , the Hochschild homology of LaTeXMLMath with coefficients in LaTeXMLMath is the homology of LaTeXMLMath .
A direct computation shows that LaTeXMLMath is a morphism of complexes .
The inverse map of LaTeXMLMath is the map LaTeXMLMath ∎ Note that when LaTeXMLMath takes its values in LaTeXMLMath , then LaTeXMLMath is the total complex of the double complex LaTeXMLMath .
For each LaTeXMLMath , we have the morphism LaTeXMLMath , defined by LaTeXMLMath .
For each LaTeXMLMath the endomorphisms of LaTeXMLMath induced by LaTeXMLMath and by LaTeXMLMath coincide .
Consequently LaTeXMLMath is a left LaTeXMLMath -module .
By a standard argument it is sufficient to prove it for LaTeXMLMath , and in this case the result is immediate∎ The chain complex LaTeXMLMath has a filtration LaTeXMLMath , where LaTeXMLMath .
The spectral sequence of this filtration is isomorphic to the one obtained in Corollary 2.1.2 .
From Proposition 3.1.2 it follows that if LaTeXMLMath is a flat LaTeXMLMath -module , then LaTeXMLMath and LaTeXMLMath .
Given an LaTeXMLMath -bimodule LaTeXMLMath we let LaTeXMLMath denote the LaTeXMLMath -submodule of LaTeXMLMath generated by the commutators LaTeXMLMath ( LaTeXMLMath and LaTeXMLMath ) .
From Corollary 3.1.3 it follows immediately that if LaTeXMLMath is separable , then LaTeXMLMath , and if LaTeXMLMath is quasi-free , then there is a long exact sequence LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be a separable subalgebra of LaTeXMLMath .
Next we prove that if the LaTeXMLMath -cocycle LaTeXMLMath takes its values in LaTeXMLMath , then the Hochschild homology of LaTeXMLMath with coefficients in LaTeXMLMath is the homology of LaTeXMLMath with coefficients in a chain complex .
When LaTeXMLMath equals LaTeXMLMath we recover the first part of Remark 3.1.4 .
Assume that LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath -times ) for LaTeXMLMath , and let LaTeXMLMath be the cyclic tensor product over LaTeXMLMath of LaTeXMLMath and LaTeXMLMath ( see LaTeXMLCite or LaTeXMLCite ) .
Using the fact that LaTeXMLMath takes its values in LaTeXMLMath , it is easy to see that LaTeXMLMath acts on LaTeXMLMath via LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
The Hochschild homology LaTeXMLMath , of LaTeXMLMath with coefficients in LaTeXMLMath , is the homology of LaTeXMLMath with coefficients in LaTeXMLMath .
Let LaTeXMLMath be the double complex with horizontal differentials LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the total complex of LaTeXMLMath .
We must prove that LaTeXMLMath is the homology of LaTeXMLMath .
Let LaTeXMLMath be the map LaTeXMLMath .
Consider the filtration LaTeXMLMath of LaTeXMLMath , where LaTeXMLMath .
From Theorem 1.2 of LaTeXMLCite , it follows that LaTeXMLMath is a morphism of filtered complexes inducing an quasi-isomorphism between the graded complexes associated to the filtrations of LaTeXMLMath and LaTeXMLMath .
Consequently LaTeXMLMath is a quasi-isomorphism .
The proof can be finished by applying Theorem 3.1.1∎ Here we freely use the notations of Subsection 2.1.3 .
Suppose LaTeXMLMath is a Hopf bimodule .
A direct computation shows that the LaTeXMLMath -coaction of LaTeXMLMath , obtained transporting the one of LaTeXMLMath through LaTeXMLMath , is given by LaTeXMLEquation .
For each subcoalgebra LaTeXMLMath of LaTeXMLMath , we consider the subcomplex LaTeXMLMath of LaTeXMLMath with modules LaTeXMLMath , and we let LaTeXMLMath denote its homology .
If LaTeXMLMath decomposes as a direct sum of subcoalgebras LaTeXMLMath ( LaTeXMLMath ) , then LaTeXMLMath .
Consequently LaTeXMLMath .
From LaTeXMLMath it follows that if LaTeXMLMath is cocommutative , then LaTeXMLMath .
Finally , the filtration of LaTeXMLMath induces a filtration on LaTeXMLMath .
Hence , when LaTeXMLMath is cocommutative and LaTeXMLMath is a flat LaTeXMLMath -module , we have a convergent spectral sequence LaTeXMLEquation where LaTeXMLMath is a left LaTeXMLMath -module via the action introduced in Proposition 3.1.2 .
Let LaTeXMLMath be a field , LaTeXMLMath an arbitrary LaTeXMLMath -algebra , LaTeXMLMath a right LaTeXMLMath -module and LaTeXMLMath a left LaTeXMLMath -module .
It is well known that LaTeXMLMath ( here LaTeXMLMath is an LaTeXMLMath -bimodule via LaTeXMLMath ) .
This fact and Corollary 3.1.3 show that if LaTeXMLMath is a field , LaTeXMLMath is a right LaTeXMLMath -module and LaTeXMLMath is a left LaTeXMLMath -module , then there is a convergent spectral sequence LaTeXMLEquation .
Let LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath ) be the morphisms defined by : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the complex LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
The map LaTeXMLMath , given by LaTeXMLEquation is an isomorphism of complexes .
Consequently , the Hochschild cohomology of LaTeXMLMath with coefficients in LaTeXMLMath is the homology of LaTeXMLMath .
It is similar to the proof of Theorem 3.1.1∎ Note that when LaTeXMLMath takes its values in LaTeXMLMath , then LaTeXMLMath is the total complex of the double complex LaTeXMLMath .
For each LaTeXMLMath we have the map LaTeXMLMath , defined by LaTeXMLMath .
For each LaTeXMLMath the endomorphisms of LaTeXMLMath induced by LaTeXMLMath and by LaTeXMLMath coincide .
Consequently LaTeXMLMath is a right LaTeXMLMath -module .
By a standard argument it is sufficient to prove it for LaTeXMLMath , and in this case the result is immediate∎ The cochain complex LaTeXMLMath has a filtration LaTeXMLMath , where LaTeXMLMath .
The spectral sequence of this filtration is isomorphic to the one obtained in Corollary 2.2.2 .
From Proposition 3.2.2 it follows that LaTeXMLMath and LaTeXMLMath .
Given an LaTeXMLMath -bimodule LaTeXMLMath , we let LaTeXMLMath denote the LaTeXMLMath -submodule of LaTeXMLMath consisting of the elements LaTeXMLMath verifying LaTeXMLMath for all LaTeXMLMath .
From Corollary 3.2.3 , it follows immediately that if LaTeXMLMath is separable , then LaTeXMLMath and if LaTeXMLMath is quasi-free , then there is a long exact sequence LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be a separable subalgebra of LaTeXMLMath and let LaTeXMLMath ( LaTeXMLMath ) be as in 3.1.5 .
Suppose LaTeXMLMath for all LaTeXMLMath .
Using the fact that LaTeXMLMath takes its values in LaTeXMLMath it is easy to see that LaTeXMLMath acts on LaTeXMLMath via LaTeXMLMath .
The Hochschild cohomology LaTeXMLMath , of LaTeXMLMath with coefficients in LaTeXMLMath , is the cohomology of LaTeXMLMath with coefficients in LaTeXMLMath .
It is similar to the proof of Theorem 3.1.5.1∎ Let LaTeXMLMath be a field , LaTeXMLMath an arbitrary LaTeXMLMath -algebra and LaTeXMLMath , LaTeXMLMath two left LaTeXMLMath -modules .
It is well known that LaTeXMLMath ( here LaTeXMLMath is an LaTeXMLMath -bimodule via LaTeXMLMath ) .
This fact and Corollary 3.2.3 show that if LaTeXMLMath is a field and LaTeXMLMath and LaTeXMLMath are left LaTeXMLMath -modules , then there is a convergent spectral sequence LaTeXMLEquation .
As a corollary we obtain that LaTeXMLMath , where LaTeXMLMath denotes the left global dimension .
Note that this result implies Maschke ’ s Theorem for crossed product , as it was established in LaTeXMLCite .
Assume that LaTeXMLMath is a crossed product with invertible cocycle .
In this case another two spectral sequences converging to LaTeXMLMath and with LaTeXMLMath -term LaTeXMLMath can be considered .
They are the Cartan-Leray and the Grothendieck spectral sequences .
The last one was introduced for the more general setting of Galois extension in LaTeXMLCite .
In this Section we recall these constructions and we prove that both coincide with the Hochschild-Serre spectral sequence .
Similar results are valid in the cohomological setting .
Let LaTeXMLMath be the canonical resolution of LaTeXMLMath as a right LaTeXMLMath -module and LaTeXMLMath .
Consider LaTeXMLMath as an LaTeXMLMath -bimodule via LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
It is clear that LaTeXMLEquation where LaTeXMLMath , is a complex of LaTeXMLMath -bimodules .
Moreover LaTeXMLMath is contractible as a complex of left LaTeXMLMath -modules , with contracting homotopy LaTeXMLMath ( LaTeXMLMath ) given by LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
Since the map LaTeXMLEquation given by LaTeXMLMath , is an isomorphism of LaTeXMLMath -bimodules ( the inverse of LaTeXMLMath is the map LaTeXMLMath ) , LaTeXMLMath is a relative projective resolution of LaTeXMLMath .
Let LaTeXMLMath be an LaTeXMLMath -bimodule .
The groups LaTeXMLMath are left LaTeXMLMath -modules via LaTeXMLMath , where LaTeXMLMath .
There is an isomorphism LaTeXMLEquation .
Let LaTeXMLMath .
It is immediate that LaTeXMLMath , is a filtration of the last complex .
The spectral sequence associate to this filtration converges to LaTeXMLMath and has LaTeXMLMath -term LaTeXMLMath .
This spectral sequence is called the homological Cartan-Leray spectral sequence .
Similarly the groups LaTeXMLMath are right LaTeXMLMath modules via LaTeXMLMath and there is an isomorphism LaTeXMLEquation .
This complex has a filtration LaTeXMLMath , defined by LaTeXMLMath .
The spectral sequence associate to this filtration converges to LaTeXMLMath and has LaTeXMLMath -term LaTeXMLMath .
This spectral sequence is called the cohomological Cartan-Leray spectral sequence .
Let LaTeXMLMath and LaTeXMLMath be the morphisms of LaTeXMLMath -bimodule complexes , recursively defined by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It is hold that LaTeXMLMath and that LaTeXMLMath is homotopically equivalent to the identity map .
The homotopy LaTeXMLMath is recursively defined by LaTeXMLMath and LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
It is easy to see that LaTeXMLMath and LaTeXMLMath are morphisms of complexes .
Arguing as in Proposition 1.2.1 we get that LaTeXMLMath is an homotopy from LaTeXMLMath to the identity map .
It remains to prove that LaTeXMLMath .
It is clear that LaTeXMLMath .
Assume that LaTeXMLMath .
Since LaTeXMLMath , we have that on LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation Next , we consider the normalized Hochschild resolution LaTeXMLMath filtered as in Proposition 1.2.2 and the resolution LaTeXMLMath filtered by LaTeXMLMath , where LaTeXMLMath .
We have that LaTeXMLEquation .
LaTeXMLEquation Consequently the map LaTeXMLMath preserve filtrations .
It follows by induction on LaTeXMLMath , using the recursive definition of LaTeXMLMath ∎ The map LaTeXMLMath induces an homotopy equivalence of LaTeXMLMath -bimodule complexes between the graded complexes associated to the filtrations of LaTeXMLMath and LaTeXMLMath .
Note that LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the boundary map introduced in Subsection 1.1 .
By Proposition 1.2.2 it suffices to check that LaTeXMLMath induces an homotopy equivalence LaTeXMLMath of LaTeXMLMath -bimodules complexes , from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be as in Subsection 1.1 and LaTeXMLMath endowed with the structure of LaTeXMLMath -bimodule given by LaTeXMLMath , where LaTeXMLMath .
Consider the diagram LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
We assert that LaTeXMLMath , where LaTeXMLMath , with LaTeXMLMath .
To prove this it suffices to check that LaTeXMLEquation which follows by induction on LaTeXMLMath , using that LaTeXMLMath .
Now , it is immediate that LaTeXMLMath .
Since LaTeXMLMath is an isomorphism and the rows of LaTeXMLMath are relative projective resolutions of LaTeXMLMath and LaTeXMLMath respectively , it follows that LaTeXMLMath is an homotopy equivalence∎ The ( co ) homological Cartan-Leray spectral sequence is isomorphic to the ( co ) homological Hochschild-Serre spectral sequence .
Sonoluminescence is the phenomena of light emission from a collapsing gas bubble in a liquid .
Theoretical explanations of this extreme energy focusing are controversial and difficult to validate experimentally .
We propose to use molecular dynamics simulations of the collapsing gas bubble to clarify the energy focusing mechanism , and provide insight into the mechanism of light emission .
In this paper , we model the interior of a collapsing noble gas bubble as a hard sphere gas driven by a spherical piston boundary moving according to the Rayleigh-Plesset equation .
We also include a simple treatment of ionization effects in the gas at high temperatures .
By using fast , tree-based algorithms , we can exactly follow the dynamics of million particle systems during the collapse .
Our results clearly show strong energy focusing within the bubble , including the formation of shocks , strong ionization , and temperatures in the range of 50,000—500,000 degrees Kelvin .
Our calculations show that the gas-liquid boundary interaction has a strong effect on the internal gas dynamics .
We also estimate the duration of the light pulse from our model , which predicts that it scales linearly with the ambient bubble radius .
As the number of particles in a physical sonoluminescing bubble is within the foreseeable capability of molecular dynamics simulations we also propose that fine scale sonoluminescence experiments can be viewed as excellent test problems for advancing the art of molecular dynamics .
As a gas bubble in a liquid collapses , the potential energy stored during its prior expansion is released and strongly focused .
The extent of focusing can be so great that a burst of light is emitted at the final stage of collapse .
This process can be driven repeatedly by exciting bubbles with a sound field , and the resulting transduction of sound into light is known as sonoluminescence ( SL ) LaTeXMLCite .
Sonoluminescence can be observed in dense fields of transient cavitation bubbles produced by applying intense sound to a liquid , or in a periodic single bubble mode which allows more detailed experimental observations .
In single bubble SL , a single gas bubble in the liquid is created and periodically driven to expand and collapse by an applied sound field .
The bubble begins its cycle of evolution as the low pressure phase of the sound field arrives , causing it to expand to a maximal radius .
As the applied acoustic pressure increases , the bubble begins to collapse , first reaching its ambient radius and corresponding ambient pressure and internal temperature , and then radially collapsing further , with the bubble walls falling inward driven by the rising external fluid pressure .
The collapse accelerates rapidly , until gas trapped inside the bubble is compressed and heated to a pressure that ultimately halts and reverses the motion of the bubble walls .
Thus the bubble reaches a minimum radius , and then rapidly “ bounces ” back to a much larger size .
At some point near the minimum radius , the resulting internal “ hot spot ” can release a burst of light .
While the basic bubble collapse dynamics can be observed for a variety of gas and liquid combinations , light emission typically requires the bubble to contain sufficient noble gas , and works particularly well in water .
The mechanism of light emission from the gas is not understood , nor is much known about related quantities such as peak temperatures , pressures or levels of ionization .
It is known that the mechanical conditions during collapse of a common gas bubble in water are quite extreme : as the bubble reaches sub-micron diameter , the bubble wall experiences accelerations that exceed LaTeXMLMath g and supersonic changes in velocity that occur on picosecond time scales .
In order to understand how this affects the state of the internal gas , the standard approach is to apply continuum fluid mechanics .
Some models assume that pressure and temperature are uniform inside of the collapsed bubble LaTeXMLCite while other theories calculate the effects of imploding shock waves LaTeXMLCite .
Various fluid models have been applied both at non-dissipative ( Euler Equations ) and dissipative ( Navier-Stokes Equations ) levels of description LaTeXMLCite .
All these fluid approaches are limited in their predictive power by the need to represent transport processes and the equation of state .
Under such extreme flow conditions , little is known about these effects and one is forced to extrapolate from known forms .
The net result is that the modeling predictions directly reflect these assumptions .
This is not satisfactory for the purpose of understanding what actually occurs within the bubble .
An especially fundamental limitation of continuum mechanical approaches is the assumption of local thermodynamic equilibrium , i.e .
it is assumed that the macroscopic fluid variables do not change much over molecular length and time scales .
Although the bubble starts out in such a state , its subsequent runaway collapse ultimately leads to a regime where this clearly does not hold .
In this state , from which the ultraviolet picosecond flash of light is emitted , one can question the basic applicability of hydrodynamic models .
We propose to remove the assumption of thermodynamic equilibrium , and also eliminate any controversy over the correct equation of state , by using molecular dynamics ( MD ) simulations of the gas dynamics within the bubble .
In this approach , we directly apply Newton ’ s laws of motion to the gas molecules , including as much detail as is desired ( or practical ) about the molecular collisions and related atomic physics .
While this approach is computationally intensive , it delivers to us a clear , physical picture of what actually transpires during the bubble collapse .
With the inclusion of sufficient detail and efficient programming , it could ultimately allow the simulation of of the light emission process itself .
While the small length and time scales of sonoluminescence present major obstacles for hydrodynamic modeling , they actually make it ideal for molecular dynamics : precisely because the final system is so small , it becomes possible to do a complete MD simulation of the collapse .
In fact , sonoluminescence is somewhat unique in this regard .
Usually the systems directly simulated with molecular dynamics are many orders of magnitude smaller—fewer particles , shorter time scales—than the corresponding systems realized in experiments or in nature , and this gap is too large to be eliminated by increases in computing power LaTeXMLCite .
In contrast , the number of particles within the interior of a small SL bubble is comparable to the number of simulation particles that can be handled with current computational facilities .
For example , a typical SL bubble driven at 30kHz has an ambient radius of 6 LaTeXMLMath and contains LaTeXMLMath particles .
At the extreme , SL bubbles containing on the order of several million particles have been observed in systems driven at Megahertz frequencies LaTeXMLCite .
This compares well with simulations , where we have been able to compute the gas dynamics of a one million particle bubble collapse using a run time of a few days on a single processor workstation-grade computer .
Parallel processing simulations would make 10 to 100 million particle simulations feasible .
As the number of simulation particles reaches that in real systems , the remaining computer power can be used to add in more complex atomic physics , and thus allow more detailed study of the processes involved .
The overall goal of the MD modeling is to generate a better understanding of the processes that result in energy focusing and light emission during SL .
This is to be accomplished through a dual approach of model prediction and model validation : we use the model to illustrate the phenomena that can not be experimentally observed during the collapse , and also to make predictions that can be experimentally validated .
The basic experimental unknown in SL is the degree of energy focusing that is achieved inside of the bubble .
For example , the spectral density of light from helium bubbles in water is still increasing at wavelengths as short as 200nm ( energy exceeding 6 eV ) where the extinction coefficient of water cuts off the measurement LaTeXMLCite .
Related to the question of energy focusing are the detailed questions of whether there is shock formation within the bubble , whether there is plasma formation , and what peak temperatures are achieved during the collapse .
For example , the most extreme theoretical estimates suggest that the interior may reach temperatures sufficient to induce deuterium-tritium fusion LaTeXMLCite .
Over the range of parameter space studied , shock formation and strong ionization appear to be typical , and the lowest peak temperatures found in our simulations are about 40,000K , with the highest approaching 500,000K .
Our findings also indicate that boundary conditions strongly affect the interior motion .
With a low , fixed temperature ( i.e .
heat bath ) condition the peak temperatures and internal gradients are higher than for adiabatic motion .
A key experimental observable in SL is the duration of the light flash , or “ flash width ” LaTeXMLCite , because knowledge of this puts constraints on the underlying mechanism of light emission .
This can be used as a validation point for any model or theory .
For example , volume radiation from a plasma will yield a different flash width than surface radiation from a black body .
Since our simulations do not include fundamental radiative mechanisms such as atomic excitation or charged particle acceleration , our current MD model can not directly determine determine the light emitting mechanism or the flash width .
However , a prediction about the flash width can be obtained from our calculation of the peak temperature as a function of time , assuming the light emission occurs while the peak temperature is high .
Our simulations for Helium show that simple adiabatic compression does not produce a sharp temperature spike in time , but the thermal boundary condition causes a spike with a duration that scales with the ambient bubble radius .
This result predicts that the flash width should scale with the ambient bubble radius .
If valid , this scaling suggests that at high acoustic frequencies LaTeXMLMath MHz ] LaTeXMLCite the duration of a SL flash could be about equal to or less than 1 ps .
The outline of the paper follows : Section 2 describes the model for the bubble collapse in detail .
Section 3 outlines the principle algorithms used to evolve the hard sphere system .
Section 4 provides detailed results from our MD simulations .
Finally , Section 5 concludes a summary of our observations , and lists interesting future areas of investigation suggested by this first attempt at molecular dynamics modeling of sonoluminescence .
In this section , we present our Molecular Dynamics model for SL bubbles .
The overall strategy is to model the system as a spherical piston that compresses a gas of hard spheres , with energy deducted from the system for ionization events at higher temperatures .
The details and motivations for this are given in the following subsections .
We want to focus on the simulation of single bubble sonoluminescence , so that results can be compared to the best studied experimental SL systems .
Such bubbles remain spherical during their collapse LaTeXMLCite , and their behavior is parameterized by their ambient radius ( the radius they have when at rest at the ambient pressure ) and their maximum radius ( the radius they attain when maximally expanded at the low pressure point of the applied sound field ) .
We can not directly simulate all such SL bubbles , since they may contain several orders of magnitude more gas particles than our computational budget can accommodate .
Typically , we can afford to do a calculation with some given number of simulation particles , LaTeXMLMath , and the question becomes how large of a bubble can we directly simulate .
The ambient radius , LaTeXMLMath , is related to the number of gas particles , LaTeXMLMath , by the ideal gas equation of state LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath atm are the ambient temperature and pressure , and LaTeXMLMath is Boltzmann ’ s constant .
Thus we see that the fewer the simulation particles we use , the smaller the ambient size of the bubble being simulated .
Once the ambient size is determined by our simulation budget , we are however free to choose any maximum radius .
For experimentally relevant simulations , the maximum radius LaTeXMLMath is chosen to yield the same ratio of LaTeXMLMath [ LaTeXMLMath ] for the MD simulation as is seen in experimental SL bubbles .
This is natural because this ratio is a measure of the available energy stored in the expansion , since the stored energy/particle due to the work done by expanding to the maximum volume LaTeXMLMath against the applied pressure LaTeXMLMath is LaTeXMLEquation .
Since the bubble remains spherical during collapse , its boundary dynamics are described entirely by the radius as a function of time , LaTeXMLMath .
We are concerned with energy focusing processes and gas dynamics inside the bubble , and in this spirit we will take LaTeXMLMath as being known .
A convenient model of the spherical piston that captures some qualitative features of the supersonic collapse is provided by Rayleigh ’ s equation LaTeXMLCite LaTeXMLEquation with a van der Waals hard core equation of state LaTeXMLEquation .
LaTeXMLMath , where LaTeXMLMath is the radius of the gas in the bubble when compressed to its van der Waals hard core ( LaTeXMLMath for He , Ar , Xe ) , LaTeXMLMath is the density of the surrounding fluid , and the initial condition for the solution to ( LaTeXMLRef ) is that LaTeXMLMath when LaTeXMLMath .
We emphasize that the derivation of Equations ( LaTeXMLRef , LaTeXMLRef ) applies only for small Mach number motion and thus they are invalid as a fundamental theory for SL LaTeXMLCite .
However , in this first attempt to simulate SL with molecular dynamics we are interested in possible focusing processes within the bubble and use of ( LaTeXMLRef , LaTeXMLRef ) as a launch condition appears appropriate since the resulting LaTeXMLMath reasonably approximates the gross bubble pulsation LaTeXMLCite .
Consistent with this approximation , viscous damping and acoustic radiation have also been neglected .
At the next level of simulation one should include a self-consistently determined boundary condition on pressure at the bubble ’ s wall .
In this way , energy loss due to acoustic radiation is properly accounted for .
As a point of comparison , it is worth noting that the adiabatic equations of state for the van der Waals pressure LaTeXMLMath and the computed equilibrium hard sphere pressure LaTeXMLMath ( the equilibrium pressure as a function of radius as the radius is decreased slowly on the hard spheres ) agree very well , except at bubble radii near the hard core , as graphed in Figure LaTeXMLRef for Helium ( with the other noble gases also in good agreement , except near the hard core ) .
At small radii , the van der Waals pressure diverges as LaTeXMLMath tends to LaTeXMLMath and the LaTeXMLMath diverges as LaTeXMLMath tends to LaTeXMLEquation which is the minimum radius for random closed packing LaTeXMLCite .
It has been observed that for SL in water , the bubble must contain sufficient amounts of a noble gas .
Thus in many single bubble SL experiments , the water is first de-gassed to remove atmospheric gases , and then re-saturated with a noble gas to produce pure noble gas bubbles .
We will focus our gas dynamic model on this system , since it is a frequent experimental model and also because it allows the simplest molecular gas dynamics models .
Because the gas is noble , it consists of isolated atoms that do not engage in chemical reactions .
Thus we can model it with simple gas particles that have no rotational or internal vibrational degrees of freedom , and which do not engage in any chemical reactions with the water walls of the bubble , even at elevated temperatures .
Molecular dynamics simulations for such simple gas particles fall into two broad categories , defined by the way they treat interatomic forces .
The forces can either be given by a potential that varies continuously with radius from the atom center ( “ soft sphere ” ) , or by a potential that is a step function of radius ( “ hard sphere ” ) .
The latter particles behave simply like billiard balls .
While the continuous potential are more physically realistic , they are also much more costly to compute with .
This is because numerical time integration methods must be used to compute the particle motions in response to the continuously varying forces , and the time step must be small enough to accurately resolve all particle trajectories in the system .
Thus the motion of a few fast moving particles will force the use a small , costly timestep for all particles in the system .
In contrast , step potentials do not experience this problem because they evolve in time by a series of discrete collision events .
No explicit numerical integration is needed since impulsive collisions are carried out only when atoms interact , and between collisions each atom follows an independent linear trajectory .
Thus each atom effectively uses its own optimally large timestep , instead of an excessively small step imposed by the the fastest particles in the system .
Moreover , there is no numerical integration error because trajectories are evaluated to within the roundoff error of the machine .
LaTeXMLCite Because of this difference in computational cost , it is desirable to use the hard sphere model if it can capture the physics of interest .
In our case , we want to get accurate gas dynamics at mid to high energies for fairly large numbers of particles .
Whether hard spheres are sufficient to model this regime in an SL bubble is an empirical question , but such models have been shown to yield accurate predictions of noble gas viscosity from room temperature up to the gas ionization temperatures LaTeXMLCite .
We take this to be a reasonable validation that a hard sphere gas provides a good model for the gas dynamics encountered during bubble collapse , at least up to ionization temperatures .
Near that point and beyond , it also seems reasonable that a hard sphere model applies , since the softer parts of the potential are all the more insignificant for high energy collisions .
The dynamics of a hard sphere system involve processing impulsive collisions at the collision times .
To illustrate , consider two particles separated by a relative position r and having a relative velocity v .
These particles collide if their separation equals the atomic diameter LaTeXMLMath at some time LaTeXMLMath in the future .
If such a collision occurs , then LaTeXMLMath is the smaller positive solution of LaTeXMLEquation which has a solution LaTeXMLEquation .
Collisions are carried out impulsively so that the change in velocities preserves energy and momentum .
Specifically , LaTeXMLEquation where LaTeXMLMath is the change in velocity of the first particle , LaTeXMLMath is the change in velocity of the second particle and LaTeXMLMath is the relative position at the time of collision .
Extensions to step potentials that consist of a hard repulsive core surrounded by an attractive well are also possible .
See LaTeXMLCite for details .
When a gas particle hits the bubble wall , it might simply be directed back into the interior by a strong collision with a liquid molecule , or it may penetrate into the liquid , undergoing multiple thermalizing collisions .
In the latter case , assuming the liquid is already saturated with gas atoms , the thermalized atom ( or an equivalent one from the saturated liquid reservoir ) will ultimately random walk its way back into the bubble interior .
For our MD model , we will idealize these two modes of boundary interaction as either as energy conserving specular collisions or as a heat bath boundary conditions .
For the case of specular collisions , particles reflect from the boundary with a speed equal to the collision speed in the local rest frame of the wall .
The direction of propagation is determined according to the law of reflection , where the angle of incidence equals the angle of reflection with respect to the local normal to the bubble surface .
For heat bath boundary conditions , when a particle hits the boundary it is assigned a thermal velocity at the ambient liquid temperature LaTeXMLMath , and the direction of propagation back into the interior is chosen according to a suitable angular distribution .
We ignore the small time lag that might exist between exit and reentry for the thermalized gas particle .
For the angular distribution , we use the cosine distribution , where the angle of reflection LaTeXMLMath is assigned randomly according to a probability density function , LaTeXMLEquation .
We have also tried a uniform distribution in angle in selected test cases .
This did not significantly change the simulation results .
( Nonetheless , we note that there may exist situations where results differ qualitatively since a uniform distribution has a greater tendency to cause reflected particles to build up near the wall . )
In reality , we expect that the physical boundary will have some characteristics of both models .
By investigating these extreme cases we hope to see the full range of effects that boundary conditions can have on the bubble dynamics .
Initially the bubble is its maximum radius , LaTeXMLMath , and particles are moving in uniformly distributed random directions , with the same thermal speed LaTeXMLMath , where LaTeXMLMath is the initial temperature and LaTeXMLMath is the mass of the particle .
Randomization of speeds is not necessary , since the particles rapidly thermalize their energies in any case .
For heat bath boundary conditions , the initial temperature is taken to be the ambient temperature , LaTeXMLMath , reflecting the thermalization with the liquid .
With specular boundary conditions this choice generates unphysically large temperatures when the bubble collapses to its ambient size , due to adiabatic heating .
In order to achieve the ambient temperature LaTeXMLMath at the ambient radius LaTeXMLMath , the initial temperature must be scaled down to LaTeXMLMath .
The factor of LaTeXMLMath approximately cancels the adiabatic heating ( since LaTeXMLMath in a LaTeXMLMath ideal gas at constant entropy ) during the initial , slow portion of the collapse .
The basic properties associated with the hard sphere model are the gas particle mass and diameter .
The mass is simply taken to be the mass of the noble gas atom being simulated .
See Table I below .
The choice of proper hard sphere diameter is a much more difficult question .
The diameter should represent the statistical average distance of approach of the particles during collisions , and thus in general it should depend on the collision energy .
In our most basic model we will neglect this temperature dependence and choose particle diameters that have been derived from the kinetic theory for the viscosity of a gas at room temperature LaTeXMLCite : Table I .
Hard Sphere Diameters and Masses To produce a more realistic model for higher temperature regimes of interest in SL , the hard sphere diameter should depend on the relative velocity of the colliding particles .
A variety of models have been proposed to take this effect into account LaTeXMLCite .
These include the variable hard sphere ( VHS ) model LaTeXMLCite , the variable soft sphere ( VSS ) model LaTeXMLCite and the generalized hard sphere ( GHS ) model LaTeXMLCite which is an extension of the VHS and VSS models .
In this report , we are mainly interested in contrasting how variable and constant hard sphere diameters affect our simulations , so the recent VSS model is chosen for its combination of simplicity and calibrated accuracy .
In fact , we find that the VSS model and constant diameter models often produce quantitatively similar results .
See Section LaTeXMLRef for details .
The viscosity based diameter of a VSS particle is LaTeXMLEquation where LaTeXMLMath is Boltzmann ’ s constant , LaTeXMLMath is the mass of the particle , LaTeXMLMath is the dimensionless viscosity index and LaTeXMLMath is a dimensionless constant for each gas .
The constant LaTeXMLMath represents the viscosity at the reference temperature LaTeXMLMath and pressure ( 1 atm ) .
Finally , LaTeXMLMath is the asymptotic kinetic energy where LaTeXMLMath is the reduced mass LaTeXMLEquation and LaTeXMLMath is the relative velocity between the particles .
Tabulated values for these new parameters are provided in LaTeXMLCite and are summarized below .
Table II .
VSS Molecular Parameters Near the minimum radius of the bubble , collisions may become sufficiently energetic to ionize the gas atoms .
Ionization exerts a very strong cooling effect on the gas , since on the order of LaTeXMLMath eV of thermal energy is removed from the gas by each ionization event .
Indeed , if such energy losses are not included , Xenon simulations can reach temperatures in excess of one million degrees Kelvin , while the inclusion of ionization cooling brings these peak temperatures down substantially ( see Section LaTeXMLRef ) .
This clearly shows that some degree of ionization must occur during collapse , and that its cooling effects must be included for proper prediction of peak temperatures .
The ions and free electrons produced by ionization will move according to coulomb forces , but the need to incorporate these effects is not as clear , and their inclusion is more difficult and expensive due to the long range effects , so they will not be included in this first treatment .
We will only consider the impact of ionization on energy accounting .
For the purpose of energy accounting , an ionization ultimately produces two losses : the energy of ionization is lost immediately , and the emitted cold electron will quickly be heated to thermal equilibrium with the gas through subsequent electron-gas collisions , thus extracting an additional one particle ’ s worth of thermal energy by the equipartition of energy .
For our model , we will simply assume ionization occurs with probability 1 whenever the collision energy exceeds the ionization potential , we deduct a suitable amount of energy from the pair .
We also we keep track of how many electrons each particle has lost , so that we can make use of the appropriate next ionization energies and calculate the local ionization levels .
The direction of gas particle propagation is updated exactly as without ionization .
See Section LaTeXMLRef for details .
More precisely , if the kinetic energy ( in the center of mass frame ) of two colliding particles is greater than the next ionization energy of either of the pair ( which may already be ionized ) , that particle loses an additional electron .
We account for the net energy loss by setting the kinetic energy of the pair to be LaTeXMLEquation where LaTeXMLMath is the original kinetic energy of the particle , LaTeXMLMath is the kinetic energy of both particles before the collision and LaTeXMLMath is the ionization potential of the minimally charged particle .
Note that the final kinetic energy of the pair is the initial energy , minus the ionization energy , with an additional 1/3 deducted to represent the subsequent energy lost to thermalizing the electron .
This is not the only possible way to include this effect , and of course in reality this process involves losses from other gas particles besides the colliding pair , but this approach is the simplest way to include the effect .
We also do not account for subsequent electron-ion recombination to neutral atoms , although this would be interesting to include at the next level of description .
In particular , this could be an interesting source of radiation as the hot spot decays .
For reference , the approximate ionization potentials used for the three noble gases are provided in Table III below .
Table III .
Ionization potential [ MJ/mol ] .
Each entry represents the energy required to ionize the indicated state .
Efficient algorithms are needed to evolve our hard sphere model for sonoluminescence since a naive coding is prohibitively slow for anything more than a few thousand particles .
To achieve this goal , we modify and extend existing methods LaTeXMLCite rather than develop new algorithms and codes from scratch .
This section outlines the principle algorithms used to evolve our hard sphere system .
Further details and basic codes are provided in LaTeXMLCite .
The hard sphere simulation proceeds according to a time ordered sequence of collision events LaTeXMLCite .
But clearly a direct determination of the next event for a given particle is impractical in our large simulations because LaTeXMLMath work is required per particle to examine all possible collision partners , where LaTeXMLMath is the total number of particles .
Fortunately this work can be reduced to a constant independent of LaTeXMLMath by dividing the bubble into a number of cells LaTeXMLCite .
By taking an edge length that is larger than the sphere diameter , LaTeXMLMath , it is obvious that collisions can only occur between particles in the same and adjacent cells .
Since we want a relatively small number of particles in each cell and we want the number of cells to be comparable to the number of particles in the simulation the size of the cells must be reduced as the bubble collapses .
We use a straightforward subdivision procedure to accomplish this task .
Initially , the bubble is subdivided into approximately LaTeXMLMath square , identical cells .
Every time the bubble declines by a factor of two in diameter , cell size is reduced by a factor of two ( keeping in mind that we must stop the procedure once the cell diameters reach the particle diameter ) .
Note that we do not need to recompute which particle belongs to which cell after each collision .
Instead , we introduce a cell crossing event and update a particle ’ s cell location only when the corresponding cell crossing event is processed ( cf .
LaTeXMLCite ) .
Also note that when a collision occurs , only particles in the immediate neighborhood of the cell need to be updated .
For this reason a ‘ personal ’ time is stored for each particle , representing the time when the particle was last updated .
The entire system configuration only needs to be updated when properties ( such as density , temperature etc . )
are evaluated .
LaTeXMLCite Because we require information on when particle collisions , hard wall collisions and cell crossings occur some sort of event calendar is needed .
This calendar will store many future events .
As collisions and cell crossings occur , newly predicted collisions and cell crossings must be added to the calendar and events that are no longer relevant must be removed .
LaTeXMLCite Specifically , whenever particles collide their velocities are changed .
This implies that future events involving these particles are no longer valid and should be removed from the calendar .
It also implies that new cell crossings and collision events need to be calculated and added to the calendar .
( Fortunately , the only particle collisions that need to be considered are those involving the current cell and its neighbors . )
On the other hand , when a particle crosses cell boundaries previous particle collisions remain valid but the newly adjacent cells contain potential collision targets that must be examined .
New cell crossing events must also be checked .
As pointed out in LaTeXMLCite , there is no way in which a collision can be missed provided all these details are taken care of correctly .
Of course , it is essential that the calendar can be managed efficiently both in terms of memory and CPU usage .
To meet this requirement , we utilize the binary tree data structure described in LaTeXMLCite .
Here , each scheduled event is represented by a node in the tree .
The information contained within the node identifies the time at which the event is scheduled and the event details .
Calendar events are added or deleted by adding or deleting the corresponding nodes from the tree .
To facilitate traversing the tree , three pointers are used to link event nodes .
These point to the left and right descendents of the node and to the node ’ s parent .
The ordering is carried out so that left hand descendents of a particular node are events scheduled to occur before the event at the current node , while right hand descendents correspond to events that occur after it .
Finally , the rapid deletion of event nodes is supported by linking event nodes into two circular lists .
See LaTeXMLCite for details .
It is interesting that estimates of the theoretical performance of the tree structure are possible in a number of instances LaTeXMLCite .
For example , if a tree is constructed from a series of events that are randomly distributed , the average number of nodal tests to insert a new node into the tree is LaTeXMLMath .
Also , the average number of cycles to delete a randomly selected node is a constant independent of LaTeXMLMath .
It is noteworthy that measurements have been performed to confirm these results in actual MD simulations LaTeXMLCite .
Our sonoluminescence simulations spend most of the CPU time on compressing the bubble from its maximum radius to the ambient radius .
Since the bubble is fairly uniform in this regime , the assumption of a random distribution of events seems plausible and we expect that this type of estimate on theoretical performance should hold .
( On the other hand , near the short-lived hot spot the behavior is far from equilibrium and this assumption on randomness may be invalid . )
A detailed study of the theoretical performance of the tree structure will be the focus of subsequent work .
We need to evaluate spatially dependent average properties of the gas at various times .
To minimize statistical fluctuations , we assume that the results are radially symmetric and average over shells that are 1/40th of the bubble radius .
We calculate dimensionless values for density , temperature , velocity and average charge as follows : The dimensionless density is given by the density divided by the average ambient density .
The dimensionless velocity is given by the velocity divided by the ambient speed of sound LaTeXMLMath where LaTeXMLMath is the ratio of heat capacities and LaTeXMLMath is the mass of a single particle .
The dimensionless temperature is given by the temperature divided by the ambient temperature , LaTeXMLMath .
Specifically , LaTeXMLEquation where the summation is over all LaTeXMLMath particles in the shell , LaTeXMLMath is the speed of the LaTeXMLMath particle and LaTeXMLMath is the normal speed of the gas in the shell .
Ionization is simply the average charge per particle .
In each case we plot properties as a function of a dimensionless bubble radius , LaTeXMLMath , which equals the physical radius LaTeXMLMath divided by a constant approximating the atomic diameter .
For Helium this constant is chosen to be 2.18Å ( see Table I ) .
For Argon and Xenon these constants are chosen to be 4.11Å and 5.65Å , respectively .
( These latter two choices represent average VSS model values at 273K and also approximate the values given in Table I ) .
See LaTeXMLCite for further details on calculating equilibrium and transport properties for hard sphere models .
In this section we simulate the collapse of a sonoluminescing bubble from its maximum radius to its hot spot .
Our focus is on how boundary conditions affect the interior dynamics of the collapse .
Results for Helium , Argon and Xenon are presented .
The section begins with a study of the collapse of million particle bubbles and concludes by addressing how simulations vary according to ambient bubble size .
We first consider evolving a bubble of one million helium atoms using specular boundary conditions .
With the constant diameter model and no ionization the temperature and density increase uniformly as the bubble collapses to the minimum radius .
After the minimum radius is attained , the temperature becomes hotter towards the center of the bubble and cools at the expanding outer boundary of the bubble , with a peak temperature of about 80000K reached at the center .
( At these temperatures , it is clear that ionization events will occur so the remainder of our simulations consider ionization . )
See Figure LaTeXMLRef for plots of the density , temperature and velocity as a function of distance from the center of the bubble at various bubble radii .
With the constant diameter model and ionization we again find that the temperature and density increase uniformly as the bubble collapses to the minimum radius .
However , after the minimum radius is attained , ionization causes the temperature to cool across the entire bubble rather than just at the boundary of the bubble ( although cooling occurs most rapidly at the bubble boundary ) .
A peak temperature of about 40000K is attained at the minimum radius .
It is particularly noteworthy that recorded properties are nearly constant throughout the bubble when the peak temperature occurs — See Figure LaTeXMLRef .
Changing to the VSS diameter model gives very similar results , except now ionization occurs less frequently because the effective size of the particles is smaller .
Because less ionization occurs , the temperature continues to increase for a short while after the minimum bubble radius leading to a peak temperature of about 45000K .
See Figure LaTeXMLRef for details .
Our next set of simulations evolve a bubble of one million helium atoms using heat bath boundary conditions and ionization .
With the constant diameter model the density increases dramatically at the edge of the bubble as the minimum radius is attained .
Temperature and velocity are also much more profiled than for specular boundary conditions , with peaks occurring about 25 percent of the way from the boundary of the bubble to the center .
No ionization has occurred at the minimum radius .
For a short time after the minimum radius LaTeXMLRef as the forcing equation for LaTeXMLMath .
, the peak temperature of the bubble continues to increase ( to a maximum of 95000K ) , and temperature and density profiles become even more pronounced — See Figure LaTeXMLRef .
At first sight , it is counterintuitive that heat bath boundaries create conditions whereby the cooling from the boundary leads to greater energy focusing and higher peak temperatures .
Perhaps cooling lowers the speed of sound and enhances the nonlinear response to the high speed LaTeXMLMath of collapse .
Changing to the VSS diameter model gives very similar results , except now ionization occurs less frequently because the effective size of the particles is smaller .
See Figure LaTeXMLRef for details .
Our next set of simulations evolve million particle Argon and Xenon bubbles using specular boundary conditions and ionization .
We start by considering an Argon bubble with the VSS diameter model .
Because the speed of sound is slower in Argon than in Helium we expect Argon simulations to exhibit much sharper profiles than Helium .
This is indeed the case .
Moreover , our simulation results are surprisingly similar to those for Helium with heat bath boundaries : Density increases at the edge of the bubble as the minimum radius is attained .
Temperature and velocity are sharply profiled , with peaks occurring closer to the boundary of the bubble than to its center .
Also , for a short time after the minimum radius , the peak temperature of the bubble continues to increase rapidly ( to a maximum of 100000K ) , and temperature and density profiles become even more pronounced — See Figure LaTeXMLRef .
Constant diameter hard sphere simulations of Argon are also possible .
These simulations are unique in that the hot spot occurs before the minimum radius value of 58.2 — See Figure LaTeXMLRef .
As expected , this simulation gives sharper profiles than the corresponding model for Helium .
However , since the minimum radius is close to the minimum radius allowed by the packing of the hard spheres the results are much more uniform than those derived using the VSS model for Argon .
Also note that as a result of the collapse of the bubble , energy stored at the maximum radius is converted into heating , ionization and kinetic energy of the local center of mass .
From Figure LaTeXMLRef one can estimate these quantities .
The average temperature of the atoms is 30000K which is a thermal energy of about 3.75 eV/atom .
As half the atoms are ionized , the ionization energy is about 8 eV/atom .
Since electrons have about the same thermal energy as the ions , their energy is about 2 eV/atom .
Taken together , these channels add to about 21 eV/atom which is less than the 25 eV/atom available in the initial state but the difference is within the accuracy of the energy estimates .
For Helium at the hot spot , the energy of the hard sphere ( plus ionization ) is substantially less than the energy stored at LaTeXMLMath .
This can be attributed to the fact that LaTeXMLMath .
For Argon , almost all the stored energy ends up in the hard sphere gas since LaTeXMLMath is much closer to LaTeXMLMath ; LaTeXMLMath .
In both cases , inclusion of a self-consistent boundary condition at the wall will account for any further energy discrepancies .
Of course , in a physical system , energy will diminish due to acoustic radiation and thermal losses through the boundary of the bubble and we expect this to constitute a strong effect .
Simulations for Xenon bubbles with the VSS diameter model were also carried out .
Because the speed of sound is slower in Xenon than in Argon we expect Xenon simulations to exhibit even sharper profiles than Argon .
Indeed , this is the case and temperatures of up to 300000K were obtained despite the occurrence of multiple ionization ( exceeding 4 per particle at the center ) — See Figure LaTeXMLRef .
A proviso for the Xenon data is that these calculations bog down before the minimum radius is attained when the constant diameter model is used , whereas the Helium data is hardly affected by this modification .
The explanation lies in the consistency of the minimum bubble radius LaTeXMLMath and the hard sphere radius for Xenon .
Specifically , the minimum radius of the bubble wall is less than the minimum radius allowed by the packing of hard spheres .
On the other hand , Xenon simulations carried out using the VSS model are relatively insensitive to changes in LaTeXMLMath .
For example , increasing LaTeXMLMath by LaTeXMLMath changes the peak temperature by about LaTeXMLMath , and leaves the qualitative features invariant .
Note that in this case , LaTeXMLMath as with Helium simulations .
We now consider the evolution of million particle Argon and Xenon bubbles using Heat Bath boundary conditions and ionization .
Applying the VSS diameter model to an Argon bubble gives results that have the same qualitative features as the corresponding Helium simulation , except that all properties are much more sharply profiled .
Indeed , temperatures of up to 300000K were obtained in this simulation showing ( once again ) that heat bath boundaries create conditions whereby the cooling from the boundary leads to greater energy focusing and higher peak temperatures than specular boundary conditions .
See Figure LaTeXMLRef for details .
Simulations for Xenon bubbles with the VSS diameter model were also carried out .
Because the speed of sound is slower in Xenon than in Argon , Xenon simulations exhibit even sharper profiles than Argon .
Indeed , temperatures of up to 500000K were obtained .
See Figure LaTeXMLRef for details .
As discussed in the previous section , the constant diameter model for Xenon is not able to compute down to the minimum radius since that radius is smaller than the minimum packing radius of the hard sphere gas .
On the other hand , Argon calculations are possible .
These simulations are qualitatively similar to the constant diameter model for Helium , but produce a more highly ionized gas ( reaching an average charge of +6 per particle near the center ) and much higher temperatures ( up to 1.5 million K ) than any other simulations that were considered .
Because such extreme values arise it seems likely the constant diameter model for Argon also experiences a significant consistency problem near the minimum radius .
An important experimental measurement for SL bubbles is the flash width , i.e .
the duration of the light emission , because this constrains the possible light emission mechanisms and thus provides a point of validation for any proposed model or theory .
Since our simulations do not include the fundamental atomic excitation or charge acceleration effects responsible for radiation , the current model does not directly yield a flashwidth .
However , an estimated flash width can be obtained from the computed temperature as a function of time .
If we assume that whatever process is responsible for the light emission is strongly dependent on the current temperature , and that it does not appreciably alter the gross gas dynamics , the flashwidth at a particular color is simply the length of time which the temperature exceeds the appropriate turn-on threshold .
In this case , the peak temperature as a function of time is our key diagnostic quantity .
Our simulations ( Figure LaTeXMLRef ) show that emission from an adiabatic compression lacks a strong , sharp temperate spike in time , and thus the associated flash from this model would be longer and would be comprised of lower energy photons .
In contrast , the heat bath boundary conditions yield sharp transient spike in temperature , and thus this model predicts a much shorter flash which is comprised of higher energy photons .
In both cases it appears that the width of the spike roughly doubles as the number of particles in the bubble increases by factors of 10 , from LaTeXMLMath to LaTeXMLMath .
Since each factor of LaTeXMLMath in particle number corresponds to a doubling of the ambient bubble radius LaTeXMLMath , this amounts to essentially a predicted linear scaling between flash width and ambient bubble radius .
In both plots , the curves for LaTeXMLMath and LaTeXMLMath particles were derived by averaging over 10 and 20 simulations respectively in order to keep statistical fluctuations to an acceptable level .
The LaTeXMLMath simulation required just a single simulation for robust statistics .
Sonoluminescence is well suited to investigation by Molecular Dynamics because the range of densities and time scales is large , yet the number of particles involved is relatively small .
Because the phenomena still poses experimentally difficult , unsolved questions regarding its mechanism and ultimate energy focusing energy potential , we feel it is an excellent subject for much more detailed MD investigations than the initial effort we have presented here .
In this paper , we introduced a simple model for the interior dynamics of single noble gas bubble sonoluminescence , as a hard sphere gas driven by a spherical piston controlled be the Rayleigh-Plesset equation .
Energy losses due to ionization were also accounted for .
We considered both constant and variable radius hard sphere models , and these lead to quantitatively similar results .
Fast , tree-based algorithms allowed us to evolve million particle systems through the entire collapse process .
Our calculations indicate that extreme energy focusing occurs within the bubble which in some cases is driven by a shock-like compression in the gas .
Peak temperatures range from 40,000 K for He to 500,000 K for Xe .
These are accompanied by high levels of ionization during the final collapse , and formation of a transient , high density plasma state seems quite likely .
The imposition of a thermal boundary condition at the wall of the bubble leads to greatly increased energy focusing and non-uniformity within a collapsing bubble .
In any case , the predicted flash width scales roughly linearly with the ambient bubble radius .
There are a variety of interesting directions for future research in this problem .
For example , our simulations simply treat the bubble wall as a piston moving in with a prescribed velocity .
A natural improvement would be to couple the internal molecular dynamics to the wall velocity to obtain a self-consistent bubble motion and internal dynamics .
This could be done by coupling to Euler or Navier Stokes models for the surrounding fluid .
This may be particularly important for accurately computing the dynamics through the point of minimum radius .
In our present model there may be over compression of the gas in some simulations as the minimum radius is approached , since the prescribed piston motion does not respond to the rapid increase in the internal gas pressure .
Conversely , when the piston retracts after this point , a nonphysical gap often develops between the bubble boundary and the outer extent of the gas , which may under-compress the gas .
Another important area for future research is adding in water vapor into the bubble interior .
This provides a potentially important cooling mechanism , which may strongly modulate the light emission and energy focusing , and may explain the strong ambient temperature dependence of the emitted light intensity .
We have done preliminary investigations of this , and these show that water evaporation from the bubble surface into the interior must be included to avoid rapid expulsion of the water vapor .
It is also possible that the water could be directly involved in the light emission , when it is properly included in the model .
Other bubble collapse geometries could also be considered , and these may have different energy focusing characteristics .
For example , one could consider a nonspherical collapse , hemi-spherical bubbles collapsing on a solid surface , or consider collapse geometries appropriate for bubble jetting scenarios .
Similarly , one could see if special collapse profiles can be used to reach much higher internal temperatures , and otherwise explore the extremes of the energy focusing potential .
Perhaps a mode could even be found in which small amounts of deuterium-deuterium fusion could be induced , assuming there is deuterium gas in the bubble as well .
Including additional atomic physics such as atomic excitation , rotational and vibrational degrees of freedom ( needed for non-noble gases or water vapor ) and electron-ion recombination would all allow for more accurate energy accounting , and may also be directly related to light emission mechanisms .
Another major direction would be to include electric field effects into the the simulation .
Algorithms for such models must treat long range electrostatic interactions to avoid incurring serious errors .
They must also be able to evaluate long range forces efficiently since calculating interactions pairwise becomes expensive for more than a few thousand particles .
For these reasons , multipole methods are particularly attractive— they use a hierarchy of spatial subdivisions and a multipole expansion to evaluate interactions with little more than linear effort in the number of particles .
See LaTeXMLCite for details and see also LaTeXMLCite for further references on methods for evaluating long range forces .
This would potentially allow direct simulation of the ions and electrons produced as well , which , along with including atomic excitation , would allow direct simulation of the light emitting processes .
With these effects included , an extremely detailed picture of the SL phenomena could be laid out .
Of course , larger scale , parallel simulations are essential to actually achieve direct comparisons with present SL experiments .
Because the simple hard sphere interactions are quite local , the system should be amenable to parallelization .
We expect that the cost ( in collision count ) for a hard sphere MD simulation scales roughly like LaTeXMLMath , where LaTeXMLMath is the number of particles , since the collapse time from the ambient to the minimum radius scales linearly with LaTeXMLMath ( and so we conjecture that the collision rate also increases roughly linearly with the LaTeXMLMath ) .
Thus simulations using one hundred times as many particles ( i.e .
LaTeXMLMath ) would require LaTeXMLMath times as much computer time assuming near optimality in the algorithm .
This is somewhat beyond the range of a single supercomputer CPU , but would become quite practical on a 100 node system of workstation-grade CPUs .
Finally , it would be of great interest to investigate where less costly continuum models and Monte Carlo simulations are appropriate for studying sonoluminescence and to develop techniques for coupling these methods to detailed molecular dynamics simulations near the light emitting hot spot , in order to produce more complete models with greater predictive validity .
There is also a great deal to explore experimentally .
One example relevant to our study is that it would be useful to measure flashwidth as a function of ambient bubble radius ( or , in practice , intensity and frequency of the driving sound field ) , for comparison with the scaling predictions of MD and other models .
We thank D.B .
Hash , A.L .
Garcia and P.H .
Roberts for valuable discussions .
Let LaTeXMLMath be the LaTeXMLMath -deformed Poisson measure in the sense of Saitoh-Yoshida LaTeXMLCite and LaTeXMLMath be the measure given by Equation ( LaTeXMLRef ) .
In this short paper , we introduce the LaTeXMLMath -deformed analogue of the Segal-Bargmann transform associated with LaTeXMLMath .
We prove that our Segal-Bargmann transform is a unitary map of LaTeXMLMath onto the LaTeXMLMath -deformed Hardy space LaTeXMLMath .
Moreover , we give the Segal-Bargmann representation of the multiplication operator by LaTeXMLMath in LaTeXMLMath , which is a linear combination of the LaTeXMLMath -creation , LaTeXMLMath -annihilation , LaTeXMLMath -number , and scalar operators .
The classical Segal-Bargmann transform in Gaussian analysis yields a unitary map of LaTeXMLMath space of the Gaussian measure on LaTeXMLMath onto the space of LaTeXMLMath holomorphic functions of the Gaussian measure on LaTeXMLMath , see papers LaTeXMLCite .
Recently , Accardi-Bożejko LaTeXMLCite showed the existence of a unitary operator between a one-mode interacting Fock space and LaTeXMLMath space of a probability measure on LaTeXMLMath by making use of the basic properties of classical orthogonal polynomials and associated recurrence formulas LaTeXMLCite .
Inspired by this work , the author LaTeXMLCite has recently extended the Segal-Bargmann transform to non-Gaussian cases .
The crucial point is to introduce a coherent state vector as a kernel function in such a way that a transformed function , which is a holomorphic function on a certain domain in general , becomes a power series expression .
Along this line , Asai-Kubo-Kuo LaTeXMLCite have considered the case of the Poisson measure compared with the case of the Gaussian measure .
However , the case of LaTeXMLMath space of Wigner ’ s semi-circle distributions in free probability theory LaTeXMLCite is beyond their scope .
On the other hand , Leeuwen-Maassen LaTeXMLCite considered a transform associated with LaTeXMLMath deformation of the Gaussian measure LaTeXMLCite and showed that for a given real number LaTeXMLMath it is a unitary map of LaTeXMLMath space of LaTeXMLMath -defomed Gaussian measure onto the LaTeXMLMath -deformed Hardy space LaTeXMLMath where LaTeXMLMath is given in ( LaTeXMLRef ) .
Biane LaTeXMLCite examined the case of LaTeXMLMath ( Free case ) .
Roughly speaking , their methods do not give the relationship between Szegö-Jacobi parameters and kernel functions for their transforms .
As observed in Section LaTeXMLRef and Appendix LaTeXMLRef , our approach clarifies the relationship between them .
In this paper , we shall consider the LaTeXMLMath -deformed version of the Segal-Bargmann transform LaTeXMLMath associated with LaTeXMLMath -deformed Poisson measure , denoted by LaTeXMLMath , in the sense of Saitoh-Yoshida LaTeXMLCite .
As a main result , we shall provide Proposition LaTeXMLRef , which claims that LaTeXMLMath is a unitary map of LaTeXMLMath onto LaTeXMLMath .
Moreover , in Theorem LaTeXMLRef we shall give the representation in LaTeXMLMath of the multiplication operator by LaTeXMLMath in LaTeXMLMath , which is a linear combination of the LaTeXMLMath -creation , LaTeXMLMath -annihilation , LaTeXMLMath -number , and scalar operators .
We remark that our representation is compatible with that on the LaTeXMLMath -Fock space by Saitoh-Yoshida LaTeXMLCite and can be viewed as the LaTeXMLMath -analogue of the Hudson-Parthasarathy LaTeXMLCite decomposition of the usual Poisson random variable on the standard Boson Fock space ( LaTeXMLMath ) .
Ito-Kubo LaTeXMLCite also studied a similar decomposition in details from the point of white noise calculus LaTeXMLCite ( For more recent formulation , see papers LaTeXMLCite ) .
The present article serves a good example to papers by Accardi-Bożejko LaTeXMLCite and Asai LaTeXMLCite .
The present paper is organized as follows .
In Section LaTeXMLRef , we recall the recurrence formula for LaTeXMLMath -Charlier polynomials .
In Section LaTeXMLRef , we introduce a LaTeXMLMath -deformed coherent state vector and a Segal-Bargmann transform associated to a LaTeXMLMath -deformed Poisson measure .
In addition , we quickly define the Hardy space as the Segal-Bargmann representation space .
In Section LaTeXMLRef , our main results are given .
In Appendix LaTeXMLRef , we give some remarks on known results LaTeXMLCite related to LaTeXMLMath -Hermite polynomials .
Notation .
Let us recall standard notation from LaTeXMLMath -analysis LaTeXMLCite .
We put for LaTeXMLMath , LaTeXMLEquation .
Then LaTeXMLMath -factorial is naturally defined as LaTeXMLEquation .
The LaTeXMLMath -exponential is given by LaTeXMLEquation whose radious of convergence is LaTeXMLMath .
In addition , another symbol used is the q-analogue of the Pochhammer symbol , LaTeXMLEquation with the convention LaTeXMLMath .
From now on , we always assume that LaTeXMLMath is fixed .
Recently , Saitoh-Yoshida LaTeXMLCite calculated the explicit form of the LaTeXMLMath -deformed Poisson measure with a parameter LaTeXMLMath for LaTeXMLMath .
We denote it by LaTeXMLMath .
The orthogonal polynomials associated to LaTeXMLMath are the LaTeXMLMath -Charlier polynomials LaTeXMLMath with the Szegö-Jacobi parameters LaTeXMLMath .
See papers LaTeXMLCite .
We also refer the book LaTeXMLCite for the standard Charlier polynomials case ( LaTeXMLMath ) .
Let LaTeXMLMath .
The following relations hold for each LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
Then , for any LaTeXMLMath -convergent decompositions LaTeXMLMath and LaTeXMLMath , the inner product LaTeXMLMath is given by the form LaTeXMLEquation .
Moreover , let LaTeXMLMath .
Let us define the q-deformed coherent state vector for LaTeXMLMath by LaTeXMLEquation .
It is easy to see that LaTeXMLMath due to LaTeXMLEquation .
Moreover , it can be shown that LaTeXMLMath is linearly independent and total in LaTeXMLMath .
Now we are in a position to introduce our key tool in this paper .
Let us consider the LaTeXMLMath -analogue of the Segal-Bargmann transform LaTeXMLMath associated to LaTeXMLMath given by LaTeXMLEquation .
Let LaTeXMLMath .
Then LaTeXMLMath converges absolutely for all LaTeXMLMath .
For LaTeXMLMath , it is quite easy to see LaTeXMLEquation .
By the Schwartz inequality , we get the inequality LaTeXMLEquation .
This shows that the LaTeXMLMath converges absolutely for all LaTeXMLMath .
∎ The completion of the space of holomorphic functions LaTeXMLMath on LaTeXMLMath with respect to the inner product , LaTeXMLEquation is nothing but the LaTeXMLMath -deformed Hardy space LaTeXMLMath .
Here LaTeXMLMath means that LaTeXMLEquation where LaTeXMLMath is the Lebesgue measure on the circle of radious LaTeXMLMath .
LaTeXMLMath forms an orthogonal basis of LaTeXMLMath .
We adopt the same idea as in the proof LaTeXMLCite .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that we have used the LaTeXMLMath -Gamma function LaTeXMLCite , LaTeXMLEquation ∎ Remark .
It can be shown by Proposition 4.4 in the recent paper LaTeXMLCite that LaTeXMLMath is a unique measure satisfying LaTeXMLMath .
Hence , for any LaTeXMLMath , LaTeXMLMath , the inner product LaTeXMLMath is written as LaTeXMLEquation and the corresponding norm of LaTeXMLMath is LaTeXMLEquation .
LaTeXMLMath is a unitary map of LaTeXMLMath onto LaTeXMLMath .
As we have seen in Lemma LaTeXMLRef , LaTeXMLEquation .
In addition , we derive by Lemma LaTeXMLRef , LaTeXMLEquation .
Therefore , we finish the proof .
∎ Let us define operators LaTeXMLMath and LaTeXMLMath in LaTeXMLMath satisfying LaTeXMLEquation and LaTeXMLEquation .
Operators LaTeXMLMath and LaTeXMLMath play the roles of the q-creation operator and q-annihilation operator respectively and satisfy the LaTeXMLMath -deformed commutation relation LaTeXMLMath .
The q-number operator acting on LaTeXMLMath is defined by LaTeXMLEquation .
In addition , the operator LaTeXMLMath acting on LaTeXMLMath is defined by LaTeXMLEquation .
Remark that LaTeXMLMath for LaTeXMLMath -Gaussian case , see Appendix LaTeXMLRef .
By the direct calculation , we have LaTeXMLMath LaTeXMLMath LaTeXMLMath The transformation of the multiplication operator LaTeXMLMath by LaTeXMLMath in LaTeXMLMath satisfies the following relation .
LaTeXMLMath By the recurrence formula ( LaTeXMLRef ) , Equation ( LaTeXMLRef ) and Lemma LaTeXMLRef , we derive LaTeXMLEquation .
LaTeXMLEquation ∎ Moreover we obtain The operators LaTeXMLMath have the following properties : ( 1 ) LaTeXMLMath ( 2 ) LaTeXMLMath .
The proof is done by Equations ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , and Lemma LaTeXMLRef .
∎ Therefore , by Theorem LaTeXMLRef and Corollary LaTeXMLRef , the multiplication operator LaTeXMLMath by LaTeXMLMath in LaTeXMLMath is represented as a linear combination of the LaTeXMLMath -creation , LaTeXMLMath -annihilation , LaTeXMLMath -number , and scalar operators in the Hardy space LaTeXMLMath .
In the LaTeXMLMath -deformed Gaussian case , due to Equation ( LaTeXMLRef ) in Appendix LaTeXMLRef , the multiplication operator LaTeXMLMath in LaTeXMLMath is represented as a linear combination of LaTeXMLMath -creation and LaTeXMLMath -annihilation operators in the “ same ” Hardy space LaTeXMLMath .
Remark .
( 1 ) The analogous results in this paper to LaTeXMLMath have been considered by Asai , et al .
LaTeXMLCite .
( 2 ) In general , if variances of two given measures LaTeXMLMath and LaTeXMLMath are the same , then the actions of creation and annihilation operators are the same .
In addition , the transformed function spaces by LaTeXMLMath and LaTeXMLMath are also the same .
However , if LaTeXMLMath is non-symmetric and LaTeXMLMath is symmetric , then the representation of multiplication in LaTeXMLMath is different from that in LaTeXMLMath .
First of all , we refer to the papers LaTeXMLCite and references cited therein for the detailed description of LaTeXMLMath -Hermite polynomials .
For LaTeXMLMath , see books LaTeXMLCite .
Let LaTeXMLMath be the LaTeXMLMath -deformed Gaussian measure with mean zero and variance LaTeXMLMath .
It is well-known that an associated orthogonal polynomial to LaTeXMLMath is the LaTeXMLMath -Hermite polynomial LaTeXMLMath with the Szegö-Jacobi parameters LaTeXMLMath .
In this case the following relations hold for each LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
Then for any LaTeXMLMath and LaTeXMLMath in LaTeXMLMath , the inner product LaTeXMLMath is given by the form LaTeXMLEquation .
Moreover , let LaTeXMLMath .
A q-deformed coherent state vector LaTeXMLMath for LaTeXMLMath is defined by LaTeXMLEquation .
It can be shown that the set LaTeXMLMath is linearly independent and total in LaTeXMLMath .
The LaTeXMLMath -analogue of the Segal-Bargmann transform LaTeXMLMath associated to LaTeXMLMath is given by LaTeXMLEquation .
With this transform , we can reproduce the same results as for LaTeXMLMath Theorem III.4 in Leeuwen-Maassen LaTeXMLCite and for LaTeXMLMath Proposition 1 in Biane LaTeXMLCite .
That is , LaTeXMLMath -transform yields a unitary isomorphism between LaTeXMLMath and LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath is the multiplication operator by LaTeXMLMath in LaTeXMLMath .
The author expresses sincere thanks to Professors T. Hida and K. Saitô for their kind arrangement of the conference .
He is grateful for research support by International Institute for Advanced Studies , Kyoto , Japan .
He also thanks Professor Bożejko for his comments .
We obtain a number of results regarding the distribution of values of a quadratic function LaTeXMLMath on the set of LaTeXMLMath permutation matrices ( identified with the symmetric group LaTeXMLMath ) around its optimum ( minimum or maximum ) .
In particular , we estimate the fraction of permutations LaTeXMLMath such that LaTeXMLMath lies within a given neighborhood of the optimal value of LaTeXMLMath .
We identify some “ extreme ” functions LaTeXMLMath ( there are 4 of those for LaTeXMLMath even and 5 for LaTeXMLMath odd ) such that the distribution of every quadratic function around its optimum is a certain “ mixture ” of the distributions of the extremes and describe a natural class of functions ( which includes , for example , the objective function in the Traveling Salesman Problem ) with a relative abundance of near-optimal permutations .
In particular , we identify a large class of functions LaTeXMLMath with the property that permutations in the vicinity of the optimal permutation ( in the Hamming metric of LaTeXMLMath ) tend to produce near optimal values of LaTeXMLMath ( such is , for example , the objective function in the symmetric Traveling Salesman Problem ) and show that for general LaTeXMLMath , just the opposite behavior may take place : an average permutation in the vicinity of the optimal permutation may be much worse than an average permutation in the whole group LaTeXMLMath .
The Quadratic Assignment Problem ( QAP for short ) is an optimization problem on the symmetric group LaTeXMLMath of LaTeXMLMath permutations of an LaTeXMLMath -element set .
The QAP is one of the hardest problems of combinatorial optimization , whose special cases include the Traveling Salesman Problem ( TSP ) among other interesting problems .
Recently the QAP has been of interest to many people .
An excellent survey of recent results is found in [ 5 ] .
Despite this work , it is still extremely difficult to solve QAP ’ s of size LaTeXMLMath to optimality , and the solution to a QAP of size LaTeXMLMath is considered noteworthy , see , for example , [ 1 ] and [ 4 ] .
Moreover , it appears that essentially no positive approximability results for the general QAP are known , although some “ bad news ” ( non-approximability ) and approximability for special classes have been established , see [ 3 ] and [ 2 ] .
The goal of this paper is to study the distribution of values of the objective function of the QAP .
We hope that our results would allow one on one hand to understand the behavior of the local search heuristic , and , on the other hand , to get guaranteed approximations to the optimum using some simple algorithms based on random or partial enumeration with guaranteed complexity bounds .
In particular , we estimate how well the sample optimum from a random sample of a given size approximates the global optimum .
Let LaTeXMLMath be the vector space of all real LaTeXMLMath matrices LaTeXMLMath , LaTeXMLMath and let LaTeXMLMath be the set of all permutations LaTeXMLMath of the set LaTeXMLMath .
There is an action of LaTeXMLMath on the space LaTeXMLMath by simultaneous permutations of rows and columns : we let LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath , provided LaTeXMLMath for all LaTeXMLMath .
One can check that LaTeXMLMath for any two permutations LaTeXMLMath and LaTeXMLMath .
There is a standard scalar product on LaTeXMLMath : LaTeXMLEquation .
Let us fix two matrices LaTeXMLMath and LaTeXMLMath and let us consider a real-valued function LaTeXMLMath defined by LaTeXMLEquation .
The problem of finding a permutation LaTeXMLMath where the maximum or minimum value of LaTeXMLMath is attained is known as the Quadratic Assignment Problem .
It is one of the hardest problems of Combinatorial Optimization .
From now on we assume that LaTeXMLMath .
In this paper , we study the distribution of values of LaTeXMLMath from the optimization perspective : LaTeXMLMath How “ steep ” or how “ flat ” can the optimum of LaTeXMLMath be ?
LaTeXMLMath How many values of LaTeXMLMath lie within a given distance to the optimum ?
LaTeXMLMath When can we hope to improve the value of LaTeXMLMath by modifying LaTeXMLMath slightly ?
To formulate the questions rigorously ( and to answer them ) , we introduce the standard Hamming metric on the symmetric group LaTeXMLMath .
For two permutations LaTeXMLMath , let the distance LaTeXMLMath be the number of indices LaTeXMLMath where LaTeXMLMath and LaTeXMLMath disagree : LaTeXMLEquation .
One can observe that the distance is invariant under the left and right actions of LaTeXMLMath : LaTeXMLEquation for all LaTeXMLMath .
For a permutation LaTeXMLMath and an integer LaTeXMLMath , we consider the “ LaTeXMLMath -th ring ” around LaTeXMLMath : LaTeXMLEquation .
In particular , we are interested in the distribution of values of LaTeXMLMath in the set LaTeXMLMath , where LaTeXMLMath is an optimal permutation .
Our approach produces essentially identical results for a more general problem , where we are given a 4-dimensional array LaTeXMLMath of LaTeXMLMath real numbers and the function LaTeXMLMath is defined by LaTeXMLEquation .
If LaTeXMLMath for some matrices LaTeXMLMath and LaTeXMLMath , in which case we write LaTeXMLMath , we get the special case ( 1.1.1 ) we started with .
The main idea of our approach is as follows .
Let LaTeXMLEquation be the average value of LaTeXMLMath on the symmetric group and let LaTeXMLMath .
Hence the average value of LaTeXMLMath is 0 and we study the distribution of values of LaTeXMLMath around its maximum ( the problem with minimum instead of maximum is completely similar ) .
Now , as long as the distribution of values of LaTeXMLMath is concerned , without loss of generality we may assume that LaTeXMLMath attains its maximum on the identity permutation LaTeXMLMath , so that LaTeXMLMath for all LaTeXMLMath .
Let us define a function LaTeXMLMath , which we call the central projection ( with the term coming from the representation theory ) of LaTeXMLMath by LaTeXMLEquation .
It turns out that LaTeXMLMath attains its maximum on the identity permutation , that the average value of LaTeXMLMath on LaTeXMLMath is 0 and , moreover , the average values of LaTeXMLMath and LaTeXMLMath on the LaTeXMLMath -th ring LaTeXMLMath coincide for all LaTeXMLMath .
In short , LaTeXMLMath captures some important information about the distribution of LaTeXMLMath .
The set of all functions LaTeXMLMath obtained by central projection ( 1.4 ) from all functions LaTeXMLMath having maximum at the identity forms a 3-dimensional convex polyhedral cone .
We describe this cone , identifying its extreme rays ( there are LaTeXMLMath for even LaTeXMLMath and LaTeXMLMath for odd LaTeXMLMath ) , which provide us with some “ extreme ” types of distribution .
Hence we study the distribution of values of LaTeXMLMath , which is a much easier problem .
Once the distribution of values of LaTeXMLMath is understood , using ( 1.4 ) , we infer various facts about the distribution of values of LaTeXMLMath .
We remark that it is easy to compute the average value LaTeXMLMath of LaTeXMLMath given by ( 1.1.1 ) or by ( 1.3.1 ) .
Let LaTeXMLMath be a function defined by LaTeXMLEquation for some matrices LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLEquation be the average value of LaTeXMLMath on the symmetric group LaTeXMLMath .
Let us define LaTeXMLEquation .
Then LaTeXMLEquation .
Similarly , if LaTeXMLMath is a function ( 1.3.1 ) of the generalized problem , then LaTeXMLEquation .
We prove Lemma 1.5 in Section 6 .
We often denote by LaTeXMLMath some positive constant whose precise value is not of particular importance to us .
If LaTeXMLMath and LaTeXMLMath are non-negative functions of a positive integer LaTeXMLMath , we write LaTeXMLMath if LaTeXMLMath for some LaTeXMLMath and all sufficiently large LaTeXMLMath .
Similarly , we write LaTeXMLMath if LaTeXMLMath for some constant LaTeXMLMath and all sufficiently large LaTeXMLMath .
We denote by LaTeXMLMath the identity permutation in LaTeXMLMath .
We denote by LaTeXMLMath the cardinality of a finite set LaTeXMLMath and by LaTeXMLMath the convex hull of the set LaTeXMLMath in Euclidean space .
Given a function LaTeXMLMath , we denote by LaTeXMLMath its average value on LaTeXMLMath : LaTeXMLEquation and by LaTeXMLEquation the “ shifted ” function with 0 average .
Our results concern the function LaTeXMLMath .
The paper is organized as follows .
In Sections 2-5 , we state our results about the number of near-optimal permutations .
In Sections 6-11 , we prove those results and describe certain “ extreme ” distributions .
We give an informal preview of our results below .
In what follows , LaTeXMLMath is an optimal permutation such that LaTeXMLMath for all LaTeXMLMath .
Since we consider the shifted function , the minimization and maximization problems are completely similar .
In Section 2 , we consider a special case of the problem where matrix LaTeXMLMath is symmetric , has constant row and column sums and a constant diagonal ( of course , LaTeXMLMath and LaTeXMLMath are interchangeable ) .
For example , the symmetric TSP belongs to this class .
The interesting feature of this special case is what we call the “ bullseye ” distribution of values of LaTeXMLMath around its maximum .
It turns out that the average value of LaTeXMLMath over the LaTeXMLMath -th ring LaTeXMLMath ( see Definitions 1.2 ) around an optimal permutation LaTeXMLMath steadily improves as the ring contracts to LaTeXMLMath .
The proof is given in Section 8 .
This is also the simplest case to analyze .
It turns out that the set of all possible central projections LaTeXMLMath ( see ( 1.4 ) ) is one-dimensional .
In Section 3 , we consider a more general case of a not necessarily symmetric matrix LaTeXMLMath with constant row and column sums and a constant diagonal .
For example , the asymmetric TSP belongs to this class .
We call this case “ pure ” since the objective function LaTeXMLMath lacks the component that can be attributed to the Linear Assignment Problem .
Although we don ’ t have the bullseye distribution of Section 2 , we can provide some guarantees for the number of reasonably good permutations LaTeXMLMath .
Thus , for any LaTeXMLMath the probability that a random permutation LaTeXMLMath satisfies LaTeXMLMath is at least LaTeXMLMath .
Furthermore , for any LaTeXMLMath the probability that a random permutation LaTeXMLMath satisfies LaTeXMLMath is “ mildly exponential ” , that is at least of the order of LaTeXMLMath for some constant LaTeXMLMath .
The proof is given in Section 9 .
It turns out that the set of all central projections LaTeXMLMath , maximized at the identity , forms a 2-dimensional cone .
The extreme rays provide us with the extreme types of distributions , which , although not as good as the “ bullseye ” distribution of Section 2 , still quite reasonable , especially compared with types of distributions we encounter in general symmetric QAP .
In Section 4 , we consider the symmetric Quadratic Assignment Problem , where matrix LaTeXMLMath ( or , equivalently LaTeXMLMath ) is symmetric .
This case turns out to be very different in many respects from the special cases of Sections 2 and 3 .
It turns out that the “ bullseye ” distribution is no longer the law .
We present a simple example of function LaTeXMLMath where the average value of LaTeXMLMath over the LaTeXMLMath -th ring LaTeXMLMath of an optimal permutation LaTeXMLMath is much worse than the average over the whole group LaTeXMLMath even for small LaTeXMLMath .
We call such a distribution a “ spike ” .
We argue that at least for the generalized problem ( 1.3 ) , the number of near-optimal permutations is much smaller than in the pure case of Section 3 .
The proofs are given in Section 10 .
It turns out that the set of all central projections ( 1.4 ) forms a 2-dimensional cone whose extreme rays provide us with the extreme types of distributions .
One of those rays turns out to have an extreme “ spike ” distribution .
In Section 5 , we consider the general Quadratic Assignment Problem .
As in Section 3 , we prove that for any LaTeXMLMath the probability that a random permutation LaTeXMLMath satisfies LaTeXMLMath is at least LaTeXMLMath , although with a worse constant than in Section 3 .
We prove that for any LaTeXMLMath there is a constant LaTeXMLMath such that the probability that a random permutation LaTeXMLMath satisfies LaTeXMLMath is at least of the order LaTeXMLMath ( mildly exponential ) .
The proofs are given in Section 11 .
It turns out that the set of central projections ( 1.4 ) forms a 3-dimensional polyhedral cone with 4 extreme rays when LaTeXMLMath is even and 5 extreme rays when LaTeXMLMath is odd .
In a sense , those extreme rays describe all “ extreme ” distributions that one may encounter in the general Quadratic Assignment Problem .
In Section 6 , we prove some preliminary technical results .
In Section 7 , we review the necessary facts from the representation theory of the symmetric group , which we use essentially in our approach .
Our analysis of the Quadratic Assignment Problem is the simplest in the following special case ( it also exhibits some features absent in the general case ) .
Suppose that the matrix LaTeXMLMath is symmetric and has constant row and column sums and a constant diagonal : LaTeXMLEquation .
For example , LaTeXMLEquation satisfies these properties and the corresponding optimization problem is the Symmetric Traveling Salesman Problem .
It turns out that the optimum has a characteristic “ bullseye ” feature in the Hamming metric on LaTeXMLMath ( see Definition 1.2 ) .
Suppose that the matrix LaTeXMLMath is symmetric and has constant row and column sums and a constant diagonal .
Let LaTeXMLMath be the function defined by ( 1.1.1 ) for LaTeXMLMath and some matrix LaTeXMLMath .
Let LaTeXMLMath be the average value of LaTeXMLMath on LaTeXMLMath , let LaTeXMLMath and let LaTeXMLMath be an optimal permutation : LaTeXMLMath .
For LaTeXMLMath let LaTeXMLEquation be the LaTeXMLMath -th “ ring ” around LaTeXMLMath and let LaTeXMLEquation .
Then LaTeXMLEquation .
We prove Theorem 2.1 in Section 8 .
It follows from our proof that we have almost equality in the formula of Theorem 2.1 .
We observe that as the ring LaTeXMLMath contracts to the optimal permutation LaTeXMLMath , the average value of LaTeXMLMath on the ring steadily improves .
LaTeXMLEquation .
It is easy to construct examples where some values of LaTeXMLMath in a very small neighborhood of the optimum are particularly bad , but as follows from Theorem 2.1 , such values are relatively rare .
In our opinion , this provides some justification for the local search heuristic , where one starts from a permutation and tries to improve the value of the objective function by searching a small neighborhood of the current solution .
Indeed , if we had the value of LaTeXMLMath for each LaTeXMLMath equal to LaTeXMLMath , then the local search would have converged to the optimum in LaTeXMLMath steps , since each step would have brought us to a smaller neighborhood of the optimal solution .
Instead , we have that the average value over LaTeXMLMath is ( almost ) equal to LaTeXMLMath .
We can no longer guarantee that the local search converges fast ( or even converges ) to the optimal solution ( after all , our problem includes the Traveling Salesman Problem as a special case and hence is NP-hard ) , but it plausible that the local search behaves reasonably well for an “ average ” optimization problem .
This agrees with the empirical evidence that the local search works well for the Traveling Salesman Problem .
Incidentally , one can prove that the same type of the “ bullseye ” behavior is observed for the Linear Assignment Problem and some other polynomially solvable problems , such as the weighted Matching Problem .
Estimating the size of the ring LaTeXMLMath , we get the following result .
Suppose that the matrix LaTeXMLMath is symmetric and has constant row and column sums and a constant diagonal .
Let LaTeXMLMath be the function defined by ( 1.1.1 ) for LaTeXMLMath and some matrix LaTeXMLMath , let LaTeXMLMath be the average value of LaTeXMLMath on LaTeXMLMath and let LaTeXMLMath .
Let LaTeXMLMath be an optimal permutation : LaTeXMLMath .
Let us choose an integer LaTeXMLMath and a number LaTeXMLMath and let LaTeXMLEquation .
The probability that a random permutation LaTeXMLMath satisfies the inequality LaTeXMLEquation is at least LaTeXMLEquation .
We prove Theorem 2.3 in Section 8 .
Our results can be generalized in a quite straightforward way to functions LaTeXMLMath defined by ( 1.3.1 ) , if we assume that for any LaTeXMLMath and LaTeXMLMath the matrix LaTeXMLMath , where LaTeXMLMath , is symmetric with constant row and column sums and has a constant diagonal .
In this Section , we consider a more general case of a not necessarily symmetric matrix LaTeXMLMath having constant row and column sums and a constant diagonal : LaTeXMLEquation .
For example , matrix LaTeXMLEquation satisfies these properties and the corresponding optimization problem is the Asymmetric Traveling Salesman Problem .
We call this case pure , because as we remark in Sections 7 and 9 , the objective function LaTeXMLMath lacks the component attributed to the Linear Assignment Problem .
More generally , an arbitrary objective function LaTeXMLMath in the Quadratic Assignment Problem can be represented as a sum LaTeXMLMath , where LaTeXMLMath is the objective function in a Linear Assignment Problem and LaTeXMLMath is the objective function in some pure case .
In this case we can no longer claim the bullseye distribution of Section 2 ( the reasons are explained in Section 9 ) , the distribution in this case is not as bad as , for example , in the general symmetric QAP ( see Section 4 ) and the estimates of the number of relatively good values we are able to prove are almost as good as those of Section 2 .
Suppose that the matrix LaTeXMLMath has constant row and column sums and a constant diagonal .
Let LaTeXMLMath be the function defined by ( 1.1.1 ) for LaTeXMLMath and some matrix LaTeXMLMath , let LaTeXMLMath be the average value of LaTeXMLMath on LaTeXMLMath and let LaTeXMLMath .
Let LaTeXMLMath be an optimal permutation , so LaTeXMLMath .
Let us choose an integer LaTeXMLMath and a number LaTeXMLMath and let LaTeXMLEquation .
The probability that a random permutation LaTeXMLMath satisfies the inequality LaTeXMLEquation is at least LaTeXMLEquation .
In particular , by choosing an appropriate LaTeXMLMath , we obtain the following corollary .
∎ We prove Theorem 3.1 in Section 9 .
From Corollary 3.2 , it follows that to get a permutation LaTeXMLMath which satisfies ( 1 ) for any fixed LaTeXMLMath , we can use the following straightforward randomized algorithm : sample LaTeXMLMath random permutations LaTeXMLMath , compute the value of LaTeXMLMath and choose the best permutation .
With the probability which tends to 1 as LaTeXMLMath , we will hit the right permutation .
The complexity of the algorithm is quadratic in LaTeXMLMath for any LaTeXMLMath , but the coefficient of LaTeXMLMath grows as LaTeXMLMath grows .
If we are willing to settle for an algorithm of a mildly exponential complexity of the type LaTeXMLMath for some LaTeXMLMath we can achieve a better approximation ( 2 ) by searching through the set of randomly selected LaTeXMLMath permutations .
We remark that no algorithm solving the Quadratic Assignment Problem ( even in the special case considered in this section ) with an exponential in LaTeXMLMath complexity LaTeXMLMath is known , although there is a dynamic programming algorithm solving the Traveling Salesman Problem in LaTeXMLMath time .
Again , our results can be generalized in a quite straightforward way to functions LaTeXMLMath defined by ( 1.3.1 ) , if we assume that for any LaTeXMLMath and LaTeXMLMath the matrix LaTeXMLMath , where LaTeXMLMath has constant row and column sums and has a constant diagonal .
In this section , we assume that the matrix LaTeXMLMath is symmetric , that is LaTeXMLEquation .
Overall , the distribution of values of LaTeXMLMath turns out to be much more complicated when in the special cases described in Sections 2 and 3 .
First , we observe that generally one can not hope for the “ bullseye ” feature described in Section 2.2 .
Let us choose an LaTeXMLMath matrix LaTeXMLMath , where LaTeXMLEquation so LaTeXMLEquation .
Let LaTeXMLEquation and let LaTeXMLMath , where LaTeXMLEquation so LaTeXMLEquation .
Let LaTeXMLMath be the function defined by ( 1.1.1 ) .
In Section 10 , we prove the following properties of LaTeXMLMath .
LaTeXMLMath We have LaTeXMLMath for the average value of LaTeXMLMath on LaTeXMLMath ; LaTeXMLMath The maximum value of LaTeXMLMath on LaTeXMLMath is 1 and is attained , in particular , on the identity permutation LaTeXMLMath ; LaTeXMLMath For the LaTeXMLMath -th ring LaTeXMLMath centered at the identity permutation LaTeXMLMath , we have LaTeXMLEquation .
We observe that already for LaTeXMLMath ( a more careful analysis yeilds LaTeXMLMath ) the average value of LaTeXMLMath over LaTeXMLMath is negative for all sufficiently large LaTeXMLMath .
Thus an average permutation in LaTeXMLMath presents us with a choice worse than an average permutation in LaTeXMLMath .
The distribution of values of LaTeXMLMath turns out to be of the opposite nature to the bullseye distribution of Figure 1 .
We call it the “ spike ” distribution .
LaTeXMLEquation .
Of course , in this particular case the optimization problem is very easy since the function LaTeXMLMath attains only two different values .
However , this may serve as an indication that complicated distributions are indeed possible and the local search may not work well for a general symmetric QAP .
Indeed , this is the case if we allow generalized functions ( 1.3.1 ) .
In Section 10 , we show that there exists a tensor LaTeXMLMath with the property that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath and all LaTeXMLMath and LaTeXMLMath such that for the corresponding function LaTeXMLMath defined by ( 1.3.1 ) , we have LaTeXMLEquation where LaTeXMLMath is the number of fixed points of the permutation and LaTeXMLMath and LaTeXMLMath is the number of 2-cycles in the permutation .
We show that LaTeXMLMath and that the maximum value 1 of LaTeXMLMath is attained at the identity permutation LaTeXMLMath ( where LaTeXMLMath and LaTeXMLMath ) and , for even LaTeXMLMath , on the permutations that consist of LaTeXMLMath transpositions ( where LaTeXMLMath and LaTeXMLMath ) .
On the other hand , for any fixed LaTeXMLMath and all LaTeXMLMath , the value of LaTeXMLMath with LaTeXMLMath is negative .
Unfortunately , we are unable to present an example of the symmetric QAP which beats the bound of Theorem 3.1 but we can construct such an example for the generalized problem ( 1.3 ) .
In Section 10 , we prove that for any LaTeXMLMath , there exists a tensor LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath and such that for the corresponding function LaTeXMLMath we have LaTeXMLEquation where LaTeXMLMath is the number of fixed points in LaTeXMLMath and LaTeXMLMath is the number of 2-cycles in LaTeXMLMath .
We show that LaTeXMLMath and that LaTeXMLMath is the maximum value of LaTeXMLMath .
Let us fix any LaTeXMLMath and let us choose some LaTeXMLMath such that LaTeXMLMath for some LaTeXMLMath .
Then , for all sufficiently large LaTeXMLMath , the value LaTeXMLMath can be achieved only on permutations LaTeXMLMath with LaTeXMLMath .
The number of such permutations LaTeXMLMath does not exceed LaTeXMLMath , that is , the probability that a random permutation LaTeXMLMath satisfies LaTeXMLMath does not exceed LaTeXMLMath for large LaTeXMLMath .
It appears that the difference between the general case and the symmetric case of Section 4 is not as substantial as the difference between the symmetric case and the special cases of Sections 2 and 3 .
Our main result is : Let LaTeXMLMath be the function defined by ( 1.1.1 ) or ( 1.3.1 ) , let LaTeXMLMath be the average value of LaTeXMLMath on LaTeXMLMath and let LaTeXMLMath .
Let LaTeXMLMath be an optimal permutation : LaTeXMLMath .
Let us choose an integer LaTeXMLMath and a number LaTeXMLMath .
Let LaTeXMLEquation .
The probability that a random permutation LaTeXMLMath satisfies LaTeXMLEquation is at least LaTeXMLEquation .
In particular , by choosing an appropriate LaTeXMLMath , we obtain the following corollary .
∎ We prove Theorem 5.1 in Section 11 .
As in Section 2 , we conclude that for any fixed LaTeXMLMath there is a randomized LaTeXMLMath algorithm which produces a permutation LaTeXMLMath satisfying ( 1 ) .
If are willing to settle for an algorithm of mildly exponential complexity , we can achieve the bound of type ( 2 ) , which is weaker than the corresponding bound of Corollary 3.2 .
In Section 11 , we construct an example of a function of type ( 1.3.1 ) with an even sharper spike distribution than in example 4.1 .
First , we prove Lemma 1.5 .
Let us choose a pair of indices LaTeXMLMath .
Then , as LaTeXMLMath ranges over the symmetric group LaTeXMLMath , the ordered pair LaTeXMLMath ranges over all ordered pairs LaTeXMLMath with LaTeXMLMath and each such a pair LaTeXMLMath appears LaTeXMLMath times .
Similarly , for each index LaTeXMLMath , the index LaTeXMLMath ranges over the set LaTeXMLMath and each LaTeXMLMath appears LaTeXMLMath times .
Therefore , LaTeXMLEquation and the proof follows .
∎ Suppose that LaTeXMLMath for some matrices LaTeXMLMath and LaTeXMLMath and all LaTeXMLMath and suppose that the maximum value of LaTeXMLMath is attained at a permutation LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath .
Then LaTeXMLMath , hence the maximum value of LaTeXMLMath is attained at the identity permutation LaTeXMLMath and the distribution of values of LaTeXMLMath and LaTeXMLMath is the same .
We observe that if LaTeXMLMath is symmetric then LaTeXMLMath is also symmetric , and if LaTeXMLMath has constant row and column sums and a constant diagonal then so does LaTeXMLMath ( see also Section 7 ) .
Hence , as long as the distribution of values of LaTeXMLMath is concerned , without loss of generality we may assume that the maximum of LaTeXMLMath is attained at the identity permutation LaTeXMLMath .
Let LaTeXMLMath be a function .
Let us define function LaTeXMLMath by LaTeXMLEquation .
We call LaTeXMLMath the central projection of LaTeXMLMath .
The following simple observation is quite important for our approach .
let LaTeXMLMath be a function such that LaTeXMLMath for all LaTeXMLMath and let LaTeXMLMath be the central projection of LaTeXMLMath .
Then LaTeXMLMath for all LaTeXMLMath and the average values of LaTeXMLMath and LaTeXMLMath are equal : LaTeXMLMath .
We observe that LaTeXMLMath for all LaTeXMLMath and hence LaTeXMLMath .
Moreover , for any LaTeXMLMath LaTeXMLEquation .
Finally , LaTeXMLEquation and the proof follows .
∎ Moreover , one can observe that the averages of LaTeXMLMath and LaTeXMLMath on the LaTeXMLMath -th ring LaTeXMLMath coincide for all LaTeXMLMath , see Definition 1.2 .
We will rely on a Markov type estimate , which asserts , roughly , that a function with a sufficiently large average takes sufficiently large values sufficiently often .
Let LaTeXMLMath be a finite set and let LaTeXMLMath be a function .
Suppose that LaTeXMLMath for all LaTeXMLMath and that LaTeXMLEquation .
Then for any LaTeXMLMath we have LaTeXMLEquation .
We have LaTeXMLEquation .
Hence LaTeXMLEquation ∎ Finally , we need some facts about the structure of the symmetric group LaTeXMLMath ( see , for example , [ 6 ] ) .
Let us fix a permutation LaTeXMLMath .
As LaTeXMLMath ranges over the symmetric group LaTeXMLMath , the permutation LaTeXMLMath ranges over the conjugacy class of LaTeXMLMath of LaTeXMLMath , that is the set of permutations that have the same cycle structure as LaTeXMLMath .
We will be using the following facts .
If LaTeXMLMath is a function and LaTeXMLMath its central projection , then LaTeXMLEquation .
If LaTeXMLMath is a set which splits into a union of conjugacy classes LaTeXMLMath , and for each such a class we have LaTeXMLEquation for some number LaTeXMLMath , then LaTeXMLEquation .
Let us fix some positive integers LaTeXMLMath and let LaTeXMLMath be the number of permutations in LaTeXMLMath that have no cycles of length LaTeXMLMath for LaTeXMLMath .
The exponential generating function for LaTeXMLMath is given by LaTeXMLEquation where we agree that LaTeXMLMath , see , for example , pp .
170–173 of [ 7 ] .
It follows that the number of permutations LaTeXMLMath without fixed points is asymptotically LaTeXMLMath and without fixed points and 2-cycles is LaTeXMLMath .
We will use that the first number exceeds LaTeXMLMath and the second number exceeds LaTeXMLMath for LaTeXMLMath .
The number of permutations LaTeXMLMath with at least LaTeXMLMath fixed points is at most LaTeXMLMath , since to choose such a permutation , we can first choose LaTeXMLMath fixed points in LaTeXMLMath ways and then choose an arbitrary permutation of the remaining LaTeXMLMath elements in LaTeXMLMath ways ( some permutations will be counted several times ) .
Similarly , the number of permutations LaTeXMLMath with at least LaTeXMLMath transpositions ( 2-cycles ) is at most LaTeXMLMath , since to choose such a permutation , we first choose some LaTeXMLMath pairs in LaTeXMLMath ways and then an arbitrary permutation of the remaining LaTeXMLMath elements in LaTeXMLMath ways ( again , some permutations will be counted several times ) .
The crucial observation for our approach is that the vector space of all central projections LaTeXMLMath of functions LaTeXMLMath defined by ( 1.1.1 ) or ( 1.3.1 ) is 4- , 3- , or 2- dimensional depending on whether we consider the general case , the cases of Sections 3 and 4 or the special case of Section 2 .
If we require , additionally , that LaTeXMLMath then the dimensions drop by 1 to 3 , 2 and 1 , respectively .
This fact is explained by the representation theory of the symmetric group ( see , for example , [ 6 ] ) .
In this section , we review some facts that we need .
Our notation is inspired by the generally accepted notation of the representation theory .
We describe some important invariant subspaces of the action of LaTeXMLMath in the space of LaTeXMLMath matrices LaTeXMLMath by simultaneous permutations of rows and columns .
We recall that LaTeXMLMath .
Let LaTeXMLMath be the space of constant matrices LaTeXMLMath : LaTeXMLEquation .
Let LaTeXMLMath be the subspace of scalar matrices LaTeXMLMath : LaTeXMLEquation .
Finally , Let LaTeXMLMath .
One can observe that LaTeXMLMath and that LaTeXMLMath is the subspace of all matrices that remain fixed under the action of LaTeXMLMath .
Let LaTeXMLMath be the subspace of matrices with identical rows and such that the sum of entries in each row is 0 : LaTeXMLEquation .
Similarly , let LaTeXMLMath be the subspace of matrices with identical columns and such that the sum of entries in each column is 0 : LaTeXMLEquation .
Finally , let LaTeXMLMath be the subspace of diagonal matrices with the zero sum on the diagonal : LaTeXMLEquation .
Let LaTeXMLMath .
One can check that the dimension of each of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is LaTeXMLMath and that LaTeXMLMath .
Moreover , the subspaces LaTeXMLMath , LaTeXMLMath and LaTeXMLMath do not contain non-trivial invariant subspaces .
The action of LaTeXMLMath in LaTeXMLMath , although non-trivial , is not very complicated .
One can show that if LaTeXMLMath , then the problem of optimizing LaTeXMLMath defined by ( 1.1.1 ) reduces to the Linear Assignment Problem .
Let us define LaTeXMLMath as the subspace of all symmetric matrices LaTeXMLMath with row and column sums equal to 0 and zero diagonal LaTeXMLEquation .
One can check that LaTeXMLMath is an invariant subspace and that LaTeXMLMath .
Besides , LaTeXMLMath contains no non-trivial invariant subspaces .
Let us define LaTeXMLMath as the subset of all skew symmetric matrices LaTeXMLMath with row and column sums equal to 0 : LaTeXMLEquation .
One can check that LaTeXMLMath is an invariant subspace and that LaTeXMLMath .
Similarly , LaTeXMLMath contains no non-trivial invariant subspaces .
One can check that LaTeXMLMath .
The importance of the subspaces ( 7.1 ) – ( 7.4 ) is explained by the fact that they are the isotypical components of the irreducible representations of the symmetric group in the space of matrices .
The following proposition follows from the representation theory of the symmetric group [ 6 ] .
For an LaTeXMLMath matrices LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath , let LaTeXMLMath be the function defined by ( 1.1 ) and let LaTeXMLMath , LaTeXMLEquation be the central projection of LaTeXMLMath .
Given a permutation LaTeXMLMath , let LaTeXMLEquation be the number of fixed points of the permutation and the number of 2-cycles in the permutation correspondingly .
The functions LaTeXMLMath and LaTeXMLMath are the characters of corresponding irreducible representations of LaTeXMLMath for LaTeXMLMath .
They are linearly independent , and , moreover orthogonal : LaTeXMLMath for two characters of different irreducible representation of LaTeXMLMath .
In particular , LaTeXMLEquation hence the average value of all but the trivial character LaTeXMLMath is 0 .
It follows [ 6 ] that each of the functions LaTeXMLMath and LaTeXMLMath is the objective function ( 1.3.1 ) in some generalized problem with a tensor LaTeXMLMath ( see Section 1.3 ) with the property that for all LaTeXMLMath and LaTeXMLMath the matrix LaTeXMLMath for LaTeXMLMath belongs to the corresponding subspace .
Since the set of all functions ( 1.3.1 ) is closed under linear combinations , it follows that every function LaTeXMLMath is an objective function in the generalized problem .
In this section , we prove Theorem 2.1 and Theorem 2.3 .
An important observation is that LaTeXMLMath satisfies the conditions of Section 2 if and only if LaTeXMLMath ( see Section 7 ) .
Without loss of generality , we may assume that the maximum of LaTeXMLMath is attained at the identity permutation LaTeXMLMath ( see Section 6 ) .
Excluding the non-interesting case of LaTeXMLMath , by scaling LaTeXMLMath , if necessary , we can assume that LaTeXMLMath .
Let LaTeXMLMath be the central projection of LaTeXMLMath .
Then by Lemma 6.2 , LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
Moreover , since LaTeXMLMath , by Parts 1 and 3 of Proposition 7.5 , LaTeXMLMath must be a linear combination of the constant function LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath should be proportional to LaTeXMLMath and since LaTeXMLMath , we have LaTeXMLEquation .
Now LaTeXMLMath if and only if LaTeXMLMath .
Hence LaTeXMLMath for all LaTeXMLMath .
The set LaTeXMLMath splits into disjoint union of conjugacy classes LaTeXMLMath and , using ( 6.4.1 ) , we conclude that for each such LaTeXMLMath LaTeXMLEquation and , therefore , LaTeXMLEquation hence the proof follows .
∎ Using estimates of ( 6.4.2 ) , one can show that the input of the number of 2-cycles LaTeXMLMath into the average of LaTeXMLMath over LaTeXMLMath is asymptotically negligible , so there is an “ almost equality ” in the formula of Theorem 2.1 .
By estimating the cardinality of the LaTeXMLMath -th ring LaTeXMLMath , we deduce Theorem 2.3 .
As in the proof of Theorem 2.1 , we assume that the maximum value of LaTeXMLMath is equal to 1 .
Let us estimate the cardinality LaTeXMLMath .
Since LaTeXMLMath if and only if LaTeXMLMath has LaTeXMLMath fixed points , to choose a LaTeXMLMath one has to choose LaTeXMLMath points in LaTeXMLMath ways and then choose a permutation of the remaining LaTeXMLMath points without fixed points .
Using ( 6.4.2 ) , we get LaTeXMLEquation .
Applying Lemma 6.3 with LaTeXMLMath and LaTeXMLMath , from Theorem 2.1 , we conclude that LaTeXMLEquation ∎ In this case , LaTeXMLMath ( see Section 7 ) .
As in Section 8 , the LaTeXMLMath component contributes just a constant to LaTeXMLMath .
Since the LaTeXMLMath component attributed to the Linear Assignment Problem ( see Section 7.2 ) is absent , we call this case “ pure ” .
We choose a more convenient basis LaTeXMLMath and LaTeXMLMath in the vector space spanned by LaTeXMLMath and LaTeXMLMath , namely : LaTeXMLEquation .
Let LaTeXMLMath ( where LaTeXMLMath stands for “ pure ” ) be the set of all functions LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath , where LaTeXMLMath is the identity permutation .
We call LaTeXMLMath the central cone .
Identifying LaTeXMLMath with two-dimensional vector space LaTeXMLMath ( plane ) , we see that the conditions LaTeXMLMath define the central cone LaTeXMLMath as a convex cone in LaTeXMLMath .
Our goal is to find the extreme rays LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , so that every function LaTeXMLMath can be written as a non-negative linear combination of LaTeXMLMath and LaTeXMLMath .
First , we prove a useful technical result .
For a permutation LaTeXMLMath , LaTeXMLMath , let LaTeXMLMath be the point LaTeXMLEquation .
Let LaTeXMLMath be the convex hull of all such points LaTeXMLMath .
If LaTeXMLMath is even , the extreme points of LaTeXMLMath are LaTeXMLEquation .
If LaTeXMLMath is odd , the extreme points of LaTeXMLMath are LaTeXMLEquation .
The set of all possible values LaTeXMLMath , where LaTeXMLMath , consists of all pairs of non-negative integers LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath and , additionally , LaTeXMLMath or LaTeXMLMath .
To find the extreme points of the set of feasible points LaTeXMLMath , we choose a generic vector LaTeXMLMath and investigate for which values of LaTeXMLMath and LaTeXMLMath the maximum of LaTeXMLEquation is attained .
Clearly , we can assume that LaTeXMLMath .
If LaTeXMLMath then we should choose the smallest possible LaTeXMLMath which would be LaTeXMLMath unless LaTeXMLMath when we have to choose LaTeXMLMath .
Depending on the sign of LaTeXMLMath , this produces the following pairs LaTeXMLEquation .
If LaTeXMLMath then the largest possible value of LaTeXMLMath is 1 .
If LaTeXMLMath this produces the ( already included ) point LaTeXMLEquation .
If LaTeXMLMath we get LaTeXMLEquation and LaTeXMLEquation .
Summarizing , the extreme points of LaTeXMLMath are LaTeXMLEquation and LaTeXMLEquation as claimed .
∎ Now we describe the central cone LaTeXMLMath .
For LaTeXMLMath let us define the functions LaTeXMLMath and LaTeXMLMath by LaTeXMLEquation .
Then A function LaTeXMLMath can be written as a linear combination LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath .
Therefore , the inequalities LaTeXMLMath can be written as LaTeXMLEquation which , for LaTeXMLMath is equivalent to LaTeXMLEquation .
Using Lemma 9.2 , we conclude that for even LaTeXMLMath , the system is equivalent to LaTeXMLEquation and for odd LaTeXMLMath , the system is equivalent to LaTeXMLEquation .
Consequently , every solution LaTeXMLMath of ( 9.3.1 ) is a non-negative linear combination of LaTeXMLMath and LaTeXMLMath and every solution of ( 9.3.2 ) is a non-negative linear combination of LaTeXMLMath and LaTeXMLMath .
The functions LaTeXMLMath and LaTeXMLMath are obtained from LaTeXMLMath and LaTeXMLMath respectively by scaling so that the value at the identity becomes equal to 1 .
Since every solution of ( 9.3.1 ) is a solution of ( 9.3.2 ) , we conclude that LaTeXMLMath for odd LaTeXMLMath as well .
∎ If LaTeXMLMath is even , then LaTeXMLMath .
Indeed , if LaTeXMLMath is a product of LaTeXMLMath commuting transpositions , so that LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath .
LaTeXMLEquation .
The functions LaTeXMLMath and LaTeXMLMath have the bullseye distribution of Section 2 .
The distribution type of LaTeXMLMath may be characterized as that of a “ damped oscillator ” with the averages over the LaTeXMLMath -ring LaTeXMLMath changing sign and going fast to 0 as LaTeXMLMath grows .
Hence a typical function from the central cone has a “ weak ” bullseye type distribution , which becomes weaker as the function becomes closer to LaTeXMLMath .
Let LaTeXMLMath be a function such that LaTeXMLMath .
For any LaTeXMLMath , let LaTeXMLMath be a permutation such that LaTeXMLMath and LaTeXMLMath and let LaTeXMLMath be a permutation such that LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
Applying Lemma 9.3 , we may assume that LaTeXMLMath is a convex combination of LaTeXMLMath and LaTeXMLMath , hence LaTeXMLMath , for some LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation and LaTeXMLEquation .
We observe that if LaTeXMLMath and LaTeXMLMath then LaTeXMLMath and if LaTeXMLMath and LaTeXMLMath then LaTeXMLMath .
Moreover , as LaTeXMLMath change from LaTeXMLMath to LaTeXMLMath function LaTeXMLMath decreases and function LaTeXMLMath increases .
Hence the minimum of LaTeXMLMath is attained when LaTeXMLMath .
This produces the system of linear equations LaTeXMLEquation and LaTeXMLEquation with the solution LaTeXMLEquation .
The corresponding value of LaTeXMLMath is LaTeXMLEquation which completes the proof .
∎ Now we are ready to prove Theorem 3.1 .
Without loss of generality , we may assume that the maximum value of LaTeXMLMath is attained at the identity permutation LaTeXMLMath ( see Section 6 ) .
Excluding an obvious case of LaTeXMLMath , by scaling LaTeXMLMath , if necessary , we may assume that LaTeXMLMath .
Let LaTeXMLMath be the central projection of LaTeXMLMath .
By Lemma 6.2 , LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
Moreover , since LaTeXMLMath , by Proposition 7.5 , LaTeXMLMath must be a linear combination of the constant function LaTeXMLMath and functions LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is a linear combination of LaTeXMLMath and LaTeXMLMath alone .
Therefore , LaTeXMLMath lies in the central cone : LaTeXMLMath , see Definition 9.1 .
Let us choose a LaTeXMLMath and let LaTeXMLMath be the set of permutations LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath and let LaTeXMLMath be the set of permutations LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
To choose a permutation LaTeXMLMath , one has to choose LaTeXMLMath fixed points in LaTeXMLMath ways and then a permutation without fixed points or 2-cycles on the remaining LaTeXMLMath points .
Then , by ( 6.4.2 ) LaTeXMLEquation .
Similarly , to choose a permutation LaTeXMLMath , one has to choose a 2-cycle in LaTeXMLMath ways , LaTeXMLMath fixed points in LaTeXMLMath ways and a permutation without fixed points or 2-cycles on the remaining LaTeXMLMath points .
Then , by ( 6.4.2 ) LaTeXMLEquation .
Let us choose a permutation LaTeXMLMath and a permutation LaTeXMLMath and let LaTeXMLMath if LaTeXMLMath and LaTeXMLMath otherwise .
Then LaTeXMLEquation and by Lemma 9.5 , LaTeXMLEquation .
The set LaTeXMLMath is a disjoint union of some conjugacy classes LaTeXMLMath and for each LaTeXMLMath by ( 6.4.1 ) , we have LaTeXMLEquation and hence LaTeXMLEquation .
Applying Lemma 6.3 with LaTeXMLMath and LaTeXMLMath , we get that LaTeXMLEquation .
In this case , LaTeXMLMath ( see Section 7 ) .
As in Sections 8 and 9 , the LaTeXMLMath component contributes a just a constant to LaTeXMLMath .
We choose a more convenient basis LaTeXMLMath and LaTeXMLMath in the vector space spanned by LaTeXMLMath and LaTeXMLMath , namely LaTeXMLEquation where LaTeXMLMath is the number of fixed points of LaTeXMLMath and LaTeXMLMath is the number of 2-cycles in LaTeXMLMath .
Let LaTeXMLMath ( where LaTeXMLMath stands for “ symmetric ” ) be the set of all functions LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath , where LaTeXMLMath is the identity permutation .
We call LaTeXMLMath the central cone .
Identifying LaTeXMLMath with two-dimensional vector space LaTeXMLMath ( plane ) , we see that the conditions LaTeXMLMath define the central cone LaTeXMLMath as a convex cone in LaTeXMLMath .
Our immediate goal is to find the extreme rays LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , so that every function LaTeXMLMath can be written as a non-negative linear combination of LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath let us define the functions LaTeXMLMath and LaTeXMLMath by LaTeXMLEquation .
Then A function LaTeXMLMath can be written as a linear combination LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath .
Therefore , the inequalities LaTeXMLMath can be written as LaTeXMLEquation which , for LaTeXMLMath , is equivalent to LaTeXMLEquation .
Applying Lemma 9.2 , we observe that ( 10.2.1 ) is equivalent to the system of two inequalities : LaTeXMLEquation and LaTeXMLEquation .
Thus every pair LaTeXMLMath satisfying ( 10.2.1 ) can be written as a non-negative linear combination of LaTeXMLMath and LaTeXMLMath when LaTeXMLMath is even and LaTeXMLMath and LaTeXMLMath when LaTeXMLMath is odd .
The generators LaTeXMLMath and LaTeXMLMath are obtained from LaTeXMLMath , LaTeXMLMath and LaTeXMLMath respectively by scaling so that the value at the identity becomes equal to 1 .
It remains to check that LaTeXMLMath for LaTeXMLMath odd as well .
Indeed , using that LaTeXMLMath we have LaTeXMLEquation ∎ The average value of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath on LaTeXMLMath is 0 .
The function LaTeXMLMath provides an example of the “ bullseye ” distribution ( see Section 2.2 ) .
The maximum value of 1 is attained at the identity and at any transposition .
The positive values of LaTeXMLMath occur on permutations with at least two fixed points and LaTeXMLMath if LaTeXMLMath .
In contrast , LaTeXMLMath and LaTeXMLMath exhibit a spike type distribution of Section 4.1 .
The maximum value of 1 is attained at the identity and , for LaTeXMLMath , on the product of LaTeXMLMath transpositions , or , for LaTeXMLMath , on the product of LaTeXMLMath transpositions .
On the other hand , no permutation other than LaTeXMLMath with at least 2 fixed points yields a positive value .
One can observe that if LaTeXMLMath is even then LaTeXMLMath .
Indeed , if LaTeXMLMath is a product of LaTeXMLMath transpositions then LaTeXMLMath .
LaTeXMLEquation .
The picture of LaTeXMLMath is very similar to that of LaTeXMLMath , see Section 9.4 .
Let us consider Example 4.1 .
It is seen that LaTeXMLMath and hence the maximum value of LaTeXMLMath is indeed 1 and obtained , in particular , on the identity permutation LaTeXMLMath .
Applying Lemma 1.5 , we get LaTeXMLEquation .
Let us prove that the central projection of LaTeXMLMath is the function LaTeXMLMath of Lemma 10.2 .
Suppose that LaTeXMLMath is the central projection of LaTeXMLMath .
It follows that LaTeXMLMath can be written as a linear combination LaTeXMLMath .
Since LaTeXMLMath , we must have LaTeXMLMath .
Let LaTeXMLMath be a transposition , hence LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath , hence LaTeXMLMath .
Denoting by LaTeXMLMath the set of all transpositions in LaTeXMLMath , by ( 6.4.1 ) we get LaTeXMLEquation .
Therefore , LaTeXMLMath .
Let LaTeXMLMath be a conjugacy class with LaTeXMLMath .
Then LaTeXMLEquation .
Since the LaTeXMLMath -th ring LaTeXMLMath splits into a disjoint union of conjugacy classes LaTeXMLMath with LaTeXMLMath , we conclude by ( 6.4.1 ) that LaTeXMLEquation as claimed .
More generally , one can prove that for any function LaTeXMLMath there is a function LaTeXMLMath of type ( 1.1.1 ) with symmetric LaTeXMLMath , such that LaTeXMLMath , LaTeXMLMath attains its maximum at the identity and the central projection of LaTeXMLMath is LaTeXMLMath .
Let us consider the function LaTeXMLMath of Example 4.2 .
We observe that LaTeXMLEquation for LaTeXMLEquation .
Thus LaTeXMLMath is a convex combination of LaTeXMLMath and LaTeXMLMath , hence LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
Remark 7.6 implies that LaTeXMLMath is a generalized function ( 1.3.1 ) of the required type .
In this Section , we prove Theorem 5.1 and describe the “ extreme ” distributions .
Let us choose a convenient basis in LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be the set of all functions LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
We call LaTeXMLMath the central cone .
Identifying LaTeXMLMath with a 3-dimensional vector space LaTeXMLMath , we see that conditions LaTeXMLMath define the central cone LaTeXMLMath as a convex polyhedral cone in LaTeXMLMath .
The condition LaTeXMLMath defines a plane LaTeXMLMath in LaTeXMLMath and the intersection LaTeXMLMath is a base of LaTeXMLMath , that is , a polygon such that every LaTeXMLMath can be uniquely represented in the form LaTeXMLMath for some LaTeXMLMath .
Our goal is to determine the structure of LaTeXMLMath .
This is somewhat more complicated than in the 2-dimensional situations of Sections 9-10 .
Let us define functions LaTeXMLEquation .
Then A function LaTeXMLMath can be written as a linear combination LaTeXMLMath .
Then LaTeXMLMath and the conditions LaTeXMLMath are written as LaTeXMLEquation which , for LaTeXMLMath are equivalent to LaTeXMLEquation .
Applying Lemma 9.2 , we see that for even LaTeXMLMath , the system is equivalent to LaTeXMLEquation whereas for odd LaTeXMLMath , the system is equivalent to LaTeXMLEquation .
The set of all feasible 3-tuples LaTeXMLMath is a polyhedral cone , which , for even LaTeXMLMath , has at most 4 extreme rays and for odd LaTeXMLMath has at most 5 extreme rays .
We call an inequality of ( 11.2.1 ) – ( 11.2.2 ) active on a particular tuple if it holds with equality .
It is readily verified that for even LaTeXMLMath the following tuples span the extreme rays of the set of solutions to ( 11.2.1 ) : LaTeXMLEquation and that for odd LaTeXMLMath the following tuples span the extreme rays of the set of solutions to ( 11.2.1 ) : LaTeXMLEquation .
We obtain LaTeXMLMath and LaTeXMLMath by scaling the corresponding linear combinations LaTeXMLMath so that the value at the identity is equal to 1 and hence LaTeXMLMath and LaTeXMLMath lie on the same plane in LaTeXMLMath .
∎ One can observe that if LaTeXMLMath is even then LaTeXMLMath , for if LaTeXMLMath is a product of LaTeXMLMath commuting transpositions , so that LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath .
LaTeXMLEquation .
We observe that function LaTeXMLMath coincides with function LaTeXMLMath of Lemma 10.2 ( the symmetric QAP ) and that function LaTeXMLMath coincides with function LaTeXMLMath of Lemma 9.3 ( the pure QAP ) .
Function LaTeXMLMath has a bullseye type distribution ( see Section 2.2 ) whereas LaTeXMLMath is a sharp spike ( see Section 4.1 ) .
We have LaTeXMLMath if and only if LaTeXMLMath or LaTeXMLMath and LaTeXMLMath for all other LaTeXMLMath .
LaTeXMLEquation .
Function LaTeXMLMath resembles a spike , but diluted .
Now we are getting ready to prove Theorem 5.1 .
Let LaTeXMLMath be a function such that LaTeXMLMath .
For a LaTeXMLMath , let LaTeXMLMath be a permutation such that LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath be a permutation such that LaTeXMLMath and LaTeXMLMath and let LaTeXMLMath be permutation such that LaTeXMLMath .
Then LaTeXMLEquation .
We can write LaTeXMLEquation for some LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLEquation .
We observe that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are linear functions of LaTeXMLMath and LaTeXMLMath and hence LaTeXMLEquation is a convex function on the plane LaTeXMLMath .
Moreover , for LaTeXMLEquation we have LaTeXMLEquation .
Let us prove that the minimum of LaTeXMLMath on the plane LaTeXMLMath is attained at ( 11.4.1 ) .
Let LaTeXMLEquation .
Then LaTeXMLEquation .
Comparing this with ( 11.4.2 ) , we conclude that there is no point LaTeXMLMath with LaTeXMLMath such that LaTeXMLEquation ∎ Now we are ready to prove Theorem 5.1 .
Without loss of generality , we may assume that the maximum value of LaTeXMLMath is attained at the identity permutation LaTeXMLMath .
Excluding an obvious case of LaTeXMLMath , by scaling LaTeXMLMath , if necessary , we may assume that LaTeXMLMath .
Let LaTeXMLMath be the central projection of LaTeXMLMath .
By Lemma 6.2 , LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
By Proposition 7.5 , LaTeXMLMath must be a linear combination of the functions LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is a linear combination of non-trivial characters LaTeXMLMath , LaTeXMLMath and LaTeXMLMath alone .
Therefore , LaTeXMLMath lies in the central cone : LaTeXMLMath , see Definition 11.1 .
Let LaTeXMLMath be the set of all permutations LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
As in the proof of Theorem 4.1 , we conclude that LaTeXMLEquation .
Let LaTeXMLMath be the set of all permutations LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
To choose a permutation LaTeXMLMath , one has to choose a transpositions in LaTeXMLMath ways and then an arbitrary permutation of the remaining LaTeXMLMath symbols without fixed points and 2-cycles .
Using ( 6.4.2 ) , we estimate LaTeXMLEquation .
Let us choose a permutation LaTeXMLMath , a permutation LaTeXMLMath and a permutation LaTeXMLMath .
Let us choose LaTeXMLMath to be one of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , depending where the maximum value of LaTeXMLMath , LaTeXMLMath or LaTeXMLMath is attained .
Hence LaTeXMLEquation .
The set LaTeXMLMath is a disjoint union of some conjugacy classes LaTeXMLMath and for each LaTeXMLMath by ( 6.4.1 ) and Lemma 11.4 , we have LaTeXMLEquation and hence LaTeXMLEquation .
Applying Lemma 6.3 with LaTeXMLMath and LaTeXMLMath , we conclude that LaTeXMLEquation for all LaTeXMLMath .
∎ 1 .
K. Anstreicher , N. Brixius , J.-P. Goux and J. Linderoth , Solving large quadratic assignment problems on computational grids , preprint , 2000 .
2 .
E. Arkin , R. Hassin and M. Sviridenko , Approximating the maximum quadratic assignment problem , Inform .
Process .
Lett .
, 77 ( 2001 ) , no .
1 , 13–16 .
3 .
G. Ausiello , P. Crescenzi , G. Gambosi , V. Kann , A. Marchetti-Spaccamela , and M. Protasi , Complexity and Approximation .
Combinatorial optimization problems and their approximability properties , Springer-Verlag , Berlin , 1999 .
4 .
A. Brüngger , A. Marzetta , J. Clausen and M. Perregaard , Solving large scale quadratic assignment problems in parallel with the search library ZRAM , Journal of Parallel and Distributed Computing , 50 , pp .
157-66 , 1998 .
5 .
R. Burkard , E. Çela , P. Pardalos and L. Pitsoulis , The quadratic assignment problem , in : Handbook of Combinatorial Optimization ( D.-Z .
Du and P.M. Pardalos , eds .
) , Kluwer Academic Publishers , pp .
75-149 , 1999 .
6 .
W. Fulton and J. Harris , Representation Theory , Springer-Verlag , New York , 1991 .
7 .
I.P .
Goulden and D.M .
Jackson , Combinatorial Enumeration , Wiley-Interscience Series in Discrete Mathematics , John Wiley LaTeXMLMath Sons , Inc. , New York , 1983 .
We use Padoa ’ s principle of independence of primitive symbols in axiomatic systems in order to show that time is dispensable in continuum thermodynamics , according to the axiomatic formulation of Gurtin and Williams .
We also show how to define time by means of the remaining primitive concepts of Gurtin and Williams system .
Finally , we introduce thermodynamics without time and briefly discuss some physical and philosophical consequences of our main results .
The present paper has a philosophical aspect in the sense that it copes with philosophical questions regarding the foundations of thermodynamics .
On the other hand , it is also a work on mathematical physics , in the sense that we are concerned with the mathematical foundations of thermodynamics .
Many branches of mathematics provide useful tools for theoretical physics , like probabilities , distributions , special functions , functional analysis , differential calculus , and so on .
But few people use logic in theoretical physics .
We consider this paper as a work on mathematical physics because we use mathematical logic in order to answer some fundamental questions concerning the mathematical role of time in thermodynamics .
In a recent work LaTeXMLCite we have shown , by using Padoa ’ s principle of independence of primitive concepts , that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics , Maxwell ’ s electromagnetic theory , the Dirac electron , classical gauge fields , and general relativity .
Nevertheless , in all these theories physical phenomena are reversible with respect to time .
Thermodynamics is a theory which talks about irreversible physical phenomena and so it seems to point out to an “ arrow of time ” .
There are many physical and philosophical discussions , in the liteature , about the objective existence of an arrow of time .
For some authors there is , indeed , an objective “ passage ” of time LaTeXMLCite .
Nevertheless any concept of an objective passage of time seems to entail logical loops or to invoke “ absolute time ” , which contradicts relativity theory LaTeXMLCite .
Other authors believe that most physicists prefer to regard time ’ s passage as an illusion , since in theories like Einstein-Minkowski spacetime , all events ( past , present , and future ) have the same level of reality LaTeXMLCite .
In LaTeXMLCite ( p. 203 ) Prigogine makes reference to a private letter written by A. Einstein where it is written : For us who are convinced physicists , the distinction between past , present , and future is only an illusion , however persistent .
Prigogine himself ( op .
cit .
p. 213 ) says that The distinction between past and future is a kind of primitive concept that in a sense precedes scientific activity .
But some paragraphs later he remarks that classical science is trying “ to go beyond the world of appearances , to reach a timeless world of supreme rationality. ” In this paper we study the role of time in thermodynamics from the mathematical point of view .
Since the theory is supposed to give a picture of physical phenomena , we hope that our results may be useful for the understanding of the physical and philosophical meaning of time in a theory which admits the existence of irreversible phenomena .
Some decades ago M. E. Gurtin and W. O. Williams presented an axiomatic framework for continuum thermodynamics LaTeXMLCite .
In the present paper we rewrite Gurtin and Williams axiomatic system as a Bourbaki species of structures and prove that time is definable , and so is dispensable , in thermodynamics .
For a brief review on the use of the axiomatic method in physics as well as other proposals for the interpretation of time and spacetime in physics see LaTeXMLCite .
This section is essentially based on our previous work LaTeXMLCite .
In an axiomatic system LaTeXMLMath a primitive term or concept LaTeXMLMath is definable by means of the remaining primitive ones if there is an appropriate formula , provable in the system , that fixes the meaning of LaTeXMLMath in function of the other primitive terms of LaTeXMLMath .
This formulation of definability is not rigorous but is enough here .
When LaTeXMLMath is not definable in LaTeXMLMath , it is said to be independent of the the other primitive terms .
There is a method , introduced by A. Padoa LaTeXMLCite , which can be employed to show the independence of concepts .
In fact , Padoa ’ s method gives a necessary and sufficient condition for independence LaTeXMLCite .
In order to present Padoa ’ s method , some preliminary remarks are necessary .
Loosely speaking , if we are working in set theory , as our basic theory , an axiomatic system LaTeXMLMath characterizes a species of mathematical structures in the sense of Bourbaki LaTeXMLCite .
Actually there is a close relationship between Bourbaki ’ s species of structures and Suppes predicates LaTeXMLCite ; for details see LaTeXMLCite .
On the other hand , if our underlying logic is higher-order logic ( type theory ) , LaTeXMLMath determines a usual higher-order structure LaTeXMLCite .
In the first case , our language is the first order language of set theory , and , in the second , it is the language of ( some ) type theory .
Tarski showed that Padoa ’ s method is valid in the second case LaTeXMLCite , and Beth that it is applicable in the first LaTeXMLCite .
From the point of view of applications of the axiomatic method , for example in the foundations of physics , it is easier to assume that our mathematical systems and structures are contructed in set theory LaTeXMLCite .
A simplified and sufficiently rigorous formulation of the method , adapted to our exposition , is described in the next paragraphs .
Let LaTeXMLMath be an axiomatic system whose primitive concepts are LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath .
One of these concepts , say LaTeXMLMath , is independent from the remaining if and only if there are two models of LaTeXMLMath in which LaTeXMLMath , … , LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath have the same interpretation , but the interpretations of LaTeXMLMath in such models are different .
Of course a model of LaTeXMLMath is a set-theoretical structure in which all axioms of LaTeXMLMath are true , according to the interpretation of its primitive terms LaTeXMLCite .
It is important to recall that , according to the theory of definition LaTeXMLCite , a definition should satisfy the criterion of eliminability .
That means that a defined symbol should always be eliminable from any formula of the theory .
In the sequel we apply Padoa ’ s method to thermodynamics ( in an axiomatic form ) , in order to prove that time is eliminable ( or dispensable ) .
In this section we present the axiomatic system of thermodynamics due to Gurtin and Williams LaTeXMLCite .
We present it as a Bourbaki ’ s species of structure .
This demands some technical adaptations which do not affect the physical meaning of the theory .
But such adaptations are necessary in order to use Padoa ’ s method .
We use the same notation as in LaTeXMLCite .
It is interesting to settle our mathematical notation .
We denote by LaTeXMLMath the Euclidian three dimensional space .
LaTeXMLMath is the set of real numbers .
If LaTeXMLMath is a subset of LaTeXMLMath then its boundary is denoted by LaTeXMLMath , the interior by LaTeXMLMath , and the closure by LaTeXMLMath .
The letters LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , … , always denote subsets of LaTeXMLMath .
A standard region or body of LaTeXMLMath is the closure of a bounded open set LaTeXMLMath whose boundary is the union of a closed set of zero area and a countable number of class LaTeXMLMath two-dimensional manifolds , each of the manifolds having the open set LaTeXMLMath on just one side .
If LaTeXMLMath is a subset of LaTeXMLMath , its exterior is LaTeXMLEquation .
If LaTeXMLMath , the relative exterior of LaTeXMLMath in LaTeXMLMath is LaTeXMLEquation .
LaTeXMLEquation Straightforward from definitions .
A surface is the ( relative ) closure of an oriented class LaTeXMLMath two-dimensional differentiable manifold or the countable union of such ( closed ) manifolds .
According to LaTeXMLCite : “ The boundary of a standard region is taken to be oriented in the positive sense with respect to that region , i.e .
with the orientation corresponding to the external normal vector .
A surface contained ( in the sense of set-inclusion ) in another surface is a positive segment of that surface if it has the same orientation ; if it has the opposite orientation it is called a negative segment . ” A surface LaTeXMLMath contained in a body LaTeXMLMath is a material surface if it is a positive segment of the boundary of a subbody of LaTeXMLMath .
A part of a body LaTeXMLMath is a Borel subset of LaTeXMLMath .
The set of Borel subsets of LaTeXMLMath is denoted by LaTeXMLMath .
By measure we mean a finite real-valued Borel signed measure LaTeXMLCite .
The species of structures ( à la Bourbaki ) of a Gurtin-Williams System for Continuum Thermodynamics is the ordered LaTeXMLMath -tuple LaTeXMLEquation such that the following axioms are satisfied : LaTeXMLMath is the Euclidean three dimensional space .
For every body LaTeXMLMath there is a class of subbodies LaTeXMLMath such that Every element of LaTeXMLMath is a subset of LaTeXMLMath ; Every element of LaTeXMLMath is a body ; LaTeXMLMath implies LaTeXMLMath ; If LaTeXMLMath is a solid circular cylinder or a solid prism in LaTeXMLMath , then LaTeXMLEquation .
If LaTeXMLMath is a material surface , there is a monotone sequence LaTeXMLMath of elements of LaTeXMLMath such that LaTeXMLEquation .
If LaTeXMLMath and LaTeXMLMath is a vector in LaTeXMLMath , then LaTeXMLMath .
LaTeXMLMath is an interval of real numbers , which we interpret as time .
For each body LaTeXMLMath there is an energy function LaTeXMLMath such that LaTeXMLMath is a measure on LaTeXMLMath .
This function corresponds to the internal energy of each part of the body LaTeXMLMath .
For each body LaTeXMLMath and for each part LaTeXMLMath there exists the derivative LaTeXMLEquation .
For all LaTeXMLMath there exists a scalar LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation where LaTeXMLMath is the Lebesgue volume measure in LaTeXMLMath .
The material universe for a given body LaTeXMLMath is the set LaTeXMLEquation .
For each body LaTeXMLMath of LaTeXMLMath and for each element LaTeXMLMath of the material universe for LaTeXMLMath there is a heat flux function LaTeXMLMath such that LaTeXMLMath is a measure on LaTeXMLMath for a fixed LaTeXMLMath .
A real-valued function LaTeXMLMath defined on LaTeXMLMath or LaTeXMLMath is separately additive , or s-additive for short , if LaTeXMLEquation for every pair of separate elements LaTeXMLMath , LaTeXMLMath in the domain of LaTeXMLMath .
For a fixed part LaTeXMLMath of a given body LaTeXMLMath the heat flux LaTeXMLMath is s-additive on all elements of LaTeXMLMath separate from LaTeXMLMath .
There exist scalar functions LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath which are separate , where LaTeXMLMath stands for the Lebesgue surface measure on manifolds in LaTeXMLMath .
For every body LaTeXMLMath LaTeXMLEquation .
For each body LaTeXMLMath there is an internal entropy function LaTeXMLMath such that LaTeXMLMath is a measure on LaTeXMLMath .
The derivative LaTeXMLEquation exists for each LaTeXMLMath .
For all LaTeXMLMath there exists a scalar LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation where , like in axiom T6 , LaTeXMLMath is the Lebesgue volume measure in LaTeXMLMath .
For each body LaTeXMLMath of LaTeXMLMath and for each element LaTeXMLMath of the material universe for LaTeXMLMath there is an entropy flux function LaTeXMLMath such that LaTeXMLMath is a measure on LaTeXMLMath for a fixed LaTeXMLMath .
For a fixed part LaTeXMLMath of a given body LaTeXMLMath the entropy flux LaTeXMLMath is s-additive on all elements of LaTeXMLMath separate from LaTeXMLMath .
A part LaTeXMLMath of a body LaTeXMLMath is thermally isolated from LaTeXMLMath if for each part LaTeXMLMath LaTeXMLEquation .
For every subbody LaTeXMLMath of a given body LaTeXMLMath LaTeXMLEquation .
If a part LaTeXMLMath is thermally isolated from LaTeXMLMath LaTeXMLEquation .
There exist scalars LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath , where LaTeXMLEquation is the radiative entropy flux and LaTeXMLEquation is the conductive entropy flux .
It is worth to remark that in LaTeXMLCite there is a theorem that says that for any LaTeXMLMath , LaTeXMLMath admits the unique decomposition LaTeXMLEquation .
A very detailed analysis on these axioms is made in LaTeXMLCite .
At the present paper we are mainly concerned with the results introduced in the next section .
In this section we : ( i ) prove that time is definable ; ( ii ) define time from the remaining primitive concepts of Gurtin-Williams system for thermodynamics ; and ( iii ) rephrase thermodynamics without time .
So , our starting point is the next theorem : Time is dispensable in Gurtin-Williams system for continuum thermodynamics .
Proof : Padoa ’ s Principle says that the primitive concept LaTeXMLMath in LaTeXMLMath is independent from the remaining primitive concepts iff there are two models of LaTeXMLMath such that LaTeXMLMath has two interpretations and all the other primitive symbols have the same interpretation .
But these two interpretations are not possible , since all the remaining concepts , except LaTeXMLMath , depend on time LaTeXMLMath as functions .
Any change of interpretation related to LaTeXMLMath will imply a change of interpretation of LaTeXMLMath , and LaTeXMLMath .
Therefore , time is not independent and hence it can be defined .
So , according to the criterion of eliminability of definitions , time is dispensable within the scope of this axiomatic framework for thermodynamics .
LaTeXMLMath Since time is definable in continuum thermodynamics , the natural question is : how to define it ?
From the logical point of view the answer to this question is not difficult .
But first we need the next definition : If the domain of a function LaTeXMLMath is LaTeXMLMath , then we call each LaTeXMLMath a component of the domain of LaTeXMLMath .
For short , we say that LaTeXMLMath is a component of LaTeXMLMath .
Now we are able to define time in Gurtin-Williams system for thermodynamics : Time is the last component of the functions LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath and is the domain of the scalar functions LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Now , the final question is : how to rephrase thermodynamics with no explicit mention to time ?
Our answer is the definition given below .
Some axioms of this ‘ new ’ system are the same as in LaTeXMLMath , mainly those axioms that make no reference to time .
We write the word ‘ new ’ because actually it is not a new system .
It is the same theory , but with no explicit mention to a definable ( eliminable ) term called ‘ time ’ .
The species of structures ( à la Bourbaki ) of a Gurtin-Williams System for Continuum Thermodynamics Without Time is the ordered LaTeXMLMath -tuple LaTeXMLEquation such that the following axioms are satisfied : T1 .
T2 .
For each body LaTeXMLMath there is an energy function LaTeXMLMath with two components .
The first component is LaTeXMLMath , such that LaTeXMLMath is a measure on LaTeXMLMath .
The co-domain of this function is LaTeXMLMath .
The function LaTeXMLMath is differentiable with respect to the elements of the last ( second ) component of LaTeXMLMath .
We denote the derivative as LaTeXMLEquation .
This same derivative operator is also defined with respect to the last ( second ) component of LaTeXMLMath on the space of functions LaTeXMLMath .
We denote it by LaTeXMLEquation .
T6 .
For each body LaTeXMLMath of LaTeXMLMath and for each element LaTeXMLMath of the material universe for LaTeXMLMath there is a function LaTeXMLMath such that LaTeXMLMath has three components .
The first component is LaTeXMLMath , and the second component is the material universe LaTeXMLMath .
LaTeXMLMath is a measure on LaTeXMLMath for a fixed LaTeXMLMath .
The co-domain of the function LaTeXMLMath is LaTeXMLMath .
T8 .
T9 .
T10 .
For each body LaTeXMLMath there is a function LaTeXMLMath with two components .
The first one is LaTeXMLMath .
LaTeXMLMath is a measure on LaTeXMLMath .
The co-domain of LaTeXMLMath is LaTeXMLMath .
T13 .
For each body LaTeXMLMath of LaTeXMLMath and for each element LaTeXMLMath of the material universe for LaTeXMLMath there is a function LaTeXMLMath with three components .
The first component is LaTeXMLMath and the second is LaTeXMLMath .
LaTeXMLMath is a measure on LaTeXMLMath for a fixed LaTeXMLMath .
The co-domain of LaTeXMLMath is LaTeXMLMath .
T15 .
T16 .
T17 .
It seems to be clear that this picture for thermodynamics is not very operational .
So , for all practical purposes , it is still interesting to keep the notion of time .
The eliminability of time should be rather regarded as a logical consequence from the foundations of thermodynamics .
In this section we make some final remarks concerning our main results : The reader can verify that time is also eliminable from continuum mechanics , at least within the scope of the axiomatic system presented by W. Noll in LaTeXMLCite .
In that system some primitive notions like motion LaTeXMLMath , linear momentum LaTeXMLMath , and angular momentum LaTeXMLMath of a body are functions with time LaTeXMLMath as one component .
So , following Padoa ’ s principle , any change of interpretation on LaTeXMLMath entails a change of interpretation on other primitive concepts , like LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
In LaTeXMLCite we did not define space-time in classical field theories or time in classical particle mechanics .
Nevertheless , the same ideas presented in the previous section of the present paper may be used for that purpose .
We keep this as an exercise for the reader .
We can easily see that space is also definable in Gurtin-Williams system .
That is ledt as an exercise for the reader .
Some neurologists point out that blind people may have great difficulty to develop the notion of physical space LaTeXMLCite .
They live in a spaceless world where time is their main reference frame .
It seems that the vision in healthy people helps them to create an intuitive notion of space , since the visual perception allows people to be aware of many different objects and places at the same time .
On the other hand , U. Mohrhoff LaTeXMLCite presents a novel interpretation of quantum mechanics , where objective probabilities are assigned to counterfactuals and are calculated on the basis of all relevant facts , including those that are still in the future .
According to this proposal , the intuitive distinction between here and there , past and future , has nothing to do with any physical reality .
His starting point is a paper by N. D. Mermin LaTeXMLCite , where this author considers that conscious perception “ should be viewed as a mystery about us and should not be confused with the problem of understanding quantum mechanics. ” In the present paper we are not concerned with quantum mechanics .
But we believe that the definability of time ( and even space ) could be interpreted as it follows : from our physical experience with the world , and our conscious perception of it , we develop some sort of spacetime interpretation about what we see and feel with all our senses .
Much more should be said about our conscious perception of space and time .
But that is a task that we leave for future works .
In the present paper we want to point out the mathematical role of time in thermodynamics .
The tautological rings LaTeXMLMath are natural subrings of the Chow rings of the Deligne-Mumford moduli spaces of pointed curves : LaTeXMLEquation ( the Chow rings are taken with LaTeXMLMath -coefficients ) .
The system of tautological subrings ( LaTeXMLRef ) is defined to be the set of smallest LaTeXMLMath -subalgebras satisfying the following three properties [ FP ] : LaTeXMLMath contains the cotangent line classes LaTeXMLEquation .
The system is closed under push-forward via all maps forgetting markings : LaTeXMLEquation .
The system is closed under push-forward via all gluing maps : LaTeXMLEquation .
LaTeXMLEquation The tautological rings possess remarkable algebraic and combinatorial structures with basic connections to topological gravity .
A discussion of these properties together with a conjectural framework for the study of LaTeXMLMath can be found in [ F ] , [ FP ] .
In genus 0 , the equality LaTeXMLEquation for LaTeXMLMath , is well-known from Keel ’ s study [ K ] .
Denote the image of LaTeXMLMath under the canonical map to the ring of even cohomology classes by : LaTeXMLEquation .
In genus 1 , Getzler has claimed the isomorphisms : LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath , see [ G1 ] .
For LaTeXMLMath , complete results are known only in codimension 1 .
The equality LaTeXMLEquation for LaTeXMLMath , is a consequence of Harer ’ s cohomological calculations [ H ] .
It is natural to ask whether all algebraic cycles classes on LaTeXMLMath are tautological .
The existence of nontautological cycles defined over LaTeXMLMath may be deduced from the odd cohomology of LaTeXMLMath .
There are two arguments which may be used : By a theorem of Jannsen , since the map to cohomology LaTeXMLEquation is not surjective , the map is not injective .
We may then deduce the existence of a nontautological Chow class in LaTeXMLMath from Getzler ’ s claims LaTeXMLCite .
More precisely , the existence of a holomorphic 11-form and a theorem of Srinivas together imply LaTeXMLMath is an infinite dimensional vector space , while LaTeXMLMath , see LaTeXMLCite .
These arguments do not produce an explicit algebraic cycle which is not tautological .
Several further questions are also left open .
Are there nontautological cycles defined over LaTeXMLMath ?
Are there algebraic cycles with cohomological image not contained in LaTeXMLMath ?
Are there nontautological classes on the noncompact spaces LaTeXMLMath ?
We answer all these questions in the affirmative by explicit constructions of integrally defined algebraic cycles .
Our basic criterion for detecting nontautological cycles is the following Proposition .
Let LaTeXMLMath be the gluing map to a boundary divisor .
If LaTeXMLMath , then LaTeXMLMath has a tautological Künneth decomposition : LaTeXMLEquation .
While the above result is well known to experts , we know of no adequate reference , so we give a proof in the Appendix .
Our strategy for finding nontautological classes combines Proposition LaTeXMLRef with the existence of odd cohomology on the moduli spaces of curves .
We find loci in moduli space which restrict to diagonal loci of symmetric boundary divisors .
By the existence of odd cohomology in certain cases , the Künneth decomposition of the diagonal is not tautological .
Let LaTeXMLMath be an odd integer and set LaTeXMLMath .
Let LaTeXMLMath denote the closure of the set of nonsingular curves of genus LaTeXMLMath which admit a degree 2 map to a nonsingular curve of genus LaTeXMLMath .
Intersecting LaTeXMLMath with the boundary map from LaTeXMLMath yields the diagonal .
Pikaart has proven , for sufficiently large LaTeXMLMath , LaTeXMLMath has odd cohomology [ P ] .
Hence , we can conclude LaTeXMLMath is not a tautological class , even in homology .
For all sufficiently large odd LaTeXMLMath , LaTeXMLEquation .
Our other examples are loci in the moduli space of pointed genus 2 curves .
We will use the odd cohomology of LaTeXMLMath to find nontautological Künneth decompositions .
Let LaTeXMLMath in LaTeXMLMath be a product of 10 disjoint 2-cycles , LaTeXMLEquation inducing an involution on LaTeXMLMath .
Let LaTeXMLMath denote the component of the fixed locus of the involution corresponding generically to a 20-pointed , nonsingular , bielliptic curve of genus 2 with the 10 pairs of conjugate markings .
LaTeXMLMath is of codimension 11 in LaTeXMLMath .
The intersection of LaTeXMLMath with the boundary map LaTeXMLEquation yields the diagonal .
LaTeXMLMath Although the methods used to prove Theorems LaTeXMLRef and LaTeXMLRef depend crucially on the structure of the boundary of the moduli space , in Section LaTeXMLRef we use Getzler ’ s results on the cohomology of LaTeXMLMath to show the class LaTeXMLMath is nontautological even on the interior .
LaTeXMLMath Finally , although the diagonal loci were used in our deductions of the above results , we could not conclude the diagonals themselves were nontautological .
We show a diagonal locus is nontautological in at least one case .
Let LaTeXMLMath denote the boundary inclusion , LaTeXMLEquation .
Let LaTeXMLMath denote the class of the diagonal in LaTeXMLMath .
The push-forward LaTeXMLMath is not a tautological class : LaTeXMLEquation .
While it seems likely the image of the diagonal by ( LaTeXMLRef ) in LaTeXMLMath is not tautological , we do not have a proof .
To our knowledge there are still no ( proven ) examples of nontautological classes on LaTeXMLMath .
While the methods of our paper could perhaps be used to find such a class , in particular the class of Theorem 1 may be nontautological when restricted to LaTeXMLMath , our techniques are unlikely to produce nontautological classes of low codimension .
As the tautological ring of LaTeXMLMath vanishes in codimension LaTeXMLMath and higher , the question of nontautological classes on LaTeXMLMath of codimension less than LaTeXMLMath is particularly interesting .
The authors thank C. Faber and E. Getzler for several conversations about the cohomology of LaTeXMLMath , and R. Vakil for discussions about hunting for nontautological classes .
T. G. was partially supported by an NSF post-doctoral fellowship .
R. P. was partially supported by DMS-0071473 and fellowships from the Sloan and Packard foundations .
Part of the research reported here was pursued during a visit by R. P. to the Royal Technical Institute in Stockholm .
For the proofs of Theorems LaTeXMLRef and LaTeXMLRef we will require certain moduli spaces of double covers .
Choose LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
We let LaTeXMLMath denote the ( open ) space parameterizing double covers , LaTeXMLEquation of curves of genus LaTeXMLMath and LaTeXMLMath respectively together with an ordering of the branch points of the morphism LaTeXMLMath .
The space LaTeXMLMath is a finite étale cover of LaTeXMLMath where LaTeXMLMath is the number of branch points of LaTeXMLMath .
The map , LaTeXMLEquation is simply defined by LaTeXMLEquation where LaTeXMLMath are the ordered branch points .
There is a natural compactification by admissible double covers , LaTeXMLEquation over LaTeXMLMath .
An admissible double cover LaTeXMLMath of a stable curve is branched over the marked points and possibly the nodes .
Over the nodes of the target , the map LaTeXMLMath is either étale or étale locally of the form , LaTeXMLEquation .
LaTeXMLEquation By construction , the space LaTeXMLMath is equipped with maps to both LaTeXMLMath and LaTeXMLMath .
The latter map involves a stabilization process , since the source curve of an admissible covering need not be stable .
We will require additionally pointed moduli spaces of admissible covers , LaTeXMLMath .
These pointed spaces are finite covers of LaTeXMLMath which parametrizes admissible double covers of a LaTeXMLMath pointed nodal curve of genus LaTeXMLMath by a curve of genus LaTeXMLMath , with the ramification over the first LaTeXMLMath marked points and possibly the nodes of the target curve together with an ordering of the fibers of the last LaTeXMLMath marked points .
The pointed spaces are equipped with natural morphisms to LaTeXMLMath and LaTeXMLMath .
For the latter map , we adopt the ordering convention that the two points in the fiber over the LaTeXMLMath th marked point of the target curve have markings LaTeXMLMath and LaTeXMLMath on the source .
Essentially , we require only one fact about the moduli spaces of admissible covers : LaTeXMLMath is dense ( and similarly for the open subset LaTeXMLMath ) .
Over the complex numbers , the density is easily proven analytically .
One can simply locally smooth the double cover of a small neighborhood of the node , and glue the result together with the restriction of the original cover away from the node .
The local description of our double covers ensures that they can be smoothed locally .
A treatment of the theory of admissible covers can be found in [ HaM ] .
Let LaTeXMLMath be an odd positive integer , and let LaTeXMLMath .
Consider the morphism LaTeXMLEquation .
The image cycle , LaTeXMLEquation consists of those curves of genus LaTeXMLMath which are admissible double covers of a curve of genus LaTeXMLMath .
Equivalently , LaTeXMLMath is the closure of the set of nonsingular curves of genus LaTeXMLMath which admit a degree 2 map to a nonsingular curve of genus LaTeXMLMath .
We want to apply Proposition LaTeXMLRef to conclude that LaTeXMLMath is not tautological .
We will look at the pullback of LaTeXMLMath under the gluing map LaTeXMLEquation .
LaTeXMLMath for some positive constant LaTeXMLMath .
Proof .
We first prove LaTeXMLMath .
Let LaTeXMLMath .
We will construct an admissible double cover with target LaTeXMLMath union a a rational tail glued at LaTeXMLMath carrying the two branch markings .
A double cover is given by two disjoint copies of LaTeXMLMath joined by a rational curve with a degree 2 mapping to the rational tail of the target branched over the two markings .
Under stabilization , the domain is mapped to the diagonal point LaTeXMLEquation .
An easy count shows LaTeXMLMath is an irreducible component of LaTeXMLMath of expected dimension .
To prove the Lemma , we need only show LaTeXMLMath .
Suppose there were another irreducible component LaTeXMLMath .
Let LaTeXMLEquation be an admissible double cover corresponding to a general point of LaTeXMLMath .
Then LaTeXMLMath may be expressed as a union of two curves of arithmetic genus LaTeXMLMath joined at a single node .
The chosen node of LaTeXMLMath must map to a node of LaTeXMLMath .
Since the space of admissible coverings is a finite cover of LaTeXMLMath , the preimage of the locus of curves with 2 or more nodes is not a divisor .
Hence , we conclude LaTeXMLMath has exactly 1 node .
The node of LaTeXMLMath must be disconnecting since there are no reducible admissible double covers of an irreducible curve with branch points .
We write LaTeXMLMath .
Since LaTeXMLMath is odd , we may assume , without loss of generality , LaTeXMLMath has genus greater than LaTeXMLMath .
Since LaTeXMLMath has 1 node , LaTeXMLMath must have either 1 or 2 nodes .
Since any cover of LaTeXMLMath by a curve of genus LaTeXMLMath must be unramified , LaTeXMLMath can not have exactly 1 node .
The domain LaTeXMLMath must therefore have 2 nodes lying over the node of LaTeXMLMath .
If the induced cover of LaTeXMLMath were connected , the neither node of LaTeXMLMath could be disconnecting Hence , the cover of LaTeXMLMath must be disconnected .
Then , each component of the cover of LaTeXMLMath must map isomorphically to LaTeXMLMath .
The cover of LaTeXMLMath must be connected of genus 0 in order for the assumed decomposition of LaTeXMLMath into curves of arithmetic genus LaTeXMLMath to exist .
Therefore , we find we are in the component LaTeXMLMath of LaTeXMLMath .
∎ Pikaart LaTeXMLCite has shown for all sufficiently large values of LaTeXMLMath , LaTeXMLEquation .
Hence , the diagonal in LaTeXMLMath does not have tautological Künneth decomposition .
By Proposition LaTeXMLRef , the proof of Theorem LaTeXMLRef is complete .
The argument for the nontautological cycle on LaTeXMLMath is similar .
Let LaTeXMLMath be the image of LaTeXMLMath in LaTeXMLMath .
Consider the boundary stratum , LaTeXMLEquation obtained by attaching at the last point on each marked curve , and numbering the markings of the glued curve in order with the first 10 markings from the first factor and the last 10 from the second factor .
LaTeXMLMath for some positive constant LaTeXMLMath .
The proof of the Lemma is essentially identical to the proof of Lemma LaTeXMLRef above .
Theorem LaTeXMLRef is then a consequence of Proposition LaTeXMLRef and the existence of odd cohomology on LaTeXMLMath .
To deduce Theorem LaTeXMLRef from Theorem LaTeXMLRef , we will need the following results announced by Getzler : LaTeXMLEquation and for all odd LaTeXMLMath , LaTeXMLEquation .
The statement ( LaTeXMLRef ) is equivalent to the generation of even cohomology by the classes of boundary strata for LaTeXMLMath .
Actually , we require the following consequences of Getzler ’ s results .
Three properties for LaTeXMLMath : Every algebraic cycle on LaTeXMLMath of complex codimension less than 11 is homologous to a tautological class .
Every algebraic cycle on LaTeXMLMath is homologous to a tautological class for LaTeXMLMath .
Every algebraic cycle on LaTeXMLMath is homologous to a tautological class .
Proof .
Let LaTeXMLMath be an algebraic cycle on LaTeXMLMath of complex codimension less than 11 .
Consider the Künneth decomposition of LaTeXMLMath .
There can be no odd terms by ( LaTeXMLRef ) .
Thus , by ( LaTeXMLRef ) , we can write LaTeXMLMath as a sum of products of tautological classes proving ( i ) .
By ( LaTeXMLRef ) and Poincaré duality , all the cohomology of LaTeXMLMath is tautological when LaTeXMLMath .
By Keel ’ s results , all the cohomology of LaTeXMLMath is tautological .
Hence , in the Künneth decomposition of our cycle in parts ( ii ) and ( iii ) , none of the odd cohomology of LaTeXMLMath can appear .
∎ Consider the class LaTeXMLMath on LaTeXMLMath constructed in Theorem LaTeXMLRef .
We claim the image of LaTeXMLMath in LaTeXMLMath is not tautological .
The argument is by contradiction .
Suppose the image is tautological .
There must exist a collection of cycles LaTeXMLMath of codimension equal to 11 in LaTeXMLMath , supported on boundary strata , for which LaTeXMLMath is tautological .
Then , LaTeXMLMath is not homologous to a tautological class when intersected with LaTeXMLMath .
By Lemma LaTeXMLRef part ( i ) , if any cycle LaTeXMLMath is supported on the image stratum of LaTeXMLMath , then LaTeXMLMath is homologous to a tautological class ( since the codimension of LaTeXMLMath is less than 11 in the divisor ) .
We discard all LaTeXMLMath contained in the image of LaTeXMLMath .
Let LaTeXMLMath be the union of boundary divisors supporting the remaining LaTeXMLMath .
The sum of the remaining LaTeXMLMath is homologically nontautological when pushed into LaTeXMLMath and restricted to LaTeXMLMath .
However , it is clear the push-pull will produce an algebraic cycle class supported on LaTeXMLEquation .
Since LaTeXMLMath does not contain the image of LaTeXMLMath , the locus ( LaTeXMLRef ) is contained in boundary strata which either have a genus 1 factor with fewer than 11 points , or have less than two genus 1 factors .
Parts ( ii ) and ( iii ) of Lemma LaTeXMLRef show that there are no homologically nontautological classes supported on these loci .
The contradiction completes the proof of Theorem LaTeXMLRef .
We will require several properties of the odd cohomology of the moduli spaces LaTeXMLMath for the proof of Theorem LaTeXMLRef .
The first is a well-known specialization of ( LaTeXMLRef ) .
The odd cohomology groups of LaTeXMLMath vanish in case LaTeXMLMath .
Cusp forms of weight LaTeXMLMath may be used to construct cohomology classes in LaTeXMLMath .
The discriminant form LaTeXMLMath , the unique cusp form of weight 12 , yields a canonical non-zero element LaTeXMLMath .
The odd cohomology of LaTeXMLMath is concentrated in LaTeXMLEquation .
LaTeXMLEquation Moreover , the LaTeXMLMath -module in both cases is the alternating representation .
By the LaTeXMLMath -module identification , the class LaTeXMLMath is not LaTeXMLMath -invariant .
Let LaTeXMLMath denote the uniquely defined Poincaré dual class to LaTeXMLMath : LaTeXMLEquation .
Both Proposition LaTeXMLRef and LaTeXMLRef are well-known .
Proofs can be found , for example , in [ G2 ] where the LaTeXMLMath -equivariant Hodge polynomials of LaTeXMLMath are calculated for all LaTeXMLMath .
We will need a dimension calculation in the LaTeXMLMath pointed case [ G2 ] : The dimension of LaTeXMLMath is 11 .
In fact , the odd cohomology of LaTeXMLMath is concentrated in LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath ( all 11 dimensional ) .
Let LaTeXMLMath .
For each index LaTeXMLMath , let LaTeXMLEquation denote the forgetful map .
Since we consider LaTeXMLEquation as an ordered set , the last isomorphism above is canonical .
Define the classes LaTeXMLMath and LaTeXMLMath by : LaTeXMLEquation .
LaTeXMLEquation For each index LaTeXMLMath , let LaTeXMLMath be the map defined by the inclusion LaTeXMLEquation .
Here , an LaTeXMLMath pointed curve is obtained from an LaTeXMLMath pointed curve by attaching a rational tail containing the markings LaTeXMLMath to the point LaTeXMLMath of the latter curve .
The map LaTeXMLMath is simply the inclusion of the boundary divisor LaTeXMLMath with genus splitting LaTeXMLMath and point splitting LaTeXMLEquation .
Define the classes LaTeXMLMath and LaTeXMLMath by : LaTeXMLEquation .
LaTeXMLEquation Here , the cohomological push-forward is defined by the equivalent equalities LaTeXMLEquation .
LaTeXMLEquation The sets LaTeXMLMath and LaTeXMLMath form a pair of Poincaré dual bases of LaTeXMLMath and LaTeXMLMath .
Proof .
By the dimension result of Proposition LaTeXMLRef , it suffices to prove LaTeXMLEquation .
By definition of the cohomological push-forward , LaTeXMLEquation .
The first equality in ( LaTeXMLRef ) is true exactly ( not up to sign ) by the precise ordering conventions used .
The vanishing of ( LaTeXMLRef ) when LaTeXMLMath is a direct consequence of Proposition LaTeXMLRef .
We find LaTeXMLEquation .
The composition LaTeXMLMath has image isomorphic to LaTeXMLMath .
As the image supports no odd cohomology , the integral ( LaTeXMLRef ) vanishes .
∎ An identical argument proves the duality result for the classes LaTeXMLMath and LaTeXMLMath .
The sets LaTeXMLMath and LaTeXMLMath form a pair of Poincaré dual bases of LaTeXMLMath and LaTeXMLMath .
The intersection form is : LaTeXMLEquation .
Let LaTeXMLMath denote the cotangent line class at the point LaTeXMLMath .
Multiplication by LaTeXMLMath defines linear maps : LaTeXMLEquation .
LaTeXMLEquation These maps are completely determined by the following result .
For all LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Proof .
Consider the morphism LaTeXMLMath .
A standard comparison result governing the cotangent line class is : LaTeXMLEquation where LaTeXMLMath denotes the pull-back of the cotangent class on LaTeXMLMath .
We then find : LaTeXMLEquation .
Since LaTeXMLMath has odd cohomology only in degree 11 , the first summand of ( LaTeXMLRef ) vanishes .
The second summand is exactly equal to LaTeXMLMath ( using the ordering conventions ) .
We conclude LaTeXMLMath .
The derivation of LaTeXMLMath is identical .
∎ Set LaTeXMLMath .
Consider the boundary map LaTeXMLEquation defined by attaching LaTeXMLMath to LaTeXMLMath ( and ordering the markings arbitrarily ) .
Define LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath is the diagonal subvariety of LaTeXMLMath ( under the canonical isomorphism LaTeXMLMath ) .
Here , LaTeXMLMath is easily seen to define an embedding .
The normal bundle to LaTeXMLMath in LaTeXMLMath has top Chern class LaTeXMLMath .
By the self-intersection formula , LaTeXMLEquation .
Let LaTeXMLMath be a basis of LaTeXMLMath .
Let LaTeXMLMath denote the corresponding basis of LaTeXMLMath .
The Künneth decomposition of LaTeXMLMath is determined by : LaTeXMLEquation where LaTeXMLEquation .
In particular , if LaTeXMLMath is a self dual basis , then LaTeXMLEquation where LaTeXMLMath are the degrees of LaTeXMLMath and LaTeXMLMath respectively .
We are interested in the Künneth components of LaTeXMLMath of odd type — that is Künneth components lying in LaTeXMLEquation .
By Proposition LaTeXMLRef and LaTeXMLRef , the odd type summands of LaTeXMLMath are : LaTeXMLEquation .
Hence , the odd summands of LaTeXMLMath are : LaTeXMLEquation .
By Proposition LaTeXMLRef , we find the odd summands of LaTeXMLMath equal : LaTeXMLEquation .
As the odd summands ( LaTeXMLRef ) do not vanish , LaTeXMLEquation by Proposition LaTeXMLRef .
The proof of Theorem LaTeXMLRef is complete .
∎ The boundary strata of the moduli space of curves correspond to stable graphs LaTeXMLEquation satisfying the following properties : LaTeXMLMath is a vertex set with a genus function LaTeXMLMath , LaTeXMLMath is an half-edge set equipped with a vertex assignment LaTeXMLMath and fixed point free involution LaTeXMLMath , LaTeXMLMath , the edge set , is defined by the orbits of LaTeXMLMath in LaTeXMLMath ( self-edges at vertices are permitted ) , LaTeXMLMath define a connected graph , LaTeXMLMath is a set of numbered legs attached to the vertices , For each vertex LaTeXMLMath , the stability condition holds : LaTeXMLEquation where LaTeXMLMath is the valence of LaTeXMLMath at LaTeXMLMath including both half-edges and legs .
The genus of LaTeXMLMath is defined by : LaTeXMLEquation .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath denote the cardinalities of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath respectively .
A boundary stratum of LaTeXMLMath naturally determines a stable graph of genus LaTeXMLMath with LaTeXMLMath legs by considering the dual graph of a generic pointed curve parameterized by the stratum .
Let LaTeXMLMath be a stable graph .
Define the moduli space LaTeXMLMath by the product : LaTeXMLEquation .
Let LaTeXMLMath denote the projection from LaTeXMLMath to LaTeXMLMath associated to the vertex LaTeXMLMath .
There is a canonical morphism LaTeXMLMath with image equal to the boundary stratum associated to the graph LaTeXMLMath .
To construct LaTeXMLMath , a family of stable pointed curves over LaTeXMLMath is required .
Such a family is easily defined by attaching the pull-backs of the universal families over each of the LaTeXMLMath along the sections corresponding to half-edges .
Our main goal in the Appendix is to understand the fiber product : LaTeXMLEquation .
To this end , we will require additional terminology .
A stable graph LaTeXMLMath is a specialization of a stable graph LaTeXMLMath if LaTeXMLMath is obtained from LaTeXMLMath by replacing each vertex LaTeXMLMath of LaTeXMLMath with a stable graph of genus LaTeXMLMath with LaTeXMLMath legs .
Specialization of graphs corresponds to specialization of stable curves .
There is a subtlety involved in the notion of specialization : a given graph LaTeXMLMath may arise as a specialization of LaTeXMLMath in more than one way .
An LaTeXMLMath -graph structure on a stable graph LaTeXMLMath is a choice of subgraphs of LaTeXMLMath in bijective correspondence with LaTeXMLMath such that LaTeXMLMath can be constructed by replacing each vertex of LaTeXMLMath by the corresponding subgraph .
If LaTeXMLMath has an LaTeXMLMath -structure , then every half-edge of LaTeXMLMath corresponds to a particular half-edge of LaTeXMLMath , and every vertex of LaTeXMLMath is associated to a particular vertex of LaTeXMLMath .
A point of LaTeXMLMath is given by a stable curve together with a choice of LaTeXMLMath -structure on its dual graph .
In fact , we can naturally identify the stack LaTeXMLMath with a stack defined in terms of LaTeXMLMath -structures .
This identification will be useful for the analyzing the fiber products of strata .
Define a stable LaTeXMLMath -curve over a connected base LaTeXMLMath , LaTeXMLEquation to be a stable LaTeXMLMath -pointed curve of genus LaTeXMLMath over LaTeXMLMath together with : LaTeXMLMath sections LaTeXMLMath of LaTeXMLMath with image in the singular locus of LaTeXMLMath , LaTeXMLMath sections of the normalization of LaTeXMLMath along the sections LaTeXMLMath corresponding to the nodal separations , LaTeXMLMath disjoint LaTeXMLMath -relative components of LaTeXMLMath whose union is LaTeXMLMath , An isomorphism between LaTeXMLMath and the canonical stable graph defined by the dual graph of the LaTeXMLMath LaTeXMLMath -relative components and LaTeXMLMath sections of the normalization ( corresponding to half-edges ) .
Here , a LaTeXMLMath -relative component is a connected component of of LaTeXMLMath which remains connected upon pullback under an arbitrary morphism of connected schemes LaTeXMLMath .
The data of a stable LaTeXMLMath -curve can be pulled back under any morphism of base schemes .
After pull-back to a geometric point , an LaTeXMLMath -curve is exactly an LaTeXMLMath -structure on the dual graph of the corresponding curve .
A stack LaTeXMLMath of curves with LaTeXMLMath -structure morphisms and respecting the LaTeXMLMath -structure may be defined .
However , we find the following result .
There is a natural isomorphism between LaTeXMLMath and LaTeXMLMath .
Proof .
A natural morphism from LaTeXMLMath to LaTeXMLMath is obtained by assigning the canonical LaTeXMLMath -structure to the universal curve over LaTeXMLMath .
In the other direction , given an LaTeXMLMath -valued point of LaTeXMLMath , we naturally obtain a collection of LaTeXMLMath stable curves by analyzing the LaTeXMLMath -relative components of LaTeXMLMath normalized at the LaTeXMLMath nodes .
Since we have a bijection between these curves and LaTeXMLMath , and a bijection between the new markings and the LaTeXMLMath sections , we obtain an LaTeXMLMath -valued point of LaTeXMLMath .
This correspondence induces a bijection on the space of morphisms between corresponding objects .
∎ By definition , an LaTeXMLMath -valued point of LaTeXMLMath is an LaTeXMLMath -valued point of LaTeXMLMath , an LaTeXMLMath -valued point of LaTeXMLMath , and a choice of isomorphism between the two pull-backs of the universal curve over LaTeXMLMath under the boundary inclusions .
If LaTeXMLMath is LaTeXMLMath , we find the dual graph LaTeXMLMath of the curve over LaTeXMLMath defined by the map to LaTeXMLMath is naturally equipped with both an LaTeXMLMath -structure and a LaTeXMLMath -structure .
Conversely , given a curve LaTeXMLMath together with two such structures on the dual graph , we naturally obtain a point of LaTeXMLMath .
A graph LaTeXMLMath equipped with both LaTeXMLMath and LaTeXMLMath -structures will be called an LaTeXMLMath -graph .
An LaTeXMLMath -graph LaTeXMLMath is generic if every half-edge of LaTeXMLMath corresponds to a half-edge of LaTeXMLMath or a half-edge of LaTeXMLMath .
The irreducible components of LaTeXMLMath will correspond to generic LaTeXMLMath -graphs .
A graph with an LaTeXMLMath -structure is canonically a specialization of a unique generic LaTeXMLMath -graph : the generic graph is obtained by contracting all those edges which do not correspond to edges of LaTeXMLMath or LaTeXMLMath .
Associated to an LaTeXMLMath -graph LaTeXMLMath , we obtain a moduli space LaTeXMLMath which naturally maps to LaTeXMLMath .
The moduli space may be described either as LaTeXMLMath or in stack terms analogous to the definition of LaTeXMLMath above ( the stack does not depend on the LaTeXMLMath -structure on LaTeXMLMath , although the map to LaTeXMLMath does ) .
We find the following result .
There is a canonical isomorphism between LaTeXMLMath and the disjoint union of LaTeXMLMath over all generic LaTeXMLMath -graphs LaTeXMLMath .
Proof .
It will suffice to identify the categories involved over connected base schemes LaTeXMLMath .
We will give the morphisms in both directions .
If LaTeXMLMath is an LaTeXMLMath -graph , then we clearly have a morphism from LaTeXMLMath to both LaTeXMLMath and LaTeXMLMath , and a choice of isomorphism between the induced maps to LaTeXMLMath .
In the other direction , suppose data corresponding to LaTeXMLMath is given over LaTeXMLMath .
In particular , we have a stable curve over LaTeXMLMath : LaTeXMLEquation .
Consider the LaTeXMLMath -fiber over a geometric point of LaTeXMLMath .
The LaTeXMLMath -fiber has a dual graph which is equipped with an LaTeXMLMath -structure by virtue of the maps to LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the unique generic LaTeXMLMath -graph which specializes to the LaTeXMLMath -structure found at the geometric point .
There is a canonical LaTeXMLMath -structure on LaTeXMLMath .
The half-edges of LaTeXMLMath are already naturally identified with half-edges of the graphs LaTeXMLMath or LaTeXMLMath , and since LaTeXMLMath has LaTeXMLMath and LaTeXMLMath -structures , LaTeXMLMath is equipped with sections associated to all of the half-edges .
The LaTeXMLMath -structure is constant on LaTeXMLMath because of the connectedness of LaTeXMLMath .
The morphisms in the two categories are the same by a straightforward check .
However , it is important to note that an automorphism of an object of LaTeXMLMath must induce a trivial automorphism of the graph LaTeXMLMath , because each half-edge corresponds to an edge of either LaTeXMLMath or LaTeXMLMath .
∎ The pull-backs of tautological classes to the boundary may now be explicitly determined .
The basic calculation is the pull-back of the fundamental class of one boundary stratum to another .
In terms of the diagram of Section LaTeXMLRef , we want to compute LaTeXMLMath .
As we have identified LaTeXMLMath explicitly as a smooth stack , the pull-back will be straightforward to compute .
The intersection product is a sum of contributions of each component of LaTeXMLMath , and each contribution is the Euler class of an excess bundle on the component .
The components of LaTeXMLMath have been identified in Proposition LaTeXMLRef .
Let LaTeXMLMath be a generic LaTeXMLMath -graph , and let LaTeXMLMath be the corresponding component of LaTeXMLMath .
The excess bundle is easily identified on LaTeXMLMath .
First , we observe that the normal bundle to LaTeXMLMath naturally splits as a copy of LaTeXMLMath line bundles .
Let the edge LaTeXMLMath be the join of the distinct half-edges LaTeXMLMath incident to the vertices LaTeXMLMath ( which may coincide ) .
The line bundle associated to LaTeXMLMath is LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the tangent lines at LaTeXMLMath and LaTeXMLMath of the factors LaTeXMLMath and LaTeXMLMath respectively .
The normal bundle to LaTeXMLMath in LaTeXMLMath is a sum of the analogous line bundles for those edges of LaTeXMLMath which do not correspond to edges of LaTeXMLMath .
Precisely the same situation holds with respect to LaTeXMLMath .
We can conclude that the excess normal bundle of LaTeXMLMath thought of as a component of LaTeXMLMath is exactly the sum of the line bundles corresponding to those edges of LaTeXMLMath which correspond to edges of both LaTeXMLMath and LaTeXMLMath .
We have deduced the following formula : LaTeXMLEquation .
The sum is over all generic LaTeXMLMath -graphs LaTeXMLMath .
The product is over all edges LaTeXMLMath of LaTeXMLMath which come from both an edge of LaTeXMLMath and an edge of LaTeXMLMath , and LaTeXMLMath are the vertices joined by LaTeXMLMath .
The morphism LaTeXMLMath denotes the natural map from LaTeXMLMath to LaTeXMLMath .
Formula ( LaTeXMLRef ) yields an explicit tautological Künneth decomposition of the pull-back class since the morphism LaTeXMLMath is simply the product of various boundary strata maps over the factors of LaTeXMLMath .
We will compute a simple example to illustrate the formula .
Consider the boundary divisor LaTeXMLMath in LaTeXMLMath corresponding to the morphism LaTeXMLEquation .
The graph LaTeXMLMath of LaTeXMLMath has one vertex of genus LaTeXMLMath and one self-edge .
We will compute the self intersection of the stratum : LaTeXMLMath .
We first write down all generic LaTeXMLMath -graphs .
They are LaTeXMLMath itself with the obvious LaTeXMLMath -structure , and then one graph for each integer from LaTeXMLMath to LaTeXMLMath .
LaTeXMLMath is the graph with one vertex of genus LaTeXMLMath and two loops .
LaTeXMLMath has two distinct isomorphism classes of LaTeXMLMath -structures , but only one of them is generic : the LaTeXMLMath -structure where the edge contracted for the first LaTeXMLMath -structure is different from the edge contracted for the second LaTeXMLMath -structure .
Similarly , LaTeXMLMath is the graph with a vertex of genus LaTeXMLMath and another vertex of genus LaTeXMLMath connected to each other by two edges .
The unique generic LaTeXMLMath -structure is obtained by contracting a different edge for the two LaTeXMLMath -structures .
Applying formula ( LaTeXMLRef ) , we find : LaTeXMLEquation with hopefully evident notations .
Notice the boundary strata corresponding to LaTeXMLMath do not appear in the above formula because the corresponding dual graphs do not admit a generic LaTeXMLMath -structure .
In more geometric terms , these strata do not contribute an extra term because they have only one non-disconnecting node .
We observe that our calculations easily generalize to computing pull-backs of arbitrary tautological classes to boundary strata .
Define the tautological LaTeXMLMath classes by : LaTeXMLEquation where LaTeXMLMath is the map forgetting the last marking LaTeXMLMath .
The first observation is the following result concerning the push-forwards of the LaTeXMLMath and LaTeXMLMath classes .
Let LaTeXMLMath be the map forgetting the last LaTeXMLMath points .
The LaTeXMLMath push-forward of any element of the subring of LaTeXMLMath generated by LaTeXMLEquation lies in the subring of LaTeXMLMath generated by LaTeXMLEquation .
A proof can be found in [ AC ] .
We can now describe a set of additive generators for LaTeXMLMath .
Let LaTeXMLMath be a stable graph of genus LaTeXMLMath with LaTeXMLMath legs .
For each vertex LaTeXMLMath of LaTeXMLMath , let LaTeXMLEquation be an arbitrary monomial in the cotangent line and LaTeXMLMath classes of the vertex moduli space .
LaTeXMLMath is generated additively by classes of the form LaTeXMLEquation .
Proof .
By the definition of LaTeXMLMath , the claimed generators lie in the tautological ring .
We first show the span of the generators is closed under the intersection product .
The closure follows from : the pull-back formula ( LaTeXMLRef ) for strata classes , the trivial pull-back formula for cotangent lines under boundary maps , the pull-back formula for LaTeXMLMath classes under boundary maps [ AC ] LaTeXMLEquation .
To prove the claimed generators span LaTeXMLMath , we must prove the system defined by the generators is closed under push-forward by the forgetting maps and the gluing maps .
Closure under push-forward by the forgetting maps is a consequence of Proposition LaTeXMLRef .
Closure under push-forward by the gluing maps is a trivial condition .
∎ LaTeXMLMath is a finite dimensional LaTeXMLMath -vector space .
Proof .
The set of stable graphs LaTeXMLMath for fixed LaTeXMLMath and LaTeXMLMath is finite , and there are only finitely many non-vanishing monomials LaTeXMLMath for each vertex LaTeXMLMath .
∎ Let LaTeXMLMath .
Let LaTeXMLMath be a stable graph .
Let LaTeXMLEquation .
Then , LaTeXMLMath has a tautological Künneth decomposition with respect to the product structure of LaTeXMLMath .
Proof .
The proposition follows from Proposition LaTeXMLRef together with the pull-back formulas .
The pull-back formulas for the three types of classes all yield tautological Künneth decompositions .
∎ Department of Mathematics Harvard University Cambridge , MA 02138 graber @ @ math.harvard.edu Department of Mathematics California Institute of Technology Pasadena , CA 91125 rahulp @ @ cco.caltech.edu
The two–parametric quantum superalgebra LaTeXMLMath and its representations are considered .
All finite–dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and non–typical ones according to a proposition proved in the present paper .
This proposition is a nontrivial deformation from the one for the classical superalgebra LaTeXMLMath , unlike the case of one–parametric deformations .
IRREDUCIBLE REPRESENTATIONS OF U LaTeXMLMath [ gl ( 2/2 ) ] Nguyen Anh Ky [ 2mm ] Institute of Physics , P.O .
Box 429 , Bo Ho , Hanoi 10000 , Vietnam [ 1mm ] PACS numbers : 02.20Tw , 11.30Pb .
MSC–class .
: 81R50 ; 17A70 .
I .
Introduction The quantum groups LaTeXMLCite – LaTeXMLCite were introduced in 80 ’ s as a result of the study on quantum integrable systems and Yang–Baxter equations ( YBE ’ s ) LaTeXMLCite .
It turns out that they are related to unrelated , at first sight , areas of both physics and mathematics and therefore , have been intensively investigated in various aspects including their applications ( see Refs .
LaTeXMLCite – LaTeXMLCite and references therein ) .
For applications of quantum groups , as in the non–deformed cases , we often need their explicit representations , in particular , the finite–dimensional ones which in many cases are connected with rational and trigonometric solutions of the quantum YBE ’ s LaTeXMLCite – LaTeXMLCite .
However , in spite of efforts and remarkable results in this direction the problem of investigating and constructing explicit representations of quantum groups , especially those for quantum superalgebras , is still far from being satisfactorily solved .
Even in the case of one–parametric quantum superalgebras , explicit representations are mainly known for quantum Lie superalgebras of lower ranks and of particular types like LaTeXMLMath and LaTeXMLMath ( Refs .
LaTeXMLCite ) .
So far , finite–dimensional representations of some bigger quantum superalgebras such as LaTeXMLMath and LaTeXMLMath with LaTeXMLMath have been considered but have not been explicitly constructed ( see , for example , LaTeXMLCite ) .
At the moment , detailed results in this aspect are known only for the cases with both LaTeXMLMath considered in LaTeXMLCite , while for LaTeXMLMath with arbitrary LaTeXMLMath and LaTeXMLMath not all finite–dimensional representations but only a , although big , class of representations called essentially typical is known LaTeXMLCite .
As far as the multi–parametric deformations ( first considered in LaTeXMLCite ) are concerned , this area is even less covered and results are much poorer .
Some kinds of two–parametric deformations have been considered by a number of authors from different points of view ( see LaTeXMLCite , LaTeXMLCite and references therein ) but , to our knowledge , explicit representations are known and/or classified in a few lower rank cases such as LaTeXMLMath and LaTeXMLMath only LaTeXMLCite .
The latter two–parametric quantum superalgebra LaTeXMLMath was consistently introduced and investigated in LaTeXMLCite where all its finite–dimensional irreducible representations were explicitly constructed and classified at generic deformation parameters .
This LaTeXMLMath , however , is still a small quantum superalgebra which can be defined without the so–called extra–Serre defining relations LaTeXMLCite representing additional constraints on odd Chevalley generators in higher rank cases .
In order to include the extra–Serre relations on examination we introduced and considered a bigger two–parametric quantum superalgebra , namely LaTeXMLMath , and its representations LaTeXMLCite .
Another our motivation for considering this quantum superalgebra is that already in the non–deformed case , the superalgebras LaTeXMLMath , especially , their subalgebras LaTeXMLMath and LaTeXMLMath , have special properties ( in comparison with other LaTeXMLMath , LaTeXMLMath ) and , therefore , attract interest LaTeXMLCite .
Additionally , structures of two–parameter deformations investigated in LaTeXMLCite and here are , of course , richer than those of one–parameter deformations .
Every deformation parameter can be independently chosen to take a separate generic value ( including zero ) or to be a root of unity .
Combining the advantages of the previously developed methods LaTeXMLCite for LaTeXMLMath and LaTeXMLMath we described in LaTeXMLCite how to construct finite–dimensional representations of the two–parametric quantum Lie superalgebra LaTeXMLMath .
In this paper we consider when these representations constructed are irreducible .
It turns out that they can be classified again into typical and nontypical representations which , even at generic deformation parameters , however , are nontrivial deformations from the classical analogues LaTeXMLCite , unlike many cases of one-parametric deformations .
II .
The quantum superalgebra LaTeXMLMath The quantum superalgebra LaTeXMLMath as a two–parametric deformation of the universal enveloping algebra LaTeXMLMath of the Lie superalgebra LaTeXMLMath can be completely generated by the operators LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) called again Cartan–Chevalley generators subjects to the following ( defining ) relations LaTeXMLCite : a ) the super–commutation relations ( LaTeXMLMath ) : LaTeXMLMath = 0 , ( 1a ) LaTeXMLMath = LaTeXMLMath ( 1b ) LaTeXMLMath = LaTeXMLMath , ( 1c ) [ even generator , LaTeXMLMath ] = 0 , LaTeXMLMath , ( 1d ) LaTeXMLMath = LaTeXMLMath , ( 1e ) b ) the Serre–relations : LaTeXMLMath = LaTeXMLMath = 0 , ( 2a ) LaTeXMLMath = LaTeXMLMath = 0 , ( 2b ) LaTeXMLMath = LaTeXMLMath = LaTeXMLMath = LaTeXMLMath , ( 2c ) and c ) the extra–Serre relations : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ( with LaTeXMLMath and LaTeXMLMath explained later ) , LaTeXMLMath is a so-called LaTeXMLMath –deformation of LaTeXMLMath being a number or an operator and , finally , [ , } is a notation for the supercommutators .
Here , the operators LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and the operators composed in the following way LaTeXMLMath : = LaTeXMLMath LaTeXMLMath , ( 5a ) LaTeXMLMath : = LaTeXMLMath LaTeXMLMath ( 5b ) are defined as new generators , where LaTeXMLMath .
These generators , like LaTeXMLMath and LaTeXMLMath , are all odd and have vanishing squares .
The generators LaTeXMLMath , LaTeXMLMath , are two–parametric deformation analogues ( LaTeXMLMath –analogues ) of the Weyl generators LaTeXMLMath of the superalgebra LaTeXMLMath whose universal enveloping algebra LaTeXMLMath is a classical limit of LaTeXMLMath when LaTeXMLMath .
The so–called maximal–spin operator LaTeXMLMath ( or LaTeXMLMath ) is a constant within a finite–dimensional irreducible module ( fidirmod ) of a LaTeXMLMath ( defined below ) and are different for different LaTeXMLMath –fidirmods .
Therefore , commutators between these operators with the odd generators intertwining LaTeXMLMath –fidirmods take concrete forms on concrete basis vectors .
Other commutation relations between LaTeXMLMath follow from the relations ( 1 ) – ( 3 ) and the definitions ( 4 ) and ( 5 ) .
III .
Representations of LaTeXMLMath The subalgebra LaTeXMLMath is even and isomorphic to LaTeXMLMath which can be completely generated by LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , LaTeXMLEquation .
In order to distinguish two components LaTeXMLMath of LaTeXMLMath we set LaTeXMLEquation .
LaTeXMLEquation that is LaTeXMLEquation .
We see that every of the odd spaces LaTeXMLMath and LaTeXMLMath spanned on the positive and negative odd roots ( generators ) LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , respectively LaTeXMLEquation .
LaTeXMLEquation is a representation space of the even subalgebra LaTeXMLMath which , as seen from ( 1 ) – ( 2 ) , is a stability subalgebra of LaTeXMLMath .
Therefore , we can construct representations of LaTeXMLMath induced from some ( finite–dimensional irreducible , for example ) representations of LaTeXMLMath which are realized in some representation spaces ( modules ) LaTeXMLMath being tensor products of LaTeXMLMath –modules LaTeXMLMath and LaTeXMLMath –modules LaTeXMLMath LaTeXMLEquation where LaTeXMLMath ’ s are some signatures ( such as highest weights , respectively ) characterizing the modules ( highest weight modules , respectively ) .
Here LaTeXMLMath and LaTeXMLMath are referred to as the left and the right components of LaTeXMLMath , respectively , LaTeXMLEquation .
If we demand LaTeXMLEquation hence LaTeXMLEquation we turn the LaTeXMLMath –module LaTeXMLMath into a LaTeXMLMath –module where LaTeXMLEquation .
The LaTeXMLMath –module LaTeXMLMath induced from the LaTeXMLMath –module LaTeXMLMath is the factor–space LaTeXMLEquation which , of course , depends on LaTeXMLMath , where LaTeXMLEquation while LaTeXMLMath is the subspace LaTeXMLEquation .
Using the commutation relations ( 1 ) – ( 3 ) and the definitions ( 4 ) and ( 5 ) we can prove the an analogue of the Poincaré–Birkhoff–Witt theorem .
Consequently , a basis of LaTeXMLMath can be constituted by taking all the vectors of the form LaTeXMLEquation where LaTeXMLMath is a ( Gel ’ fand–Zetlin , for example ) basis of LaTeXMLMath .
This basis of LaTeXMLMath called the induced LaTeXMLMath –basis ( or simply , the induced basis ) , however , is not convenient for investigating the module structure of LaTeXMLMath .
It was the reason the so-called reduced basis was introduced LaTeXMLCite .
It is obvious that if the module LaTeXMLMath is finite–dimensional so is the module LaTeXMLMath .
In this case LaTeXMLMath can be characterized by a signature LaTeXMLMath and is decomposed into a direct sum of ( sixteen , at most ) LaTeXMLMath –fidirmod ’ s LaTeXMLMath of signatures LaTeXMLMath : LaTeXMLEquation .
Thus , the reduced basis of LaTeXMLMath is a union of the bases of all LaTeXMLMath ’ s which can be presented by the quasi–Gel ’ fand–Zetlin patterns LaTeXMLCite , corresponding to the branching rule LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath are complex numbers such that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath The second row LaTeXMLMath in ( 22 ) is fixed for a given LaTeXMLMath , as for LaTeXMLMath it takes the value of the first row LaTeXMLMath which is fixed for all LaTeXMLMath .
Now , a signature LaTeXMLMath of a LaTeXMLMath is identified with a second row , LaTeXMLEquation while the signature LaTeXMLMath single in the whole LaTeXMLMath ( i.e. , the same for all LaTeXMLMath ’ s ) is indentified with the first row , LaTeXMLEquation .
The actions of the generators LaTeXMLMath on the basis ( 22 ) are given in LaTeXMLCite or can be calculated by using the method explained there .
The basis vector ( 22 ) with LaTeXMLMath and LaTeXMLMath LaTeXMLEquation annihilated by LaTeXMLMath and LaTeXMLMath is , by definition , the highest weight vector of the submodule LaTeXMLMath .
For LaTeXMLMath the highest weight vector of the submodule LaTeXMLMath LaTeXMLEquation is , in addition , also annihilated by the odd genrator LaTeXMLMath and , therefore , simultaneously represents the highest weight vector of both LaTeXMLMath and LaTeXMLMath .
A monomial of the form LaTeXMLEquation would shift a subspace LaTeXMLMath to another subspace LaTeXMLMath with LaTeXMLMath .
So here we would call the former a higher ( weight ) subspace with respect to the latter called a lower ( weight ) subspace .
Proposition : The induced module LaTeXMLMath constructed is irreducible if and only if LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath The irreducible module LaTeXMLMath constructed with keeping the condition ( 26 ) valid is called typical , otherwise , we say it is an indecomposable module .
In the latter case , however , there always exists a maximal invariant submodule LaTeXMLMath ( of class LaTeXMLMath , LaTeXMLMath ) of LaTeXMLMath and the compliment to LaTeXMLMath subspace of LaTeXMLMath is not invariant under LaTeXMLMath transformations .
The representation carried in the factor module LaTeXMLMath is irreducible and called a nontypical representation of LaTeXMLMath .
It can be shown that these typical and nontypical representations contain all classes of finite–dimensional irreducible representations of LaTeXMLMath .
As every subspace LaTeXMLMath , k=0,1 , … , 15 , is close and already irreducible under the even subalgebra LaTeXMLMath , to see if LaTeXMLMath is an irreducible module of LaTeXMLMath it remains to consider the action of its odd generators only .
By construction ( see Eqs .
( 17 ) – ( 21 ) ) the module LaTeXMLMath is at least indecomposable since any its subspace LaTeXMLMath , LaTeXMLMath , including the lowest one LaTeXMLMath , can be always reached from higher subspaces LaTeXMLMath , LaTeXMLMath , including the highest one LaTeXMLMath , acted by the monomials LaTeXMLMath given in ( 25 ) .
Contrarily , the monomials LaTeXMLEquation send us to the opposite direction : from lower subspaces to higher ones .
Thus , the module LaTeXMLMath is irreducible if and only if LaTeXMLMath is reachable from the lowest subspace LaTeXMLMath under the action of the operators ( 27 ) .
The most optimal way to see that is to act on a vector of the subspace LaTeXMLMath by the monomial LaTeXMLMath , i.e. , the monomial ( 27 ) with all LaTeXMLMath ’ s = 1 but not less ( an action of a shorter monomial on LaTeXMLMath should not reach LaTeXMLMath ) .
Since LaTeXMLMath is an irreducible module of LaTeXMLMath , it is simplest but enough to consider when the highest weight vector LaTeXMLMath of LaTeXMLMath under the action of LaTeXMLMath reaches ( or we can say , returns to ) LaTeXMLMath .
In other words , the module LaTeXMLMath is irreducible if and only if the condition LaTeXMLEquation holds .
This condition in turn can be proved ( for LaTeXMLMath ) to be equivalent to the condition LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation which is nothing but ( 26 ) with LaTeXMLMath being eigenvalues of LaTeXMLMath on the highest weight vector LaTeXMLMath .
The proposition is , thus , proved .
IV .
Conclusion The two-parametric quantum superalgebra LaTeXMLMath was introduced in LaTeXMLCite .
Its representations constructed by the method described in LaTeXMLCite are either irreducible ( when the condition ( 26 ) is kept ) or indecomposable ( when the condition ( 26 ) is violated ) .
The irreducible representations in the former case are called typical .
In the case of indecomposable representations , however , irreducible representations can be always extracted .
One such irreducible representation called nontypical is simply a factor–representation in a factor–subspace of the original indecomposable module factorized by its maximal invariant subspace .
All the typical and nontypical representations constructed in such a way contain all classes of finite–dimensional irreducible representations of LaTeXMLMath .
For conclusion , let us emphasize that the condition ( 26 ) and the representations become more interesting at roots of unity but they , even at generic deformation parameters , are nontrivial deformations from the classical analogues LaTeXMLCite in the sense that the former can not be found from the latter by replacing in appropriate places the ordinary brackets with the quantum deformation ones , unlike many one–parametric cases .
Acknowledgements This work was supported in part by the National Research Programme for Natural Sciences of Vietnam under grant number KT – 04.1.2 .
In this paper we consider dynamical r-matrices over a nonabelian base .
There are two main results .
First , corresponding to a fat reductive decomposition of a Lie algebra LaTeXMLMath , we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations .
Second , we prove that a triangular dynamical r-matrix LaTeXMLMath corresponds to a Poisson manifold LaTeXMLMath .
A special type of quantizations of this Poisson manifold , called compatible star products in this paper , yields a generalized version of the quantum dynamical Yang-Baxter equation ( or Gervais-Neveu-Felder equation ) .
As a result , the quantization problem of a general dynamical r-matrix is proposed .
Recently , there has been growing interest in the so called quantum dynamical Yang-Baxter equation : LaTeXMLEquation .
This equation arises naturally from various contexts in mathematical physics .
It first appeared in the work of Gervais-Neveu in their study of quantum Liouville theory LaTeXMLCite .
Recently it reappeared in Felder ’ s work on the quantum Knizhnik-Zamolodchikov-Bernard equation LaTeXMLCite .
It also has been found to be connected with the quantum Caloger-Moser systems LaTeXMLCite .
As the quantum Yang-Baxter equation is connected with quantum groups , the quantum dynamical Yang-Baxter equation is known to be connected with elliptic quantum groups LaTeXMLCite , as well as with Hopf algebroids or quantum groupoids LaTeXMLCite .
The classical counterpart of the quantum dynamical Yang-Baxter equation was first considered by Felder LaTeXMLCite , and then studied by Etingof and Varchenko LaTeXMLCite .
This is the so called classical dynamical Yang-Baxter equation , and a solution to such an equation ( plus some other reasonable conditions ) is called a classical dynamical r-matrix .
More precisely , given a Lie algebra LaTeXMLMath over LaTeXMLMath ( or over LaTeXMLMath ) with an Abelian Lie subalgebra LaTeXMLMath , a classical dynamical r-matrix is a smooth ( or meromorphic ) function LaTeXMLMath satisfying the following conditions : ( zero weight condition ) LaTeXMLMath ; ( normal condition ) LaTeXMLMath , where LaTeXMLMath is a Casimir element ; ( classical dynamical Yang-Baxter equation LaTeXMLCite for the definition of a classical dynamical LaTeXMLMath -matrix in order to be consistent with the quantum dynamical Yang-Baxter equation ( LaTeXMLRef ) .
This differs in a sign from the one used in LaTeXMLCite . )
LaTeXMLEquation where LaTeXMLMath .
A fundamental question is whether a classical dynamical r-matrix is always quantizable .
There has appeared a lot of work in this direction , for example , see LaTeXMLCite .
In the triangular case ( i.e. , LaTeXMLMath is skew-symmetric : LaTeXMLMath ) , a general quantization scheme was developed by the author using the Fedosov method , which works for a vast class of dynamical r-matrices , called splittable triangular dynamical r-matrices LaTeXMLCite .
Recently , Etingof and Nikshych , using the vertex-IRF transformation method , proved the existence of quantizations for the so called completely degenerate triangular dynamical r-matrices LaTeXMLCite .
Interestingly , although the quantum dynamical Yang-Baxter equation in LaTeXMLCite only makes sense when the base Lie algebra LaTeXMLMath is Abelian , its classical counterpart admits an immediate generalization for any base Lie algebra LaTeXMLMath which is not necessarily Abelian .
Indeed , all one needs to do is to change the first condition ( i ) to : ( i ’ ) .
LaTeXMLMath is LaTeXMLMath -equivariant , where LaTeXMLMath acts on LaTeXMLMath by coadjoint action and on LaTeXMLMath by adjoint action .
There exist many examples of such classical dynamical r-matrices .
For instance , when LaTeXMLMath is a simple Lie algebra and LaTeXMLMath is a reductive Lie subalgebra containing the Cartan subalgebra , there is a classification due to Etingof-Varchenko LaTeXMLCite .
In particular , when LaTeXMLMath , an explicit formula was discovered by Alekseev and Meinrenken in their study of non-commutative Weil algebras LaTeXMLCite .
Later , this was generalized by Etingof and Schiffermann LaTeXMLCite to a more general context .
Moreover , under some regularity condition , they showed that the moduli space of dynamical r-matrices essentially consists of a single point once the initial value of the dynamical r-matrices is fixed .
A natural question arises as to what should be the quantum counterpart of these r-matrices .
And more generally , is any classical dynamical r-matrix ( with nonabelian base ) quantizable ?
A basic question is what the quantum dynamical Yang-Baxter equation should look like when LaTeXMLMath is nonabelian .
In this paper , as a toy model , we consider the special case of triangular dynamical r-matrices and their quantizations .
As in the Abelian case , these r-matrices naturally correspond to some invariant Poisson structures on LaTeXMLMath .
It is standard that quantizations of Poisson structures correspond to star products LaTeXMLCite .
The special form of the Poisson bracket relation on LaTeXMLMath suggests a specific form that their star products should take .
This leads to our definition of compatible star products .
The compatibility condition ( which , in this case , is just the associativity ) naturally leads to a quantum dynamical Yang-Baxter equation : Equation ( LaTeXMLRef ) .
As we shall see , this equation indeed resembles the usual quantum dynamical Yang-Baxter equation ( unsymmetrized version ) .
The only difference is that the usual pointwise multiplication on LaTeXMLMath is replaced by the PBW-star product , which is indeed the deformation quantization of the canonical Lie-Poisson structure on LaTeXMLMath .
Although Equation ( LaTeXMLRef ) is derived by considering triangular dynamical r-matrices , it makes perfect sense for non-triangular ones as well .
This naturally leads to our definition of quantization of dynamical r-matrices over an arbitrary base Lie subalgebra which is not necessary Abelian .
The problem is that such an equation only makes sense for LaTeXMLMath .
In the Abelian case , it appears that one may consider LaTeXMLMath valued in a deformed universal enveloping algebra LaTeXMLMath , but in most cases LaTeXMLMath is isomorphic to LaTeXMLMath as an algebra .
So Equation ( LaTeXMLRef ) , in a certain sense , is general enough to include all the interesting cases .
However , the physical meaning of this equation remains mysterious .
Another main result of the paper is to give a geometric construction of triangular dynamical r-matrices .
More precisely , we give an explicit construction of a triangular dynamical r-matrix from a fat reductive decomposition of a Lie algebra LaTeXMLMath ( see Section 2 for the definition ) .
This includes those examples of triangular dynamical r-matrices considered in LaTeXMLCite .
Our main purpose is to show that triangular dynamical r-matrices ( with nonabelian base ) do rise naturally from symplectic geometry .
This gives us another reason why it is important to consider their quantizations .
Discussion on this part occupies Section 2 .
Section 3 is devoted to the discussion of compatible star products , whose associativity leads to a “ twisted-cocycle ” condition .
In Section 4 , we will derive the quantum dynamical Yang-Baxter equation from this twisted-cocycle condition .
The last section contains some concluding remarks and open questions .
Finally , we note that in this paper , by a dynamical r-matrix , we always mean a dynamical r-matrix over a general base Lie subalgebra unless specified .
Also Lie algebras are normally assumed to be over LaTeXMLMath , although most results can be easily modified for complex Lie algebras .
Acknowledgments .
The author would like to thank Philip Boalch , Pavel Etingof , Boris Tsygan and David Vogan for fruitful discussions and comments .
He is especially grateful to Pavel Etingof for explaining the paper LaTeXMLCite , which inspired his interest on this topic .
He also wishes to thank Simone Gutt and Stefan Waldmann for providing him some useful references on star products of cotangent symplectic manifolds .
In this section , we will give a geometric construction of triangular dynamical r-matrices .
As we shall see , these r-matrices do arise naturally from symplectic geometry .
We will show some interesting examples , which include triangular dynamical r-matrices for simple Lie algebras constructed by Etingof-Varchenko LaTeXMLCite .
Below let us recall the definition of a classical triangular dynamical LaTeXMLMath -matrix .
Let LaTeXMLMath be a Lie algebra over LaTeXMLMath ( or LaTeXMLMath ) and LaTeXMLMath a Lie subalgebra .
A classical dynamical r-matrix LaTeXMLMath is said to be triangular if it is skew symmetric : LaTeXMLMath .
In other words , a classical triangular dynamical LaTeXMLMath -matrix is a smooth function ( or meromorphic function in the complex case ) LaTeXMLMath such that LaTeXMLMath is LaTeXMLMath -equivariant , where LaTeXMLMath acts on LaTeXMLMath by coadjoint action and acts on LaTeXMLMath by adjoint action .
LaTeXMLEquation where the bracket LaTeXMLMath refers to the Schouten type bracket : LaTeXMLMath induced from the Lie algebra bracket on LaTeXMLMath , LaTeXMLMath is a basis of LaTeXMLMath , and LaTeXMLMath its induced coordinate system on LaTeXMLMath .
The following proposition gives an alternative description of a classical triangular dynamical r-matrix .
A smooth function LaTeXMLMath is a triangular dynamical LaTeXMLMath -matrix iff LaTeXMLEquation is a Poisson tensor on LaTeXMLMath , where LaTeXMLMath denotes the standard Lie ( also known as Kirillov-Kostant ) Poisson tensor on the Lie algebra dual LaTeXMLMath , LaTeXMLMath is the left invariant vector field on LaTeXMLMath generated by LaTeXMLMath , and similarly LaTeXMLMath is the left invariant bivector field on LaTeXMLMath corresponding to LaTeXMLMath .
Proof .
Set LaTeXMLEquation .
Then LaTeXMLMath .
Note that , for any LaTeXMLMath , LaTeXMLMath is tangent to LaTeXMLMath , on which it is isomorphic to the standard Poisson ( symplectic ) structure on the cotangent bundle LaTeXMLMath ( see , e.g. , LaTeXMLCite ) .
Here LaTeXMLMath is identified with LaTeXMLMath ( hence with LaTeXMLMath ) via left translations .
It thus follows that LaTeXMLMath .
Therefore LaTeXMLEquation .
Now LaTeXMLEquation .
Hence LaTeXMLMath , where LaTeXMLEquation .
LaTeXMLEquation With respect to the natural bigrading on LaTeXMLMath , LaTeXMLMath and LaTeXMLMath correspond to the LaTeXMLMath and LaTeXMLMath -terms of LaTeXMLMath , respectively .
It thus follows that LaTeXMLMath iff LaTeXMLMath and LaTeXMLMath .
It is simple to see that LaTeXMLEquation .
Hence LaTeXMLMath is equivalent to Equation ( LaTeXMLRef ) .
To find out the meaning of LaTeXMLMath , let us write LaTeXMLMath .
A simple computation yields that LaTeXMLEquation .
Thus LaTeXMLMath is equivalent to LaTeXMLEquation which exactly means that LaTeXMLMath is LaTeXMLMath -equivariant .
This concludes the proof .
LaTeXMLMath Remark .
Note that LaTeXMLMath admits a left LaTeXMLMath -action and a right LaTeXMLMath -action defined as follows : LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation It is clear that the Poisson structure LaTeXMLMath is invariant under both actions .
And , in short , we will say that LaTeXMLMath is LaTeXMLMath -invariant .
A classical triangular dynamical LaTeXMLMath -matrix LaTeXMLMath is said to be non-degenerate if the corresponding Poisson structure LaTeXMLMath on LaTeXMLMath is non-degenerate , i.e. , symplectic .
In what follows , we will give a geometric construction of non-degenerate dynamical r-matrices .
To this end , let us first recall a useful construction of a symplectic manifold from a fat principal bundle LaTeXMLCite .
A principal bundle LaTeXMLMath with a connection is called fat on an open submanifold LaTeXMLMath if the scalar-valued forms LaTeXMLMath is non-degenerate on each horizontal space in LaTeXMLMath for LaTeXMLMath .
Here LaTeXMLMath is the curvature form , which is a tensorial form of type LaTeXMLMath on LaTeXMLMath ( i.e. , it is horizontal , LaTeXMLMath -valued , and LaTeXMLMath -equivariant ) .
Given a fat bundle LaTeXMLMath with a connection , one has a decomposition of the tangent bundle LaTeXMLMath .
We may identify LaTeXMLMath with a trivial bundle with fiber LaTeXMLMath .
Thus LaTeXMLEquation .
On the other hand , LaTeXMLMath .
Thus , by pulling back the canonical symplectic structure on LaTeXMLMath , one can equip LaTeXMLMath , hence LaTeXMLMath , an LaTeXMLMath -invariant presymplectic structure , where LaTeXMLMath acts on LaTeXMLMath by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is an open submanifold on which LaTeXMLMath is fat , then we obtain an LaTeXMLMath -invariant symplectic manifold LaTeXMLMath .
In fact , the presymplectic form LaTeXMLMath can be described explicitly .
Note that LaTeXMLMath admits a natural fibration with LaTeXMLMath being the fibers , and the connection on LaTeXMLMath induces a connection on this fiber bundle .
In other words , LaTeXMLMath is a symplectic fibration in the sense of Guillemin-Lerman-Sternberg LaTeXMLCite .
At any point LaTeXMLMath , the presymplectic form LaTeXMLMath can be described as follows : it restricts to the canonical two-form on the fiber ; the vertical subspace is LaTeXMLMath -orthogonal to the horizontal subspace ; and the horizontal subspace is isomorphic to the horizontal subspace of LaTeXMLMath and the restriction of LaTeXMLMath to this subspace is the two form LaTeXMLMath obtained by pairing the curvature form with LaTeXMLMath ( see Examples 2.2-2.3 in LaTeXMLCite ) .
Now assume that LaTeXMLEquation is a reductive decomposition of a Lie algebra LaTeXMLMath , i.e. , LaTeXMLMath is a Lie subalgebra and LaTeXMLMath is stable under the adjoint action of LaTeXMLMath : LaTeXMLMath .
By LaTeXMLMath , we denote a Lie group with Lie algebra LaTeXMLMath , and LaTeXMLMath the Lie subgroup corresponding to LaTeXMLMath .
It is standard LaTeXMLCite that the decomposition ( LaTeXMLRef ) induces a left LaTeXMLMath -invariant connection on the principal bundle LaTeXMLMath , where the curvature is given by LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath are arbitrary left invariant vector fields on LaTeXMLMath belonging to LaTeXMLMath .
A reductive decomposition LaTeXMLMath is said to be fat if the corresponding principal bundle LaTeXMLMath is fat on an open submanifold LaTeXMLMath .
As a consequence , a fat decomposition LaTeXMLMath gives rise to a LaTeXMLMath -invariant symplectic structure on LaTeXMLMath , where the symplectic structure is the restriction of the canonical symplectic form on LaTeXMLMath .
In other words , LaTeXMLMath is a symplectic submanifold of LaTeXMLMath .
Here the embedding LaTeXMLMath is given by the natural inclusion LaTeXMLMath , where LaTeXMLMath is the projection along the decomposition LaTeXMLMath .
Since the symplectic structure LaTeXMLMath on LaTeXMLMath is left invariant , in order to describe LaTeXMLMath explicitly , it suffices to specify it at a point LaTeXMLMath .
Now LaTeXMLMath .
Under this identification , we have LaTeXMLMath , where LaTeXMLMath is the canonical symplectic two-form on LaTeXMLMath at the point LaTeXMLMath , and LaTeXMLMath is given by LaTeXMLEquation .
Let LaTeXMLMath be the inverse of LaTeXMLMath , which always exists for LaTeXMLMath since LaTeXMLMath is assumed to be non-degenerate on LaTeXMLMath .
It thus follows that the Poisson structure on LaTeXMLMath is LaTeXMLEquation .
According to Proposition LaTeXMLRef , LaTeXMLMath is a non-degenerate triangular dynamical r-matrix .
Thus we have proved Assume that LaTeXMLMath is a reductive decomposition which is fat on an open submanifold LaTeXMLMath .
Then the dual of the linear map LaTeXMLMath defines a non-degenerate triangular dynamical LaTeXMLMath -matrix LaTeXMLMath , LaTeXMLMath .
Here LaTeXMLMath is identified with LaTeXMLMath using the non-degenerate bilinear form LaTeXMLMath .
It is often more useful to express LaTeXMLMath explicitly in terms of a basis .
To this end , let us choose a basis LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath .
By LaTeXMLMath we denote the inverse of the matrix LaTeXMLMath .
Then one has LaTeXMLEquation .
Remark .
After the completion of the first draft , we learned that a similar formula is also obtained independently by Etingof LaTeXMLCite .
Note that this dynamical r-matrix LaTeXMLMath is always singular at LaTeXMLMath .
To remove this singularity , one needs to make a shift of the dynamical parameter LaTeXMLMath .
It would be interesting to compare our formula with Theorem 3 in LaTeXMLCite .
We end this section with some examples .
Example 2.1 Let LaTeXMLMath be a simple Lie algebra over LaTeXMLMath and LaTeXMLMath a Cartan subalgebra .
Let LaTeXMLEquation be the root space decomposition , where LaTeXMLMath is the set of positive roots with respect to LaTeXMLMath .
Take LaTeXMLMath .
Then LaTeXMLMath is clearly a reductive decomposition .
Let LaTeXMLMath and LaTeXMLMath be dual vectors with respect to the Killing form : LaTeXMLMath .
For any LaTeXMLMath , set LaTeXMLMath .
It is then clear that LaTeXMLMath , whenever LaTeXMLMath ; and LaTeXMLEquation .
Therefore , from Theorem LaTeXMLRef and Equation ( LaTeXMLRef ) , it follows that LaTeXMLEquation is a non-degenerate triangular dynamical r-matrix , so we have recovered this standard example in LaTeXMLCite .
Example 2.2 As in the above example , let LaTeXMLMath be a simple Lie algebra over LaTeXMLMath with a fixed Cartan subalgebra LaTeXMLMath , and LaTeXMLMath a reductive Lie subalgebra containing LaTeXMLMath .
There is a subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLEquation .
Let LaTeXMLMath , and LaTeXMLMath , and denote by LaTeXMLMath the subspace of LaTeXMLMath : LaTeXMLEquation .
It is simple to see that LaTeXMLMath is indeed a fat reductive decomposition , and therefore induces a non-degenerate triangular dynamical r-matrix LaTeXMLMath .
To describe LaTeXMLMath explicitly , we note that the dual space LaTeXMLMath admits a natural decomposition LaTeXMLEquation .
Hence any element LaTeXMLMath can be written as LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath .
It is easy to see that LaTeXMLEquation .
By LaTeXMLMath , we denote the inverse matrix of LaTeXMLMath .
According to Equation ( LaTeXMLRef ) , LaTeXMLEquation is a non-degenerate triangular dynamical r-matrix over LaTeXMLMath .
In particular , if LaTeXMLMath , it follows immediately that LaTeXMLEquation .
Equation ( LaTeXMLRef ) was first obtained by Etingof-Varchenko in LaTeXMLCite .
The following example was pointed out to us by D. Vogan .
Example 2.3 Let LaTeXMLMath be a LaTeXMLMath dimensional Heisenberg Lie algebra and LaTeXMLMath its standard Heisenberg Lie subalgebra .
By LaTeXMLMath , we denote the standard generators of LaTeXMLMath and LaTeXMLMath , the generators of LaTeXMLMath .
Let LaTeXMLMath be the subspace of LaTeXMLMath generated by LaTeXMLMath .
It is then clear that LaTeXMLMath is a reductive decomposition .
Let LaTeXMLMath , be the dual basis corresponding to the standard generators of LaTeXMLMath .
For any LaTeXMLMath , write LaTeXMLMath .
This induces a coordinate system on LaTeXMLMath , and therefore a function on LaTeXMLMath can be identified with a function with variables LaTeXMLMath .
It is clear that LaTeXMLEquation .
LaTeXMLEquation It thus follows that LaTeXMLEquation is a non-degenerate triangular dynamical r-matrix .
From Proposition LaTeXMLRef , we know that a triangular dynamical r-matrix LaTeXMLMath is equivalent to a special type of Poisson structures on LaTeXMLMath .
It is thus very natural to expect that quantization of LaTeXMLMath can be derived from a certain special type of star-products on LaTeXMLMath .
It is simple to see that the Poisson brackets on LaTeXMLMath can be described as follows : for any LaTeXMLMath , LaTeXMLMath ; for any LaTeXMLMath and LaTeXMLMath , LaTeXMLMath ; for any LaTeXMLMath , LaTeXMLMath .
These Poisson bracket relations naturally motivate the following : A star product LaTeXMLMath on LaTeXMLMath is called a compatible star product if for any LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath is a smooth function LaTeXMLMath such that LaTeXMLMath .
Here LaTeXMLMath denotes the standard PBW-star product on LaTeXMLMath quantizing the canonical Lie-Poisson structure ( see LaTeXMLCite ) , whose definition is recalled below .
Let LaTeXMLMath be a Lie algebra with the Lie bracket LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation be the Poincaré-Birkhoff-Witt map , which is a vector space isomorphism .
Thus the multiplication on LaTeXMLMath induces a multiplication on LaTeXMLMath , hence on LaTeXMLMath , which is denoted by LaTeXMLMath .
It is easy to check that LaTeXMLMath satisfies LaTeXMLEquation where LaTeXMLMath ’ s are bidifferential operators .
In other words , LaTeXMLMath is indeed a star product on LaTeXMLMath , which is called the PBW-star product .
The following proposition is quite obvious .
The classical limit of a compatible star product is the Poisson structure LaTeXMLMath , where LaTeXMLMath .
Below we will study some important properties of compatible star products .
A compatible star product is always invariant under the left LaTeXMLMath -action .
It is right LaTeXMLMath -invariant iff LaTeXMLMath is LaTeXMLMath -equivariant , where LaTeXMLMath acts on LaTeXMLMath by the coadjoint action and on LaTeXMLMath by the adjoint action .
Proof .
First of all , note that Equations ( LaTeXMLRef - LaTeXMLRef ) completely determine a star product .
It is clear , from these equations , that LaTeXMLMath is left LaTeXMLMath -invariant .
As for the right LaTeXMLMath -action , it is obvious from Equation ( LaTeXMLRef ) that LaTeXMLMath is invariant for LaTeXMLMath .
It is standard that LaTeXMLMath is invariant under the coadjoint action , so it follows from Equation ( LaTeXMLRef ) that LaTeXMLMath is also LaTeXMLMath -invariant .
For any LaTeXMLMath , LaTeXMLMath and any fixed LaTeXMLMath , LaTeXMLEquation .
Thus it follows that LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath is a dual basis for LaTeXMLMath .
Let LaTeXMLMath be its corresponding induced coordinates on LaTeXMLMath .
Then LaTeXMLEquation .
Hence LaTeXMLEquation .
From Equation ( LaTeXMLRef ) , it follows that for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
I.e. , LaTeXMLMath is also right LaTeXMLMath -invariant .
Finally , LaTeXMLMath , LaTeXMLEquation .
On the other hand , LaTeXMLEquation .
Therefore LaTeXMLMath iff LaTeXMLMath .
The latter is equivalent to that LaTeXMLMath , or LaTeXMLMath is LaTeXMLMath -equivariant .
This concludes the proof .
LaTeXMLMath In order to give an explicit formula for LaTeXMLMath , let us write LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
Using this expression , indeed one can describe LaTeXMLMath explicitly .
Given a compatible star product LaTeXMLMath as in Definition LaTeXMLRef , for any LaTeXMLMath , LaTeXMLEquation .
We need a couple of lemmas first .
Under the same hypothesis as in Theorem LaTeXMLRef , for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
Proof .
( i ) .
It suffices to show this identity for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Now LaTeXMLEquation ( ii ) .
Similarly , we may assume that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLEquation ( iii ) .
Assume that LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
This concludes the proof of the lemma .
LaTeXMLMath Now we are ready to prove the main result of this section .
Proof of Theorem LaTeXMLRef Again , we may assume that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLMath As a consequence of Theorem LaTeXMLRef , we will see that if a function LaTeXMLMath defines a compatible star product , it must satisfy a “ twisted-cocycle ” type condition .
To describe this condition explicitly , we need to introduce some notations .
For any LaTeXMLMath , define LaTeXMLMath by LaTeXMLEquation .
The correspondence LaTeXMLMath extends naturally to a linear map from LaTeXMLMath to LaTeXMLMath , which is denoted by LaTeXMLMath .
More explicitly , assume that LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
By a suitable permutation , one may define LaTeXMLMath and LaTeXMLMath similarly .
Note that LaTeXMLMath is a Hopf algebra .
By LaTeXMLMath and LaTeXMLMath , we denote its co-multiplication and co-unit , respectively .
Then LaTeXMLMath naturally extends to a map LaTeXMLMath , which will be denoted by the same symbol .
Assume that LaTeXMLMath defines a compatible star product LaTeXMLMath as in Definition LaTeXMLRef .
Then LaTeXMLEquation .
LaTeXMLEquation Proof .
Equation ( LaTeXMLRef ) follows from the fact that LaTeXMLMath .
As for Equation ( LaTeXMLRef ) , note that for any LaTeXMLMath and LaTeXMLMath , according to Equation ( LaTeXMLRef ) , we have LaTeXMLEquation .
Now LaTeXMLEquation .
It thus follows that LaTeXMLEquation .
On the other hand , LaTeXMLEquation .
Now Equation ( LaTeXMLRef ) follows from the associativity of LaTeXMLMath .
LaTeXMLMath To end this section , as a special case , let us consider LaTeXMLMath , which is equipped with the canonical cotangent symplectic structure .
The following proposition describes an explicit formula for a compatible star-product on it .
For any LaTeXMLMath , the following equation LaTeXMLEquation defines a compatible star product on LaTeXMLMath , which is in fact a deformation quantization of its canonical cotangent symplectic structure .
Proof .
As earlier in this section , let LaTeXMLMath be equipped with the Lie bracket LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath the PBW-map .
Note that LaTeXMLMath is isomorphic to LaTeXMLMath as a Lie algebra .
Hence LaTeXMLMath is canonically isomorphic to LaTeXMLMath , whose elements can be considered as left invariant ( formal ) differential operators on LaTeXMLMath .
To each polynomial function on LaTeXMLMath , we assign a ( formal ) differential operator on LaTeXMLMath according to the following rule .
For LaTeXMLMath , we assign the operator multiplying by LaTeXMLMath ; for LaTeXMLMath , we assign the left invariant differential operator LaTeXMLMath ; in general , for LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , we assign the differential operator LaTeXMLMath .
Then the multiplication on the algebra of differential operators induces an associative product LaTeXMLMath on LaTeXMLMath , hence a star product on LaTeXMLMath .
It is simple to see , from the above construction , that for any LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation for any LaTeXMLMath , LaTeXMLEquation .
In other words , this is indeed a compatible star product with LaTeXMLMath .
Equation ( LaTeXMLRef ) thus follows immediately from Theorem LaTeXMLRef .
LaTeXMLMath Remark .
It would be interesting to compare Equation ( LaTeXMLRef ) with the general construction of star products on cotangent symplectic manifolds in LaTeXMLCite .
Equation ( LaTeXMLRef ) implies that the element LaTeXMLMath , being considered as a left invariant differential operator on LaTeXMLMath , admits the following expression : LaTeXMLEquation .
Thus we have : For any LaTeXMLMath , LaTeXMLEquation where the LaTeXMLMath on the left hand side stands for the PBW-star product on LaTeXMLMath , while on the right hand side it refers to the multiplication on the algebra tensor product of LaTeXMLMath with LaTeXMLMath .
Proof .
Let LaTeXMLMath denote the star product on LaTeXMLMath as in Proposition LaTeXMLRef .
For any LaTeXMLMath , LaTeXMLEquation .
On the other hand , LaTeXMLEquation .
The conclusion thus follows from the associativity of LaTeXMLMath .
LaTeXMLMath For any LaTeXMLMath , LaTeXMLEquation .
In particular , if LaTeXMLMath is invertible , we have LaTeXMLEquation .
The main purpose of this section is to derive the quantum dynamical Yang-Baxter equation over a nonabelian base LaTeXMLMath from the “ twisted-cocycle ” condition ( LaTeXMLRef ) .
This was standard when LaTeXMLMath is Abelian ( e.g. , see LaTeXMLCite ) .
The proof was based on the Drinfel ’ d theory of quasi-Hopf algebras LaTeXMLCite .
In our situation , however , the quasi-Hopf algebra approach does not work any more .
Nevertheless , one can carry out a proof in a way completely parallel to the ordinary case .
The main result of this section is the following : Assume that LaTeXMLMath satisfies the “ twisted-cocycle ” condition ( LaTeXMLRef ) .
Then LaTeXMLEquation satisfies the following generalized quantum dynamical Yang-Baxter equation ( or Gervais-Neveu-Felder equation ) : LaTeXMLEquation .
Here LaTeXMLMath denotes the natural multiplication on LaTeXMLMath , LaTeXMLMath , with LaTeXMLMath being equipped with the PBW-star product .
It is simple to see that the usual relation LaTeXMLEquation still holds for any LaTeXMLMath .
Define LaTeXMLMath by LaTeXMLEquation .
It is simple to see , using the associativity of LaTeXMLMath , that LaTeXMLEquation .
The following is immediate from Corollary LaTeXMLRef .
LaTeXMLEquation .
Remark .
Equation ( LaTeXMLRef ) is trivial when LaTeXMLMath is Abelian .
It , however , does not seem obvious in general .
We can see from the proof of Corollary LaTeXMLRef that this equation essentially follows from the associativity of the star product given by Equation ( LaTeXMLRef ) .
For any given LaTeXMLMath , introduce LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Proof .
By applying the permutation LaTeXMLMath on Equation ( LaTeXMLRef ) , one obtains that LaTeXMLEquation since LaTeXMLMath is cocommutative .
Similarly , applying the permutation LaTeXMLMath on Equation ( LaTeXMLRef ) , one obtains that LaTeXMLEquation .
On the other hand , by definition , LaTeXMLEquation .
LaTeXMLEquation It thus follows that LaTeXMLEquation .
Equation ( LaTeXMLRef ) can be proved similarly .
LaTeXMLMath Proof of Theorem LaTeXMLRef .
From Equation ( LaTeXMLRef ) , it follows that LaTeXMLEquation .
According to Equation ( LaTeXMLRef ) , this is equivalent to LaTeXMLEquation .
Thus , LaTeXMLEquation .
Now the twisted-cocycle condition ( LaTeXMLRef ) implies that LaTeXMLEquation .
It thus follows that LaTeXMLEquation .
Applying the permutations : LaTeXMLMath , and LaTeXMLMath respectively to the equation above , one obtains LaTeXMLEquation .
LaTeXMLEquation Equation ( LaTeXMLRef ) thus follows immediately .
LaTeXMLMath Even though our discussion so far has been mainly confined to triangular dynamical r-matrices , we should point out that there do exist many interesting examples of non-triangular ones .
For instance , when the Lie algebra LaTeXMLMath admits an ad-invariant bilinear form and the base Lie algebra LaTeXMLMath equals LaTeXMLMath , Alekseev and Meinrenken found an explicit construction of an interesting non-triangular dynamical r-matrix LaTeXMLCite in connection with their study of the non-commutative Weil algebra .
In fact , for simple Lie algebras , the existence of AM-dynamical r-matrices was already proved by Etingof and Varchenko LaTeXMLCite .
The construction of Alekseev and Meinrenken was later generalized by Etingof and Schiffmann to a more general context LaTeXMLCite .
So there is no doubt that there are abundant non-trivial examples of dynamical r-matrices with a nonabelian base .
It is therefore desirable to know how they can be quantized .
Inspired by the above discussion in the triangular case , we are ready to propose the following quantization problem along the line of Drinfel ’ LaTeXMLMath d ’ s naive LaTeXMLMath -matrix in LaTeXMLMath for an associative algebra LaTeXMLMath .
Here one can consider LaTeXMLMath as the universal enveloping algebra LaTeXMLMath , and LaTeXMLMath quantization LaTeXMLCite .
Given a classical dynamical r-matrix LaTeXMLMath , a quantization of LaTeXMLMath is LaTeXMLMath which is LaTeXMLMath -equivariant and satisfies the generalized quantum dynamical Yang-Baxter equation ( or Gervais-Neveu-Felder equation ) : LaTeXMLEquation .
Combining Proposition LaTeXMLRef , Proposition LaTeXMLRef , Corollary LaTeXMLRef and Theorem LaTeXMLRef , we may summarize the main result of this paper in the following : A triangular dynamical LaTeXMLMath -matrix LaTeXMLMath is quantizable if there exists a compatible star product on the corresponding Poisson manifold LaTeXMLMath .
We conclude this paper with a list of questions together with some thoughts .
Question 1 : Is every classical triangular dynamical LaTeXMLMath -matrix quantizable ?
According to Theorem LaTeXMLRef , this question is equivalent to asking whether a compatible star product always exists for the corresponding Poisson manifold LaTeXMLMath .
When the base Lie algebra is Abelian , a quantization procedure was found for splittable classical triangular dynamical r-matrices using Fedosov ’ s method LaTeXMLCite .
Recently Etingof and Nikshych LaTeXMLCite , using the vertex-IRF transformation method , showed the existence of quantization for the so called completely degenerate triangular dynamical r-matrices , which leads to the hope that the existence of quantization could be possibly settled by combing both methods in LaTeXMLCite and LaTeXMLCite .
However , when the base Lie algebra LaTeXMLMath is nonabelian , the method in LaTeXMLCite does not admit a straightforward generalization .
One of the main difficulties is that the Fedosov method uses Weyl quantization , while our quantization here is in normal ordering .
Nevertheless , for the dynamical r-matrices constructed in Theorem LaTeXMLRef , under some mild assumptions a quantization seems feasible by using the generalized Karabegov ’ s method LaTeXMLCite .
This problem will be discussed in a separate publication .
Question 2 : What is the symmetrized version of the quantum dynamical Yang-Baxter equation ( LaTeXMLRef ) ?
We derived Equation ( LaTeXMLRef ) from a compatible star product , which is a normal ordering star product .
The reason for us to choose the normal ordering here is that one can obtain a very explicit formula for the star product : Equation ( LaTeXMLRef ) .
A Weyl ordering compatible star product may exist , but it may be more difficult to work with .
For the canonical cotangent symplectic structure LaTeXMLMath , a Weyl ordering star product was found by Gutt LaTeXMLCite , but it is rather difficulty to write down an explicit formula LaTeXMLCite .
As we can see from the previous discussion , how a quantum dynamical Yang-Baxter equation looks is closely related to the choice of a star product on LaTeXMLMath .
When LaTeXMLMath is Abelian , there is a very simple operator establishing an isomorphism between these two quantizations , which is indeed the transformation needed to transform a unsymmetrized QDYBE into a symmetrized one .
Such an operator also exists for a general cotangent bundle LaTeXMLMath LaTeXMLCite , but it is much more complicated .
Nevertheless , this viewpoint may still provide a useful method to obtain the symmetrized version of a QDYBE .
Question 3 : Is every classical dynamical LaTeXMLMath -matrix quantizable ?
This question may be a bit too general .
As a first step , it should be already quite interesting to find a quantum analogue of Alekseev-Meinrenken dynamical r-matrices .
Question 4 : What is the deformation theory controlling the quantization problem as proposed in Definition LaTeXMLRef ?
If LaTeXMLMath , where LaTeXMLMath , is a solution to the QDYBE , the LaTeXMLMath -term LaTeXMLMath must be a solution of the classical dynamical Yang-Baxter equation .
Indeed the quantum dynamical Yang-Baxter equation implies a sequence of equations of LaTeXMLMath in terms of lower order terms .
One should expect some cohomology theory here just as for any deformation theory LaTeXMLCite .
However , in our case , the equation seems very complicated .
On the other hand , it is quite surprising that such a theory does not seem to exist in the literature even in the case of quantization of a usual LaTeXMLMath -matrix .
Finally , we would like to point out that perhaps a more useful way of thinking of quantization of a dynamical r-matrix is to consider the quantum groupoids as defined in LaTeXMLCite .
This is in some sense an analogue of the “ sophisticated ” quantization in terms of Drinfel ’ d LaTeXMLCite .
A classical dynamical r-matrix gives rise to a Lie bialgebroid LaTeXMLMath LaTeXMLCite .
Its induced Poisson structure on the base space LaTeXMLMath is the Lie-Poisson structure LaTeXMLMath , which admits the PBW-star product as a standard deformation quantization .
This leads to the following Question 5 : Does the Lie bialgebroid LaTeXMLMath corresponding to a classical dynamical r-matrix always admit a quantization in the sense of LaTeXMLCite , with the base algebra being the PBW-star algebra LaTeXMLMath ?
To connect the quantization problem in Definition LaTeXMLRef with that of Lie bialgebroids , it is clear that one needs to consider preferred quantization of Lie bialgebroids : namely , a quantization where the total algebra is undeformed and remains to be LaTeXMLMath .
Question 6 : Does the Lie bialgebroid LaTeXMLMath admit a preferred quantization ?
How is such a preferred quantization related to the quantization of LaTeXMLMath as proposed in Definition LaTeXMLRef ?
When LaTeXMLMath , namely for usual r-matrices , the answer to Question 6 is positive , due to a remarkable theorem of Etingof-Kazhdan LaTeXMLCite .
We establish a new group-theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras of ADE types in the same spirit of our new approach to the McKay correspondence .
Our vertex operator construction provides a unified description to the character tables for the spin cover of the wreath product of the twisted hyperoctahedral groups and an arbitrary finite group .
The connection between the homogeneous vertex representations of affine Lie algebras and the wreath products LaTeXMLMath associated to finite subgroups LaTeXMLMath of LaTeXMLMath was first pointed out in LaTeXMLCite and subsequently established fully in LaTeXMLCite .
This initiates a new approach to the McKay correspondence ( cf .
LaTeXMLCite ) , which classically relates the finite subgroups of LaTeXMLMath in a bijective manner with affine Dynkin diagrams of ADE type .
These results have been further extended in LaTeXMLCite to realize the vertex representations of twisted affine Lie algebras LaTeXMLMath and its toroidal counterpart by using the spin representations of a double cover LaTeXMLMath of the wreath product LaTeXMLMath .
Here LaTeXMLMath is a double cover of the symmetric group LaTeXMLMath whose representation theory was developed by Schur LaTeXMLCite and reformulated in terms of twisted vertex operators in LaTeXMLCite .
The algebraic construction of the vertex representation of the twisted affine algebra LaTeXMLMath was obtained first in LaTeXMLCite in the case of LaTeXMLMath , and in LaTeXMLCite for general LaTeXMLMath as one ingredient in the vertex ( operator ) algebra construction of the Monster group .
The goal of this paper is to provide a new finite-group-theoretic realization of the vertex representations of the twisted affine Lie algebra LaTeXMLMath and its toroidal counterpart .
Instead of LaTeXMLMath we will use a semi-direct product LaTeXMLMath , where LaTeXMLMath is a double cover of the hyperoctahedral group LaTeXMLMath .
The finite group LaTeXMLMath can also be thought as a double cover of the wreath product LaTeXMLMath .
Our present construction recaptures all the constructions in LaTeXMLCite with additional advantages .
First , this has a natural generalization given in a companion paper LaTeXMLCite , which is analogous to LaTeXMLCite , to the LaTeXMLMath -theory setup and it is intimately related to geometry .
Secondly , the present constructions of the twisted vertex operators , Heisenberg algebra , and the characteristic map etc are essentially over LaTeXMLMath while those in LaTeXMLCite involve an inevitable square root of LaTeXMLMath which originates in the theory of spin representations of LaTeXMLMath .
Our present construction in principle paves the way to study the quantum twisted vertex representations at roots of unity which we will investigate elsewhere .
More explicitly , we take the supermodule approach of Jozéfiak LaTeXMLCite to study the spin representations of LaTeXMLMath for an arbitrary finite group LaTeXMLMath .
We give a description with a complete proof of the split conjugacy classes in LaTeXMLMath which play an important role in understanding the spin supermodules of LaTeXMLMath .
This generalizes earlier works on LaTeXMLMath and its spin representations ( cf .
LaTeXMLCite ) .
This result was also stated in Read LaTeXMLCite when LaTeXMLMath is cyclic .
Given a finite group LaTeXMLMath , we consider a direct sum over all LaTeXMLMath , denoted by LaTeXMLMath , of the Grothendieck groups of spin supermodules of LaTeXMLMath .
We show that LaTeXMLMath carries a natural Hopf algebra structure .
Associated to a self-dual virtual character LaTeXMLMath of LaTeXMLMath we define a LaTeXMLMath -weighted symmetric bilinear form on LaTeXMLMath ( compare LaTeXMLCite ) .
The space LaTeXMLMath can be shown to be isomorphic to a Fock space of a twisted Heisenberg algebra .
The twisted vertex operators also make a natural appearance .
One can further interpret the Fock space as a distinguished space of symmetric functions parametrized by the set of irreducible characters of LaTeXMLMath , and in this way the irreducible characters of LaTeXMLMath correspond essentially to the Schur LaTeXMLMath -functions .
When LaTeXMLMath is a subgroup of LaTeXMLMath and the weight LaTeXMLMath is chosen suitably , the Fock space LaTeXMLMath leads to a construction of the twisted vertex representations of LaTeXMLMath and its toroidal counterpart .
On the other hand , when choosing the weight LaTeXMLMath to be trivial , the twisted vertex operator approach allows us to compute spin characters of all irreducible supermodules of the group LaTeXMLMath , as done for LaTeXMLMath in LaTeXMLCite and for LaTeXMLMath in LaTeXMLCite .
The layout of this paper is as follows .
In Sect .
LaTeXMLRef we studied in detail the conjugacy classes of the group LaTeXMLMath .
In Sect .
LaTeXMLRef , we introduce the Grothendieck group LaTeXMLMath and a weighted bilinear form on it .
We construct the Hopf algebra structure on LaTeXMLMath .
In Sect .
LaTeXMLRef we identify LaTeXMLMath with a Fock space of a twisted Heisenberg algebra .
In Sect .
LaTeXMLRef we construct twisted vertex operators ( essentially on LaTeXMLMath ) in terms of group theoretic operators .
When LaTeXMLMath is a finite subgroup of LaTeXMLMath , this leads to a group theoretic construction of the vertex representation of twisted affine and toroidal Lie algebras .
In Sect .
LaTeXMLRef we construct the irreducible spin super characters of LaTeXMLMath and recover their character table from vertex operator viewpoint .
When the statements can be proved similarly as in LaTeXMLCite , we often sketch only or omit the proofs and refer the reader to loc .
cit .
for more detail .
In this section we introduce the finite group LaTeXMLMath and a natrual LaTeXMLMath -grading on it .
We also classify the so-called split conjugacy classes in LaTeXMLMath which will play a key role in the study of spin supermodules of LaTeXMLMath in later sections .
Let LaTeXMLMath be the finite group generated by LaTeXMLMath ( LaTeXMLMath ) and the central element LaTeXMLMath subject to the relations LaTeXMLEquation .
The symmetric group LaTeXMLMath acts on LaTeXMLMath by LaTeXMLMath , LaTeXMLMath .
The semidirect product LaTeXMLMath is called the twisted hyperoctahedral group .
Explicitly the multiplication in LaTeXMLMath is given by LaTeXMLEquation .
Since LaTeXMLMath , the group LaTeXMLMath is a double cover of the hyperoctahedral group LaTeXMLMath .
Let LaTeXMLMath be a finite group .
The twisted hyperoctahedral group LaTeXMLMath acts on the product group LaTeXMLMath by letting LaTeXMLMath act trivially on LaTeXMLMath and letting LaTeXMLMath act by LaTeXMLMath The finite group LaTeXMLMath is then defined to be the semi-direct product of LaTeXMLMath and LaTeXMLMath .
Alternatively , the symmetric group LaTeXMLMath naturally acts on LaTeXMLMath by simutaneous permutations of elements in LaTeXMLMath and LaTeXMLMath , and we may regard LaTeXMLMath as the semi-direct product of the symmetric group LaTeXMLMath and LaTeXMLMath .
The double covering LaTeXMLMath of LaTeXMLMath extends to a double covering of the wreath product LaTeXMLMath by LaTeXMLMath : LaTeXMLEquation .
The order LaTeXMLMath is clearly LaTeXMLMath , where LaTeXMLMath denotes the order of LaTeXMLMath .
The group LaTeXMLMath contains several distinguished subgroups : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and the wreath product LaTeXMLMath etc .
We define a LaTeXMLMath -grading on the group LaTeXMLMath by setting the degree of LaTeXMLMath to be LaTeXMLMath and the degree of elements in LaTeXMLMath and LaTeXMLMath to be LaTeXMLMath , and denote by LaTeXMLMath ( resp .
LaTeXMLMath ) the degree zero ( resp .
one ) part .
This induces an epimorphism LaTeXMLMath ¿from LaTeXMLMath or the group algebra LaTeXMLMath to LaTeXMLMath .
The homomorphism LaTeXMLMath descends to a homomorphism on LaTeXMLMath which will be denoted by LaTeXMLMath again .
We say LaTeXMLMath or LaTeXMLMath even ( resp .
odd ) if LaTeXMLMath is LaTeXMLMath ( resp .
LaTeXMLMath ) .
A partition LaTeXMLMath of a non-negative integer LaTeXMLMath is a monotonic non-increasing sequence of integers LaTeXMLMath called parts such that LaTeXMLMath .
Here LaTeXMLMath is the length of LaTeXMLMath .
We may also write LaTeXMLMath , where LaTeXMLMath is the number of times that LaTeXMLMath appears in LaTeXMLMath .
For two partitions LaTeXMLMath and LaTeXMLMath the dominance order LaTeXMLMath is defined by LaTeXMLMath , LaTeXMLMath , etc .
A partition LaTeXMLMath is called strict if its parts are distinct integers .
We will use partitions indexed by LaTeXMLMath and LaTeXMLMath .
For a finite set LaTeXMLMath and LaTeXMLMath a family of partitions indexed by LaTeXMLMath , we write LaTeXMLMath It is convenient to regard LaTeXMLMath as a partition-valued function on LaTeXMLMath .
We denote by LaTeXMLMath the set of all partitions indexed by LaTeXMLMath and by LaTeXMLMath the subset consisting of LaTeXMLMath such that LaTeXMLMath .
The total number of parts , denoted by LaTeXMLMath , in the partition-valued function LaTeXMLMath is called the length of LaTeXMLMath .
The dominance order on LaTeXMLMath is defined naturally by LaTeXMLMath if LaTeXMLMath for each LaTeXMLMath .
We also write LaTeXMLMath if LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath .
For LaTeXMLMath we define LaTeXMLMath by LaTeXMLMath , where LaTeXMLMath denotes the conjugacy class LaTeXMLMath .
Let LaTeXMLMath be the set of partition-valued functions LaTeXMLMath in LaTeXMLMath such that all parts of LaTeXMLMath are odd integers for each LaTeXMLMath , and let LaTeXMLMath be the set of LaTeXMLMath such that each LaTeXMLMath is strict .
When LaTeXMLMath consists of a single element , we will omit LaTeXMLMath and simply write LaTeXMLMath for LaTeXMLMath , thus LaTeXMLMath or LaTeXMLMath will be used accordingly .
A variant of Euler ’ s theorem says that the number of strict partition-valued functions on a set LaTeXMLMath is equal to the number of partition-valued functions on LaTeXMLMath with odd integer parts .
We denote by LaTeXMLEquation .
LaTeXMLEquation and define LaTeXMLMath for LaTeXMLMath .
We will also need another parity LaTeXMLMath on partition-valued functions .
For a partition-valued function LaTeXMLMath we define LaTeXMLEquation .
Let LaTeXMLMath be a finite group with LaTeXMLMath conjugacy classes .
We denote by LaTeXMLMath the set of complex irreducible characters where LaTeXMLMath is the trivial character , and by LaTeXMLMath the set of conjugacy classes where LaTeXMLMath is the identity conjugacy class .
Let LaTeXMLMath be the order of the conjugacy class LaTeXMLMath , and then LaTeXMLMath is the order of the centralizer of an element in the class LaTeXMLMath .
For a subset LaTeXMLMath of the set LaTeXMLMath , we denote LaTeXMLMath .
It follows that LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
Also we can easily show by induction that LaTeXMLEquation for a permutation LaTeXMLMath of LaTeXMLMath .
Clearly the morphism LaTeXMLMath sends LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath are the generators of LaTeXMLMath .
The conjugacy classes of a wreath product is well understood , cf .
LaTeXMLCite .
In particular this gives us the following description of conjugacy classes of the wreath product LaTeXMLMath .
Given a cycle LaTeXMLMath , we call the set LaTeXMLMath the support of LaTeXMLMath , denoted by LaTeXMLMath .
Given LaTeXMLMath , every element LaTeXMLMath can be uniquely written as a product ( up to order ) LaTeXMLEquation where LaTeXMLMath is a product of disjoint cycles LaTeXMLMath , and LaTeXMLMath so that LaTeXMLMath , and we call LaTeXMLMath a signed cycle of LaTeXMLMath with the sign LaTeXMLMath .
For each signed cycle LaTeXMLMath with LaTeXMLMath , the signed cycle-product of LaTeXMLMath is the element LaTeXMLMath with the sign LaTeXMLMath .
For LaTeXMLMath let LaTeXMLMath ( resp .
LaTeXMLMath ) be the number of cycles of LaTeXMLMath whose signed cycle-product lies in LaTeXMLMath and has LaTeXMLMath ( resp .
LaTeXMLMath ) sign .
Then LaTeXMLMath with LaTeXMLMath and LaTeXMLMath is a pair of partition-valued functions on LaTeXMLMath such that LaTeXMLMath , and will be called the type of the element LaTeXMLMath .
Two elements of LaTeXMLMath are conjugate if and only if their types are the same .
We say that the conjugacy class LaTeXMLMath is even ( resp .
odd ) if it consists of even ( resp .
odd ) elements .
More precisely if LaTeXMLMath , then LaTeXMLMath is even ( resp .
odd ) if LaTeXMLMath is even ( resp .
odd ) .
We can write a general element of LaTeXMLMath as LaTeXMLMath where LaTeXMLEquation and LaTeXMLMath is a cycle decomposition of LaTeXMLMath and LaTeXMLMath .
We denote by LaTeXMLMath the complement of a subset LaTeXMLMath .
Let LaTeXMLMath be an element of LaTeXMLMath in its cycle decomposition .
Let LaTeXMLMath , then LaTeXMLEquation .
Consequently LaTeXMLMath .
Observe that LaTeXMLMath for any subset LaTeXMLMath .
For LaTeXMLMath we have LaTeXMLMath .
Therefore it reduces to see that LaTeXMLEquation where we have used the fact that LaTeXMLMath .
∎ If two elements of LaTeXMLMath are conjugate , then clearly their images are conjugate in LaTeXMLMath .
On the other hand , for any conjugacy class LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is either a conjugacy class of LaTeXMLMath or it splits into two conjugacy classes of LaTeXMLMath ( Indeed this holds in a more general setup , cf .
e.g .
LaTeXMLCite ) .
A conjugacy class LaTeXMLMath of LaTeXMLMath is called split if the preimage LaTeXMLMath splits into two conjugacy classes in LaTeXMLMath .
Equivalently , an element LaTeXMLMath is called split if LaTeXMLMath is not conjugate to LaTeXMLMath in LaTeXMLMath , then LaTeXMLMath is split if and only if LaTeXMLMath consists of split elements .
A conjugacy class of LaTeXMLMath splits if it consists of split elements .
The following theorem in the case when LaTeXMLMath was known in literature ( cf .
LaTeXMLCite ) .
It was also stated in LaTeXMLCite for LaTeXMLMath cyclic .
The conjugacy class LaTeXMLMath in LaTeXMLMath splits if and only if ( 1 ) For even LaTeXMLMath , we have LaTeXMLMath and LaTeXMLMath , ( 2 ) For odd LaTeXMLMath , we have LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath , it follows by definition that LaTeXMLEquation .
LaTeXMLEquation ( LaTeXMLMath ) i ) The conjugacy class LaTeXMLMath is even and split .
Suppose on the contrary there is a part of even integer in LaTeXMLMath .
Without loss of generality we can assume that LaTeXMLMath contains a representative element LaTeXMLMath with the signed cycle decomposition LaTeXMLEquation where LaTeXMLMath is even and LaTeXMLMath is empty ( we can take all LaTeXMLMath empty corresponding to parts in LaTeXMLMath ) and LaTeXMLMath is even .
Consider the element LaTeXMLEquation where LaTeXMLMath with LaTeXMLMath , for LaTeXMLMath and LaTeXMLMath otherwise .
We claim that LaTeXMLEquation .
In fact the LaTeXMLMath th component of LaTeXMLMath equals LaTeXMLMath for LaTeXMLMath and it also equals LaTeXMLMath for LaTeXMLMath .
Noting that LaTeXMLMath we have LaTeXMLEquation .
Moreover by Lemma LaTeXMLRef we have ( recall LaTeXMLMath ) LaTeXMLEquation .
Thus LaTeXMLMath is conjugate to LaTeXMLMath in view of Eqn .
( LaTeXMLRef ) .
Therefore if the even-parity conjugacy class LaTeXMLMath splits then LaTeXMLMath .
Now suppose that LaTeXMLMath .
Then LaTeXMLMath contains at least two parts since we assume that LaTeXMLMath is even .
Without loss of generality we can assume that LaTeXMLMath contains an element LaTeXMLMath such that LaTeXMLEquation where LaTeXMLMath .
If both LaTeXMLMath and LaTeXMLMath are of cycle length LaTeXMLMath , then LaTeXMLMath .
Assume that LaTeXMLMath , so LaTeXMLMath .
Consider LaTeXMLMath , where LaTeXMLMath for LaTeXMLMath and LaTeXMLMath otherwise .
Then LaTeXMLEquation .
Subsequently LaTeXMLMath .
Hence if LaTeXMLMath of even parity splits then LaTeXMLMath is empty .
Together with the above we have shown that split conjugacy class of even parity should have property ( 1 ) .
ii ) The conjugacy class LaTeXMLMath is odd and split .
If on the contrary LaTeXMLMath , we can assume that LaTeXMLMath contains an element LaTeXMLMath with the signed cycle decomposition LaTeXMLEquation where LaTeXMLMath is empty and LaTeXMLMath is odd .
Take the element LaTeXMLMath where LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath otherwise .
Similarly we can verify that LaTeXMLMath by using Lemma LaTeXMLRef .
In fact LaTeXMLEquation since LaTeXMLMath is odd .
Hence LaTeXMLMath does not split if LaTeXMLMath is odd and LaTeXMLMath .
Next we assume on the contrary that LaTeXMLMath contains two identical parts , then by taking conjugation if necessary we can assume that LaTeXMLMath contains an element LaTeXMLMath such that LaTeXMLEquation and LaTeXMLMath for some LaTeXMLMath .
Consider the element LaTeXMLMath , where LaTeXMLMath .
Then we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation which is a contradiction .
( LaTeXMLMath ) Suppose that ( 1 ) holds .
If on the contrary the even conjugacy class LaTeXMLMath does not split , then we can assume that LaTeXMLMath contains an element LaTeXMLMath , where LaTeXMLMath with each LaTeXMLMath being odd cycle , and LaTeXMLMath .
Therefore LaTeXMLMath , i.e. , LaTeXMLMath , which in particular implies that LaTeXMLMath .
Then LaTeXMLMath by Eq .
( LaTeXMLRef ) , and so LaTeXMLMath which contradicts with the assumption on LaTeXMLMath .
Suppose that ( 2 ) holds .
Assume on the contrary that the odd conjugacy class LaTeXMLMath does not split .
Since an identification of two elements in LaTeXMLMath implies that their respective components in LaTeXMLMath are already equal , we can assume that LaTeXMLMath consists of one strict partition LaTeXMLMath for some LaTeXMLMath .
Thus LaTeXMLMath contains a non-split element LaTeXMLMath , where LaTeXMLMath is odd and LaTeXMLMath .
Let LaTeXMLMath for some element LaTeXMLMath .
It follows that LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath is a product of disjoint cycles with mutually distinct orders , the permutation LaTeXMLMath equals LaTeXMLMath for LaTeXMLMath .
Thus we can write LaTeXMLMath with LaTeXMLMath .
As in the proof of Lemma LaTeXMLRef we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation which must equal LaTeXMLMath up to a power of LaTeXMLMath .
Set LaTeXMLMath where LaTeXMLMath is LaTeXMLMath or LaTeXMLMath .
We claim that LaTeXMLMath is always LaTeXMLMath .
Without loss of generality we let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , then LaTeXMLEquation implies that LaTeXMLMath , which in turn implies the exponent LaTeXMLMath is equal to LaTeXMLMath .
Therefore LaTeXMLMath and similarly we have LaTeXMLEquation .
LaTeXMLEquation since LaTeXMLMath is odd .
This is a contradiction .
∎ For LaTeXMLMath we let LaTeXMLMath be the split conjugacy class in LaTeXMLMath containing the elements LaTeXMLMath of type LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLMath is the other conjugacy class lying in LaTeXMLMath , which will be denoted by LaTeXMLMath .
For each partition-valued function LaTeXMLMath we define LaTeXMLEquation which is the order of the centralizer of an element of conjugacy type LaTeXMLMath in LaTeXMLMath ( cf .
LaTeXMLCite ) .
It follows from Theorem LaTeXMLRef that the order of the centralizer of an element in the conjugacy class LaTeXMLMath is LaTeXMLEquation .
In this section , we first recall some general facts about spin supermodules over a superalgebra .
This will be applied to ( the group algebra of ) LaTeXMLMath .
We introduce the Grothendieck group LaTeXMLMath of spin supermodules over LaTeXMLMath , and construct the so-called basic spin supermodules of LaTeXMLMath .
We further define a natural Hopf algebra structure on LaTeXMLMath , and introduce a weighted bilinear form on LaTeXMLMath associated to any given self-dual virtual character LaTeXMLMath of LaTeXMLMath .
When LaTeXMLMath is trivial , the weighted bilinear form reduces to the standard one .
A complex superalgebra LaTeXMLMath is a LaTeXMLMath -graded complex vector space with a binary product LaTeXMLMath such that LaTeXMLMath .
A LaTeXMLMath -graded vector space LaTeXMLMath is a supermodule for a superalgebra LaTeXMLMath if LaTeXMLMath .
A linear map LaTeXMLMath between two LaTeXMLMath -supermodules is a morphism of degree LaTeXMLMath if LaTeXMLMath and for any homogeneous element LaTeXMLMath and any homogeneous vector LaTeXMLMath we have LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath be two supermodules .
The tensor product LaTeXMLMath is also a supermodule with LaTeXMLMath .
The notions of submodules , irreducible supermodules etc are defined similarly as usual .
Let LaTeXMLMath be the LaTeXMLMath -graded vector space LaTeXMLMath .
The algebra LaTeXMLMath consisting of all linear transformations on LaTeXMLMath inherits a natural LaTeXMLMath grading from LaTeXMLMath .
It is easily seen that LaTeXMLMath is a simple superalgebra .
Another example of simple superalgebra is LaTeXMLMath , which is the subalgebra of LaTeXMLMath consisting of matrices of the form LaTeXMLEquation .
The left multiplication of LaTeXMLMath on LaTeXMLMath gives an irreducible supermodule structure on it .
A well-known result due to C. T. C. Wall says that these superalgebras are the only simple superalgebras over LaTeXMLMath .
In the sequel we will say the supermodules LaTeXMLMath and LaTeXMLMath are of type LaTeXMLMath and LaTeXMLMath respectively .
Let LaTeXMLMath be a finite group and let LaTeXMLMath be a group epimorphism .
We denote by LaTeXMLMath the kernel of LaTeXMLMath which is a subgroup of LaTeXMLMath of index LaTeXMLMath .
We regard LaTeXMLMath as a parity function on LaTeXMLMath by letting the degree of elements in LaTeXMLMath be LaTeXMLMath and letting the degree of elements in the complementary LaTeXMLMath be LaTeXMLMath .
Elements in LaTeXMLMath ( resp .
LaTeXMLMath ) will be called even ( resp .
odd ) .
In addition we assume that LaTeXMLMath contains a distinguished even central element LaTeXMLMath of order LaTeXMLMath .
A spin supermodule over LaTeXMLMath is a supermodule over the group superalgebra LaTeXMLMath such that LaTeXMLMath acts as LaTeXMLMath .
The group superalgebra is semisimple ( cf .
LaTeXMLCite ) , i.e .
decomposes into a direct sum of simple superalgebras .
We will refer the corresponding supermodules as of type LaTeXMLMath and type LaTeXMLMath .
Now let us return to our main example LaTeXMLMath .
It is easy to see that the characters of spin supermodules vanish on nonsplit classes .
Let LaTeXMLMath be the one-dimensional representation of LaTeXMLMath given by LaTeXMLMath .
A representation LaTeXMLMath of LaTeXMLMath is called a double spin representation if LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath are called associate spin representations of LaTeXMLMath .
By the general theory of supermodules ( cf .
LaTeXMLCite ) and Theorem LaTeXMLRef we obtain the following proposition .
The number of irreducible double spin representations over LaTeXMLMath is equal to LaTeXMLMath , and the number of pairs of irreducible associate spin representations is LaTeXMLMath .
The number of irreducible spin supermodules of LaTeXMLMath is LaTeXMLMath .
If LaTeXMLMath is an irreducible LaTeXMLMath -supermodule of type LaTeXMLMath , then its underlying module LaTeXMLMath ( by forgetting the LaTeXMLMath -grading structure ) is an irreducible double spin LaTeXMLMath -module .
If LaTeXMLMath is an irreducible LaTeXMLMath -supermodule of type LaTeXMLMath , then LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are a pair of irreducible associate spin LaTeXMLMath -modules .
It has been observed ( cf .
LaTeXMLCite ) that there is a very close connection between the representations of LaTeXMLMath and those of a spin cover LaTeXMLMath of the symmetric group LaTeXMLMath .
Yamaguchi LaTeXMLCite explains such an phenomenon in an elegant way by establishing an isomorphism between the group superalgebra LaTeXMLMath and the ( outer ) tensor product of the group superalgebra LaTeXMLMath with the complex Clifford algebra of LaTeXMLMath variables .
( Note that a Clifford algebra admits a unique irreducible supermodule ) .
In view of this , there is also an isomorphism by substituting LaTeXMLMath and LaTeXMLMath with LaTeXMLMath and LaTeXMLMath respectively .
This isomorphism provides a direct connection between the constructions in the present paper and LaTeXMLCite .
Given a LaTeXMLMath -graded finite group LaTeXMLMath and a LaTeXMLMath -graded subgroup LaTeXMLMath that contain an even central element LaTeXMLMath of order LaTeXMLMath , we can define the induction and restriction of supermodules similarly as usual .
In particular the induced supermodule ( or a restriction ) of a spin supermodule remains to be a spin supermodule .
The Mackey theorem remains true in this setup ( cf .
LaTeXMLCite , Sect .
3 ) .
Let LaTeXMLMath be the space of complex-valued class functions on LaTeXMLMath .
The usual bilinear form on LaTeXMLMath is defined as follows : LaTeXMLEquation .
Then LaTeXMLMath becomes an integral lattice in LaTeXMLMath under the standard bilinear form : LaTeXMLMath .
A spin class function on LaTeXMLMath is class function LaTeXMLMath such that LaTeXMLMath , hence it vanishes on non-split conjugacy classes .
A spin super class function LaTeXMLMath on LaTeXMLMath is a spin class function such that it vanishes further on odd conjugacy classes .
In other words , LaTeXMLMath corresponds to a complex functional on LaTeXMLMath in view of of Theorem LaTeXMLRef .
Let LaTeXMLMath be the LaTeXMLMath -span of spin super class functions on LaTeXMLMath .
The standard bilinear form on LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , and Eqn .
( LaTeXMLRef ) is used here .
The following holds for a general LaTeXMLMath -graded finite group LaTeXMLCite .
The characters of irreducible spin supermodules over LaTeXMLMath form a LaTeXMLMath -basis of LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be two irreducible characters of spin supermodules , then LaTeXMLEquation .
Conversely , if LaTeXMLMath for LaTeXMLMath , then LaTeXMLMath or LaTeXMLMath affords an irreducible spin LaTeXMLMath -supermodule of type LaTeXMLMath .
Let LaTeXMLMath be the Clifford algebra generated by LaTeXMLMath with relations : LaTeXMLEquation .
The group LaTeXMLMath acts on LaTeXMLMath by LaTeXMLMath and LaTeXMLMath .
The Clifford algebra LaTeXMLMath has a natural LaTeXMLMath -grading given by the parity LaTeXMLMath such that LaTeXMLMath .
The set LaTeXMLMath form a basis of LaTeXMLMath .
We observe that LaTeXMLMath LaTeXMLMath if and only if LaTeXMLMath is a union of LaTeXMLMath , where LaTeXMLMath is a cycle decomposition .
LaTeXMLCite The module LaTeXMLMath is an irreduicble LaTeXMLMath -supermodule .
The value of its character LaTeXMLMath at the conjugacy class LaTeXMLMath is given by LaTeXMLEquation .
Let LaTeXMLMath be a LaTeXMLMath -module afforded by the character LaTeXMLMath , the tensor product LaTeXMLMath is a LaTeXMLMath -module by the direct product action of LaTeXMLMath combined with permutation action of the symmetric group LaTeXMLMath .
More explicitly the action is given by LaTeXMLEquation where LaTeXMLMath .
We denote the resulting character by LaTeXMLMath .
Indeed one can extend ( cf .
LaTeXMLCite ) this construction to define a map LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -module afforded by the character LaTeXMLMath , and let LaTeXMLMath be a spin supermodule of LaTeXMLMath with the character LaTeXMLMath .
The tensor product LaTeXMLMath has a canonical spin supermodule structure for LaTeXMLMath as follows .
For any LaTeXMLMath , and an element LaTeXMLMath in LaTeXMLMath , the action is given by LaTeXMLEquation .
It is easy to see that if LaTeXMLMath and LaTeXMLMath are irreducible , then so is LaTeXMLMath .
We denote by LaTeXMLMath the character of this spin supermodule .
The following result can be proved as in LaTeXMLCite for similar results .
Let LaTeXMLMath be the character of a spin LaTeXMLMath -supermodule .
Then the value of the character LaTeXMLMath at an element LaTeXMLMath in the even split conjugacy class LaTeXMLMath is given by LaTeXMLEquation .
The map LaTeXMLMath can be extended LaTeXMLMath to LaTeXMLMath ( compare LaTeXMLCite ) : LaTeXMLEquation .
In particular , when LaTeXMLMath is the spin LaTeXMLMath -supermodule LaTeXMLMath we denote by LaTeXMLMath the character of the LaTeXMLMath -supermodule LaTeXMLMath .
We will refer to LaTeXMLMath ( associated to irreducible LaTeXMLMath ) as the basic spin supermodules over LaTeXMLMath .
The character values of LaTeXMLMath on the conjugacy classes LaTeXMLMath are given by LaTeXMLEquation .
Our main object of this paper is the space LaTeXMLEquation .
Let LaTeXMLMath be the direct product of LaTeXMLMath and LaTeXMLMath with a twisted multiplication LaTeXMLMath where LaTeXMLMath are homogeneous .
We define the spin product of LaTeXMLMath and LaTeXMLMath by LaTeXMLEquation which can be embedded into the spin group LaTeXMLMath canonically by letting LaTeXMLMath LaTeXMLMath , LaTeXMLMath .
We will identify LaTeXMLMath with its image in LaTeXMLMath .
The subgroup LaTeXMLMath has a distinguished subgroup of index LaTeXMLMath consisting of even elements with LaTeXMLMath .
Therefore we can define LaTeXMLMath to be the space of super spin class functions .
For two spin modules LaTeXMLMath and LaTeXMLMath of LaTeXMLMath and LaTeXMLMath we define the ( outer ) -tensor product by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are homogeneous elements .
Since LaTeXMLMath , the tensor LaTeXMLMath is a spin LaTeXMLMath -supermodule .
The following is a straightforward generalization of a result in LaTeXMLCite for LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be simple supermodules for LaTeXMLMath and LaTeXMLMath respectively .
Then 1 ) If both LaTeXMLMath and LaTeXMLMath are of type M , then LaTeXMLMath is a simple LaTeXMLMath -supermodule of type M. 2 ) If LaTeXMLMath and LaTeXMLMath are of different type , then LaTeXMLMath is a simple LaTeXMLMath -supermodule of type Q .
3 ) If both LaTeXMLMath and LaTeXMLMath are type Q , then LaTeXMLMath for some simple LaTeXMLMath -supermodule LaTeXMLMath of type M. We will denote LaTeXMLMath by LaTeXMLMath For a simple supermodule LaTeXMLMath we define LaTeXMLMath if LaTeXMLMath is type M and LaTeXMLMath if LaTeXMLMath is type Q .
Set LaTeXMLMath for two simple supermodules of LaTeXMLMath and LaTeXMLMath respectively .
It is easy to see that LaTeXMLMath satisfies the cocycle condition LaTeXMLEquation .
It follows from Proposition LaTeXMLRef and Eqn .
( LaTeXMLRef ) that the tensor product defines an isomorphism : LaTeXMLEquation where LaTeXMLMath for simple modules LaTeXMLMath and LaTeXMLMath .
We now define a multiplication on LaTeXMLMath by the composition LaTeXMLEquation and a comultiplication on LaTeXMLMath by the composition LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath denote the induction and restriction of spin supermodules respectively , and the isomorphism LaTeXMLMath is given by LaTeXMLMath .
The above operations define a Hopf algebra structure for LaTeXMLMath .
The associativity follows from the obervation that the two different embeddings are conjugate : LaTeXMLEquation .
Using the cocycle condition ( LaTeXMLRef ) we can check the coassociativity as we did in LaTeXMLCite .
Using the cocycle LaTeXMLMath again and super analog of Mackey ’ s theorem we can check that LaTeXMLMath preserves the multiplication structure , for details see LaTeXMLCite in a similar situation ( also compare LaTeXMLCite for Hopf algebra structures in different but related setups ) .
∎ Let LaTeXMLMath be a self-dual virtual character in LaTeXMLMath , i.e .
LaTeXMLMath , the weighted bilinear form on LaTeXMLMath ( cf .
LaTeXMLCite ) is defined by LaTeXMLEquation where LaTeXMLMath the product of two characters .
The self-duality of LaTeXMLMath is equivalent to the condition that the matrix of the weighted bilinear form is symmetric .
When LaTeXMLMath is the trivial character LaTeXMLMath of LaTeXMLMath , the weighted bilinear form becomes the standard one .
One extends the weighted bilinear form to LaTeXMLMath by bilinearity .
Let LaTeXMLMath be a spin supermodule for LaTeXMLMath and LaTeXMLMath a module for LaTeXMLMath .
The tensor product LaTeXMLMath carries a natural LaTeXMLMath -grading and admits a natural spin LaTeXMLMath -supermodule structure by letting LaTeXMLEquation where LaTeXMLMath This gives rise to a morphism : LaTeXMLEquation .
Recall we have defined LaTeXMLMath associated to LaTeXMLMath .
Its character value at the class LaTeXMLMath is given by ( cf .
LaTeXMLCite ) LaTeXMLEquation .
Thus LaTeXMLMath is self-dual as long as LaTeXMLMath is .
We now introduce a weighted bilinear form on LaTeXMLMath by letting LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
The self-duality of LaTeXMLMath implies that the bilinear form LaTeXMLMath is symmetric .
When LaTeXMLMath is taken to be the trivial character LaTeXMLMath , then it reduces to the standard bilinear form on LaTeXMLMath .
The bilinear form on LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
When LaTeXMLMath is finite subgroup of LaTeXMLMath , an importance choice for LaTeXMLMath is LaTeXMLMath where LaTeXMLMath is the character afforded by the LaTeXMLMath -dimensional representation of LaTeXMLMath given by the embedding of LaTeXMLMath in LaTeXMLMath .
We shall see that the weighted bilinear form LaTeXMLMath on LaTeXMLMath becomes positive semi-definite .
This will play an important role in the later part of this paper .
In this section , we first recall a twisted Heisenberg algebra LaTeXMLMath and its Fock space LaTeXMLMath together with a bilinear form .
We define an action of LaTeXMLMath on LaTeXMLMath in terms of group-theoretic maps .
We further show that there is a natural isometric isomorphism from LaTeXMLMath to LaTeXMLMath which is compatible with the Hopf algebra structure and Heisenberg algebra action on both spaces .
Associated with a finite group LaTeXMLMath and a self-dual class function LaTeXMLMath , a twisted Heisenberg algebra LaTeXMLMath ( cf .
LaTeXMLCite ) is generated by LaTeXMLMath and a central element LaTeXMLMath , subject to the relations : LaTeXMLEquation .
For LaTeXMLMath LaTeXMLMath we write LaTeXMLMath .
The center of LaTeXMLMath is spanned by LaTeXMLMath together with LaTeXMLMath , the radical of the bilinear form LaTeXMLMath in LaTeXMLMath .
We introduce another basis for LaTeXMLMath : LaTeXMLEquation .
Equivalently LaTeXMLMath Then the commutation relations ( LaTeXMLRef ) imply that LaTeXMLEquation where LaTeXMLMath .
The Fock space LaTeXMLMath is defined to be the symmetric algebra generated by LaTeXMLMath .
There is a natural grading on LaTeXMLMath by letting LaTeXMLEquation which makes LaTeXMLMath into a LaTeXMLMath -graded space .
We define an action of LaTeXMLMath on LaTeXMLMath as follows : LaTeXMLMath acts as multiplication operator on LaTeXMLMath and LaTeXMLMath as the identity operator ; LaTeXMLMath LaTeXMLMath acts as a derivation of algebra : LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath means the very term is deleted .
Note that LaTeXMLMath is not an irreducible representation over LaTeXMLMath in general since the bilinear form LaTeXMLMath may be degenerate .
Denote by LaTeXMLMath the ideal in the symmetric algebra LaTeXMLMath generated by LaTeXMLMath .
Denote by LaTeXMLMath the quotient LaTeXMLMath .
It follows from the definition that LaTeXMLMath is a submodule of LaTeXMLMath over the Heisenberg algebra LaTeXMLMath .
In particular , this induces a Heisenberg algebra action on LaTeXMLMath which is irreducible .
The unit LaTeXMLMath in the symmetric algebra LaTeXMLMath is the highest weight vector .
We will also denote by LaTeXMLMath its image in the quotient LaTeXMLMath .
The Fock space LaTeXMLMath admits a bilinear form LaTeXMLMath determined by LaTeXMLEquation .
Here LaTeXMLMath denotes the adjoint of LaTeXMLMath .
For LaTeXMLMath we write LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath .
For LaTeXMLMath , we define LaTeXMLEquation and similarly LaTeXMLMath for LaTeXMLMath .
It is clear that both LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -bases for LaTeXMLMath .
It follows from Eqn .
( LaTeXMLRef ) and ( LaTeXMLRef ) that LaTeXMLEquation .
The bilinear form LaTeXMLMath induces a bilinear form on LaTeXMLMath which will be denoted by the same notation .
We define the characteristic map LaTeXMLMath by letting ( compare with LaTeXMLCite ) LaTeXMLEquation where LaTeXMLMath .
In the case when LaTeXMLMath is trivial , this is essentially the same as defined in LaTeXMLCite , and a different approach is given in LaTeXMLCite .
For LaTeXMLMath and LaTeXMLMath , we let LaTeXMLMath be the split conjugacy class LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Denote by LaTeXMLMath the super class function on LaTeXMLMath which takes value LaTeXMLMath on elements in the conjugacy class LaTeXMLMath and LaTeXMLMath elsewhere .
For LaTeXMLMath , LaTeXMLMath is the class function on split conjugacy classes in LaTeXMLMath which takes value LaTeXMLMath on the conjugacy class LaTeXMLMath and LaTeXMLMath elsewhere .
Given LaTeXMLMath , we denote by LaTeXMLMath the class function on LaTeXMLMath which takes value LaTeXMLMath on elements in the class LaTeXMLMath , and LaTeXMLMath elsewhere .
The following lemma follows from definitions .
The map LaTeXMLMath sends LaTeXMLMath to LaTeXMLMath .
In particular , it sends LaTeXMLMath to LaTeXMLMath in LaTeXMLMath while sending LaTeXMLMath to LaTeXMLMath .
We define LaTeXMLMath to be a map from LaTeXMLMath to itself by the following composition LaTeXMLEquation .
We also define LaTeXMLMath to be a map from LaTeXMLMath to itself as the composition LaTeXMLEquation .
We denote by LaTeXMLMath the radical of the bilinear form LaTeXMLMath in LaTeXMLMath and denote by LaTeXMLMath the quotient LaTeXMLMath , which inherits the bilinear form LaTeXMLMath from LaTeXMLMath .
The following theorem is implied by the identification of the Hopf algebra structures on LaTeXMLMath and LaTeXMLMath ( see Proposition LaTeXMLRef below ) .
LaTeXMLMath is a module over the twisted Heisenberg algebra LaTeXMLMath by letting LaTeXMLMath LaTeXMLMath act as LaTeXMLMath and LaTeXMLMath as LaTeXMLMath .
LaTeXMLMath is a submodule of LaTeXMLMath over LaTeXMLMath and the quotient LaTeXMLMath is irreducible .
The characteristic map ch is an isomorphism of LaTeXMLMath ( resp .
LaTeXMLMath , LaTeXMLMath ) and LaTeXMLMath ( resp .
LaTeXMLMath , LaTeXMLMath ) as supermodules over LaTeXMLMath .
Recall that we have defined a map from LaTeXMLMath to LaTeXMLMath by sending LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath is the character of the basic spin supermodule LaTeXMLMath .
The image of LaTeXMLMath has the following elegant description under the characteristic map in terms of a generating function in a formal variable LaTeXMLMath .
For any LaTeXMLMath , we have LaTeXMLEquation .
Let LaTeXMLMath be a character of LaTeXMLMath .It follows from Corollary LaTeXMLRef that LaTeXMLEquation .
It follows from Eq .
( LaTeXMLRef ) that LaTeXMLMath is multiplicative on LaTeXMLMath .
Thus given two characters LaTeXMLMath of LaTeXMLMath , we have LaTeXMLEquation .
Therefore the proposition is proved .
∎ The formula ( LaTeXMLRef ) holds for any LaTeXMLMath .
In particular LaTeXMLMath is self-dual if LaTeXMLMath is self-dual .
Component-wise , we obtain LaTeXMLEquation where the sum is over all the partitions LaTeXMLMath of LaTeXMLMath into odd integers .
In Sect .
LaTeXMLRef we introduced the Hopf algebra structure on LaTeXMLMath .
On the other hand it is well known that there exists a natural Hopf algebra structure on the symmetric algebra LaTeXMLMath with the usual multiplication and the comultiplication LaTeXMLMath given by LaTeXMLEquation .
The characteristic map LaTeXMLMath is an isomorphism of Hopf algebras .
First the map LaTeXMLMath is a vector space isomorphism by comparing dimension .
The algebra isomorphism follows from the Frobenius reciprocity .
On the other hand , one can check directly that LaTeXMLEquation .
Since LaTeXMLMath ( resp .
LaTeXMLMath ) for LaTeXMLMath generate LaTeXMLMath ( resp .
LaTeXMLMath ) as an algebra , we conclude that LaTeXMLMath is a Hopf algebra isomorphism by ( LaTeXMLRef ) .
∎ Recall that we have defined a bilinear form LaTeXMLMath on LaTeXMLMath and a bilinear form LaTeXMLMath on LaTeXMLMath , and thus an induced one on LaTeXMLMath .
The lemma below follows from our definition of LaTeXMLMath and the comultiplication LaTeXMLMath .
The bilinear form LaTeXMLMath on LaTeXMLMath can be characterized by the following two properties : 1 ) .
LaTeXMLMath 2 ) .
LaTeXMLMath where LaTeXMLMath .
The characteristic map ch is an isometry from the space LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be any two super class functions in LaTeXMLMath .
By definition of the characteristic map ( LaTeXMLRef ) it follows that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we have used the inner product identity ( LaTeXMLRef ) and ( LaTeXMLRef ) .
∎ From now on the bilinear form LaTeXMLMath on LaTeXMLMath is identified with the one LaTeXMLMath on LaTeXMLMath .
In this section , we construct twisted vertex operators on a space LaTeXMLMath obtained by LaTeXMLMath tensored with a certain group algebra constructed from the finite group LaTeXMLMath .
In a most important case when LaTeXMLMath is a finite subgroup of LaTeXMLMath , we obtain by vertex operator techniques a twisted affine Lie algebra and toroidal Lie algebra acting on LaTeXMLMath in terms of group-theoretic operators .
¿From now on we assume that LaTeXMLMath is a self-dual virtual character of LaTeXMLMath , then LaTeXMLMath is an integral lattice with the symmetric bilinear form LaTeXMLMath .
The quotient lattice LaTeXMLMath is an LaTeXMLMath -dimensional vector space over the field LaTeXMLMath .
Write LaTeXMLMath .
Then LaTeXMLEquation is a natural ( even ) alternating form on LaTeXMLMath and let LaTeXMLMath be its rank over LaTeXMLMath .
The alternating form LaTeXMLMath defines a central extension LaTeXMLMath of the abelian group LaTeXMLMath by the two-element group LaTeXMLMath ( cf .
LaTeXMLCite ) : LaTeXMLEquation such that LaTeXMLMath , LaTeXMLMath .
It is easily seen that LaTeXMLMath form a basis for LaTeXMLMath .
We note that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a subgroup of LaTeXMLMath which is maximal such that the alternating form LaTeXMLMath vanishes on LaTeXMLMath .
LaTeXMLCite There are LaTeXMLMath irreducible LaTeXMLMath -module structures on the space LaTeXMLMath such that LaTeXMLMath acts faithfully and LaTeXMLEquation as operators on LaTeXMLMath .
Moreover , LaTeXMLMath .
We will denote the elements of LaTeXMLMath by LaTeXMLMath , where LaTeXMLMath .
Clearly LaTeXMLEquation .
For LaTeXMLMath we write the action of LaTeXMLMath on LaTeXMLMath as LaTeXMLEquation .
Then one can check that LaTeXMLMath is a well-defined cocycle map from LaTeXMLMath .
One also has LaTeXMLMath .
We fix an irreducible LaTeXMLMath -module structure on LaTeXMLMath described in Eqn .
( LaTeXMLRef ) .
We extend the actions of LaTeXMLMath to the space of tensor product LaTeXMLEquation by letting them act on the LaTeXMLMath part trivially .
Introduce the operators LaTeXMLMath as the following compositions of maps : LaTeXMLEquation .
LaTeXMLEquation We define LaTeXMLEquation where LaTeXMLMath is a formal variable .
We define the twisted vertex operators LaTeXMLMath , LaTeXMLMath by the following generating functions : LaTeXMLEquation .
We note that LaTeXMLMath .
The operators LaTeXMLMath are well-defined operators acting on the space LaTeXMLMath We extend the bilinear form LaTeXMLMath on LaTeXMLMath to LaTeXMLMath by letting LaTeXMLEquation .
We extend the LaTeXMLMath -gradation ¿from LaTeXMLMath to LaTeXMLMath by letting LaTeXMLEquation .
Similarly we extend the bilinear form LaTeXMLMath to the space LaTeXMLEquation and extend the LaTeXMLMath -gradation on LaTeXMLMath to a LaTeXMLMath -gradation on LaTeXMLMath .
The characteristic map ch will be extended to an isometry from LaTeXMLMath to LaTeXMLMath by fixing the subspace LaTeXMLMath .
We will denote this map again by ch .
We extend the characteristic map LaTeXMLMath to a linear map LaTeXMLMath : LaTeXMLMath by LaTeXMLEquation .
The relation between the vertex operators defined in ( LaTeXMLRef ) and the Heisenberg algebra LaTeXMLMath is revealed in the following theorem .
For any LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation The first identity follows from Proposition LaTeXMLRef and the second one is obtained by noting that LaTeXMLMath is the adjoint operator of LaTeXMLMath with respect to the bilinear form LaTeXMLMath .
∎ As a consequence we have LaTeXMLEquation .
Thus the characteristic map identifies the twisted vertex operators LaTeXMLMath defined via finite groups LaTeXMLMath with the usual twisted vertex operators of LaTeXMLCite .
The normal ordered product LaTeXMLMath , LaTeXMLMath of two vertex operators is defined as follows : LaTeXMLEquation .
The following theorem can be verified using the standard vertex operator claculus ( see e.g .
LaTeXMLCite ) , where the term LaTeXMLMath is understood as the power series expansion in the variable LaTeXMLMath .
For LaTeXMLMath one has the following operator product expansion identity for twisted vertex operators .
LaTeXMLEquation .
The next proposition follows readily from Theorem LaTeXMLRef .
Given LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
Let LaTeXMLMath be a rank LaTeXMLMath complex simple Lie algebra of ADE type , and let LaTeXMLMath be the root system generated by a set of simple roots LaTeXMLMath .
Let LaTeXMLMath be the highest root .
The Lie algebra is generated by the Chevalley generators LaTeXMLMath .
We normalize the invariant bilinear form on LaTeXMLMath by LaTeXMLMath .
The twisted toroidal algebra LaTeXMLMath ( associated to LaTeXMLMath ) is the associative algebra generated by ( LaTeXMLCite ) LaTeXMLEquation subject to the relations : LaTeXMLMath is central , LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath .
The twisted affine algebra ( cf .
LaTeXMLCite ) , denoted by LaTeXMLMath , can be identified with the subalgebra of LaTeXMLMath generated by LaTeXMLEquation .
The basic twisted representation LaTeXMLMath of LaTeXMLMath is the irreducible highest weight representation generated by a highest weight vector which is annihilated by LaTeXMLMath and LaTeXMLMath acts on LaTeXMLMath as the identity operator .
Let LaTeXMLMath be a finite subgroup of LaTeXMLMath and the virtual character LaTeXMLMath to be twice the trivial character minus the character of the two-dimensional defining representation of LaTeXMLMath .
The following is the well-known list of finite groups of LaTeXMLMath : the cyclic , binary dihedral , tetrahedral , octahedral and icosahedral groups .
McKay observed that the associated matrix to the weighted bilinear form LaTeXMLMath on LaTeXMLMath with respect to the basis of irreducible characters can be identified with an affine Dynkin diagram of ADE type LaTeXMLCite .
The following theorem provides a finite group realization of the vertex representation of the twisted toroidal Lie algebra LaTeXMLMath on LaTeXMLMath .
A vertex representation of the twisted toroidal Lie algebra LaTeXMLMath is defined on the space LaTeXMLMath by letting LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
All the commutation relations without binomial coefficients are easy consequences of Proposition LaTeXMLRef and Theorem LaTeXMLRef by the usual vertex operator calculus in the twisted picture ( see LaTeXMLCite ) .
The corresponding relations with binomial coefficients in LaTeXMLMath are equivalent to LaTeXMLEquation .
LaTeXMLEquation This is proved by using Theorem LaTeXMLRef by a standard method in the theory of vertex algebras ( cf .
e.g .
LaTeXMLCite ) .
∎ Note that the quotient lattice LaTeXMLMath inherits a positive definite integral bilinear form from that of LaTeXMLMath .
Let LaTeXMLMath be the set of non-trivial irreducible characters of LaTeXMLMath .
Let LaTeXMLMath be the sublattice of LaTeXMLMath generated by LaTeXMLMath .
Denote by LaTeXMLMath the symmetric algebra generated by LaTeXMLMath LaTeXMLMath LaTeXMLMath , which is isometric to LaTeXMLMath and LaTeXMLMath as well .
The irreducible LaTeXMLMath -module LaTeXMLMath induces an irreducible LaTeXMLMath -module structure on LaTeXMLMath given by the restriction of LaTeXMLMath .
Denote by LaTeXMLMath its rank .
In this case if the determinant of the Cartan matrix is an odd integer ( see Lemma LaTeXMLRef ) , then LaTeXMLMath and the space LaTeXMLMath is trivial .
We define LaTeXMLEquation .
LaTeXMLEquation Obviously LaTeXMLMath restricted to LaTeXMLMath is an isometric isomorphism onto LaTeXMLMath .
We remark that LaTeXMLMath is isomorphic to the tensor product of the space LaTeXMLMath associated to LaTeXMLMath and the space associated to the rank LaTeXMLMath lattice LaTeXMLMath equipped with the zero bilinear form .
The identity for a product of vertex operators LaTeXMLMath associated to LaTeXMLMath ( cf .
Theorem LaTeXMLRef ) implies that LaTeXMLMath provides a realization of the vertex representation of LaTeXMLMath on LaTeXMLMath ( cf .
LaTeXMLCite ) .
The following theorem establishes a direct link ¿from the finite group LaTeXMLMath to the affine Lie algebra LaTeXMLMath .
The operators LaTeXMLMath define an irreducible representation of the affine Lie algebra LaTeXMLMath on LaTeXMLMath which is isomorphic to the twisted basic representation .
In this section we outline how to use the specialization of LaTeXMLMath to obtain the character table for the spin supermodules of LaTeXMLMath from our vertex operator approach .
The proofs are similar to those given in LaTeXMLCite ( also cf .
LaTeXMLCite ) which we omit .
Set LaTeXMLMath .
The weighted bilinear form reduces to the standard one and LaTeXMLMath .
Let LaTeXMLMath , the matrix of the alternating form LaTeXMLMath over LaTeXMLMath , then LaTeXMLMath .
Here LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd .
Consequently Lemma LaTeXMLRef implies there are exactly LaTeXMLMath irreducible LaTeXMLMath -module structures on the LaTeXMLMath -dimensional space LaTeXMLMath , where LaTeXMLMath denotes the smallest integer LaTeXMLMath .
One of the ( at most ) two irreducible module structures is given by the cocycle LaTeXMLMath , for LaTeXMLMath , and LaTeXMLMath , for LaTeXMLMath .
In the following result the bracket LaTeXMLMath denotes the anti-commutator .
The operators LaTeXMLMath LaTeXMLMath generate a Clifford algebra : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The relations can be established using Theorem LaTeXMLRef by means of standard techniques in the theory of vertex algebras , see LaTeXMLCite for detail for a similar circumstance .
∎ Let LaTeXMLMath be the lattice generated by the characters of spin irreducible LaTeXMLMath -supermodules .
Then LaTeXMLMath .
For an LaTeXMLMath -tuple index LaTeXMLMath we denote LaTeXMLEquation .
The space LaTeXMLMath is spanned by LaTeXMLMath .
However LaTeXMLMath if LaTeXMLMath .
Using the Clifford algebra structure we know that LaTeXMLMath is spanned by LaTeXMLMath , where LaTeXMLMath runs through LaTeXMLMath -tuples LaTeXMLMath and LaTeXMLMath .
We define the raising operator LaTeXMLMath by LaTeXMLEquation .
Then we define the action of the raising operator LaTeXMLMath on LaTeXMLMath or LaTeXMLMath by LaTeXMLMath or LaTeXMLMath respectively .
Given LaTeXMLMath , we define LaTeXMLEquation .
Similarly we write LaTeXMLMath LaTeXMLCite The vectors LaTeXMLMath for LaTeXMLMath and LaTeXMLMath form an orthogonal basis in the vector space LaTeXMLMath with LaTeXMLMath .
Moreover , we have that LaTeXMLEquation where LaTeXMLMath , and LaTeXMLMath is the raising operator .
We remark that the basis elements in LaTeXMLMath corresponding to LaTeXMLMath are the classical symmetric functions LaTeXMLMath called Schur ’ s Q-functions LaTeXMLCite .
If we take LaTeXMLMath as the LaTeXMLMath -th power sum in a sequence of variables LaTeXMLMath , then the space LaTeXMLMath becomes a distinguished subspace of symmetric functions generated by odd degree power sums indexed by LaTeXMLMath .
In particular LaTeXMLMath is the Schur ’ s Q-function in the variables LaTeXMLMath .
For LaTeXMLMath , we denote LaTeXMLEquation .
For LaTeXMLMath , we define LaTeXMLMath .
For a partition LaTeXMLMath and an irreducible character LaTeXMLMath of LaTeXMLMath , we define the spin character LaTeXMLMath of LaTeXMLMath to be LaTeXMLMath ( see Corollary LaTeXMLRef ) .
For each strict partition-valued function LaTeXMLMath , the vector LaTeXMLMath corresponds , under the characteristic map LaTeXMLMath , to the irreducible character LaTeXMLMath of the spin LaTeXMLMath -supermodule given by a LaTeXMLMath -linear combination of LaTeXMLEquation where LaTeXMLMath and the first summand is LaTeXMLMath with multiplicity one .
Its character at the conjugacy class of type LaTeXMLMath , where LaTeXMLMath , is equal to the matrix coefficient LaTeXMLEquation where LaTeXMLMath .
Moreover the degree of the character is equal to LaTeXMLEquation .
All the irreducible spin LaTeXMLMath -supermodules can be described easily as follows .
For each irreducible character LaTeXMLMath let LaTeXMLMath be the irreducible LaTeXMLMath -module affording LaTeXMLMath .
For each strict partition LaTeXMLMath of LaTeXMLMath let LaTeXMLMath be the corresponding irreducible spin supermodule of LaTeXMLMath ( cf .
LaTeXMLCite ) .
We have seen earlier that LaTeXMLMath is an irreducible spin LaTeXMLMath -supermodule .
For each strict partition-valued function LaTeXMLMath LaTeXMLMath LaTeXMLMath , the tensor product LaTeXMLEquation decomposes completely into LaTeXMLMath copies of an irreducible spin LaTeXMLMath -supermodule , where LaTeXMLMath denotes the number of the partitions of odd length among LaTeXMLMath .
Denote this irreducible module by LaTeXMLMath .
Then the induced supermodule LaTeXMLMath is the irreducible spin LaTeXMLMath -supermodule corresponding to LaTeXMLMath , and it is of type LaTeXMLMath or LaTeXMLMath according to LaTeXMLMath is even or odd .
0 ] Canonical Commutation Relation Preserving Maps C. Chryssomalakos and A. Turbiner ] 0 ] ICN-UNAM-01/06 LPT-ORSAY 01-29 March 31 , 2001 [ 3cm ] Canonical Commutation Relation Preserving Maps C. Chryssomalakos and A. Turbiner Present address ( on sabbatical leave ) : Laboratoire de Physique Theorique , Université Paris Sud , Orsay 91405 , France .
On leave of absence from the Institute for Theoretical and Experimental Physics , Moscow 117259 , Russia .
XX Instituto de Ciencias Nucleares Universidad Nacional Autónoma de México Apdo .
Postal 70-543 , 04510 México , D.F. , MEXICO chryss @ nuclecu.unam.mx , turbiner @ nuclecu.unam.mx X Instituto de Ciencias Nucleares Universidad Nacional Autónoma de México Apdo .
Postal 70-543 , 04510 México , D.F. , MEXICO chryss @ nuclecu.unam.mx , turbiner @ nuclecu.unam.mx X Abstract : We study maps preserving the Heisenberg commutation relation LaTeXMLMath .
We find a one-parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative .
It involves a non-local “ coordinate ” operator while the dual “ derivative ” is just the Jackson finite-difference operator .
Substitution of this realization into any differential operator involving LaTeXMLMath and LaTeXMLMath , results in an isospectral deformation of a continuous differential operator into a finite-difference one .
We extend our results to the deformed Heisenberg algebra LaTeXMLMath .
As an example of potential applications , various deformations of the Hahn polynomials are briefly discussed .
The Heisenberg algebra LaTeXMLRef ) in the sequel as the canonical commutation relation ( CCR ) .
LaTeXMLEquation made its first appearance in physics , long before the birth of Quantum Mechanics , through its realization involving a continuous coordinate LaTeXMLMath and a dual derivative LaTeXMLMath , the latter being the basic differential operator of analysis .
General differential operators , in one dimension , are then expressed in terms of powers of LaTeXMLMath , multiplied by functions of LaTeXMLMath — a wide class of physical problems leads to an eigenvalue equation for such operators .
The reason underlying the predominance of this particular realization in physics is the continuous nature of most spaces under study .
Recently , however , there has been a growing interest in discretized versions of spacetime or other , internal spaces .
This sometimes originates in the need for numerical computation , as in , e.g .
, lattice QCD , but has also been proposed as a model of small scale structure .
On another front , alternative realizations of ( LaTeXMLRef ) have emerged in string theory ( see , e.g .
, LaTeXMLCite , LaTeXMLCite ) .
In either case , LaTeXMLMath , LaTeXMLMath ceases to be the realization of choice and , in several cases , discrete ( finite difference ) operators acquire preferred status .
Maintaining the validity of ( LaTeXMLRef ) makes the transition from the continuous to the discrete non-trivial .
The need then arises for new realizations of the Heisenberg algebra in terms of discrete operators .
Given such realizations , the differential operators mentioned above can be deformed by replacing the continuous realization by a discrete one — the non-trivial feature of such deformations is that they are isospectral .
The process may be regarded as a quantum canonical transformation .
There has been already a considerable amount of research in this direction ( see , e.g .
, LaTeXMLCite and references therein ) , with the discrete derivative LaTeXMLMath , defined by LaTeXMLEquation receiving traditionally most of the attention .
It can be argued that this is due , in part , to the fact that the form of the canonically conjugate “ coordinate ” variable LaTeXMLMath is known ( see LaTeXMLCite and Ex .
LaTeXMLRef ) .
It is clear , from its definition , that LaTeXMLMath can be restricted to the ( equally spaced ) points of a lattice .
A second natural choice would be an exponential lattice , the corresponding finite difference operator being the Jackson derivative ( or Jackson symbol , see LaTeXMLCite ) , defined by LaTeXMLEquation .
The problem with this choice , and the main motivation for this work , is that the form of the canonically conjugate “ coordinate ” operator LaTeXMLMath seems to be unknown .
We solve this problem in Sec .
LaTeXMLRef below while in Sec .
LaTeXMLRef we study representations , in a very general setting .
Sec .
LaTeXMLRef shows how to generate new canonical commutation relation preserving maps from known ones and Sec .
LaTeXMLRef briefly extends the above results to the case of LaTeXMLMath -canonical commutation relations .
In Sec .
LaTeXMLRef two isospectral deformations of the Hahn operator are presented as a concrete application — the corresponding polynomial eigenfunctions are also supplied .
Given the Jackson derivative LaTeXMLMath , satisfying LaTeXMLEquation .
One finds LaTeXMLEquation where LaTeXMLEquation .
The notation used in the r.h.s .
of the above equation is as follows .
LaTeXMLMath denotes the “ vacuum ” , a ket annihilated by derivatives , LaTeXMLMath .
The subscript of LaTeXMLMath is an instruction : express all variables to its left in terms of LaTeXMLMath , LaTeXMLMath ( already in this form , in this particular example ) , then use the commutation relation ( LaTeXMLRef ) to bring the LaTeXMLMath ’ s to the right of the LaTeXMLMath ’ s .
There they are annihilated by LaTeXMLMath leaving a function of LaTeXMLMath only — this function serves to define the l.h.s. , i.e .
, the action of LaTeXMLMath on LaTeXMLMath .
For a general function LaTeXMLMath , defined as a Taylor series in LaTeXMLMath , the above relation leads to the alternative definition ( LaTeXMLRef ) , which makes it clear that LaTeXMLMath acts on the exponential lattice LaTeXMLMath .
LaTeXMLMath can be realized as a pseudodifferential operator LaTeXMLEquation where LaTeXMLMath is the partial derivative w.r.t .
LaTeXMLMath , LaTeXMLMath , also annihilating the vacuum , LaTeXMLMath .
For the reasons mentioned in the introduction , one would like to realize also an operator LaTeXMLMath , such that LaTeXMLMath .
Using the commutation relation LaTeXMLMath we find LaTeXMLMath , so that ( LaTeXMLMath ) LaTeXMLEquation .
LaTeXMLEquation It will prove convenient in what follows to use the notation LaTeXMLMath , with LaTeXMLMath .
Notice that LaTeXMLMath .
We rewrite ( LaTeXMLRef ) LaTeXMLEquation .
LaTeXMLMath is of the form LaTeXMLMath , LaTeXMLMath .
We look for LaTeXMLMath in the form LaTeXMLMath .
Then LaTeXMLEquation .
The r.h.s .
above should be equal to LaTeXMLMath .
One concludes that LaTeXMLMath , i.e .
, LaTeXMLEquation .
The action of LaTeXMLMath on monomials is LaTeXMLEquation .
LaTeXMLMath above acts on power series as a discrete derivative .
We examine the corresponding interpretation of the action of LaTeXMLMath .
To this end , we introduce the Jackson integral operator LaTeXMLMath ( see , e.g .
, LaTeXMLCite ) given by LaTeXMLEquation .
Notice that LaTeXMLMath is invertible on the image of LaTeXMLMath , i.e .
, on LaTeXMLMath , LaTeXMLMath .
Comparison with ( LaTeXMLRef ) shows that LaTeXMLMath while LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Using the expansion LaTeXMLEquation one finds LaTeXMLEquation i.e .
, LaTeXMLMath gives the area under the dotted lines in Fig .
LaTeXMLRef and converges to LaTeXMLMath in the limit LaTeXMLMath .
Using the second of ( LaTeXMLRef ) , we find LaTeXMLEquation .
In other words , the action of LaTeXMLMath on LaTeXMLMath consists in first producing the function LaTeXMLMath and then multiplying this latter by the classical coordinate LaTeXMLMath .
Aside : We derive an alternative expression for LaTeXMLMath .
In classical calculus one has ( Rolle theorem ) LaTeXMLEquation where LaTeXMLMath denotes ( classical ) averaging in the interval LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is the ( classical ) derivative w.r.t .
LaTeXMLMath .
Let now LaTeXMLMath denote quantum averaging , LaTeXMLEquation .
Then LaTeXMLEquation .
LaTeXMLEquation which is the LaTeXMLMath -deformed ( “ quantum ” ) analogue of ( LaTeXMLRef ) .
LaTeXMLMath Since LaTeXMLMath , LaTeXMLMath obey the CCR , one can define a quantum action LaTeXMLMath , in complete analogy to the classical one , LaTeXMLEquation where , in the r.h.s. , LaTeXMLMath is commuted past LaTeXMLMath until it reaches the vacuum , where it gets annihilated – the remaining function of LaTeXMLMath is , by definition , LaTeXMLMath ( notice that the subscript of LaTeXMLMath instructs to express everything in terms of LaTeXMLMath , LaTeXMLMath ) .
This extends to arbitrary operators LaTeXMLMath acting on functions of LaTeXMLMath , just like in the classical case .
It follows trivially that We have LaTeXMLMath — the ordering of the LaTeXMLMath ’ s and LaTeXMLMath ’ s in LaTeXMLMath is immaterial , as long as the same ordering is used in LaTeXMLMath .
LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , is the LaTeXMLMath -deformation map .
We note in passing that the quantum averaging operator , LaTeXMLMath , is the inverse of LaTeXMLMath .
Notice also that the product LaTeXMLMath is constant in LaTeXMLMath ( i.e .
, invariant under the deformation ) , LaTeXMLEquation .
This implies that , when dealing with the LaTeXMLMath -deformation of a general differential operator , terms of the form LaTeXMLMath , with LaTeXMLMath , can be expressed entirely in terms of LaTeXMLMath and LaTeXMLMath , the action of which is simpler .
The resulting LaTeXMLMath -deformed operator is a differential-difference operator .
This occurs , for example , in the study of the hypergeometric operator .
Notice also that the invariance of LaTeXMLMath permits the exponentiation of the infinitesimal generator of LaTeXMLMath in a trivial manner .
Finally , it is worth pointing out that the map LaTeXMLMath admits a non-trivial classical limit , which preserves Poisson brackets .
Indeed , with LaTeXMLMath , LaTeXMLMath satisfying LaTeXMLMath , where LaTeXMLMath is the Poisson bracket , one can easily verify that LaTeXMLMath , where LaTeXMLMath , giving rise to a wide class of classical canonical transformations that might in itself be worth exploring .
Consider the Heisenberg-Weyl universal enveloping algebra LaTeXMLMath , generated by LaTeXMLMath , LaTeXMLMath , with LaTeXMLMath .
In the sequel we work with a certain completion LaTeXMLMath of LaTeXMLMath , that allows us to deal with e.g .
, exponentials in LaTeXMLMath , LaTeXMLMath .
From the discussion of the previous section ( see ( LaTeXMLRef ) ) , we abstract a map LaTeXMLMath that preserves the CCR LaTeXMLRef ) LaTeXMLEquation where LaTeXMLMath now is LaTeXMLMath .
All equations in the previous section depend only on LaTeXMLMath , LaTeXMLMath satisfying the CCR and are therefore valid for LaTeXMLMath .
Although we use the particular map LaTeXMLMath given above as an example , we emphasize that our results below are general .
For any pair LaTeXMLMath of abstract generators that satisfy the CCR , we say that the set LaTeXMLMath is an adapted basis for LaTeXMLMath if LaTeXMLMath , LaTeXMLMath act on it as lowering and raising operators respectively LaTeXMLEquation .
A classical adapted basis One particular representation of the Heisenberg algebra is supplied by the subalgebra generated by LaTeXMLMath , an adapted basis being given by LaTeXMLMath .
The action of the Heisenberg algebra generators on an arbitrary power series LaTeXMLMath is LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath Suppose now we are given a CCR-preserving map LaTeXMLMath , where LaTeXMLMath denotes any parameters LaTeXMLMath might depend on , and we wish to find an adapted basis for the deformed generators it produces .
A general solution to this problem is possible if we further impose the restriction that LaTeXMLMath be counit preserving ( i.e .
, LaTeXMLMath ) .
It is worth emphasizing that this requirement , although rather natural , excludes nevertheless a number of familiar CCR-preserving maps , like the rotation from LaTeXMLMath , LaTeXMLMath to LaTeXMLMath , LaTeXMLMath in the simple harmonic oscillator .
Keeping this observation in mind , we proceed to the following statement : given any CCR and counit-preserving map LaTeXMLMath , one can find in general an induced map LaTeXMLMath that maps any adapted basis LaTeXMLMath for LaTeXMLMath to an adapted basis LaTeXMLMath for LaTeXMLMath .
Indeed , to any function LaTeXMLMath one can associate its LaTeXMLMath - projection LaTeXMLMath given by LaTeXMLMath -projection of LaTeXMLMath depends on the particular deformation LaTeXMLMath used — for simplicity of notation we do not show this dependence explicitly .
LaTeXMLEquation .
We now show that LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
We have LaTeXMLEquation .
We comment briefly on the steps that lead to ( LaTeXMLRef ) .
The first equality follows from ( LaTeXMLRef ) , taking into account that LaTeXMLMath is counit preserving , so that we may put LaTeXMLMath in place of LaTeXMLMath in the subscript of the vacuum .
The second equality follows from ( LaTeXMLRef ) .
In the expression LaTeXMLMath we are instructed to express LaTeXMLMath , LaTeXMLMath in terms of LaTeXMLMath , LaTeXMLMath , and then bring the LaTeXMLMath ’ s to the right etc .
.. One can do this though in several ways .
The one shown above involves first bringing all LaTeXMLMath ’ s to the right of the LaTeXMLMath ’ s , then substituting LaTeXMLMath and bringing the LaTeXMLMath ’ s to the right ( this is equivalent to annihilating the LaTeXMLMath ’ s themselves , since LaTeXMLMath preserves the counit ) .
At this point one is left with a function of LaTeXMLMath which is clearly LaTeXMLMath .
Finally , one substitutes LaTeXMLMath and brings the LaTeXMLMath ’ s to the right .
Given an adapted basis LaTeXMLMath for LaTeXMLMath , we construct the set LaTeXMLMath , where LaTeXMLMath is the ket LaTeXMLMath .
LaTeXMLEquation and claim that it is an adapted basis for LaTeXMLMath .
Indeed , LaTeXMLEquation .
Also , LaTeXMLEquation .
A quantum adapted basis Continuing our earlier classical example , we now turn to the realization of the Heisenberg algebra provided by the map LaTeXMLMath .
We take LaTeXMLMath and find for LaTeXMLMath LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath A second discrete realization Consider the pair of operators LaTeXMLEquation .
One easily verifies that LaTeXMLMath — this is the LaTeXMLMath -realization of the CCR mentioned in the introduction .
We take again LaTeXMLMath and compute LaTeXMLMath LaTeXMLEquation .
For the falling LaTeXMLMath -factorial polynomials LaTeXMLMath , defined by the next-to-last line above ( also known as LaTeXMLMath -quasi-monomials LaTeXMLCite ) , it holds LaTeXMLEquation where LaTeXMLMath are the Stirling numbers of the first kind .
LaTeXMLMath The LaTeXMLMath -exponential as a LaTeXMLMath -projection Consider the spectral problem LaTeXMLEquation .
Relying on ( LaTeXMLRef ) , we look instead at the equation LaTeXMLMath and compute LaTeXMLMath above from LaTeXMLMath .
We get LaTeXMLMath and , using LaTeXMLMath , we find LaTeXMLEquation i.e .
, the standard LaTeXMLMath -deformed exponential LaTeXMLCite is just LaTeXMLMath .
More generally , if LaTeXMLEquation the eigenfunctions of LaTeXMLMath are LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath It is important to emphasize at this point the formal character of the above results .
In particular , a given eigenfunction LaTeXMLMath of some differential operator may converge for all LaTeXMLMath , while its LaTeXMLMath -projection LaTeXMLMath might have a finite ( or even zero ) radius of convergence as it happens , for example , with the LaTeXMLMath -exponential LaTeXMLEquation that appears as the projection LaTeXMLMath for the map LaTeXMLMath of Ex .
LaTeXMLRef .
If LaTeXMLMath , LaTeXMLMath are CCR-preserving maps then so is their composition LaTeXMLMath .
Considering only smooth maps , with a smooth inverse , one arrives at the notion of the group of CCR-preserving maps ( CCR-PM ) .
In the sequel we impose further the requirement that our maps preserve the counit .
We can then use the fact that LaTeXMLEquation to compute the induced map of a composition of maps .
We illustrate this in the following Composition of LaTeXMLMath , LaTeXMLMath Consider the map LaTeXMLMath discussed earlier .
For each value of LaTeXMLMath , LaTeXMLMath is an element of the CCR-PM .
However , LaTeXMLMath is not a one-parameter subgroup since , in general , there is no LaTeXMLMath such that LaTeXMLMath .
Notice also that LaTeXMLMath , for finite LaTeXMLMath , LaTeXMLMath .
Similar remarks hold for LaTeXMLMath .
For the composition LaTeXMLMath we find LaTeXMLEquation .
LaTeXMLEquation while for LaTeXMLMath we get LaTeXMLEquation .
LaTeXMLEquation For the adapted bases that correspond to the above compositions we find LaTeXMLEquation .
For example , LaTeXMLEquation .
Also , LaTeXMLEquation so that , for example , LaTeXMLEquation which should be compared with ( LaTeXMLRef ) .
LaTeXMLMath Given LaTeXMLMath , LaTeXMLMath satisfying the LaTeXMLMath -Heisenberg algebra LaTeXMLEquation with LaTeXMLMath .
We say that LaTeXMLMath is the quantum canonical conjugate ( QCC ) of LaTeXMLMath ( and vice versa ) .
To complete our treatment of the map LaTeXMLMath of Ex .
LaTeXMLRef , we undertake here the determination of the QCC of LaTeXMLMath .
We work again with abstract operators LaTeXMLMath , LaTeXMLMath and remark that LaTeXMLMath ( given in ( LaTeXMLRef ) ) and LaTeXMLMath satisfy ( LaTeXMLRef ) , LaTeXMLMath .
Notice that , the ( classical ) Heisenberg algebra admits the LaTeXMLMath -involution LaTeXMLEquation which we extend as complex conjugation to the parameter LaTeXMLMath , LaTeXMLMath .
Then , taking the LaTeXMLMath of ( LaTeXMLRef ) ( with LaTeXMLMath expressed in terms of LaTeXMLMath , LaTeXMLMath , as in ( LaTeXMLRef ) ) , we find LaTeXMLEquation .
LaTeXMLEquation Up to now we disposed of the deforming map LaTeXMLMath , LaTeXMLMath , which , applied to a pair of operators satisfying the classical Heisenberg algebra , produces a pair satisfying the quantum Heisenberg algebra .
Notice that it does so by leaving LaTeXMLMath invariant and only deforming LaTeXMLMath .
What we have achieved in ( LaTeXMLRef ) , is to produce a second similar map LaTeXMLMath , which instead leaves LaTeXMLMath invariant and deforms only LaTeXMLMath : LaTeXMLMath , LaTeXMLMath .
We only need apply LaTeXMLMath to ( LaTeXMLRef ) to get LaTeXMLEquation which identifies LaTeXMLMath as the QCC of LaTeXMLMath .
As an example of potential applications of our results , we present here various deformations of the Hahn operator and its eigenfunctions , the Hahn polynomials .
We start with some definitions .
The action of the Hahn operator LaTeXMLMath on a function LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath are parameters .
What distinguishes LaTeXMLMath is that it is the most general three-point finite-difference operator with infinitely many polynomial eigenfunctions LaTeXMLCite .
The latter are called Hahn polynomials ( of continuous argument ) and we denote them by LaTeXMLMath , where LaTeXMLEquation and we have set , without loss of generality , LaTeXMLMath and LaTeXMLMath .
The corresponding eigenvalues are LaTeXMLEquation .
For particular values of their parameters and/or arguments , LaTeXMLMath reduce to the Meixner , Charlier , Tschebyschev , Krawtchouk or ( discrete argument ) Hahn polynomials ( for details and their rôle in finite difference equations , see LaTeXMLCite and references therein ) .
We will use the form LaTeXMLEquation where LaTeXMLMath is as in ( LaTeXMLRef ) and the LaTeXMLMath are known coefficients .
It has been shown in LaTeXMLCite that LaTeXMLMath belongs to LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath are given in ( LaTeXMLRef ) , with LaTeXMLMath .
We are now at a setting where our earlier results may be applied directly .
First , we deform isospectrally LaTeXMLMath to LaTeXMLMath , by effecting the substitution LaTeXMLMath in ( LaTeXMLRef ) — a little bit of algebra gives LaTeXMLEquation .
The corresponding polynomial eigenfunctions are obtained directly from ( LaTeXMLRef ) , using the results of Ex .
LaTeXMLRef , LaTeXMLEquation ( the eigenvalues are , of course , still given by ( LaTeXMLRef ) ) .
A second isospectral deformation , involving the substitution LaTeXMLMath in ( LaTeXMLRef ) , leads to LaTeXMLEquation ( LaTeXMLMath ) with polynomial eigenfunctions LaTeXMLEquation .
Notice the ease with which LaTeXMLMath are computed , despite the highly non-trivial complexity of the differential-difference operator LaTeXMLMath .
Finally , we point out that the spectrum of LaTeXMLMath in ( LaTeXMLRef ) may also be LaTeXMLMath -deformed by effecting the substitution LaTeXMLMath in ( LaTeXMLRef ) — one gets a finite-difference operator LaTeXMLEquation with infinitely many polynomial eigenfunctions — the corresponding eigenvalues are LaTeXMLEquation .
Note Added After this work was sent for publication , Professor C. Zachos , to whom we express our gratitude , pointed out to us that LaTeXMLMath , LaTeXMLMath can be related to LaTeXMLMath , LaTeXMLMath via a similarity transformation .
Indeed , starting with the ansatz LaTeXMLMath , it follows that LaTeXMLMath and , using ( LaTeXMLRef ) , LaTeXMLMath , from which the formal expression LaTeXMLMath can be derived .
Here LaTeXMLMath denotes the LaTeXMLMath -deformed gamma function , LaTeXMLMath ( see , e.g .
, LaTeXMLCite ) .
Then , LaTeXMLMath is computed as LaTeXMLMath and the constancy of LaTeXMLMath in LaTeXMLMath follows trivially .
Methods of Lie group analysis of differential equations are extended to weak solutions of ( linear and nonlinear ) PDEs , where the term “ weak solution ” comprises the following settings : Distributional solutions .
Solutions in generalized function algebras .
Solutions in the sense of association ( corresponding to a number of weak or integral solution concepts in classical analysis ) .
Factorization properties and infinitesimal criteria are developed that allow to treat all three settings simultaneously , thereby unifying and extending previous work in this area .
Key words .
Algebras of generalized functions , Lie symmetries of differential equations , group analysis , Colombeau algebras .
Mathematics Subject Classification ( 2000 ) .
46F30 , 22E70 , 35Dxx , 35A30 .
Local symmetries for equations with weak type solutions , such as , e.g. , conservation laws , involve different constraints depending on the framework in which the equations are analyzed .
The aim of this paper is to study symmetry properties of differential equations involving singular ( in particular : distributional ) terms through an analysis of symmetries in distribution spaces and generalized function algebras , as well as associated ( i.e. , weak type ) symmetries .
A main ingredient in our analysis will be the determination of infinitesimal criteria for these solution concepts .
Investigations in this direction have been initiated by Methé ( LaTeXMLCite ) , Tengstrand ( LaTeXMLCite ) , Szmydt and Ziemian ( LaTeXMLCite ) and have been systematically pursued by Berest and Ibragimov ( LaTeXMLCite ) in the distributional setting .
More recently , in LaTeXMLCite , an extension of the purely distributional methods applied so far has been given that allows to also consider nonlinear equations involving singularities .
The basic tool allowing for such an extension is Colombeau ’ s theory of algebras of generalized functions .
In what follows , on the one hand we are going to continue the analysis of LaTeXMLCite , and on the other hand we shall establish connections between the distributional criteria developed in LaTeXMLCite and the Colombeau-type methods given in LaTeXMLCite .
Moreover , the framework of generalized function algebras enables us to study symmetry properties of associated solutions ( i.e .
of weak or integral solutions ) by the same methods .
To begin with , let us fix some notations concerning group analysis of differential equations and Colombeau ’ s theory of generalized functions .
Our principal reference for symmetries of differential equations is LaTeXMLCite , whose terminology we shall follow closely .
Let LaTeXMLMath be an open subset of LaTeXMLMath ( in what follows , LaTeXMLMath will be the number of independent variables of a system of differential equations , LaTeXMLMath the number of dependent variables ) and LaTeXMLMath a Lie group acting regularly on LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath we write LaTeXMLEquation .
If LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath is called projectable .
Elements of the Lie algebra of LaTeXMLMath as well as the corresponding vector fields on LaTeXMLMath will typically be denoted by LaTeXMLMath .
Put LaTeXMLMath , LaTeXMLMath .
By identifying a function LaTeXMLMath with its graph LaTeXMLMath , the action of LaTeXMLMath onto LaTeXMLMath is defined ( locally ) by LaTeXMLEquation where LaTeXMLMath is the identity mapping on LaTeXMLMath ; in the projectable case this specializes to LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath .
Here LaTeXMLMath , with coordinates LaTeXMLMath , LaTeXMLMath , LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath , is the number of different partial derivatives of order LaTeXMLMath of a scalar valued smooth function of LaTeXMLMath variables .
Elements of LaTeXMLMath are denoted by LaTeXMLMath .
LaTeXMLMath is called the LaTeXMLMath -jet space of LaTeXMLMath and we set LaTeXMLMath The coordinates on LaTeXMLMath will also be written as LaTeXMLMath .
For any LaTeXMLMath , the LaTeXMLMath -th prolongation or LaTeXMLMath -jet of LaTeXMLMath is the function LaTeXMLMath formed by LaTeXMLMath and its derivatives up to order LaTeXMLMath .
The LaTeXMLMath -th prolongation of a group action LaTeXMLMath or vector field LaTeXMLMath is written as LaTeXMLMath or LaTeXMLMath , respectively .
Let LaTeXMLMath be a system of differential equations with LaTeXMLMath variables and LaTeXMLMath unknown functions of the form LaTeXMLEquation ( with LaTeXMLMath smooth for all LaTeXMLMath ) .
We shall henceforth assume that ( LaTeXMLRef ) is nondegenerate ( i.e .
locally solvable and of maximal rank , see LaTeXMLCite ) .
Any LaTeXMLMath which solves the system on its domain will be called a solution .
This amounts to saying that the graph of the LaTeXMLMath -jet of LaTeXMLMath , LaTeXMLMath is contained in the zero-set LaTeXMLMath of LaTeXMLMath .
A symmetry group of LaTeXMLMath is a local transformation group on LaTeXMLMath such that if LaTeXMLMath is a solution of the system , LaTeXMLMath and LaTeXMLMath is defined then also LaTeXMLMath is a solution of LaTeXMLMath .
Next , let us shortly recall some basic definitions from Colombeau ’ s theory of generalized functions ( LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
For notational simplicity we are going to work in the so-called “ special ” Colombeau algebra LaTeXMLMath , defined as the quotient algebra LaTeXMLMath , where LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath .
LaTeXMLMath is a differential algebra ( with componentwise operations ) and LaTeXMLMath is a fine sheaf on LaTeXMLMath .
The equivalence class of LaTeXMLMath in LaTeXMLMath will be denoted by LaTeXMLMath or LaTeXMLMath for short .
We shall make use of the function spaces LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLMath is the space of test functions on LaTeXMLMath , elements of LaTeXMLMath and LaTeXMLMath are called rapidly decreasing and slowly increasing , respectively .
The algebra LaTeXMLMath of tempered generalized functions is defined by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By componentwise insertion , elements of LaTeXMLMath and LaTeXMLMath can be composed with slowly increasing functions .
Next , choose LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLMath ( the space of compactly supported distributions ) is linearly embedded into LaTeXMLMath via LaTeXMLMath ( where LaTeXMLMath ) .
Moreover LaTeXMLMath coincides with the identical embedding LaTeXMLMath on LaTeXMLMath , so LaTeXMLMath becomes a subalgebra of LaTeXMLMath via LaTeXMLMath .
Finally , there is a unique sheaf morphism LaTeXMLMath extending LaTeXMLMath to LaTeXMLMath ( where LaTeXMLMath denotes the space of distributions ) .
LaTeXMLMath commutes with partial derivatives , and its restriction to LaTeXMLMath is a sheaf morphism of algebras .
The map LaTeXMLMath defined above also provides a linear embedding of LaTeXMLMath into LaTeXMLMath commuting with partial derivatives and making LaTeXMLEquation a faithful subalgebra .
So far , Colombeau algebras are the only known differential algebras enjoying these optimal embedding properties .
Moreover , an intrinsic global formulation of Colombeau ’ s construction on differentiable manifolds retaining all the characteristics of the local theory has recently been achieved ( LaTeXMLCite ) .
Of the various variants of Colombeau algebras we shall also use the subalgebra LaTeXMLMath of LaTeXMLMath consisting of those elements of LaTeXMLMath possessing a representative LaTeXMLMath such that LaTeXMLMath .
Finally , we shall employ the “ mixed type ” -algebra LaTeXMLMath whose elements satisfy LaTeXMLMath -bounds with respect to LaTeXMLMath and LaTeXMLMath -bounds with respect to LaTeXMLMath .
The ring of constants in LaTeXMLMath is denoted by LaTeXMLMath , its elements are called generalized numbers .
Elements of LaTeXMLMath may be used to model infinitesimal numbers ( e.g. , LaTeXMLMath is a representative of an infinitely small yet nonzero generalized number ) , which may be viewed as a “ nonstandard ” aspect of the theory .
The support of a generalized function LaTeXMLMath , LaTeXMLMath , is defined as the complement of the largest open subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath .
This notion is coherent with the embedding LaTeXMLMath , i.e .
for any LaTeXMLMath we have LaTeXMLMath .
Finally , we mention the concept of association in the algebra LaTeXMLMath : Two elements LaTeXMLMath , LaTeXMLMath are associated LaTeXMLMath if there exist representatives LaTeXMLMath and LaTeXMLMath of LaTeXMLMath and LaTeXMLMath , respectively , such that LaTeXMLEquation .
Clearly , this definition does not depend on the choice of representatives .
The concept of association ( resp .
strong association , cf .
LaTeXMLRef below ) plays a central role in the Colombeau framework as in many cases it allows for a distributional interpretation of results achieved in LaTeXMLMath .
Particularly in applications to physics and numerics it is of fundamental importance ( cf .
e.g .
LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
With this terminology at hand we can now formulate the main goals of this article : Let LaTeXMLMath be a transformation group acting on the space of independent and dependent variables of a system ( LaTeXMLRef ) of ( linear or nonlinear ) differential equations .
We are looking for criteria for LaTeXMLMath to transform weak solutions of ( LaTeXMLRef ) into weak solutions .
More precisely , we shall develop conditions under which LaTeXMLMath transforms LaTeXMLMath -solutions to LaTeXMLMath -solutions ( in case ( LaTeXMLRef ) is linear ) , LaTeXMLMath -solutions to LaTeXMLMath -solutions , or solutions in the sense of association into solutions in the sense of association ( in which case ( LaTeXMLRef ) is to be replaced by LaTeXMLMath ) .
We shall see that these questions are in fact closely linked and that criteria for one situation are often useful in other cases as well .
As an application we study the symmetries of the quasilinear hyperbolic system LaTeXMLMath where LaTeXMLMath is an LaTeXMLMath matrix with LaTeXMLMath entries ( cf .
LaTeXMLCite , LaTeXMLCite ) .
For treating this system distribution type spaces are not convenient , while LaTeXMLMath provides a quite satisfactory solution concept ( cf .
LaTeXMLCite , LaTeXMLCite and the literature cited therein ) .
We consider strongly associated solutions and calculate symmetry transformations of such solutions .
Also , we discuss infinitesimal criteria for symmetries of this system .
Let LaTeXMLMath be a projectable group action on some open set LaTeXMLMath .
As was already pointed out in the introduction , composition in the framework of Colombeau generalized functions requires polynomial growth restrictions on the smooth functions .
Thus we first single out those group actions which can be applied to elements of LaTeXMLMath ( cf .
LaTeXMLCite ) .
An element LaTeXMLMath with LaTeXMLMath , is called slowly increasing if LaTeXMLMath is slowly increasing , uniformly for LaTeXMLMath in compact sets ; LaTeXMLMath is called strictly slowly increasing if LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is slowly increasing , the action of LaTeXMLMath on LaTeXMLMath is the element of LaTeXMLMath given by LaTeXMLEquation .
Also , in order to be able to insert elements of LaTeXMLMath into ( LaTeXMLRef ) we will from now on suppose that LaTeXMLMath is slowly increasing , uniformly for LaTeXMLMath in compact sets .
A symmetry group of ( LaTeXMLRef ) in LaTeXMLMath is a local transformation group acting on LaTeXMLMath such that if LaTeXMLMath is a solution of the system in LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is defined , then also LaTeXMLMath is a solution of LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be a slowly increasing symmetry group of some differential equation LaTeXMLEquation and let LaTeXMLMath be a generalized solution to ( LaTeXMLRef ) .
Then for any representative LaTeXMLMath of LaTeXMLMath there exists some LaTeXMLMath such that for all LaTeXMLMath and all LaTeXMLMath we have LaTeXMLEquation .
Due to the nontrivial right hand side of ( LaTeXMLRef ) it is clear that a direct ( componentwise ) application of classical symmetry methods to Colombeau solutions is not feasible .
A transfer of classical symmetry groups into the LaTeXMLMath -setting therefore has to rely on properties of symmetry transformations that are better suited to the algebraic structure of LaTeXMLMath .
The key concept serving this purpose ( and , as we shall see shortly , at the same time applicable to symmetries of LaTeXMLMath - and other weak solutions ) is that of factorization : Let LaTeXMLMath be a ( classical ) one parameter symmetry group of ( LaTeXMLRef ) LaTeXMLEquation .
Then by LaTeXMLCite , Prop .
2.10 there exists a smooth map LaTeXMLMath ( LaTeXMLMath open in LaTeXMLMath , LaTeXMLMath ) such that LaTeXMLEquation .
Throughout this paper , LaTeXMLMath will denote an open set as specified above .
By LaTeXMLCite , Th .
3.4 for any smooth LaTeXMLMath such that LaTeXMLMath exists we have LaTeXMLEquation .
In particular , LaTeXMLMath is a symmetry group of ( LaTeXMLRef ) in LaTeXMLMath if LaTeXMLMath LaTeXMLMath is slowly increasing with respect to LaTeXMLMath , uniformly for LaTeXMLMath varying in compact sets ( LaTeXMLCite , Proposition 3.5 ) .
Thus the need for conditions ensuring that the LaTeXMLMath remain well behaved ( in the above sense ) arises .
Theorem 3.8 of LaTeXMLCite shows that for scalar differential equations possessing a stand alone term ( i.e .
LaTeXMLMath for some LaTeXMLMath ) this is indeed always the case .
Our first aim is to generalize this result .
Since ( LaTeXMLRef ) is nondegenerate it follows that the Jacobian LaTeXMLMath of LaTeXMLMath has rank LaTeXMLMath on the zero-set of LaTeXMLMath .
The following result uses a mild strengthening of this assumption to derive a factorization property adapted to the polynomial growth restrictions necessary for applying nonlinearities to elements of LaTeXMLMath .
Theorem Let LaTeXMLMath be a slowly increasing symmetry group of system ( LaTeXMLRef ) and suppose that there exist LaTeXMLMath such that , setting LaTeXMLMath , the following conditions are satisfied : LaTeXMLMath is injective , where LaTeXMLMath .
LaTeXMLMath is defined globally and is slowly increasing , uniformly for LaTeXMLMath varying in compact sets .
Then there exists a smooth mapping LaTeXMLMath which is slowly increasing in LaTeXMLMath , uniformly for LaTeXMLMath varying in compact sets such that ( LaTeXMLRef ) holds .
In particular , ( LaTeXMLRef ) holds for any smooth LaTeXMLMath such that LaTeXMLMath exists .
Proof .
Without loss of generality we may suppose LaTeXMLMath .
We set LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath and define LaTeXMLMath by LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath is a diffeomorphism by ( i ) .
Moreover , LaTeXMLEquation .
Now set LaTeXMLMath .
By LaTeXMLCite , Proposition 3.7 and our general assumption on LaTeXMLMath , LaTeXMLMath is slowly increasing in LaTeXMLMath , uniformly for LaTeXMLMath varying in compact sets .
Since LaTeXMLMath is a symmetry group of ( LaTeXMLRef ) we have LaTeXMLMath if LaTeXMLMath .
Thus LaTeXMLEquation .
Inserting LaTeXMLMath into ( LaTeXMLRef ) we arrive at LaTeXMLEquation .
In order to establish the claimed growth properties of LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath are slowly increasing , by the chain rule it suffices to estimate LaTeXMLMath .
The determinant of this Jacobian is precisely LaTeXMLMath , so the claim follows from ( ii ) .
LaTeXMLMath Remarks By dropping the growth restrictions on LaTeXMLMath , LaTeXMLMath ( in particular , allowing for LaTeXMLMath to be nonprojectable and merely supposing that LaTeXMLMath exists globally ) , the same proof as above still provides the explicit form ( LaTeXMLRef ) of the factorization property of general symmetry groups of ( LaTeXMLRef ) , which reads LaTeXMLEquation .
A necessary and sufficient condition for ( ii ) in LaTeXMLRef is given by LaTeXMLEquation .
An extensive compilation of sufficient conditions for global injectivity of smooth maps ( LaTeXMLMath in our case ) can be found in LaTeXMLCite .
As an example we mention a result of Gale and Nikaido ( LaTeXMLCite , ch .
3 ) stating that any LaTeXMLMath ( with LaTeXMLMath a rectangular region in LaTeXMLMath ) is injective if its Jacobian is a LaTeXMLMath -matrix on all of LaTeXMLMath ( i.e .
all principal minors are positive ) .
Moreover , global invertibility results for Sobolev functions are given in LaTeXMLCite .
Corollary Under the assumptions of Theorem LaTeXMLRef , LaTeXMLMath is a symmetry group of ( LaTeXMLRef ) in LaTeXMLMath .
Proof .
Let LaTeXMLMath be a solution of ( LaTeXMLRef ) .
Then by ( LaTeXMLRef ) , for any representative LaTeXMLMath of LaTeXMLMath we have , taking into account the projectability of LaTeXMLMath , LaTeXMLEquation .
Here , LaTeXMLMath is negligible by definition and Theorem LaTeXMLRef shows LaTeXMLMath to be moderate .
It follows that LaTeXMLMath is negligible , which precisely means that LaTeXMLMath is a solution in LaTeXMLMath .
LaTeXMLMath Example Suppose that system ( LaTeXMLRef ) satisfies LaTeXMLMath for some LaTeXMLMath and nonzero constants LaTeXMLMath .
Then ( LaTeXMLRef ) satisfies conditions ( i ) and ( ii ) of Theorem LaTeXMLRef .
In fact , LaTeXMLMath is constant and since the underlying domain is convex the above assumption on ( LaTeXMLRef ) is equivalent with LaTeXMLEquation with LaTeXMLMath smooth and LaTeXMLMath as in LaTeXMLRef ( i ) .
From this , injectivity of LaTeXMLMath is immediate .
Thus by LaTeXMLRef any slowly increasing classical symmetry group of ( LaTeXMLRef ) is a LaTeXMLMath -symmetry group as well .
This observation applies e.g .
to the system LaTeXMLEquation considered in LaTeXMLCite , Example 3.6 .
Also , Theorem 3.8 of LaTeXMLCite is a special case of this setup ( for LaTeXMLMath ) .
In this section we are going to investigate symmetries of system ( LaTeXMLRef ) in the sense of association , i.e .
we shall be concerned with group actions that transform solutions of LaTeXMLEquation into solutions in the sense of association .
Such group actions will be called symmetries in the sense of association or , by slight abuse of terminology , associated symmetries .
Also , we will give a rather general criterion for transferring classical symmetry groups of linear systems to the distributional setting .
In what follows we will only consider projectable symmetry groups .
Definition LaTeXMLMath is a solution to ( LaTeXMLRef ) ( also called an associated solution to ( LaTeXMLRef ) ) if LaTeXMLMath has a representative LaTeXMLMath such that for every LaTeXMLMath LaTeXMLEquation .
The set of all associated solutions to ( LaTeXMLRef ) will be denoted by LaTeXMLMath .
The set of all LaTeXMLMath satisfying ( LaTeXMLRef ) will be termed LaTeXMLMath .
Let LaTeXMLMath .
A symmetry group LaTeXMLMath of ( LaTeXMLRef ) will be called an LaTeXMLMath -symmetry group if LaTeXMLMath whenever LaTeXMLMath and LaTeXMLMath is defined .
In what follows , LaTeXMLMath LaTeXMLMath , the space of compactly supported LaTeXMLMath -functions on LaTeXMLMath is equipped with the inductive limit topology of its subspaces LaTeXMLMath ( LaTeXMLMath compact in LaTeXMLMath ) .
Definition Let LaTeXMLMath ( resp .
LaTeXMLMath and let LaTeXMLMath .
LaTeXMLMath is called a LaTeXMLMath -strongly associated ( LaTeXMLMath -associated ) solution to ( LaTeXMLRef ) if it has a representative LaTeXMLMath such that for every set LaTeXMLMath which is bounded in LaTeXMLMath we have LaTeXMLEquation .
The space of LaTeXMLMath -strongly associated solutions to ( LaTeXMLRef ) is denoted by LaTeXMLMath and we set LaTeXMLMath .
The corresponding symmetry groups are called LaTeXMLMath - and LaTeXMLMath -symmetry groups , respectively .
Remark Associated solutions ( i.e. , solutions of type LaTeXMLMath ) play a central role in applications to numerics ( cf .
the remarks in Section LaTeXMLRef ) .
Moreover , by imposing slight changes in their definition , Colombeau type algebras can be adapted to a wide variety of solution types .
Thus LaTeXMLMath is particularly useful for the investigation of shock wave solutions of conservation laws ( cf .
LaTeXMLRef , LaTeXMLRef , LaTeXMLRef below ) , which fall into this class of generalized functions .
It will turn out in Theorem LaTeXMLRef ( i ) that the uniform bounddedness of solutions is essential for the characterization of LaTeXMLMath -symmetry groups .
Example Consider the Riemann problem LaTeXMLEquation where LaTeXMLMath denotes the Heaviside function .
For LaTeXMLMath the unique weak solution to this problem is LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and set LaTeXMLMath , LaTeXMLMath .
We are looking for LaTeXMLMath -strongly associated solutions to ( LaTeXMLRef ) of the form LaTeXMLMath , with LaTeXMLMath to be determined .
Thus let LaTeXMLMath be a bounded subset of LaTeXMLMath .
This means that there exists LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLEquation .
Let LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation by dominated convergence , uniformly for LaTeXMLMath .
Thus LaTeXMLMath as above is a LaTeXMLMath -strongly associated solution if and only if LaTeXMLMath , i.e .
if and only if the Rankine-Hugoniot jump condition is satisfied .
The transfer of classical symmetry groups into symmetry groups in the sense of association is again governed by factorization properties .
Sufficient conditions for this transfer are provided by the following result : Theorem Let LaTeXMLMath be a slowly increasing symmetry group of ( LaTeXMLRef ) admitting a global factorization of the form ( LaTeXMLRef ) .
Then If LaTeXMLMath depends exclusively on LaTeXMLMath , LaTeXMLMath and LaTeXMLMath then LaTeXMLMath is an LaTeXMLMath -symmetry group of ( LaTeXMLRef ) .
If LaTeXMLMath depends exclusively on LaTeXMLMath and LaTeXMLMath then for any LaTeXMLMath LaTeXMLMath is an LaTeXMLMath -symmetry group of ( LaTeXMLRef ) .
Proof .
( i ) Let LaTeXMLMath and LaTeXMLMath .
It follows from ( LaTeXMLRef ) that LaTeXMLMath .
Moreover , by ( LaTeXMLRef ) , for any LaTeXMLMath we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
If LaTeXMLMath varies in a subset of LaTeXMLMath which is bounded in LaTeXMLMath then due to the fact that the LaTeXMLMath are globally bounded , LaTeXMLMath varies in a LaTeXMLMath -bounded subset of LaTeXMLMath .
Hence the above expression tends to zero , uniformly in LaTeXMLMath and LaTeXMLMath .
( ii ) With LaTeXMLMath as in ( i ) , LaTeXMLEquation .
LaTeXMLEquation which tends to zero since LaTeXMLMath is LaTeXMLMath -bounded if LaTeXMLMath varies in a LaTeXMLMath -bounded subset of LaTeXMLMath .
LaTeXMLMath Example Let LaTeXMLMath be an LaTeXMLMath matrix with LaTeXMLMath -entries LaTeXMLMath LaTeXMLMath LaTeXMLMath on LaTeXMLMath .
We consider the quasilinear system LaTeXMLEquation ( for solvability resp .
unique solvability of ( LaTeXMLRef ) we refer to LaTeXMLCite and LaTeXMLCite ) and the action of a classical symmetry group LaTeXMLMath LaTeXMLEquation where LaTeXMLMath .
We are going to show that LaTeXMLMath is an LaTeXMLMath -symmetry group of ( LaTeXMLRef ) for each LaTeXMLMath provided that LaTeXMLMath does not depend on LaTeXMLMath , i.e .
LaTeXMLMath acts linearly on the dependent variables .
LaTeXMLMath does not depend on LaTeXMLMath .
Denoting as above the Jacobian of LaTeXMLMath by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are given by the first and second column of the matrix valued function LaTeXMLMath , where LaTeXMLMath is defined by LaTeXMLEquation .
Moreover , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLMath does not depend on LaTeXMLMath by our assumption on LaTeXMLMath and the explicit form of LaTeXMLMath .
Now LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath identity matrix .
This implies LaTeXMLEquation .
By assumptions ( i ) and ( ii ) it follows that LaTeXMLMath does not depend on LaTeXMLMath .
Thus our claim follows from LaTeXMLRef ( ii ) .
Example Let LaTeXMLMath and let LaTeXMLMath be a primitive of LaTeXMLMath .
The generalized Burgers equation LaTeXMLEquation has a weak solution of the form LaTeXMLEquation where LaTeXMLMath .
( If LaTeXMLMath and LaTeXMLMath , then this solution is unique . )
The same arguments as in the case of Burgers ’ equation in LaTeXMLRef imply that ( LaTeXMLRef ) is the LaTeXMLMath -strongly associated solution to ( LaTeXMLRef ) if and only if LaTeXMLMath is of the given form .
In this special case of ( LaTeXMLRef ) , conditions ( i ) and ( ii ) of LaTeXMLRef are in fact necessary for shock wave solutions to be transformed into LaTeXMLMath -strongly associated solutions .
In fact , if ( i ) or ( ii ) is violated then from the explicit calculations in LaTeXMLRef it follows that the integrand in the proof of LaTeXMLRef ( ii ) will contain unbounded terms .
More complex factorizations arise in case LaTeXMLMath is invertible .
Indeed , by a straightforward explicit calculation , the infinitesimal generators LaTeXMLEquation give rise to the group actions LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath , LaTeXMLMath , LaTeXMLMath with factors LaTeXMLEquation .
LaTeXMLEquation Theorem LaTeXMLRef raises the question of finding criteria for classical symmetry groups to display the favorable factorization properties used above .
As an important case where a general result is available we now turn to systems of linear PDEs .
With a view to applications in distribution theory ( cf .
section LaTeXMLRef ) we also restrict our attention to group actions which act linearly on the dependent variables .
Theorem Suppose that in ( LaTeXMLRef ) , LaTeXMLMath is a linear differential operator : LaTeXMLEquation .
Furthermore , let LaTeXMLMath be a one parameter symmetry group of ( LaTeXMLRef ) which acts linearly in the dependent variables : LaTeXMLEquation .
If there exist LaTeXMLMath such that LaTeXMLMath is globally nonsingular then conditions ( i ) and ( ii ) of LaTeXMLRef are satisfied and LaTeXMLMath in ( LaTeXMLRef ) depends exclusively on LaTeXMLMath and LaTeXMLMath .
Proof .
Using the same notations and conventions as in the proof of Theorem LaTeXMLRef we have LaTeXMLMath where the LaTeXMLMath matrix LaTeXMLMath is of the form LaTeXMLMath with LaTeXMLMath an LaTeXMLMath matrix and LaTeXMLMath invertible .
Thus LaTeXMLEquation is affine linear in LaTeXMLMath for LaTeXMLMath .
By ( LaTeXMLRef ) and LaTeXMLCite , ( 2.18 ) we have LaTeXMLEquation where LaTeXMLEquation with certain smooth functions LaTeXMLMath ( Actually , the upper limit LaTeXMLMath in these sums is only attained for terms corresponding to highest order derivatives but for the following argument only the general form of ( LaTeXMLRef ) is of interest ) .
Hence both LaTeXMLMath and LaTeXMLMath are ( affine ) linear in LaTeXMLMath for LaTeXMLMath .
It follows that the matrix LaTeXMLMath is independent of LaTeXMLMath for LaTeXMLMath .
This observation , together with ( LaTeXMLRef ) , finishes the proof .
LaTeXMLMath Remark The conclusion of Theorem LaTeXMLRef remains valid for a semilinear system LaTeXMLEquation provided that LaTeXMLMath correspond to indices of highest order derivatives of the dependent variables .
This follows immediately from an inspection of the above proof ( only the form of LaTeXMLMath in ( LaTeXMLRef ) changes ) .
Corollary Let LaTeXMLMath be a symmetry group of the linear ( resp .
semilinear ) system ( LaTeXMLRef ) ( resp .
( LaTeXMLRef ) ) such that the assumptions of Theorem LaTeXMLRef ( resp .
of Remark LaTeXMLRef ) are satisfied .
Then LaTeXMLMath is an LaTeXMLMath -symmetry group of ( LaTeXMLRef ) ( resp .
( LaTeXMLRef ) ) .
Proof .
Immediate from LaTeXMLRef ( ii ) and LaTeXMLRef .
LaTeXMLMath Example Let LaTeXMLEquation then setting LaTeXMLMath and using the notation of LaTeXMLRef we have LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
Hence LaTeXMLEquation .
Turning now to the distributional setting , we first note that the most general group actions applicable to distributions are those which are projectable and act linearly on the dependent variables , i.e .
those which are of the form ( LaTeXMLRef ) .
If LaTeXMLMath then the action of LaTeXMLMath is defined by LaTeXMLEquation where LaTeXMLMath denotes ( componentwise ) distributional pullback , i.e. , LaTeXMLEquation for LaTeXMLMath a diffeomorphism .
We will sometimes also write LaTeXMLMath instead of LaTeXMLMath .
Definition Suppose that ( LaTeXMLRef ) is linear and let LaTeXMLMath be a local transformation group acting linearly on the dependent variables .
LaTeXMLMath is called a distributional ( or LaTeXMLMath - ) symmetry group of ( LaTeXMLRef ) if it transforms distributional solutions of ( LaTeXMLRef ) into distributional solutions .
LaTeXMLRef is the most general definition of distributional symmetry groups .
More restrictive notions ( as introduced e.g .
in LaTeXMLCite ) will be discussed in the following section .
Theorem Let LaTeXMLMath be a symmetry group of the linear system ( LaTeXMLRef ) such that the assumptions of Theorem LaTeXMLRef are satisfied .
Then LaTeXMLMath is a distributional symmetry group of ( LaTeXMLRef ) .
Proof .
By LaTeXMLRef and ( LaTeXMLRef ) , for any smooth LaTeXMLMath we have LaTeXMLEquation .
Now suppose that LaTeXMLMath is a solution to ( LaTeXMLRef ) .
Choose some LaTeXMLMath with LaTeXMLMath and set LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath is smooth and converges to LaTeXMLMath in LaTeXMLMath for LaTeXMLMath .
Since LaTeXMLMath acts linearly on the dependent variables we also have that LaTeXMLMath in LaTeXMLMath .
Hence LaTeXMLEquation .
LaTeXMLEquation in LaTeXMLMath , which concludes the proof .
LaTeXMLMath In this section we develop infinitesimal criteria for finding symmetries applicable to all three settings of interest ( distributional , weak , Colombeau ) .
The results introduced here also establish a direct connection of our approach to symmetries of distributional and weak solutions to that given in LaTeXMLCite ( cf .
also LaTeXMLCite ) .
In LaTeXMLCite , systems of the form LaTeXMLEquation where LaTeXMLEquation ( LaTeXMLMath the order of LaTeXMLMath ) and LaTeXMLMath , LaTeXMLMath , are examined .
The form ( LaTeXMLRef ) of writing a system of linear PDEs ( which , of course , is equivalent to ( LaTeXMLRef ) ) provides the advantage of allowing to derive very concise forms of infinitesimal criteria for factorization properties , cf .
( LaTeXMLRef ) below .
In Berest ’ s approach , symmetry groups of ( LaTeXMLRef ) are defined via factorization properties : Let LaTeXMLMath be a projectable one parameter group acting linearly on the dependent variables .
Thus LaTeXMLMath is of the form ( LaTeXMLMath , LaTeXMLMath ) LaTeXMLEquation .
LaTeXMLMath is called a symmetry group of ( LaTeXMLRef ) if there exists a smooth matrix valued map LaTeXMLMath such that ( see LaTeXMLCite , ( 1.6 ) ) : LaTeXMLEquation for all LaTeXMLMath Note that this form corresponds exactly to ( LaTeXMLRef ) with LaTeXMLMath depending on LaTeXMLMath and LaTeXMLMath exclusively .
LaTeXMLCite , ( 1.7 ) gives the following infinitesimal criterion for the validity of ( LaTeXMLRef ) : LaTeXMLEquation where LaTeXMLEquation .
The equivalence of ( LaTeXMLRef ) and ( LaTeXMLRef ) raises the question whether factorization properties for general systems ( LaTeXMLRef ) of differential equations can always be characterized by infinitesimal conditions similar to ( LaTeXMLRef ) .
For LaTeXMLMath smooth , an affirmative answer is given by the following result , contained implicitly in LaTeXMLCite ( cf .
LaTeXMLRef ( ii ) below ) .
Since its method of proof will form the basis for our generalizations to the LaTeXMLMath - resp .
LaTeXMLMath -settings we state it explicitly .
Proposition Let LaTeXMLMath be a one parameter group with infinitesimal generator LaTeXMLMath acting on LaTeXMLMath .
Then the following are equivalent There exists a smooth mapping LaTeXMLMath such that LaTeXMLEquation .
There exists a smooth mapping LaTeXMLMath such that LaTeXMLEquation .
Proof .
( i ) LaTeXMLMath ( ii ) : Noting that LaTeXMLMath ( LaTeXMLMath the infinitesimal generator of LaTeXMLMath ) is a vectorfield on LaTeXMLMath whose flow is precisely LaTeXMLMath , differentiation of ( LaTeXMLRef ) with respect to LaTeXMLMath at LaTeXMLMath gives ( LaTeXMLRef ) ( with LaTeXMLMath ) .
( ii ) LaTeXMLMath ( i ) : ( LaTeXMLRef ) yields the following linear ODE for LaTeXMLMath : LaTeXMLEquation .
Let LaTeXMLMath be a principal matrix solution to ( LaTeXMLRef ) ( i.e .
the LaTeXMLMath -th column of LaTeXMLMath is precisely the solution with initial value LaTeXMLMath ) .
Then we immediately obtain the unique solution to ( LaTeXMLRef ) in the form LaTeXMLEquation .
LaTeXMLMath Remarks From the explicit formulae given in the above proof it follows that LaTeXMLMath depends exclusively on LaTeXMLMath for LaTeXMLMath ( e.g .
exclusively on LaTeXMLMath ) if and only if the same is true of LaTeXMLMath : indeed it suffices to note that by LaTeXMLCite , ( 2.20 ) LaTeXMLMath ( where LaTeXMLMath is the natural projection ) .
By LaTeXMLCite , eq .
( 2.26 ) , for any nondegenerate system ( LaTeXMLRef ) , ( LaTeXMLRef ) is equivalent to LaTeXMLMath on the zero-set of LaTeXMLMath ( LaTeXMLMath ) which in turn ( by LaTeXMLCite , Th .
2.71 ) is necessary and sufficient for LaTeXMLMath to generate a one-parameter symmetry group of ( LaTeXMLRef ) .
Thus ( LaTeXMLRef ) is precisely the infinitesimal version of the global factorization ( LaTeXMLRef ) .
It follows immediately from the definition of LaTeXMLMath that ( LaTeXMLRef ) is equivalent with LaTeXMLEquation where LaTeXMLMath runs through all open sets .
A similar reformulation is valid for ( LaTeXMLRef ) .
In the Colombeau setting the general form of ( LaTeXMLRef ) allows for LaTeXMLMath itself to be a generalized function .
More precisely , we suppose that LaTeXMLMath .
Thereby , the admissible symmetry transformations themselves will become generalized functions , so-called ( projectable ) generalized group actions LaTeXMLMath .
Thus LaTeXMLMath is supposed to satisfy LaTeXMLMath -bounds with respect to the group parameter LaTeXMLMath and LaTeXMLMath -bounds with respect to LaTeXMLMath ( LaTeXMLCite , Def .
4.8 ) .
Many of the infinitesimal methods of classical group analysis can be recovered in this setting .
For a detailed analysis we refer to LaTeXMLCite .
The proof of the analogue to LaTeXMLRef in the present situation requires the following auxiliary result on solutions of linear ODEs in LaTeXMLMath : Lemma Let LaTeXMLMath such that for all LaTeXMLMath , LaTeXMLEquation .
Then for each LaTeXMLMath the initial value problem LaTeXMLEquation has a unique solution LaTeXMLMath in LaTeXMLMath .
Setting LaTeXMLMath the matrix with columns the unique solutions with initial conditions LaTeXMLMath ( LaTeXMLMath ) , the solution to ( LaTeXMLRef ) is given by LaTeXMLMath .
We call LaTeXMLMath a principal matrix solution to ( LaTeXMLRef ) .
Proof .
We only sketch the argument ( for details , cf .
LaTeXMLCite ) .
Choosing representatives LaTeXMLMath of LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , by the corresponding result in the LaTeXMLMath -setting we obtain representatives LaTeXMLMath , LaTeXMLMath satisfying the claimed properties for each fixed LaTeXMLMath .
Then LaTeXMLEquation .
Thus Gronwall ’ s inequality and the supposed growth restriction on LaTeXMLMath imply moderateness of the representatives and unique solvability .
LaTeXMLMath Note that the corresponding statement for initial value problems in the sense of association ( i.e .
replacing LaTeXMLMath by LaTeXMLMath in ( LaTeXMLRef ) ) is false since unique solvability of linear ODEs breaks down in that context .
As an easy example take LaTeXMLMath ( LaTeXMLMath ) .
Then LaTeXMLMath but LaTeXMLMath is not associated to any constant .
An element LaTeXMLMath of LaTeXMLMath is called globally of LaTeXMLMath -log-type ( cf .
LaTeXMLCite ) if it possesses a representative LaTeXMLMath with LaTeXMLMath .
After these preparations we can state ( for the notion of LaTeXMLMath -n-completeness , see LaTeXMLCite , Def .
4.15 ) : Proposition Let LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -n-complete group action on LaTeXMLMath .
Consider the statements There exists LaTeXMLMath with LaTeXMLEquation .
There exists LaTeXMLMath LaTeXMLEquation .
Then ( i ) implies ( ii ) .
If LaTeXMLMath is an element of LaTeXMLMath satisfying the growth property given in LaTeXMLRef then ( ii ) implies ( i ) .
If LaTeXMLMath and LaTeXMLMath are supposed to be globally of LaTeXMLMath -log-type then ( i ) and ( ii ) are equivalent .
Proof .
Using LaTeXMLRef , the proof proceeds along the lines of LaTeXMLRef .
LaTeXMLMath From the pointvalue characterization of Colombeau generalized functions given in LaTeXMLCite it follows that ( LaTeXMLRef ) is equivalent with LaTeXMLEquation ( cf .
LaTeXMLCite , Lemma 4.13 and Prop .
4.14 ) .
Remark If in LaTeXMLRef , LaTeXMLMath and LaTeXMLMath , then the solution to ( LaTeXMLRef ) , LaTeXMLMath belongs to LaTeXMLMath .
Thus , if we suppose LaTeXMLMath in LaTeXMLRef then we have LaTeXMLMath , LaTeXMLMath and ( i ) and ( ii ) in LaTeXMLRef are equivalent Turning now to the distributional setting , we first note the following result on linear ODEs in LaTeXMLMath .
Lemma Let LaTeXMLMath and let LaTeXMLMath .
Then the initial value problem LaTeXMLEquation has the unique solution LaTeXMLMath where LaTeXMLMath is the smooth principal matrix solution of ( LaTeXMLRef ) .
Proof .
That LaTeXMLMath is a solution follows easily by regularizing the initial data via convolution with a standard mollifier and then taking the distributional limit of the solutions to the resulting smooth problems .
Uniqueness follows from uniqueness of the corresponding smooth initial value problem by observing that for any solution LaTeXMLMath of ( LaTeXMLRef ) with LaTeXMLMath we have LaTeXMLMath .
But then LaTeXMLMath with LaTeXMLMath a constant vector which is necessarily LaTeXMLMath .
LaTeXMLMath Let us suppose that LaTeXMLMath is of the form ( LaTeXMLRef ) with LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath ) .
This is the most general form of differential operators applicable to elements LaTeXMLMath of LaTeXMLMath .
Moreover , we suppose that the group action LaTeXMLMath is of the form ( LaTeXMLRef ) ( also the most general action applicable to distributions ) .
We consider LaTeXMLMath as an element of LaTeXMLMath by embedding LaTeXMLMath as LaTeXMLMath into LaTeXMLMath .
For the following result , to simplify notations we suppose that LaTeXMLMath and that LaTeXMLMath is defined on all of LaTeXMLMath .
Proposition Under the assumptions formulated before LaTeXMLRef , the following are equivalent There exists a smooth mapping LaTeXMLMath such that LaTeXMLEquation .
There exists a smooth mapping LaTeXMLMath such that LaTeXMLEquation .
There exists a smooth mapping LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
Proof .
( i ) LaTeXMLMath ( ii ) : Using LaTeXMLRef , the proof is again identical to that of LaTeXMLRef ( for the distributional identity LaTeXMLMath used in the argument , see e.g. , LaTeXMLCite , Th .
3.7 ) .
( i ) LaTeXMLMath ( iii ) : With the notations introduced in ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) and LaTeXMLMath LaTeXMLMath we have LaTeXMLEquation .
It follows that ( LaTeXMLRef ) can be written in the form LaTeXMLEquation .
LaTeXMLEquation Hence , introducing suitable smooth functions LaTeXMLMath and distributions LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath ) , the proof reduces to establishing the equivalence of LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation This last assertion naturally splits into a ( distributional ) invariance property of LaTeXMLMath resp .
LaTeXMLMath and a smooth part , both of which are easily seen to be equivalent .
LaTeXMLMath Thus ( LaTeXMLRef ) corresponds precisely to ( LaTeXMLRef ) .
In particular , the specific form of the infinitesimal criterion for the validity of ( LaTeXMLRef ) follows from an explicit calculation of LaTeXMLMath in the notation ( due to Berest ) introduced at the beginning of this section .
In fact , we have LaTeXMLEquation .
Then by LaTeXMLCite , Th .
2.36 , LaTeXMLMath , where LaTeXMLEquation ( LaTeXMLMath ) .
Thus the LaTeXMLMath -th component of LaTeXMLMath is calculated as follows : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation This calculation , combined with LaTeXMLRef and ( LaTeXMLRef ) provides a rigorous proof of the equivalence of ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Moreover , LaTeXMLRef allows to derive infinitesimal criteria for factorization properties even for systems that are not necessarily linear and thereby to obtain workable criteria for finding symmetries of weak or Colombeau solutions of such systems .
For example , let us consider the semilinear system LaTeXMLEquation where we shall suppose LaTeXMLMath ( to allow for an insertion of Colombeau functions , note however that LaTeXMLRef below does not use this assumption ) .
Furthermore , let us assume that the group action LaTeXMLMath is of the form ( LaTeXMLRef ) .
Then we have Proposition Under the above assumptions , the following are equivalent : There exists a smooth mapping LaTeXMLMath such that for all LaTeXMLMath LaTeXMLMath and all LaTeXMLMath LaTeXMLEquation .
There exists a smooth mapping LaTeXMLMath such that for all LaTeXMLMath LaTeXMLMath and all LaTeXMLMath LaTeXMLEquation ( with LaTeXMLMath the Jacobian of LaTeXMLMath ) .
Proof .
By LaTeXMLRef and LaTeXMLRef ( iii ) it suffices to calculate LaTeXMLMath for LaTeXMLMath as above .
Noting that LaTeXMLMath , the result follows exactly as in the above calculation .
LaTeXMLMath In LaTeXMLCite , it was shown that a certain splitting of ( LaTeXMLRef ) is advantageous for establishing a connection between determining symmetry groups of PDEs and group invariance of the solutions themselves , in particular with a view to determining group invariant fundamental solutions of linear systems .
In our more general setup , we first note that ( using obvious abbreviations ) setting LaTeXMLMath LaTeXMLMath LaTeXMLMath and LaTeXMLMath , ( LaTeXMLRef ) is equivalent to ( LaTeXMLRef ) as well as to ( LaTeXMLRef ) .
Also , ( LaTeXMLRef ) is equivalent to ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We first note the following immediate consequence of ( LaTeXMLRef ) : Proposition Let LaTeXMLMath be a one-parameter symmetry group of the form ( LaTeXMLRef ) of the homogeneous system LaTeXMLMath satisfying ( LaTeXMLRef ) .
If LaTeXMLMath is a solution , resp .
associated solution resp .
strongly associated solution then so is LaTeXMLMath .
LaTeXMLMath Definition Let LaTeXMLMath be a slowly increasing one-parameter group .
We say that LaTeXMLMath is LaTeXMLMath -invariant under LaTeXMLMath , LaTeXMLMath -invariant , or LaTeXMLMath -invariant , respectively , if LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath for all LaTeXMLMath .
If LaTeXMLMath is of the form ( LaTeXMLRef ) and LaTeXMLMath then LaTeXMLMath is called LaTeXMLMath -invariant under LaTeXMLMath if LaTeXMLMath in LaTeXMLMath .
Proposition Let LaTeXMLMath be of the form ( LaTeXMLRef ) .
Let LaTeXMLMath resp .
LaTeXMLMath resp .
LaTeXMLMath .
A necessary and sufficient condition for LaTeXMLMath to be invariant resp .
LaTeXMLMath -invariant resp .
LaTeXMLMath -invariant under LaTeXMLMath is that LaTeXMLMath equals LaTeXMLMath in LaTeXMLMath resp .
in LaTeXMLMath resp .
in LaTeXMLMath .
Then LaTeXMLMath is LaTeXMLMath -invariant , resp .
LaTeXMLMath -invariant , if LaTeXMLMath resp .
LaTeXMLMath ( LaTeXMLMath ) .
Proof .
( i ) We first note that in each of the possible settings we have LaTeXMLEquation which is immediate from the chain rule .
Thus the conditions are necessary .
Conversely , for fixed LaTeXMLMath set LaTeXMLEquation .
Then LaTeXMLMath and LaTeXMLMath .
Thus the claim follows from unique solvability of linear ODEs in each of the respective settings ( cf .
LaTeXMLRef with LaTeXMLMath smooth ( hence automatically satisfying the necessary growth restrictions ) and LaTeXMLRef ) .
( ii ) Setting LaTeXMLMath we have LaTeXMLEquation .
LaTeXMLEquation Also , LaTeXMLMath resp .
LaTeXMLMath for all LaTeXMLMath follows from our assumption ( by substituting for LaTeXMLMath in the respective integrals ) .
With LaTeXMLMath the principal matrix solution to the corresponding homogeneous system we obtain the solution to this initial value problem in the form LaTeXMLEquation .
Thus for LaTeXMLMath resp .
LaTeXMLMath bounded in LaTeXMLMath , we have LaTeXMLEquation as LaTeXMLMath , resp .
this limit is uniform for LaTeXMLMath .
It follows that LaTeXMLMath , resp .
LaTeXMLMath i.e. , that LaTeXMLMath resp .
LaTeXMLMath LaTeXMLMath Note that the converse assertion in ( ii ) does not follow since LaTeXMLMath resp .
LaTeXMLMath for all LaTeXMLMath does not imply any information on LaTeXMLMath .
Example We derive infinitesimal criteria for symmetries ( LaTeXMLRef ) of the quasilinear system ( LaTeXMLRef ) whose infinitesimal generators we write as LaTeXMLEquation .
Applying LaTeXMLMath to LaTeXMLMath which is just the short form of ( LaTeXMLRef ) , we obtain LaTeXMLEquation .
Thus , LaTeXMLEquation .
LaTeXMLEquation Hence by ( LaTeXMLRef ) we get the following expression for LaTeXMLMath : LaTeXMLMath ( where LaTeXMLMath is a reminder that LaTeXMLMath is LaTeXMLMath matrix ) .
Note that in this expression the coefficient of LaTeXMLMath equals LaTeXMLMath .
Thus the determining system LaTeXMLMath for an infinitesimal projectable symmetry reads LaTeXMLEquation .
LaTeXMLEquation or LaTeXMLEquation .
LaTeXMLEquation Example We continue to analyze the last determining system in the case of a strictly hyperbolic conservation law LaTeXMLEquation .
LaTeXMLEquation corresponding to ( LaTeXMLRef ) with LaTeXMLEquation with the characteristic values LaTeXMLMath where LaTeXMLMath is open in LaTeXMLMath ( cf .
LaTeXMLCite ; for generalized solutions , see LaTeXMLCite ) .
We denote by LaTeXMLMath the characteristic vectors : LaTeXMLMath We will calculate the coefficients of an infinitesimal symmetry LaTeXMLMath for ( LaTeXMLRef ) , ( LaTeXMLRef ) .
By LaTeXMLRef we have LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Differentiating ( LaTeXMLRef ) with respect to LaTeXMLMath and LaTeXMLMath we have LaTeXMLEquation .
LaTeXMLEquation Putting both together we obtain the matrix equation LaTeXMLEquation .
Taking the derivative of ( LaTeXMLRef ) with respect to LaTeXMLMath and subtracting from ( LaTeXMLRef ) we arrive at LaTeXMLEquation or in shorter form LaTeXMLEquation .
Thus we obtain the system ( LaTeXMLRef ) , ( LaTeXMLRef ) which , while not equivalent to ( LaTeXMLRef ) , ( LaTeXMLRef ) ( due to the differentiations used in deriving it ) , considerably facilitates the determination of infinitesimal symmetries .
By LaTeXMLCite , Section 16 , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In order to determine LaTeXMLMath and LaTeXMLMath as well as LaTeXMLMath we use LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLRef resp .
LaTeXMLRef , solutions of this system in LaTeXMLMath resp .
LaTeXMLMath determine infinitesimal symmetries in each of the respective settings .
Note that the above considerations also hold for systems of order LaTeXMLMath even in non-conservative form with the additional assumption that the coefficients of LaTeXMLMath do not depend on LaTeXMLMath and LaTeXMLMath .
Self–scaled barrier functions are fundamental objects in the theory of interior–point methods for linear optimization over symmetric cones , of which linear and semidefinite programming are special cases .
We are classifying all self–scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier function .
Together with a decomposition theorem for self–scaled barriers this concludes the algebraic classification theory of these functions .
After introducing the reader to the concepts relevant to the problem and tracing the history of the subject , we start by deriving our result from first principles in the important special case of semidefinite programming .
We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras .
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports SELF–SCALED BARRIERS FOR IRREDUCIBLE SYMMETRIC CONES Raphael Hauser and Yongdo Lim DAMTP 2001/NA04 April 2001 Department of Applied Mathematics and Theoretical Physics Silver Street Cambridge England CB3 9EW April 2 , 2001 Key Words Semidefinite programming , self–scaled barrier functions , interior–point methods , symmetric cones , Euclidean Jordan algebras .
AMS 1991 Subject Classification Primary 90C25 , 52A41 , 90C60 .
Secondary 90C05 , 90C20 .
Contact Information and Credits Raphael Hauser , Department of Applied Mathematics and Theoretical Physics , University of Cambridge , Silver Street , Cambridge CB3 9EW , England .
rah48 @ damtp.cam.ac.uk .
Research supported in part by the Norwegian Research Council through project No .
127582/410 “ Synode II ” , by the Engineering and Physical Sciences Research Council of the UK under grant No .
GR/M30975 , and by NSERC of Canada grants of J. Borwein and P. Borwein .
Yongdo Lim , Department of Mathematics , Kyungpook National University , Taegu 702–701 , Korea .
ylim @ knu.ac.kr .
Research supported in part by the Basic Research Program of the Korea Science and Engineering Foundation through project No .
2000-1-10100-007-3 .
Self–scaled barriers are a special class of self–concordant barrier functions LaTeXMLCite introduced by Nesterov and Todd LaTeXMLCite for the purpose of extending long–step primal–dual symmetric interior–point methods from linear and semidefinite programming to more general convex optimization problems LaTeXMLCite , see Definition LaTeXMLRef below .
The domain of definition of a self–scaled barrier LaTeXMLMath is an proper open convex cone LaTeXMLMath lying in a real Euclidean space LaTeXMLMath .
By abuse of language one often refers to LaTeXMLMath as a self–scaled barrier for the topological closure LaTeXMLMath of LaTeXMLMath .
Not every proper open convex cone LaTeXMLMath allows a self–scaled barrier , and for those who do LaTeXMLMath is called a self–scaled cone in the terminology of Nesterov and Todd LaTeXMLCite .
Güler LaTeXMLCite found that the family of interiors of self–scaled cones is identical to the set of symmetric cones studied in the theory of Euclidean Jordan algebras .
Due to this discovery , Jordan algebra theory became an important analytic tool in the theory of semidefinite programming and its natural generalisation , self–scaled programming .
Self–scaled barriers were studied by Nesterov–Todd LaTeXMLCite , Güler LaTeXMLCite , Güler–Tunçel LaTeXMLCite ( see p. 124 and related material ) , Hauser LaTeXMLCite and others .
Though an axiomatic theory of these functions exists , the only known examples are trivially related to the characteristic function of the cone LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath is the polar of LaTeXMLMath with respect to LaTeXMLMath .
This was first discovered by Güler who showed that the universal barrier LaTeXMLMath LaTeXMLCite of a symmetric cone is self–scaled and is a homothetic transformation LaTeXMLMath of the characteristic function LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are constants .
More generally , every symmetric cone LaTeXMLMath has a decomposition , unique up to indexing , into a direct sum of irreducible symmetric cones LaTeXMLEquation where the LaTeXMLMath lie in subspaces LaTeXMLMath decomposing LaTeXMLMath into a direct sum LaTeXMLMath .
The irreducible summands LaTeXMLMath can be classified into five different types , see LaTeXMLCite and references therein .
All known self–scaled barrier functions for LaTeXMLMath are of the form LaTeXMLEquation where LaTeXMLMath .
We use the direct sum notation for LaTeXMLMath in Equation ( LaTeXMLRef ) and elsewhere to indicate that each LaTeXMLMath is mapped to LaTeXMLMath .
It is also well–known that each function of the form ( LaTeXMLRef ) is a self–scaled barrier for LaTeXMLMath .
The natural question arises as to whether all self–scaled barrier functions are of the form ( LaTeXMLRef ) .
Early dents into this question were made by Güler and Tunçel when considering invariant barriers , see LaTeXMLCite page 124 and related material .
In a chapter of his thesis LaTeXMLCite and in a subsequent report LaTeXMLCite , Hauser showed that any self–scaled barrier LaTeXMLMath over a symmetric cone LaTeXMLMath decomposes into a direct sum LaTeXMLMath of self–scaled barriers LaTeXMLMath over the irreducible components LaTeXMLMath , and that any isotropic , i.e. , rotationally invariant , self–scaled barrier LaTeXMLMath on LaTeXMLMath is of the form LaTeXMLMath with LaTeXMLMath .
Hauser also observed that any self–scaled barrier LaTeXMLMath on LaTeXMLMath is invariant under a rich class of rotations of LaTeXMLMath , i.e. , elements of LaTeXMLMath , see ( LaTeXMLRef ) below , where these particular rotations are defined in terms of the Hessians of LaTeXMLMath , see Lemma 2.2.19 LaTeXMLCite .
Suspecting that in the case where LaTeXMLMath is irreducible this family of rotations is rich enough to generate all of LaTeXMLMath , Hauser LaTeXMLCite conjectured that all self–scaled barriers on irreducible symmetric cones are isotropic .
This conjecture , whose correctness to prove is the primary objective of this paper , shows that all self–scaled barriers are indeed of the form ( LaTeXMLRef ) and concludes their algebraic classification .
In a second report LaTeXMLCite Hauser showed the correctness of the isotropy conjecture for the special case of the cone of positive semidefinite symmetric matrices .
The proof follows exactly the path suggested by Lemma 2.2.19 LaTeXMLCite , and as outlined above .
Shortly after Hauser ’ s report LaTeXMLCite was announced , Lim LaTeXMLCite generalised Hauser ’ s arguments to general irreducible symmetric cones and settled the isotropy conjecture .
This report is a joint publication consisting of a revision of Hauser ’ s report LaTeXMLCite and of Lim ’ s generalisation LaTeXMLCite .
Subsequently , both Schmieta LaTeXMLCite and Güler LaTeXMLCite independently of each other and independently of Lim also proved the isotropy conjecture .
Schmieta ’ s report LaTeXMLCite was the first publication where the full classification result became available .
Güler ’ s approach LaTeXMLCite was later incorporated in a joint publication with Hauser LaTeXMLCite which started as a major revision of the report LaTeXMLCite .
It is interesting to note that , though the approaches of Lim , Güler and Schmieta differ in important details , all three involve two key mechanisms : The so–called fundamental formula on the one hand , see ( LaTeXMLRef ) below , and Koecher ’ s Theorem 4.9 ( b ) LaTeXMLCite on the other hand .
Already Hauser ’ s approach LaTeXMLCite to solving the special case of the positive semidefinite cone was based on fundamentally the same ideas , as his Proposition 3.3 was essentially an independent rediscovery of Koecher ’ s theorem in this particular case ( c.f .
Corollary 4.3 LaTeXMLCite ) .
The main part of this paper is organised in two sections addressing different communities : Section LaTeXMLRef treats the case of the positive semidefinite cone only .
Readers interested in semidefinite programming and lacking a background in Jordan algebra theory will find it easy to read this section , the results being derived from first principles .
All the qualitative features of the general approach already appear in the restricted framework , and it is possible to understand some of the essential ideas behind Koecher ’ s theorem by reading the proof of Proposition LaTeXMLRef .
Section LaTeXMLRef on the other hand treats the general case and addresses primarily readers with a background in Jordan algebra theory .
It is possible to understand this section without prior lecture of Section LaTeXMLRef .
We conclude this section by giving the essential definitions and identities that form the basis of the theory of self–scaled barriers .
Recall that we introduced the notation LaTeXMLMath and LaTeXMLMath above .
The set of vector space automorphisms of LaTeXMLMath that leave LaTeXMLMath invariant is called the symmetry group of LaTeXMLMath , and we denote it by LaTeXMLEquation .
The inner product on LaTeXMLMath defines a notion of adjoint of an endomorphism and hence an orthogonal group LaTeXMLMath .
The subgroup LaTeXMLEquation is called the orthogonal group of LaTeXMLMath .
A LaTeXMLMath –self–concordant logarithmically homogeneous LaTeXMLCite barrier functional LaTeXMLMath is said to be self–scaled if the following conditions are satisfied : LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
The function LaTeXMLMath is a self–scaled barrier defined on LaTeXMLMath LaTeXMLCite .
It is assumed in the definition of a self–concordant function , see LaTeXMLCite , that the Hessians LaTeXMLMath are non–singular for all LaTeXMLMath .
The next theorem is a compilation of several separate results of Nesterov and Todd LaTeXMLCite : Let LaTeXMLMath be self–scaled and LaTeXMLMath .
Then there exists a unique scaling point LaTeXMLMath such that LaTeXMLMath .
Furthermore , the following properties hold : LaTeXMLMath LaTeXMLMath ∎ It is customary to change the inner product LaTeXMLMath to LaTeXMLMath where LaTeXMLMath is a fixed element in LaTeXMLMath .
We will always assume that LaTeXMLMath is already of this kind , i.e. , that there exists an element LaTeXMLMath such that LaTeXMLMath is the identity when the Hessian is computed with respect to this inner product .
Under this assumption it is easy to show that LaTeXMLMath and LaTeXMLMath .
Hence , in this framework we can reformulate Parts LaTeXMLMath and LaTeXMLMath of Theorem LaTeXMLRef as follows : LaTeXMLEquation .
LaTeXMLEquation Theorem LaTeXMLRef reveals that LaTeXMLMath can allow a self–scaled barrier only if it is a symmetric cone , c.f .
Section LaTeXMLRef .
As mentioned above , this fact was first observed by Güler LaTeXMLCite who showed that the relation also holds in the opposite direction .
Equation ( LaTeXMLRef ) is a reformulation of an identity which is called fundamental formula in Jordan algebra theory , see Equation ( LaTeXMLRef ) below .
This identity is one of the keys to proving the isotropy conjecture , as it allows to express all the Hessians of LaTeXMLMath in terms of the Hessian LaTeXMLMath at a single element LaTeXMLMath and of the Hessians of the standard logarithmic barrier function LaTeXMLEquation see Equation ( LaTeXMLRef ) , which is self–scaled LaTeXMLCite .
Rothaus LaTeXMLCite proved the following result that will be important for our purposes : For every LaTeXMLMath there exists a unique LaTeXMLMath and a unique LaTeXMLMath such that LaTeXMLMath has the polar decomposition LaTeXMLMath .
∎ Since LaTeXMLMath and LaTeXMLMath is a self–adjoint positive definite automorphism of LaTeXMLMath , Rothaus ’ polar decomposition is identical to Cartan ’ s polar decomposition .
Theorem LaTeXMLRef shows the non–trivial fact that both factors lie in LaTeXMLMath .
The uniqueness of Cartan ’ s polar decomposition trivially implies the following lemma which will be useful in later sections : The set of self–adjoint positive definite automorphisms of LaTeXMLMath that preserve LaTeXMLMath coincides with LaTeXMLMath .
∎ This section is limited to semidefinite programming and provides readers who are unfamiliar with Jordan algebra terminology access to the main ideas behind the mechanism that forces self–scaled barriers to be essentially unique .
In this framework it is customary to write variable names with capitalised letters .
LaTeXMLMath is the space LaTeXMLMath of symmetric LaTeXMLMath matrices endowed with the trace inner product LaTeXMLMath which corresponds to the Frobenius norm .
LaTeXMLMath is the cone LaTeXMLMath of positive definite symmetric LaTeXMLMath matrices .
The following identities hold for the standard logarithmic barrier function LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Let us assume that LaTeXMLMath is an arbitrary self–scaled barrier function for LaTeXMLMath .
Applying Equation ( LaTeXMLRef ) to both LaTeXMLMath and LaTeXMLMath , we can derive a series of expressions that will allow us to relate Hessians of LaTeXMLMath to Hessians of LaTeXMLMath .
Since LaTeXMLMath is a self–adjoint positive definite automorphism of LaTeXMLMath it follows from Lemma LaTeXMLRef that there exists a well–defined mapping LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
We claim that LaTeXMLMath is a scalar function , i.e. , there exists a number LaTeXMLMath such that LaTeXMLMath .
The proof of this claim is going to occupy us until Corollary LaTeXMLRef .
Let LaTeXMLMath and LaTeXMLMath denote the scaling points of LaTeXMLMath defined by LaTeXMLMath and LaTeXMLMath , see Theorem LaTeXMLRef .
Equation ( LaTeXMLRef ) implies that LaTeXMLMath , and it follows from the uniqueness part of Theorem LaTeXMLRef that LaTeXMLMath .
Therefore , Equation ( LaTeXMLRef ) applied to LaTeXMLMath shows that LaTeXMLEquation .
LaTeXMLEquation Note that LaTeXMLMath .
Therefore , Equation ( LaTeXMLRef ) applied to LaTeXMLMath and LaTeXMLMath shows that LaTeXMLMath .
Using this fact in conjunction with Equations ( LaTeXMLRef ) and ( LaTeXMLRef ) we get LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation and by virtue of ( LaTeXMLRef ) this implies that LaTeXMLEquation for all LaTeXMLMath .
Clearly , this condition is equivalent to LaTeXMLEquation where LaTeXMLMath is the set of LaTeXMLMath matrices that can be written as the product of two symmetric positive definite matrices .
The following characterisation of this set is due to Mike Todd LaTeXMLCite : LaTeXMLMath coincides with the set of non–defective LaTeXMLMath matrices with real coefficients , all of whose eigenvalues are strictly positive real numbers .
If LaTeXMLMath , then LaTeXMLEquation where LaTeXMLMath is the spectral decomposition of the symmetric positive definite matrix LaTeXMLMath , and LaTeXMLMath .
This gives the eigenvalue decomposition of LaTeXMLMath , with eigenvalues the positive entries of LaTeXMLMath and eigenvectors the columns of LaTeXMLMath .
Conversely , suppose LaTeXMLMath , where LaTeXMLMath is diagonal with positive diagonal entries .
Then we can write LaTeXMLMath .
∎ Note that LaTeXMLMath for all LaTeXMLMath .
In the next proposition we characterise the closed subgroup of LaTeXMLMath generated by matrices of this form .
This constitutes the mathematical core mechanism of our proof and is a close relative of Koecher ’ s Theorem 4.9 ( b ) LaTeXMLCite .
The closed subgroup of LaTeXMLMath generated by the set of orthogonal matrices of the form LaTeXMLMath with LaTeXMLMath coincides with LaTeXMLMath .
Let this closed subgroup be denoted by LaTeXMLMath , and let LaTeXMLMath be its Lie Algebra .
Since LaTeXMLMath is connected , it suffices to show that LaTeXMLMath , or in other words that the tangent space of the manifold LaTeXMLMath at the point LaTeXMLMath coincides with the set of LaTeXMLMath skew–symmetric matrices over the reals .
In fact , use of the exponential mapping LaTeXMLMath shows that LaTeXMLMath and LaTeXMLMath share a neighbourhood LaTeXMLMath of LaTeXMLMath , and parallel transport by left trivialisation shows that LaTeXMLMath and LaTeXMLMath share the neighbourhood LaTeXMLMath of any element LaTeXMLMath .
Therefore , LaTeXMLMath is both open and closed as a subset of LaTeXMLMath , and since LaTeXMLMath is connected , the result follows .
It remains to show that LaTeXMLMath : Let LaTeXMLMath have eigenvalues LaTeXMLMath .
Then LaTeXMLMath for all LaTeXMLMath , since the LaTeXMLMath linearly independent eigenvectors of LaTeXMLMath are also eigenvectors of LaTeXMLMath and correspond to the strictly positive eigenvalues LaTeXMLMath , LaTeXMLMath .
The Neumann–series development of LaTeXMLMath shows that LaTeXMLEquation .
Upon taking squares on both sides of the ansatz LaTeXMLEquation we get LaTeXMLMath , and hence LaTeXMLMath .
Equations ( LaTeXMLRef ) and ( LaTeXMLRef ) thus yield the identity LaTeXMLEquation .
LaTeXMLEquation and this shows that LaTeXMLMath for all LaTeXMLMath .
Clearly , we have LaTeXMLMath as expected .
Now , for LaTeXMLMath and LaTeXMLMath we get LaTeXMLMath .
Hence , LaTeXMLMath .
Let LaTeXMLMath be the permutation matrix that permutes the LaTeXMLMath and LaTeXMLMath variables , and let LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath –th element in the canonical basis of LaTeXMLMath .
Then consider LaTeXMLEquation .
Clearly , LaTeXMLMath and LaTeXMLMath .
But since LaTeXMLMath forms a basis of LaTeXMLMath we find that the elements of LaTeXMLMath span this whole space .
This shows the claim .
∎ There exists a positive constant LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
The invariance property ( LaTeXMLRef ) is clearly preserved when taking compositions and limits .
Hence , Lemma LaTeXMLRef implies that the symmetric positive definite matrix LaTeXMLMath satisfies the condition LaTeXMLMath for all LaTeXMLMath .
It is a trivial matter to prove that this forces LaTeXMLMath to be a scalar , and the result now follows from ( LaTeXMLRef ) .
∎ If LaTeXMLMath is a self–scaled barrier functional for the cone of symmetric positive semidefinite LaTeXMLMath matrices then there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
It follows from Corollary LaTeXMLRef and the fundamental theorem of differential and integral calculus that LaTeXMLMath is of the form LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is a linear form on LaTeXMLMath , i.e .
there exists a symmetric matrix LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
One of the conditions in the definition of a LaTeXMLMath –self–concordant barrier functional LaTeXMLMath for a convex open domain LaTeXMLMath is that the length of the Newton step LaTeXMLMath at LaTeXMLMath , measured in the Riemannian metric LaTeXMLMath defined by LaTeXMLMath be uniformly bounded by LaTeXMLMath , see e.g .
LaTeXMLCite , i.e. , LaTeXMLEquation .
LaTeXMLEquation In particular , in the case of LaTeXMLMath this means that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for all LaTeXMLMath .
But clearly , this implies that LaTeXMLMath .
∎ It is possible to prove Theorem LaTeXMLRef by invoking Lemma 2.2.19 LaTeXMLCite , see LaTeXMLCite for details .
This approach is interesting , since it follows the path traced by the intuition that first led to the isotropy conjecture .
However , the proof we gave above fits better into the mainstream literature and is therefore easier to understand .
An open convex cone LaTeXMLMath in a real Euclidean space LaTeXMLMath that is self–dual with respect to the given inner product and is homogeneous in the sense that the group LaTeXMLEquation acts transitively on LaTeXMLMath is called a symmetric cone .
The theory of symmetric cones is closely tied to that of Euclidean Jordan algebras .
We recall certain basic notions and well–known facts concerning Jordan algebras from the book LaTeXMLCite by J. Faraut and A. Korányi .
A Jordan algebra LaTeXMLMath over the field LaTeXMLMath or LaTeXMLMath is a commutative algebra satisfying LaTeXMLMath for all LaTeXMLMath We also assume the existence of a multiplicative identity LaTeXMLMath Denote by LaTeXMLMath the left translation LaTeXMLMath and LaTeXMLMath by the quadratic representation LaTeXMLMath for LaTeXMLMath An alternate statement of the Jordan algebra law is LaTeXMLMath a weak associativity condition that is strong enough to ensure that the subalgebra generated by LaTeXMLMath in LaTeXMLMath is associative .
An element LaTeXMLMath is said to be invertible if there exists an element LaTeXMLMath in the subalgebra generated by LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath It is known that an element LaTeXMLMath in LaTeXMLMath is invertible if and only if LaTeXMLMath is invertible .
In this case , LaTeXMLMath If LaTeXMLMath and LaTeXMLMath are invertible , then LaTeXMLMath is invertible and LaTeXMLMath Furthermore , the fundamental formula LaTeXMLEquation holds for any elements LaTeXMLMath , see Proposition II.3.2 ( iii ) LaTeXMLCite .
A Euclidean Jordan algebra is a finite–dimensional real Jordan algebra LaTeXMLMath equipped with an associative inner product LaTeXMLMath , i.e. , satisfying LaTeXMLMath for all LaTeXMLMath .
The space LaTeXMLMath of LaTeXMLMath real symmetric matrices is a Euclidean Jordan algebra with Jordan product LaTeXMLMath and inner product LaTeXMLMath The spectral theorem ( see Theorem III.1.2 of LaTeXMLCite ) of a Euclidean Jordan algebra LaTeXMLMath states that for LaTeXMLMath there exist a Jordan frame , i.e. , a complete system of orthogonal primitive idempotents LaTeXMLMath , where LaTeXMLMath is the rank of LaTeXMLMath , and real numbers LaTeXMLMath , the eigenvalues of LaTeXMLMath , such that LaTeXMLMath Due to the the power associative property LaTeXMLMath , see LaTeXMLCite , the exponential map LaTeXMLMath LaTeXMLEquation is well defined .
Likewise as for the special case LaTeXMLMath , the Jordan algebra exponential map is a bijection between its domain of definition LaTeXMLMath and its image LaTeXMLMath In fact , LaTeXMLMath coincides with the interior of the set of square elements of LaTeXMLMath , and this is equal to the set of invertible squares of LaTeXMLMath .
A fundamental theorem of Euclidean Jordan algebras asserts that ( i ) LaTeXMLMath is a symmetric cone , and ( ii ) every symmetric cone in a real Euclidean space arises in this way .
In the case of the Jordan algebra LaTeXMLMath the Jordan algebra exponential is simply the matrix exponential map , and hence the corresponding symmetric cone is LaTeXMLMath , the open convex cone of positive definite LaTeXMLMath matrices .
Irreducible symmetric cones have been completely classified , and the remaining cases consist of ( i ) the cones of positive definite Hermitian and Hermitian quaternion LaTeXMLMath matrices , ( ii ) the Lorentzian cones , and ( iii ) a 27–dimensional exceptional cone .
General symmetric cones are Cartesian products of these .
The connected component LaTeXMLMath of the identity LaTeXMLMath in the Jordan algebra automorphism group LaTeXMLMath is a subgroup of LaTeXMLMath .
LaTeXMLMath is irreducible if and only if LaTeXMLMath is simple , and in this case we have LaTeXMLMath .
For all of these statements , see LaTeXMLCite and the references therein .
In the special case LaTeXMLMath the general formula LaTeXMLMath reduces to LaTeXMLMath , where the multiplication in the right hand side of this equation is the usual matrix multiplication , not the Jordan multiplication .
A key tool for generalising LaTeXMLMath to arbitrary Euclidean Jordan algebras is to consistently replace expressions of the form LaTeXMLMath by LaTeXMLMath Throughout this section we will assume that LaTeXMLMath is a simple Euclidean Jordan algebra with the associative inner product LaTeXMLMath and that LaTeXMLMath is the symmetric cone associated with LaTeXMLMath .
The symmetric cone LaTeXMLMath carries a LaTeXMLMath –invariant Riemannian metric defined by LaTeXMLEquation for which the Jordan inversion LaTeXMLMath on LaTeXMLMath is an involutive isometry fixing LaTeXMLMath The curve LaTeXMLMath is the unique geodesic that joins LaTeXMLMath to LaTeXMLMath in LaTeXMLMath , and the Riemannian distance LaTeXMLMath is given by LaTeXMLMath , where the LaTeXMLMath are the eigenvalues of LaTeXMLMath The geometric mean LaTeXMLMath of two elements LaTeXMLMath is defined by LaTeXMLEquation .
This is the unique midpoint – or geodesic middle – of LaTeXMLMath and LaTeXMLMath for the Riemannian distance LaTeXMLMath The metric LaTeXMLMath is known as a Bruhat–Tits metric , i.e. , a complete metric satisfying the semi–parallelogram law , with midpoint LaTeXMLMath .
See LaTeXMLCite for more details .
If LaTeXMLMath then the geometric mean LaTeXMLMath of positive definite matrices LaTeXMLMath and LaTeXMLMath is given by LaTeXMLMath .
The following basic properties of geometric means will be useful for our purpose : Let LaTeXMLMath and LaTeXMLMath be elements of LaTeXMLMath .
Then LaTeXMLMath is a unique solution belonging to LaTeXMLMath of the quadratic equation LaTeXMLMath .
( The commutativity property ) LaTeXMLMath ( The inversion property ) LaTeXMLMath LaTeXMLMath ( The transformation property ) LaTeXMLMath for all LaTeXMLMath Let LaTeXMLMath be the standard logarithmic barrier functional on the symmetric cone LaTeXMLMath , where LaTeXMLMath is the determinant function of the Jordan algebra LaTeXMLMath , see LaTeXMLCite .
Then one can see that LaTeXMLMath and the Hessian of LaTeXMLMath is given by LaTeXMLEquation .
Proposition LaTeXMLRef implies that the geometric mean LaTeXMLMath is the scaling point of LaTeXMLMath and LaTeXMLMath defined by LaTeXMLMath .
Indeed , LaTeXMLMath , that is , LaTeXMLMath .
Let LaTeXMLMath be an irreducible symmetric cone and LaTeXMLMath a function such that LaTeXMLEquation for all LaTeXMLMath Then LaTeXMLMath for some positive real number LaTeXMLMath Upon exchanging the roles of LaTeXMLMath and LaTeXMLMath , Proposition LaTeXMLRef ( a ) implies that Condition ( LaTeXMLRef ) is equivalent to LaTeXMLEquation .
Setting LaTeXMLMath and using LaTeXMLMath in ( LaTeXMLRef ) , we get LaTeXMLEquation for all LaTeXMLMath .
Let us show that LaTeXMLMath for all LaTeXMLMath .
Applying ( LaTeXMLRef ) and ( LaTeXMLRef ) both to LaTeXMLMath and LaTeXMLMath we get LaTeXMLEquation and hence we obtain the identity LaTeXMLMath for all LaTeXMLMath , which generalises ( LaTeXMLRef ) .
Set LaTeXMLEquation .
It follows from the definition of the geometric mean and from the fundamental formula that LaTeXMLEquation .
Together with Proposition LaTeXMLRef ( b ) this implies LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , the set LaTeXMLMath can be written as LaTeXMLMath .
By Koecher ’ s Theorem 4.9 ( b ) LaTeXMLCite , LaTeXMLMath generates LaTeXMLMath This implies that the point LaTeXMLMath is fixed by all Jordan automorphisms LaTeXMLMath Finally , Corollary IV.2.7 LaTeXMLCite ( in which the assumption of irreducibility for LaTeXMLMath is essential ) says that the group LaTeXMLMath acts transitively on the set of primitive idempotents of LaTeXMLMath .
The spectral theorem applied to LaTeXMLMath therefore implies that LaTeXMLMath for some positive real number LaTeXMLMath .
Together with ( LaTeXMLRef ) this implies that LaTeXMLMath for all LaTeXMLMath .
∎ Let LaTeXMLMath be an arbitrary self–scaled barrier for the irreducible symmetric cone LaTeXMLMath .
Then there exists a positive constant LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath Since the Hessian LaTeXMLMath are positive definite cone automorphisms , Lemma LaTeXMLRef implies that there exists a well–defined function LaTeXMLMath such that LaTeXMLEquation .
Since LaTeXMLMath is self-scaled , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for all LaTeXMLMath where LaTeXMLMath denotes the scaling point of LaTeXMLMath and LaTeXMLMath for the self–scaled barrier LaTeXMLMath .
The quadratic representation LaTeXMLMath is injective on LaTeXMLMath , see Lemma 2.3 LaTeXMLCite .
Therefore , the identity above shows that LaTeXMLEquation for all LaTeXMLMath By Proposition LaTeXMLRef , we have LaTeXMLEquation for all LaTeXMLMath .
Now , LaTeXMLMath by definition of LaTeXMLMath , and Proposition LaTeXMLRef ( a ) shows that we have LaTeXMLMath , which together with ( LaTeXMLRef ) shows that LaTeXMLEquation for all LaTeXMLMath The proof is now completed by Lemma LaTeXMLRef .
∎ If LaTeXMLMath is a self–scaled barrier functional for LaTeXMLMath then there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
Similar to that of Theorem LaTeXMLRef .
∎ Both authors would like to thank Professor Leonid Faybusovich for suggesting this collaboration and establishing the contact between them .
Raphael Hauser also wishes to express his warmest thanks to Syvert Nørsett for inviting him to NTNU Trondheim , and to Peter and Jonathan Borwein for inviting him to SFU in Vancouver , where part of this research was done .
Let LaTeXMLMath be a nonsingular algebraic curve of genus LaTeXMLMath , and let LaTeXMLMath denote the moduli space of stable vector bundles of rank LaTeXMLMath and degree LaTeXMLMath with fixed determinant LaTeXMLMath over LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are coprime and LaTeXMLMath .
We assume that if LaTeXMLMath then LaTeXMLMath and if LaTeXMLMath then LaTeXMLMath .
Let LaTeXMLMath denote the vector bundle over LaTeXMLMath defined by the direct image LaTeXMLMath where LaTeXMLMath is a universal vector bundle over LaTeXMLMath and LaTeXMLMath is a line bundle over LaTeXMLMath of degree zero .
The space of infinitesimal deformations of LaTeXMLMath is proved to be isomorphic to LaTeXMLMath .
This construction gives a complete family of vector bundles over LaTeXMLMath parametrized by the Jacobian LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath is the vector bundle corresponding to LaTeXMLMath .
The connected component of the moduli space of stable sheaves with the same Hilbert polynomial as LaTeXMLMath over LaTeXMLMath containing LaTeXMLMath is in fact isomorphic to LaTeXMLMath as a polarised variety .
Let LaTeXMLMath be a connected nonsingular projective algebraic curve of genus LaTeXMLMath defined over the complex numbers .
Let LaTeXMLMath denote the Jacobian ( Picard variety ) of LaTeXMLMath and LaTeXMLMath the variety of line bundles of degree LaTeXMLMath over LaTeXMLMath ; thus in particular LaTeXMLMath .
Suppose LaTeXMLMath and let LaTeXMLMath be a Poincaré ( universal ) bundle over LaTeXMLMath .
If we denote by LaTeXMLMath the natural projection from LaTeXMLMath to LaTeXMLMath , the direct image LaTeXMLMath is then locally free and is called the Picard bundle of degree LaTeXMLMath .
These bundles have been investigated by a number of authors over at least the last 40 years .
It may be noted that the projective bundle corresponding to LaTeXMLMath can be identified with the LaTeXMLMath -fold symmetric product LaTeXMLMath .
Picard bundles were studied in this light by A. Mattuck LaTeXMLCite and I. G. Macdonald LaTeXMLCite among others ; both Mattuck and Macdonald gave formulae for their Chern classes .
Somewhat later R. C. Gunning LaTeXMLCite gave a more analytic treatment involving theta-functions .
Later still , and of especial relevance to us , G. Kempf LaTeXMLCite and S. Mukai LaTeXMLCite independently studied the deformations of the Picard bundle ; the problem then is to obtain an inversion formula showing that all deformations of LaTeXMLMath arise in a natural way .
Kempf and Mukai proved that LaTeXMLMath is simple and that , if LaTeXMLMath is not hyperelliptic , the space of infinitesimal deformations of LaTeXMLMath has dimension given by LaTeXMLEquation .
Moreover , all the infinitesimal deformations arise from genuine deformations .
In fact there is a complete family of deformations of LaTeXMLMath parametrised by LaTeXMLMath , the two factors corresponding respectively to translations in LaTeXMLMath and deformations of LaTeXMLMath ( LaTeXMLCite , LaTeXMLCite ) .
( The deformations of LaTeXMLMath are given by LaTeXMLMath for LaTeXMLMath . )
Since LaTeXMLMath is a principally polarised abelian variety and LaTeXMLMath , LaTeXMLMath can be identified with LaTeXMLMath ( strictly speaking LaTeXMLMath is the dual abelian variety , but the principal polarisation allows the identification ) .
Mukai ’ s paper LaTeXMLCite was set in a more general context involving a transform which provides an equivalence between the derived category of the category of LaTeXMLMath -modules over an abelian variety LaTeXMLMath and the corresponding derived category on the dual abelian variety LaTeXMLMath .
This technique has come to be known as the Fourier–Mukai transform and has proved very useful in studying moduli spaces of sheaves on abelian varieties and on some other varieties .
Our object in this paper is to generalise the results of Kempf and Mukai on deformations of Picard bundles to the moduli spaces of higher rank vector bundles over LaTeXMLMath with fixed determinant .
In particular we compute the space of infinitesimal deformations of the Picard bundle in this context and also identify a complete family of deformations .
While the analogy is not precise , this identification can be seen as a type of Fourier–Mukai transform .
We fix a holomorphic line bundle LaTeXMLMath over LaTeXMLMath of degree LaTeXMLMath .
Let LaTeXMLMath be the moduli space of stable vector bundles LaTeXMLMath over LaTeXMLMath with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
We assume that LaTeXMLMath and LaTeXMLMath are coprime , ensuring the smoothness and completeness of LaTeXMLMath , and that LaTeXMLMath .
We assume also that if LaTeXMLMath then LaTeXMLMath and if LaTeXMLMath then LaTeXMLMath .
The case LaTeXMLMath together with the three special cases LaTeXMLMath with LaTeXMLMath and LaTeXMLMath with LaTeXMLMath are omitted in our main result since the method of proof does not cover these cases .
It is known that there is a universal vector bundle over LaTeXMLMath .
Two such universal bundles differ by tensoring with the pullback of a line bundle on LaTeXMLMath .
However , since LaTeXMLMath , it is possible to choose canonically a universal bundle .
Let LaTeXMLMath be the smallest positive number such that LaTeXMLMath mod LaTeXMLMath .
There is a unique universal vector bundle LaTeXMLMath over LaTeXMLMath such that LaTeXMLMath LaTeXMLCite , where LaTeXMLMath and LaTeXMLMath is the ample generator of LaTeXMLMath .
Henceforth , by a universal bundle we will always mean this canonical one .
We denote by LaTeXMLMath and LaTeXMLMath the natural projections of LaTeXMLMath onto the two factors .
Now suppose that LaTeXMLMath .
For any LaTeXMLMath , let LaTeXMLEquation be the direct image .
The assumption on LaTeXMLMath ensures that LaTeXMLMath is a locally free sheaf on LaTeXMLMath and all the higher direct images of LaTeXMLMath vanish .
The rank of LaTeXMLMath is LaTeXMLMath and LaTeXMLMath .
By analogy with the case LaTeXMLMath , we shall refer to the bundles LaTeXMLMath as Picard bundles .
Our first main result concerns the infinitesimal deformations of LaTeXMLMath .
We prove Theorem LaTeXMLRef .
For any line bundle LaTeXMLMath , the space of infinitesimal deformations of the vector bundle LaTeXMLMath , namely LaTeXMLMath , is canonically isomorphic to LaTeXMLMath .
In particular , LaTeXMLEquation .
In the special case where LaTeXMLMath and LaTeXMLMath , this was proved by V. Balaji and P. A. Vishwanath in LaTeXMLCite using a construction of M. Thaddeus LaTeXMLCite .
For all LaTeXMLMath , it is known that LaTeXMLMath is simple LaTeXMLCite and indeed that it is stable ( with respect to the unique polarisation of LaTeXMLMath ) LaTeXMLCite ; in fact the proof of stability generalises easily to show that LaTeXMLMath is stable .
In this context , note that Y. Li LaTeXMLCite has proved a stability result for Picard bundles over the non-fixed determinant moduli space LaTeXMLMath , but this does not imply the result for LaTeXMLMath .
As a byproduct of our proof of Theorem LaTeXMLRef , we obtain a new proof that LaTeXMLMath is simple ( Corollary LaTeXMLRef ) .
We can consider the bundles LaTeXMLMath as a family of bundles over LaTeXMLMath , parametrized by LaTeXMLMath .
We prove that this is a complete family of deformations , both globally and in the local sense that the infinitesimal deformation map LaTeXMLEquation is an isomorphism for all LaTeXMLMath ( Corollary LaTeXMLRef ) .
As we noted , the bundles LaTeXMLMath are stable .
We denote by LaTeXMLMath the connected component of the moduli space of stable sheaves with the same Hilbert polynomial as LaTeXMLMath on LaTeXMLMath containing LaTeXMLMath .
We prove Theorem LaTeXMLRef .
The morphism LaTeXMLEquation given by LaTeXMLMath is an isomorphism of polarised varieties .
A further consequence is the following Torelli theorem : if LaTeXMLMath and LaTeXMLMath are smooth projective curves , LaTeXMLMath , LaTeXMLMath are line bundles of degree LaTeXMLMath over LaTeXMLMath , LaTeXMLMath respectively and LaTeXMLMath as polarised varieties , then LaTeXMLMath ( Corollary LaTeXMLRef ) .
Notation and assumptions .
We work throughout over the complex numbers and suppose that LaTeXMLMath is a connected nonsingular projective algebraic curve of genus LaTeXMLMath .
We suppose that LaTeXMLMath and that if LaTeXMLMath then LaTeXMLMath and if LaTeXMLMath then LaTeXMLMath .
We assume moreover that LaTeXMLMath and LaTeXMLMath .
In general , we denote the natural projections of LaTeXMLMath onto its factors by LaTeXMLMath , LaTeXMLMath , writing LaTeXMLMath for LaTeXMLMath for simplicity .
For a variety LaTeXMLMath , we denote by LaTeXMLMath the projection onto the LaTeXMLMath -th factor and by LaTeXMLMath the projection onto the Cartesian product of the LaTeXMLMath -th and the LaTeXMLMath -th factors .
Finally , for any LaTeXMLMath , we denote by LaTeXMLMath the bundle over LaTeXMLMath obtained by restricting LaTeXMLMath to LaTeXMLMath .
Acknowledgements .
Part of this work was done during a visit of the authors to ICTP , Italy .
The authors thank ICTP for its hospitality .
The third author would like also to thank the Isaac Newton Institute , Cambridge and the organisers of the HDG programme for their hospitality during the completion of work on this paper .
To compute the cohomology groups LaTeXMLMath we need the following propositions .
If LaTeXMLMath and LaTeXMLMath are two line bundles in LaTeXMLMath then LaTeXMLEquation for any LaTeXMLMath .
Moreover , LaTeXMLEquation for LaTeXMLMath , where LaTeXMLMath is the canonical line bundle over LaTeXMLMath .
By the assumption that LaTeXMLMath we have that LaTeXMLMath for all LaTeXMLMath .
Using the projection formula and the Leray spectral sequence we have LaTeXMLEquation .
This proves the first part .
For every stable vector bundle LaTeXMLMath of rank LaTeXMLMath and degree LaTeXMLMath , LaTeXMLEquation .
Consequently , the projection formula gives LaTeXMLEquation for LaTeXMLMath , and we have by relative Serre duality LaTeXMLEquation .
Finally , using the projection formula and the Leray spectral sequence , it follows that LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath .
Thus , LaTeXMLEquation as asserted in the proposition .
∎ Proposition LaTeXMLRef can be formulated in a more general context .
Let LaTeXMLMath , LaTeXMLMath be flat families of vector bundles over LaTeXMLMath parametrised by a complete irreducible variety LaTeXMLMath such that for each LaTeXMLMath we have LaTeXMLMath for LaTeXMLMath .
Under this assumption LaTeXMLEquation .
The proof is the same as for Proposition LaTeXMLRef .
Denote by LaTeXMLMath the LaTeXMLMath -th direct image of LaTeXMLMath for the projection LaTeXMLMath , that is LaTeXMLEquation .
From the Leray spectral sequence we obtain the exact sequences LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath .
Note that for any LaTeXMLMath LaTeXMLEquation .
From LaTeXMLCite and LaTeXMLCite we know that for generic LaTeXMLMath , the two vector bundles LaTeXMLMath and LaTeXMLMath are non-isomorphic and stable .
Hence LaTeXMLMath This implies that LaTeXMLEquation .
So ( LaTeXMLRef ) gives LaTeXMLEquation and the proof is complete .
∎ In the following sections we will use Hecke transformations to compute the fibre of LaTeXMLMath for LaTeXMLMath , and will prove the following propositions .
LaTeXMLMath LaTeXMLMath is a line bundle .
Moreover , LaTeXMLMath Assume for the moment Propositions LaTeXMLRef and LaTeXMLRef .
From the exact sequence ( LaTeXMLRef ) and the previous propositions we have the following theorem .
LaTeXMLMath Combining ( LaTeXMLRef ) , Proposition LaTeXMLRef and the second part of Proposition LaTeXMLRef it follows that LaTeXMLEquation .
From Proposition LaTeXMLRef it follows immediately that LaTeXMLEquation and hence the proof is complete .
∎ If LaTeXMLMath , then LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLEquation .
Consequently , LaTeXMLMath .
∎ The following is also a corollary of Theorem LaTeXMLRef .
The vector bundle LaTeXMLMath is simple for any LaTeXMLMath .
In other words , LaTeXMLMath .
The following theorem gives the infinitesimal deformations of LaTeXMLMath .
For any line bundle LaTeXMLMath , the space of infinitesimal deformations of the vector bundle LaTeXMLMath , namely LaTeXMLMath , is canonically isomorphic to LaTeXMLMath .
In particular , LaTeXMLEquation .
Let LaTeXMLMath .
From ( LaTeXMLRef ) and Proposition LaTeXMLRef we obtain an isomorphism LaTeXMLEquation .
From Proposition LaTeXMLRef we have LaTeXMLEquation .
Combining this observation with Proposition LaTeXMLRef and ( LaTeXMLRef ) we get LaTeXMLEquation ∎ From the proof of Theorem LaTeXMLRef and Proposition LaTeXMLRef we see that if LaTeXMLMath and LaTeXMLMath are not isomorphic then LaTeXMLEquation .
Hence LaTeXMLEquation .
In this section we will use Hecke transformations to compute the cohomology groups LaTeXMLMath for any LaTeXMLMath and prove Proposition LaTeXMLRef .
The details of the Hecke transformation and its properties can be found in LaTeXMLCite .
We will briefly describe it and note those properties that will be needed here .
Fix a point LaTeXMLMath .
Let LaTeXMLMath denote the projective bundle over LaTeXMLMath consisting of lines in LaTeXMLMath .
If LaTeXMLMath denotes the natural projection of LaTeXMLMath to LaTeXMLMath and LaTeXMLMath the tautological line bundle then LaTeXMLEquation and LaTeXMLMath for all LaTeXMLMath From the commutative diagram LaTeXMLEquation and the base change theorem , we deduce that LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath .
Moreover , since LaTeXMLMath , there is a canonical isomorphism LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath .
To compute the cohomology groups LaTeXMLMath we use Hecke transformations .
A point in LaTeXMLMath represents a vector bundle LaTeXMLMath and a line LaTeXMLMath in the fiber LaTeXMLMath at LaTeXMLMath , or equivalently a non-trivial exact sequence LaTeXMLEquation determined up to a scalar multiple ; here LaTeXMLMath denotes the torsion sheaf supported at LaTeXMLMath with stalk LaTeXMLMath .
The sequences ( LaTeXMLRef ) fit together to form a universal sequence LaTeXMLEquation on LaTeXMLMath .
If LaTeXMLMath denotes the line bundle LaTeXMLMath over LaTeXMLMath and LaTeXMLMath the moduli space of stable bundles LaTeXMLMath then from ( LaTeXMLRef ) and ( LaTeXMLRef ) we get a rational map LaTeXMLEquation which sends any pair LaTeXMLMath to LaTeXMLMath .
This map is not everywhere defined since the bundle LaTeXMLMath in ( LaTeXMLRef ) need not be stable .
Our next object is to find a Zariski-open subset LaTeXMLMath of LaTeXMLMath , over which LaTeXMLMath is defined and is a projective fibration , such that the complement of LaTeXMLMath in LaTeXMLMath has codimension at least 4 .
The construction and calculations are similar to those of LaTeXMLCite , but our results do not seem to follow directly from that proposition .
As in LaTeXMLCite or LaTeXMLCite , we define a bundle LaTeXMLMath to be LaTeXMLMath - stable if , for every proper subbundle LaTeXMLMath of LaTeXMLMath , LaTeXMLEquation .
Clearly every LaTeXMLMath -stable bundle is stable .
We denote by LaTeXMLMath the subset of LaTeXMLMath consisting of LaTeXMLMath -stable bundles .
( i ) LaTeXMLMath is a Zariski-open subset of LaTeXMLMath whose complement has codimension at least LaTeXMLMath .
( ii ) LaTeXMLMath is a projective fibration over LaTeXMLMath and LaTeXMLMath is a Zariski-open subset of LaTeXMLMath whose complement has codimension at least LaTeXMLMath .
( i ) The fact that LaTeXMLMath is Zariski-open is standard ( see LaTeXMLCite ) .
The bundle LaTeXMLMath of rank LaTeXMLMath and degree LaTeXMLMath fails to be LaTeXMLMath -stable if and only if it has a subbundle LaTeXMLMath of rank LaTeXMLMath and degree LaTeXMLMath such that LaTeXMLMath , i.e. , LaTeXMLEquation .
By considering the extensions LaTeXMLEquation we can estimate the codimension of LaTeXMLMath and show that it is at least LaTeXMLEquation ( compare the proof of LaTeXMLCite ) .
Note that , since LaTeXMLMath , ( LaTeXMLRef ) implies that LaTeXMLMath .
Given that LaTeXMLMath , we see that LaTeXMLMath only if LaTeXMLMath , LaTeXMLMath or LaTeXMLMath , LaTeXMLMath .
These are exactly the cases that were excluded in the introduction .
( ii ) LaTeXMLMath consists of all pairs LaTeXMLMath for which the bundle LaTeXMLMath in ( LaTeXMLRef ) is LaTeXMLMath -stable .
As in ( i ) , this is a Zariski-open subset .
It follows at once from ( LaTeXMLRef ) that , if LaTeXMLMath is LaTeXMLMath -stable , then LaTeXMLMath is stable .
So , if LaTeXMLMath , it follows from ( LaTeXMLRef ) that LaTeXMLMath can be identified with the projective space LaTeXMLMath .
Using the universal projective bundle on LaTeXMLMath , we see that LaTeXMLMath is a projective fibration over LaTeXMLMath ( not necessarily locally trivial ) .
Suppose now that LaTeXMLMath belongs to the complement of LaTeXMLMath in LaTeXMLMath .
This means that the bundle LaTeXMLMath in ( LaTeXMLRef ) is not LaTeXMLMath -stable and therefore possesses a subbundle LaTeXMLMath satisfying ( LaTeXMLRef ) .
If LaTeXMLMath , this contradicts the stability of LaTeXMLMath .
So there exists an exact sequence LaTeXMLEquation with LaTeXMLMath a subbundle of LaTeXMLMath of rank LaTeXMLMath and degree LaTeXMLMath .
Moreover , since LaTeXMLMath is a subbundle of LaTeXMLMath , LaTeXMLMath must contain the line LaTeXMLMath .
For fixed LaTeXMLMath , these conditions determine a subvariety of LaTeXMLMath of dimension at most LaTeXMLEquation .
Since LaTeXMLMath , a simple calculation shows that the codimension is at least the number LaTeXMLMath given by ( LaTeXMLRef ) .
As in ( i ) , this gives the required result .
∎ By Lemma LaTeXMLRef ( ii ) and a Hartogs-type theorem ( see LaTeXMLCite ) we have an isomorphism LaTeXMLEquation for LaTeXMLMath .
Now let LaTeXMLMath .
As in the proof of Lemma LaTeXMLRef , we identify LaTeXMLMath with LaTeXMLMath and denote it by LaTeXMLMath .
On LaTeXMLMath there is a universal exact sequence LaTeXMLEquation .
The restriction of ( LaTeXMLRef ) to any point of LaTeXMLMath is isomorphic to the corresponding sequence ( LaTeXMLRef ) .
Let LaTeXMLMath be defined by the universal sequence ( LaTeXMLRef ) .
Then LaTeXMLEquation .
Restricting ( LaTeXMLRef ) to LaTeXMLMath gives LaTeXMLEquation .
This must coincide with the universal sequence ( LaTeXMLRef ) up to tensoring by some line bundle lifted from LaTeXMLMath .
The result follows .
∎ Next we tensor ( LaTeXMLRef ) by LaTeXMLMath , restrict it to LaTeXMLMath and take the direct image on LaTeXMLMath .
This gives LaTeXMLEquation .
LaTeXMLMath for all LaTeXMLMath .
It is sufficient to show that LaTeXMLMath has trivial cohomology .
By Proposition LaTeXMLRef LaTeXMLEquation and the result follows .
∎ LaTeXMLMath for LaTeXMLMath .
Moreover , LaTeXMLEquation .
This follows at once from ( LaTeXMLRef ) and Proposition LaTeXMLRef .
∎ Now we are in a position to compute the cohomology groups of LaTeXMLMath for LaTeXMLMath LaTeXMLMath for any LaTeXMLMath The combination of ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) yields LaTeXMLEquation .
Using Corollary LaTeXMLRef and Lemma LaTeXMLRef ( i ) , the Leray spectral sequence for the map LaTeXMLMath gives LaTeXMLEquation .
It is known that LaTeXMLMath LaTeXMLCite .
Therefore , LaTeXMLEquation ∎ Proof of Proposition LaTeXMLRef .
Proposition LaTeXMLRef is an immediate consequence of Proposition LaTeXMLRef .
LaTeXMLMath For any point LaTeXMLMath , LaTeXMLMath .
As in the proof of Proposition LaTeXMLRef we conclude that LaTeXMLEquation .
Now LaTeXMLMath is just the non-singular part of the moduli space of semistable bundles of rank LaTeXMLMath and determinant LaTeXMLMath , and the latter space is complete and normal .
So LaTeXMLMath .
∎ Since the fibres of LaTeXMLMath are projective spaces , we have LaTeXMLMath and all the higher direct images of LaTeXMLMath are LaTeXMLMath .
Hence LaTeXMLEquation for all LaTeXMLMath .
Similarly LaTeXMLEquation for LaTeXMLMath since LaTeXMLMath is a smooth projective rational variety .
It follows from the proof of Lemma LaTeXMLRef that , if we define LaTeXMLMath as in ( LaTeXMLRef ) , LaTeXMLEquation .
The proof of Proposition LaTeXMLRef now gives LaTeXMLEquation .
This in turn implies that LaTeXMLMath .
Proposition LaTeXMLRef and the Leray spectral sequence of LaTeXMLMath ( cf .
( LaTeXMLRef ) and ( LaTeXMLRef ) ) now give LaTeXMLEquation .
In particular , taking LaTeXMLMath and LaTeXMLMath , and using ( LaTeXMLRef ) , we obtain LaTeXMLEquation except possibly when LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
Let LaTeXMLMath be the diagonal divisor in LaTeXMLMath .
Pull back the exact sequence LaTeXMLEquation to LaTeXMLMath and tensor it with LaTeXMLMath .
Now , the direct image sequence for the projection LaTeXMLMath gives the following exact sequence over LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation The following propositions will be used in computing the direct images LaTeXMLMath For any LaTeXMLMath , the direct images of LaTeXMLEquation have the following description : LaTeXMLMath LaTeXMLMath LaTeXMLMath where LaTeXMLMath is the tangent bundle of LaTeXMLMath .
Identifying LaTeXMLMath with LaTeXMLMath we have LaTeXMLEquation .
The proposition follows from a result of Narasimhan and Ramanan LaTeXMLCite that says LaTeXMLEquation .
For LaTeXMLMath the isomorphism is given by the obvious inclusion of LaTeXMLMath in LaTeXMLMath and therefore globalises to give LaTeXMLMath .
Similarly for LaTeXMLMath the isomorphism is given by the infinitesimal deformation map of LaTeXMLMath regarded as a family of bundles over LaTeXMLMath parametrised by LaTeXMLMath ; this globalises to LaTeXMLMath .
∎ Propositions LaTeXMLRef , LaTeXMLRef and the exact sequence ( LaTeXMLRef ) together give the following exact sequence of direct images LaTeXMLEquation .
LaTeXMLEquation For any LaTeXMLMath we have the cohomology exact sequence LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the fiber of LaTeXMLMath at LaTeXMLMath .
By ( LaTeXMLRef ) , LaTeXMLMath .
So the Leray spectral sequence for LaTeXMLMath gives LaTeXMLEquation and LaTeXMLEquation .
Since LaTeXMLMath is simple LaTeXMLCite , ( LaTeXMLRef ) gives the exact sequence LaTeXMLEquation .
LaTeXMLEquation This implies that LaTeXMLMath has torsion at LaTeXMLMath .
Now from ( LaTeXMLRef ) we conclude that LaTeXMLMath is a torsion sheaf , and hence LaTeXMLEquation .
The Leray spectral sequence for LaTeXMLMath now yields LaTeXMLEquation and LaTeXMLEquation .
Now from ( LaTeXMLRef ) it follows that LaTeXMLMath is a torsion sheaf , and from Proposition LaTeXMLRef and ( LaTeXMLRef ) that its space of sections is LaTeXMLMath .
So LaTeXMLMath and by ( LaTeXMLRef ) LaTeXMLEquation .
So , we have the following proposition : LaTeXMLMath Now Proposition LaTeXMLRef is easy to derive .
Proof of Proposition LaTeXMLRef .
By Proposition LaTeXMLRef LaTeXMLEquation .
Therefore , LaTeXMLMath is a line bundle .
Moreover Proposition LaTeXMLRef implies that the map LaTeXMLMath in the exact sequence ( LaTeXMLRef ) is surjective .
Therefore LaTeXMLMath must be an isomorphism , and the proof is complete .
LaTeXMLMath Earlier Theorems LaTeXMLRef and LaTeXMLRef were proved assuming Propositions LaTeXMLRef and LaTeXMLRef .
Therefore Theorems LaTeXMLRef and LaTeXMLRef are now established .
Fix a point LaTeXMLMath .
Let LaTeXMLMath be the Poincaré line bundle over LaTeXMLMath which is trivial on LaTeXMLMath .
It was proved in LaTeXMLCite that the family LaTeXMLMath over LaTeXMLMath is a complete family of deformations of LaTeXMLMath parametrised by LaTeXMLMath .
In this section we will prove that the direct image LaTeXMLEquation is a complete family of deformations of LaTeXMLMath for any LaTeXMLMath .
First note that LaTeXMLMath is locally free .
Indeed , LaTeXMLMath for LaTeXMLMath Moreover , for each LaTeXMLMath , LaTeXMLEquation .
In LaTeXMLCite it was proved that LaTeXMLMath is stable with respect to the unique polarisation of LaTeXMLMath .
From the proof of the Theorem in LaTeXMLCite we can see that for any line bundle LaTeXMLMath , LaTeXMLMath is also stable .
Since LaTeXMLMath is irreducible , the Hilbert polynomial LaTeXMLMath of LaTeXMLMath is well defined and independent of the choice of LaTeXMLMath .
Denote by LaTeXMLMath the connected component of the moduli space of stable sheaves , with the same Hilbert polynomial as LaTeXMLMath , on LaTeXMLMath containing LaTeXMLMath .
As in LaTeXMLCite the determinant line bundle LaTeXMLMath on LaTeXMLMath defines a polarisation on LaTeXMLMath .
Define the morphism LaTeXMLEquation by LaTeXMLMath The morphism LaTeXMLEquation is an isomorphism of polarised varieties .
By Corollary LaTeXMLRef LaTeXMLMath is injective , so its image has dimension LaTeXMLMath .
On the other hand , by Theorem LaTeXMLRef , the Zariski tangent space of LaTeXMLMath also has dimension LaTeXMLMath at every point of the image of LaTeXMLMath .
It follows that LaTeXMLMath is smooth of dimension LaTeXMLMath at every point of the image of LaTeXMLMath .
Hence , by Zariski ’ s Main Theorem , LaTeXMLMath is an isomorphism onto an open subset of LaTeXMLMath .
Finally LaTeXMLMath is complete and LaTeXMLMath is connected and separated ( since LaTeXMLMath is stable ) , so LaTeXMLMath is an isomorphism .
Let LaTeXMLMath be the polarisation on LaTeXMLMath given by the determinant line bundle LaTeXMLCite .
Let LaTeXMLMath denote the principal polarisation on LaTeXMLMath defined by a theta divisor .
We wish to show that the isomorphism LaTeXMLMath takes LaTeXMLMath to a nonzero constant scalar multiple ( independent of the curve LaTeXMLMath ) of LaTeXMLMath .
Take any family of pairs LaTeXMLMath , where LaTeXMLMath is a connected non-singular projective curve of genus LaTeXMLMath and LaTeXMLMath is a line bundle on LaTeXMLMath of degree LaTeXMLMath , parametrized by a connected space LaTeXMLMath .
Consider the corresponding family of moduli spaces LaTeXMLMath ( respectively , Jacobians LaTeXMLMath ) over LaTeXMLMath , where LaTeXMLMath runs over the family .
Using the map LaTeXMLMath an isomorphism between these two families is obtained .
The polarisation LaTeXMLMath ( respectively , LaTeXMLMath ) defines a constant section of the second direct image over LaTeXMLMath of the constant sheaf LaTeXMLMath over the family .
It is known that for the general curve LaTeXMLMath of genus LaTeXMLMath , the Neron-Severi group of LaTeXMLMath is LaTeXMLMath .
Therefore , for such a curve , LaTeXMLMath takes LaTeXMLMath to a nonzero constant scalar multiple of LaTeXMLMath .
Since LaTeXMLMath is connected , if LaTeXMLMath contains a curve with LaTeXMLMath , then LaTeXMLMath takes LaTeXMLMath to the same nonzero constant scalar multiple of LaTeXMLMath for every curve in the family .
Since the moduli space of smooth curves of genus LaTeXMLMath is connected , the proof is complete .
∎ The family LaTeXMLMath parametrised by LaTeXMLMath is complete .
Moreover , the infinitesimal deformation map of this family at any point of LaTeXMLMath is an isomorphism .
This follows at once from the theorem .
∎ In the proof of Theorem LaTeXMLRef , smoothness follows from the fact that the dimension of LaTeXMLMath is equal to the dimension of its Zariski tangent space .
So we do not need to know that LaTeXMLMath ( see Remark LaTeXMLRef ) or even that LaTeXMLMath is reduced .
Finally we have our Torelli theorem .
Let LaTeXMLMath and LaTeXMLMath be two non-singular algebraic curves of genus LaTeXMLMath and let LaTeXMLMath ( respectively LaTeXMLMath ) be a line bundle of degree LaTeXMLMath on LaTeXMLMath ( respectively LaTeXMLMath ) .
If LaTeXMLMath as polarised varieties then LaTeXMLMath .
This follows at once from Theorem LaTeXMLRef and the classical Torelli theorem .
∎
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski ’ s projective frame .
In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic terms .
Keywords : Lie sphere geometry , Wilczynski ’ s frame , twistor methods , Schrödinger operators with magnetic terms .
Mathematical Subject Classification : 53A40 , 53A20 .
Let LaTeXMLMath be a surface in LaTeXMLMath parametrized by coordinates LaTeXMLMath of curvature lines , with the radius-vector LaTeXMLMath and the unit normal LaTeXMLMath satisfying the Weingarten equations LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the radii of principal curvature , LaTeXMLMath .
Let us recall the construction of the Lie sphere map LaTeXMLCite .
With any sphere LaTeXMLMath having radius LaTeXMLMath and center LaTeXMLMath this map associates the 6-vector LaTeXMLMath with hexaspherical coordinates LaTeXMLEquation which obey the relation LaTeXMLEquation .
This equation defines the so-called Lie quadric .
Thus , with any sphere LaTeXMLMath in LaTeXMLMath we associate a point on the Lie quadric ( LaTeXMLRef ) .
The Lie sphere map linearises the action of the Lie sphere group which is a group of contact transformations in LaTeXMLMath generated by conformal transformations and normal shifts .
In hexaspherical coordinates , the action of the Lie sphere group coincides with the linear action of LaTeXMLMath which preserves the Lie quadric ( LaTeXMLRef ) .
The reader may consult LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite for further properties of this construction .
Applying Lie sphere map to the curvature spheres LaTeXMLMath and LaTeXMLMath of the surface LaTeXMLMath , we obtain a pair of two-dimensional submanifolds LaTeXMLEquation and LaTeXMLEquation of the Lie quadric .
Blaschke ’ s approach LaTeXMLCite to the Lie sphere geometry of surfaces in 3-space is based on the following two simple facts : ( a ) By construction , vectors LaTeXMLMath and LaTeXMLMath have zero norm LaTeXMLEquation in the scalar product defined by ( LaTeXMLRef ) .
( b ) The triple LaTeXMLMath is orthogonal to the triple LaTeXMLMath .
Using appropriate linear combinations among these triples , one can construct an invariant 6-frame canonically associated with a surface LaTeXMLMath ( Appendix B , see also LaTeXMLCite , LaTeXMLCite ) .
Conversely , given a pair of 6-vectors LaTeXMLMath and LaTeXMLMath satisfying ( a ) and ( b ) , the surface can be reconstructed uniquely as the envelope of the corresponding family of curvature spheres .
Our construction of vectors LaTeXMLMath and LaTeXMLMath which satisfy the above properties was borrowed from projective differential geometry .
In 1907 , Wilczynski LaTeXMLCite proposed the approach to surfaces in projective space based on a linear system LaTeXMLEquation where LaTeXMLMath are real functions of LaTeXMLMath and LaTeXMLMath .
Cross-differentiating ( LaTeXMLRef ) and assuming LaTeXMLMath to be independent , we arrive at the compatibility conditions LaTeXMLEquation .
For any LaTeXMLMath satisfying ( LaTeXMLRef ) the linear system ( LaTeXMLRef ) has rank 4 , so that LaTeXMLMath can be viewed as a 4-component real vector of homogeneous coordinates of a surface LaTeXMLMath in a projective space LaTeXMLMath .
The compatibility conditions ( LaTeXMLRef ) can be interpreted as the ‘ Gauss-Codazzi ’ equations in projective differential geometry .
The independent variables LaTeXMLMath play the role of asymptotic coordinates on the surface LaTeXMLMath .
Introducing the vectors LaTeXMLEquation in LaTeXMLMath , one readily verifies that ( a ) vectors LaTeXMLMath and LaTeXMLMath have zero norm LaTeXMLEquation in the natural scalar product in LaTeXMLMath defined by the Plücker formula ; ( b ) the triple LaTeXMLMath is orthogonal to the triple LaTeXMLMath .
The passage from a projective surface to a pair of submanifolds LaTeXMLMath and LaTeXMLMath in the Plücker quadric in LaTeXMLMath plays an important role in projective differential geometry .
We refer to LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) for a further discussion .
Some more details on Wilczynski ’ s approach are included in Appendix A .
The main observation of this paper is that a similar approach , based on the linear system LaTeXMLEquation applies to Lie sphere geometry .
Here LaTeXMLMath and LaTeXMLMath denote partial derivatives with respect to the independent variables LaTeXMLMath , and LaTeXMLMath are real potentials .
This system is a complex analogue of the linear system ( LaTeXMLRef ) ( indeed , the complex transformation LaTeXMLMath identifies both systems ) .
Again , cross-differentiation produces the compatibility conditions LaTeXMLEquation .
For any fixed LaTeXMLMath satisfying ( LaTeXMLRef ) , the linear system ( LaTeXMLRef ) is compatible and its solution space has complex dimension 4 , so that we can view LaTeXMLMath as an element of the twistor space LaTeXMLMath .
In what follows , equations ( LaTeXMLRef ) will be identified with the ‘ Gauss-Codazzi ’ equations in Lie sphere geometry , while the independent variables LaTeXMLMath will play the role of curvature line coordinates .
An important property of linear system ( LaTeXMLRef ) is the existence of the invariant pseudo-Hermitian scalar product LaTeXMLMath of the signature LaTeXMLMath such that LaTeXMLEquation ( all other scalar products being zero ) .
Fixing LaTeXMLMath satisfying both ( LaTeXMLRef ) and ( LaTeXMLRef ) , we define two real vectors LaTeXMLMath and LaTeXMLMath in LaTeXMLMath as LaTeXMLEquation .
Introducing in LaTeXMLMath a pseudo-Hermitian scalar product LaTeXMLMath induced by ( LaTeXMLRef ) ( the wedge product LaTeXMLMath and the scalar product LaTeXMLMath in LaTeXMLMath are explicitly defined in Appendix C ) , we show that the restriction of LaTeXMLMath to the 6-dimensional invariant real subspace in LaTeXMLMath spanned by LaTeXMLMath and LaTeXMLMath is a real scalar product of the signature LaTeXMLMath .
Moreover , ( a ) vectors LaTeXMLMath and LaTeXMLMath have zero norm LaTeXMLEquation ( b ) the triple LaTeXMLMath is orthogonal to the triple LaTeXMLMath .
Thus , LaTeXMLMath and LaTeXMLMath are hexaspherical coordinates of curvature spheres of a surface LaTeXMLMath parametrized by curvature line coordinates LaTeXMLMath .
In fact , our construction is based on the isomorphism LaTeXMLMath which is a basis of twistor theory LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
The main motivation for the construction described above comes from the study of Lie-applicable surfaces .
We recall that two surfaces are called Lie-applicable if , being non-equivalent under Lie sphere transformations , they have the same coefficients LaTeXMLMath and LaTeXMLMath in the appropriate curvature line parametrization LaTeXMLMath LaTeXMLCite .
Analytically , Lie-applicable surfaces are described by equations ( LaTeXMLRef ) which , for given LaTeXMLMath and LaTeXMLMath , are not uniquely solvable for LaTeXMLMath and LaTeXMLMath .
Examples presented in section 3 demonstrate that for some particularly interesting classes of Lie-applicable surfaces , equations ( LaTeXMLRef ) describe joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields .
These examples include nonsingular doubly periodic operators on a two-dimensional torus and the Dirac monopole on a two-dimensional sphere LaTeXMLCite .
It should be emphasized that this important structure is not visible in the standard approach based on Blaschke ’ s 6-frame , becoming apparent only after applying the twistor construction .
In section 4 we give a separate treatment of canal surfaces ( which are characterized by a condition LaTeXMLMath ( or LaTeXMLMath ) ) , since the construction of the Lie sphere frame adopted in section 2 does not automatically carry over to this case .
As an example , we explicitly construct the surfaces of revolution for which equations ( LaTeXMLRef ) reduce to eigenfunction equations of the Schrödinger operator in a homogeneous magnetic field .
The case of the first nontrivial Landau level is worked out in detail .
In this section we propose an approach to surfaces in Lie sphere geometry based on the linear system ( LaTeXMLRef ) satisfying the compatibility conditions ( LaTeXMLRef ) .
Our further constructions will follow those from projective differential geometry — see Appendix A .
Notice first that system ( LaTeXMLRef ) is covariant under transformations of the form LaTeXMLEquation which act on the potentials LaTeXMLMath as follows LaTeXMLEquation .
Here LaTeXMLMath is the Schwarzian derivative LaTeXMLEquation so that LaTeXMLMath and LaTeXMLMath can be interpreted as projective connections .
Transformation formulae ( LaTeXMLRef ) , ( LaTeXMLRef ) imply that the symmetric 2-form LaTeXMLEquation and the conformal class of the cubic form LaTeXMLEquation are invariant .
In what follows they will play the roles of the Lie-invariant metric and the Lie-invariant cubic form , respectively .
In this section we assume both LaTeXMLMath and LaTeXMLMath to be nonzero .
Let us introduce the four vectors LaTeXMLEquation which are straightforward analogues of vertices of Wilczynski ’ s moving tetrahedral — see formulae ( LaTeXMLRef ) in Appendix A .
Notice that under transformations ( LaTeXMLRef ) vectors ( LaTeXMLRef ) acquire nonzero multiples which do not change them as points in the complex projective space .
Using ( LaTeXMLRef ) and ( LaTeXMLRef ) , we readily derive for LaTeXMLMath the linear system LaTeXMLEquation where we introduced the notation LaTeXMLEquation .
The compatibility conditions of equations ( LaTeXMLRef ) imply LaTeXMLEquation which is just an equivalent form of the ‘ Gauss-Codazzi ’ equations ( LaTeXMLRef ) .
An important property of system ( LaTeXMLRef ) is the existence of the quadratic integral LaTeXMLEquation which defines an invariant pseudo-Hermitian scalar product of the signature LaTeXMLMath on the space of solutions of system ( LaTeXMLRef ) .
Using ( LaTeXMLRef ) , this integral can be rewritten in the form LaTeXMLEquation .
The existence of the invariant pseudo-Hermitian scalar product ( LaTeXMLRef ) allows to choose a basis of solutions of ( LaTeXMLRef ) such that LaTeXMLEquation all other scalar products being zero .
Notice that formulae ( LaTeXMLRef ) are equivalent to ( LaTeXMLRef ) .
Here LaTeXMLMath denotes the pseudo-Hermitian scalar product in LaTeXMLMath LaTeXMLEquation of the signature LaTeXMLMath .
Equations ( LaTeXMLRef ) also imply that the determinant LaTeXMLMath is invariant : LaTeXMLEquation ( indeed , both matrices in ( LaTeXMLRef ) are traceless ) .
Thus , besides ( LaTeXMLRef ) , we can impose the additional constraint LaTeXMLEquation .
From now on , we fix a null-tetrad LaTeXMLMath satisfying both ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Notice that this basis LaTeXMLMath described above is defined up to a natural linear action of the group LaTeXMLMath which preserves both ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Introducing the basis in LaTeXMLMath as follows ( notice the analogy with self-dual and anti-self-dual forms in twistor theory ) LaTeXMLEquation we arrive at the equations LaTeXMLEquation which identically coincide with equations of the Lie sphere frame — see Appendix B .
Let us define the wedge product LaTeXMLMath as in Appendix C and introduce the pseudo-Hermitian scalar product LaTeXMLMath in LaTeXMLMath LaTeXMLEquation ( we hope that the same notation LaTeXMLMath for pseudo-Hermitian scalar products in LaTeXMLMath and LaTeXMLMath will not lead to a confusion ) .
A direct computation shows that the only nonzero products among the vectors LaTeXMLMath are LaTeXMLEquation .
This invariant pseudo-Hermitian scalar product corresponds to the quadratic integral LaTeXMLEquation of system ( LaTeXMLRef ) .
Similarly , we can define the complex scalar product LaTeXMLMath in LaTeXMLMath ( see Appendix C ) : LaTeXMLEquation .
A direct computation shows that the only nonzero products among the vectors LaTeXMLMath , LaTeXMLMath are LaTeXMLEquation .
This invariant complex scalar product corresponds to the quadratic integral LaTeXMLEquation of system ( LaTeXMLRef ) .
Notice the important relation between the pseudo-Hermitian scalar product LaTeXMLMath and the complex scalar product LaTeXMLMath in LaTeXMLMath : LaTeXMLEquation for any LaTeXMLMath in LaTeXMLMath ( see Appendix 3 ) .
Let us finally introduce in LaTeXMLMath the real vectors LaTeXMLEquation .
Vectors LaTeXMLMath and LaTeXMLMath have zero norm : LaTeXMLEquation .
Moreover , the triple LaTeXMLMath is orthogonal to the triple LaTeXMLMath .
Hence , LaTeXMLMath and LaTeXMLMath are curvature spheres of a surface .
Proof : It readily follows from ( LaTeXMLRef ) that the triple LaTeXMLMath is equivalent to LaTeXMLMath .
Similarly , the triple LaTeXMLMath is equivalent to LaTeXMLMath .
Conditions LaTeXMLMath and the orthogonality of both triples follow by virtue of ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Let us show , for instance , that LaTeXMLMath : LaTeXMLEquation which is zero by virtue of ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Remark 1 .
It is straightforward to show that any surface can be obtained ( locally ) by a construction described above .
Remark 2 .
In view of ( LaTeXMLRef ) the surface LaTeXMLMath defines a Legendre submanifold of the quadric LaTeXMLMath equipped with a real contact form LaTeXMLMath .
I would like to thank L. Mason for clarifying this point .
In contrast with the Euclidean geometry , where a surface is uniquely determined by its first and second fundamental forms , there exist examples of surfaces in Lie sphere geometry which are not uniquely specified by the Lie-invariant metric LaTeXMLEquation and the conformal class of the cubic form LaTeXMLEquation .
Such surfaces are called Lie-applicable ( Lie-deformable ) .
In this section we consider examples of surfaces possessing 3-parameter families of Lie deformations .
A calculation similar to the one done by Finikov in LaTeXMLCite shows that these surfaces are characterized by the constraints LaTeXMLEquation where LaTeXMLMath is a constant .
There are different cases to distinguish depending on the value of LaTeXMLMath .
Here we discuss the two simplest cases LaTeXMLMath and LaTeXMLMath ( for LaTeXMLMath the formulae become more complicated ) .
Case c=0 Utilizing transformations ( LaTeXMLRef ) and ( LaTeXMLRef ) , we can represent LaTeXMLMath and LaTeXMLMath in the form LaTeXMLEquation implying , after the substitution into ( LaTeXMLRef ) and elementary integration , the following expressions for LaTeXMLMath and LaTeXMLMath LaTeXMLEquation .
Moreover , LaTeXMLMath and LaTeXMLMath satisfy the ODE ’ s LaTeXMLEquation implying that LaTeXMLMath and LaTeXMLMath are elliptic functions .
Here LaTeXMLMath are arbitrary constants ( if LaTeXMLMath is nonzero one can always reduce LaTeXMLMath and LaTeXMLMath to zero by adding constants to LaTeXMLMath ) .
Notice that for given LaTeXMLMath and LaTeXMLMath the corresponding LaTeXMLMath and LaTeXMLMath are determined up to three arbitrary constants LaTeXMLMath , which are thus responsible for Lie deformations .
It is important to emphasize the linear dependence of LaTeXMLMath and LaTeXMLMath on the deformation parameters .
This readily follows from ( LaTeXMLRef ) , indeed , for given LaTeXMLMath and LaTeXMLMath these equations are linear in LaTeXMLMath and LaTeXMLMath .
The corresponding system ( LaTeXMLRef ) takes the form LaTeXMLEquation which , upon the addition and subtraction , readily rewrites in the form LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath are commuting Schrödinger operators with magnetic terms LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath and LaTeXMLMath are the eigenvalues , and the potentials LaTeXMLMath and LaTeXMLMath are given by LaTeXMLEquation .
LaTeXMLEquation For generic values of constants , operators LaTeXMLMath and LaTeXMLMath will be non-singular and doubly periodic .
The spectral theory of these operators will be discussed elsewhere .
Case c=1 Here LaTeXMLEquation implying LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are functions of LaTeXMLMath and LaTeXMLMath , respectively .
The corresponding LaTeXMLMath and LaTeXMLMath are given by LaTeXMLEquation where the constants LaTeXMLMath are responsible for Lie deformations .
These surfaces are known to have both families of curvature lines in linear complexes LaTeXMLCite .
Introducing the rescaled vector LaTeXMLMath by the formula LaTeXMLEquation we readily rewrite equations ( LaTeXMLRef ) in the equivalent form LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath denote the derivatives of LaTeXMLMath and LaTeXMLMath , respectively .
Solving for LaTeXMLMath and LaTeXMLMath , we arrive at the eigenfunction equations LaTeXMLEquation where the operator LaTeXMLMath is of the form LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath are the components of a diagonal metric of Stäckel type LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath are the components of the magnetic vector potential LaTeXMLEquation and LaTeXMLMath is the scalar potential LaTeXMLEquation .
Geometrically , LaTeXMLMath represents the Laplace-Beltrami operator corresponding to the metric LaTeXMLMath in the magnetic potential LaTeXMLMath and the scalar potential LaTeXMLMath .
Notice that the scalar potential LaTeXMLMath can be represented in a simple coordinate-free form LaTeXMLEquation where LaTeXMLMath is the Gaussian curvature of the metric LaTeXMLMath .
The second term LaTeXMLMath is nothing but the trace of the Killing tensor of the Stäckel metric LaTeXMLMath , and hence also makes an invariant sense .
Computation of the magnetic field implies LaTeXMLEquation where LaTeXMLEquation is the area form of the metric LaTeXMLMath .
Thus , the magnetic field also makes an invariant sense .
Notice that in the case when LaTeXMLEquation are cubic polynomials , the Gaussian curvature LaTeXMLMath and the operator LaTeXMLMath represents Dirac monopole on the unit sphere in the spherical-conical coordinates LaTeXMLMath .
The scalar potential LaTeXMLMath ( which in this case is proportional to LaTeXMLMath ) has a meaning of the external quadratic potential .
We refer to LaTeXMLCite for the discussion of some algebraic aspects of spectral theory of such operators in the particular case LaTeXMLMath .
The general situation will be discussed elsewhere .
The examples discussed in this section clearly demonstrate that there exists a one-to-one correspondence between commuting Schrödinger operators with magnetic fields and Lie-applicable surfaces which possess multi-parameter families of Lie deformations .
Our approach to the canal surfaces will be based on a linear system LaTeXMLEquation which is a specialization of ( LaTeXMLRef ) corresponding to LaTeXMLMath .
The compatibility conditions of system ( LaTeXMLRef ) take the form LaTeXMLEquation .
Since LaTeXMLMath , we can not use formulae ( LaTeXMLRef ) .
Instead , we introduce the four vectors LaTeXMLEquation which satisfy the linear system LaTeXMLEquation where the notation LaTeXMLEquation is introduced .
Compatibility conditions of equations ( LaTeXMLRef ) imply LaTeXMLEquation .
An important property of system ( LaTeXMLRef ) is the existence of the quadratic integral LaTeXMLEquation which defines an invariant pseudo-Hermitian scalar product of the signature LaTeXMLMath on the space of solutions of system ( LaTeXMLRef ) .
Using ( LaTeXMLRef ) , this integral can be rewritten in the form LaTeXMLEquation .
The invariant pseudo-Hermitian scalar product ( LaTeXMLRef ) implies the existence of a basis of solutions of ( LaTeXMLRef ) such that LaTeXMLEquation all other scalar products being zero .
Equations ( LaTeXMLRef ) are obviously equivalent to LaTeXMLEquation .
Here LaTeXMLMath denotes the pseudo-Hermitian scalar product in LaTeXMLMath of the signature LaTeXMLMath as in Appendix C. Equations ( LaTeXMLRef ) also imply that the determinant LaTeXMLMath is invariant : LaTeXMLEquation .
Thus , besides ( LaTeXMLRef ) , we can impose the additional constraint LaTeXMLEquation .
From now on , we fix a null-tetrad LaTeXMLMath satisfying both ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Notice that such a basis is defined up to the action of the group LaTeXMLMath which preserves both ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Introducing the basis in LaTeXMLMath as follows LaTeXMLEquation we arrive at the equations LaTeXMLEquation .
Let us define the pseudo-Hermitian scalar product LaTeXMLMath and the complex scalar product LaTeXMLMath in LaTeXMLMath as in Appendix 3 .
A direct computation shows that the only nonzero products among the vectors LaTeXMLMath are LaTeXMLEquation and LaTeXMLEquation respectively .
Thus , LaTeXMLEquation for any LaTeXMLMath in LaTeXMLMath .
Introducing in LaTeXMLMath the real vectors LaTeXMLEquation we can formulate the main result of this section Vectors LaTeXMLMath and LaTeXMLMath have zero norm : LaTeXMLEquation .
Moreover , the triple LaTeXMLMath is orthogonal to the triple LaTeXMLMath .
Hence , LaTeXMLMath and LaTeXMLMath are curvature spheres of a surface .
This surface will be a canal surface since LaTeXMLMath .
Any canal surface can be obtained ( locally ) by this construction .
The proof of this theorem copies the proof of Theorem 1 from section 2 : it readily follows from ( LaTeXMLRef ) that the triple LaTeXMLMath is equivalent to LaTeXMLMath .
Similarly , the triple LaTeXMLMath is equivalent to LaTeXMLMath .
The conditions LaTeXMLMath and the orthogonality of both triples follow by virtue of ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Example .
Let us consider the Landau operator LaTeXMLEquation describing a quantum particle in the homogeneous magnetic field ( LaTeXMLMath ) .
Operator LaTeXMLMath obviously commutes with the operator LaTeXMLEquation so that the equations for their joint eigenfunctions LaTeXMLEquation can be rewritten in the form LaTeXMLEquation .
System ( LaTeXMLRef ) is obviously of the form ( LaTeXMLRef ) under the identification LaTeXMLMath .
In what follows we assume LaTeXMLMath ( this can be achieved by a rescaling LaTeXMLMath ) .
The corresponding linear system for LaTeXMLMath possesses 4 linearly independent solutions LaTeXMLEquation where LaTeXMLMath form a basis of solutions of Hermite ’ s equation LaTeXMLEquation .
Let us introduce the complex 4-vector LaTeXMLEquation where LaTeXMLMath is the Wronskian .
Then LaTeXMLEquation all other products being zero .
Moreover , LaTeXMLEquation .
A direct computation implies LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Obviously , LaTeXMLMath .
Dividing by LaTeXMLMath , we obtain the normalised vector LaTeXMLEquation .
Since the centers of the corresponding spheres lie on the LaTeXMLMath -axis , our surface is a surface of revolution .
Parametric equations of centers LaTeXMLMath and radii LaTeXMLMath are LaTeXMLEquation .
LaTeXMLEquation The Landau levels correspond to LaTeXMLMath .
For LaTeXMLMath we have LaTeXMLEquation .
I would like to thank K.R .
Khusnutdinova for the investigation of this and other examples .
The details will be given elsewhere .
Based on LaTeXMLCite ( see also LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) , let us briefly recall the standard way of defining surfaces LaTeXMLMath in projective space LaTeXMLMath in terms of solutions of a linear system ( LaTeXMLRef ) satisfying the compatibility conditions ( LaTeXMLRef ) .
For any fixed LaTeXMLMath satisfying ( LaTeXMLRef ) , the linear system ( LaTeXMLRef ) is compatible and possesses a solution LaTeXMLMath where LaTeXMLMath can be regarded as homogeneous coordinates of a surface in projective space LaTeXMLMath .
In what follows , we assume that our surfaces are hyperbolic and the corresponding asymptotic coordinates LaTeXMLMath and LaTeXMLMath are real .
Even though the coefficients LaTeXMLMath define a surface LaTeXMLMath uniquely up to projective equivalence via ( LaTeXMLRef ) , it is not entirely correct to regard LaTeXMLMath as projective invariants .
Indeed , the asymptotic coordinates LaTeXMLMath are only defined up to an arbitrary reparametrization of the form LaTeXMLEquation which induces a scaling of the surface vector according to LaTeXMLEquation .
Thus LaTeXMLCite , the form of equations ( LaTeXMLRef ) is preserved by the above transformation with the new coefficients LaTeXMLMath given by LaTeXMLEquation where LaTeXMLMath is the Schwarzian derivative , that is LaTeXMLEquation .
The transformation formulae ( LaTeXMLRef ) imply that the symmetric 2-form LaTeXMLEquation and the conformal class of the cubic form LaTeXMLEquation are absolute projective invariants .
They are known as the projective metric and the Darboux cubic form , respectively , and play an important role in projective differential geometry .
The vanishing of the Darboux cubic form is characteristic for quadrics : indeed , in this case LaTeXMLMath so that asymptotic curves of both families are straight lines .
The vanishing of the projective metric ( which is equivalent to either LaTeXMLMath or LaTeXMLMath ) characterises ruled surfaces .
In what follows we exclude these two degenerate situations and require LaTeXMLMath , LaTeXMLMath .
Using ( LaTeXMLRef ) - ( LaTeXMLRef ) , one can verify that the four points LaTeXMLEquation are defined in an invariant way , that is , under the transformation formulae ( LaTeXMLRef ) - ( LaTeXMLRef ) they acquire a nonzero multiple which does not change them as points in projective space LaTeXMLMath .
These points form the vertices of the so-called Wilczynski moving tetrahedral LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Since the lines passing through LaTeXMLMath and LaTeXMLMath are tangential to the LaTeXMLMath - and LaTeXMLMath -asymptotic curves , respectively , the three points LaTeXMLMath span the tangent plane of the surface LaTeXMLMath .
The line through LaTeXMLMath lying in the tangent plane is known as the directrix of Wilczynski of the second kind .
The line through LaTeXMLMath is transversal to LaTeXMLMath and is known as the directrix of Wilczynski of the first kind .
It plays the role of a projective ‘ normal ’ .
The Wilczynski tetrahedral proves to be the most convenient tool in projective differential geometry .
Using ( LaTeXMLRef ) and ( LaTeXMLRef ) , we easily derive for LaTeXMLMath the linear equations LaTeXMLCite LaTeXMLEquation where we introduced the notation LaTeXMLEquation .
The compatibility conditions of equations ( LaTeXMLRef ) imply LaTeXMLEquation which is just the equivalent form of the projective ‘ Gauss-Codazzi ’ equations ( LaTeXMLRef ) .
Equations ( LaTeXMLRef ) can be rewritten in the Plücker coordinates .
For a convenience of the reader we briefly recall this construction .
Let us consider a line LaTeXMLMath in LaTeXMLMath passing through the points LaTeXMLMath and LaTeXMLMath with the homogeneous coordinates LaTeXMLMath and LaTeXMLMath .
With the line LaTeXMLMath we associate a point LaTeXMLMath in projective space LaTeXMLMath with the homogeneous coordinates LaTeXMLEquation where LaTeXMLEquation .
The coordinates LaTeXMLMath satisfy the well-known quadratic Plücker relation LaTeXMLEquation .
Instead of LaTeXMLMath and LaTeXMLMath we may consider an arbitrary linear combinations thereof without changing LaTeXMLMath as a point in LaTeXMLMath .
Hence , we arrive at the well-defined Plücker coorrespondence LaTeXMLMath between lines in LaTeXMLMath and points on the Plücker quadric in LaTeXMLMath .
Plücker correspondence plays an important role in the projective differential geometry of surfaces and often sheds some new light on those properties of surfaces which are not ‘ visible ’ in LaTeXMLMath but acquire a precise geometric meaning only in LaTeXMLMath .
Thus , let us consider a surface LaTeXMLMath with the Wilczynski tetrahedral LaTeXMLMath satisfying equations ( LaTeXMLRef ) .
Since the two pairs of points LaTeXMLMath and LaTeXMLMath generate two lines in LaTeXMLMath which are tangential to the LaTeXMLMath - and LaTeXMLMath -asymptotic curves , respectively , the formulae LaTeXMLEquation define the images of these lines under the Plücker embedding .
Hence , with any surface LaTeXMLMath there are canonically associated two surfaces LaTeXMLMath and LaTeXMLMath in LaTeXMLMath lying on the Plücker quadric ( LaTeXMLRef ) .
In view of the formulae LaTeXMLEquation we conclude that the line in LaTeXMLMath passing through a pair of points LaTeXMLMath can also be generated by the pair of points LaTeXMLMath ( and hence is tangential to the LaTeXMLMath -coordinate line on the surface LaTeXMLMath ) or by a pair of points LaTeXMLMath ( and hence is tangential to the LaTeXMLMath -coordinate line on the surface LaTeXMLMath ) .
Consequently , the surfaces LaTeXMLMath and LaTeXMLMath are two focal surfaces of the congruence of straight lines LaTeXMLMath or , equivalently , LaTeXMLMath is the Laplace transform of LaTeXMLMath with respect to LaTeXMLMath and LaTeXMLMath is the Laplace transform of LaTeXMLMath with respect to LaTeXMLMath .
We emphasize that the LaTeXMLMath - and LaTeXMLMath -coordinate lines on the surfaces LaTeXMLMath and LaTeXMLMath are not asymptotic but conjugate .
Continuation of the Laplace sequence in both directions , that is taking the LaTeXMLMath -transform of LaTeXMLMath , the LaTeXMLMath -transform of LaTeXMLMath , etc. , leads , in the generic case , to an infinite Laplace sequence in LaTeXMLMath known as the Godeaux sequence of a surface LaTeXMLMath LaTeXMLCite .
The surfaces of the Godeaux sequence carry important geometric information about the surface LaTeXMLMath itself .
Introducing LaTeXMLEquation we arrive at the following equations for the Plücker coordinates : LaTeXMLEquation .
Equations ( LaTeXMLRef ) are consistent with the following table of scalar products : LaTeXMLEquation all other scalar products being equal to zero .
This defines a scalar product of the signature ( 3 , 3 ) which is the same as that of the quadratic form ( LaTeXMLRef ) .
Different types of surfaces can be defined by imposing additional constraints on LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ( respectively , LaTeXMLMath ) , so that , in a sense , projective differential geometry is the theory of ( integrable ) reductions of the underdetermined system ( LaTeXMLRef ) ( respectively , ( LaTeXMLRef ) ) .
Although the three linear systems ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) are in fact equivalent , some of them prove to be more suitable for studying particular classes of projective surfaces — see LaTeXMLCite , LaTeXMLCite for the further discussion .
Here we describe the construction of the so-called Lie sphere frame canonically associated with a surface in Lie sphere geometry ( see LaTeXMLCite ) .
Although the construction follows essentially that of Blaschke LaTeXMLCite , our final formulae prove to be more suitable for the purposes of this paper .
Let LaTeXMLMath be a surface in LaTeXMLMath parametrized by coordinates LaTeXMLMath of curvature lines , with the radius-vector LaTeXMLMath and the unit normal LaTeXMLMath satisfying the Weingarten equations ( LaTeXMLRef ) .
Introducing the 6-vectors LaTeXMLEquation and LaTeXMLEquation we readily verify that LaTeXMLEquation where the scalar product of 6-vectors is defined by the indefinite quadratic form ( LaTeXMLRef ) .
In what follows we use the same notation LaTeXMLMath for both the scalar product defined by ( LaTeXMLRef ) as well as for the standard Euclidean scalar product in LaTeXMLMath ; however , the dimension of vectors will clearly indicate which one has to be choosen .
A direct computation gives LaTeXMLEquation implying LaTeXMLEquation .
Differentiating ( LaTeXMLRef ) and taking into account ( LaTeXMLRef ) and ( LaTeXMLRef ) , we conclude that the only nonzero scalar products among the vectors LaTeXMLMath are the following : LaTeXMLEquation .
Here LaTeXMLMath are the components of the third fundamental form of the surface LaTeXMLMath .
Differentiating the zero scalar products among LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and keeping in mind ( LaTeXMLRef ) , one can show that the triple LaTeXMLMath is orthogonal to the triple LaTeXMLMath .
In order to complete the vectors LaTeXMLMath and LaTeXMLMath to a frame in LaTeXMLMath with the simplest possible table of scalar products , we will choose appropriate combinations among the triples LaTeXMLMath and LaTeXMLMath , separately .
Up to a certain normalization , the choice described below coincides with that from LaTeXMLCite .
Let us introduce the normalized vectors LaTeXMLEquation .
This normalization is convenient for several reasons : first of all , equations ( LaTeXMLRef ) reduce to the Dirac equation LaTeXMLEquation with the coefficients LaTeXMLMath and LaTeXMLMath given by LaTeXMLEquation .
It is important that both LaTeXMLMath are LaTeXMLMath are Lie-invariant ( we emphasize that coefficients in ( LaTeXMLRef ) are not Lie-invariant ) .
The reparametrization of coordinates LaTeXMLEquation induces the transformation of LaTeXMLMath and LaTeXMLMath as follows : LaTeXMLEquation so that we can introduce the Lie-invariant metric LaTeXMLEquation and the Lie-invariant cubic form LaTeXMLEquation ( notice that only the conformal class of the cubic form does make an invariant sense ) .
There exist one more important property of the normalized vector LaTeXMLMath ( resp. , LaTeXMLMath ) .
It turns out that the action of the Lie sphere group in LaTeXMLMath induces linear transformations of the coordinates of LaTeXMLMath ( resp. , LaTeXMLMath ) .
Since this linear action should necessarily preserve the Lie quadric ( LaTeXMLRef ) , we arrive at the well-known isomorphism of the Lie sphere group and LaTeXMLMath .
Thus , the normalization ( LaTeXMLRef ) linearises the action of the Lie sphere group ( see LaTeXMLCite for the details ) .
The only nonzero scalar products among normalized vectors LaTeXMLMath are the following : LaTeXMLEquation .
Obviously , the normalized triples LaTeXMLMath and LaTeXMLMath remain mutually orthogonal .
Let us introduce the following vectors LaTeXMLMath from the first triple : LaTeXMLEquation which we require to have the following nonzero scalar products : LaTeXMLEquation .
This uniquely specifies LaTeXMLEquation .
Similarly , we can choose the vectors LaTeXMLEquation with the nonzero scalar products LaTeXMLEquation which fixes LaTeXMLEquation .
Vectors LaTeXMLMath and LaTeXMLMath constitute the Lie sphere frame with the following simple table of scalar products LaTeXMLEquation all other scalar products are zero , which is of the desired signature ( 4 , 2 ) .
Equations of motion of the Lie sphere frame can be conveniently represented in matrix form ( LaTeXMLRef ) ( see LaTeXMLCite ) LaTeXMLEquation .
The compatibility conditions of ( LaTeXMLRef ) produce the equations ( LaTeXMLRef ) LaTeXMLEquation which can be viewed as the ” Gauss-Codazzi ” equations in Lie sphere geometry .
Another ( equivalent ) form of equations ( LaTeXMLRef ) can be obtained by introducing LaTeXMLMath and LaTeXMLMath LaTeXMLEquation which , upon the substitution into ( LaTeXMLRef ) , implies ( LaTeXMLRef ) : LaTeXMLEquation .
Let us consider a space LaTeXMLMath equiped with the pseudo-Hermitian scalar product of the signature LaTeXMLMath LaTeXMLEquation and define the wedge product LaTeXMLMath of the vectors LaTeXMLMath and LaTeXMLMath by the formula LaTeXMLEquation where LaTeXMLEquation ( LaTeXMLMath ) .
Let LaTeXMLEquation be the wedge product of any other two vectors LaTeXMLMath and LaTeXMLMath .
The pseudo-Hermitian scalar product LaTeXMLMath in LaTeXMLMath induces the pseudo-Hermitian scalar product LaTeXMLMath in LaTeXMLMath as follows : LaTeXMLEquation .
In terms of LaTeXMLMath and LaTeXMLMath this pseudo-Hermitian scalar product takes the form LaTeXMLEquation which is of the signature LaTeXMLMath .
Let us also define the complex scalar product LaTeXMLMath in LaTeXMLMath LaTeXMLEquation ( here the right hand side is undestood as a half of the determinant of the LaTeXMLMath matrix with the rows LaTeXMLMath ) .
In terms of LaTeXMLMath and LaTeXMLMath this scalar product takes the form LaTeXMLEquation .
Clearly , LaTeXMLEquation for any LaTeXMLMath in LaTeXMLMath .
This research was supported by the EPSRC grant GR/N30941 .
I would like to thank F. Burstall for drawing my attention to the twistor theory and L. Mason for usefull references .
I would also like to thank A. Pushnitski and A. Veselov for the discussions on Schrödinger operators with magnetic fields .
I am particularly grateful to K.R .
Khusnutdinova for a detailed investigation of the surfaces of revolution appearing in section 4 .
We show that the indices of certain twisted Dirac operators vanish on a LaTeXMLMath -manifold LaTeXMLMath of positive sectional curvature if the symmetry rank of LaTeXMLMath is LaTeXMLMath or if the symmetry rank is one and LaTeXMLMath is two connected .
We also give examples of simply connected manifolds of positive Ricci curvature which do not admit a metric of positive sectional curvature and positive symmetry rank .
An important application of index theory in Riemannian geometry is in the study of manifolds of positive scalar curvature .
Soon after Atiyah and Singer proved the index theorem , Lichnerowicz used a Bochner type formula to show that the index of the Dirac operator vanishes on closed LaTeXMLMath -manifolds of positive scalar curvature .
Whereas the relation between index theory and positive scalar curvature ( for high dimensional simply connected manifolds ) is well understood LaTeXMLCite possible relations to stronger curvature conditions such as positive Ricci or positive sectional curvature remain obscure ( see however the fascinating conjecture in LaTeXMLCite ) .
In this paper we give obstructions to metrics of positive sectional curvature ( positive curvature for short ) with symmetry .
We show that the indices of certain twisted Dirac operators vanish on a positively curved closed LaTeXMLMath -manifold LaTeXMLMath provided that the symmetry rank ( i.e .
the rank of the isometry group of LaTeXMLMath ) is at least two and the dimension of LaTeXMLMath is sufficiently large .
These indices occur as coefficients in an expansion of the elliptic genus .
A similar result holds if LaTeXMLMath is LaTeXMLMath -connected and the symmetry rank is LaTeXMLMath .
The elliptic genus LaTeXMLMath is a ring homomorphism from the oriented bordism ring to the ring of modular forms for LaTeXMLEquation ( in particular , LaTeXMLMath vanishes in all dimensions not divisible by LaTeXMLMath ) .
On the complex projective spaces LaTeXMLMath it is given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are modular forms of weight LaTeXMLMath and LaTeXMLMath , respectively .
The normalized elliptic genus LaTeXMLMath of an oriented LaTeXMLMath -dimensional manifold LaTeXMLMath expands in one of the cusps of LaTeXMLMath as a series of twisted signatures .
Following Witten LaTeXMLCite this series is best thought of as the index of a hypothetical signature operator on the free loop space of LaTeXMLMath .
In the other cusp of LaTeXMLMath the elliptic genus expands as a series LaTeXMLMath of characteristic numbers .
If LaTeXMLMath is LaTeXMLMath the coefficients of this expansion are indices of twisted Dirac operators LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath denotes the index of the Dirac operator twisted with the complexification LaTeXMLMath of a real vector bundle LaTeXMLMath over LaTeXMLMath .
For a LaTeXMLMath -manifold LaTeXMLMath the first coefficient of the series LaTeXMLMath , the LaTeXMLMath -genus , vanishes if LaTeXMLMath admits a metric of positive scalar curvature LaTeXMLCite or if LaTeXMLMath admits a non-trivial smooth LaTeXMLMath -action LaTeXMLCite .
Our main result asserts that additional coefficients vanish if LaTeXMLMath admits a metric of positive curvature with symmetry rank LaTeXMLMath .
Let LaTeXMLMath be a closed connected LaTeXMLMath -manifold of dimension LaTeXMLMath .
If LaTeXMLMath admits a metric of positive curvature and symmetry rank LaTeXMLMath then the indices of twisted Dirac operators occurring as the first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
We remark that all simply connected manifolds known to carry a metric of positive curvature have a lot of symmetry .
Besides the biquotients found by Eschenburg LaTeXMLCite and Bazaikin LaTeXMLCite , all other examples admit a homogeneous metric of positive curvature .
The latter were classified by Berger LaTeXMLCite , Aloff , Wallach LaTeXMLCite and Bérard Bergery LaTeXMLCite ( for recent progress on cohomogeneity one manifolds see LaTeXMLCite ) .
In the case that the symmetry rank of LaTeXMLMath is at least two Theorem LaTeXMLRef states : Let LaTeXMLMath be a closed connected LaTeXMLMath -manifold of dimension LaTeXMLMath .
If LaTeXMLMath admits a metric of positive curvature and symmetry rank LaTeXMLMath then LaTeXMLMath and LaTeXMLMath vanish .
Note that the index LaTeXMLMath does not vanish for the quaternionic plane .
Since the symmetry rank of LaTeXMLMath ( with its standard metric ) is three the lower bound on the dimension of LaTeXMLMath is necessary .
We believe that the vanishing of LaTeXMLMath also holds under weaker symmetry assumptions .
For LaTeXMLMath -connected manifolds we show Let LaTeXMLMath be a closed LaTeXMLMath -connected manifold of dimension LaTeXMLMath .
If LaTeXMLMath admits a metric of positive curvature with effective isometric LaTeXMLMath -action then LaTeXMLMath and LaTeXMLMath vanish .
The proofs of Theorem LaTeXMLRef and Theorem LaTeXMLRef are rather indirect .
For both statements we study the fixed point manifold of isometric cyclic subactions .
The Bott-Taubes-Witten rigidity theorem LaTeXMLCite for elliptic genera implies that the codimension of the fixed point manifold is bounded from above by a constant which depends on the pole order of the expansion LaTeXMLMath LaTeXMLCite .
Further restrictions arise from the curvature assumption .
A component of the fixed point manifold is a totally geodesic submanifold of the positively curved manifold LaTeXMLMath .
By an old result of Frankel LaTeXMLCite totally geodesic submanifolds of sufficiently large dimension must intersect .
This property imposes additional restrictions on the fixed point manifold .
The consequences of the rigidity theorem and Frankel ’ s result indicated above are the main ingredients in the proofs of Theorem LaTeXMLRef and Theorem LaTeXMLRef which also rely on the work of Grove and Searle on isometric LaTeXMLMath -actions of codimension two LaTeXMLCite and recent work of Wilking on the connectivity of the inclusion of totally geodesic submanifolds LaTeXMLCite .
We don ’ t know how to prove Theorem LaTeXMLRef and Theorem LaTeXMLRef by more direct methods such as the Bochner formula for twisted Dirac operators .
Already for the proof of the vanishing of LaTeXMLMath we need to use the entire elliptic genus .
Note that in view of the above discussion for LaTeXMLMath a Bochner type argument for the vanishing of LaTeXMLMath would not apply in dimension eight !
Manifolds of positive curvature ( no assumptions on the symmetry ) are classified in dimension LaTeXMLMath LaTeXMLCite .
In dimension LaTeXMLMath the only known obstructions to positive curvature are given by restrictions for the fundamental group ( cf .
LaTeXMLCite , see also LaTeXMLCite ) , Gromov ’ s Betti number theorem LaTeXMLCite and the Lichnerowicz-Hitchin vanishing theorem LaTeXMLCite for the LaTeXMLMath -invariant of LaTeXMLMath -manifolds .
Further progress concerning obstructions and classification has been obtained for positively curved manifolds with a lot of symmetry , e.g .
manifolds with large isometry dimension , large ( discrete ) symmetry rank or small cohomogeneity LaTeXMLCite .
All these results require that the dimension of the manifold is bounded from above by a constant depending on the symmetry .
In contrast Theorem LaTeXMLRef and Theorem LaTeXMLRef only require a lower bound on the dimension of the manifold .
Theorem LaTeXMLRef allows to distinguish positive curvature from weaker curvature properties under assumptions on the symmetry rank .
For example , consider the product of LaTeXMLMath and a Ricci-flat LaTeXMLMath -surface .
The Riemannian manifold LaTeXMLMath ( equipped with the product metric ) has symmetry rank three and positive scalar curvature as well as non-negative Ricci curvature .
The index LaTeXMLMath does not vanish .
Hence , if one restricts to metrics with symmetry rank LaTeXMLMath it follows from Theorem LaTeXMLRef that LaTeXMLMath admits a metric of positive scalar curvature but no metric of positive curvature .
This kind of reasoning can be pushed further to yield examples of simply connected manifolds of positive Ricci curvature which do not admit a metric of positive curvature if one restrict to metrics with a prescribed lower bound on the symmetry rank ( see Section LaTeXMLRef for precise statements ) .
Using different arguments ( based on LaTeXMLCite ) it is possible to distinguish positive Ricci from positive curvature under rather mild assumptions on the symmetry .
We shall call an LaTeXMLMath -action on a Riemannian manifold finite-order-isometric of order LaTeXMLMath if the cyclic subgroup of order LaTeXMLMath acts effectively and isometrically .
For every LaTeXMLMath and every LaTeXMLMath there exists a simply connected closed manifold LaTeXMLMath of dimension greater than LaTeXMLMath such that : LaTeXMLMath admits a metric of positive Ricci curvature with finite-order-isometric LaTeXMLMath -action of order LaTeXMLMath .
LaTeXMLMath does not admit a metric of positive curvature with finite-order-isometric LaTeXMLMath -action of order LaTeXMLMath .
We note that the examples ( given in Section LaTeXMLRef ) admit metrics of positive Ricci curvature and symmetry rank LaTeXMLMath .
In particular , one obtains simply connected manifolds of positive Ricci curvature and positive symmetry rank which do not admit a metric of positive curvature with positive symmetry rank .
The paper is structured in the following way .
In the next section we review basic properties of positive curvature used in the proofs of Theorem LaTeXMLRef , Theorem LaTeXMLRef and Theorem LaTeXMLRef .
In Section LaTeXMLRef we recall the rigidity theorem for elliptic genera and discuss applications to cyclic actions .
The proofs of Theorem LaTeXMLRef and Theorem LaTeXMLRef are given in Section LaTeXMLRef and Section LaTeXMLRef .
In Section LaTeXMLRef we also discuss related results for positive LaTeXMLMath th Ricci curvature , finite isometric actions and integral cohomology LaTeXMLMath ’ s .
In the final section we show Theorem LaTeXMLRef .
In this section we review basic properties of positively curved manifolds used in the proofs of Theorem LaTeXMLRef , Theorem LaTeXMLRef and Theorem LaTeXMLRef .
A main ingredient is an old result of Frankel on the intersection property for totally geodesic submanifolds .
Let LaTeXMLMath and LaTeXMLMath be totally geodesic submanifolds of a positively curved connected manifold LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath and LaTeXMLMath have non-empty intersection .
LaTeXMLMath The proof uses a Synge type argument for the parallel transport along a geodesic from LaTeXMLMath to LaTeXMLMath which minimizes the distance .
Whereas it is difficult to find totally geodesic submanifolds for generic metrics they do occur naturally as fixed point components in the presence of symmetry .
Theorem LaTeXMLRef clearly imposes restrictions on the fixed point manifold of isometric actions .
The following consequence is immediate .
Let LaTeXMLMath be an isometry of a positively curved connected manifold LaTeXMLMath and let LaTeXMLMath be a connected component of the fixed point manifold LaTeXMLMath of minimal codimension .
Then the dimension of every other component is less than the codimension of LaTeXMLMath .
LaTeXMLMath In LaTeXMLCite Frankel applied Theorem LaTeXMLRef to show that the inclusion LaTeXMLMath of a totally geodesic submanifold of codimension LaTeXMLMath is LaTeXMLMath -connected provided LaTeXMLMath is not larger than half of the dimension of the positively curved manifold LaTeXMLMath .
Recently , Wilking generalized this result significantly using Morse theory .
Let LaTeXMLMath be an LaTeXMLMath -dimensional connected Riemannian manifold and LaTeXMLMath a connected totally geodesic submanifold of codimension LaTeXMLMath .
If LaTeXMLMath is positively curved then the inclusion LaTeXMLMath is LaTeXMLMath -connected .
LaTeXMLMath The theorem imposes severe restrictions on the topology .
For example , if LaTeXMLMath is simply connected and the codimension of LaTeXMLMath is two then either LaTeXMLMath is homeomorphic to an odd dimensional sphere or all odd Betti numbers of LaTeXMLMath and LaTeXMLMath vanish LaTeXMLCite .
The case where LaTeXMLMath is fixed under an isometric LaTeXMLMath -action was studied before by Grove and Searle .
Let LaTeXMLMath be a simply connected positively curved manifold with isometric LaTeXMLMath -action .
If LaTeXMLMath then LaTeXMLMath is diffeomorphic to a sphere or a complex projective space .
LaTeXMLMath In this section we recall the rigidity theorem for elliptic genera and discuss applications to cyclic actions .
For more information on elliptic genera we refer to LaTeXMLCite .
A genus is a ring homomorphism from the oriented bordism ring LaTeXMLMath to a LaTeXMLMath -algebra LaTeXMLMath LaTeXMLCite .
The genus is called elliptic ( of level LaTeXMLMath ) if its logarithm LaTeXMLMath is given by a formal elliptic integral LaTeXMLEquation where LaTeXMLMath LaTeXMLCite .
Classical examples of elliptic genera are the signature ( LaTeXMLMath ) and the LaTeXMLMath -genus ( LaTeXMLMath , LaTeXMLMath ) .
The ring of modular forms LaTeXMLMath is a polynomial ring with generators LaTeXMLMath and LaTeXMLMath of weight LaTeXMLMath and LaTeXMLMath , respectively LaTeXMLCite .
The corresponding elliptic genus LaTeXMLEquation is universal since LaTeXMLMath and LaTeXMLMath are algebraically independent .
As in LaTeXMLCite we shall consider for a LaTeXMLMath -dimensional oriented manifold LaTeXMLMath the normalized elliptic genus LaTeXMLMath which is a modular function of weight LaTeXMLMath ( with LaTeXMLMath -character ) .
In one of the cusps ( the signature cusp ) LaTeXMLMath has an expansion which is equal to the following series of twisted signatures LaTeXMLEquation .
Here LaTeXMLMath denotes the index of the signature operator twisted with the complexified vector bundle LaTeXMLMath .
The series LaTeXMLMath describes the “ signature ” of the free loop space LaTeXMLMath localized at the manifold LaTeXMLMath of constant loops LaTeXMLCite .
In the other cusp ( the LaTeXMLMath -cusp ) LaTeXMLMath expands as a series of characteristic numbers LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath , where LaTeXMLMath denotes the multiplicative sequence for the LaTeXMLMath -genus , LaTeXMLMath is the complexification of the vector bundle LaTeXMLMath , LaTeXMLMath denotes the fundamental cycle and LaTeXMLMath is the Kronecker pairing .
If LaTeXMLMath is LaTeXMLMath LaTeXMLMath is equal to the index of the Dirac operator twisted with LaTeXMLMath by the Atiyah-Singer index theorem LaTeXMLCite .
In this case LaTeXMLMath has an interpretation as a series of indices of twisted Dirac operators ( twisted Dirac-indices for short ) .
The main feature of the elliptic genus is its rigidity under actions of compact connected Lie groups .
The rigidity was explained by Witten in LaTeXMLCite using heuristic arguments from quantum field theory and proved rigorously by Taubes and Bott-Taubes LaTeXMLCite ( cf .
also LaTeXMLCite ) .
Assume the LaTeXMLMath -manifold LaTeXMLMath carries an action by a compact Lie group LaTeXMLMath preserving the LaTeXMLMath -structure ( note that any smooth LaTeXMLMath -action lifts to the LaTeXMLMath -structure after passing to a two-fold covering action , if necessary ) .
Then the indices occurring in the expansions of LaTeXMLMath refine to virtual LaTeXMLMath -representations which we identify with their characters .
If LaTeXMLMath is connected the elliptic genus is rigid , i.e .
the characters do not depend on LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -equivariant LaTeXMLMath -manifold .
If LaTeXMLMath is connected then each twisted signature ( resp .
each twisted Dirac-index ) occurring as coefficient in the expansion of LaTeXMLMath in the signature cusp ( resp .
in the LaTeXMLMath -cusp ) is constant as a character of LaTeXMLMath .
LaTeXMLMath The rigidity of LaTeXMLMath also holds for certain non- LaTeXMLMath manifolds such as LaTeXMLMath -manifolds with first Chern class a torsion class LaTeXMLCite or orientable manifolds with finite second homotopy group LaTeXMLCite .
In the remaining part of this section we discuss consequences of Theorem LaTeXMLRef for cyclic actions which are used in the proofs of Theorem LaTeXMLRef , Theorem LaTeXMLRef and Theorem LaTeXMLRef .
Assume LaTeXMLMath acts on the LaTeXMLMath -manifold LaTeXMLMath ( not necessarily preserving the LaTeXMLMath -structure ) .
Let LaTeXMLMath be the element of order LaTeXMLMath .
In LaTeXMLCite Hirzebruch and Slodowy showed that the expansion of the elliptic genus in the signature cusp can be expressed in terms of the transversal self-intersection LaTeXMLMath of the fixed point manifold LaTeXMLMath LaTeXMLEquation .
The formula is equivalent to LaTeXMLMath which implies the following generalization of the Atiyah-Hirzebruch vanishing theorem LaTeXMLCite for the LaTeXMLMath -genus .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be the element of order two in LaTeXMLMath .
If LaTeXMLMath then the first LaTeXMLMath coefficients of LaTeXMLMath vanish .
LaTeXMLMath Here LaTeXMLMath denotes the minimal codimension of the connected components of LaTeXMLMath in LaTeXMLMath .
Recall that the LaTeXMLMath -action is called even if it lifts to the LaTeXMLMath -structure ( otherwise the action is called odd ) .
In the even case the codimension of all fixed point components of LaTeXMLMath is divisible by LaTeXMLMath whereas in the odd case the codimensions are always LaTeXMLMath ( cf .
LaTeXMLCite , Lemma 2.4 ) .
Note that for an odd action the series LaTeXMLMath is an element in LaTeXMLMath whereas LaTeXMLMath .
Thus formula ( LaTeXMLRef ) implies Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action .
If the action is odd then LaTeXMLMath vanishes identically .
LaTeXMLMath In the remaining part of this section we recall a generalization of Theorem LaTeXMLRef to cyclic actions of arbitrary order LaTeXMLCite .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be of order LaTeXMLMath .
At a connected component LaTeXMLMath of the fixed point manifold LaTeXMLMath the tangent bundle LaTeXMLMath splits equivariantly as the direct sum of LaTeXMLMath and the normal bundle LaTeXMLMath .
The latter splits ( non-canonically ) as a direct sum LaTeXMLMath corresponding to the irreducible real LaTeXMLMath -dimensional LaTeXMLMath -representations LaTeXMLMath , LaTeXMLMath .
We fix such a decomposition of LaTeXMLMath .
For each LaTeXMLMath choose LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath .
On each vector bundle LaTeXMLMath introduce a complex structure such that LaTeXMLMath acts on LaTeXMLMath by scalar multiplication with LaTeXMLMath .
Finally define LaTeXMLEquation where LaTeXMLMath runs over the connected components of LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action .
If LaTeXMLMath then the first LaTeXMLMath coefficients of LaTeXMLMath vanish .
LaTeXMLMath To prove this result one analysis the expansion of the equivariant elliptic genus in the LaTeXMLMath -cusp using the Lefschetz fixed point formula LaTeXMLCite and Theorem LaTeXMLRef ( see LaTeXMLCite for details ) .
Since LaTeXMLMath Theorem LaTeXMLRef implies the following corollary which generalizes Theorem LaTeXMLRef to cyclic actions of arbitrary finite order .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be of order LaTeXMLMath .
If LaTeXMLMath then the first LaTeXMLMath coefficients of LaTeXMLMath vanish .
LaTeXMLMath In this section we prove the vanishing results for positively curved manifolds stated in the introduction .
Recall from Section LaTeXMLRef that positive curvature restricts the dimension of fixed point components of isometric actions .
This property is an essential ingredient in the proofs of Theorem LaTeXMLRef and Theorem LaTeXMLRef .
Let LaTeXMLMath be a closed connected manifold with smooth action by a torus LaTeXMLMath .
We say LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath if for all cyclic subgroups LaTeXMLMath of order a power of LaTeXMLMath and every connected component LaTeXMLMath the dimension of two different connected components LaTeXMLMath and LaTeXMLMath of LaTeXMLMath is restricted by LaTeXMLMath .
By Theorem LaTeXMLRef a positively curved manifold with isometric LaTeXMLMath -action has restricted fixed point dimension for any prime LaTeXMLMath .
Other examples are given by LaTeXMLMath -manifolds with the same integral cohomology ring as a projective space ( see LaTeXMLCite , Chapter VII ) .
A main step in the study of the elliptic genus for these manifolds is the following lemma which is a consequence of Theorem LaTeXMLRef and Theorem LaTeXMLRef ( see the next section for the proof ) .
Let LaTeXMLMath be a torus of rank LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -manifold with effective LaTeXMLMath -action .
If LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath and LaTeXMLMath then at least one of the following possibilities holds : The first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
For some subgroup LaTeXMLMath with involution LaTeXMLMath the fixed point manifolds LaTeXMLMath and LaTeXMLMath are orientable and connected of codimension LaTeXMLMath and LaTeXMLMath , respectively .
With this information at hand we now prove Theorem LaTeXMLRef stated in the introduction .
Let LaTeXMLMath be a closed connected LaTeXMLMath -manifold of dimension LaTeXMLMath .
If LaTeXMLMath admits a metric of positive curvature and symmetry rank LaTeXMLMath then the indices of twisted Dirac operators occurring as the first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
Proof : For LaTeXMLMath the theorem follows from LaTeXMLCite .
So assume LaTeXMLMath .
Let LaTeXMLMath denote a torus of rank LaTeXMLMath acting isometrically and effectively on LaTeXMLMath .
Assume the first LaTeXMLMath coefficients in the expansion LaTeXMLMath do not vanish .
In particular , LaTeXMLMath is of even dimension and simply connected LaTeXMLCite .
The LaTeXMLMath -action has the properties given in the second part of Lemma LaTeXMLRef , i.e .
LaTeXMLMath and LaTeXMLMath are orientable connected submanifolds of codimension LaTeXMLMath and LaTeXMLMath , respectively .
Note that LaTeXMLMath , being a totally geodesic submanifold of LaTeXMLMath inherits positive curvature from LaTeXMLMath .
Since the submanifold LaTeXMLMath of LaTeXMLMath has codimension two it follows from Theorem LaTeXMLRef that LaTeXMLMath is diffeomorphic to a sphere or a complex projective space .
It is well-known that a non-trivial LaTeXMLMath -action on a complex projective space has more than one connected fixed point component ( cf .
LaTeXMLCite , Chapter VII , Theorem 5.1 ) .
Thus LaTeXMLMath is diffeomorphic to a sphere of codimension LaTeXMLMath .
Since LaTeXMLMath the Euler class LaTeXMLMath of the normal bundle LaTeXMLMath of LaTeXMLMath is trivial ( we fix compatible orientations for LaTeXMLMath and LaTeXMLMath ) .
This implies that the expansion of the elliptic genus in the signature cusp vanishes as we will explain next ( see LaTeXMLCite for details ) .
Recall that the coefficients of this expansion are twisted signatures LaTeXMLMath , where LaTeXMLMath is a virtual complex vector bundle associated to the tangent bundle LaTeXMLMath .
The LaTeXMLMath -action on LaTeXMLMath induces an action on each LaTeXMLMath .
Let LaTeXMLMath denote the LaTeXMLMath -equivariant twisted signature and LaTeXMLMath the equivariant expansion of the elliptic genus .
Recall from Theorem LaTeXMLRef that LaTeXMLMath for any LaTeXMLMath .
By the Lefschetz fixed point formula LaTeXMLCite LaTeXMLMath is equal to LaTeXMLEquation .
Here LaTeXMLMath ( resp .
LaTeXMLMath ) denote the formal roots of LaTeXMLMath ( resp .
the normal bundle LaTeXMLMath ) , LaTeXMLMath denotes the equivariant Chern character of LaTeXMLMath , LaTeXMLMath the fundamental cycle and LaTeXMLMath the Kronecker pairing .
Since LaTeXMLMath is trivial LaTeXMLMath for every LaTeXMLMath .
By Theorem LaTeXMLRef LaTeXMLMath vanishes identically .
Thus LaTeXMLMath contradicting the initial assumption of the proof .
LaTeXMLMath Theorem LaTeXMLRef also follows from Lemma LaTeXMLRef and Theorem LaTeXMLRef .
The statement for LaTeXMLMath in Theorem LaTeXMLRef can be strengthened to : If LaTeXMLMath admits a metric of positive curvature and symmetry rank LaTeXMLMath then the first three coefficients in the expansion LaTeXMLMath vanish , i.e .
LaTeXMLMath .
Under stronger assumptions on the bounds for the symmetry rank and the dimension the conclusion of Theorem LaTeXMLRef holds if one only assumes that LaTeXMLMath has positive LaTeXMLMath th Ricci curvature and an elementary LaTeXMLMath -abelian subgroup of the torus acts by isometries .
Recall that a manifold LaTeXMLMath has positive LaTeXMLMath th Ricci curvature ( or LaTeXMLMath -positive Ricci curvature ) if for any LaTeXMLMath mutually orthogonal unit tangent vectors LaTeXMLMath ( at any point of LaTeXMLMath ) the sum of curvatures LaTeXMLMath is positive LaTeXMLCite .
Thus , LaTeXMLMath -positive Ricci curvature is equivalent to positive curvature and LaTeXMLMath -positive Ricci curvature is equivalent to positive Ricci curvature .
Assume that LaTeXMLMath has positive LaTeXMLMath th Ricci curvature and assume that a torus LaTeXMLMath of rank LaTeXMLMath acts smoothly on LaTeXMLMath such that the induced action of the LaTeXMLMath -torus LaTeXMLMath , LaTeXMLMath a prime , is isometric and effective .
To keep the exposition simple we shall assume the generous bounds LaTeXMLMath and LaTeXMLMath .
For a connected LaTeXMLMath -manifold LaTeXMLMath as above the indices of twisted Dirac operators occurring as the first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
Sketch of proof : First note that the intersection property for totally geodesic submanifolds in positive curvature ( Theorem LaTeXMLRef ) extends to positive LaTeXMLMath th Ricci curvature LaTeXMLCite : Two totally geodesic submanifolds LaTeXMLMath and LaTeXMLMath of a manifold LaTeXMLMath of positive LaTeXMLMath th Ricci curvature intersect if LaTeXMLMath .
In particular , if LaTeXMLMath and LaTeXMLMath are two different connected fixed point components of an isometry LaTeXMLMath then LaTeXMLMath .
Assume the first LaTeXMLMath coefficients in the expansion LaTeXMLMath do not vanish .
Consider the action of LaTeXMLMath on LaTeXMLMath .
By Corollary LaTeXMLRef the codimension of LaTeXMLMath is LaTeXMLMath .
Hence , a connected component LaTeXMLMath of LaTeXMLMath has either “ small codimension ” , i.e .
LaTeXMLMath , or “ small dimension ” , i.e .
LaTeXMLMath .
Consider a LaTeXMLMath -fixed point LaTeXMLMath ( which exists since LaTeXMLMath ) and let LaTeXMLMath denote the component which contains LaTeXMLMath .
It is an elementary exercise to show that the LaTeXMLMath -torus LaTeXMLMath has a basis LaTeXMLMath such that LaTeXMLMath .
This implies that the codimension of LaTeXMLMath is small .
Since LaTeXMLMath the codimension of LaTeXMLMath is small provided this holds for LaTeXMLMath and LaTeXMLMath .
Hence , the codimension of LaTeXMLMath is small for every LaTeXMLMath , i.e .
LaTeXMLMath for every LaTeXMLMath .
However , it follows from elementary linear algebra that for some LaTeXMLMath the codimension of LaTeXMLMath is at least LaTeXMLMath .
This gives the desired contradiction .
Hence , the first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
LaTeXMLMath The next result implies Theorem LaTeXMLRef stated in the introduction .
Let LaTeXMLMath be a closed LaTeXMLMath -connected manifold of dimension LaTeXMLMath with effective smooth LaTeXMLMath -action .
Assume LaTeXMLMath admits a metric of positive curvature such that the subgroup LaTeXMLMath acts by isometries .
Then LaTeXMLMath and LaTeXMLMath vanish .
Proof : We fix an orientation and a LaTeXMLMath -structure for LaTeXMLMath .
Note that LaTeXMLMath vanishes by LaTeXMLCite .
Let LaTeXMLMath be an element of order LaTeXMLMath and let LaTeXMLMath .
Assume LaTeXMLMath does not vanish .
Then LaTeXMLMath has dimension LaTeXMLMath and the fixed point manifold LaTeXMLMath is the union of a connected component LaTeXMLMath of codimension LaTeXMLMath and ( a possible empty set of ) isolated fixed points by Corollary LaTeXMLRef , Theorem LaTeXMLRef and Corollary LaTeXMLRef .
Next apply Theorem LaTeXMLRef ( for LaTeXMLMath ) to conclude that either LaTeXMLMath or the codimension of LaTeXMLMath in LaTeXMLMath is two .
We claim that LaTeXMLMath .
If the codimension of LaTeXMLMath is two then the inclusion LaTeXMLMath is LaTeXMLMath -connected by Theorem LaTeXMLRef .
This implies that LaTeXMLMath is equal to LaTeXMLMath ( use Poincaré duality for LaTeXMLMath and LaTeXMLMath ) .
Note that LaTeXMLMath is LaTeXMLMath -connected by Theorem LaTeXMLRef .
Hence , LaTeXMLMath vanishes .
In particular , the Euler class of the normal bundle of LaTeXMLMath in LaTeXMLMath vanishes rationally .
Now argue as in the proof of Theorem LaTeXMLRef to conclude that LaTeXMLMath contradicting the assumption on LaTeXMLMath .
Hence , LaTeXMLMath .
Since LaTeXMLMath is fixed by LaTeXMLMath the action of LaTeXMLMath on the normal bundle LaTeXMLMath induces a complex structure such that LaTeXMLMath acts by multiplication with LaTeXMLMath .
We fix the orientation of LaTeXMLMath which is compatible with the orientations of LaTeXMLMath and LaTeXMLMath .
Also the action of LaTeXMLMath induces a complex structure on the normal bundle of any of the isolated LaTeXMLMath -fixed points .
We shall now compute the local contributions in the Lefschetz fixed point formula for LaTeXMLMath : LaTeXMLEquation .
Here LaTeXMLMath ( resp .
LaTeXMLMath ) denotes the local contribution at LaTeXMLMath ( resp .
at an isolated fixed point LaTeXMLMath ) .
The term LaTeXMLMath is given by LaTeXMLCite LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
Here LaTeXMLMath ( resp .
LaTeXMLMath ) denote the formal roots of LaTeXMLMath ( resp .
LaTeXMLMath ) .
Since LaTeXMLMath is LaTeXMLMath -connected the first Chern class LaTeXMLMath of LaTeXMLMath vanishes .
This implies LaTeXMLEquation .
Hence , the expression for LaTeXMLMath simplifies to LaTeXMLEquation .
The term LaTeXMLMath is given by LaTeXMLCite LaTeXMLEquation .
By Theorem LaTeXMLRef LaTeXMLMath , where LaTeXMLMath is the integer obtained by summing up LaTeXMLMath .
Equivalently , LaTeXMLEquation .
Note that LaTeXMLMath , whereas LaTeXMLMath .
By comparing the expansions in the LaTeXMLMath -cusp of both sides of ( LaTeXMLRef ) it follows that LaTeXMLMath ( in fact , depending on the parity of LaTeXMLMath , LaTeXMLMath is equal to LaTeXMLMath or LaTeXMLMath ) .
Since LaTeXMLMath this implies that LaTeXMLMath .
LaTeXMLMath In the remaining part of this section we use Lemma LaTeXMLRef to study the Pontrjagin numbers of LaTeXMLMath -dimensional LaTeXMLMath -manifolds with symmetry which have the same integral cohomology ring as a projective space .
Such manifolds are either integral cohomology LaTeXMLMath ’ s or integral cohomology Cayley planes ( for recent progress in the study of integral cohomology LaTeXMLMath ’ s with symmetry we refer to LaTeXMLCite and references therein ) .
Note that an integral cohomology Cayley plane with smooth non-trivial LaTeXMLMath -action has the same Pontrjagin numbers as the Cayley plane since these are completely determined by the signature and the LaTeXMLMath -genus ( the latter vanishes by LaTeXMLCite ) .
The same argument applies to an integral cohomology LaTeXMLMath with smooth non-trivial LaTeXMLMath -action .
We shall now apply Lemma LaTeXMLRef to integral cohomology LaTeXMLMath ’ s for LaTeXMLMath .
Let LaTeXMLMath be an integral cohomology LaTeXMLMath , i.e .
LaTeXMLMath , where LaTeXMLMath has degree LaTeXMLMath .
If a torus LaTeXMLMath of rank LaTeXMLMath acts effectively and smoothly on LaTeXMLMath and LaTeXMLMath then the first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
Proof : Note that LaTeXMLMath is a LaTeXMLMath -manifold with restricted fixed point dimension for any prime .
By Lemma LaTeXMLRef we may assume that for some LaTeXMLMath -subgroup of LaTeXMLMath the fixed point manifold LaTeXMLMath is connected of codimension LaTeXMLMath .
It is well known that LaTeXMLMath is an integral cohomology projective space of the form LaTeXMLMath where LaTeXMLMath has even degree LaTeXMLMath ( cf .
LaTeXMLCite , Chapter VII , Theorem 5.1 ) .
Also LaTeXMLMath by the Lefschetz fixed point formula for the Euler characteristic .
Since the codimension of LaTeXMLMath is equal to LaTeXMLMath it follows that LaTeXMLMath and LaTeXMLMath .
Hence , LaTeXMLMath is an integral cohomology LaTeXMLMath .
For a LaTeXMLMath -dimensional manifold the elliptic genus is a linear combination of the signature and the LaTeXMLMath -genus .
The signature of LaTeXMLMath vanishes for trivial reasons .
The LaTeXMLMath -genus of LaTeXMLMath vanishes since LaTeXMLMath is a LaTeXMLMath -manifold with non-trivial LaTeXMLMath -action LaTeXMLCite .
Hence , LaTeXMLMath .
This completes the proof .
LaTeXMLMath Finally we point out the following consequence of Lemma LaTeXMLRef .
Let LaTeXMLMath be a torus of rank LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action of dimension LaTeXMLMath .
Assume the signature of LaTeXMLMath does not vanish .
If LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath then the first LaTeXMLMath coefficients in the expansion LaTeXMLMath vanish .
Proof : By the rigidity of the signature LaTeXMLMath for any LaTeXMLMath -action .
Since the signature of LaTeXMLMath does not vanish LaTeXMLMath if LaTeXMLMath is connected .
In particular , LaTeXMLMath and the statement follows from Lemma LaTeXMLRef .
LaTeXMLMath In this section we prove Lemma LaTeXMLRef .
Let LaTeXMLMath be a torus of rank LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -manifold with effective LaTeXMLMath -action .
Assume LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath and LaTeXMLMath .
Assume also that the first LaTeXMLMath coefficients in the expansion LaTeXMLMath do not vanish .
This implies that LaTeXMLMath .
Our goal is to show that for some subgroup LaTeXMLMath with involution LaTeXMLMath the fixed point manifolds LaTeXMLMath and LaTeXMLMath are orientable and connected of codimension LaTeXMLMath and LaTeXMLMath , respectively .
To this end we first examine the action of LaTeXMLMath at a fixed point LaTeXMLMath ( which exists since LaTeXMLMath ) .
The tangent space LaTeXMLMath of LaTeXMLMath at LaTeXMLMath decomposes ( non-canonically ) into LaTeXMLMath complex one-dimensional LaTeXMLMath -representations LaTeXMLMath .
With respect to such a decomposition the action of LaTeXMLMath on LaTeXMLMath is given by a homomorphism LaTeXMLMath .
We denote by LaTeXMLMath the induced homomorphism of integral lattices LaTeXMLMath .
Since the LaTeXMLMath -action is effective the mod LaTeXMLMath reduction LaTeXMLMath of LaTeXMLMath is injective for every LaTeXMLMath .
For a finite covering homomorphism LaTeXMLMath we denote by LaTeXMLMath the homomorphism of integral lattices LaTeXMLMath for the covering action on LaTeXMLMath .
Note that an element LaTeXMLMath , LaTeXMLMath , determines a homomorphism LaTeXMLMath , LaTeXMLMath , and an LaTeXMLMath -subgroup LaTeXMLMath .
Let LaTeXMLMath be the image of the standard basis of LaTeXMLMath under LaTeXMLMath .
Let LaTeXMLMath be a prime .
There exists a finite covering homomorphism LaTeXMLMath of degree coprime to LaTeXMLMath such that LaTeXMLEquation where each LaTeXMLMath is coprime to LaTeXMLMath .
The claim is an elementary fact from linear algebra .
For convenience we give a proof at the end of the section .
In the following let LaTeXMLMath and let LaTeXMLMath denote a finite covering action of odd degree with the properties described in Claim LaTeXMLRef .
The next claim gives information on the action of involutions of LaTeXMLMath on LaTeXMLMath ( again the proof is postponed to the end of the section ) .
This information will be used to exhibit involutions for which the fixed point manifold is connected of codimension LaTeXMLMath ( since LaTeXMLMath has odd degree the same property holds for involutions of LaTeXMLMath ) .
For the involution LaTeXMLMath corresponding to LaTeXMLMath ( LaTeXMLMath acts on LaTeXMLMath by LaTeXMLMath ) the component of LaTeXMLMath which contains the LaTeXMLMath -fixed point LaTeXMLMath has codimension LaTeXMLMath , i.e .
LaTeXMLMath has two odd entries .
The involution LaTeXMLMath acts trivially on each representation space LaTeXMLMath , LaTeXMLMath , i.e .
each of the last LaTeXMLMath columns of LaTeXMLMath has an even number of odd entries .
The LaTeXMLMath -torus of LaTeXMLMath ( i.e .
the subgroup generated by the involutions LaTeXMLMath ) acts non-trivially on at most LaTeXMLMath of the representation spaces LaTeXMLMath , i.e .
at most LaTeXMLMath of the last LaTeXMLMath columns of LaTeXMLMath are non-zero modulo LaTeXMLMath .
Hence , for every involution LaTeXMLMath and every LaTeXMLMath -fixed point the connected component of LaTeXMLMath which contains the fixed point has codimension LaTeXMLMath .
Since LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath it follows that LaTeXMLMath is the union of a connected component of codimension LaTeXMLMath with LaTeXMLMath -fixed point ( in fact the codimension is LaTeXMLMath by Theorem LaTeXMLRef ) and ( a possible empty set of ) components with fixed point free LaTeXMLMath -action .
Recall that LaTeXMLMath .
Since any isolated LaTeXMLMath -fixed point is also fixed by LaTeXMLMath it follows that LaTeXMLMath is connected of codimension LaTeXMLMath for every LaTeXMLMath .
Let LaTeXMLMath denote the LaTeXMLMath th LaTeXMLMath -factor of LaTeXMLMath .
Below we shall use Theorem LaTeXMLRef to show The fixed point manifold LaTeXMLMath is connected of codimension LaTeXMLMath .
In particular , LaTeXMLMath has two odd entries and all other entries vanish .
Note that Claim LaTeXMLRef , Claim LaTeXMLRef and Claim LaTeXMLRef imply that the codimension of any connected component of LaTeXMLMath is LaTeXMLMath .
With this information at hand we shall now complete the proof of Lemma LaTeXMLRef .
By the second part of Claim LaTeXMLRef one can choose LaTeXMLMath such that LaTeXMLEquation .
Consider the LaTeXMLMath -subgroup LaTeXMLMath determined by LaTeXMLMath .
The fixed point manifold LaTeXMLMath of the involution LaTeXMLMath is equal to LaTeXMLMath which is connected of codimension LaTeXMLMath .
The LaTeXMLMath -fixed point manifold has codimension LaTeXMLMath in LaTeXMLMath .
Since LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath the fixed point manifold LaTeXMLMath is connected .
Also , LaTeXMLMath is orientable .
Since LaTeXMLMath is a LaTeXMLMath -manifold LaTeXMLMath is orientable as well ( see for example LaTeXMLCite , Lemma 10.1 ) .
Finally note that the same properties hold for the LaTeXMLMath -subgroup of LaTeXMLMath which is the image of LaTeXMLMath under the covering homomorphism LaTeXMLMath .
LaTeXMLMath Proof of Claim LaTeXMLRef : It is an elementary fact from linear algebra that LaTeXMLMath admits a basis LaTeXMLMath such that LaTeXMLEquation after permuting columns ( i.e .
after permuting the representation spaces LaTeXMLMath ) if necessary .
Here each LaTeXMLMath is coprime to LaTeXMLMath .
Note that the choice of a basis of LaTeXMLMath is equivalent to the choice of an isomorphism LaTeXMLMath .
Using suitable row operations of the form LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , one obtains a basis LaTeXMLMath of LaTeXMLMath such that the matrix LaTeXMLMath has the properties given in the claim .
Each LaTeXMLMath determines a homomorphism LaTeXMLMath .
Since LaTeXMLMath is a basis of LaTeXMLMath the homomorphism LaTeXMLMath is a finite covering homomorphism .
The induced homomorphism LaTeXMLMath of integral lattices maps the standard basis to LaTeXMLMath .
In view of the properties of the matrix LaTeXMLMath the covering homomorphism has degree coprime to LaTeXMLMath .
LaTeXMLMath Proof of Claim LaTeXMLRef : First note that by Theorem LaTeXMLRef and Corollary LaTeXMLRef for every involution LaTeXMLMath the fixed point manifold LaTeXMLMath has codimension LaTeXMLMath and the dimension of each connected component of LaTeXMLMath is divisible by LaTeXMLMath .
Since the dimension of LaTeXMLMath is LaTeXMLMath and LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath it follows that for every connected component LaTeXMLMath of LaTeXMLMath either LaTeXMLMath or LaTeXMLMath .
Let LaTeXMLMath denote the mod LaTeXMLMath reduction of LaTeXMLMath .
For the binary linear code LaTeXMLMath we conclude that each code word LaTeXMLMath has either weight LaTeXMLMath is defined as the number of entries equal to LaTeXMLMath .
LaTeXMLMath or co-weight LaTeXMLMath .
In particular , the mod LaTeXMLMath reduction of LaTeXMLMath , denoted by LaTeXMLMath , has weight LaTeXMLMath .
Since the weight function is sublinear , i.e .
LaTeXMLMath , and LaTeXMLMath it follows that the subset of code words with weight LaTeXMLMath is closed under addition .
Hence LaTeXMLMath for every LaTeXMLMath .
In particular , this inequality holds for LaTeXMLMath and LaTeXMLMath which implies LaTeXMLMath ( i.e .
LaTeXMLMath has two odd entries ) and implies that each of the last LaTeXMLMath columns of LaTeXMLMath has an even number of odd entries .
Finally note that if LaTeXMLMath of the last LaTeXMLMath columns of LaTeXMLMath are non-zero modulo LaTeXMLMath then the weight of some code word is LaTeXMLMath .
Hence LaTeXMLMath .
This completes the proof of the claim .
LaTeXMLMath Proof of Claim LaTeXMLRef : To show that LaTeXMLMath is connected of codimension LaTeXMLMath it suffices to prove this property for all cyclic subgroups LaTeXMLMath of order LaTeXMLMath , LaTeXMLMath .
We know already that LaTeXMLMath is connected of codimension LaTeXMLMath .
Assume LaTeXMLMath is connected of codimension LaTeXMLMath .
To show the corresponding property for LaTeXMLMath we will use the following consequence of Theorem LaTeXMLRef which we prove first : Let LaTeXMLMath be a connected LaTeXMLMath -manifold with LaTeXMLMath -action .
Assume the cyclic subgroup LaTeXMLMath of order LaTeXMLMath acts non-trivially on LaTeXMLMath and the fixed point manifold LaTeXMLMath is connected .
If the first LaTeXMLMath coefficients in the expansion LaTeXMLMath do not vanish then the codimension of the submanifold LaTeXMLMath of LaTeXMLMath is LaTeXMLMath .
Let LaTeXMLMath .
Note that for some connected component LaTeXMLMath of the LaTeXMLMath -fixed point manifold LaTeXMLMath by Theorem LaTeXMLRef .
Let LaTeXMLMath be the connected component which contains LaTeXMLMath .
Note that LaTeXMLMath is strictly larger than LaTeXMLMath since LaTeXMLMath is a proper connected submanifold of LaTeXMLMath .
Hence the codimension of LaTeXMLMath in LaTeXMLMath must be less than LaTeXMLMath which implies that the codimension of the submanifold LaTeXMLMath of LaTeXMLMath is LaTeXMLMath .
We shall now continue with the study of the action of LaTeXMLMath .
By the statement above the codimension of the submanifold LaTeXMLMath of LaTeXMLMath is LaTeXMLMath .
Since LaTeXMLMath has restricted fixed point dimension for the prime LaTeXMLMath a connected component of LaTeXMLMath has either codimension LaTeXMLMath in LaTeXMLMath or has dimension LaTeXMLMath .
Next consider the action of the LaTeXMLMath -subgroup LaTeXMLMath of LaTeXMLMath which is determined by LaTeXMLMath .
Let LaTeXMLMath denote the cyclic subgroup of order LaTeXMLMath .
By the statement above the codimension of the submanifold LaTeXMLMath of LaTeXMLMath is LaTeXMLMath .
It follows that either the number of entries of LaTeXMLMath which are LaTeXMLMath is LaTeXMLMath or the number of entries of LaTeXMLMath which are LaTeXMLMath is LaTeXMLMath .
The same property holds for LaTeXMLMath .
Also the mod LaTeXMLMath -reductions of LaTeXMLMath and LaTeXMLMath have the same last LaTeXMLMath entries by the second part of Claim LaTeXMLRef .
This implies that the mod LaTeXMLMath -reduction of LaTeXMLMath has only two non-zero entries .
In other words LaTeXMLMath acts trivially on the tangent bundle of LaTeXMLMath at the LaTeXMLMath -fixed point LaTeXMLMath .
Thus LaTeXMLMath and LaTeXMLMath is connected of codimension LaTeXMLMath .
This completes the induction step .
It follows that the fixed point manifold LaTeXMLMath is connected of codimension LaTeXMLMath .
LaTeXMLMath Apparently the only known topological property which allows to distinguish positive Ricci curvature from positive curvature on simply connected manifolds is based on Gromov ’ s Betti number theorem LaTeXMLCite : With respect to any field of coefficients the sum of Betti numbers of a positively curved LaTeXMLMath -dimensional manifold is less than a constant LaTeXMLMath depending only on the dimension .
The bound in LaTeXMLCite which depends doubly exponentially on the dimension LaTeXMLMath has been improved by Abresch LaTeXMLCite who showed that the Betti number theorem holds for a bound LaTeXMLMath depending exponentially on LaTeXMLMath ( the bound LaTeXMLMath may be chosen to satisfy LaTeXMLMath ) .
Sha and Yang LaTeXMLCite constructed metrics of positive Ricci curvature on the LaTeXMLMath -fold connected sum of LaTeXMLMath , LaTeXMLMath , for every LaTeXMLMath .
By the Betti number theorem these manifolds do not admit positively curved metrics if LaTeXMLMath is sufficiently large .
In this section we present two methods to exhibit manifolds with “ small ” Betti numbers ( i.e .
the sum is less than LaTeXMLMath ) which admit metrics of positive Ricci curvature but do not admit metrics of positive curvature under assumptions on the symmetry .
The first method which relies on Theorem LaTeXMLRef leads to For every LaTeXMLMath and every LaTeXMLMath there exists a simply connected closed manifold LaTeXMLMath of dimension greater than LaTeXMLMath such that : LaTeXMLMath admits a metric of positive Ricci curvature with symmetry rank LaTeXMLMath .
LaTeXMLMath does not admit a metric of positive curvature with symmetry rank LaTeXMLMath .
The manifold LaTeXMLMath may be chosen to have small Betti numbers .
The second method which relies on Theorem LaTeXMLRef and recent work of Wilking ( see Theorem LaTeXMLRef ) gives stronger information .
Recall from the introduction that an LaTeXMLMath -action is called finite-order-isometric of order LaTeXMLMath if the cyclic subgroup of order LaTeXMLMath acts effectively and isometrically .
For every LaTeXMLMath and every LaTeXMLMath there exists a simply connected closed manifold LaTeXMLMath of dimension greater than LaTeXMLMath such that : LaTeXMLMath admits a metric of positive Ricci curvature with finite-order-isometric LaTeXMLMath -action of order LaTeXMLMath .
LaTeXMLMath does not admit a metric of positive curvature with finite-order-isometric LaTeXMLMath -action of order LaTeXMLMath .
Again the manifold LaTeXMLMath may be chosen to have small Betti numbers .
The examples occurring in both theorems are given by Riemannian products of the form LaTeXMLMath .
The first factor LaTeXMLMath is a LaTeXMLMath -manifold of positive Ricci curvature with large symmetry rank and non-vanishing elliptic genus ( below we shall choose LaTeXMLMath to be a product of quaternionic projective planes ) .
The second factor LaTeXMLMath is a LaTeXMLMath -manifold of positive Ricci curvature for which the index LaTeXMLMath does not vanish .
The next proposition shows that these properties hold for a hypersurface in LaTeXMLMath of degree LaTeXMLMath .
A non-singular hypersurface LaTeXMLMath in LaTeXMLMath of degree LaTeXMLMath has the following properties : LaTeXMLMath is LaTeXMLMath and admits a metric of positive Ricci curvature .
The Betti numbers of LaTeXMLMath are small .
The index LaTeXMLMath does not vanish .
Proof : For a proof of some of the properties of hypersurfaces used below we refer to LaTeXMLCite and references therein .
Let LaTeXMLMath denote a non-singular hypersurface in LaTeXMLMath of degree LaTeXMLMath .
The tangent bundle of LaTeXMLMath is stably the restriction of LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath denotes the complex line bundle over LaTeXMLMath with first Chern class LaTeXMLMath dual to LaTeXMLMath ( in the following we shall denote the restriction of LaTeXMLMath to LaTeXMLMath also by LaTeXMLMath ) .
Note that LaTeXMLMath .
The total Chern class of LaTeXMLMath is given by LaTeXMLEquation .
In particular , LaTeXMLMath .
Taking LaTeXMLMath we conclude that LaTeXMLMath is a LaTeXMLMath -manifold with positive first Chern class .
By Yau ’ s solution LaTeXMLCite of the Calabi conjecture LaTeXMLMath admits a metric of positive Ricci curvature .
The Euler characteristic of LaTeXMLMath is equal to the Chern number LaTeXMLMath which can be computed via the formula for LaTeXMLMath given above .
In turns out that LaTeXMLEquation .
By the Lefschetz theorem on hyperplane sections the odd Betti numbers of LaTeXMLMath vanish for any field of coefficients ( cf .
LaTeXMLCite , §22 ) .
Hence the Betti numbers of LaTeXMLMath are small .
Finally we compute LaTeXMLMath .
By the cohomological version of the index theorem LaTeXMLCite LaTeXMLMath is equal to LaTeXMLMath , where LaTeXMLEquation .
It follows that LaTeXMLMath is the residue of LaTeXMLMath at LaTeXMLMath .
Changing variables , LaTeXMLMath , one computes that LaTeXMLMath is equal to the coefficient of LaTeXMLMath in LaTeXMLEquation .
Hence , LaTeXMLMath which does not vanish for LaTeXMLMath .
This completes the proof of the proposition .
LaTeXMLMath Proof of Theorem LaTeXMLRef : Let LaTeXMLMath be the Riemannian product of LaTeXMLMath -copies of LaTeXMLMath , let LaTeXMLMath be a non-singular hypersurface in LaTeXMLMath of degree LaTeXMLMath ( equipped with a metric of positive Ricci curvature ) and let LaTeXMLMath be the Riemannian product LaTeXMLMath .
Since the symmetry rank of LaTeXMLMath is three and LaTeXMLMath the symmetry rank of the positively Ricci curved LaTeXMLMath -manifold LaTeXMLMath is LaTeXMLMath .
By Proposition LaTeXMLRef LaTeXMLMath .
Thus the expansion LaTeXMLMath has a pole of order LaTeXMLMath .
For a homogeneous LaTeXMLMath -manifold the elliptic genus is equal to the signature LaTeXMLCite .
In particular , LaTeXMLMath which can also be shown by a direct computation .
This gives LaTeXMLEquation .
It follows that LaTeXMLMath has a pole of order LaTeXMLMath .
Thus , the first LaTeXMLMath coefficient in the expansion LaTeXMLMath do not vanish .
By Theorem LaTeXMLRef the manifold LaTeXMLMath does not admit a metric of positive curvature with symmetry rank LaTeXMLMath .
LaTeXMLMath Proof of Theorem LaTeXMLRef : Let LaTeXMLMath be the Riemannian product of LaTeXMLMath and LaTeXMLMath ( the latter shall be equipped with a metric of positive Ricci curvature ) .
Since the positively Ricci curved manifold LaTeXMLMath has symmetry rank LaTeXMLMath it admits a finite-order-isometric LaTeXMLMath -action of any order .
Now assume LaTeXMLMath carries a metric of positive curvature with finite-order-isometric LaTeXMLMath -action of order LaTeXMLMath .
We shall derive a contradiction for LaTeXMLMath .
Let LaTeXMLMath be of order LaTeXMLMath .
Since LaTeXMLMath has a pole of order LaTeXMLMath the codimension of LaTeXMLMath is bounded from above by a constant which depends on LaTeXMLMath but does not depend on LaTeXMLMath ( see Corollary LaTeXMLRef ) .
Let LaTeXMLMath be a fixed point component of minimal codimension LaTeXMLMath .
By Theorem LaTeXMLRef the inclusion LaTeXMLMath is LaTeXMLMath -connected , where LaTeXMLMath .
Using Poincaré duality for LaTeXMLMath and LaTeXMLMath it follows that LaTeXMLMath and LaTeXMLMath are isomorphic in the range LaTeXMLMath .
Hence , for LaTeXMLMath the cohomology groups LaTeXMLMath and LaTeXMLMath are isomorphic .
Recall from LaTeXMLCite , §22 , that the even Betti numbers LaTeXMLMath are one except for LaTeXMLMath and the odd Betti numbers vanish .
This implies LaTeXMLMath .
However LaTeXMLMath is LaTeXMLMath ( compare with the formula for LaTeXMLMath in the proof of the proposition above ) .
This gives the desired contradiction .
LaTeXMLMath Acknowledgements : I like to thank Wilderich Tuschmann , Burkhard Wilking and Wolfgang Ziller for many stimulating discussions .
Anand Dessai e-mail : dessai @ math.uni-augsburg.de http : //www.math.uni-augsburg.de/geo/dessai/homepage.html Department of Mathematics , University of Augsburg , D-86159 Augsburg
In this paper we establish a commutator estimate which allows one to concretely identify the product LaTeXMLMath space , LaTeXMLMath , of Chang and R. Fefferman , as an operator space on LaTeXMLMath .
The one parameter analogue of this result is a well–known theorem of Nehari LaTeXMLCite .
The novelty of this paper is that we discuss a situation governed by a two parameter family of dilations , and so the spaces LaTeXMLMath and LaTeXMLMath have a more complicated structure .
Here LaTeXMLMath denotes the upper half-plane and LaTeXMLMath is defined to be the dual of the real-variable Hardy space LaTeXMLMath on the product domain LaTeXMLMath .
There are several equivalent ways to define this latter space and the reader is referred to LaTeXMLCite for the various characterizations .
We will be more interested in the biholomorphic analogue of LaTeXMLMath , which can be defined in terms of the boundary values of biholomorphic functions on LaTeXMLMath and will be denoted throughout by LaTeXMLMath , cf LaTeXMLCite .
In one variable , the space LaTeXMLMath decomposes as the direct sum LaTeXMLMath where LaTeXMLMath is defined as the boundary values of functions in LaTeXMLMath and LaTeXMLMath denotes the space of complex conjugate of functions in LaTeXMLMath .
The space LaTeXMLMath , therefore , decomposes as the direct sum of the four spaces LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath where the tensor products are the Hilbert space tensor products .
Let LaTeXMLMath denote the orthogonal projection of LaTeXMLMath onto the holomorphic/anti-holomorphic subspaces , in the first and second variables , respectively , and let LaTeXMLMath denote the one–dimensional Hilbert transform in the LaTeXMLMath variable , LaTeXMLMath .
In terms of the projections LaTeXMLMath , LaTeXMLEquation .
The nested commutator determined by the function LaTeXMLMath is the operator LaTeXMLMath acting on LaTeXMLMath where , for a function LaTeXMLMath on the plane , we define LaTeXMLMath .
In terms of the projections LaTeXMLMath , it takes the form LaTeXMLEquation .
Ferguson and Sadosky LaTeXMLCite established the inequality LaTeXMLMath .
The main result is the converse inequality .
There is a constant LaTeXMLMath such that LaTeXMLMath for all functions LaTeXMLMath in LaTeXMLMath .
As A. Chang and R. Fefferman have established for us , the structure of the space LaTeXMLMath is more complicated in the two parameter setting , requiring a more subtle approach to this theorem , despite the superficial similarity of the results to the one parameter setting .
The proof relies on three key ideas .
The first is the dyadic characterization of the LaTeXMLMath norm given in LaTeXMLCite .
The second is a variant of Journé ’ s lemma , LaTeXMLCite , ( whose proof is included in the appendix . )
The third idea is that we have the estimates , the second of which was shown in LaTeXMLCite , LaTeXMLEquation .
An unpublished example of L. Carleson shows that the rectangular LaTeXMLMath norm is not comparable to the LaTeXMLMath norm , LaTeXMLCite .
We may assume that the rectangular LaTeXMLMath norm of the function LaTeXMLMath is small .
Indeed , this turns out to be an essential aspect of the argument .
From theorem LaTeXMLRef we deduce a weak factorization for the ( biholomorphic ) space LaTeXMLMath .
The idea is that if the function LaTeXMLMath has biholomorphic extension to LaTeXMLMath then for functions LaTeXMLMath , LaTeXMLEquation .
So in this case , the operator norm of the nested commutator LaTeXMLMath is comparable to the dual norm LaTeXMLEquation where the supremum above is over all pairs LaTeXMLMath in the unit ball of LaTeXMLMath .
On the other hand , since LaTeXMLMath and LaTeXMLMath are comparable , the dual norm above satisfies LaTeXMLEquation where the supremum is over all functions LaTeXMLMath in the unit ball of LaTeXMLMath .
Let LaTeXMLMath be in LaTeXMLMath with LaTeXMLMath .
Then there exists functions LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath We remark that the weak factorization above implies the analogous factorization for LaTeXMLMath of the bidisk .
Indeed , for all LaTeXMLMath , the map LaTeXMLMath defined by LaTeXMLEquation is an isometry with isometric inverse LaTeXMLEquation .
The dual formulation of weak factorization for LaTeXMLMath is a Nehari theorem for the bidisk .
Specifically , if LaTeXMLMath then the little Hankel operator with symbol LaTeXMLMath is densely defined on LaTeXMLMath by the formula LaTeXMLEquation .
By ( LaTeXMLRef ) , LaTeXMLMath and thus , by theorem LaTeXMLRef , LaTeXMLMath is comparable to LaTeXMLMath which , by definition , is just the norm of LaTeXMLMath acting on LaTeXMLMath .
So the boundedness of the Hankel operator LaTeXMLMath implies that there is a function LaTeXMLMath such that LaTeXMLMath .
Several variations and complements on these themes in the one parameter setting have been obtained by Coifman , Rochberg and Weiss LaTeXMLCite .
The paper is organized as follows .
Section 2 gives the one-dimensional preliminaries for the proof of theorem LaTeXMLRef and Section 3 is devoted to the proof of theorem LaTeXMLRef .
The appendix contains the variant of Journé ’ s lemma we use in our proof in Section 3 .
One final remark about notation .
LaTeXMLMath means that there is an absolute constant LaTeXMLMath for which LaTeXMLMath .
LaTeXMLMath means that LaTeXMLMath and LaTeXMLMath .
We are endebited to the anonymous referee .
Several factors conspire to make the one dimensional case easier than the higher dimensional case .
Before proceeding with the higher dimensional case , we make several comments about the one dimensional case , comments that extend and will be useful in the subsequent section .
Let LaTeXMLMath denote the Hilbert transform in one variable , LaTeXMLMath be the projection of LaTeXMLMath onto the positive frequencies , and LaTeXMLMath is LaTeXMLMath , the projection onto the negative frequencies .
We shall in particular rely upon the following basic computation .
LaTeXMLEquation .
The frequency distribution of LaTeXMLMath is symmetric since it is real valued .
Thus , LaTeXMLEquation .
LaTeXMLEquation Moreover , if LaTeXMLMath is supported on an interval LaTeXMLMath , we see that LaTeXMLEquation which is the LaTeXMLMath estimate on LaTeXMLMath .
We seek an extension of this estimate in the two parameter setting .
We use a wavelet proof of theorem LaTeXMLRef , and specifically use a wavelet with compact frequency support constructed by Y. Meyer LaTeXMLCite .
There is a Schwarz function LaTeXMLMath with these properties .
LaTeXMLMath .
LaTeXMLMath is supported on LaTeXMLMath together with the symmetric interval about LaTeXMLMath .
LaTeXMLMath is a Schwartz function .
More particularly , we have LaTeXMLEquation .
Let LaTeXMLMath denote a collection of dyadic intervals on LaTeXMLMath .
For any interval LaTeXMLMath , let LaTeXMLMath denote it ’ s center , and define LaTeXMLEquation .
Set LaTeXMLMath .
The central facts that we need about the functions LaTeXMLMath are these .
First , that these functions are an orthonormal basis on LaTeXMLMath .
Second , that we have the Littlewood–Paley inequalities , valid on all LaTeXMLMath , though LaTeXMLMath will be of special significance for us .
These inequalities are LaTeXMLEquation .
Third , that the functions LaTeXMLMath have good localization properties in the spatial variables .
That is , LaTeXMLEquation where LaTeXMLMath .
We find the compact localization of the wavelets in frequency to be very useful .
The price we pay for this utility below is the careful accounting of “ Schwartz tails ” we shall make in the main argument .
Fifth , we have the identity below for the commutator of one LaTeXMLMath with a LaTeXMLMath .
Observe that since LaTeXMLMath is one half of LaTeXMLMath , it suffices to replace LaTeXMLMath by LaTeXMLMath in the definition of the commutator .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
From this we see a useful point concerning orthogonality .
For intervals LaTeXMLMath and LaTeXMLMath , assume LaTeXMLMath and likewise for LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
Indeed , this follows from a direct calculation .
The positive frequency support of LaTeXMLMath is contained in the interval LaTeXMLMath .
Under the conditions on LaTeXMLMath and LaTeXMLMath , the frequency supports are disjoint .
LaTeXMLMath will denote the LaTeXMLMath of two parameters ( or product LaTeXMLMath ) defined as the dual of ( real ) LaTeXMLMath .
The following characterization of the space LaTeXMLMath is due to Chang and R. Fefferman LaTeXMLCite .
The relevant class of rectangles is LaTeXMLMath , all rectangles which are products of dyadic intervals .
These are indexed by LaTeXMLMath .
For such a rectangle , write it as a product LaTeXMLMath and then define LaTeXMLEquation .
A function LaTeXMLMath iff LaTeXMLEquation .
Here , the sum extends over those rectangles LaTeXMLMath and the supremum is over all open sets in the plane of finite measure .
Note that the supremum is taken over a much broader class of sets than merely rectangles in the plane .
We denote this supremum as LaTeXMLMath .
In this definition , if the supremum over LaTeXMLMath is restricted to just rectangles , this defines the “ rectangular LaTeXMLMath ” space , and we denote this restricted supremum as LaTeXMLMath .
Let us make a further comment on the LaTeXMLMath condition .
Suppose that for LaTeXMLMath , we have non–negative constants LaTeXMLMath for which LaTeXMLEquation for all open sets LaTeXMLMath in the plane of finite measure .
Then , we have the John–Nirenberg inequality LaTeXMLEquation .
See LaTeXMLCite .
This , with the Littlewood–Paley inequalities , will be used several times below , and referred to as the John–Nirenberg inequalities .
We begin the principle line of the argument .
The function LaTeXMLMath may be taken to be of Schwarz class .
By multiplying LaTeXMLMath by a constant , we can assume that the LaTeXMLMath norm of LaTeXMLMath is LaTeXMLMath .
Set LaTeXMLMath to be the operator norm of LaTeXMLMath .
Our purpose is to provide a lower bound for LaTeXMLMath .
Let LaTeXMLMath be an open set of finite measure for which we have the equality LaTeXMLEquation .
As LaTeXMLMath is of Schwarz class , such a set exists .
By invariance under dilations by a factor of two , we can assume that LaTeXMLMath .
In several estimates below , the measure of LaTeXMLMath enters in , a fact which we need not keep track of .
An essential point is that we may assume that the rectangular LaTeXMLMath norm of LaTeXMLMath is at most LaTeXMLMath .
The reason for this is that we have the estimate LaTeXMLMath .
See LaTeXMLCite .
Therefore , for a small constant LaTeXMLMath to be chosen below , we can assume that LaTeXMLMath , for otherwise we have a lower bound on LaTeXMLMath .
Associated to the set LaTeXMLMath is the set LaTeXMLMath , defined below , which is an expansion of the set LaTeXMLMath .
In defining this expansion , it is critical that the measure of LaTeXMLMath be only slightly larger than the measure of LaTeXMLMath , and so in particular we do not use the strong maximal function to define this expansion .
In the definition of LaTeXMLMath , the parameter LaTeXMLMath will be specified later and LaTeXMLMath is the one dimensional maximal function applied in the direction LaTeXMLMath .
Define LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The quantity LaTeXMLMath measures how deeply a rectangle LaTeXMLMath is inside LaTeXMLMath .
This quantity enters into the essential Journé ’ s Lemma , see LaTeXMLCite or the variant we prove in the appendix .
In the argument below , we will be projecting LaTeXMLMath onto subspaces spanned by collections of wavelets .
These wavelets are in turn indexed by collections of rectangles .
Thus , for a collection LaTeXMLMath , let us denote LaTeXMLEquation .
The relevant collections of rectangles are defined as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
For functions LaTeXMLMath and LaTeXMLMath , we set LaTeXMLMath .
We will demonstrate that for all LaTeXMLMath there is a constant LaTeXMLMath so that LaTeXMLMath LaTeXMLMath Furthermore , we will show that LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are arbitrary , a lower bound on LaTeXMLMath will follow from an appropriate choice of LaTeXMLMath and LaTeXMLMath .
To be specific , one concludes the argument by estimating LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Implied constants are absolute .
Choosing LaTeXMLMath first and then LaTeXMLMath appropriately small supplies a lower bound on LaTeXMLMath .
The estimate LaTeXMLMath relies on the John–Nirenberg inequality and the two parameter version of ( LaTeXMLRef ) , namely LaTeXMLEquation .
This identity easily follows from the one-variable identities .
Here LaTeXMLMath denotes the projection onto the positive/negative frequencies in the first and second variables .
These projections are orthogonal and moreover , since LaTeXMLMath is real valued we have that LaTeXMLMath .
Therefore , LaTeXMLMath .
It follows by the John–Nirenberg inequality that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The estimate LaTeXMLMath relies on the fact that the one dimensional maximal function maps LaTeXMLMath into weak LaTeXMLMath with norm one .
Thus , for all LaTeXMLMath , LaTeXMLEquation .
Now , if LaTeXMLMath , then LaTeXMLMath and since LaTeXMLMath has LaTeXMLMath norm one , it follows that LaTeXMLEquation .
Hence LaTeXMLMath .
Yet the LaTeXMLMath norm of LaTeXMLMath can be no more than that of LaTeXMLMath , which is to say LaTeXMLMath .
Interpolating norms we see that LaTeXMLMath , and so LaTeXMLEquation .
We now turn to the estimate LaTeXMLMath .
Roughly speaking LaTeXMLMath and LaTeXMLMath live on disjoint sets .
But in this argument we are trading off precise Fourier support of the wavelets for imprecise spatial localization , that is the ” Schwartz tails ” problem .
Accounting for this requires a careful analysis , invoking several subcases .
A property of the commutator that we will rely upon is that it controls the geometry of LaTeXMLMath and LaTeXMLMath .
Namely , LaTeXMLMath iff writing LaTeXMLMath and likewise for LaTeXMLMath , we have for both LaTeXMLMath , LaTeXMLMath .
This follows immediately from our one–dimensional calculations , in particular ( LaTeXMLRef ) .
We abbreviate this condition on LaTeXMLMath and LaTeXMLMath as LaTeXMLMath and restrict our attention to this case .
Orthogonality also enters into the argument .
Observe the following .
For rectangles LaTeXMLMath , LaTeXMLMath , with LaTeXMLMath , and for LaTeXMLMath or LaTeXMLMath LaTeXMLEquation .
This follows from applying ( LaTeXMLRef ) in the LaTeXMLMath th coordinate .
Therefore , there are different partial orders on rectangles that are relevant to our argument .
They are LaTeXMLMath iff LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath or LaTeXMLMath , define LaTeXMLMath iff LaTeXMLMath and LaTeXMLMath but LaTeXMLMath .
LaTeXMLMath iff LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
These four partial orders divide the collection LaTeXMLMath into four subclasses which require different arguments .
In each of these four arguments , we have recourse to this definition .
Set LaTeXMLMath , for LaTeXMLMath to be those rectangles in LaTeXMLMath with LaTeXMLMath .
Journé ’ s Lemma enters into the considerations .
Let LaTeXMLMath be a collection of rectangles which are pairwise incomparable with respect to inclusion .
For this latter collection , we have the inequality LaTeXMLEquation .
See Journé LaTeXMLCite , also see the appendix .
This together with the assumption that LaTeXMLMath has small rectangular LaTeXMLMath norm gives us LaTeXMLEquation .
This interplay between the small rectangular LaTeXMLMath norm and Journé ’ s Lemma is a decisive feature of the argument .
Essentially , the decomposition into the collections LaTeXMLMath is a spatial decomposition of the collection LaTeXMLMath .
A corresponding decomposition of LaTeXMLMath enters in .
Yet the definition of this class differs slightly depending on the partial order we are considering .
For LaTeXMLMath and LaTeXMLMath the term LaTeXMLMath is a linear combination of LaTeXMLEquation .
Consider the last term .
As we are to estimate an LaTeXMLMath norm , the leading operators LaTeXMLMath can be ignored .
Moreover , the essential properties of wavelets used below still hold for the conjugates and Hilbert transforms of the same .
These properties are Fourier localization and spatial localization .
Similar comments apply to the other three terms , and so the arguments below applies to each type of term seperately .
We consider the case of LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
The sums we considering are related to the following definition .
Set LaTeXMLEquation .
Note that we consider the maximal truncation of the sum over all choices of dimensions of the rectangles in the sum .
Thus , this sum is closely related to the strong maximal function LaTeXMLMath applied to LaTeXMLMath , so that in particular we have the estimate below , which relies upon ( LaTeXMLRef ) .
LaTeXMLEquation ( By a suitable definition of the strong maximal function LaTeXMLMath , one can deduce this inequality from the LaTeXMLMath bounds for LaTeXMLMath . )
We apply this inequality far away from the set LaTeXMLMath .
For the set LaTeXMLMath , LaTeXMLMath , we have the inequality LaTeXMLEquation .
We shall need a refined decomposition of the collection LaTeXMLMath , the motivation for which is the following calculation .
Let LaTeXMLMath .
For LaTeXMLMath , set LaTeXMLMath .
In addition , let LaTeXMLEquation .
And set LaTeXMLMath .
Then , in view of ( LaTeXMLRef ) , we see that LaTeXMLMath and LaTeXMLMath are orthogonal if LaTeXMLMath and LaTeXMLMath differ by at least LaTeXMLMath in either coordinate .
Thus , LaTeXMLEquation .
The rectangles LaTeXMLMath are all translates of one another .
Thus , taking advantage of the rapid spatial decay of the wavelets , we can estimate LaTeXMLEquation .
In this display , we let LaTeXMLMath and for rectangles LaTeXMLMath , LaTeXMLMath .
Note that LaTeXMLMath depends only on the dimensions of LaTeXMLMath and not its location .
Continuing , note the trivial inequality LaTeXMLMath .
We can estimate LaTeXMLEquation .
LaTeXMLEquation Here we take LaTeXMLMath .
The terms LaTeXMLMath are essentially of the order of magnitude LaTeXMLMath times a the scaled distance between LaTeXMLMath and the open set LaTeXMLMath .
To make this precise requires a decomposition of the collection LaTeXMLMath .
For integers LaTeXMLMath and LaTeXMLMath , set LaTeXMLMath to be those LaTeXMLMath which satisfy these three conditions .
First , LaTeXMLMath if LaTeXMLMath and LaTeXMLMath if LaTeXMLMath .
Second , there is an LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Third , for every LaTeXMLMath with LaTeXMLMath , we have LaTeXMLMath .
Certainly , this collection of rectangles is empty if LaTeXMLMath .
We see that LaTeXMLEquation .
The first estimate follows since the rectangles LaTeXMLMath are contained in the set LaTeXMLMath .
The second estimate follows from ( LaTeXMLRef ) .
But then from ( LaTeXMLRef ) we see that for LaTeXMLMath , LaTeXMLEquation .
In the case that LaTeXMLMath , we have the bound LaTeXMLMath .
This is obtained by taking the minimum to be LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath take the minimum to be LaTeXMLMath with LaTeXMLMath .
This last estimate is summable over LaTeXMLMath and LaTeXMLMath to at most LaTeXMLMath , and so completes this case .
We treat the case of LaTeXMLMath , while the case of LaTeXMLMath is same by symmetry .
The structure of this partial order provides some orthogonality in the first variable , leaving none in the second variable .
Bounds for the expressions from the second variable are derived from a cognate of a Carleson measure estimate .
There is a basic calculation that we perform for a subset LaTeXMLMath .
For an integer LaTeXMLMath define LaTeXMLMath , and LaTeXMLEquation .
Recalling ( LaTeXMLRef ) , if LaTeXMLMath and LaTeXMLMath differ by more than LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath are orthogonal .
Observe that for LaTeXMLMath and LaTeXMLMath as in the sum defining LaTeXMLMath , we have the estimate LaTeXMLEquation .
In this display , we are using the same notation as before , LaTeXMLMath and for rectangles LaTeXMLMath , LaTeXMLMath .
In addition , LaTeXMLMath , with LaTeXMLMath being the center of LaTeXMLMath .
[ This “ distance ” is more properly the inverse of a distance that takes into account the scale of the rectangle LaTeXMLMath . ]
Now define LaTeXMLEquation .
The main point of these observations and definitions is this .
For the function LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
At this point , it occurs to one to appeal to the Carleson measure property associated to the coefficients LaTeXMLMath .
This necessitates that one proves that the coefficients LaTeXMLMath satisfy a similar condition , which doesn ’ t seem to be true in general .
A slightly weaker condition is however true .
To get around this difficulty , we make a further diagonalization of the terms LaTeXMLMath above .
For integers LaTeXMLMath , LaTeXMLMath and a rectangle LaTeXMLMath , consider rectangles LaTeXMLMath such that LaTeXMLEquation [ The quantity LaTeXMLMath depends upon the particular subcollection LaTeXMLMath we are considering . ]
We denote one of these rectangles as LaTeXMLMath .
An important geometrical fact is this .
We have LaTeXMLMath .
And in particular , this last rectangle has measure LaTeXMLMath .
Therefore , there are at most LaTeXMLMath possible choices for LaTeXMLMath .
[ Small integral powers of LaTeXMLMath are completely harmless because of the large power of LaTeXMLMath that appears in ( LaTeXMLRef ) . ]
Our purpose is to bound this next expression by a term which includes a power of LaTeXMLMath , a small power of LaTeXMLMath and a power of LaTeXMLMath .
Define LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The innermost sum can be bounded this way .
First LaTeXMLMath , so that LaTeXMLEquation .
Second , by our geometrical observation about LaTeXMLMath , LaTeXMLEquation .
In particular , the factor LaTeXMLMath does not enter into this estimate .
This means that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The point of these computations is that a further trivial application of the Cauchy–Schwartz inequality proves that LaTeXMLEquation where LaTeXMLMath is the largest integer such that for all LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath .
We shall complete this section by decomposing LaTeXMLMath into subcollections for which this last estimate summable to LaTeXMLMath .
Indeed , take LaTeXMLMath to be those LaTeXMLMath with LaTeXMLMath for all LaTeXMLMath with LaTeXMLMath .
And there is an LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Certainly , we need only consider LaTeXMLMath .
It is clear that this decomposition of LaTeXMLMath will conclude the treatment of this partial order .
We now consider the case of LaTeXMLMath , which is less subtle as there is no orthogonality to exploit and the Carleson measure estimates are more directly applicible .
We prove the bound LaTeXMLEquation .
The diagonalization in space takes two different forms .
For LaTeXMLMath and LaTeXMLMath set LaTeXMLMath to be a choice of LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
( The definition is vacuous for LaTeXMLMath . )
It is clear that we need only consider LaTeXMLMath choices of these functions LaTeXMLMath .
There is an LaTeXMLMath estimate which allows one to take advantage of the spatial separation between LaTeXMLMath and LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This estimate uses ( LaTeXMLRef ) and is a very small estimate .
To complete this case we need to provide an estimate in LaTeXMLMath .
Here , we can be quite inefficient .
By Cauchy–Schwartz and the Littlewood–Paley inequalities , LaTeXMLEquation .
LaTeXMLEquation This follows directly from the LaTeXMLMath assumption on LaTeXMLMath .
Our proof is complete .
Let LaTeXMLMath be an open set of finite measure in the plane .
Let LaTeXMLMath be those maximal dyadic rectangles in LaTeXMLMath that are contained in LaTeXMLMath .
Define for each LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation and LaTeXMLMath .
For each LaTeXMLMath set LaTeXMLEquation .
The form of Journé ’ s Lemma we need is For each LaTeXMLMath and each open set LaTeXMLMath in the plane of finite measure , LaTeXMLEquation .
Journé ’ s Lemma is the central tool in verifying the Carleson measure condition , and points to the central problem in two dimensions : That there can be many rectangles close to the boundary of an open set .
Among the references we could find in the literature LaTeXMLCite , the form of Journé ’ s Lemma cited and proved is relative to a strictly larger quantity than the one we use , LaTeXMLMath above .
To define it , for any rectangle LaTeXMLMath , denote it as a product of intervals LaTeXMLMath .
Set LaTeXMLMath to be the strong maximal function .
Then , for open set LaTeXMLMath of finite measure and LaTeXMLMath , set ( taking LaTeXMLMath for simplicity ) LaTeXMLEquation .
Thus , one only measures how deeply LaTeXMLMath is in the enlarged set in one direction .
The Lemma above then holds for LaTeXMLMath , with however a slightly sharper form of the sum than we prove here .
In addition , note that one measures the depth of LaTeXMLMath with respect to a simpler set , LaTeXMLMath .
We did not use this simplification in our proof as the strong maximal function LaTeXMLMath does not act boundedly on LaTeXMLMath of the plane .
There are however examples which show that that the quantity LaTeXMLMath can be much larger than LaTeXMLMath .
Indeed , consider a horizontal row of evenly spaced squares .
For a square LaTeXMLMath in the middle of this row , LaTeXMLMath will be quite big , while LaTeXMLMath will be about LaTeXMLMath for all LaTeXMLMath .
Thus we give a proof of our form of Journé ’ s Lemma .
We can assume that LaTeXMLMath as the terms LaTeXMLMath decrease as LaTeXMLMath increases .
Fix LaTeXMLMath .
Set LaTeXMLMath to be those rectangles in LaTeXMLMath with LaTeXMLMath .
It suffices to show that LaTeXMLEquation .
For then this estimate is summed over LaTeXMLMath .
In showing this estimate , we can further assume that for all LaTeXMLMath , writing LaTeXMLMath and likewise for LaTeXMLMath , that if for LaTeXMLMath , LaTeXMLMath then LaTeXMLMath , where we set LaTeXMLMath .
This is done by restricting LaTeXMLMath to be in an arithmetic progression of difference LaTeXMLMath .
This neccessitates the division of all rectangles into LaTeXMLMath subclasses and so we prove the bound above without the logarithmic term .
We define a “ bad ” class of rectangles LaTeXMLMath as follows .
For LaTeXMLMath , let LaTeXMLMath be those rectangles LaTeXMLMath for which there are rectangles LaTeXMLEquation so that for each LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
Thus LaTeXMLMath if it is nearly completely covered in the LaTeXMLMath th direction of the plane .
Set LaTeXMLMath .
It follows that if LaTeXMLMath , it is not covered in both the vertical and horizontal directions , hence LaTeXMLEquation .
And , since all LaTeXMLMath , it follows that LaTeXMLEquation .
Thus , it remains to consider seperately the set of rectangles LaTeXMLMath and LaTeXMLMath .
Observe that for any collection LaTeXMLMath , LaTeXMLMath as follows immediately from the definition .
Hence LaTeXMLMath .
And we argue that this last set is empty .
As our definition of LaTeXMLMath and LaTeXMLMath is symmetric with respect to the coordinate axes , this is enough to finish the proof .
We argue that LaTeXMLMath is empty by contradiction .
Assume that LaTeXMLMath .
Consider those rectangles LaTeXMLMath in LaTeXMLMath for which LaTeXMLMath LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Then LaTeXMLEquation .
Fix a one of these rectangles LaTeXMLMath with LaTeXMLMath being minimal .
We then claim that LaTeXMLMath , which contradicts the assumption that LaTeXMLMath is no more than LaTeXMLMath .
Indeed , all the rectangles in LaTeXMLMath are themselves covered in the first coordinate axis .
We see that the the set LaTeXMLMath contains the rectangle LaTeXMLMath , in which LaTeXMLMath is the second coordinate interval for the rectangle LaTeXMLMath and LaTeXMLMath is the dyadic interval that contains LaTeXMLMath and has measure LaTeXMLMath .
But then the rectangle LaTeXMLMath is contained in LaTeXMLMath .
And since LaTeXMLMath is contained in this last rectangle , we have contradicted the assumption that LaTeXMLMath .
∎ Sarah H. Ferguson Department of Mathematics Wayne State University Detroit MI 48202 sarah @ math.wayne.edu http : //www.math.wayne.edu/~sarah Michael T. Lacey School of Mathematics Georgia Institute of Technology Atlanta GA 30332 lacey @ math.gatech.edu http : //www.math.gatech.edu/~lacey
We introduce a class of cycles , called nondegenerate , strictly decomposable cycles , and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed Hodge structures does not vanish .
This class contains certain cycles in the kernel of the Abel-Jacobi map .
The construction gives a refinement of Nori ’ s argument in the case of a self-product of a curve .
As an application , we show that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1-form .
Introduction Let LaTeXMLMath be a smooth projective complex variety , and let LaTeXMLMath be the Chow group of codimension LaTeXMLMath cycles modulo rational equivalence .
Consider the subgroup LaTeXMLMath consisting of cycles homologically equivalent to zero .
We have the Abel-Jacobi map to the intermediate Jacobian [ 24 ] .
If LaTeXMLMath , it is known that this map is neither surjective nor injective in general .
D. Mumford [ 30 ] showed that the kernel can be very large , see also [ 7 ] .
However it is not necessarily easy to construct a nonzero element in the kernel , because the rational equivalence is rather complicated for explicit calculations .
An argument constructing such an example was found by M. Nori in the case of a self-product of a curve , see [ 38 ] .
His argument involves the transcendental part of the second cohomology , and requires some more calculations to verify the hypothesis on cycles .
In this paper , we simplify and generalize it by introducing the notion of a nondegenerate , strictly decomposable cycle .
Let LaTeXMLMath be smooth projective varieties over LaTeXMLMath , and set LaTeXMLMath .
Take divisors LaTeXMLMath which are homologically equivalent to zero .
We assume that they are not torsion .
Consider LaTeXMLMath in LaTeXMLMath .
This belongs to the kernel of the Abel-Jacobi map .
It may be zero , for example , if LaTeXMLMath and LaTeXMLMath are the same elliptic curve and LaTeXMLMath and LaTeXMLMath are the same cycle of the form LaTeXMLMath , see ( 2.9.2 ) .
So we are interested in the following question : When is LaTeXMLMath not rationally equivalent to zero ( with rational coefficients ) ?
This is nontrivial even in the case of a self-product of an elliptic curve .
In this paper we give a partial answer as follows .
0.1 .
Theorem .
Assume there exists an algebraically closed subfield LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are defined over LaTeXMLMath , but LaTeXMLMath is not .
Then the cycle LaTeXMLMath does not vanish in LaTeXMLMath .
This is an example of an exterior product of cycles [ 39 ] although the varieties here are all defined over LaTeXMLMath .
In the curve case , Theorem ( 0.1 ) implies ( by replacing LaTeXMLMath ) that LaTeXMLMath does not vanish in LaTeXMLMath if there exists an algebraically closed subfield LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are defined over LaTeXMLMath , but LaTeXMLMath , LaTeXMLMath are not , and if the field of definition LaTeXMLMath of LaTeXMLMath is algebraically independent of LaTeXMLMath ( i.e .
if they generate a subfield of transcendence degree LaTeXMLMath ) .
Here LaTeXMLMath is the image of the morphism LaTeXMLMath defined by LaTeXMLMath , see ( 2.9 ) .
The proof of ( 0.1 ) uses a refined cycle map and the associated Leray filtration on the Chow group , which follow from the theory of ( arithmetic ) mixed sheaves [ 34 ] , [ 36 ] , [ 37 ] ( see also [ 1 ] ) as conjectured in [ 4 ] , [ 7 ] .
This construction was originally considered in order to define an analogue of “ spreading out ” of algebraic cycles [ 7 ] for mixed Hodge Modules ( see [ 34 ] , 1.9 ) , and was greatly inspired by earlier work of M. Green [ 23 ] , C. Voisin [ 40 ] , [ 41 ] and others .
We remark that our Leray filtration seems to coincide with a filtration which has been studied recently by M. Green in the case the variety is defined over LaTeXMLMath as explained in his talk at Azumino , July 2000 .
We can use also Green ’ s formulation for the proof of Theorem ( 0.1 ) , see Remark ( ii ) of ( 1.5 ) .
The theory of mixed Hodge modules is essential in the case the variety is not defined over LaTeXMLMath .
Note that LaTeXMLMath in Theorem ( 0.1 ) can be a cycle of any codimension provided that its image by the Abel-Jacobi map is not torsion .
Indeed , we have a more general assertion ( 2.6 ) whose proof is reduced to a simple lemma on Hodge structures ( 2.7 ) .
We can extend Theorem ( 2.6 ) to the case of higher Chow groups LaTeXMLMath [ 9 ] as in ( 3.3 ) , which implies 0.2 .
Theorem .
Let LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath does not vanish if there exists an algebraically closed subfield LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath are defined over LaTeXMLMath and the following condition is satisfied for LaTeXMLMath or LaTeXMLMath ( i ) LaTeXMLMath is defined over LaTeXMLMath , but LaTeXMLMath is not , ( ii ) the image of LaTeXMLMath by the cycle map ( 3.1.2 ) does not vanish , ( iii ) LaTeXMLMath .
Here LaTeXMLMath for LaTeXMLMath and LaTeXMLMath by loc .
cit .
So we may assume LaTeXMLMath if the above hypothesis is satisfied for LaTeXMLMath and LaTeXMLMath .
Originally this theory was considered in the case when both LaTeXMLMath and LaTeXMLMath are nonzero with the notation of ( 2.1 ) in order to show the nonvanishing of some extension classes .
In the study of Nori ’ s construction [ 38 ] , a special case of ( 2.6 ) was obtained for a self-product of a curve of certain type [ 32 ] .
Then these were generalized to ( 2.6 ) .
As an application , we have the following .
For a smooth complex projective variety LaTeXMLMath , let LaTeXMLMath denote the group of indecomposable higher cycles with rational coefficients , see ( 4.1 ) .
We have the reduced higher Abel-Jacobi map ( 4.1.2 ) which is analogous to Griffiths ’ Abel-Jacobi map .
0.3 .
Theorem .
Let LaTeXMLMath as above .
If LaTeXMLMath and the reduced higher Abel-Jacobi map ( 4.1.2 ) for LaTeXMLMath is not zero , then LaTeXMLMath is uncountable .
More precisely , let LaTeXMLMath be an element of LaTeXMLMath such that its image by ( 4.1.2 ) does not vanish .
Let LaTeXMLMath denote the Picard variety of LaTeXMLMath .
Assume LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are defined over LaTeXMLMath .
Then for any LaTeXMLMath , the image of LaTeXMLMath in LaTeXMLMath is nonzero .
This does not contradicts Voisin ’ s conjecture [ 42 ] on the countability of LaTeXMLMath , because LaTeXMLMath .
It is known that LaTeXMLMath is uncountable for certain varieties LaTeXMLMath , see [ 13 ] .
The assumption on the nonvanishing of the image of LaTeXMLMath by ( 4.1.2 ) is satisfied for some examples , see [ 14 ] , [ 35 ] .
The paper is organized as follows .
In Sect .
1 we recall the notion of arithmetic mixed Hodge structures .
In Sect .
2 we prove Theorem ( 0.1 ) .
Then Sect .
3 treats the case of higher cycles , and the application to indecomposable higher cycles is given in Sect .
4 .
In this paper , a variety means a separated scheme of finite type over a field .
For a complex variety LaTeXMLMath and a ring LaTeXMLMath , we denote LaTeXMLMath by LaTeXMLMath to simplify the notation .
1 .
Arithmetic mixed Hodge structure 1.1 .
Mixed Hodge structure .
For a subfield LaTeXMLMath of LaTeXMLMath , let LaTeXMLMath denote the category of graded-polarizable mixed LaTeXMLMath -Hodge structure with LaTeXMLMath -structure , see [ 17 ] , [ 18 ] , [ 26 ] , etc .
( In this paper , LaTeXMLMath -adic cohomology is not used , because LaTeXMLMath is assumed to be algebraically closed and the embedding of LaTeXMLMath into LaTeXMLMath is fixed . )
An object LaTeXMLMath of LaTeXMLMath consists of a bifiltered LaTeXMLMath -vector space LaTeXMLMath and a filtered LaTeXMLMath -vector space LaTeXMLMath together with an filtered isomorphism LaTeXMLMath , and they define a graded-polarizable mixed LaTeXMLMath -Hodge structure in the sense of [ 16 ] after tensoring LaTeXMLMath with LaTeXMLMath .
Here the polarizations on the graded pieces of LaTeXMLMath are assumed to be defined over LaTeXMLMath so that the graded pieces are semisimple .
Morphisms are pairs of morphisms of bifiltered or filtered vector spaces compatible with LaTeXMLMath .
This is an abelian category such that every morphism is bistrictly compatible with LaTeXMLMath .
We have naturally a constant object LaTeXMLMath .
We can define the Tate twist LaTeXMLMath for LaTeXMLMath as in loc .
cit .
For LaTeXMLMath , we have LaTeXMLEquation by an argument similar to [ 12 ] , see [ 36 ] .
Let LaTeXMLMath be a LaTeXMLMath -variety .
Then we have naturally the cohomology LaTeXMLMath in LaTeXMLMath by [ 16 ] together with the compatibility of de Rham cohomology with base change .
Furthermore , we can define canonically LaTeXMLMath in the derived category LaTeXMLMath such that its cohomology LaTeXMLMath is isomorphic to LaTeXMLMath in LaTeXMLMath .
This can be done by applying the construction of the direct image for perverse sheaves [ 5 ] to mixed Hodge Modules .
In the case LaTeXMLMath is smooth and quasiprojective , we may assume that the complement of LaTeXMLMath in a smooth projective compactification of LaTeXMLMath is a divisor .
Then we can take two sets of general hyperplane sections LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath , and use the Cech and co-Cech complexes associated to the affine coverings LaTeXMLMath and LaTeXMLMath together with the generalization of the weak Lefschetz theorem in [ 6 ] , so that we get a complex whose LaTeXMLMath -th component is LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
By ( 1.1.1 ) we have a noncanonical isomorphism LaTeXMLEquation 1.2 .
Deligne cohomology .
We define an analogue of Deligne cohomology by LaTeXMLEquation .
If LaTeXMLMath , this coincides with the absolute LaTeXMLMath -Hodge cohomology of Beilinson [ 3 ] which is denoted by LaTeXMLMath in this paper , see also [ 20 ] , [ 21 ] , [ 25 ] , [ 33 ] .
In general , if we put LaTeXMLMath , then the forgetful functor induces a natural morphism LaTeXMLEquation .
More generally , for LaTeXMLMath , we define LaTeXMLEquation .
By ( 1.1.1 ) we have a natural short exact sequence LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLMath be the Chow group of codimension LaTeXMLMath cycles with rational coefficients modulo rational equivalence .
Assume LaTeXMLMath is smooth .
Then we have a cycle map ( see e.g .
[ 34 ] ) LaTeXMLEquation .
This is compatible with the cycle map LaTeXMLEquation for LaTeXMLMath via ( 1.2.1 ) .
The cycle map ( 1.2.4 ) induces Griffiths ’ Abel-Jacobi map [ 24 ] tensored with LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath denotes the subgroup consisting of homologically equivalent to zero cycles , and the last isomorphism is due to [ 12 ] .
1.3 .
Leray filtration .
Let LaTeXMLMath be a LaTeXMLMath -variety , and put LaTeXMLMath .
Then we have a canonical isomorphism LaTeXMLEquation .
There is an increasing Leray filtration LaTeXMLMath on LaTeXMLMath induced by the canonical truncation LaTeXMLMath on LaTeXMLMath .
( In general , LaTeXMLMath for a complex LaTeXMLMath is defined by LaTeXMLMath for LaTeXMLMath , LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath . )
We have a canonical isomorphism LaTeXMLEquation .
Since the filtration LaTeXMLMath splits by ( 1.1.2 ) , it induces a decreasing Leray filtration LaTeXMLMath on LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation Here the filtration LaTeXMLMath is shifted as in [ 16 ] .
We have a natural short exact sequence LaTeXMLEquation .
LaTeXMLEquation The filtration LaTeXMLMath and this short exact sequence are compatible with the pull-back induced by the base change under a morphism of LaTeXMLMath -varieties LaTeXMLMath .
Assume LaTeXMLMath , LaTeXMLMath are smooth .
By the cycle map ( 1.2.3 ) for LaTeXMLMath , we get the induced Leray filtration LaTeXMLMath on LaTeXMLMath such that LaTeXMLEquation is injective .
1.4 .
Arithmetic mixed Hodge structure .
Let LaTeXMLMath be a subfield of LaTeXMLMath .
We assume LaTeXMLMath is algebraically closed for simplicity ( hence LaTeXMLMath -adic sheaves are not used in this paper ) .
Then the category of arithmetic mixed Hodge structures LaTeXMLMath is defined to be the inductive limit of the categories of mixed Hodge Modules on LaTeXMLMath ( i.e .
mixed Hodge Modules on LaTeXMLMath whose underlying bifiltered LaTeXMLMath -Modules and polarizations are defined over LaTeXMLMath , where LaTeXMLMath runs over the finitely generated smooth LaTeXMLMath -subalgebra of LaTeXMLMath .
Although we can also define it to be the inductive limit of the categories of admissible variations of mixed Hodge structures on LaTeXMLMath , we need mixed Hodge Modules in order to calculate higher extension groups , because the adjunction relation between direct images and pull-backs does not hold for the derived category of admissible variations .
There is a canonical pull-back functor LaTeXMLEquation induced by the projection LaTeXMLMath for any finitely generated smooth LaTeXMLMath -subalgebra LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath .
We have also the forgetful functor LaTeXMLEquation by restricting to the geometric generic point of LaTeXMLMath defined by the inclusion LaTeXMLMath .
For a complex algebraic variety LaTeXMLMath , the cohomology LaTeXMLMath is naturally defined in LaTeXMLMath by taking an LaTeXMLMath -scheme LaTeXMLMath such that LaTeXMLMath , because the direct image of the constant sheaf by LaTeXMLMath is defined by the theory of mixed Hodge Modules .
Furthermore , it can be lifted to a complex LaTeXMLMath having a canonical isomorphism LaTeXMLEquation .
We define an analogue of Deligne cohomology by LaTeXMLEquation .
From now on , we assume LaTeXMLMath for a smooth LaTeXMLMath -variety LaTeXMLMath .
Then the argument is very much simplified , and we have natural isomorphisms LaTeXMLEquation .
LaTeXMLEquation By ( 1.3 ) we have a Leray filtration LaTeXMLMath on LaTeXMLMath such that LaTeXMLEquation .
Since LaTeXMLMath , we have a cycle map LaTeXMLEquation which induces the ( decreasing ) Leray filtration LaTeXMLMath on LaTeXMLMath such that LaTeXMLEquation is injective .
This cycle map is compatible with ( 1.2.4 ) .
1.5 .
Remarks .
( i ) By the compatibility of the cycle maps ( 1.2.4 ) and ( 1.4.5 ) , LaTeXMLMath coincides with the subgroup LaTeXMLMath of homologically equivalent to zero cycles , and LaTeXMLMath is contained in the kernel of the Abel-Jacobi map LaTeXMLMath .
Here we have equality if LaTeXMLMath ( see e.g .
[ 36 ] , 3.6 ) .
This follows from Murre ’ s result on Albanese motives [ 31 ] together with the compatibility of the action of a correspondence .
Restricting to the subgroup LaTeXMLMath of algebraically equivalent to zero cycles , we have more generally the equality LaTeXMLMath for any LaTeXMLMath , see [ 37 ] , 3.9 .
We can show LaTeXMLMath for LaTeXMLMath .
In the case LaTeXMLMath , it is expected that the filtration LaTeXMLMath gives the conjectural filtration of Bloch [ 7 ] and Beilinson [ 4 ] , see also [ 27 ] .
The problem is the injectivity of the cycle map ( 1.4.5 ) .
In the case LaTeXMLMath and LaTeXMLMath , this can be reduced to a conjecture of Beilinson [ 4 ] and Bloch on the injectivity of Abel-Jacobi map for codimension LaTeXMLMath cycles defined over number fields ( see [ 11 ] for a special case where the conjecture is verified ) .
( ii ) It is also possible to consider a variant of Deligne cohomology by replacing LaTeXMLMath with LaTeXMLMath in ( 1.4.3 ) ( i.e .
by forgetting LaTeXMLMath -structure ) .
Then we get a filtration similar to the one studied recently by M. Green and also to the one treated in [ 28 ] .
( However , the definition given in loc .
cit .
in terms of currents does not seem to be correct even in the product case , because the Hodge filtration LaTeXMLMath on some graded pieces of LaTeXMLMath on the complex of currents is filtered quasi-isomorphic to the filtration LaTeXMLMath on the localization of holomorphic differential forms , and not the pole order filtration of Griffiths [ 24 ] and Deligne [ 15 ] . )
In view of the above conjecture on the injectivity of Abel-Jacobi map , it may be expected that we get still the same filtration on the Chow group after the modification .
Note that we can forget the LaTeXMLMath -structure for the proof of Theorems ( 0.1 ) and ( 0.2 ) .
( iii ) For a subfield LaTeXMLMath of LaTeXMLMath containing LaTeXMLMath , we can repeat the arguments in ( 1.4 ) by restricting LaTeXMLMath to those contained in LaTeXMLMath .
2 .
Strictly decomposable cycles 2.1 .
Let LaTeXMLMath and LaTeXMLMath be smooth complex projective varieties defined over an algebraically closed subfield LaTeXMLMath of LaTeXMLMath , i.e .
there are smooth projective LaTeXMLMath -varieties LaTeXMLMath such that LaTeXMLMath .
Put LaTeXMLMath , LaTeXMLMath as in the introduction .
We say that a cycle LaTeXMLMath on LaTeXMLMath is strictly decomposable if there exist subfields LaTeXMLMath of LaTeXMLMath finitely generated over LaTeXMLMath , together with cycles LaTeXMLMath on LaTeXMLMath for LaTeXMLMath such that the canonical morphism LaTeXMLMath is injective , and LaTeXMLMath coincides with the base change of the cycle LaTeXMLMath on LaTeXMLEquation by LaTeXMLMath ( replacing LaTeXMLMath with a finite extension if necessary ) .
We say that a strictly decomposable cycle is of bicodimension LaTeXMLMath if LaTeXMLMath .
Put LaTeXMLMath .
Let LaTeXMLMath be a finitely generated smooth LaTeXMLMath -subalgebra of LaTeXMLMath such that the fraction field of LaTeXMLMath is LaTeXMLMath , and LaTeXMLMath is defined over LaTeXMLMath .
Set LaTeXMLMath .
We denote by LaTeXMLMath the base change of LaTeXMLMath by LaTeXMLMath .
Note that LaTeXMLMath , and its restriction to LaTeXMLMath coincides with LaTeXMLMath , where LaTeXMLMath is defined by the inclusion LaTeXMLMath .
Let LaTeXMLEquation denote the Künneth components of the cycle class of LaTeXMLMath in LaTeXMLMath .
Define LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath belongs to LaTeXMLMath by ( 1.3 ) together with Remark ( iii ) in ( 1.5 ) , and LaTeXMLMath gives LaTeXMLEquation by forgetting the LaTeXMLMath -structure , see ( 1.3.3 ) .
2.2 .
Proposition .
With the above notation , LaTeXMLMath if and only if LaTeXMLMath comes from LaTeXMLMath by the pull-back under LaTeXMLMath .
In the case LaTeXMLMath , these conditions are further equivalent to that LaTeXMLMath comes from LaTeXMLMath .
Proof .
This follows from the short exact sequence ( 1.2.2 ) for LaTeXMLMath and LaTeXMLMath , where the first term is isomorphic to LaTeXMLMath , and the first morphism is induced by the pull-back functor for LaTeXMLMath .
Indeed , LaTeXMLMath coincides with the image of LaTeXMLMath in the last term , and we get the first equivalence .
Then the second is clear because the cycle map to Deligne cohomology is an isomorphism in the divisor case and the base change by LaTeXMLMath induces an injective morphism of Chow groups with rational coefficients .
2.3 .
Remark .
By ( 2.2 ) the cohomology class LaTeXMLMath measures how the cycle LaTeXMLMath varies along the parameter space LaTeXMLMath .
If the first equivalent conditions of ( 2.2 ) are satisfied , we may assume LaTeXMLMath as long as LaTeXMLMath is concerned , e.g .
if LaTeXMLMath .
( Indeed , we can restrict to a LaTeXMLMath -valued point of LaTeXMLMath . )
2.4 .
Definition .
We say that LaTeXMLMath is degenerate if LaTeXMLMath .
A strictly decomposable cycle LaTeXMLMath is called separately degenerate if one of LaTeXMLMath is degenerate , and cohomologically degenerate if LaTeXMLMath for both LaTeXMLMath , and degenerate if it is separately degenerate or cohomologically degenerate .
In other words , LaTeXMLMath is nondegenerate if one of LaTeXMLMath and LaTeXMLMath does not vanish for each LaTeXMLMath and one of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath does not vanish .
2.5 .
Remark .
Assume LaTeXMLMath .
Then LaTeXMLMath is degenerate if and only if it is zero in the Chow group with rational coefficients .
If LaTeXMLMath is cohomologically degenerate , it comes from a cycle on LaTeXMLMath by the pull-back under the projection to LaTeXMLMath , see ( 2.2 ) .
If furthermore LaTeXMLMath is a number field , it is expected that LaTeXMLMath is rationally equivalent to zero according to the conjecture of Beilinson [ 2 ] and Bloch ( see also [ 8 ] , [ 26 ] ) .
If LaTeXMLMath is nondegenerate , however , we can detect it by the refined cycle map due to the following : 2.6 .
Theorem .
Let LaTeXMLMath be a nondegenerate , strictly decomposable cycle of codimension LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath be as in ( 2.1 ) .
Then LaTeXMLMath .
More precisely , if LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
Proof .
Let LaTeXMLMath and LaTeXMLMath .
Then the cycle map ( 1.2.3 ) gives LaTeXMLEquation and LaTeXMLMath is the external product of LaTeXMLMath and LaTeXMLMath using ( 1.3.1 ) .
Since LaTeXMLMath if LaTeXMLMath , we see that LaTeXMLMath and hence LaTeXMLMath for LaTeXMLMath as above .
Then the assertion is clear if one of LaTeXMLMath does not vanish .
Indeed , it is reduced to the case LaTeXMLMath ( replacing LaTeXMLMath with a finite extension if necessary ) , and follows from the nonvanishing of LaTeXMLMath or LaTeXMLMath for LaTeXMLMath .
Thus we may assume LaTeXMLMath and LaTeXMLMath .
By ( 1.3.1 ) we have the product LaTeXMLEquation in the notation of ( 2.1.3 ) , and it is induced by LaTeXMLMath .
We have to show that the pull-back of LaTeXMLMath by any dominant morphism LaTeXMLMath does not vanish .
It is enough to show that the restriction of LaTeXMLMath to any nonempty open subvariety LaTeXMLMath of LaTeXMLMath does not vanish by the same argument as in [ 36 ] , 2.6 .
( Indeed , we can restrict to a subvariety of LaTeXMLMath which is finite étale over an open subvariety of LaTeXMLMath , and then take the direct image . )
If both LaTeXMLMath and LaTeXMLMath are nonzero , the assertion is easy , and follows from the same argument as in [ 36 ] , 4.4 .
Indeed , if LaTeXMLMath denotes the dual of LaTeXMLMath , then LaTeXMLMath corresponds to a morphism of mixed Hodge structures LaTeXMLEquation and its image has level LaTeXMLMath by hypothesis .
( Here the level of a Hodge structure is the difference between the maximal and minimal numbers LaTeXMLMath such that LaTeXMLMath . )
This level does not change by replacing LaTeXMLMath with any nonempty open subvariety LaTeXMLMath of LaTeXMLMath , see [ 16 ] .
So we get the assertion in this case .
Now it remains to consider the case LaTeXMLMath and LaTeXMLMath ( hence LaTeXMLMath .
By ( 2.2 ) , LaTeXMLMath is the pull-back of LaTeXMLMath by LaTeXMLMath , and we may replace LaTeXMLMath with LaTeXMLMath .
So the assertion is reduced to the case LaTeXMLMath .
Then by ( 1.2.2 ) it is sufficient to show the nonvanishing of LaTeXMLEquation after replacing LaTeXMLMath with any nonempty open subvariety of LaTeXMLMath .
Let LaTeXMLEquation .
Then LaTeXMLMath comes from an element of LaTeXMLMath , because LaTeXMLMath is pure of weight LaTeXMLMath .
We denote by LaTeXMLMath the extension class associated to the canonical short exact sequence .
Assume LaTeXMLMath .
Using the long exact sequence associated with LaTeXMLEquation this means that LaTeXMLMath coincides with the composition of some LaTeXMLMath with LaTeXMLMath .
We have to show that this implies LaTeXMLMath .
Since LaTeXMLMath is isomorphic to a direct sum of LaTeXMLMath , LaTeXMLMath is identified with a direct sum of LaTeXMLEquation and LaTeXMLMath with LaTeXMLMath , where LaTeXMLMath .
In particular , LaTeXMLMath factors through the inclusion LaTeXMLMath .
On the other hand , LaTeXMLMath factors through LaTeXMLMath .
Thus , by using a direct sum decomposition of LaTeXMLMath compatible with LaTeXMLMath and LaTeXMLMath , the assertion is reduced to the following : 2.7 .
Lemma .
Let LaTeXMLMath be a polarizable Hodge structure of weight LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath , and LaTeXMLMath .
Then LaTeXMLEquation .
Proof .
It is sufficient to show the assertion for the underlying LaTeXMLMath -Hodge structure .
Using the projections to the direct factors , we may assume that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are simple ( i.e .
LaTeXMLMath -dimensional ) , and furthermore LaTeXMLMath is isomorphic to LaTeXMLMath ( because LaTeXMLMath or LaTeXMLMath otherwise ) .
Let LaTeXMLMath be a basis of LaTeXMLMath over LaTeXMLMath such that LaTeXMLMath generates LaTeXMLMath , and LaTeXMLMath is the complex conjugate of LaTeXMLMath .
Then LaTeXMLEquation for LaTeXMLMath .
In particular , LaTeXMLMath is contained in the subspace generated by LaTeXMLMath and LaTeXMLMath .
Thus we get ( 2.7.1 ) .
This completes the proof of Theorem ( 2.6 ) .
2.8 .
Proof of Theorem ( 0.1 ) .
The assertion follows from ( 2.6 ) and ( 2.2 ) .
2.9 .
Examples .
( i ) Let LaTeXMLMath be smooth proper curves over LaTeXMLMath , and assume they are defined over LaTeXMLMath ( i.e .
LaTeXMLMath for a curve LaTeXMLMath over LaTeXMLMath .
Let LaTeXMLMath , and assume that LaTeXMLMath and LaTeXMLMath are defined over LaTeXMLMath , but LaTeXMLMath is not , and that LaTeXMLMath is not torsion in the Jacobian .
Then Theorem ( 2.6 ) implies LaTeXMLEquation .
Note that the condition on LaTeXMLMath is satisfied if there exists an algebraically closed subfield LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath are defined over LaTeXMLMath , but not LaTeXMLMath ( using the embedding of LaTeXMLMath in the Jacobian ) .
Replacing LaTeXMLMath , ( 2.9.1 ) holds if LaTeXMLMath are defined over LaTeXMLMath , and if the field of definition of LaTeXMLMath ( i.e .
the image of LaTeXMLMath to LaTeXMLMath by the morphism defined by LaTeXMLMath and that of LaTeXMLMath are algebraically independent ( i.e .
if there exists an algebraically closed subfield LaTeXMLMath of LaTeXMLMath which contains LaTeXMLMath and such that LaTeXMLMath is defined over LaTeXMLMath , but LaTeXMLMath is not ) .
See [ 1 ] for the case LaTeXMLMath and LaTeXMLMath , where it is assumed that the rank of the Néron-Severi group of LaTeXMLMath is LaTeXMLMath and LaTeXMLMath is not torsion in the Jacobian of LaTeXMLMath ( here LaTeXMLMath is the canonical line bundle ) .
Note that the assertion does not hold for an elliptic curve by the following .
( ii ) Let LaTeXMLMath be an elliptic curve over LaTeXMLMath with the origin LaTeXMLMath .
Then for any LaTeXMLMath LaTeXMLEquation .
This can be verified by taking LaTeXMLMath such that LaTeXMLMath , and using the diagonal and antidiagonal curves passing LaTeXMLMath ( which is viewed as a new origin ) .
3 .
The higher cycle case 3.1 .
For a smooth complex variety LaTeXMLMath and a positive integer LaTeXMLMath , let LaTeXMLMath denote the higher Chow group of LaTeXMLMath with rational coefficients [ 9 ] .
By [ 36 ] we have the refined cycle map LaTeXMLEquation which is compatible with the cycle map in [ 35 ] LaTeXMLEquation .
The latter is expected to be compatible with the one in [ 10 ] , [ 19 ] .
( For LaTeXMLMath , it is easy to verify this by reducing to the case LaTeXMLMath . )
We assume LaTeXMLMath and LaTeXMLMath is smooth proper .
Then LaTeXMLMath and LaTeXMLMath have the Leray filtration LaTeXMLMath as before , and ( 3.1.1 ) induces LaTeXMLEquation .
LaTeXMLEquation because LaTeXMLMath is pure of weight LaTeXMLMath .
This is the inductive limit of LaTeXMLEquation .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath be as in ( 2.1 ) .
In this paper we define the notion of a strictly decomposable higher cycle ( in the strong sense ) by allowing a higher cycle for LaTeXMLMath in the notation of ( 2.1 ) .
More precisely , we say that a higher cycle LaTeXMLMath is strictly decomposable if there exist subfields LaTeXMLMath of LaTeXMLMath and cycles LaTeXMLMath , LaTeXMLMath such that the LaTeXMLMath satisfy the conditions in ( 2.1 ) , LaTeXMLMath , and LaTeXMLMath is the base change of LaTeXMLMath as before .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be as in ( 2.1 ) .
For LaTeXMLMath , we have the Künneth component of the cohomology class LaTeXMLEquation .
Note that LaTeXMLMath in this case .
Let LaTeXMLMath .
Then , applying ( 3.1.4 ) to LaTeXMLMath and forgetting the LaTeXMLMath -structure , we get LaTeXMLEquation .
By the exact sequence ( 1.2.2 ) , it induces LaTeXMLMath , and comes from LaTeXMLMath if LaTeXMLMath .
Thus we can define the notion of a nondegenerate , strictly decomposable higher cycle in the same way as in ( 2.4 ) .
3.2 .
Remark .
We have LaTeXMLMath if LaTeXMLMath , see the proof of ( 3.3 ) below .
3.3 .
Theorem .
The assertion of Theorem ( 2.6 ) holds also for nondegenerate , strictly decomposable higher cycles LaTeXMLMath .
Proof .
The argument is similar to ( 2.6 ) .
We can identify LaTeXMLMath with a morphism of mixed Hodge structures LaTeXMLEquation where the source is the dual of LaTeXMLMath , and the target is pure of weight LaTeXMLMath .
Therefore the image of ( 3.3.1 ) is a direct sum of LaTeXMLMath if LaTeXMLMath , and vanishes otherwise .
For a nonempty open subvariety LaTeXMLMath of LaTeXMLMath , we have the injectivity of the restriction morphism LaTeXMLEquation because LaTeXMLMath for LaTeXMLMath .
Then , taking the tensor with LaTeXMLMath , we get the assertion in the case LaTeXMLMath and LaTeXMLMath ( where we may assume LaTeXMLMath ) .
If LaTeXMLMath and LaTeXMLMath , we may assume LaTeXMLMath , and the assertion is clear .
Thus the assertion is reduced to the case LaTeXMLMath .
The argument is similar if both LaTeXMLMath and LaTeXMLMath are nonzero ( in particular , LaTeXMLMath .
For any nonempty open subset LaTeXMLMath of LaTeXMLMath , we have the injectivity of the restriction morphism LaTeXMLEquation because the local cohomology LaTeXMLMath is pure of type LaTeXMLMath , where LaTeXMLMath .
Thus we get the assertion in this case , and we may assume either LaTeXMLMath or LaTeXMLMath vanishes .
If LaTeXMLMath and LaTeXMLMath ( hence LaTeXMLMath , then the assertion follows immediately from the long exact sequence associated to ( 2.6.2 ) , because LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath ( hence LaTeXMLMath and LaTeXMLMath , then we may assume LaTeXMLMath and LaTeXMLMath is a direct sum of LaTeXMLMath because the image of ( 3.3.1 ) is a direct factor of the target ( and the similar assertion holds over LaTeXMLMath .
Furthermore , it is sufficient to show the nonvanishing of the composition of LaTeXMLMath with the morphism induced by the projection LaTeXMLMath after replacing LaTeXMLMath with any nonempty open subvariety .
Thus the assertion follows from the injectivity of ( 3.3.2 ) .
This finishes the proof of ( 3.3 ) .
3.4 .
Proof of Theorem ( 0.2 ) .
This follows from ( 3.3 ) and ( 2.2 ) .
3.5 .
Remarks .
( i ) Theorems ( 2.6 ) and ( 3.3 ) hold also for a variety LaTeXMLMath over a subfield LaTeXMLMath of LaTeXMLMath , where the LaTeXMLMath are assumed to be subfields of LaTeXMLMath , see Remark ( iii ) of ( 1.5 ) .
( ii ) Let LaTeXMLMath be a curve of positive genus , and assume LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then ( 2.6 ) simplifies Nori ’ s construction explained in [ 38 ] ( because ( 2.6 ) does not assume the condition on the relation to the transcendental part of the second cohomology ) .
So Theorems ( 2.6 ) and ( 3.3 ) may be viewed as a refinement of Nori ’ s construction .
4 .
Application to indecomposable higher cycles 4.1 .
Let LaTeXMLMath be a smooth complex projective variety .
We have a canonical morphism LaTeXMLEquation see [ 22 ] , [ 29 ] .
Its cokernel ( resp .
image ) is denoted by LaTeXMLMath ( resp .
LaTeXMLMath .
An element of this group is called an indecomposable ( resp .
a decomposable ) higher cycle .
Let LaTeXMLMath be the maximal subobject of LaTeXMLMath which is isomorphic to a direct sum of copies of LaTeXMLMath ( i.e .
the subgroup of Hodge cycles ) .
It is a direct factor of LaTeXMLMath by semisimplicity .
The usual cycle map induces the reduced higher Abel-Jacobi map LaTeXMLEquation since the image of LaTeXMLMath is contained in LaTeXMLMath see loc .
cit .
4.2 .
Theorem .
With the notation of ( 3.1 ) and ( 4.1 ) , assume LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and the image of LaTeXMLMath by ( 4.1.2 ) for LaTeXMLMath is nonzero .
Then LaTeXMLMath in LaTeXMLMath .
Proof .
Assume LaTeXMLMath is decomposable , i.e .
LaTeXMLMath coincides with the image of LaTeXMLMath where LaTeXMLMath and LaTeXMLMath .
Here we may assume that the LaTeXMLMath are linearly independent over LaTeXMLMath in LaTeXMLMath , and hence LaTeXMLMath , because the image of LaTeXMLMath by the usual cycle map ( 3.1.2 ) vanishes due to the assumption LaTeXMLMath ( indeed , the last assumption implies LaTeXMLMath in ( 3.3 ) ) .
To simplify the notation , let LaTeXMLMath and LaTeXMLMath .
We may assume that the LaTeXMLMath and LaTeXMLMath are defined over LaTeXMLMath replacing LaTeXMLMath if necessary .
We denote by LaTeXMLMath the base change of LaTeXMLMath by LaTeXMLMath as before , and similarly for LaTeXMLMath etc .
We will induce a contradiction by showing that the image of LaTeXMLMath by ( 4.1.2 ) vanishes .
Since LaTeXMLMath , we see that the image of LaTeXMLMath in LaTeXMLMath vanishes , and so does that of LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath as before .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) denote the image of LaTeXMLMath ( resp .
LaTeXMLMath ) in LaTeXMLEquation using the cycle map ( 3.1.2 ) together with the Künneth decomposition .
We choose direct sum decompositions ( see ( 1.1.2 ) ) LaTeXMLEquation .
Applying this to the image of LaTeXMLMath by the cycle map to LaTeXMLMath , and restricting to the relevant Künneth component , we get LaTeXMLEquation .
LaTeXMLEquation because LaTeXMLMath .
We see that the other Künneth components do not contribute to ( 4.2.1 ) , because the LaTeXMLMath , which are viewed as sections of LaTeXMLMath , induce LaTeXMLEquation by using ( 4.2.2 ) .
Thus LaTeXMLEquation .
On the other hand , LaTeXMLMath with the notation of ( 3.2 ) .
We have a direct sum decomposition LaTeXMLEquation such that LaTeXMLMath is a direct sum of copies of LaTeXMLMath , and LaTeXMLEquation because the composition LaTeXMLMath splits by semisimplicity .
Clearly LaTeXMLMath , and LaTeXMLMath come respectively from LaTeXMLEquation .
LaTeXMLEquation because LaTeXMLMath .
So we get LaTeXMLMath by the equality LaTeXMLMath together with the decomposition ( 4.2.3 ) .
Furthermore , we may replace LaTeXMLMath with LaTeXMLMath , using the long exact sequence associated to ( 2.6.2 ) as in the proof of ( 3.3 ) ( because LaTeXMLMath .
Thus the image of LaTeXMLMath by ( 4.1.2 ) vanishes by the following lemma which we apply to LaTeXMLEquation .
LaTeXMLEquation 4.3 .
Lemma .
Let LaTeXMLMath be a polarizable Hodge structure of weight LaTeXMLMath for LaTeXMLMath .
Assume LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath for LaTeXMLMath .
If LaTeXMLMath in LaTeXMLMath , then LaTeXMLMath .
Proof .
Since LaTeXMLMath is a direct factor of LaTeXMLMath , we may assume LaTeXMLMath by using the projection to the direct factor LaTeXMLMath .
Then LaTeXMLMath , and the assertion is clear .
This completes the proof of ( 4.2 ) .
4.4 .
Proof of Theorem ( 0.3 ) .
The last assertion of ( 0.3 ) follows from ( 4.2 ) combined with ( 2.2 ) .
If LaTeXMLMath is countable , there are countably many LaTeXMLMath such that for any LaTeXMLMath , there exists LaTeXMLMath such that the images of LaTeXMLMath and LaTeXMLMath in LaTeXMLMath coincide .
We may assume LaTeXMLMath by replacing LaTeXMLMath with a countably generated subfield of LaTeXMLMath .
Take any LaTeXMLMath , and let LaTeXMLMath such that the image of LaTeXMLMath in LaTeXMLMath is zero .
But this contradicts the above assertion , because LaTeXMLMath .
This terminates the proof of ( 0.3 ) .
4.5 .
Remark .
It is known that LaTeXMLMath is uncountable in some cases , see e.g .
[ 13 ] for the case of the Jacobian of a curve , and also [ 14 ] , [ 35 ] for examples such that the hypothesis of ( 0.3 ) is satisfied .
( We can show that a higher cycle on a product of elliptic curves in [ 22 ] is decomposable by restricting to the generic fiber of the first projection . )
References [ 1 ] Asakura , M. , Motives and algebraic de Rham cohomology , in : The arithmetic and geometry of algebraic cycles ( Banff ) , CRM Proc .
Lect .
Notes , 24 , AMS , 2000 , pp .
133–154 .
[ 2 ] Beilinson , A. , Higher regulators and values of LaTeXMLMath -functions , J. Soviet Math .
30 ( 1985 ) , 2036–2070 .
[ 3 ] , Notes on absolute Hodge cohomology , Contemporary Math .
55 ( 1986 ) 35–68 .
[ 4 ] , Height pairing between algebraic cycles , Lect .
Notes in Math. , vol .
1289 , Springer , Berlin , 1987 , pp .
1–26 .
[ 5 ] , On the derived category of perverse sheaves , ibid .
pp .
27–41 .
[ 6 ] Beilinson , A. , Bernstein , J. and Deligne , P. , Faisceaux pervers , Astérisque , vol .
100 , Soc .
Math .
France , Paris , 1982 .
[ 7 ] Bloch , S. , Lectures on algebraic cycles , Duke University Mathematical series 4 , Durham , 1980 .
[ 8 ] , Algebraic cycles and values of LaTeXMLMath -functions , J. Reine Angew .
Math .
350 ( 1984 ) , 94–108 .
[ 9 ] , Algebraic cycles and higher LaTeXMLMath -theory , Advances in Math. , 61 ( 1986 ) , 267–304 .
[ 10 ] , Algebraic cycles and the Beilinson conjectures , Contemporary Math .
58 ( 1 ) ( 1986 ) , 65–79 .
[ 11 ] Bloch , S. , Kas , A. and Lieberman , D. , Zero cycles on surfaces with LaTeXMLMath , Compos .
Math .
33 ( 1976 ) , 135–145 .
[ 12 ] Carlson , J. , Extensions of mixed Hodge structures , in : Journées de Géométrie Algébrique d ’ Angers 1979 , Sijthoff-Noordhoff Alphen a/d Rijn , 1980 , pp .
107–128 .
[ 13 ] Collino , A. and Fakhruddin , N. , Indecomposable higher Chow cycles on Jacobians , preprint .
[ 14 ] del Angel , P. and Müller-Stach , S. , The transcendental part of the regulator map for LaTeXMLMath on a mirror family of LaTeXMLMath surfaces , preprint .
[ 15 ] Deligne , P. , Equations Différentielles à Points Singuliers Réguliers , Lect .
Notes in Math .
vol .
163 , Springer , Berlin , 1970 .
[ 16 ] , Théorie de Hodge I , Actes Congrès Intern .
Math. , 1970 , vol .
1 , 425-430 ; II , Publ .
Math .
IHES , 40 ( 1971 ) , 5–57 ; III ibid. , 44 ( 1974 ) , 5–77 .
[ 17 ] , Valeurs de fonctions LaTeXMLMath et périodes d ’ intégrales , Proc .
Symp .
in pure Math. , 33 ( 1979 ) part 2 , pp .
313–346 .
[ 18 ] Deligne , P. , Milne , J. , Ogus , A. and Shih , K. , Hodge Cycles , Motives , and Shimura varieties , Lect .
Notes in Math. , vol 900 , Springer , Berlin , 1982 .
[ 19 ] Deninger , C. and Scholl , A. , The Beilinson conjectures , Proceedings Cambridge Math .
Soc .
( eds .
Coats and Taylor ) 153 ( 1992 ) , 173–209 .
[ 20 ] El Zein , F. and Zucker , S. , Extendability of normal functions associated to algebraic cycles , in : Topics in transcendental algebraic geometry , Ann .
Math .
Stud. , 106 , Princeton Univ .
Press , Princeton , N.J. , 1984 , pp .
269–288 .
[ 21 ] Esnault , H. and Viehweg , E. , Deligne-Beilinson cohomology , in : Beilinson ’ s conjectures on Special Values of LaTeXMLMath -functions , Academic Press , Boston , 1988 , pp .
43–92 .
[ 22 ] Gordon , B. and Lewis , J. , Indecomposable higher Chow cycles on products of elliptic curves , J. Alg .
Geom .
8 ( 1999 ) , 543–567 .
[ 23 ] Green , M. , Griffiths ’ infinitesimal invariant and the Abel-Jacobi map , J. Diff .
Geom .
29 ( 1989 ) , 545–555 .
[ 24 ] Griffiths , P. , On the period of certain rational integrals I , II , Ann .
Math .
90 ( 1969 ) , 460–541 .
[ 25 ] Jannsen , U. , Deligne homology , Hodge LaTeXMLMath -conjecture , and motives , in Beilinson ’ s conjectures on Special Values of LaTeXMLMath -functions , Academic Press , Boston , 1988 , pp .
305–372 .
[ 26 ] , Mixed motives and algebraic LaTeXMLMath -theory , Lect .
Notes in Math. , vol .
1400 , Springer , Berlin , 1990 .
[ 27 ] , Motivic sheaves and filtrations on Chow groups , Proc .
Symp .
Pure Math .
55 ( 1994 ) , Part 1 , pp .
245–302 .
[ 28 ] Lewis , J.D. , A filtration on the Chow groups of a complex projective variety , Compos .
Math .
128 ( 2001 ) , 299–322 .
[ 29 ] Müller-Stach , S. , Constructing indecomposable motivic cohomology classes on algebraic surfaces , J. Alg .
Geom .
6 ( 1997 ) , 513–543 .
[ 30 ] Mumford , D. , Rational equivalence of LaTeXMLMath -cycles on surfaces , J .
Math .
Kyoto Univ .
9 ( 1969 ) , 195–204 .
[ 31 ] Murre , J.P. , On the motive of an algebraic surface , J. Reine Angew .
Math .
409 ( 1990 ) , 190–204 .
[ 32 ] Rosenschon , A. , A Note on arithmetic mixed Hodge structures , preprint 2001 .
[ 33 ] Saito , M. , Mixed Hodge Modules , Publ .
RIMS , Kyoto Univ. , 26 ( 1990 ) , 221–333 .
[ 34 ] , On the formalism of mixed sheaves , RIMS-preprint 784 , Aug. 1991 .
[ 35 ] , Bloch ’ s conjecture , Deligne cohomology and higher Chow groups , preprint RIMS-1284 ( or math.AG/9910113 ) .
[ 36 ] , Arithmetic mixed sheaves , Inv .
Math .
144 ( 2001 ) , 533–569 .
[ 37 ] , Refined cycle maps , preprint ( math.AG/0103116 ) .
[ 38 ] Schoen , C. , Zero cycles modulo rational equivalence for some varieties over fields of transcendence degree one , Proc .
Symp .
Pure Math .
46 ( 1987 ) , part 2 , pp .
463–473 .
[ 39 ] , On certain exterior product maps of Chow groups , Math .
Res .
Let .
7 ( 2000 ) , 177–194 , [ 40 ] Voisin , C. , Variations de structures de Hodge et zéro-cycles sur les surfaces générales , Math .
Ann .
299 ( 1994 ) , 77–103 .
[ 41 ] , Transcendental methods in the study of algebraic cycles , Lect .
Notes in Math .
vol .
1594 , pp .
153–222 .
[ 42 ] , Remarks on zero-cycles of self-products of varieties , in : Moduli of Vector Bundles , Lect .
Notes in Pure and Applied Mathematics , vol .
179 , M. Dekker , New York , 1996 , pp .
265–285 .
Andreas Rosenschon Department of Mathematics Duke University Durham , NC 27708 U.S.A E-Mail : axrmath.duke.edu Morihiko Saito RIMS Kyoto University , Kyoto 606–8502 Japan E-Mail : msaitokurims.kyoto-u.ac.jp Nov. 12 , 2001 , v.4
These notes hopefully provide an aid to the comprehension of the Connes-Moscovici and Connes-Kreimer works , by isolating common mathematical features of the Connes-Moscovici , rooted trees , and Feynman-graph Hopf algebras ( as a new special branch of the theory of Hopf algebras expected to become important ) .
We discuss in particular the dual Milnor-Moore situation .
definition Dedicated to Sergio Doplicher and John Roberts In what follows LaTeXMLMath is a commutative field of zero characteristic ( e.g .
LaTeXMLMath .
( the algebraïc dual of Hopf algebras ) .
Let LaTeXMLMath be a Hopf algebra over LaTeXMLMath .
Consider the algebraic dual LaTeXMLMath of LaTeXMLMath endowed with the topology of simple convergence on LaTeXMLMath .
the LaTeXMLMath -topology .
The value of LaTeXMLMath for LaTeXMLMath is denoted by LaTeXMLMath .
Then : As the dual of the coalgebra LaTeXMLMath is a genuine algebra with the following product LaTeXMLMath because it is the convolution product of End LaTeXMLMath , cf .
definition ( 2,1 ) in LaTeXMLCite , namely , denoting the covalue LaTeXMLMath by LaTeXMLMath : LaTeXMLMath , where we can omit LaTeXMLMath owing to LaTeXMLMath .
LaTeXMLMath and unit LaTeXMLMath : LaTeXMLEquation .
LaTeXMLMath is however not a Hopf algebra : we do have LaTeXMLMath and LaTeXMLMath : LaTeXMLEquation but in general LaTeXMLMath .
We now investigate two subsets of LaTeXMLMath which LaTeXMLMath maps into LaTeXMLMath : The characters and infinitesimal characters of LaTeXMLMath : Let LaTeXMLMath be a Hopf LaTeXMLMath -algebra .
a character of LaTeXMLMath is a non vanishing LaTeXMLMath -linear map LaTeXMLMath which is multiplicative : LaTeXMLEquation ( note that , as a consequence , one has LaTeXMLEquation indeed ( LaTeXMLRef ) entails that LaTeXMLMath vanishes , whence ( LaTeXMLRef ) if LaTeXMLMath does not vanish ) .
We denote by Char LaTeXMLMath the set of characters of LaTeXMLMath .
Note that the counit is a character of LaTeXMLMath .
An infinitesimal character of LaTeXMLMath is a LaTeXMLMath -linear map LaTeXMLMath fulfilling : LaTeXMLRef ) defines LaTeXMLMath as a derivation of the LaTeXMLMath -bimodule LaTeXMLMath consisting of LaTeXMLMath with multiplication by LaTeXMLMath from the left and from the right multiplication by LaTeXMLMath : LaTeXMLMath ) LaTeXMLEquation ( note that , as a consequence , one has LaTeXMLEquation indeed one has LaTeXMLEquation and that LaTeXMLMath vanishes on the square LaTeXMLMath of the augmentation ideal LaTeXMLMath Ker LaTeXMLMath ) .
Note also that ( LaTeXMLRef ) is the infinitesimal form of ( LaTeXMLRef ) written for LaTeXMLMath with LaTeXMLMath considered as small ) .
We denote by LaTeXMLMath Char LaTeXMLMath the set of infinitesimal characters of LaTeXMLMath .
With the definitions in LaTeXMLRef we have that : Equipped with the convolution product ( LaTeXMLRef ) , the unit LaTeXMLMath : LaTeXMLEquation the inverse : LaTeXMLEquation and the LaTeXMLMath -topology , LaTeXMLMath Char LaTeXMLMath is a topological group , the group of characters of LaTeXMLMath .
LaTeXMLMath with product LaTeXMLMath .
LaTeXMLMath is dually definable as the group of “ group-like elements ” of LaTeXMLMath : LaTeXMLMath is not a Hopf algebra cf .
( ii ) .
LaTeXMLEquation .
Equipped with the bracket : LaTeXMLEquation where LaTeXMLMath is the convolution product ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLMath Char LaTeXMLMath is a Lie algebra , the Lie algebra of infinitesimal characters of LaTeXMLMath .
LaTeXMLMath associated with the algebra LaTeXMLMath .
LaTeXMLMath is dually definable as the Lie algebra of “ primitive elements ” of LaTeXMLMath : LaTeXMLMath is not a Hopf algebra cf .
( ii ) .
LaTeXMLEquation .
The product first line ( LaTeXMLRef ) is known to be associative .
To assert that LaTeXMLMath Char LaTeXMLMath we need to check multiplicativity : now , for LaTeXMLMath : LaTeXMLEquation where we used the multiplicativity of LaTeXMLMath due to the commutativity of LaTeXMLMath : indeed , one has , for LaTeXMLMath : LaTeXMLEquation .
Check that LaTeXMLMath : we have by the first line ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation Check of ( LaTeXMLRef ) we have : LaTeXMLEquation .
LaTeXMLEquation The facts that product and inverse are continuous in the LaTeXMLMath -topology is clear .
( ia ) For the proof we write LaTeXMLMath Char LaTeXMLMath and LaTeXMLMath .
Check of LaTeXMLMath : for LaTeXMLMath , LaTeXMLMath , we have : LaTeXMLEquation .
Check of LaTeXMLMath : LaTeXMLMath implies : LaTeXMLEquation hence we have LaTeXMLMath with LaTeXMLMath .
The Lie algebra property of LaTeXMLMath can be checked directly from ( LaTeXMLRef ) , but will immediately result from ( iia ) , which we now check .
( iia ) For the proof we write LaTeXMLMath Char LaTeXMLMath and LaTeXMLMath .
Check of LaTeXMLMath : LaTeXMLMath implies for LaTeXMLMath : LaTeXMLEquation .
Check of LaTeXMLMath : for LaTeXMLMath , LaTeXMLMath one has , LaTeXMLEquation hence we have LaTeXMLMath with LaTeXMLMath .
Check of ( ii ) : it is clear that LaTeXMLMath is a Lie-subalgebra of the Lie algebra Lie LaTeXMLMath .
Let LaTeXMLMath be the subalgebra of LaTeXMLMath generated by LaTeXMLMath and LaTeXMLMath .
One has LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath vanishes on LaTeXMLMath .
One has LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
One has LaTeXMLMath and LaTeXMLMath is multiplicative : LaTeXMLEquation and cocommutative in the sense : LaTeXMLEquation ( o ) One has LaTeXMLMath , for LaTeXMLMath , LaTeXMLMath .
For LaTeXMLMath one has LaTeXMLMath .
( i ) We first notice the property LaTeXMLEquation transposed of the known property LaTeXMLMath .
Check of LaTeXMLMath : for LaTeXMLMath one has LaTeXMLMath , indeed one has : LaTeXMLEquation .
It follows that LaTeXMLMath .
Now LaTeXMLMath turns the unit LaTeXMLMath into LaTeXMLMath .
And , for LaTeXMLMath and LaTeXMLMath , one has using the known property LaTeXMLMath : LaTeXMLMath : LaTeXMLEquation ( ii ) ( LaTeXMLRef ) reads LaTeXMLMath , transposed of the axiom LaTeXMLMath .
Check of the cocommutativity property ( LaTeXMLRef ) : holds obviously on the generators LaTeXMLMath and LaTeXMLMath .
Now ( LaTeXMLRef ) propagates multiplicatively : if LaTeXMLMath fulfill LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , owing to multiplicatively of LaTeXMLMath .
LaTeXMLMath is a cocommutative Hopf algebra .
In the graded connected case isomorphic to the enveloping algebra LaTeXMLMath of the Lie algebra LaTeXMLMath of its primitive elements ( special case of the Milnor-Moore Theorem ) .
The Hopf algebras LaTeXMLMath and LaTeXMLMath are set in duality through the bilinear form LaTeXMLMath .
duality generally not separating LaTeXMLMath – certainly not for LaTeXMLMath non-commutative. , separating however in the case of CKM-Hopf algebras , cf .
LaTeXMLRef .
LaTeXMLMath is a unital algebra with unit LaTeXMLMath by definition .
Using the terminology of Appendix A , we have that the coalgebra axioms LaTeXMLMath , LaTeXMLMath for LaTeXMLMath result by transposition from the algebra axioms LaTeXMLMath , resp .
LaTeXMLMath for LaTeXMLMath .
Bialgebra axioms for LaTeXMLMath : we already met LaTeXMLMath in ( LaTeXMLRef ) , LaTeXMLMath is the known fact that LaTeXMLMath , we have met LaTeXMLMath and LaTeXMLMath in Definition-Lemma LaTeXMLRef ( o ) above .
Finally the axiom LaTeXMLMath for LaTeXMLMath and LaTeXMLMath are transposed of each other .
With LaTeXMLMath a LaTeXMLMath -vector space direct sum of LaTeXMLMath -vector subspaces : LaTeXMLEquation we let LaTeXMLMath , LaTeXMLMath , resp .
LaTeXMLMath , be the LaTeXMLMath -linear maps : LaTeXMLMath respectively specified by : LaTeXMLEquation .
LaTeXMLEquation and set , for LaTeXMLMath homogeneous : LaTeXMLMath is homogeneous whenever LaTeXMLMath ( LaTeXMLMath LaTeXMLMath is an eigenvector of LaTeXMLMath – then with eigenvalue LaTeXMLMath ) .
LaTeXMLEquation ( LaTeXMLMath -graded algebras , coalgebras and bialgebras ) .
( i ) The LaTeXMLMath -algebra LaTeXMLMath is LaTeXMLMath - graded ( with algebra LaTeXMLMath - grading LaTeXMLMath ) whenever one of the following equivalent requirements ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) or ( LaTeXMLRef ) prevails : LaTeXMLEquation .
LaTeXMLEquation ( saying that LaTeXMLMath is a one-parameter automorphism group of the algebra LaTeXMLMath , and implying LaTeXMLMath , LaTeXMLMath ) LaTeXMLEquation ( saying that LaTeXMLMath is a derivation of the algebra LaTeXMLMath , and implying LaTeXMLMath ) , LaTeXMLEquation for LaTeXMLMath homogeneous , which implies LaTeXMLMath homogeneous .
( ii ) The LaTeXMLMath -coalgebra LaTeXMLMath LaTeXMLMath is LaTeXMLMath - graded ( with coalgebra LaTeXMLMath - grading LaTeXMLMath ) whenever one of the following equivalent requirements ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , or ( LaTeXMLRef ) prevails : Note that ( LaTeXMLRef ) obviously implies LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLEquation ( saying that LaTeXMLMath is a one-parameter automorphism group of the coalgebra LaTeXMLMath , and implying LaTeXMLMath , LaTeXMLMath ) LaTeXMLEquation ( saying that LaTeXMLMath is a coalgebra coderivation , and implying LaTeXMLMath , LaTeXMLMath ) LaTeXMLEquation ( iii ) The LaTeXMLMath -bialgebra LaTeXMLMath is LaTeXMLMath - graded ( with bialgebra LaTeXMLMath - grading LaTeXMLMath ) whenever it is LaTeXMLMath -graded both as an algebra and as a coalgebra .
We say that LaTeXMLMath is LaTeXMLMath - graded connected whenever one has in addition : LaTeXMLEquation ( iv ) A Hopf algebra LaTeXMLMath is called LaTeXMLMath - graded whenever it is LaTeXMLMath -graded as a bialgebra ( cf .
Definition-Lemma LaTeXMLRef ( i ) ) .
LaTeXMLMath then automatically commutes with LaTeXMLMath , LaTeXMLMath , and with LaTeXMLMath : LaTeXMLEquation .
For the LaTeXMLMath - graded algebra ( resp .
coalgebra , bialgebra ) LaTeXMLMath we refer to LaTeXMLMath as the automorphism group , to LaTeXMLMath as the derivation ( resp .
coderivation , biderivation ) , and to LaTeXMLMath as the degree , of the LaTeXMLMath - grading .
( i ) ( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) : LaTeXMLMath : it suffices to take LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath , then : LaTeXMLEquation .
LaTeXMLMath : for LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation thus LaTeXMLMath .
( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) : LaTeXMLMath : it suffices to take LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLMath : for LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation thus LaTeXMLMath .
( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) : LaTeXMLMath , whilst LaTeXMLEquation ( ii ) ( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) : LaTeXMLMath : it suffices to take LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLMath : for LaTeXMLMath , LaTeXMLEquation implying LaTeXMLMath .
LaTeXMLMath such that LaTeXMLMath , resp .
the LaTeXMLMath , are linearly independent .
( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) : LaTeXMLMath : it suffices to take LaTeXMLMath , then : LaTeXMLEquation .
LaTeXMLMath : for LaTeXMLMath , LaTeXMLEquation implying LaTeXMLMath as above .
( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) : LaTeXMLMath , whilst LaTeXMLEquation ( iv ) Check of the first line ( LaTeXMLRef ) : we check that LaTeXMLMath is a convolution inverse of LaTeXMLMath : LaTeXMLEquation where we used ( LaTeXMLRef ) , ( LaTeXMLRef ) , LaTeXMLMath , and LaTeXMLMath .
Check of the second line ( LaTeXMLRef ) : equate the derivations at 0 of LaTeXMLMath .
In practice recognizing a biderivation often occurs by first establishing the derivation property , and afterwards the coderivation property , using Assume that one knows that a bialgebra LaTeXMLMath is LaTeXMLMath -graded as an algebra .
For investigating the possible coderivation property of LaTeXMLMath one needs only look at the restriction of LaTeXMLMath on a set of generators .
From LaTeXMLEquation .
LaTeXMLMath homogeneous , we conclude for the product LaTeXMLMath with coproduct LaTeXMLMath that LaTeXMLMath equals LaTeXMLEquation .
It is instructive to check instead the property ( LaTeXMLRef ) , showing that LaTeXMLEquation .
We next show that the algebraïc dual of a LaTeXMLMath -graded coalgebra is a LaTeXMLMath -graded algebra .
( the dual N-grading ) .
With LaTeXMLMath , LaTeXMLMath , a LaTeXMLMath - graded bialgebra , let LaTeXMLMath be the algebraïc dual of LaTeXMLMath endowed with the algebra-structure dual of the coalgebra-structure of LaTeXMLMath , thus with product : Here the convolution product is that of End LaTeXMLMath , and LaTeXMLMath is the duality bilinear form , cf .
( LaTeXMLRef ) LaTeXMLEquation .
The definitions : LaTeXMLEquation then yield a LaTeXMLMath -grading LaTeXMLMath of the dual algebra LaTeXMLMath where LaTeXMLMath .
One has the inclusions LaTeXMLMath “ LaTeXMLMath ” LaTeXMLMath “ LaTeXMLMath ” and LaTeXMLMath “ LaTeXMLMath ” LaTeXMLMath “ LaTeXMLMath ” , LaTeXMLMath .
We define “ LaTeXMLMath ” LaTeXMLMath , “ LaTeXMLMath ” LaTeXMLMath ( written with quotation marks because LaTeXMLMath is not a bona-fide coalgebra – this notation becomes bona-fide in LaTeXMLMath , cf .
Section LaTeXMLRef ) .
LaTeXMLMath thus becomes a graded bialgebra .
We check that one has LaTeXMLMath for LaTeXMLMath .
With LaTeXMLMath the decomposition of the unit associated to the direct sum LaTeXMLMath , we have LaTeXMLMath , thus for LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation hence LaTeXMLMath .
The fact that ( LaTeXMLRef ) and ( LaTeXMLRef ) are transposed of each other then implies that LaTeXMLMath is a derivation of the algebra LaTeXMLMath .
The proof of LaTeXMLMath for LaTeXMLMath is analogous .
Check of the two last claims : – for LaTeXMLMath “ LaTeXMLMath ” : LaTeXMLEquation – for LaTeXMLMath “ LaTeXMLMath ” and LaTeXMLMath : LaTeXMLEquation ( LaTeXMLMath -graded connected bialgebras . )
Let LaTeXMLEquation be a LaTeXMLMath -graded connected bialgebra ( i.e .
LaTeXMLMath ) and let LaTeXMLMath Ker LaTeXMLMath ( called the augmentation ideal – it is known that LaTeXMLMath cf .
Proposition LaTeXMLRef ) .
Then : ( i ) one has : LaTeXMLEquation ( ii ) one has the implication : LaTeXMLEquation with LaTeXMLMath a Sweedler-type sum of terms fulfilling either ( and thus both ) of the equivalent requirements : all LaTeXMLMath , LaTeXMLMath are strictly smaller than LaTeXMLMath .
Note that the property LaTeXMLMath of the Hopf LaTeXMLMath -grading merely entails that LaTeXMLMath , LaTeXMLMath , whilst progressivity requires strict inequalities .
Note also that applying LaTeXMLMath and LaTeXMLMath to both sides of ( LaTeXMLRef ) one proves LaTeXMLMath , hence that all LaTeXMLMath and LaTeXMLMath belong to LaTeXMLMath .
all LaTeXMLMath , LaTeXMLMath are strictly positive .
( Equivalence of ( a ) and ( b ) : since the bialgebra LaTeXMLMath -grading entails that one has LaTeXMLEquation and LaTeXMLEquation are simultaneously strictly positive ) .
The consequence ( LaTeXMLRef ) of LaTeXMLMath -graded connectedness plays a prominent technical role both in the Connes-Moscovici index theory and in the Connes-Kreimer theory of renormalization .
LaTeXMLRef .
We therefore cast the special name of progressiveness for this property , LaTeXMLRef ( ii ) above then says that a LaTeXMLMath -graded connected bialgebra is progressive .
( i ) Let LaTeXMLMath , LaTeXMLMath : by ( LaTeXMLRef ) we have LaTeXMLEquation .
Hence : LaTeXMLEquation implying LaTeXMLMath .
We proved that LaTeXMLMath Ker LaTeXMLMath .
( In fact we found that : LaTeXMLEquation first subscript indicating the LaTeXMLMath -grade ) .
Conversely let LaTeXMLMath Ker LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath .
By what precedes we have LaTeXMLMath , hence LaTeXMLMath , proving that LaTeXMLMath .
( ii ) For LaTeXMLMath , ( LaTeXMLRef ) implies ( LaTeXMLRef ) .
However ( LaTeXMLRef ) is linear w.r.t .
LaTeXMLMath , hence holds for LaTeXMLMath by ( i ) .
( i ) Progressiveness at large follows from progressiveness for generators of LaTeXMLMath .
Indeed : ( ii ) Let LaTeXMLMath : if ( LaTeXMLRef ) holds for LaTeXMLMath and LaTeXMLMath , it holds for the product LaTeXMLMath .
( i ) follows from ( ii ) and the multiplicativity of LaTeXMLMath .
( ii ) assume that LaTeXMLMath with all LaTeXMLMath , LaTeXMLMath strictly positive , and LaTeXMLMath with all LaTeXMLMath , LaTeXMLMath strictly positive .
It follows that LaTeXMLEquation where all the tensor products but the two first have both factors of strictly positive degree .
We now present two important consequences of progressiveness .
The first is that it makes automatic the existence of a ( recursively defined ) antipode : Assume that the LaTeXMLMath -graded bialgebra LaTeXMLMath is LaTeXMLMath -graded connected ( thus progressive ) .
Then LaTeXMLMath is a Hopf algebra .
The requirements , for the LaTeXMLMath -linear LaTeXMLMath , resp .
LaTeXMLMath : LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation resp .
LaTeXMLEquation .
LaTeXMLEquation determine LaTeXMLMath and LaTeXMLMath by induction w.r.t .
the degree LaTeXMLMath as respective right- and left convolution inverses of id LaTeXMLMath for the convolution product of End LaTeXMLMath : but these then coincide because LaTeXMLMath : thus LaTeXMLMath is a bilateral convolution inverse of id LaTeXMLMath , thus the ( unique ) antipode .
The second consequence of progressiveness of LaTeXMLMath is that , given LaTeXMLMath , there is a drastic limitation ( depending on LaTeXMLMath ) of non-vanishing values LaTeXMLMath , LaTeXMLMath .
We recall that ( cf .
Section LaTeXMLRef ) : – the algebraïc dual LaTeXMLMath of LaTeXMLMath becomes a topological algebra if equipped with the LaTeXMLMath -topology and the algebra-structure ( LaTeXMLMath , LaTeXMLMath ) stemming from the coalgebra-structure of LaTeXMLMath , – the subalgebra LaTeXMLMath generated in LaTeXMLMath by the unit LaTeXMLMath and the Lie-algebra LaTeXMLMath of infinitesimal characters becomes by transposition a Hopf algebra LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Let the bialgebra LaTeXMLMath be LaTeXMLMath -graded connected ( thus progressive ) hence a Hopf algebra , cf .
Proposition LaTeXMLRef and let LaTeXMLMath be the projection on LaTeXMLMath parallel to LaTeXMLMath in LaTeXMLMath .
One has LaTeXMLMath , cf .
( LaTeXMLRef ) , and LaTeXMLMath , cf .
Proposition LaTeXMLRef .
Correlatively LaTeXMLMath projects LaTeXMLMath onto LaTeXMLMath .
Then : ( i ) One has for LaTeXMLMath the implication : LaTeXMLEquation ( ii ) One has consequently for LaTeXMLMath the implication : LaTeXMLEquation alternatively formulated as : LaTeXMLEquation where LaTeXMLMath indicates the annihilator in the dual ) .
( i ) Since by ( LaTeXMLRef ) a general element of LaTeXMLMath is of the form LaTeXMLMath with LaTeXMLMath , and we have obviously LaTeXMLMath , it is no restriction for the proof of the implication ( LaTeXMLRef ) to assume that LaTeXMLMath belongs to LaTeXMLMath .The projection LaTeXMLMath applied to LaTeXMLMath then suppresses all the products LaTeXMLMath containing at least one factor in LaTeXMLMath : the remaining products LaTeXMLMath stem from the successive Sweedler-type summations LaTeXMLMath in ( LaTeXMLRef ) , thus have by progressiveness all their factors LaTeXMLMath such that LaTeXMLMath ( observe , as noticed in footnote 6 , that all LaTeXMLMath belong to LaTeXMLMath ) .
The equality LaTeXMLEquation requires LaTeXMLMath , otherwise LaTeXMLMath vanishes , whence ( LaTeXMLRef ) and ( LaTeXMLRef ) for LaTeXMLMath , the extension to LaTeXMLMath being trivial .
( ii ) The expression of the n-fold convolution product of LaTeXMLMath : LaTeXMLEquation yields for products of elements of LaTeXMLMath vanishing on LaTeXMLMath : LaTeXMLEquation ( LaTeXMLRef ) thus follows from ( LaTeXMLRef ) .
Note that ( LaTeXMLRef ) holds for LaTeXMLMath ( obvious directly - embodying this into our proof would require the convention LaTeXMLMath ) .
Assume that the LaTeXMLMath -graded connected bialgebra LaTeXMLMath has a countable vector space-basis LaTeXMLMath consisting of homogeneous elements of non-decreasing degree , and let LaTeXMLMath be the closure of LaTeXMLMath in LaTeXMLMath .
Then : ( i ) LaTeXMLMath is metrizable with the distance LaTeXMLMath is a translation-invariant distance : one has LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath implies LaTeXMLMath .
These properties follow from the properties LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath plus the implication LaTeXMLMath for all LaTeXMLMath .
The equality of topologies arises as follows : the LaTeXMLMath are a basis of neighbourhoods of zero in the LaTeXMLMath -topology , with LaTeXMLMath LaTeXMLEquation ( ii ) LaTeXMLMath is a closed topological subalgebra of LaTeXMLMath .
( iii ) LaTeXMLMath is stable under LaTeXMLMath acting in LaTeXMLMath .
LaTeXMLMath Note that at this point the question of whether LaTeXMLMath , and whether , if so , LaTeXMLEquation is a Hopf algebra , remains open .
( i ) is the known device for proving metrizability of a topological vector-space whose topology is specified by a countable set of semi-norms .
One checks all the properties stated in footnote 9 .
( ii ) follows from the known fact that the closure of a topological algebra is an algebra .
( iii ) Check of LaTeXMLMath : let LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , we have , for LaTeXMLMath : LaTeXMLEquation hence LaTeXMLMath , hence LaTeXMLMath .
Assuming that the LaTeXMLMath -graded connected bialgebra LaTeXMLMath has a countable vector space-basis LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath be the closure of LaTeXMLMath in LaTeXMLMath , we have that : ( i ) with LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , the sequence LaTeXMLMath is a Cauchy-sequence of LaTeXMLMath which converges for LaTeXMLMath towards an element LaTeXMLMath .
( ia ) the same holds for multi-variable power-series of elements of LaTeXMLMath .
( ii ) in particular the Mac-Laurin power-series of LaTeXMLMath , LaTeXMLMath , yields an element , denoted by LaTeXMLMath , of LaTeXMLMath which is a character of LaTeXMLMath .
LaTeXMLMath Note that at this point the question of whether the LaTeXMLMath , LaTeXMLMath , generate the group LaTeXMLMath of characters of LaTeXMLMath remains open .
( i ) we have from ( LaTeXMLRef ) , with LaTeXMLMath : LaTeXMLEquation whence our claim since LaTeXMLMath for LaTeXMLMath .
( ia ) the proof is the same up to notation .
( ii ) Particular case of what precedes .
Now the exponential of an infinitesimal character , if it makes sense , is a character .
We cast a special name for a type of Hopf algebras playing a prevalent role in index theory and in renormalized quantum field theory .
In most of what follows LaTeXMLMath could be replaced by any commutative field LaTeXMLMath of characteristics 0 .
LaTeXMLMath is a Connes-Moscovici-Kreimer Hopf algebra ( for shortness a CMK Hopf algebra ) whenever ( i ) LaTeXMLMath is the LaTeXMLMath -algebra of ( commutative ) polynomials of countably many variables LaTeXMLMath : in other words LaTeXMLMath is the symmetric algebra over the ( countably-dimensional ) LaTeXMLMath -vector space LaTeXMLMath with a specified basis LaTeXMLMath : LaTeXMLEquation ( we shall write indifferently LaTeXMLMath ( symmetrized tensor product ) or LaTeXMLMath ( product of polynomials ) for the product of elements LaTeXMLMath ) .
The algebra LaTeXMLMath is also definable as the universal envelope LaTeXMLMath of the Lie-algebra LaTeXMLMath with trivial brackets LaTeXMLMath , LaTeXMLMath .
The degree of monomials yields a LaTeXMLMath -grading of the algebra LaTeXMLMath called its polynomial grading .
An essential feature for the sequel of the polynomial grading is that LaTeXMLMath is generated in degrees 0 and 1 .
The upper index LaTeXMLMath of the LaTeXMLMath th-grade summand LaTeXMLMath r.h. of ( LaTeXMLRef ) indicates a LaTeXMLMath -fold power of the product of LaTeXMLMath .
We denote the corresponding degree by deg : deg LaTeXMLMath ( thus deg LaTeXMLMath ) .
( ii ) This algebra-structure is part of a Hopf structure LaTeXMLMath such that LaTeXMLEquation ( the second line implying that Ker LaTeXMLMath ) .
( iii ) In addition LaTeXMLMath is endowed with a Hopf LaTeXMLMath -grading LaTeXMLEquation ( distinct from the ( not Hopf ) polynomial grading ) for which all LaTeXMLMath ( thus all monomials ) are homogeneous of strictly positive degree .
Denoting respectively by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , the corresponding biderivation : LaTeXMLEquation degree : LaTeXMLEquation and automorphism group : LaTeXMLEquation we thus have that LaTeXMLEquation hence LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , leave stable each of the subspaces LaTeXMLMath , LaTeXMLMath ( the Hopf grading is subordinate to the polynomial grading ) .
As a consequence ( cf .
( LaTeXMLRef ) in Section LaTeXMLRef ) , and using a previous remark , we have : LaTeXMLEquation ( iv ) the LaTeXMLMath -graded Hopf algebra LaTeXMLMath is LaTeXMLMath -graded-connected : LaTeXMLEquation thus progressive , i.e. , such that , using a Sweedler-like notation , we have the implication : LaTeXMLEquation whereby all LaTeXMLMath , LaTeXMLMath are strictly positive and strictly smaller than LaTeXMLMath .
The Hopf LaTeXMLMath -grading is such that LaTeXMLMath , entailing merely LaTeXMLMath , LaTeXMLMath .
Note that we have that all LaTeXMLMath ( cf .
footnote 6 of Section LaTeXMLRef ) .
We could replace in ( ii ) “ Hopf algebra ” by “ bialgebra ” since ( LaTeXMLRef ) makes the existence of the antipode automatic , cf .
Proposition LaTeXMLRef .
Let LaTeXMLMath be a CMK Hopf algebra .
The Lie algebra LaTeXMLMath Char LaTeXMLMath is in linear bijection with the algebraïc dual of LaTeXMLMath : a LaTeXMLMath is of the type : LaTeXMLEquation all elements of LaTeXMLMath being obtained in this way ( we then write LaTeXMLMath ) .
Equality LaTeXMLMath : Since LaTeXMLMath , cf .
( LaTeXMLRef ) , the inclusion LaTeXMLMath is obvious , the inclusion LaTeXMLMath following from the fact that the algebra LaTeXMLMath is generated in polynomial grade 0 and 1 .
The definition property LaTeXMLEquation .
LaTeXMLMath entails for LaTeXMLMath that LaTeXMLMath , and that LaTeXMLMath for LaTeXMLMath Ker LaTeXMLMath .
Conversely for LaTeXMLMath as in ( LaTeXMLRef ) one has LaTeXMLMath Char LaTeXMLMath indeed : – LaTeXMLMath whilst LaTeXMLMath , – for LaTeXMLMath : LaTeXMLMath whilst LaTeXMLMath , – for LaTeXMLMath : LaTeXMLMath hence LaTeXMLMath whilst LaTeXMLMath since LaTeXMLMath and LaTeXMLMath both vanish .
The next result , a partial LaTeXMLMath -fold generalization of Proposition LaTeXMLRef , is a tool for proving Theorem LaTeXMLRef ( i ) Let LaTeXMLMath be a CMK Hopf algebra with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , we have that : LaTeXMLEquation ( the sum LaTeXMLMath is over the set LaTeXMLMath of permutations of the LaTeXMLMath first integers ) , consequently LaTeXMLMath is separated by the LaTeXMLMath -fold product LaTeXMLMath : one has the implication : LaTeXMLEquation ( ii ) We have , for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath : LaTeXMLEquation ( in other terms : LaTeXMLEquation ( Warning : ( LaTeXMLRef ) with LaTeXMLMath generally does not hold : for instance , for LaTeXMLMath one has LaTeXMLMath . )
( i ) Before proving ( LaTeXMLRef ) and ( LaTeXMLRef ) for general LaTeXMLMath , we look for orientation at the cases LaTeXMLMath , and 4 .
Case LaTeXMLMath : ( LaTeXMLRef ) and ( LaTeXMLRef ) amount to the second , resp .
third line ( LaTeXMLRef ) .
Case LaTeXMLMath : We first prove LaTeXMLEquation .
LaTeXMLEquation In order to simultaneously prepare the next steps we compute LaTeXMLMath for general LaTeXMLMath : computation indicated by the table ( with the terms of LaTeXMLMath ) in the first column , the terms of LaTeXMLMath ) in the first line , and the products at the intersection of the other lines and columns . )
Table LaTeXMLMath : Computation of LaTeXMLMath yields 9 tensor products : – the LaTeXMLMath indicated by # belong to LaTeXMLMath , thus vanish under LaTeXMLMath , – the LaTeXMLMath indicated by $ belong to LaTeXMLMath , thus also vanish under LaTeXMLMath , – the remaining 2 : LaTeXMLMath and LaTeXMLMath , yield ( LaTeXMLRef ) upon making LaTeXMLMath , LaTeXMLMath .
We next prove LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath , is of the form LaTeXMLMath with LaTeXMLMath , we compute the relevant part of LaTeXMLMath whereby the # -terms of LaTeXMLMath can be discarded since multiplications by LaTeXMLMath leave LaTeXMLMath invariant : we thus have the table : Table LaTeXMLMath : Computation of LaTeXMLMath yielding 15 tensor products , all in LaTeXMLMath or LaTeXMLMath , thus vanishing under LaTeXMLMath .
Case LaTeXMLMath : We first prove LaTeXMLEquation .
LaTeXMLEquation We need to classify the tensor products in Table LaTeXMLMath : They consist of : – the LaTeXMLMath indicated by # belong to LaTeXMLMath , thus vanish under LaTeXMLMath LaTeXMLMath , – the LaTeXMLMath indicated by $ which belong to LaTeXMLMath , turned by LaTeXMLMath into LaTeXMLMath whose first factor vanishes under LaTeXMLMath , – the remaining 3 : LaTeXMLMath , LaTeXMLMath in the set LaTeXMLMath of cyclic permutations of LaTeXMLMath , whose sum is turned by LaTeXMLMath into LaTeXMLMath , by Table LaTeXMLMath equal modulo LaTeXMLMath to LaTeXMLMath ( in other terms LaTeXMLMath : we proved ( LaTeXMLRef ) .
We next prove LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath , is of the form LaTeXMLMath with LaTeXMLMath , we compute the relevant part of LaTeXMLMath whereby the # -terms of LaTeXMLMath can be discarded since multiplications by LaTeXMLMath leave LaTeXMLMath invariant : we thus have the table : Table LaTeXMLMath : Computation of LaTeXMLMath yielding 21 tensor products , all in LaTeXMLMath or LaTeXMLMath , thus vanishing under LaTeXMLMath , because LaTeXMLMath vanishes under LaTeXMLMath by ( LaTeXMLRef ) .
A more detailed analysis of this table registrates , amongst its tensor products : – the LaTeXMLMath indicated by # which belong to LaTeXMLMath turned by LaTeXMLMath into LaTeXMLMath vanishing under LaTeXMLMath LaTeXMLMath , – the LaTeXMLMath indicated by $ which belong to LaTeXMLMath turned by LaTeXMLMath into LaTeXMLMath vanishing under LaTeXMLMath because LaTeXMLMath vanishes under LaTeXMLMath : indeed LaTeXMLMath acting on the tensor products in LaTeXMLMath contained in LaTeXMLMath , and LaTeXMLMath vanishes under LaTeXMLMath by ( LaTeXMLRef ) .
– the remaining 4 : LaTeXMLMath , LaTeXMLMath in the set LaTeXMLMath of cyclic permutations of LaTeXMLMath .
We now reexamine ( LaTeXMLRef ) .
We have , LaTeXMLMath denoting equality up to negligible terms : LaTeXMLEquation hence LaTeXMLMath , hence LaTeXMLEquation .
It should now be clear how things propagate recursively to yield a proof for general LaTeXMLMath .
We assume that Table LaTeXMLMath is as follows : its LaTeXMLMath tensor products consist of : – the LaTeXMLMath indicated by # vanishing under LaTeXMLMath – the LaTeXMLMath indicated by $ vanishing under LaTeXMLMath – the remaining LaTeXMLMath with sum LaTeXMLMath , LaTeXMLMath in the set LaTeXMLMath of cyclic permutations of LaTeXMLMath , with ( LaTeXMLRef ) holding for LaTeXMLMath .
From what precedes it is clear that these features propagate from LaTeXMLMath to LaTeXMLMath .
In particular we have : LaTeXMLEquation and LaTeXMLEquation .
Our final result elucidates the structure of LaTeXMLMath by displaying it as in strict Hopf-algebra-duality with the enveloping ( Hopf ) algebra LaTeXMLMath of the Lie algebra LaTeXMLMath , the latter isomorphic to the Hopf algebra LaTeXMLMath by the Milnor-Moore theorem .
Let LaTeXMLMath be a CMK Hopf algebra .
LaTeXMLMath is separated by LaTeXMLMath : for LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath entails LaTeXMLMath .
Each LaTeXMLMath is of the form LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
We assume that LaTeXMLMath vanishes under all elements LaTeXMLMath and show that LaTeXMLMath : indeed , let LaTeXMLMath , LaTeXMLMath : then : – since by ( LaTeXMLRef ) all the terms of h but the first lie in LaTeXMLMath ker LaTeXMLMath one has LaTeXMLMath : the requirement LaTeXMLMath thus entails the vanishing of the first term LaTeXMLMath .
– next by Corollary LaTeXMLRef ( ii ) one has LaTeXMLMath : asking this to vanish for all LaTeXMLMath thus entails the vanishing of the second term LaTeXMLMath .
– assume that we have shown that LaTeXMLMath vanishes for LaTeXMLMath we have , by Corollary LaTeXMLRef ( ii ) and ( iii ) : LaTeXMLEquation whose vanishing under all LaTeXMLMath entails LaTeXMLMath : we thus proved Theorem LaTeXMLRef inductively .
In what follows LaTeXMLMath is a CMK Hopf algebra , cf .
Sections LaTeXMLRef and LaTeXMLRef of which we adopt the notation .
In what follows the ground field is LaTeXMLMath and we will handle algebras with and without unit .
Since our general practice was to use the word : algebra to mean : unital algebra , we shall write “ algebra ” ( quote-unquote ) to mean : not necessarily unital algebra over LaTeXMLMath .
( i ) We consider the following “ algebras ” of analytic functions on the Riemann sphere PC LaTeXMLMath with north pole LaTeXMLMath and south pole LaTeXMLMath .
Notation using the chart of PC LaTeXMLMath obtained by stereographic projection from the north pole on the equator coordinatized by LaTeXMLMath : in phrasing the Definitions ( LaTeXMLRef ) below , this chart is used to identify PC LaTeXMLMath – LaTeXMLMath with LaTeXMLMath .
LaTeXMLEquation ( ii ) A loop is a map LaTeXMLMath PC LaTeXMLMath .
The set of loops is denoted by LaTeXMLMath .
( iii ) With LaTeXMLMath one of the algebras LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , a LaTeXMLMath - loop is a map LaTeXMLMath PC LaTeXMLMath such that the functions PC LaTeXMLMath , LaTeXMLMath Ker LaTeXMLMath , belongs to LaTeXMLMath .
Definition implying that LaTeXMLMath for all LaTeXMLMath .
The set of LaTeXMLMath -loops is denoted by LaTeXMLMath .
We recall that a LaTeXMLMath -loop LaTeXMLMath has a unique Birkhoff decomposition : LaTeXMLEquation where LaTeXMLEquation ( i ) The LaTeXMLMath -loops build a group under the operations : LaTeXMLEquation with the unit the constant loop : LaTeXMLMath .
( ii ) The LaTeXMLMath -loops LaTeXMLMath are one-to-one with elements LaTeXMLMath Hom LaTeXMLMath through the bijection : Hom LaTeXMLMath is the set of homomorphisms between “ algebras ” , i.e .
LaTeXMLMath -linear multiplicative maps : LaTeXMLMath .
LaTeXMLEquation a group isomorphism with the following correspondence of group products : LaTeXMLEquation where the product on the left is that of the group LaTeXMLMath ( first line ( LaTeXMLRef ) ) , whilst LaTeXMLMath on the right is the convolution product of Hom LaTeXMLMath : LaTeXMLEquation .
One has for LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In the sequel we are concerned with the “ special loops ” defined as follows : The LaTeXMLMath -loop LaTeXMLMath , LaTeXMLMath or LaTeXMLMath , is special whenever its inverse LaTeXMLMath : To alleviate writing we write the complex variable of the loop LaTeXMLMath as an index .
Note that the limit in ( LaTeXMLRef ) is taken in the topology of LaTeXMLMath : simple convergence on LaTeXMLMath .
LaTeXMLEquation is such that LaTeXMLMath , ( product in the group LaTeXMLMath has a limit for LaTeXMLMath for all LaTeXMLMath : LaTeXMLEquation and moreover LaTeXMLMath is differentiable in LaTeXMLMath for LaTeXMLMath .
In the case of LaTeXMLMath -loops , as soon as condition ( LaTeXMLRef ) is fulfilled , LaTeXMLMath is differentiable as shown by the following computation .
Continuity of the limit needs only be examined for the values on the kernel of LaTeXMLMath since LaTeXMLMath is constant .
Now for LaTeXMLMath with LaTeXMLMath : LaTeXMLEquation .
This makes it clear that the limit LaTeXMLMath is a polynomial in LaTeXMLMath if it exists .
Let LaTeXMLMath be a special loop as in Definition LaTeXMLRef .
Then LaTeXMLMath in ( LaTeXMLRef ) is a flow , one has : LaTeXMLEquation thus LaTeXMLEquation with the LaTeXMLMath - function terminology borrowed from renormalization theory .
the following element of LaTeXMLMath : LaTeXMLEquation ( i ) Let LaTeXMLMath .
First LaTeXMLEquation indeed , for LaTeXMLMath of degree LaTeXMLMath : first LaTeXMLEquation since LaTeXMLMath .
Then : LaTeXMLEquation .
Let LaTeXMLMath be a special LaTeXMLMath -loop with : LaTeXMLEquation ( we set LaTeXMLMath Res LaTeXMLMath , called the residue of LaTeXMLMath ) .
One has : LaTeXMLEquation rewritten LaTeXMLEquation .
LaTeXMLEquation explicitly : LaTeXMLEquation ( using LaTeXMLEquation .
The morale of this is that the loop LaTeXMLMath is determined by its residue , thus by LaTeXMLMath , with dependence expressed explicitly by ( LaTeXMLRef ) .
Performing a licit exchange of limits in : LaTeXMLEquation we first show that we have LaTeXMLEquation .
Indeed , the horizontal limit comes from ( LaTeXMLRef ) with LaTeXMLMath ; the right vertical limit is ( LaTeXMLRef ) .
As for the left vertical limit , we have that LaTeXMLMath applied to : LaTeXMLEquation yields LaTeXMLEquation .
Next the function LaTeXMLMath is holomorphic on the whole Riemann sphere , thus must be constant : we have LaTeXMLMath , LaTeXMLMath , i.e .
: LaTeXMLEquation .
Feeding ( LaTeXMLRef ) into this yields : LaTeXMLEquation equating coefficients of LaTeXMLMath then yields ( LaTeXMLRef ) .
Passage from ( LaTeXMLRef ) to ( LaTeXMLRef ) : LaTeXMLMath a is injective in restriction to the augmentation ideal LaTeXMLMath .
Thus LaTeXMLMath ( thus also Res LaTeXMLMath determines LaTeXMLMath , the key to the explicit dependence being formula ( LaTeXMLRef ) , itself obtained as follows : integrating from 0 to LaTeXMLMath , LaTeXMLEquation yields LaTeXMLMath , it suffices now to inverse LaTeXMLMath in restriction to LaTeXMLMath .
Applying ( LaTeXMLRef ) to LaTeXMLMath yields ( LaTeXMLRef ) for LaTeXMLMath .
Inductive iteration then yields ( LaTeXMLRef ) : assume it holds for LaTeXMLMath : applying LaTeXMLMath on both sides yields : LaTeXMLEquation up to numbering of variables identical to ( LaTeXMLRef ) .
Extension of the Lie algebra LaTeXMLMath by an additional element LaTeXMLMath enables us now to give a compact expression of the last result , and , as a byproduct , a characterization of “ specialty ” for LaTeXMLMath -loops .
( i ) LaTeXMLMath is LaTeXMLMath with the Lie-brackets of LaTeXMLMath plus : LaTeXMLEquation ( ii ) The so defined LaTeXMLMath is a Lie algebra .
We need only check the additional cases of the Jacobi requirement .
Now , for LaTeXMLMath , we have LaTeXMLEquation and LaTeXMLEquation by the fact that the algebra-derivation LaTeXMLMath of LaTeXMLMath is a Lie algebra-derivation of the Lie algebra Lie LaTeXMLMath : for LaTeXMLMath we have LaTeXMLMath and LaTeXMLMath thus LaTeXMLMath .
With LaTeXMLMath as in Definition-Lemma LaTeXMLRef , one has for each LaTeXMLMath -loop LaTeXMLMath LaTeXMLEquation .
We apply the expansional formula LaTeXMLEquation to LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , yielding : LaTeXMLEquation .
LaTeXMLEquation We thus have LaTeXMLEquation whence ( LaTeXMLRef ) LaTeXMLEquation .
A Birkhoff sum ( over LaTeXMLMath ) is a commutative ( unital ) LaTeXMLMath -algebra LaTeXMLMath which is the direct sum LaTeXMLMath of two LaTeXMLMath -linear multiplicative subspaces LaTeXMLMath and LaTeXMLMath .
We do not say : subalgebras because according to our general terminological practice a subalgebra contains the unit .
Note that Definition ( LaTeXMLRef ) follows from the fact that both LaTeXMLMath and LaTeXMLMath are closed under multiplications .
The projection LaTeXMLMath parallel to LaTeXMLMath thus fulfills : LaTeXMLEquation ( Algebraïc Birkhoff decomposition ) .
With LaTeXMLMath a LaTeXMLMath -graded , connected , progressive , i.e. , LaTeXMLMath has a LaTeXMLMath -grading commuting with the Hopf structure , such that LaTeXMLMath , and moreover such that LaTeXMLMath with max ( degh LaTeXMLMath , degh LaTeXMLMath , LaTeXMLMath .
commutative Hopf algebra over LaTeXMLMath , and LaTeXMLMath a Birkhoff sum , denote by Hom LaTeXMLMath ) the set of unital LaTeXMLMath -algebra homomorphisms : LaTeXMLMath , considered as a LaTeXMLMath -algebra under the convolution product : The multiplicativity of LaTeXMLMath and LaTeXMLMath implies that of LaTeXMLMath owing to the commutativity of LaTeXMLMath .
LaTeXMLEquation .
Let LaTeXMLMath be given .
Requiring , for a LaTeXMLMath -linear LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath Ker LaTeXMLMath , The ( commutative ) product LaTeXMLMath r.h.s .
is in LaTeXMLMath .
specifies inductively a LaTeXMLMath having its range in LaTeXMLMath : LaTeXMLEquation such that the product LaTeXMLMath has its range in LaTeXMLMath : LaTeXMLEquation this yielding the “ algebraïc Birkhoff decomposition ” : The product and the inverse r.h.s .
are taken w.r.t .
the convolution product ( LaTeXMLRef ) .
LaTeXMLEquation .
Progressiveness of LaTeXMLMath implies that ( LaTeXMLRef ) and ( LaTeXMLRef ) specifies inductively a LaTeXMLMath -linear map LaTeXMLMath : LaTeXMLMath .
Check of ( LaTeXMLRef ) : obvious from ( LaTeXMLRef ) and LaTeXMLMath .
Check of ( LaTeXMLRef ) : we have , by ( LaTeXMLRef ) and ( LaTeXMLRef ) , ( LaTeXMLRef ) , for LaTeXMLMath , omitting parentheses : LaTeXMLEquation .
LaTeXMLEquation whence ( LaTeXMLRef ) owing to LaTeXMLMath .
What is not obvious from ( LaTeXMLRef ) is that LaTeXMLMath is multiplicative .
We prove this by induction : for LaTeXMLMath , LaTeXMLMath we have , still omitting parentheses : LaTeXMLEquation .
LaTeXMLEquation hence , by the induction hypothesis : LaTeXMLEquation .
On the other hand we have , using ( LaTeXMLRef ) and the commutativity of LaTeXMLMath : LaTeXMLEquation the two expressions containing the same terms .
In what follows LaTeXMLMath is a commutative field of characteristics zero ( e.g .
LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath ) .
( Algebras , coalgebras , bialgebras , Hopf algebras ) .
Let LaTeXMLMath be a LaTeXMLMath -vector space : with the following notation for the maps LaTeXMLMath : Notation : LaTeXMLEquation the following axioms successively define LaTeXMLMath as an algebra , coalgebra , bialgebra , and Hopf algebra : Axioms : Algebra LaTeXMLEquation .
Coalgebra LaTeXMLEquation .
Bialgebra : add to the previous axioms : LaTeXMLEquation .
Hopf algebra : add to the previous axioms : LaTeXMLEquation .
Let LaTeXMLMath be a Hopf algebra over LaTeXMLMath .
One has the following properties : LaTeXMLEquation [ Proof of Proposition LaTeXMLRef ] ( i ) resp .
( ii ) see below LaTeXMLRef resp .
B.4 .
( iii ) LaTeXMLMath : by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath : LaTeXMLEquation .
LaTeXMLMath : for LaTeXMLMath : LaTeXMLEquation moreover , by LaTeXMLMath , LaTeXMLMath , LaTeXMLMath : LaTeXMLEquation .
LaTeXMLMath : by LaTeXMLMath for LaTeXMLMath : LaTeXMLEquation hence LaTeXMLMath .
( The short exact sequence of LaTeXMLMath . )
( i ) The short exact sequence of LaTeXMLMath is split with lift LaTeXMLMath : LaTeXMLEquation ( where LaTeXMLMath is a notation for Ker LaTeXMLMath ) corresponding to the following direct sum splitting into a “ vertical ” and a “ horizontal ” part : LaTeXMLEquation .
The splitting of short exact sequence ( 1,1 ) can alternatively be specified by : – the “ horizontal ” projection LaTeXMLMath – the “ vertical ” projection LaTeXMLMath ( with Imp LaTeXMLMath Ker LaTeXMLMath ) .
[ Proof of Proposition LaTeXMLRef ] ( i ) one has LaTeXMLMath , indeed , for LaTeXMLMath : LaTeXMLEquation .
The other claims follow from known facts about split short exact sequences .
In what follows the ground field is LaTeXMLMath ( with possible choices LaTeXMLMath ) .
We write LaTeXMLMath , Hom , and End for LaTeXMLMath , Hom LaTeXMLMath , and End LaTeXMLMath .
With LaTeXMLMath a coalgebra , LaTeXMLMath an algebra ; and Hom LaTeXMLMath the vector space of homomomorphisms : LaTeXMLMath ( as LaTeXMLMath -vector spaces ) , define as follows LaTeXMLMath Hom LaTeXMLMath : LaTeXMLEquation .
Under this product called convolution and with LaTeXMLMath as a unit , Hom LaTeXMLMath becomes an algebra .
Specifically one has : LaTeXMLEquation ( more generally LaTeXMLEquation in Sweedlers ’ notation : LaTeXMLEquation ( more generally LaTeXMLEquation .
LaTeXMLEquation finally LaTeXMLEquation .
We have LaTeXMLMath because LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Check of ( LaTeXMLRef ) , we have : LaTeXMLEquation whilst LaTeXMLEquation .
Check of ( LaTeXMLRef ) , we have : LaTeXMLEquation .
Check of ( LaTeXMLRef ) , LaTeXMLEquation and LaTeXMLEquation ( i ) Result LaTeXMLRef applies in particular to the case where LaTeXMLMath is a bialgebra ( with unit element LaTeXMLMath ) .
Hom LaTeXMLMath then possesses an algebra structures , with product LaTeXMLMath for which LaTeXMLMath acting as LaTeXMLMath , LaTeXMLMath , is the unit .
( ii ) The further specialization LaTeXMLMath with LaTeXMLMath a Hopf algebra interprets the Hopf axiom as the fact that LaTeXMLMath : LaTeXMLMath ( this immediately implying uniqueness of the antipode , LaTeXMLMath and LaTeXMLMath implying LaTeXMLMath ) .
With LaTeXMLMath a Hopf algebra , taking for LaTeXMLMath the tensor product coalgebra LaTeXMLMath and for LaTeXMLMath the algebra LaTeXMLMath , we get for Hom LaTeXMLMath an algebra-structure with product LaTeXMLMath and unit LaTeXMLMath .
For the latter LaTeXMLMath is a left , and LaTeXMLMath is a right inverse of LaTeXMLMath ( they thus coincide , this proving property ( Hm ) ) .
With LaTeXMLMath a Hopf algebra , taking for LaTeXMLMath the coalgebra LaTeXMLMath and for LaTeXMLMath the tensor product algebra LaTeXMLMath we get for Hom LaTeXMLMath an algebra structure with product LaTeXMLMath and unit LaTeXMLMath .
For the latter LaTeXMLMath is a left , and LaTeXMLMath is a right inverse of LaTeXMLMath ( they thus coincide , this proving property ( H LaTeXMLMath ) ) .
Let LaTeXMLMath be a bialgebra over the field LaTeXMLMath .
( i ) The element LaTeXMLMath is called primitive whenever LaTeXMLEquation .
The set of primitive elements of LaTeXMLMath is denoted by LaTeXMLMath .
( ii ) The element LaTeXMLMath is called group-like whenever LaTeXMLEquation .
The set of group-like elements of LaTeXMLMath is denoted by LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -bialgebra .
( i ) LaTeXMLMath is a LaTeXMLMath -linear subspace of LaTeXMLMath .
Moreover if LaTeXMLMath , then LaTeXMLMath .
Consequently LaTeXMLMath is a Lie subalgebra of the Lie algebra Lie LaTeXMLMath .
( ii ) One has LaTeXMLMath : LaTeXMLEquation ( iii ) LaTeXMLMath is multiplicative : if LaTeXMLMath , then LaTeXMLMath .
( iv ) One has LaTeXMLEquation .
If furthermore LaTeXMLMath is a Hopf algebra with antipode LaTeXMLMath , then : LaTeXMLEquation .
LaTeXMLEquation showing that LaTeXMLMath is then a group .
( i ) Let LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath .
Furthermore : LaTeXMLEquation ( ii ) LaTeXMLMath entails LaTeXMLEquation ( iii ) LaTeXMLMath entails LaTeXMLEquation ( iv ) LaTeXMLMath entails LaTeXMLEquation .
Assume that LaTeXMLMath is a Hopf algebra with antipode LaTeXMLMath .
Check of ( LaTeXMLRef ) : LaTeXMLMath entails LaTeXMLEquation .
Check of ( LaTeXMLRef ) : LaTeXMLMath entails LaTeXMLEquation whilst LaTeXMLMath entails LaTeXMLEquation
On a Two-Variable Zeta Function for Number Fields Jeffrey C. Lagarias Eric Rains AT & T Labs - Research Florham Park , NJ 07932 ( July 6 , 2002 version ) Abstract This paper studies a two-variable zeta function LaTeXMLMath attached to an algebraic number field LaTeXMLMath , introduced by van der Geer and Schoof LaTeXMLCite , which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors .
When LaTeXMLMath this function becomes the completed Dedekind zeta function LaTeXMLMath of the field LaTeXMLMath .
The function is an meromorphic function of two complex variables with polar divisor LaTeXMLMath , and it satisfies the functional equation LaTeXMLMath .
We consider the special case LaTeXMLMath , where for LaTeXMLMath this function is LaTeXMLMath .
The function LaTeXMLMath is shown to be an entire function on LaTeXMLMath , to satisfy the functional equation LaTeXMLMath and to have LaTeXMLMath We study the location of the zeros of LaTeXMLMath for various real values of LaTeXMLMath .
For fixed LaTeXMLMath the zeros are confined to a vertical strip of width at most LaTeXMLMath and the number of zeros LaTeXMLMath to height LaTeXMLMath has similar asymptotics to the Riemann zeta function .
For fixed LaTeXMLMath these functions are strictly positive on the “ critical line ” LaTeXMLMath .
This phenomenon is associated to a positive convolution semigroup with parameter LaTeXMLMath , which is a semigroup of infinitely divisible probability distributions .
having densities LaTeXMLMath for real LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath AMS Subject Classification ( 2000 ) : 11M41 ( Primary ) 14G40 , 60E07 ( Secondary ) Keywords : Arakelov divisors , functional equation , zeta functions Recently van der Geer and Schoof LaTeXMLCite formulated an “ exact ” analogue of the Riemann-Roch theorem valid for an algebraic number field LaTeXMLMath , based on Arakelov divisors .
They used this result to formally express the completed zeta function LaTeXMLMath of LaTeXMLMath as an integral over the Arakelov divisor class group LaTeXMLMath of LaTeXMLMath .
They introduced a two-variable zeta function attached to a number field LaTeXMLMath , also given as an integral over the Arakelov class group , which we call either the Arakelov zeta function or the two-variable zeta function .
This zeta function was modelled after a two-variable zeta function attached to a function field over a finite field , introduced in 1996 by Pellikaan LaTeXMLCite .
For convenience we review the Arakelov divisor interpretation of the two-variable zeta function and the Riemann-Roch theorem for number fields in an appendix .
In this paper we study in detail the two-variable zeta function attached to the rational field LaTeXMLMath .
Then in the final section we consider two-variable zeta functions for general algebraic number fields LaTeXMLMath .
The results are derived starting from an integral representation of this function , and if one takes it as given , then the paper is independent of the Arakelov divisor interpretation .
The Arakelov class group of LaTeXMLMath can be identified with the positive real line ( with multiplication as the group operation ) and van der Geer and Schoof ’ s integral becomes , formally , LaTeXMLEquation in which LaTeXMLEquation is the theta function LaTeXMLMath , where LaTeXMLEquation is a Jacobi theta function .
The LaTeXMLMath used in ( LaTeXMLRef ) reflects the fact that the integral on its right side converges nowhere ; a regularization is needed to assign it a meaning .
Such a regularization can be obtained using the Arakelov two-variable zeta function LaTeXMLMath attached to LaTeXMLMath , which we define to be LaTeXMLEquation .
Our definition here differs from the one in van der Geer and Schoof LaTeXMLCite by a linear change of variable , setting their second variable LaTeXMLMath The integral on the right side of ( LaTeXMLRef ) has a region of absolute convergence in LaTeXMLMath , which is the open cone LaTeXMLEquation .
The function LaTeXMLMath meromorphically continues from the cone LaTeXMLMath to all of LaTeXMLMath , with polar divisor consisting of the ( complex ) hyperplanes LaTeXMLMath , a set of real-codimension two , see §2 .
On restricting LaTeXMLMath to the line LaTeXMLMath , the resulting function is the completed Riemann zeta function LaTeXMLMath , which is LaTeXMLEquation .
Thus the two-variable zeta function LaTeXMLMath defined via ( LaTeXMLRef ) provides a method to regularize the integral ( LaTeXMLRef ) , and the same can be done for arbitrary number fields .
We are motivated by several questions about this function .
( 1 ) What are the properties of the function as a meromorphic function of two complex variables ?
In particular , determine information about its zero divisor .
( 2 ) What is the meaning of the additional variable LaTeXMLMath and what arithmetic information does it encode ?
( 3 ) What properties of this two-variable zeta function reflect Arakelov geometry ?
( 4 ) Is there any connection between zeta functions encoding information based on Arakelov geometry and zeta functions coming from automorphic representations and the Langlands program ?
This paper mainly addresses question ( 1 ) , obtaining information on the zero set of the Arakelov zeta function .
Concerning question ( 2 ) , we observe in §2 that the function LaTeXMLMath is representable by the integral LaTeXMLEquation which expresses it as a Mellin transform of LaTeXMLMath .
The function LaTeXMLMath is a modular form of weight LaTeXMLMath ( with multiplier system ) on a congruence subgroup of the modular group , and the complex variable LaTeXMLMath parametrizes the weight of this modular form .
The arithmetic information it encodes includes the invariants LaTeXMLMath and LaTeXMLMath introduced in van der Geer and Schoof LaTeXMLCite , defined in the appendix .
Concerning question ( 3 ) , we observe that there is an extra structure associated to LaTeXMLMath , which is a holomorphic convolution semigroup of complex-valued measures on lines LaTeXMLMath in the real-codimension one cone LaTeXMLEquation see §7.2 .
This cone is contained in the region of absolute convergence LaTeXMLMath of the integral representation ( LaTeXMLRef ) .
Of particular interest for LaTeXMLMath is the real-codimension two subcone LaTeXMLEquation which generalizes the “ critical line ” of the zeta function , and on which the measures are real-valued .
Perhaps this semigroup structure is associated in some way with Arakelov geometry , since various constants associated with the semigroup on the subcone LaTeXMLMath have arithmetic interpretations in the framework of van der Geer and Schoof , see §7 .
Concerning question ( 4 ) , the subject of Arakelov geometry was developed in part to answer Diophantine questions and has a completely different origin from automorphic representations .
Any connection between these two subjects could potentially be of great interest .
However we do not find any obvious connection , and note only that the LaTeXMLMath variable interpolates between modular forms of different weights , and when LaTeXMLMath is a positive even integer these are holomorphic modular forms of the type appearing in automorphic representations .
In general these forms are not eigenforms for Hecke operators , and in §3.4 we show these forms have associated Euler products exactly when LaTeXMLMath and LaTeXMLMath .
Besides giving information on questions ( 1 ) - ( 4 ) above , the analysis of this paper may be useful for other purposes .
This function provides an interesting example of an entire function in two complex variables of finite order , see Ronkin LaTeXMLCite and Stoll LaTeXMLCite .
The information about the zero locus of LaTeXMLMath that we obtain mainly concerns the region where where the variable LaTeXMLMath is real ; these are Mellin transforms of modular forms of real weight , which have been extensively studied .
The movements of zeros in the LaTeXMLMath -plane as the ( real ) parameter LaTeXMLMath is varied may be compared with movement of zeros under milder deformations such as those in linear combinations of L-functions , see Bombieri and Hejhal LaTeXMLCite and Hejhal LaTeXMLCite .
The function LaTeXMLMath shares many properties of the Riemann zeta function .
It satisfies the functional equation LaTeXMLEquation .
When LaTeXMLMath is real then LaTeXMLMath retains several familiar symmetries of the Riemann zeta function : it is real on the real axis LaTeXMLMath , and it is real on the “ critical line ” LaTeXMLMath , which is the line of symmetry of the functional equation .
Thus for fixed real LaTeXMLMath , the zeros of LaTeXMLMath which do not lie on the critical line or the real axis must occur in sets of four : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
This extra symmetry can be used to extract information about the components of the zero locus , see Lemma LaTeXMLRef .
On the other hand , for most real LaTeXMLMath the function LaTeXMLMath fails to satisfy a Riemann hypothesis , as we describe below .
The Riemann zeta function appears when LaTeXMLMath and it is interesting to note that it also appears in terms of data at LaTeXMLMath , given in the following result , which is proved in §5 .
The function LaTeXMLEquation is an entire function in two complex variables .
At LaTeXMLMath it is LaTeXMLEquation where LaTeXMLMath is the completed Riemann zeta function .
In particular , for all real LaTeXMLMath , LaTeXMLEquation is strictly positive , with LaTeXMLMath The Jacobi triple product formula plays an essential role in our derivation of the formula ( LaTeXMLRef ) .
Note that the function LaTeXMLMath coincides with the Riemann LaTeXMLMath -function , and the functional equation LaTeXMLMath is inherited from LaTeXMLMath The most striking result of this paper appears in §7 , and concerns for negative real LaTeXMLMath the behavior of the function LaTeXMLMath on the “ critical line ” LaTeXMLMath .
This result makes a connection with probability theory , involving infinitely divisible distributions .
For negative real LaTeXMLMath , with LaTeXMLMath LaTeXMLMath the function LaTeXMLMath is given as the Fourier transform LaTeXMLEquation in which LaTeXMLEquation and LaTeXMLEquation .
The function LaTeXMLMath is the characteristic function of an infinitely divisible probability measure with finite second moment , whose associated Khintchine canonical measure LaTeXMLMath has LaTeXMLMath with LaTeXMLEquation in which LaTeXMLMath is the completed Riemann zeta function .
In particular , LaTeXMLEquation .
Many interesting connections between zeta and theta functions and probability theory are known ; see Biane , Pitman and Yor LaTeXMLCite for a comprehensive survey .
Theorem LaTeXMLRef appears structurally different from any of the known results .
The positivity property ( LaTeXMLRef ) can be called an “ anti-Riemann hypothesis ” , because it shows there are no zeros on the “ critical line ” LaTeXMLMath for fixed real LaTeXMLMath .
There are a number of different canonical forms used to specify infinitely divisible distributions .
Feller LaTeXMLCite uses the canonical measure LaTeXMLMath , which we term the Feller canonical measure , while the Khintchine canonical measure LaTeXMLMath is often used , see LaTeXMLCite .
An infinitely divisible distribution is a member of a positive convolution semigroup of measures , and the Feller canonical measure is related to the infinitesimal generator of the semigroup .
The measure LaTeXMLMath above involves the values of the Riemann zeta function on the boundary of its critical strip , noting that the functional equation gives LaTeXMLEquation .
Theorem LaTeXMLRef follows from two results proved in §7 , Theorem LaTeXMLRef and Theorem LaTeXMLRef .
We also note that the value LaTeXMLMath appearing in Theorem LaTeXMLRef equals LaTeXMLMath where LaTeXMLMath is the “ genus of LaTeXMLMath ” as defined by van der Geer and Schoof LaTeXMLCite , see the appendix .
The positive holomorphic convolution semigroup structure associated to this two-variable zeta function merits further study .
It seems an interesting question to determine the generality of this positivity property .
All algebraic number fields LaTeXMLMath have an associated holomorphic convolution semigroup of complex-valued measures , which are real-valued measures on the “ critical line ” .
However the positivity fails to hold in general , and perhaps is true only for a few specific number fields , see §9 .
We comment on related work .
There is precedent for studying two-variable functions given by integrals of the form ( LaTeXMLRef ) with LaTeXMLMath replaced with some other modular form .
Conrey and Ghosh LaTeXMLCite considered a Fourier integral associated to powers of the modular form LaTeXMLMath of weight 12 , which is a cusp form .
The integral they consider can be transformed to a constant multiple of the integral LaTeXMLEquation where they take LaTeXMLMath , and LaTeXMLMath .
They note that associated Dirichlet series has an Euler product for LaTeXMLMath and LaTeXMLMath .
Bruggeman LaTeXMLCite studied properties of families of automorphic forms of variable weight ; he considers powers of the Dedekind eta function family LaTeXMLMath in LaTeXMLCite .
This family appears in ( LaTeXMLRef ) since LaTeXMLMath , see LaTeXMLCite .
We now summarize the contents of the paper .
In §2 we give the analytic continuation and functional equation for LaTeXMLMath essentially following Riemann ’ s second proof of the functional equation for LaTeXMLMath .
We derive integral formulas for LaTeXMLMath which converge on LaTeXMLMath off certain hyperplanes .
In §3 , as a preliminary to later results , we study the Fourier coefficients LaTeXMLMath of the modular form LaTeXMLEquation .
We show that LaTeXMLMath is a polynomial of degree LaTeXMLMath with nonnegative integer coefficients .
For LaTeXMLMath on the positive real axis we obtain the estimate LaTeXMLMath whose merit is that it is uniform in LaTeXMLMath .
For general LaTeXMLMath there is an upper bound LaTeXMLEquation which follows from classical estimates .
We show that the Dirichlet series LaTeXMLMath for LaTeXMLMath has an Euler product if and only if LaTeXMLMath and LaTeXMLMath .
In §4 we study growth properties of the entire function LaTeXMLMath We first show that LaTeXMLMath is an entire function of order one and infinite type in two complex variables , in the sense that it satisfies the growth bound : There is a constant LaTeXMLMath such that for any LaTeXMLMath , if LaTeXMLMath , then LaTeXMLEquation .
Thus any linear slice function LaTeXMLMath , has at most LaTeXMLMath zeros in the disk of radius LaTeXMLMath as LaTeXMLMath , provided it is not identically zero .
We then show that for fixed LaTeXMLMath and fixed LaTeXMLMath , the function LaTeXMLEquation has rapid decrease , is in the Schwartz class LaTeXMLMath and is uniformly bounded in vertical strips LaTeXMLMath , for finite LaTeXMLMath .
In §5 we treat the case LaTeXMLMath and prove Theorem LaTeXMLRef .
In §6 we treat the case when LaTeXMLMath is a fixed positive real number , and study the zeros of LaTeXMLMath .
We show that these zeros are confined to the vertical strip LaTeXMLMath Then we show that the number LaTeXMLMath of zeros LaTeXMLMath having LaTeXMLMath has similar asymptotics to that of the Riemann zeta function , namely LaTeXMLEquation with LaTeXMLEquation and the constant LaTeXMLMath is absolute .
The zeros of LaTeXMLMath appear to lie on the “ critical line ” LaTeXMLMath only for special values LaTeXMLMath and LaTeXMLMath ; we observe that only an infinitesimal fraction of zeros are on this line for LaTeXMLMath and LaTeXMLMath .
In §7 we consider LaTeXMLMath where LaTeXMLMath is a fixed negative real number ( LaTeXMLMath ) .
We prove Theorem LaTeXMLRef , that the function LaTeXMLMath , which is necessarily real on the critical line LaTeXMLMath , is always positive there .
The proof of this result makes essential use of the Jacobi product formula , which is applicable because the constant term in the theta function is present in the integral representation ( LaTeXMLRef ) .
The associated structure behind Theorem LaTeXMLRef is a holomorphic convolution semigroup LaTeXMLMath of complex-valued measures on the real line , defined for LaTeXMLMath real in the cone LaTeXMLMath and LaTeXMLMath , and these measures are positive real on the line LaTeXMLMath .
We derive formulae for the cumulants and moments of these measures .
We also list a number of open questions concerning the location of zeros for negative real LaTeXMLMath .
For example , for real LaTeXMLMath , are the asympotics of the number of zeros LaTeXMLMath with LaTeXMLMath as LaTeXMLMath the same for negative real LaTeXMLMath as they are for positive real LaTeXMLMath ?
In §8 we consider general complex LaTeXMLMath , and the zero locus LaTeXMLMath of LaTeXMLMath .
The set LaTeXMLMath viewed geometrically is a one-dimensional complex manifold , having more than one irreducible analytic component ( possibly infinitely many components ) , each one of which is a Riemann surface embedded in LaTeXMLMath .
We show that the zeta zeros LaTeXMLMath and LaTeXMLMath are on the same irreducible component , and raise the question whether the zeta zeros ( for LaTeXMLMath , LaTeXMLMath varying ) are all on a single irreducible component of the zero locus .
In §9 we briefly consider Arakelov zeta functions attached to general algebraic number fields LaTeXMLMath .
All results of this paper extend to the Arakelov zeta function attached to the Gaussian field LaTeXMLMath and many of the results extend to general LaTeXMLMath , with similar proofs .
However the positivity property of Theorem LaTeXMLRef for LaTeXMLMath , our proof used a product formula for the modular form and does not extend to general number fields LaTeXMLMath .
Numerical experiments show that the positivity property does not hold for several imaginary quadratic fields , with discriminants LaTeXMLMath and LaTeXMLMath .
Our computations allows the possibility that it might hold for some fields whose modular forms do not have a product formula , including the imaginary quadratic fields with discriminants LaTeXMLMath and LaTeXMLMath .
In the appendix we review the Arakelov divisor framework of van der Geer and Schoof LaTeXMLCite , and derive formulas for the two-variable zeta function for LaTeXMLMath and LaTeXMLMath .
The authors thank E. Bombieri , J .
B. Conrey , C. Deninger , J. Pitman , J .
A. Reeds , M. Yor and the reviewer for helpful comments and references .
The variables LaTeXMLMath denote complex variables with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath always denote real variables .
We use two versions of the Fourier transform , differing in their scaling , because the usual conventions for the Fourier transform differ in probability theory and number theory .
The Fourier transform LaTeXMLMath is given by LaTeXMLEquation with inverse LaTeXMLEquation .
In probability theory the characteristic function LaTeXMLMath of a Borel measure LaTeXMLMath of unit mass on the line is LaTeXMLEquation .
In the case where LaTeXMLMath we write LaTeXMLMath as an inverse Fourier transform , and the corresponding Fourier transform is LaTeXMLEquation .
We now obtain the meromorphic continuation of LaTeXMLMath , which determines its polar divisor and part of its zero divisor .
Using the theta function transformation formula LaTeXMLEquation we can rewrite LaTeXMLEquation in the form LaTeXMLEquation .
Then , after a change of variable LaTeXMLMath , followed by LaTeXMLMath , one obtains LaTeXMLEquation .
Note that LaTeXMLMath rapidly as LaTeXMLMath , hence LaTeXMLMath rapidly as LaTeXMLMath .
This implies that ( LaTeXMLRef ) converges absolutely on the open domain LaTeXMLMath in LaTeXMLMath .
The convergence is uniform on compact subsets of this domain , which defines LaTeXMLMath as an analytic function there .
The function LaTeXMLMath analytically continues to an entire function on LaTeXMLMath , and satisfies the functional equation LaTeXMLEquation .
For LaTeXMLMath we have LaTeXMLMath and LaTeXMLMath where LaTeXMLMath is Riemann ’ s LaTeXMLMath -function , and we recover the functional equation for LaTeXMLMath .
We give a Fourier-Laplace transform integral representation for LaTeXMLMath in Theorem LaTeXMLRef .
We split the integral ( LaTeXMLRef ) into two pieces LaTeXMLMath and LaTeXMLMath and consider them separately .
Using the transformation law yields LaTeXMLEquation .
Both sides are defined and converge LaTeXMLMath and LaTeXMLMath .
On the right side the first integral converges for all LaTeXMLMath , because for LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation as LaTeXMLMath , where the constant in the O-symbol depends only on LaTeXMLMath .
This uniformity of convergence shows that this integral is an entire function on LaTeXMLMath .
The second integral in ( LaTeXMLRef ) converges , for LaTeXMLMath , to the function LaTeXMLMath .
Similarly , LaTeXMLEquation with both sides convergent for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
This region overlaps the region of convergence of ( LaTeXMLRef ) in an open domain in LaTeXMLMath .
The first integral on the right side of ( LaTeXMLRef ) defines an entire function on LaTeXMLMath , while the second integral in ( LaTeXMLRef ) is LaTeXMLMath for LaTeXMLMath .
We obtain LaTeXMLEquation which is valid for LaTeXMLMath , for LaTeXMLMath .
Since the right side of this equation is invariant under LaTeXMLMath , we obtain the functional equation LaTeXMLEquation .
Now ( LaTeXMLRef ) implies that LaTeXMLMath is an entire function on LaTeXMLMath , and on setting LaTeXMLMath in ( LaTeXMLRef ) we see that LaTeXMLMath is identically zero .
Thus LaTeXMLMath is also an entire function on LaTeXMLMath , and satisfies the same functional equation .
Viewed as a modular form , LaTeXMLMath in ( LaTeXMLRef ) is not a cusp form , due to its nonzero constant term .
One consequence is that the Mellin transform LaTeXMLMath fails to converge anywhere .
Riemann ’ s second proof of the functional equation ( for LaTeXMLMath ) circumvents this problem by removing the constant term , using LaTeXMLMath in the integrand , and in this case the Mellin transform integral converges for LaTeXMLMath .
In Theorem LaTeXMLRef , the constant term “ evaporates ” because , formally , LaTeXMLEquation .
More precisely LaTeXMLEquation .
LaTeXMLEquation One convention for “ regularization ” of the integral is to analytically continue these two pieces separately and then add them , which results in ( LaTeXMLRef ) .
Theorem LaTeXMLRef justifies this convention by introducing the extra variable LaTeXMLMath , finding a common domain LaTeXMLMath in the LaTeXMLMath -plane where the integral converges , and then analytically continuing in both variables to the line LaTeXMLMath .
We next give modified integral formulas for LaTeXMLMath valid on most of LaTeXMLMath .
We define Heaviside ’ s function LaTeXMLMath for complex LaTeXMLMath with LaTeXMLMath to be LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation where both integrals converge absolutely .
This result expresses LaTeXMLMath by a convergent integral formula with integrand LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation These regions are pictured in Figure LaTeXMLRef ; the dotted line is the “ critical line ” LaTeXMLMath The second integral ( Laplace transform ) follows by the change of variable LaTeXMLMath , so it suffices to consider the first integral .
We recall LaTeXMLEquation .
LaTeXMLEquation Now , we observe that for LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation whichever integral would diverge is killed by the factor of LaTeXMLMath .
By replacing LaTeXMLEquation .
LaTeXMLEquation and similarly for LaTeXMLMath , the formula for LaTeXMLMath simplifies to give the desired result .
Since LaTeXMLMath has singularities at LaTeXMLMath and LaTeXMLMath when LaTeXMLMath is real , we can not obtain quite as nice an expression for LaTeXMLMath along vertical lines LaTeXMLMath .
Indeed , the Heaviside functions are precisely the contributions of the poles as we move the integral through those points ; the poles are also reflected in the fact that for LaTeXMLMath , the integral diverges .
However , if we renormalize the integrals : LaTeXMLEquation .
LaTeXMLEquation then the formula is in fact valid for all LaTeXMLMath ; to prove this , use the identity LaTeXMLEquation valid for LaTeXMLMath , and proceed as before .
The function LaTeXMLMath is a modular form of weight LaTeXMLMath in the variable LaTeXMLMath for LaTeXMLMath in the upper half-plane , with a multiplier system with respect to the theta group LaTeXMLMath is the set of LaTeXMLMath or LaTeXMLMath LaTeXMLMath in LaTeXMLMath .
LaTeXMLMath , a non-normal subgroup of index LaTeXMLMath in the modular group LaTeXMLMath which contains LaTeXMLMath , the principal congruence subgroup of level LaTeXMLMath .
Thus LaTeXMLMath is a modular form of ( complex ) weight LaTeXMLMath with ( non-unitary ) multiplier system on the same group .
We consider its Fourier expansion at the cusp LaTeXMLMath ( of width LaTeXMLMath ) , given by LaTeXMLEquation ( The theta group has two cusps , with the second cusp at LaTeXMLMath , see Bruggeman LaTeXMLCite ; we do not consider the other cusp here . )
In this section our object is to obtain estimates for the size of the Fourier coefficients LaTeXMLMath as LaTeXMLMath for fixed LaTeXMLMath .
At the end of the section we give explicit formulas for a few integer values of LaTeXMLMath where the Fourier coefficients have arithmetic significance , namely LaTeXMLMath and LaTeXMLMath .
We establish basic properties of the Fourier coefficients LaTeXMLMath as a function of LaTeXMLMath .
The Fourier coefficient LaTeXMLMath is a polynomial in LaTeXMLMath of degree LaTeXMLMath .
For each LaTeXMLMath , the polynomial LaTeXMLEquation has nonnegative integer coefficients , lead term LaTeXMLMath , and vanishing constant term .
To prove this result we will need the triple product formula of the Jacobi theta function LaTeXMLMath , see Andrews LaTeXMLCite or Andrews , Askey and Roy LaTeXMLCite .
LaTeXMLMath Jacobi Triple Product Formula LaTeXMLMath The Jacobi theta function LaTeXMLEquation is given by LaTeXMLEquation .
This formula is valid for LaTeXMLMath and all LaTeXMLMath The Fourier coefficients LaTeXMLMath are computable using the expansion LaTeXMLEquation .
Terms involving LaTeXMLMath can appear only for LaTeXMLMath , hence we find that LaTeXMLMath is a polynomial of degree LaTeXMLMath in LaTeXMLMath with rational coefficients and leading term LaTeXMLMath .
Clearly LaTeXMLEquation and LaTeXMLMath Multiplication by LaTeXMLMath clears denominators LaTeXMLMath for LaTeXMLMath hence LaTeXMLMath .
It remains to show nonnegativity .
We have LaTeXMLEquation .
The Jacobi triple product formula gives LaTeXMLEquation hence LaTeXMLEquation .
For LaTeXMLMath real , we have LaTeXMLEquation .
Now LaTeXMLEquation .
LaTeXMLEquation is a bivariate power series with all LaTeXMLMath .
This nonnegativity property is preserved under multiplication of power series , hence LaTeXMLMath inherits this property by ( LaTeXMLRef ) .
Thus all the coefficients of LaTeXMLMath are nonnegative .
One has LaTeXMLEquation in which LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , with LaTeXMLMath .
Let LaTeXMLMath .
In this case LaTeXMLMath is a modular form of real weight LaTeXMLMath , on the theta group LaTeXMLMath , with a unitary multiplier system .
Classical estimates of Petersson LaTeXMLCite and Lehner LaTeXMLCite for the Fourier coefficients of arbitrary modular forms of positive real weight ( with multiplier systems ) show they grow polynomially in LaTeXMLMath , with LaTeXMLEquation with LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , where the LaTeXMLMath -symbol constants depend on LaTeXMLMath in an unspecified manner .
Here we establish some weaker estimates , whose merit is that the dependence on LaTeXMLMath is completely explicit , for use in §6 .
Suppose LaTeXMLMath is real .
( i ) For LaTeXMLMath , LaTeXMLEquation ( ii ) For LaTeXMLMath , LaTeXMLEquation ( i ) Write LaTeXMLMath , so that LaTeXMLMath .
Using Cauchy ’ s theorem we obtain the following formula : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for any choice LaTeXMLMath .
We take LaTeXMLMath , and thus LaTeXMLEquation .
For the other term , we first observe that , since LaTeXMLMath , LaTeXMLEquation .
Since the coefficients of LaTeXMLMath are positive , the maximum occurs for LaTeXMLMath ; thus we must estimate LaTeXMLMath .
Using the functional equation of LaTeXMLMath , we find , for LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation using LaTeXMLEquation .
The bound ( LaTeXMLRef ) follows immediately .
( ii ) For LaTeXMLMath we have LaTeXMLMath , so we may restrict our attention to LaTeXMLMath .
Differentiating LaTeXMLMath by LaTeXMLMath ( denoted by LaTeXMLMath ) and dividing by LaTeXMLMath , we find : LaTeXMLEquation .
Using Cauchy ’ s theorem we obtain the following formula : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Taking LaTeXMLMath as before , we need only estimate the third factor .
The Jacobi triple product formula implies that the coefficients of LaTeXMLMath are alternating in sign ; in other words , using LaTeXMLMath the power series expansion in LaTeXMLMath of LaTeXMLEquation has positive coefficients .
In particular , its maximum is given by LaTeXMLEquation .
LaTeXMLEquation Writing LaTeXMLEquation we estimate LaTeXMLEquation .
LaTeXMLEquation We conclude that LaTeXMLEquation .
Substituting this in ( LaTeXMLRef ) , we conclude LaTeXMLEquation .
Now ( LaTeXMLRef ) follows after multiplying by LaTeXMLMath .
Theorem LaTeXMLRef implies that for real positive LaTeXMLMath the Dirichlet series LaTeXMLEquation converges absolutely on the half-plane LaTeXMLMath The estimate of Petersson LaTeXMLCite for the Fourier coefficients implies that the Dirichlet series converges absolutely in the half-plane LaTeXMLMath for LaTeXMLMath .
It seems likely that for general LaTeXMLMath the Dirichlet series has no half-plane of absolute convergence , except when LaTeXMLMath , because the Fourier coefficients grow too rapidly off the positive real axis .
We do not address this question in this paper .
By Theorem LaTeXMLRef the maximium size of LaTeXMLMath on the circle LaTeXMLMath occurs on the negative real axis LaTeXMLMath Convergent series are known for Fourier coefficients of modular forms of negative real integer weight , see Petersson LaTeXMLCite and Lehner LaTeXMLCite ; the convergent series of Rademacher for the partition function is an example .
The following proposition extracts the main term in that convergent expansion , as given in Lehner LaTeXMLCite ; one can also prove it following the proof for the partition function in Apostol LaTeXMLCite .
For real LaTeXMLMath , there holds LaTeXMLEquation where LaTeXMLMath is the modified Bessel function of the first kind , given by LaTeXMLEquation .
For fixed LaTeXMLMath , this function satisfies LaTeXMLEquation .
The formula ( LaTeXMLRef ) implies a general upper bound of the form LaTeXMLEquation for all complex LaTeXMLMath For a few special values of LaTeXMLMath the associated Dirichlet series has an Euler product , and the Fourier coefficients have an explicit description .
For LaTeXMLMath we define the function LaTeXMLMath by LaTeXMLEquation .
The function LaTeXMLMath can be assigned a ( formal ) Dirichlet series which for LaTeXMLMath is LaTeXMLEquation and for LaTeXMLMath is LaTeXMLEquation in which LaTeXMLMath Here the use of LaTeXMLMath indicates that the ( formal ) Dirichlet series expansion based on Fourier coefficients need not have any region of absolute convergence .
However for real nonnegative LaTeXMLMath it does converge on a half-plane , as follows from Theorem LaTeXMLRef .
It is easy to determine which values LaTeXMLMath give ( formal ) Dirichlet series that have an Euler product .
For LaTeXMLMath the formal Dirichlet series assigned to LaTeXMLMath has an Euler product expansion if and only if LaTeXMLMath and LaTeXMLMath .
To see that LaTeXMLMath and LaTeXMLMath are the only complex values for which LaTeXMLMath can have an Euler product , we consider the necessary conditon LaTeXMLEquation .
This gives a polynomial of degree five in LaTeXMLMath whose roots are LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath the Dirichlet series LaTeXMLMath has an Euler product .
For LaTeXMLMath we have already seen that LaTeXMLEquation .
For LaTeXMLMath , LaTeXMLEquation .
For LaTeXMLMath LaTeXMLEquation .
For LaTeXMLMath , LaTeXMLEquation .
These Dirichlet series are scaled multiples of the zeta functions for the rational field , the Gaussian integers , the integral quaternions and the integral octaves , respectively .
For LaTeXMLMath the Dirichlet series also has an Euler product , which is LaTeXMLEquation .
This follows from Theorem LaTeXMLRef below ; note that it converges absolutely for LaTeXMLMath One immediately sees from the expression for LaTeXMLMath as a product of shifted Riemann zeta functions that most of its zeros can not be on “ critical line ” LaTeXMLMath The only zeros on the line come from the Euler factor at the prime LaTeXMLMath and have are of number LaTeXMLMath up to height LaTeXMLMath , while there are LaTeXMLMath zeros off the line coming from the Riemann zeta zeros .
Exactly the same thing happens for LaTeXMLMath There are other positive integer values of LaTeXMLMath where the Dirichlet series can be determined explicitly , without an Euler product .
For LaTeXMLMath the Dirichlet series is a linear combination of two Dirichlet series with Euler products , namely LaTeXMLEquation in which LaTeXMLMath with LaTeXMLMath being the Jacobi symbol .
It is well known that the Riemann LaTeXMLMath -function LaTeXMLMath is an entire function of order one and infinite type .
It is bounded in vertical strips .
In this section we show that both these properties generalize to the function LaTeXMLMath .
We prove the following bound for the the two-variable zeta function .
There is a constant LaTeXMLMath such that the entire function LaTeXMLMath satisfies the growth bound : For all LaTeXMLMath , if LaTeXMLMath then LaTeXMLEquation .
Set LaTeXMLEquation .
Then ( 2.9 ) gives LaTeXMLEquation .
For LaTeXMLMath LaTeXMLEquation and LaTeXMLMath for all LaTeXMLMath since all of the terms are positive .
It follows that for all positive LaTeXMLMath and all LaTeXMLMath , LaTeXMLEquation .
Moreover , the right-hand-side is increasing in LaTeXMLMath , so we find that for all LaTeXMLMath LaTeXMLEquation using LaTeXMLMath for LaTeXMLMath .
Similarly , the function LaTeXMLMath is decreasing in LaTeXMLMath , so we find that for LaTeXMLMath , we have : LaTeXMLEquation where LaTeXMLMath .
In particular , for LaTeXMLMath , LaTeXMLEquation .
We thus have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath , and LaTeXMLMath on this range , it follows that there exists a constant LaTeXMLMath such that LaTeXMLEquation .
Since both LaTeXMLMath and LaTeXMLMath are LaTeXMLMath , the theorem follows .
A notion of entire function of finite order for functions of several complex variables is described in Ronkin LaTeXMLCite , LaTeXMLCite and Stoll LaTeXMLCite .
In particular , there is a Weierstrass factorization theorem for such functions in terms of their zero locus .
The zero locus LaTeXMLMath of LaTeXMLMath is a one-dimensional complex analytic manifold , possibly with singular points , which may have many connected components .
One way to study it is to take “ linear slices ” to obtain functions LaTeXMLMath of one complex variable , whose zero sets consist of isolated points which can be counted .
If LaTeXMLMath and LaTeXMLMath with LaTeXMLMath , the linear slice function LaTeXMLMath of LaTeXMLMath , is LaTeXMLEquation where we assume LaTeXMLMath to avoid constant functions .
Any linear slice function LaTeXMLMath is an entire function of order at most LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLEquation where the implied constant in the LaTeXMLMath -symbol depends on LaTeXMLMath .
This follows from the growth estimate in Theorem LaTeXMLRef , using Jensen ’ s formula .
This result applies in particular to linear slices where LaTeXMLMath is held fixed , i.e .
LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
which gives the function LaTeXMLMath with LaTeXMLMath regarded as constant , e. g. in §7 .
We next consider growth bounds for LaTeXMLMath on vertical lines LaTeXMLMath with LaTeXMLMath held fixed .
Recall that a function LaTeXMLMath is in the Schwartz space LaTeXMLMath if and only if for each LaTeXMLMath there is a finite constant LaTeXMLMath such that LaTeXMLEquation with a similar definition for functions defined on a closed half-line LaTeXMLMath or LaTeXMLMath .
For each LaTeXMLMath , LaTeXMLMath , and real LaTeXMLMath , the function LaTeXMLEquation belongs to the Schwartz space LaTeXMLMath Furthermore , the implied Schwartz constants can be chosen uniformly on any compact subset LaTeXMLMath of LaTeXMLMath In particular , these functions are bounded in vertical strips , i.e .
there is a finite constant LaTeXMLMath such that LaTeXMLEquation .
We will deduce this result from an integral representation of LaTeXMLMath which we prove first .
We define the function LaTeXMLMath for LaTeXMLMath and LaTeXMLMath with LaTeXMLMath by LaTeXMLEquation .
Next we define LaTeXMLEquation .
We have the following Fourier-Laplace transform formula for LaTeXMLMath valid on LaTeXMLMath Let LaTeXMLMath , and real LaTeXMLMath with LaTeXMLMath Then for all LaTeXMLMath , LaTeXMLEquation .
Regarded as a Fourier transform , with LaTeXMLMath with fixed LaTeXMLMath , the integrand LaTeXMLEquation belongs to the Schwartz space LaTeXMLMath and the implied Schwartz constants are uniform on compact subsets of LaTeXMLMath To prove Theorem LaTeXMLRef we formulate several estimates as preliminary lemmas .
Define an analytic function LaTeXMLMath for LaTeXMLMath , with LaTeXMLMath , by LaTeXMLEquation .
For any integer LaTeXMLMath , and for any region of the form LaTeXMLEquation there is a finite constant LaTeXMLMath such that LaTeXMLEquation .
For LaTeXMLMath the results of §3 show that LaTeXMLMath has a Fourier expansion : LaTeXMLEquation .
Note that LaTeXMLMath is well-defined as a polynomial in LaTeXMLMath , hence this expansion makes sense for LaTeXMLMath as well , with Fourier coefficient LaTeXMLMath Now , for LaTeXMLMath , we have the inequality LaTeXMLEquation this follows from the fact that LaTeXMLMath has positive coefficients , and is 0 at LaTeXMLMath .
Moreover , this upper bound is monotonically increasing in LaTeXMLMath .
In particular , since LaTeXMLMath has radius of convergence 1 , and no zeros in the unit disk , we find that for fixed LaTeXMLMath , LaTeXMLEquation has radius of convergence 1 , and thus LaTeXMLEquation .
In particular , since LaTeXMLMath , the condition LaTeXMLMath makes LaTeXMLMath analytic in the region LaTeXMLMath Similarly , for the LaTeXMLMath th derivative , we have : LaTeXMLEquation and , using term-by-term absolute value estimates , LaTeXMLEquation .
Over the region LaTeXMLMath , LaTeXMLMath , this is bounded by its value for LaTeXMLMath , LaTeXMLMath ; as the sum converges for those values , we obtain the required uniform bound .
Let LaTeXMLMath , and for fixed LaTeXMLMath and fixed LaTeXMLMath , the function LaTeXMLEquation where LaTeXMLMath is defined as in Theorem LaTeXMLRef , lies in the Schwartz space LaTeXMLMath .
Moreover , the implied Schwartz bounds are uniform over compact regions in LaTeXMLMath .
By definition LaTeXMLMath , in which LaTeXMLMath so that LaTeXMLMath We claim that for any LaTeXMLMath , the function LaTeXMLMath is a Schwartz function on the half-line LaTeXMLMath , and that the implied Schwartz bounds are uniform over compact regions in LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
To see this , note that for LaTeXMLMath , we find LaTeXMLMath and since LaTeXMLMath is bounded away from LaTeXMLMath , LaTeXMLMath is bounded away from 0 .
Now using Lemma LaTeXMLRef , we have LaTeXMLEquation as LaTeXMLMath , uniformly in a compact region in LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , proving the claim .
( The function LaTeXMLMath and its derivatives go to zero at a super-exponential rate as LaTeXMLMath , all other variables fixed . )
Now applying the operator LaTeXMLMath to LaTeXMLMath , the claim implies that LaTeXMLMath is a Schwartz function in LaTeXMLMath on LaTeXMLMath The functional equation LaTeXMLMath then yields LaTeXMLEquation so that LaTeXMLMath is a Schwartz function in LaTeXMLMath on LaTeXMLMath as well .
Thus it is in LaTeXMLMath , and the uniformity of the estimates in compact regions in LaTeXMLMath , LaTeXMLMath is inherited .
To prove Theorem LaTeXMLRef , we will use repeated integration by parts starting from the integral representation for LaTeXMLMath in Theorem LaTeXMLRef .
To justify this step we show that the integrand of that representation is a Schwarz function for LaTeXMLMath Let LaTeXMLMath , and LaTeXMLMath be fixed , with LaTeXMLMath .
Then for LaTeXMLMath and LaTeXMLMath , the functions of LaTeXMLMath , LaTeXMLEquation all belong to the Schwartz space LaTeXMLMath It suffices to show that LaTeXMLMath and LaTeXMLMath are Schwartz functions on the half-line LaTeXMLMath .
We have LaTeXMLEquation using the relation LaTeXMLMath We also have LaTeXMLEquation using the functional equation for LaTeXMLMath In each case , the first term has been shown to be Schwartz , and to be uniform in the parameters , in the proof of Lemma LaTeXMLRef .
It remains only to consider the “ correction ” terms , on the half-line LaTeXMLMath .
Suppose LaTeXMLMath , it suffices to observe that for all LaTeXMLMath with LaTeXMLMath and fixed LaTeXMLMath the function LaTeXMLMath is Schwartz on the half-line LaTeXMLMath ; indeed , LaTeXMLEquation .
When LaTeXMLMath , the exponential dominates , and thus the function is bounded ; when LaTeXMLMath , LaTeXMLMath , so the function is 0 .
Since LaTeXMLMath , we can safely divide by LaTeXMLMath without affecting boundedness .
Suppose LaTeXMLMath .
Then the only new “ correction ” term is the function LaTeXMLMath , which again on a half-line LaTeXMLMath is in the Schwartz space .
With some further work it can be shown that in Lemma LaTeXMLRef the implied constants for the Schwartz functions are uniform over compact regions in LaTeXMLMath -space that avoid the two lines LaTeXMLMath and LaTeXMLMath However we do not need this result .
We take LaTeXMLMath .
When LaTeXMLMath , LaTeXMLMath , Theorem LaTeXMLRef gives LaTeXMLEquation where LaTeXMLMath is given in Lemma LaTeXMLRef .
Now LaTeXMLMath is a Schwartz function in LaTeXMLMath by Lemma LaTeXMLRef , so we may integrate by parts twice , using LaTeXMLMath , obtaining LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation This integral agrees with LaTeXMLMath off the lines LaTeXMLMath Since the integrand is uniformly Schwartz by Lemma LaTeXMLRef , this integral gives an analytic function of LaTeXMLMath and LaTeXMLMath , and must therefore agree with LaTeXMLMath everywhere .
View the integral representation of LaTeXMLMath in Theorem LaTeXMLRef as a Fourier transform , with LaTeXMLMath with fixed LaTeXMLMath Since the Fourier transform maps Schwartz space LaTeXMLMath to itself , it follows that for fixed LaTeXMLMath , LaTeXMLEquation belongs to LaTeXMLMath .
The uniformity of the Schwartz constants on compact subsets LaTeXMLMath of LaTeXMLMath -space is inherited from the corresponding uniformity property in Theorem LaTeXMLRef .
We conclude this section with another consequence of the Fourier-Laplace integral representation of LaTeXMLMath by ( uniform ) Schwartz functions .
Let LaTeXMLMath be any polynomial .
Then for any LaTeXMLMath with LaTeXMLMath and LaTeXMLMath with LaTeXMLMath , LaTeXMLEquation .
Here the integrand is a Schwartz function of LaTeXMLMath .
Using Lemma LaTeXMLRef , the integral is a Fourier transform with integrand in LaTeXMLMath , viewing LaTeXMLMath as fixed .
The case LaTeXMLMath follows from Theorem LaTeXMLRef by taking the Fourier transform , since the right side of that formula can be viewed as an inverse Fourier transform .
Now use the fact that the Fourier transform leaves LaTeXMLMath invariant and transforms multiplication by LaTeXMLMath to differentiation , and apply LaTeXMLMath to both sides of the identity with polynomial LaTeXMLMath We evaluate the entire function LaTeXMLMath in the plane LaTeXMLMath .
The entire function LaTeXMLMath of two complex variables has LaTeXMLEquation in which LaTeXMLMath ( 1 ) It is evident from ( LaTeXMLRef ) that LaTeXMLMath satisfies the functional equation LaTeXMLEquation ( 2 ) Comparing the formula LaTeXMLMath with Theorem LaTeXMLRef , and using the functional equation for LaTeXMLMath leads to LaTeXMLEquation .
It is evident that this Dirichlet series has an Euler product , already stated in §3.4 .
The proof of Theorem LaTeXMLRef depends on the Jacobi triple product formula , which is used in evaluating the Fourier coefficients of LaTeXMLMath .
We state this as a preliminary lemma .
The coefficients LaTeXMLMath of LaTeXMLEquation are given by LaTeXMLEquation where LaTeXMLEquation .
We use the eta product LaTeXMLEquation .
By the Jacobi triple product formula , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
If we define LaTeXMLMath by LaTeXMLEquation then we have LaTeXMLEquation with the convention that LaTeXMLMath if LaTeXMLMath .
Now , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation so that LaTeXMLMath as required .
Since LaTeXMLMath is an entire function of two variables we have , for positive real LaTeXMLMath , LaTeXMLEquation .
Fix LaTeXMLMath ; for LaTeXMLMath , we have LaTeXMLEquation .
Suppose LaTeXMLMath Letting LaTeXMLMath , and using LaTeXMLMath we eventually have LaTeXMLMath and so we legitimately obtain LaTeXMLEquation .
Now expand LaTeXMLMath in Fourier series and integrate term-by-term to obtain LaTeXMLEquation and this converges for LaTeXMLMath since LaTeXMLMath by Lemma LaTeXMLRef .
Setting LaTeXMLEquation and then using ( LaTeXMLRef ) gives LaTeXMLEquation .
However we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation valid whenever LaTeXMLMath ( since LaTeXMLMath ) .
We thus conclude that for LaTeXMLMath LaTeXMLEquation .
Using the duplication formula LaTeXMLMath and the functional equation for the zeta function , we obtain for LaTeXMLMath that LaTeXMLEquation .
Since both sides are analytic in LaTeXMLMath , the formula is valid for all LaTeXMLMath .
In this section we suppose LaTeXMLMath is fixed , and study the zeros of LaTeXMLMath .
We will first show that the zeros of LaTeXMLMath are localized in a vertical strip centered on the “ critical line ” LaTeXMLMath whose width depends on LaTeXMLMath , and then we shall derive an estimate for the number of zeros with imaginary part of height at most LaTeXMLMath .
We show for real LaTeXMLMath that the zeros of LaTeXMLMath are confined to a vertical strip of width LaTeXMLMath .
Theorem LaTeXMLRef implies that this bound is valid for LaTeXMLMath as well .
Let LaTeXMLMath be a fixed real number .
Then the entire function LaTeXMLMath has all its zeros in the vertical strip LaTeXMLEquation .
We have LaTeXMLEquation where LaTeXMLEquation converges absolutely for LaTeXMLMath by Theorem LaTeXMLRef ( i ) , and meromorphically continues to LaTeXMLMath All zeros of LaTeXMLMath with LaTeXMLMath must come from those of the Dirichlet series LaTeXMLMath .
The Dirichlet series LaTeXMLMath has no zeros in any half plane LaTeXMLMath for any LaTeXMLMath with LaTeXMLEquation .
Since LaTeXMLMath , for LaTeXMLMath we may rewrite this as LaTeXMLEquation .
Now Theorem LaTeXMLRef ( ii ) gives LaTeXMLEquation .
Choosing LaTeXMLMath yields LaTeXMLEquation as required .
Thus LaTeXMLMath has no zeros in LaTeXMLMath , hence LaTeXMLMath also has no zeros there .
Finally the functional equation LaTeXMLMath implies that LaTeXMLMath has no zeros in the region LaTeXMLMath The width of the strip of Lemma LaTeXMLRef is qualitatively correct , in that it must grow like LaTeXMLMath for large LaTeXMLMath and it must be of positive width , at least LaTeXMLMath , as LaTeXMLMath to accomodate the zeros of LaTeXMLMath given in Theorem LaTeXMLRef .
We establish the following estimate for the number of zeros LaTeXMLMath within distance LaTeXMLMath of the real axis of LaTeXMLMath , which generalizes a similar estimate for the Riemann zeta function .
There is an absolute constant LaTeXMLMath such that , for all real LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation in which LaTeXMLMath satisfies LaTeXMLEquation .
The proof of this result generalizes the proof for LaTeXMLMath in Davenport LaTeXMLCite , with some extra work to control the dependence in LaTeXMLMath in all estimates .
We prove several preliminary lemmas .
We use the argument principle , and let LaTeXMLMath denote the change in argument LaTeXMLMath in a function LaTeXMLMath along a contour LaTeXMLMath on which LaTeXMLMath never vanishes .
For positive real LaTeXMLMath , the zeros of LaTeXMLMath are those of the analytic continuation of the Dirichlet series LaTeXMLMath given in ( LaTeXMLRef ) , possibly excluding zeros of LaTeXMLMath at negative even integers .
We consider the rectangular contour LaTeXMLMath , oriented counterclockwise , with corners at LaTeXMLMath and LaTeXMLMath , where LaTeXMLEquation and LaTeXMLMath is chosen to avoid any zeros of LaTeXMLMath ( See Figure 5.1 ) We will mainly use the quarter-contour LaTeXMLMath consisting of a vertical line LaTeXMLMath from LaTeXMLMath to LaTeXMLMath , followed by a horizontal line LaTeXMLMath from LaTeXMLMath to LaTeXMLMath .
For real LaTeXMLMath , and LaTeXMLMath , LaTeXMLEquation with LaTeXMLEquation .
We have LaTeXMLEquation on the rectangular contour LaTeXMLMath , oriented counterclockwise , which has its corners at LaTeXMLMath and LaTeXMLMath , as given above .
The functional equation LaTeXMLMath and the symmetry LaTeXMLMath imply that LaTeXMLEquation on the quarter-contour LaTeXMLMath , with each other quarter-contour of LaTeXMLMath contributing the same amount .
Now LaTeXMLEquation so we obtain LaTeXMLEquation .
The first three terms on the right contribute LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where Stirling ’ s formula is used for the last estimate .
This yields ( LaTeXMLRef ) .
Our object will be to estimate LaTeXMLMath using the formula LaTeXMLEquation starting from the endpoint LaTeXMLMath of LaTeXMLMath , where the next lemma shows LaTeXMLMath is real and positive and LaTeXMLMath is real and positive .
In the integral we analytically continue LaTeXMLMath along LaTeXMLMath , and we choose LaTeXMLMath so that the contour LaTeXMLMath encounters no zero of LaTeXMLMath .
We need information on the zeros LaTeXMLMath of LaTeXMLMath obtained from the Hadamard product .
Let LaTeXMLMath be real .
For LaTeXMLMath there holds LaTeXMLEquation .
For LaTeXMLMath , LaTeXMLEquation and there is an absolute constant LaTeXMLMath independent of LaTeXMLMath such that LaTeXMLEquation ( i ) .
We use the Hadamard product expansion LaTeXMLEquation where LaTeXMLMath is the set of zeros of LaTeXMLMath counted with multiplicity .
This formula is valid because LaTeXMLMath is an entire function of order at most one by the growth estimate of Theorem LaTeXMLRef .
Note that LaTeXMLMath because for fixed LaTeXMLMath the function LaTeXMLMath has a simple pole with residue 1 at LaTeXMLMath The derivation of LaTeXMLCite yields the formula LaTeXMLEquation where the prime in the first sum indicates that complex conjugate zeros LaTeXMLMath and LaTeXMLMath are to be summed in pairs , and the last sum converges absolutely .
We set equal the logarithmic derivatives of ( LaTeXMLRef ) and ( LaTeXMLRef ) , to obtain LaTeXMLEquation .
This yields LaTeXMLEquation .
Taking real parts yields , LaTeXMLEquation .
Now ( LaTeXMLRef ) holds in the entire plane by analytic continuation , since the functions are single-valued .
Applying the formula for LaTeXMLMath simplifies ( LaTeXMLRef ) to ( LaTeXMLRef ) , proving ( i ) .
( ii ) Suppose LaTeXMLMath .
Then the formula ( LaTeXMLRef ) for LaTeXMLMath and Theorem LaTeXMLRef ( ii ) give LaTeXMLEquation .
This implies ( LaTeXMLRef ) .
Next , applying Theorem LaTeXMLRef ( ii ) again , LaTeXMLEquation which gives ( LaTeXMLRef ) .
( i ) There is an absolute constant LaTeXMLMath , such that for all LaTeXMLMath , and all real LaTeXMLMath , LaTeXMLEquation ( ii ) There is an absolute constant LaTeXMLMath such that for all LaTeXMLMath and all LaTeXMLMath , the number of zeros LaTeXMLMath with LaTeXMLEquation counting multiplicity , is at most LaTeXMLEquation ( i ) .
Choose LaTeXMLMath with LaTeXMLEquation so LaTeXMLMath Now apply ( LaTeXMLRef ) and the bound ( LaTeXMLRef ) to obtain LaTeXMLEquation .
We recall the formula LaTeXMLEquation valid for LaTeXMLMath for any fixed LaTeXMLMath .
( We choose LaTeXMLMath . )
Now ( LaTeXMLRef ) gives LaTeXMLEquation .
Thus ( LaTeXMLRef ) becomes LaTeXMLEquation .
The bound of Lemma LaTeXMLRef gives LaTeXMLEquation .
Thus LaTeXMLEquation and ( LaTeXMLRef ) follows .
( ii ) .
Let LaTeXMLMath denote the set of zeros in LaTeXMLMath satisfying LaTeXMLMath Then , since LaTeXMLMath , LaTeXMLEquation .
Combining this with ( LaTeXMLRef ) implies ( LaTeXMLRef ) with LaTeXMLMath .
There is an absolute constant LaTeXMLMath , such that for LaTeXMLMath and LaTeXMLMath with LaTeXMLMath in the region LaTeXMLEquation there holds , for all LaTeXMLMath LaTeXMLEquation .
Set LaTeXMLMath , and LaTeXMLMath .
Then , differencing ( LaTeXMLRef ) at LaTeXMLMath and LaTeXMLMath , we obtain LaTeXMLEquation .
The bound LaTeXMLMath ensures that LaTeXMLMath has LaTeXMLMath for LaTeXMLMath , hence the bounds ( LaTeXMLRef ) applies to give LaTeXMLEquation .
Now LaTeXMLEquation .
For those zeros with LaTeXMLMath , Lemma LaTeXMLRef ( i ) gives the bound LaTeXMLEquation .
If LaTeXMLMath , then by Lemma LaTeXMLRef ( ii ) the number of such zeros is LaTeXMLMath and for each one LaTeXMLEquation .
So their total contribution is LaTeXMLMath in ( LaTeXMLRef ) .
Substituting these bounds in ( LaTeXMLRef ) yields LaTeXMLEquation as required .
Set LaTeXMLMath .
Recall that the quarter- contour LaTeXMLMath consists of the vertical segment LaTeXMLMath from LaTeXMLMath to LaTeXMLMath , and the horizontal segment LaTeXMLMath from LaTeXMLMath to LaTeXMLMath , so that LaTeXMLEquation by ( LaTeXMLRef ) .
We have LaTeXMLEquation using Lemma LaTeXMLRef ( ii ) .
Now LaTeXMLEquation and to estimate this we apply Lemma LaTeXMLRef .
For each zero LaTeXMLMath we have LaTeXMLEquation which contributes at most LaTeXMLMath .
Now Lemma LaTeXMLRef ( ii ) gives that there are at most LaTeXMLMath such zeros in the sum ( LaTeXMLRef ) , so their total contribution to the argument is at most LaTeXMLMath .
The error term in ( LaTeXMLRef ) integrated over LaTeXMLMath contributes at most a further LaTeXMLMath to the argument , since LaTeXMLMath is a path of length at most LaTeXMLMath .
Combining these estimates in ( LaTeXMLRef ) yields the bound ( LaTeXMLRef ) , completing the proof .
The zero-counting estimate of Theorem LaTeXMLRef for LaTeXMLMath is easily checked to remains valid for LaTeXMLMath by virtue of the explicit formula for LaTeXMLMath in Theorem LaTeXMLRef .
Note that the extra zeros provided by the terms LaTeXMLMath are needed to make the main term ( LaTeXMLRef ) valid .
It is interesting to examine the behavior of the zero set of LaTeXMLMath for nonnegative real LaTeXMLMath , as LaTeXMLMath is varied .
As above , consider variation LaTeXMLMath , or , more generally , over a fixed bounded range of LaTeXMLMath .
For that range of LaTeXMLMath , Theorem LaTeXMLRef asserts that the general density of zeros to height LaTeXMLMath remains almost constant , with a variation of LaTeXMLMath Since the number of zeros in a unit interval at this height is of the same order , it suggests that every zero can move vertically a distance of at most LaTeXMLMath , while the horizontal movement is certainly restricted to distance LaTeXMLMath by Lemma LaTeXMLRef .
Therefore it would appear that varying LaTeXMLMath , there is a constant LaTeXMLMath depending on LaTeXMLMath such that every zero moves at most the bounded amount LaTeXMLMath , independent of the height LaTeXMLMath of this zero in the critical strip .
We can not assert this rigorously , however , because we have not ruled out the possibility of zeros going off the line in pairs and then hop-scotching around other zeros remaining on the line .
LaTeXMLEquation .
Theorem LaTeXMLRef for LaTeXMLMath and LaTeXMLMath shows that the zero-counting functions of LaTeXMLMath and LaTeXMLMath have extemely similar asymptotics .
In Table LaTeXMLRef we compare the first LaTeXMLMath such zeros .
Since LaTeXMLMath , we have indicated the zeros of LaTeXMLMath associated to LaTeXMLMath with an asterisk , and the remainder come from LaTeXMLMath .
Note that the appearance of every zeta zero on both sides of this table shows that zeta zeros have a kind of self-similar structure by powers of two .
However this is only approximately true , because there is no precise correspondence of zeta zeros due to the phenomonon of zeros coalescing as LaTeXMLMath varies , as indicated in Lemma LaTeXMLRef in §8 .
In this section we suppose LaTeXMLMath is negative real ( LaTeXMLMath ) and fixed .
We will not find zeros , but will instead specify places in the LaTeXMLMath -plane where the the function LaTeXMLMath has no zeros .
An interesting extra structure underlying certain properties of the two-variable zeta function LaTeXMLMath for negative real LaTeXMLMath is a holomorphic convolution semigroup of complex-valued measures described in §7.2 .
On the critical line LaTeXMLMath we will show these are real-valued positive measures , normalizable to be probability measures , in §7.1 .
We will establish the following result , concerning the absence of zeros on the “ critical line ” LaTeXMLMath ( LaTeXMLMath ) , which will be deduced from Theorem LaTeXMLRef below .
LaTeXMLMath Positivity Property LaTeXMLMath For real LaTeXMLMath with LaTeXMLMath and all real LaTeXMLMath , LaTeXMLEquation .
Recall that the functional equation on the “ critical line ” LaTeXMLMath implies that LaTeXMLEquation hence LaTeXMLMath is real .
For real LaTeXMLMath , the integral representation LaTeXMLEquation converges absolutely for all LaTeXMLMath , and it gives LaTeXMLEquation since the integrand is positive .
Thus the assertion that LaTeXMLMath for all real LaTeXMLMath is equivalent to the positivity condition ( LaTeXMLRef ) .
By changing variables in the integral ( LaTeXMLRef ) , with LaTeXMLMath , we obtain LaTeXMLEquation .
This implies that the function LaTeXMLEquation has inverse Fourier transform LaTeXMLEquation .
The function LaTeXMLMath is an even function and has LaTeXMLEquation .
This equation shows that LaTeXMLMath is a signed measure of mass one .
Theorem LaTeXMLRef asserts that for LaTeXMLMath for all LaTeXMLMath holds for all LaTeXMLMath , which would imply that LaTeXMLMath is a probability measure .
In any case LaTeXMLMath is the characteristic function of the ( signed ) measure LaTeXMLMath , and ( LaTeXMLRef ) shows that LaTeXMLMath where LaTeXMLEquation where the last expression is derived using the functional equation for the theta function .
The assertion that LaTeXMLMath is a characteristic function for all real LaTeXMLMath is equivalent to the assertion that each LaTeXMLMath is an infinitely divisible probability measure ; the collection LaTeXMLMath then form a semigroup under convolution .
We note that the normalizing factor LaTeXMLEquation appearing in ( LaTeXMLRef ) is the invariant LaTeXMLMath introduced by van der Geer and Schoof LaTeXMLCite .
They define the genus of LaTeXMLMath to be the “ dimension ” LaTeXMLMath of the canonical divisor LaTeXMLMath , which is LaTeXMLEquation see the appendix .
A necessary and sufficient condition for a function to be a characteristic function of an infinitely divisible probability measure was developed by Khintchine , Levy and Kolmogrov .
We follow the treatment in Feller LaTeXMLCite .
A measure LaTeXMLMath on LaTeXMLMath is called canonical if it is nonnegative , assigns finite masses to finite intervals and if both the integrals LaTeXMLEquation converge for some ( and therefore all ) LaTeXMLMath .
A complex-valued function LaTeXMLMath on LaTeXMLMath is the characteristic function of an infinitely divisible probability measure if and only if LaTeXMLMath with LaTeXMLMath having the form LaTeXMLEquation for some canonical measure LaTeXMLMath and real constant LaTeXMLMath .
The canonical measure LaTeXMLMath and constant LaTeXMLMath are unique .
This is shown in Feller LaTeXMLCite .
The representation ( LaTeXMLRef ) implies that LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath and that LaTeXMLMath is well-defined , with its imaginary part determined by continuity starting from LaTeXMLMath We call the measure LaTeXMLMath in Proposition LaTeXMLRef , which may have infinite mass , the Feller canonical measure associated to LaTeXMLMath .
A related canonical measure is the Khintchine canonical measure LaTeXMLMath , given by LaTeXMLEquation .
It can be an arbitrary bounded nonnegative measure , see Feller LaTeXMLCite .
Biane , Pitman and Yor LaTeXMLCite consider the Levy-Khintchine canonical measure LaTeXMLMath which is defined for infinitely divisible distributions supported on LaTeXMLMath , and is related to the corresponding Feller canonical measure by LaTeXMLEquation .
Note that the Feller canonical measure LaTeXMLMath given in Theorem LaTeXMLRef has support on the whole real line , so has no associated Levy-Khintchine canonical measure .
We will use a variant of this result which characterizes infinitely divisible distributions with finite second moment .
A complex-valued function LaTeXMLMath on LaTeXMLMath is the characteristic function of an infinitely divisible probability distribution having a finite second moment if ( and only if ) LaTeXMLMath is a LaTeXMLMath -function , LaTeXMLMath , LaTeXMLMath , for all LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation has LaTeXMLMath and LaTeXMLMath is the characteristic function of a positive measure LaTeXMLMath of finite mass , LaTeXMLEquation .
If so , then LaTeXMLMath is the associated Feller canonical measure to LaTeXMLMath .
The “ if ” part of this result appears in Feller LaTeXMLCite .
We will not use the “ only if ” part of the result and omit its proof .
The function LaTeXMLEquation on LaTeXMLMath is the characteristic function of an infinitely divisible probability measure with finite second moment .
Its associated Feller canonical measure LaTeXMLMath is equal to LaTeXMLMath with LaTeXMLEquation in which LaTeXMLMath .
We apply the criterion of Proposition LaTeXMLRef .
We start from the function LaTeXMLEquation and must show that LaTeXMLMath and that LaTeXMLEquation is a nonnegative function having finite mass .
We have LaTeXMLMath and LaTeXMLEquation using ( LaTeXMLRef ) .
Now Theorem LaTeXMLRef gives , on taking LaTeXMLMath and LaTeXMLMath , that for all LaTeXMLMath , LaTeXMLEquation and it follows that LaTeXMLMath Now Theorem LaTeXMLRef , which uses the Jacobi triple product formula , gives LaTeXMLEquation .
Using LaTeXMLMath we obtain LaTeXMLEquation .
This shows that LaTeXMLMath is nonnegative , and its strict positivity follows from the well-known result that LaTeXMLMath is nonzero on the line LaTeXMLMath , using LaTeXMLMath .
The positive measure LaTeXMLMath has finite mass since LaTeXMLMath is a Schwartz function by Theorem LaTeXMLRef , so the result follows by Proposition LaTeXMLRef .
( The mass of LaTeXMLMath is explicitly determined in Theorem LaTeXMLRef below ; numerically it is about LaTeXMLMath ) For real LaTeXMLMath the function LaTeXMLMath with LaTeXMLMath given by ( LaTeXMLRef ) is by Theorem LaTeXMLRef the characteristic function of a probability density of a nonnegative measure .
The measure LaTeXMLMath is positive except on a discrete set because the density LaTeXMLMath is an analytic function of LaTeXMLMath .
Then , using the infinite divisibility property , we have LaTeXMLEquation giving positivity for all real LaTeXMLMath .
The probability density LaTeXMLMath is given by ( LaTeXMLRef ) , and LaTeXMLMath , so we conclude that LaTeXMLMath for all real LaTeXMLMath .
In §7.1 we showed that for LaTeXMLMath the family of probability measures LaTeXMLMath having the density functions LaTeXMLEquation form a semigroup under convolution , i.e .
LaTeXMLMath We can extend this to a convolution semigroup of complex-valued measures on the real line , indexed by two real parameters LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , a region which forms an open cone in LaTeXMLMath closed under addition .
Given such LaTeXMLMath we define a complex-valued measure LaTeXMLMath on the real line by LaTeXMLEquation .
In terms of the function LaTeXMLMath these values occupy the real-codimension one cone LaTeXMLEquation which is contained in the absolute convergence region LaTeXMLMath of the integral representation ( LaTeXMLRef ) .
For all LaTeXMLMath and real LaTeXMLMath , LaTeXMLMath is a complex-valued measure with LaTeXMLEquation .
The measures LaTeXMLMath form a convolution semigroup .
That is , for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we have : LaTeXMLEquation .
In particular LaTeXMLEquation .
The first formula follows from the formula LaTeXMLEquation which generalizes ( LaTeXMLRef ) .
Indeed the functions LaTeXMLEquation form a semigroup under multiplication , i.e .
LaTeXMLEquation and the inverse Fourier transform relation ( LaTeXMLRef ) implies that the measures LaTeXMLMath form a semigroup under convolution , i.e .
LaTeXMLEquation as required .
This semigroup is holomorphic in the sense that the density functions LaTeXMLMath are holomorphic functions of LaTeXMLMath in the cone where they are defined .
Below we compute the moments of the distributions LaTeXMLMath using Lemma LaTeXMLRef .
Before doing that , we derive a formula for the logarithmic derivatives of the theta function LaTeXMLMath at LaTeXMLMath .
Set LaTeXMLEquation .
For each LaTeXMLMath , LaTeXMLEquation where LaTeXMLEquation .
It is given by an even polynomial of degree at most LaTeXMLMath in LaTeXMLMath , In particular , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Consider the polynomial ring LaTeXMLMath generated by the three functions LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation These functions are all ( essentially ) modular forms of weight LaTeXMLMath on the theta group .
( The Eisenstein series LaTeXMLMath is not quite a modular form . )
One has LaTeXMLEquation where the last two are derived using the transformation laws for LaTeXMLMath and LaTeXMLMath .
It follows that if LaTeXMLMath , then LaTeXMLMath We claim that the polynomial ring LaTeXMLMath is closed under differentiation LaTeXMLMath Indeed one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
These can be deduced using properties of derivatives of modular forms ; the operator LaTeXMLMath must take both LaTeXMLMath and LaTeXMLMath into a modular form of weight 4 , the specific one being determined by the first few Fourier coefficients , and LaTeXMLMath does the same for LaTeXMLMath The procedure is that used in Lang LaTeXMLCite .
Furthermore , by symmetry , we observe that the subring LaTeXMLMath is also closed under differentiation .
Now , we observe that LaTeXMLEquation so for each LaTeXMLMath , LaTeXMLMath .
The theorem follows upon evaluation at LaTeXMLMath .
The explicit formulas were found by computation .
We recall that the cumulants LaTeXMLMath of a probability distribution LaTeXMLMath are defined in terms of the characteristic function of LaTeXMLMath by : LaTeXMLEquation .
In particular , LaTeXMLMath is the mean and LaTeXMLMath is the variance of LaTeXMLMath .
We extend this definition to the complex measures LaTeXMLMath .
The usefulness of cumulants for infinitely divisible distributions ( as opposed to moments ) is that they scale nicely with the parameter LaTeXMLMath .
For real LaTeXMLMath and real LaTeXMLMath , the mean value LaTeXMLEquation .
For all LaTeXMLMath , the LaTeXMLMath -th cumulant of the distribution LaTeXMLMath has the form LaTeXMLEquation where LaTeXMLMath in LaTeXMLMath , with LaTeXMLEquation .
Moreover , if LaTeXMLMath is odd , then LaTeXMLMath , while if LaTeXMLMath is even , then LaTeXMLMath is an even polynomial of degree LaTeXMLMath in LaTeXMLMath In particular , we have LaTeXMLEquation .
LaTeXMLEquation We have : LaTeXMLEquation .
LaTeXMLEquation Applying Lemma LaTeXMLRef with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , replacing LaTeXMLMath by LaTeXMLMath , and multiplying by LaTeXMLMath , we obtain LaTeXMLEquation and thus LaTeXMLEquation .
We expand this in a Taylor expansion in LaTeXMLMath ; clearly , LaTeXMLMath contributes only to the first cumulant .
Conversely , since LaTeXMLMath is multiplied by an even function of LaTeXMLMath , it contributes only to the even cumulants ; the claim for LaTeXMLMath follows immediately .
Since LaTeXMLEquation we find that LaTeXMLEquation and thus has Taylor coefficients in LaTeXMLMath .
Since the moments of a measure are polynomials in its cumulants , we obtain : For real LaTeXMLMath and LaTeXMLMath and integer LaTeXMLMath , the moments LaTeXMLEquation satisfy LaTeXMLMath , where LaTeXMLMath We determine the mass of the Feller canonical measure .
For all LaTeXMLMath , LaTeXMLEquation .
In particular , the mass of the Feller canonical measure LaTeXMLMath is LaTeXMLEquation which is LaTeXMLMath We deduce this using Lemma LaTeXMLRef .
From the uniform Schwartz property of LaTeXMLMath , we see that the left-hand side is entire in LaTeXMLMath .
It thus suffices to prove the theorem when LaTeXMLMath .
We then have LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Differentiating and evaluating , we obtain : LaTeXMLEquation as required .
Finally , since LaTeXMLMath we obtain the mass of the Feller canonical measure on taking LaTeXMLMath We know little about the location of the zeros of LaTeXMLMath , for negative real LaTeXMLMath , aside from the general bound given by Lemma LaTeXMLRef .
We raise the following questions .
Question 1 .
Is LaTeXMLMath in the entire real-codimension one cone LaTeXMLMath ?
If true , this would extend the result of Theorem LaTeXMLRef to exclude zeros from the open cone .
At LaTeXMLMath all the zeros are strictly outside the cone , so to prove this result it would suffice to show that there are never any zeros on the boundary of the cone .
Question 2 .
For each fixed LaTeXMLMath are the zeros of LaTeXMLMath confined to a vertical strip LaTeXMLMath for some function LaTeXMLMath ?
The result of §5 shows that this is true on the boundary plane LaTeXMLMath perhaps it persists in the region LaTeXMLMath The results of §5 also suggest for LaTeXMLMath a limited movement of zeros in the vertical direction as LaTeXMLMath varies , so we ask the following question for negative LaTeXMLMath .
Question 3 .
For fixed LaTeXMLMath let LaTeXMLMath count ( with multiplicities ) the total number of zeros LaTeXMLMath of LaTeXMLMath lying in the horizontal strip LaTeXMLMath Is LaTeXMLMath finite for each LaTeXMLMath , and if so , does it obey the same asymptotic formula as that for LaTeXMLMath in Theorem LaTeXMLRef ?
Here we only ask for a remainder term smaller than the main term , possibly with a different dependence on LaTeXMLMath .
For real LaTeXMLMath , Theorem LaTeXMLRef shows that there are no zeros on the center line LaTeXMLMath Thus the nonreal zeros always occur in quadruples LaTeXMLMath Question 4 .
The Riemann hypothesis is encoded in the location of the zeros of LaTeXMLMath asserting they are on the four lines LaTeXMLMath Indeed , the assertion that LaTeXMLMath has no zero with LaTeXMLMath is equivalent to the Riemann hypothesis .
Does the convolution semigroup structure for LaTeXMLMath play any factor in controlling the location of these zeros ?
This question can be studied by considering more general convolution semigroups .
The zero locus LaTeXMLMath of the entire function LaTeXMLMath decomposes into a countable union of Riemann surfaces embedded in LaTeXMLMath ; we call these components .
How many components are there in LaTeXMLMath ?
While we can not answer this question , we can at least show that certain zeta zeros ( on the slice LaTeXMLMath ) lie on the same component .
The Riemann zeta zeros LaTeXMLMath and LaTeXMLMath appearing as zeros of LaTeXMLMath in the slice LaTeXMLMath belong to the same component of LaTeXMLMath .
We deform LaTeXMLMath through real values in the interval LaTeXMLMath .
By numerical computation we find that at LaTeXMLMath these two zeros have moved off the critical line to assume complex conjugate values .
We use the symmetry that when LaTeXMLMath is real , if LaTeXMLMath is a zero , then so are LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
In particular , zeros can not move off the critical line except by combining in pairs .
As LaTeXMLMath changes , at some point LaTeXMLMath they must coalesce on the critical line as a double zero , then as LaTeXMLMath changes go off the line , becoming a pair of complex conjugate zeros .
The point of coalescence at LaTeXMLMath of two zeros could be either the intersection of two different components of LaTeXMLMath ( the intersection having real codimension 4 in LaTeXMLMath ) or a single component of LaTeXMLMath having a branch point of order two there ( when viewed as projected on the LaTeXMLMath - plane . )
The latter case must occur , because in the first case the movement of the zeros LaTeXMLMath would have a first derivative as a function of LaTeXMLMath which varies analytically in LaTeXMLMath at the critical point .
This manifestly does not happen , because as a function of LaTeXMLMath the zeros first move vertically on the critical line , then change directions at LaTeXMLMath to move horizontally off the line .
Thus the component forms a single Riemann surface , with a path on it from LaTeXMLMath to LaTeXMLMath .
It seems reasonable to guess that the zeta zeros LaTeXMLMath lie on the same component of LaTeXMLMath .
If so , the same would hold for LaTeXMLMath , since the zero set is invariant under complex conjugation , i.e .
LaTeXMLMath The simplest hypothesis concerning the zero set would seem to be that it is the closure of a single irreducible complex-analytic variety of multiplicity one .
However we do not have any strong evidence for this hypothesis .
We now briefly consider the Arakelov zeta function for a general algebraic number field LaTeXMLMath .
Many of the results extend to general LaTeXMLMath but the positivity property of Theorem LaTeXMLRef does not .
In the case of the Gaussian field LaTeXMLMath , we have the identity LaTeXMLEquation derived in the appendix .
Thus all the results proved here immediately apply to LaTeXMLMath For a general algebraic number field LaTeXMLMath , the Arakelov two-variable zeta function LaTeXMLMath has a functional equation .
Furthermore it can be shown that LaTeXMLMath is an entire function of order one and infinite type of two variables , by generalization of the proofs for LaTeXMLMath .
The proofs given here for the distribution of zeros of LaTeXMLMath for positive real LaTeXMLMath partially extend to general LaTeXMLMath .
The proofs of confinement of the zeros to a vertial strip of width depending on LaTeXMLMath extend to a few fields such as the Gaussian field LaTeXMLMath ; they depend on the existence of an associated Dirichlet series with a nonempty half-plane of absolute convergence .
For general LaTeXMLMath the Dirichlet series has a nonempty half-plane of convergence for LaTeXMLMath a positive integer , but perhaps not for any other values of LaTeXMLMath .
One expects to get estimates for zeros to height LaTeXMLMath for such integer values of LaTeXMLMath .
We do not know whether for fixed positive noninteger real LaTeXMLMath and general LaTeXMLMath the zeros are confined to a vertical strip of finite width , or that a generalization of the zero counting bounds in Theorem LaTeXMLRef holds , counting zeros in a horizontal strip LaTeXMLMath .
The latter seems plausible , because the zeros are confined at positive integer LaTeXMLMath , but if so , new proofs are needed .
The convolution semigroup property , of a family of complex-valued measures on a cone associated to negative real LaTeXMLMath , continues to hold for general number fields LaTeXMLMath .
The associated measures are real-valued on the “ critical line. ” However , the proof of positivity of such measures on the “ critical line ” for LaTeXMLMath given in Theorem LaTeXMLRef extends only to a few specific number fields , such as LaTeXMLMath Our proof used a product formula for the associated modular form , which permits calculations with its logarithm and yields an explicit form for the associated Feller canonical measure .
Such product forms exist only for modular forms all of whose zeros are at cusps .
The modular forms associated to most imaginary quadratic fields generally do not have a product formula , because the associated modular forms have zeros in the interior of a fundamental domain , and the logarithm of such forms are multivalued functions .
There are imaginary quadratic number fields with class number one for which the positivity property does not hold .
One can show that the positivity property holds for a field LaTeXMLMath if and only if LaTeXMLMath is nonnegative for all real LaTeXMLMath , where LaTeXMLMath For LaTeXMLMath one finds that LaTeXMLEquation which clearly has sign changes .
We also found by computer calculation that LaTeXMLMath for LaTeXMLMath changes sign between LaTeXMLMath and LaTeXMLMath , and in addition on the line LaTeXMLMath there is a sign change , with LaTeXMLMath For LaTeXMLMath there is a sign change of LaTeXMLMath between LaTeXMLMath and LaTeXMLMath .
It remains conceivable that there exist imaginary quadratic fields having the positivity property , whose associated modular form does not have a product formula .
In support of this , computer experiments for the imaginary quadratic number fields LaTeXMLMath and LaTeXMLMath did not locate any sign changes for LaTeXMLMath .
In a different direction , the positivity property of the convolution semigroup for negative real LaTeXMLMath on the “ critical line ” generalizes to certain classes of modular forms not associated to number fields , which do have a product formula , as we hope to treat elsewhere .
This appendix summarizes the framework of van der Geer and Schoof LaTeXMLCite , and obtains explicit formulas for the two-variable zeta function for LaTeXMLMath and LaTeXMLMath The expression of van der Geer and Schoof for the Arakelov two-variable zeta function is , formally , LaTeXMLEquation .
In this expression LaTeXMLMath resp .
LaTeXMLMath are analogous to the “ dimension ” of a sheaf cohomology group .
They give a direct definition of LaTeXMLMath , and then define LaTeXMLMath indirectly LaTeXMLCite has given a direct definition of LaTeXMLMath in some cases , with a proof of the formula ( LaTeXMLRef ) .
to be LaTeXMLEquation where LaTeXMLMath is the “ canonical ” Arakelov divisor for the ring of integers of LaTeXMLMath , which is what the Riemann-Roch formula predicts .
We now define LaTeXMLMath and LaTeXMLMath The value LaTeXMLMath turns out to be the logarithm of a multivariable theta function at a specific point depending on LaTeXMLMath , see ( LaTeXMLRef ) .
In what follows , let LaTeXMLMath be an algebraic number field , with LaTeXMLMath its ring of integers and LaTeXMLMath its discriminant .
Set LaTeXMLMath , with LaTeXMLMath real places and LaTeXMLMath complex places .
We denote archimedean places of LaTeXMLMath by LaTeXMLMath and non-archimedean places by LaTeXMLMath ( i ) An Arakelov divisor LaTeXMLMath is a formal finite sum over the non-archimedean places LaTeXMLMath of LaTeXMLMath and the LaTeXMLMath archimedean places LaTeXMLMath , LaTeXMLEquation in which each LaTeXMLMath is an integer and each LaTeXMLMath is a real number ( even at a complex place LaTeXMLMath . )
( ii ) An Arakelov divisor LaTeXMLMath is principal if there is an element LaTeXMLMath such that LaTeXMLEquation in which LaTeXMLMath equals LaTeXMLMath or LaTeXMLMath according as LaTeXMLMath is a real place or a complex place .
Here LaTeXMLMath runs over all embeddings of LaTeXMLMath into LaTeXMLMath , with the convention that only one out of each conjugate complex pair of complex embeddings is used .
( iii ) LaTeXMLMath denotes the group of Arakelov divisors ( under addition ) .
The Arakelov divisor class group LaTeXMLMath is the quotient group by the subgroup of principal Arakelov divisors .
The divisor class of LaTeXMLMath is denoted LaTeXMLMath .
The roots of unity LaTeXMLMath in LaTeXMLMath have Arakelov divisor zero .
They fit in the exact sequence LaTeXMLEquation .
The degree LaTeXMLMath of an Arakelov divisor is the real number LaTeXMLEquation .
Here LaTeXMLMath , where LaTeXMLMath , in which LaTeXMLMath is the number of elements in the residue field of LaTeXMLMath .
Principal divisors have degree zero , so the degree LaTeXMLMath is well-defined on Arakelov divisor classes .
The norm LaTeXMLMath of an Arakelov divisor LaTeXMLMath is LaTeXMLEquation ( i ) The ideal LaTeXMLMath associated to an Arakelov divisor LaTeXMLMath at the finite places is the fractional ideal LaTeXMLEquation where LaTeXMLMath denotes the prime ideal at LaTeXMLMath .
( ii ) The lattice structure associated to an Arakelov divisor LaTeXMLMath at the infinite places is a positive inner product on LaTeXMLMath .
defined as follows .
At a real place , LaTeXMLMath determines a scalar product on LaTeXMLMath such that LaTeXMLMath At a complex place LaTeXMLMath determines a Hermitian inner product on LaTeXMLMath such that LaTeXMLMath The combined inner product is LaTeXMLEquation .
The ( metrized ) lattice LaTeXMLMath associated to LaTeXMLMath is the fractional ideal LaTeXMLMath ( viewed as a subset of LaTeXMLMath ) embedded in LaTeXMLMath as Galois conjugates of each element LaTeXMLMath , with distance function measured by this inner product .
The number field LaTeXMLMath embeds as a dense subset of LaTeXMLMath , while each fractional ideal LaTeXMLMath embeds discretely as a lattice in this space .
The archimedean coordinates LaTeXMLMath define a metric structure at the infinite places such that LaTeXMLEquation where LaTeXMLMath is the discriminant of LaTeXMLMath .
The Arakelov class group LaTeXMLMath parametrizes isometry classes of lattices that have compatible LaTeXMLMath -structures under multiplication .
Following Szpiro , the Euler-Poincaré characteristic LaTeXMLMath of an Arakelov divisor LaTeXMLMath is defined as LaTeXMLEquation .
It is well-defined on divisor classes LaTeXMLMath .
In general the Arakelov class group LaTeXMLEquation where LaTeXMLMath denotes the ( wide ) ideal class group of LaTeXMLMath , and the second factors combined are LaTeXMLMath where LaTeXMLMath is an LaTeXMLMath dimensional lattice given by logarithms of ( absolute values of ) Galois conjugates of units .
Note that LaTeXMLMath , the group of divisor classes of degree LaTeXMLMath , is compact , and its volume is LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are the class number and regulator of LaTeXMLMath , respectively .
The canonical divisor LaTeXMLMath of a number field LaTeXMLMath is the Arakelov divisor whose associated ideal part LaTeXMLMath is the inverse different LaTeXMLMath for LaTeXMLMath , and all of whose archimedean coordinates LaTeXMLMath .
These definitions imply that LaTeXMLEquation ( i ) An Arakelov divisor LaTeXMLMath is effective if LaTeXMLMath ( ii ) The effectivity LaTeXMLMath of an effective divisor LaTeXMLMath is LaTeXMLEquation in which LaTeXMLEquation .
The effectivity LaTeXMLMath of a non-effective divisor is LaTeXMLMath .
The effective divisors are those Arakelov divisors with each LaTeXMLMath and the effectivity LaTeXMLMath of any divisor takes a value LaTeXMLMath The only “ functions ” LaTeXMLMath whose associated principal divisors LaTeXMLMath are effective are the roots of unity in LaTeXMLMath , whose associated Arakelov divisor is LaTeXMLMath , the identity element .
By convention we add a symbol LaTeXMLMath to represent a “ divisor at infinity ” corresponding to the element LaTeXMLMath , with the convention that LaTeXMLMath for all Arakelov divisors LaTeXMLMath and we define the effectivity LaTeXMLMath ( i ) The order LaTeXMLMath of the group of effective divisors associated to an Arakelov divisor LaTeXMLMath is LaTeXMLEquation .
This sum includes a term LaTeXMLMath , with the convention that LaTeXMLMath so that LaTeXMLMath ( ii ) The effectivity dimension LaTeXMLMath of LaTeXMLMath is given by LaTeXMLEquation .
One has LaTeXMLMath .
The quantities LaTeXMLMath and LaTeXMLMath are constant for all divisors in an Arakelov divisor class LaTeXMLMath and may therefore be denoted LaTeXMLMath and LaTeXMLMath , respectively .
van der Geer and Schoof LaTeXMLCite state the following result .
( Riemann-Roch Theorem for Number Fields ) For any algebraic number field LaTeXMLMath and any Arakelov divisor class LaTeXMLMath , LaTeXMLEquation in which LaTeXMLMath is the canonical Arakelov divisor for LaTeXMLMath , and LaTeXMLMath .
van der Geer and Schoof defined a new invariant LaTeXMLMath of a number field LaTeXMLMath , and an Arakelov analogue of the genus LaTeXMLMath of LaTeXMLMath .
( i ) The invariant LaTeXMLMath of LaTeXMLMath is defined by LaTeXMLEquation ( ii ) The genus LaTeXMLMath of LaTeXMLMath is defined by LaTeXMLEquation .
For the rational number field LaTeXMLEquation and LaTeXMLMath The value LaTeXMLMath , where LaTeXMLMath is the theta function ( LaTeXMLRef ) .
One also has LaTeXMLEquation see LaTeXMLCite .
The genus of a function field is usually defined to be the dimension LaTeXMLMath of the vector space of effective divisors for the canonical class LaTeXMLMath ; this motivates the definition of the genus LaTeXMLMath of a number field .
For a function field the degree LaTeXMLMath of the canonical class is LaTeXMLMath , so one may consider LaTeXMLEquation as a second analogue of genus for a number field .
This analogue appears on the right side of the Riemann-Roch theorem for number fields .
One has LaTeXMLEquation .
In particular LaTeXMLMath , and LaTeXMLMath .
The two notions of genus agree in the function field case and differ in the number field case .
Below we obtain explicit integral formulas for the two-variable zeta functions for LaTeXMLMath and LaTeXMLMath , of the form ( LaTeXMLRef ) , and also indicate the form of the two-variable zeta function for a general algebraic number field LaTeXMLMath .
To define an integral over the Arakelov class group , one must specify a measure on the group .
For compact groups it is Haar measure , and on noncompact additive groups LaTeXMLMath it is LaTeXMLMath It is convenient to replace additive groups by multiplicative group LaTeXMLMath using the change of variable LaTeXMLMath at real places and the appropriate measure becomes LaTeXMLMath For LaTeXMLMath and LaTeXMLMath the Arakelov class group is isomorphic to LaTeXMLMath There is a single real place LaTeXMLMath .
For representatives of Arakelov divisor classes LaTeXMLMath we may take LaTeXMLMath to have ideal LaTeXMLMath and with value at the infinite place LaTeXMLMath arbitrary , with measure LaTeXMLMath at the infinite place .
Thus the Arakelov class group LaTeXMLMath , the additive group , with LaTeXMLMath being the degree of the divisor .
We have LaTeXMLEquation in which one sets LaTeXMLMath .
Using the multiplicative change of variable LaTeXMLMath we identify LaTeXMLMath with the multiplicative group LaTeXMLMath with measure LaTeXMLMath Thus we have LaTeXMLEquation in terms of the theta function ( LaTeXMLRef ) .
The different LaTeXMLMath so the canonical divisor LaTeXMLMath Consequently we have LaTeXMLEquation .
We obtain LaTeXMLEquation .
There is a single complex place LaTeXMLMath .
The class number of LaTeXMLMath is one , so all Arakelov divisor classes LaTeXMLMath in LaTeXMLMath have a representative LaTeXMLMath whose associated ideal is LaTeXMLMath .
Thus LaTeXMLMath as an additive group .
Letting LaTeXMLMath , we have LaTeXMLEquation in which LaTeXMLMath Thus LaTeXMLEquation .
The different LaTeXMLMath and the canonical divisor LaTeXMLMath Consequently we have LaTeXMLEquation .
We obtain LaTeXMLEquation using the change of variables LaTeXMLMath .
Comparing this with ( LaTeXMLRef ) yields LaTeXMLEquation .
Let LaTeXMLMath be an algebraic number field , of degree LaTeXMLMath .
We follow Lang LaTeXMLCite for Hecke ’ s functional equation for the Dedekind zeta function .
One can show that LaTeXMLEquation which uses a decomposition of the Arakelov class group ( LaTeXMLRef ) .
Here LaTeXMLMath runs over a set of representatives of the ( wide ) ideal class group , LaTeXMLMath counts the number of roots of unity in LaTeXMLMath , and LaTeXMLMath is a fundamental domain in the ( logarithmic ) space of units , with Haar measure LaTeXMLMath .
The theta function LaTeXMLMath is defined in Lang LaTeXMLCite and satisfies the functional equation LaTeXMLEquation using the fact that LaTeXMLMath see Lang LaTeXMLCite .
Using this functional equation and the substitution LaTeXMLMath one obtains LaTeXMLEquation .
One has the functional equation LaTeXMLEquation .
For LaTeXMLMath one recovers the completed Dedekind zeta function LaTeXMLEquation in which LaTeXMLMath , see Lang LaTeXMLCite .
email : jcl @ research.att.com rains @ research.att.com
We study the geometry of families of hypersurfaces in Eguchi–Hanson space that arise as complex line bundles over curves in LaTeXMLMath and are three–dimensional , non–compact Riemannian manifolds , which are foliated in Hopf tori for closed curves .
They are negatively curved , asymptotically flat spaces , and we compute the complete three–dimensional curvature tensor as well as the second fundamental form , giving also some results concerning their geodesic flow .
We show the non–existence of LaTeXMLMath –harmonic functions on these hypersurfaces for every LaTeXMLMath and arbitrary curves , and determine the infima of the spectra of the Laplace and of the square of the Dirac operator in the case of closed curves .
We also show that , in this case , zero lies in the spectrum of the Dirac operator .
For circles we compute the LaTeXMLMath –kernel of the Dirac operator in the sense of spectral theory and show that it is trivial .
We consider further the Einstein–Dirac system on these spaces and construct explicit examples of LaTeXMLMath –Killing spinors on them .
In this paper we shall study certain families of hypersurfaces in Eguchi–Hanson space that arise as complex line bundles over curves on LaTeXMLMath .
They are three-dimensional , open , asymptotically flat Riemannian manifolds of non–positive scalar curvature which , in case of a closed curve , are foliated in Hopf tori .
We describe their geometry in detail , computing the complete three–dimensional curvature tensor as well as the second fundamental form , and give also some results on the structure of the geodesic flow .
Since an explicit description of the geometric properties of these hypersurfaces is possible , we are able to make precise statements about the spectra of the scalar Laplacian and the Dirac operator and also about the existence of solutions of spinorial field equations .
In particular , we show that there are no LaTeXMLMath –harmonic functions for every LaTeXMLMath and arbitrary curves , and that for curves arising by Möbius transforms from closed curves the spectra of the scalar Laplacian and the square of the Dirac operator come arbitrarily close to zero , implying that zero lies in the spectrum of the considered operators .
In the mentioned case , it also turns out that zero lies in the spectrum of the Dirac operator .
In case that the considered curves are generalized circles in LaTeXMLMath that arise by Möbius transforms from circles in LaTeXMLMath with center at the origin the LaTeXMLMath –kernel of the Dirac operator in the sense of spectral theory can be computed explicitly and we show that it is trivial .
As it turns out , these hypersurfaces do not admit solutions to the Einstein–Dirac system ; such solutions can only be obtained by deformation into a singular situation .
Nevertheless , we can construct explicit examples of LaTeXMLMath –Killing spinors , which are solutions of a generalized Killing equation for spinors .
Hopf tori have been extensively studied , see e.g .
LaTeXMLCite , and where first considered by Pinkall LaTeXMLCite .
If LaTeXMLMath denotes the Hopf fibration , the inverse image of any closed curve in LaTeXMLMath will be an immersed torus in LaTeXMLMath , which is called a Hopf torus .
Using Hopf tori Pinkall showed that every compact Riemann surface of genus one can be conformally embedded as a flat torus into the unit sphere LaTeXMLMath .
As a further application , and using elastic curves in LaTeXMLMath , he constructed new examples of compact embedded Willmore surfaces in LaTeXMLMath , which are extremal surfaces for the Willmore functional LaTeXMLMath , where LaTeXMLMath denotes the mean curvature .
The Eguchi–Hanson metric is a four–dimensional metric , which can be constructed in the total space of the fibration LaTeXMLMath , and since its holonomy is contained in LaTeXMLMath , it is Ricci flat and self–dual .
Both the Hopf fibration LaTeXMLMath and the projection LaTeXMLMath are compatible with the action of LaTeXMLMath in LaTeXMLMath , and , like the standard metric in LaTeXMLMath , the Eguchi–Hanson metric is invariant under this action .
Therefore , its restriction to the three–dimensional projective space LaTeXMLMath , which is immersed in LaTeXMLMath as the set of all cotangential vectors of unit length , corresponds exactly to the standard metric in LaTeXMLMath .
For this reason the projection LaTeXMLMath is a geometric extension of the Hopf fibration , and the preimage of any closed curve on LaTeXMLMath under the projection LaTeXMLMath gives rise to a three–dimensional non–compact Riemannian manifold foliated in Hopf tori .
Its end is of topological type LaTeXMLMath , where LaTeXMLMath is the two–dimensional torus .
Nevertheless , the corresponding Willmore functional turns out to be unbounded , so that the considered hypersurfaces are not accessible to integral geometry .
The interest in Eguchi–Hanson space itself originates from a result of Schoen and Yau LaTeXMLCite , who proved that a complete asymptotically Euclidean four–manifold whose Ricci tensor vanishes is necessarily flat .
For Ricci flat asymptotically locally Euclidean Kähler metrics this turns out not to be true , the first example of such a metric being given by the Eguchi–Hanson metric LaTeXMLCite .
We give now a description of the main results of this work .
The Sections LaTeXMLRef , LaTeXMLRef , LaTeXMLRef and LaTeXMLRef are concerned with the geometry of the hypersurfaces studied , the Sections LaTeXMLRef , LaTeXMLRef and LaTeXMLRef with the spectra of the Dirac and the Laplace operator , while Section LaTeXMLRef is devoted to the study of spinorial field equations .
The Eguchi–Hanson metric is described in Section LaTeXMLRef : it depends on a real parameter LaTeXMLMath , thus giving rise to a one–parameter family of Riemannian metrics LaTeXMLMath .
These metrics become degenerate along the zero section in case that LaTeXMLMath .
For any curve LaTeXMLMath in LaTeXMLMath we consider its preimage LaTeXMLMath and obtain a family of hypersurfaces LaTeXMLMath , where we assume that LaTeXMLMath is parametrized by arc length and LaTeXMLMath denotes the induced Riemannian metric .
Each of these hypersurfaces is a complex line bundle over LaTeXMLMath , and introducing the polar coordinates LaTeXMLMath and LaTeXMLMath in each fiber , we obtain a parametrization of LaTeXMLMath outside the zero section by the coordinates LaTeXMLMath , see Section LaTeXMLRef .
Since the coefficients of LaTeXMLMath do not depend on LaTeXMLMath , the corresponding LaTeXMLMath –symmetry is an isometry .
We determine the inner geometry of the hypersurfaces and in Theorem LaTeXMLRef , page LaTeXMLRef , the complete Ricci tensor is computed with respect to an orthonormal frame , one eigenvalue being positive , one negative and the third one becoming negative at infinity , yielding , for the scalar curvature , the expression LaTeXMLEquation .
It is negative and tends to zero for large LaTeXMLMath and LaTeXMLMath with the order LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath remains regular at LaTeXMLMath , i.e. , the scalar curvature vanishes on the zero section .
In Section LaTeXMLRef we turn to the study of the Levi–Civita connection of the Eguchi–Hanson space and determine the second fundamental form of the hypersurfaces LaTeXMLMath with respect to the above orthonormal frame , thus obtaining LaTeXMLEquation see Theorem LaTeXMLRef on page LaTeXMLRef , and Corollary LaTeXMLRef on page LaTeXMLRef , where LaTeXMLMath is the function LaTeXMLMath and LaTeXMLMath denotes the mean curvature .
It is given by the geodesic curvature LaTeXMLMath of LaTeXMLMath as a curve in LaTeXMLMath according to the formula LaTeXMLEquation .
This appears to be natural , since the geometry of the vector bundle LaTeXMLMath is determined by the elliptic geometry of LaTeXMLMath .
The above formula also implies that LaTeXMLMath is a minimal surface if and only if LaTeXMLMath , i.e. , if LaTeXMLMath is a great circle in LaTeXMLMath .
Further , since the function LaTeXMLMath corresponds to the distance in LaTeXMLMath , both the scalar curvature LaTeXMLMath and the mean curvature LaTeXMLMath , as well as the components of the Ricci tensor and of the second fundamental form , are manifestly invariant under the action of the isometry group LaTeXMLMath .
Section LaTeXMLRef contains some results concerning the geodesic flow of the hypersurfaces LaTeXMLMath .
So , in case LaTeXMLMath is a circle in LaTeXMLMath with center at the origin , we are able to compute the distance of a point in LaTeXMLMath to the curve LaTeXMLMath , i.e. , to the zero section , see Proposition LaTeXMLRef on page LaTeXMLRef , and , in this way , to calculate the exponential growth of LaTeXMLMath explicitly .
In Section LaTeXMLRef the vanishing of the LaTeXMLMath –kernel , LaTeXMLMath , of the scalar Laplacian on the hypersurfaces LaTeXMLMath is proved for every LaTeXMLMath and every curve LaTeXMLMath by showing the existence of a canonical exhaustion function on the considered hypersurfaces , see Proposition LaTeXMLRef and Corollary LaTeXMLRef on page LaTeXMLRef .
The result then follows from the work of Greene and Wu LaTeXMLCite , who studied integrals of certain generalized subharmonic functions on connected non–compact Riemannian manifolds admitting such a function , and showed that these integrals can not be bounded .
For the smallest spectral value of the scalar Laplacian we obtain , in Section LaTeXMLRef , the first estimate LaTeXMLEquation where LaTeXMLMath is a closed curve , see Corollary LaTeXMLRef on page LaTeXMLRef , since by general theory lower bounds for the Ricci tensor of open complete manifolds imply upper bounds for the smallest spectral value of the Laplace operator LaTeXMLCite .
By using the LaTeXMLMath – LaTeXMLMath principle we are then able to determine the infimum of the spectrum of the closure of the Laplacian LaTeXMLMath on LaTeXMLMath , obtaining LaTeXMLEquation for every LaTeXMLMath and arbitrary LaTeXMLMath and closed curves LaTeXMLMath .
Since , by Corollary LaTeXMLRef , zero can be no LaTeXMLMath –eigenvalue , we therefore get that zero lies in LaTeXMLMath , the essential spectrum of LaTeXMLMath .
A result of Brooks LaTeXMLCite then implies that in this case the hypersurfaces LaTeXMLMath must be of subexponential growth , generalizing the previously obtained result .
Section LaTeXMLRef is devoted to the study of spinorial field equations .
In LaTeXMLCite Friedrich and Kim showed that in dimension 3 the existence of a solution to the Einstein Dirac system is equivalent to the existence of a so–called Weak Killing or WK spinor .
For the existence of such a spinor geometric integrability conditions that are independent of the considered spin structure are known , and we show that , for LaTeXMLMath , these conditions can never be fulfilled , implying that there can not be any solutions to the Einstein Dirac system on the hypersurfaces LaTeXMLMath for any LaTeXMLMath and any curve LaTeXMLMath , see Proposition LaTeXMLRef on page LaTeXMLRef .
Nevertheless , such solutions can be constructed explicitly with respect to the trivial spin structure in case that LaTeXMLMath , the manifolds considered then being no longer complete .
As remarked above , the Eguchi–Hanson metric is self–dual and , due to this , there is a parallel spinor on Eguchi–Hanson space .
By restricting this spinor to the hypersurfaces LaTeXMLMath we show in Proposition LaTeXMLRef that there exists a LaTeXMLMath –Killing spinor on LaTeXMLMath if and only if LaTeXMLMath is a minimal surface .
The spectrum of the Dirac operator LaTeXMLMath is studied in Section LaTeXMLRef .
There we show , by estimating the Rayleigh quotient from above and using again the LaTeXMLMath – LaTeXMLMath principle , that the infimum of the spectrum of LaTeXMLMath on LaTeXMLMath becomes arbitrarily small , LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is a closed curve , see Theorem LaTeXMLRef on page LaTeXMLRef , LaTeXMLMath being arbitrary ; here the involved spin structure is again the trivial one .
In this case it also follows that LaTeXMLMath , by explicit construction of an approximating sequence .
In case that LaTeXMLMath is a circle in LaTeXMLMath with center at the origin , an isometric LaTeXMLMath –action is given and the LaTeXMLMath –kernel of the Dirac operator and of its closure decompose into the unitary representations of this action according to the spectral decomposition of the corresponding generators LaTeXMLMath , LaTeXMLMath , and with respect to the trivial spin structure one obtains LaTeXMLEquation while on LaTeXMLMath the LaTeXMLMath –kernels of the Dirac operator and its closure turn out to be trivial .
Thus , in this case , LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are isometric for every LaTeXMLMath , statements for a particular curve in LaTeXMLMath can be generalized to curves that arise from it by Möbius transforms .
Let LaTeXMLMath be a finite nontrivial subgroup of LaTeXMLMath that acts freely on LaTeXMLMath .
Then LaTeXMLMath carries an isolated quotient singularity at zero and any resolution LaTeXMLMath of LaTeXMLMath is a non–compact complex manifold .
A Kähler metric LaTeXMLMath on LaTeXMLMath is said to be asymptotic to the Euclidean metric LaTeXMLMath on LaTeXMLMath if there is a smooth surjective map LaTeXMLMath such that LaTeXMLMath is a connected , simply connected , finite union of compact submanifolds of LaTeXMLMath and LaTeXMLMath induces a diffeomorphism LaTeXMLMath .
Under this diffeomorphism LaTeXMLMath should satisfy LaTeXMLEquation for large LaTeXMLMath , where LaTeXMLMath is the distance from the origin and LaTeXMLMath is the flat connection in LaTeXMLMath .
Such a metric is called an asymptotically locally Euclidean or ALE metric .
Notice that the topological type of the end is given by a quotient of the Euclidean space .
In the following we will mainly be concerned with the case of LaTeXMLMath .
In LaTeXMLCite Schoen and Yau proved that a complete asymptotically Euclidean four–manifold whose Ricci–tensor vanishes is necessarily flat .
Nevertheless , a similar statement for Ricci–flat ALE Kähler–metrics does not hold , since , as mentioned , the topology of the end differs from the topology of Euclidean space .
An important class of Ricci–flat Kähler metrics which give rise to ALE spaces is given by the so–called hyperkähler structures .
In the case of an oriented four–dimensional smooth manifold LaTeXMLMath a hyperkähler structure is a metric whose holonomy is contained in LaTeXMLMath .
A manifold with such a structure is Ricci–flat and self–dual , and its metric is Kähler with respect to each of the three anticommuting complex structures .
Alternatively , a hyperkähler structure on LaTeXMLMath may be defined to be a triple of smooth , closed LaTeXMLMath –forms LaTeXMLMath on LaTeXMLMath that can be represented locally according to LaTeXMLEquation where LaTeXMLMath is a local oriented frame of LaTeXMLMath –forms on LaTeXMLMath .
The systematic construction of ALE metrics with holonomy LaTeXMLMath as hyperkähler quotients was initiated by Hitchin LaTeXMLCite and carried over by Kronheimer LaTeXMLCite , who studied the spaces LaTeXMLMath for general polyhedra groups LaTeXMLMath and showed the existence of hyperkähler metrics on the resolution LaTeXMLMath for the considered groups LaTeXMLMath , giving a complete classification .
For cyclic groups these metrics are explicitly known .
The first example of a hyperkähler ALE four–manifold was found by Eguchi and Hanson LaTeXMLCite .
We will now briefly proceed to describe its construction .
Let LaTeXMLMath and LaTeXMLMath and consider the mapping LaTeXMLEquation .
The image of LaTeXMLMath under LaTeXMLMath is LaTeXMLEquation and LaTeXMLMath induces a bijection LaTeXMLMath so that LaTeXMLMath becomes analytically equivalent to LaTeXMLMath .
The canonical bundle over LaTeXMLMath , LaTeXMLEquation can be described explicitly as follows .
If one introduces the homogeneous coordinates LaTeXMLMath in LaTeXMLMath , then the total space LaTeXMLMath consists of all equivalence classes of triples LaTeXMLMath with respect to the equivalence relation LaTeXMLMath where LaTeXMLMath , i. e. LaTeXMLEquation .
The one–dimensional complex tangential bundle LaTeXMLMath is biholomorphic to the square of the dual of the canonical bundle LaTeXMLCite LaTeXMLEquation from which one obtains , for the cotangential bundle LaTeXMLMath , the description LaTeXMLEquation with the equivalence relation LaTeXMLMath .
Notice that LaTeXMLMath is simply-connected .
We define now the mapping LaTeXMLEquation .
The preimage of the point LaTeXMLMath under LaTeXMLMath is the zero section of the bundle LaTeXMLMath .
Away from this set LaTeXMLMath is bijective , and hence LaTeXMLMath represents a resolution of the singularity of LaTeXMLMath at zero .
Summing up one obtains the diagram & π _1 & & π & X C ^2/ { ± 1 } & & & ≃ ^ Φ & where the mapping LaTeXMLMath is given by the formula LaTeXMLMath .
The closed holomorphic 2–form LaTeXMLMath and the function LaTeXMLMath on LaTeXMLMath are invariant under reflections at the origin , descend to LaTeXMLMath and , thus , lift to forms on LaTeXMLMath , which we will denote by LaTeXMLMath and LaTeXMLMath as well .
We come now to the description of the Eguchi–Hanson metric .
Following LaTeXMLCite we consider , on the complex manifold LaTeXMLMath , the family of real–valued functions LaTeXMLMath depending on the parameter LaTeXMLMath , LaTeXMLEquation .
Here the function LaTeXMLMath is explicitly given by LaTeXMLMath , from which it follows that , away from the exceptional curve , i.e. , the zero section , LaTeXMLMath is a smooth function , and the same holds for LaTeXMLMath .
For LaTeXMLMath the associated form LaTeXMLEquation is regular even in the exceptional curve and thus defines a Kähler form on LaTeXMLMath .
For using homogeneous coordinates we can define a complex analytic structure on LaTeXMLMath as follows .
Let LaTeXMLMath be open subsets in LaTeXMLMath and define the homeomorphisms LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath is a biholomorphic mapping , this gives a complex analytic structure on LaTeXMLMath .
We can therefore choose the functions LaTeXMLMath and LaTeXMLMath as local coordinates in LaTeXMLMath by setting LaTeXMLMath equal to LaTeXMLMath , so that LaTeXMLMath and LaTeXMLMath are given by LaTeXMLEquation on LaTeXMLMath .
The regularity of LaTeXMLMath for LaTeXMLMath then follows by noting that the derivatives of LaTeXMLMath with respect to LaTeXMLMath and LaTeXMLMath become regular ( note that LaTeXMLMath ) .
For example , in LaTeXMLMath , one has LaTeXMLEquation .
In case that LaTeXMLMath on has LaTeXMLMath , and LaTeXMLMath becomes degenerate along the zero section .
On LaTeXMLMath the Kähler form LaTeXMLMath induces a Riemannian metric through the formula LaTeXMLEquation where LaTeXMLMath denotes the complex structure of LaTeXMLMath .
For LaTeXMLMath LaTeXMLMath becomes a complete Riemannian manifold .
The complex manifold LaTeXMLMath is an open dense subset of LaTeXMLMath , so it suffices for the study of the geometric properties of LaTeXMLMath to consider LaTeXMLMath as well as the other relevant geometric objects just on LaTeXMLMath .
Further , since LaTeXMLMath maps LaTeXMLMath bijectively onto LaTeXMLMath , LaTeXMLMath can be explicitly computed on LaTeXMLMath with respect to the coordinates LaTeXMLMath .
For LaTeXMLMath as a function on LaTeXMLMath one has therefore LaTeXMLEquation .
LaTeXMLEquation see e. g. LaTeXMLCite , which yields LaTeXMLMath and thus LaTeXMLEquation .
The form LaTeXMLMath is expressed with respect to the coordinates LaTeXMLMath by LaTeXMLEquation and the action of LaTeXMLMath is given by LaTeXMLMath .
Computation of LaTeXMLMath restricted to LaTeXMLMath then gives LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLMath are the smooth functions LaTeXMLEquation .
For later use we define the smooth function LaTeXMLEquation .
From this it becomes evident that LaTeXMLMath satisfies condition ( LaTeXMLRef ) .
One further computes the volume form LaTeXMLMath to be LaTeXMLMath ) , where LaTeXMLMath .
By the general theory of Kähler manifolds LaTeXMLCite the Ricci–form LaTeXMLMath then vanishes and it follows immediately that the Riemannian curvature tensor with respect to the decomposition LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the negative and positive part of the Weyl tensor respectively , LaTeXMLMath is the trace–free part of the Ricci tensor and LaTeXMLMath denotes the scalar curvature .
The condition LaTeXMLMath implies that LaTeXMLMath is an Einstein space and the vanishing of LaTeXMLMath means that LaTeXMLMath is self–dual ; the latter is equivalent to the statement that the bundle LaTeXMLMath is flat which in turn implies that there exist three parallel forms on LaTeXMLMath .
These forms can be chosen as LaTeXMLMath and the two closed LaTeXMLMath –forms LaTeXMLMath defined by LaTeXMLMath .
One can show that the triple LaTeXMLMath may locally be written in the form ( LaTeXMLRef ) and thus forms a hyperkähler structure on LaTeXMLMath and hence on LaTeXMLMath .
We consider now the projection LaTeXMLEquation which is explicitly given by LaTeXMLMath .
The function LaTeXMLMath is invariant under the standard action of LaTeXMLMath on LaTeXMLMath resp .
LaTeXMLMath , which is given by its matrix representation .
On the other hand , LaTeXMLMath acts as a group of holomorphic transformations on LaTeXMLMath by the so–called Möbius transform LaTeXMLEquation resp .
on LaTeXMLMath by LaTeXMLEquation .
Taking the mapping LaTeXMLMath which is given by LaTeXMLMath , one therefore sees that LaTeXMLEquation which means that the diagram ∖ P ^1 ( C ) & π _1 _ ≃ & C ^2 ∖ { 0 } / { ± 1 } ~ p & P ^1 ( C ) & & is compatible with the group action of LaTeXMLMath .
Since the set LaTeXMLMath is mapped by LaTeXMLMath onto itself , it follows that , by extending the action of LaTeXMLMath to LaTeXMLMath , the exceptional curve in LaTeXMLMath must be mapped onto itself , too .
The projection LaTeXMLMath is therefore also compatible with the LaTeXMLMath –action .
Since LaTeXMLMath vanishes on the zero section , it becomes LaTeXMLMath –invariant on LaTeXMLMath .
The Kähler metric LaTeXMLMath and the Riemannian metric LaTeXMLMath , which are defined by means of the function LaTeXMLMath , are thus also invariant under LaTeXMLMath in LaTeXMLMath .
We now introduce certain hypersurfaces in LaTeXMLMath and to this end consider for any curve LaTeXMLMath in LaTeXMLMath its preimage LaTeXMLEquation obtaining a real three–dimensional hypersurface in LaTeXMLMath .
Let LaTeXMLMath be the Riemannian metric on LaTeXMLMath induced by LaTeXMLMath .
The three–manifold LaTeXMLMath is open and in case of a closed curve its end is of topological type LaTeXMLMath , where LaTeXMLMath is the two–dimensional torus .
The hypersurfaces LaTeXMLMath are asymptotically flat , but no ALE spaces , since their end is not modeled on the end of LaTeXMLMath .
Note that LaTeXMLMath is a one–dimensional complex vector bundle over LaTeXMLMath .
Since LaTeXMLMath is compatible with the action of LaTeXMLMath , and since LaTeXMLMath and hence LaTeXMLMath are invariant under this action , LaTeXMLMath is mapped isometrically onto LaTeXMLMath , where LaTeXMLMath .
Remember that under Möbius transforms generalized circles in LaTeXMLMath are mapped again into generalized circles .
We will now compute the inner geometry of the hypersurfaces LaTeXMLMath and assume from now on that LaTeXMLMath is parametrized by arc length .
Using the projection LaTeXMLMath one obtains a parametrization LaTeXMLMath of the hypersurfaces LaTeXMLMath outside the zero section LaTeXMLEquation where LaTeXMLMath is the length parameter of LaTeXMLMath and LaTeXMLMath denotes the parameter of the fiber over LaTeXMLMath .
All the following calculations will be performed in LaTeXMLMath , which is dense in LaTeXMLMath .
The vector fields on LaTeXMLMath induced by the parametrization LaTeXMLMath read LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Further , one has LaTeXMLEquation since LaTeXMLMath is the arc length parameter of LaTeXMLMath .
Note also that LaTeXMLMath and LaTeXMLMath .
Moreover , outside the zero section the following identities hold : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLMath .
Let LaTeXMLMath be the Riemannian metric on LaTeXMLMath induced by LaTeXMLMath .
In the case of a closed curve LaTeXMLMath the hypersurface LaTeXMLMath is a complete Riemannian manifold for LaTeXMLMath .
Making use of the above relations one obtains the following proposition .
On LaTeXMLMath , the coefficients of the induced Riemannian metric LaTeXMLMath with respect to the local coordinate frame LaTeXMLMath are given by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The function LaTeXMLMath is given on LaTeXMLMath by the formula LaTeXMLEquation and the functions LaTeXMLMath and LaTeXMLMath by LaTeXMLEquation .
In order to compute the relevant geometric quantities of LaTeXMLMath it turns out to be convenient to work within the framework of Cartan .
For this purpose we determine an orthonormal frame with respect to LaTeXMLMath by the ansatz LaTeXMLEquation .
The vector fields LaTeXMLMath and LaTeXMLMath are normalized to length LaTeXMLMath ; since LaTeXMLMath , they are orthogonal to each other .
From the condition LaTeXMLMath together with LaTeXMLMath one obtains LaTeXMLEquation .
Here we have introduced the function LaTeXMLMath and one computes LaTeXMLEquation .
The vector fields LaTeXMLMath are defined on LaTeXMLMath and outside the exceptional curve do represent a global section in the frame bundle of LaTeXMLMath .
Note that since LaTeXMLMath is positive , LaTeXMLMath is always positive .
The local base of LaTeXMLMath –forms LaTeXMLMath dual to the orthonormal frame is then given by LaTeXMLEquation and the connection forms LaTeXMLMath of the Levi–Civita–connection LaTeXMLMath on LaTeXMLMath as well as the components of the Riemannian curvature tensor LaTeXMLMath are uniquely determined by Cartan ’ s structure equations LaTeXMLEquation .
LaTeXMLEquation We determine now the connection forms of the considered hypersurfaces .
With respect to the orthonormal frame ( LaTeXMLRef ) , the forms LaTeXMLMath of the Levi–Civita connection on LaTeXMLMath are given by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Using the orthonormal frame LaTeXMLMath on LaTeXMLMath , the components of the Levi–Civita–connection can be obtained via the formulas LaTeXMLEquation resulting from the Koszul–formula .
A direct computation of the commutators yields LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Now , a short calculation gives LaTeXMLEquation by which one further calculates LaTeXMLEquation .
LaTeXMLEquation which shows that the first coefficient of LaTeXMLMath vanishes .
Similarly , it can be seen that the second coefficient is also zero , since LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and by using ( LaTeXMLRef ) again one sees that the commutator LaTeXMLMath vanishes completely .
In the equations ( LaTeXMLRef ) therefore only the terms LaTeXMLEquation are non–trivial , and for the forms LaTeXMLMath this gives the stated expressions .
∎ Summing up , one obtains that the structure equations ( LaTeXMLRef ) read LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We are now able to compute the components of the Riemannian curvature tensor as well as the Ricci tensor and the scalar curvature of the hypersurface LaTeXMLMath .
With respect to the section ( LaTeXMLRef ) , the components of the Riemannian curvature tensor LaTeXMLMath of the hypersurfaces LaTeXMLMath are given by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation while LaTeXMLMath , LaTeXMLMath and LaTeXMLMath vanish .
We calculate the components of LaTeXMLMath by using the structure equations ( LaTeXMLRef ) of the hypersurface LaTeXMLMath .
By Proposition LaTeXMLRef the 2–forms LaTeXMLMath and LaTeXMLMath vanish and LaTeXMLEquation .
Further , the differentials LaTeXMLMath of the connection forms are given by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and one obtains the stated formulas for the components LaTeXMLMath of the curvature tensor by using ( LaTeXMLRef ) .
Notice that LaTeXMLEquation implying that LaTeXMLMath vanishes .
∎ The components LaTeXMLMath of the Ricci tensor LaTeXMLMath of the Riemannian LaTeXMLMath –manifolds LaTeXMLMath are given with respect to the orthonormal frame ( LaTeXMLRef ) by LaTeXMLEquation and the scalar curvature is LaTeXMLEquation .
One computes LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation obtaining thus , with the previous proposition , for the components LaTeXMLMath of the Ricci tensor that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation the remaining coefficients being equal to zero .
In the same way as the components of the Riemannian curvature tensor turn out to be bounded when LaTeXMLMath , the components of LaTeXMLMath and thus LaTeXMLMath stay bounded , too .
Explicitly one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation showing that the divergent terms cancel out and the assertion follows .
∎ Hence , the scalar curvature LaTeXMLMath is negative and tends as LaTeXMLMath to zero as LaTeXMLMath and LaTeXMLMath go to infinity .
For LaTeXMLMath all components of the Riemannian and Ricci tensor as well as LaTeXMLMath remain regular at LaTeXMLMath and are therefore defined everywhere on the Riemannian manifolds LaTeXMLMath .
For LaTeXMLMath the scalar curvature degenerates at LaTeXMLMath in concordance with the fact that the hypersurfaces LaTeXMLMath are no longer complete in this case .
We proceed now studying the second fundamental form of the hypersurfaces LaTeXMLMath .
In order to do so , we need the Levi–Civita connection LaTeXMLMath of the Eguchi–Hanson space LaTeXMLMath .
It can be obtained from the Koszul formula , which reads for commuting vector fields as follows : LaTeXMLEquation .
In the following we will denote the coordinates LaTeXMLMath of the dense complex manifold LaTeXMLMath by LaTeXMLMath so that the components of LaTeXMLMath on LaTeXMLMath are given by LaTeXMLEquation .
Because of the symmetry LaTeXMLEquation of the covariant coefficients of the metric one obtains , for the Christoffel symbols , the relations LaTeXMLEquation .
Now the contravariant coefficients LaTeXMLMath of LaTeXMLMath are given by the matrix LaTeXMLEquation where LaTeXMLMath .
Further , the derivatives of the functions LaTeXMLMath and LaTeXMLMath are LaTeXMLEquation with LaTeXMLMath .
A straightforward calculation yields the Christoffel symbols of the first kind .
The Christoffel symbols of the first kind of the Eguchi–Hanson space LaTeXMLMath are given on LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation the remaining ones can be obtained from these by taking into account the symmetry LaTeXMLMath as well as the relations ( LaTeXMLRef ) together with the additional symmetries LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation The Christoffel symbols of the second kind are derived from these formulas as indicated in ( LaTeXMLRef ) .
By the symmetries of the Levi–Civita connection LaTeXMLMath it is sufficient to compute only six of them explicitly .
So one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we have introduced LaTeXMLMath In a similar way one obtains LaTeXMLEquation .
Further one checks that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation as well as LaTeXMLEquation where LaTeXMLMath .
Finally , one calculates LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation with LaTeXMLMath given by LaTeXMLMath Taking into account the relations ( LaTeXMLRef ) , ( LaTeXMLRef ) one thus obtains that the LaTeXMLMath are given as follows .
The components LaTeXMLMath of the Levi–Civita connection of the Eguchi–Hanson space LaTeXMLMath are given on LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLMath .
All remaining LaTeXMLMath can be obtained from the above by using LaTeXMLMath as well as the relations LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation In order to describe the outer geometry of the hypersurfaces LaTeXMLMath , we first determine a field of unit normal vectors LaTeXMLMath on LaTeXMLMath .
Up to orientation such a field is given by the conditions LaTeXMLEquation which are equivalent to the system of equations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By solving these equations with respect to the components LaTeXMLMath of the unit normal vectors one obtains the following proposition .
On LaTeXMLMath a field of unit normal vectors is given by LaTeXMLEquation where the functions LaTeXMLMath are LaTeXMLEquation .
LaTeXMLEquation We note that LaTeXMLMath and LaTeXMLMath can be viewed as the real and imaginary part of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath as the real and imaginary part of LaTeXMLMath respectively .
By construction the hypersurfaces LaTeXMLMath are imbedded in LaTeXMLMath .
If LaTeXMLMath denotes the field of unit normal vectors determined above , the second fundamental form of LaTeXMLMath is defined by LaTeXMLEquation .
It is symmetric and bilinear .
In the following we will write the coordinates LaTeXMLMath as LaTeXMLMath , and denote the components of LaTeXMLMath with respect to the induced frame of coordinate vector fields by LaTeXMLMath .
For shortness , we will simply write LaTeXMLMath for LaTeXMLMath in the remaining of this section .
Explicitly , LaTeXMLEquation where LaTeXMLMath .
On LaTeXMLMath the coordinates LaTeXMLMath can be expressed by the coordinates LaTeXMLMath according to LaTeXMLEquation where we have defined LaTeXMLEquation .
Thus one obtains that on LaTeXMLMath the polynomials appearing in the expressions for the LaTeXMLMath are given by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We compute now the covariant derivatives LaTeXMLMath .
To this end we first note the relations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Because of LaTeXMLEquation .
LaTeXMLEquation one has further LaTeXMLEquation .
LaTeXMLEquation and thus LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath By using the symmetries of the Christoffel symbols LaTeXMLMath one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The first component of LaTeXMLMath reads LaTeXMLEquation since by the relations ( LaTeXMLRef ) one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The second component is given by LaTeXMLEquation as can be verified by an analogous calculation .
As far as the third component is concerned , using also the relations ( LaTeXMLRef ) one computes LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and LaTeXMLEquation so that LaTeXMLEquation .
In the same way one verifies for the fourth component that LaTeXMLEquation .
The stated expression for LaTeXMLMath then follows by noting that the equalities LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation hold and that the derivatives LaTeXMLMath of the components of the normal vector with respect to LaTeXMLMath are given by the components LaTeXMLMath according to LaTeXMLEquation thus finishing the proof .
We remark that , since LaTeXMLMath , one has that LaTeXMLMath for all vector fields LaTeXMLMath ) , and a computation indeed shows that the normal part of LaTeXMLMath vanishes .
∎ LaTeXMLEquation .
One computes LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Once again we calculate the components of LaTeXMLMath separately .
By using the symmetries of the LaTeXMLMath and ( LaTeXMLRef ) , ( LaTeXMLRef ) one obtains LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and , moreover , LaTeXMLEquation .
A similar calculation gives for the second component the expression LaTeXMLEquation .
One calculates further LaTeXMLEquation as well as LaTeXMLEquation .
LaTeXMLEquation thus obtaining for the third component of LaTeXMLMath that LaTeXMLEquation .
Finally , by an analogous calculation one finds that the fourth component reads LaTeXMLEquation .
Since LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation the desired statement follows by noting that LaTeXMLMath and LaTeXMLEquation ∎ It remains to compute the covariant derivative of LaTeXMLMath with respect to LaTeXMLMath .
LaTeXMLEquation where LaTeXMLMath .
Again , LaTeXMLEquation .
By the symmetries of the LaTeXMLMath one has for LaTeXMLMath odd that LaTeXMLEquation and one obtains for the first and third component LaTeXMLEquation .
LaTeXMLEquation In an analogous way one has for LaTeXMLMath even LaTeXMLEquation the second and fourth component being given by LaTeXMLEquation .
LaTeXMLEquation and the assertion follows .
Again , one verifies that LaTeXMLMath .
∎ We are now able to compute the second fundamental form of the hypersurface LaTeXMLMath .
With respect to the coordinate frame LaTeXMLMath the components of the second fundamental form of the Riemannian LaTeXMLMath –manifolds LaTeXMLMath are given by LaTeXMLEquation .
By the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) and Proposition LaTeXMLRef one has LaTeXMLEquation where we made use of the relations ( LaTeXMLRef ) as well as LaTeXMLEquation .
LaTeXMLEquation Since further LaTeXMLMath , LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation one obtains for the expression above that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Here we made use of the relation LaTeXMLMath and LaTeXMLMath as well as LaTeXMLEquation .
Because of LaTeXMLMath one finally obtains LaTeXMLEquation since LaTeXMLMath .
By proposition LaTeXMLRef , LaTeXMLEquation and using the equalities LaTeXMLEquation .
LaTeXMLEquation in addition to the above relations , one also sees that LaTeXMLMath vanishes , since by Proposition LaTeXMLRef and ( LaTeXMLRef ) , LaTeXMLEquation .
Analogously , LaTeXMLEquation .
LaTeXMLEquation Finally , LaTeXMLEquation and the remaining components are determined by the symmetry of LaTeXMLMath .
∎ In order to compute the invariants of LaTeXMLMath we need to express the second fundamental form with respect to the orthonormal frame LaTeXMLMath .
In this case we denote its components by LaTeXMLMath .
Let LaTeXMLMath be the eigenvalues of LaTeXMLMath in this base , regarded as a symmetric transformation on LaTeXMLMath .
The mean curvature , the first elementary symmetric function associated with LaTeXMLMath , is then given by the sum LaTeXMLMath .
Now , writing LaTeXMLMath one has LaTeXMLEquation i.e. , LaTeXMLMath , where the coefficients of LaTeXMLMath are determined by the equations ( LaTeXMLRef ) .
As a consequence of the previous theorem we obtain then the following result .
The mean curvature of the hypersurfaces LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath denotes the geodesic curvature of the curve LaTeXMLMath in LaTeXMLMath .
In particular , LaTeXMLMath is a minimal surface if and only if LaTeXMLMath , i.e. , if LaTeXMLMath is a great circle in LaTeXMLMath .
One easily sees that LaTeXMLMath as well as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
So LaTeXMLMath .
We further remark that LaTeXMLMath and LaTeXMLEquation .
Let us now compute the geodesic curvature of LaTeXMLMath regarded as a curve in LaTeXMLMath using the stereographic projection .
With respect to the coordinates LaTeXMLMath the induced metric on LaTeXMLMath reads LaTeXMLEquation .
Writing LaTeXMLMath for LaTeXMLMath the geodesic curvature of LaTeXMLMath is then given by ( see e.g .
LaTeXMLCite ) LaTeXMLEquation the Christoffel symbols LaTeXMLMath being obtained from the formulas ( LaTeXMLRef ) , where now the LaTeXMLMath denote the components of LaTeXMLMath and the LaTeXMLMath should be replaced by the corresponding LaTeXMLMath .
Note that LaTeXMLMath .
We put LaTeXMLMath , LaTeXMLMath and obtain LaTeXMLEquation .
LaTeXMLEquation a direct calculation then yields LaTeXMLEquation and by noting that LaTeXMLMath and LaTeXMLMath one finally has ( up to a sign ) LaTeXMLEquation and thus the assertion .
∎ The third elementary symmetric function associated with LaTeXMLMath is the Gauss curvature ; it is given by LaTeXMLMath and equal to zero ; the second one is the so–called second order homogeneous curvature .
The second order homogeneous curvature of the hypersurfaces LaTeXMLMath is LaTeXMLEquation .
As computed in the proof of the previous corollary , the components of LaTeXMLMath with respect to the orthonormal frame ( LaTeXMLRef ) are given by LaTeXMLEquation .
The roots of the characteristic polynomial LaTeXMLEquation are then LaTeXMLMath , LaTeXMLMath .
∎ Since LaTeXMLMath , the three elementary symmetric functions associated with the second fundamental form , i.e. , essentially its trace LaTeXMLMath and the scalar curvature LaTeXMLMath , are manifestly invariant under the action of the isometry group LaTeXMLMath .
The fact that the mean curvature of the hypersurfaces LaTeXMLMath is given in terms of the geodesic curvature of LaTeXMLMath in LaTeXMLMath appears to be natural , since the geometry of the vector bundle LaTeXMLMath is determined by the elliptic geometry of LaTeXMLMath .
Note that LaTeXMLMath is the geodesic curvature of LaTeXMLMath as a curve in LaTeXMLMath with respect to the Euclidean metric .
As an immediate consequence one obtains the following statement .
Let LaTeXMLMath be a curve in LaTeXMLMath of bounded geodesic curvature .
Then the functionals LaTeXMLEquation stay bounded for LaTeXMLMath and LaTeXMLMath , respectively .
Consequently , the Willmore functional LaTeXMLMath remains unbounded , the hypersurfaces LaTeXMLMath thus being not accessible to integral geometry .
In this section we will study the structure of the geodesic flow of the hypersurfaces LaTeXMLMath and compute the exponential growth LaTeXMLMath explicitly , at least in the case where LaTeXMLMath is a generalized circle in LaTeXMLMath that arises by a Möbius transform from a circle in LaTeXMLMath with center at the origin .
In general , the exponential growth of an open , complete Riemannian manifold LaTeXMLMath is defined as LaTeXMLEquation where LaTeXMLMath is a point in LaTeXMLMath and LaTeXMLMath denotes the volume of the ball of radius LaTeXMLMath with center at LaTeXMLMath .
If LaTeXMLMath , one says that LaTeXMLMath has subexponential growth .
In case LaTeXMLMath has finite volume , this quantity is not interesting , since then one always has LaTeXMLMath , but for LaTeXMLMath the exponential growth is directly related to the infimum of the essential spectrum of the Laplace operator on LaTeXMLMath .
We will return to this point in section LaTeXMLRef .
There we will be able to calculate the exponential growth of LaTeXMLMath for arbitrary closed curves .
Let LaTeXMLMath be a smooth curve in LaTeXMLMath and LaTeXMLMath LaTeXMLMath a vector field along LaTeXMLMath .
Its covariant derivative with respect to LaTeXMLMath is given by the formula LaTeXMLEquation where LaTeXMLMath are the components of the Levi–Civita connection of LaTeXMLMath with respect to the coordinate frame LaTeXMLMath .
For a geodesic it holds that LaTeXMLMath and one obtains the system of differential equations LaTeXMLEquation .
However , it turns out to be more convenient to determine the geodesic lines of the hypersurfaces LaTeXMLMath by considering the first integrals of the geodesic flow .
Let us consider therefore the geodesic system LaTeXMLMath of LaTeXMLMath , where the Lagrangian LaTeXMLMath is given by the metric , LaTeXMLEquation .
The function LaTeXMLMath is a first integral of the geodesic flow , i. e. with respect to the coordinate reper LaTeXMLMath one has that LaTeXMLEquation is constant for any geodesic line .
Let now LaTeXMLMath and LaTeXMLMath .
Since the coefficients of the metric LaTeXMLMath do not depend on the angle variable LaTeXMLMath , the map LaTeXMLEquation represents a one–parameter family of isometries .
Consequently , using Noether ’ s theorem , the function LaTeXMLEquation is a second first integral of the geodesic flow and a computation yields the formula LaTeXMLEquation .
For LaTeXMLMath and LaTeXMLMath it can be seen immediately from the equations ( LaTeXMLRef ) for a geodesic or the relation LaTeXMLMath that , for LaTeXMLEquation the curve LaTeXMLMath must be a geodesic in LaTeXMLMath .
We will assume from now on that LaTeXMLMath is constant and , in this case , determine the distance of a point LaTeXMLMath to the set LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are constant , LaTeXMLMath , the coefficients LaTeXMLMath do also not depend on LaTeXMLMath so that LaTeXMLEquation is an additional one–parameter group of isometries and Noether ’ s Theorem gives a third first integral , LaTeXMLEquation i. e. , LaTeXMLEquation is constant for any geodesic line as well .
From the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) one obtains LaTeXMLEquation and LaTeXMLEquation .
Solving the latter two equations with respect to LaTeXMLMath and LaTeXMLMath yields LaTeXMLEquation .
LaTeXMLEquation as well as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation thus the functions LaTeXMLMath and LaTeXMLMath are determined by the function LaTeXMLMath .
Equation ( LaTeXMLRef ) now reads LaTeXMLEquation .
Note that LaTeXMLMath is non–negative .
By inserting the expressions for LaTeXMLMath and LaTeXMLMath into the previous equation one finally obtains the following ordinary differential equation for LaTeXMLMath : LaTeXMLEquation .
Thus , for LaTeXMLMath , all geodesics LaTeXMLMath in LaTeXMLMath are parametrized by the three parameters LaTeXMLMath .
We are now able to compute the distance of a point LaTeXMLMath to the curve LaTeXMLMath .
Let LaTeXMLMath be a circle in LaTeXMLMath of radius LaTeXMLMath .
The distance of a point LaTeXMLMath to the curve LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath denotes the hypergeometric function , which is defined for LaTeXMLMath by the series LaTeXMLEquation the parameters LaTeXMLMath being arbitrary complex numbers , LaTeXMLMath .
Let LaTeXMLMath be a geodesic of positive energy LaTeXMLMath from the curve LaTeXMLMath to the point LaTeXMLMath with coordinates LaTeXMLMath .
For LaTeXMLMath the geodesic LaTeXMLMath is precisely the geodesic line ( LaTeXMLRef ) already described .
If LaTeXMLMath were not equal zero , at least LaTeXMLMath would be different from zero almost everywhere ; then equation ( LaTeXMLRef ) would imply that there exists a critical value LaTeXMLMath for which LaTeXMLEquation .
For smaller values of LaTeXMLMath the right–hand side of ( LaTeXMLRef ) would become negative , implying that LaTeXMLMath must hold for all LaTeXMLMath .
This means that for LaTeXMLMath the geodesic LaTeXMLMath can never reach the curve LaTeXMLMath .
Assume therefore LaTeXMLMath , LaTeXMLMath being arbitrary .
By ( LaTeXMLRef ) we have LaTeXMLEquation .
In case that LaTeXMLMath , this expression becomes negative for small LaTeXMLMath so that LaTeXMLMath can never reach the set LaTeXMLMath .
However , for LaTeXMLMath we have that LaTeXMLMath is non–negative for all LaTeXMLMath , as well as LaTeXMLEquation so there are infinitely many geodesic lines LaTeXMLMath reaching the set LaTeXMLMath in LaTeXMLMath in a spiral motion .
In this case , equation ( LaTeXMLRef ) implies for LaTeXMLMath the relation LaTeXMLEquation i.e. , LaTeXMLMath as well as LaTeXMLMath are strictly monotone increasing as functions in LaTeXMLMath and the point LaTeXMLMath is reached earliest , that is , for smallest LaTeXMLMath , in case that LaTeXMLMath is also zero .
Since the length of a geodesic is given by LaTeXMLEquation the distance of the point LaTeXMLMath to the set LaTeXMLMath must be given by the length of the geodesic LaTeXMLMath .
The integral LaTeXMLMath can not be represented by elementary functions and one has LaTeXMLEquation where LaTeXMLMath is the hypergeometric function introduced above .
For LaTeXMLMath the defining series converges even in LaTeXMLMath ; the hypergeometric function has an analytic continuation for LaTeXMLMath and under the assumption that LaTeXMLMath it can be written for all LaTeXMLMath as the integral LaTeXMLEquation where LaTeXMLMath denotes the Gamma function and LaTeXMLMath is assumed in order to make the integrand uniquely defined .
If LaTeXMLMath is real , differentiation under the integral with respect to LaTeXMLMath gives the stated equality ( LaTeXMLRef ) if one takes the relation LaTeXMLMath , LaTeXMLMath , LaTeXMLMath arbitrary , into account additionally .
For LaTeXMLMath we finally deduce from ( LaTeXMLRef ) LaTeXMLEquation .
LaTeXMLEquation and thus LaTeXMLEquation finishing the proof .
∎ We are now in a position to compute the exponential growth of the hypersurface LaTeXMLMath in case that LaTeXMLMath is a circle in LaTeXMLMath .
Note that we can estimate the volume of the ball with radius LaTeXMLMath around a point LaTeXMLMath by the volume of the union over all LaTeXMLMath –balls around points of LaTeXMLMath , thus obtaining LaTeXMLEquation .
LaTeXMLEquation since by our previous considerations LaTeXMLEquation where LaTeXMLMath is given by the expression ( LaTeXMLRef ) for LaTeXMLMath .
The analytic continuation of LaTeXMLMath for LaTeXMLMath is given by the formula LaTeXMLEquation .
LaTeXMLEquation so that for LaTeXMLMath being big enough the distance of LaTeXMLMath to the set LaTeXMLMath is given by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation implying that LaTeXMLMath is proportional to LaTeXMLMath for LaTeXMLMath .
We obtain for LaTeXMLMath that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation the corresponding limes superior therefore being zero , too .
By isometry arguments we thus obtain the following proposition .
Let LaTeXMLMath be a circle in LaTeXMLMath with center at the origin and radius LaTeXMLMath .
Then LaTeXMLMath for all LaTeXMLMath .
We want to finish this section with some remarks concerning closed geodesics in LaTeXMLMath , where we assume again that LaTeXMLMath is a circle in LaTeXMLMath of radius LaTeXMLMath .
In this case LaTeXMLMath is foliated by the two–dimensional tori LaTeXMLMath , LaTeXMLMath being constant .
Let LaTeXMLMath be a geodesic line parametrized by arc length .
Since LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are also zero .
Relation ( LaTeXMLRef ) then reads LaTeXMLMath and equation ( LaTeXMLRef ) must hold , representing a condition on LaTeXMLMath for given values of LaTeXMLMath .
Now , by the LaTeXMLMath –symmetry of LaTeXMLMath , LaTeXMLEquation .
Writing LaTeXMLMath , LaTeXMLMath we see that LaTeXMLMath is a closed geodesic if and only if LaTeXMLMath and LaTeXMLMath are satisfied , i. e. , if LaTeXMLEquation of course , if LaTeXMLMath or LaTeXMLMath are zero , LaTeXMLMath is also a closed geodesic .
Inserting the expressions for LaTeXMLMath and LaTeXMLMath computed above we obtain for the previous condition LaTeXMLEquation .
Note that LaTeXMLMath .
Taking all together we find as solutions for LaTeXMLMath and LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath are integers .
The curve LaTeXMLMath is then a closed geodesic in LaTeXMLMath , where LaTeXMLMath depend on LaTeXMLMath as explained above .
In particular , there must be at least countably many closed geodesics in LaTeXMLMath .
In this section we will show that the LaTeXMLMath –kernel of the Laplacian on the hypersurfaces LaTeXMLMath becomes trivial for all LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are arbitrary .
We will base our considerations on the much more general work of Greene and Wu LaTeXMLCite , who studied integrals of certain generalized subharmonic functions on connected non–compact Riemannian manifolds admitting a canonical exhaustion function and showed that these integrals can not be bounded .
More precisely , they showed that the following theorem holds .
Let LaTeXMLMath be a connected non–compact oriented LaTeXMLMath Riemannian manifold .
Suppose that there exists a continuous proper function LaTeXMLMath and a compact set LaTeXMLMath such that a ) LaTeXMLMath is LaTeXMLMath .
b ) LaTeXMLMath is uniformly Lipschitz continuous .
c ) LaTeXMLMath is subharmonic .
Denote by LaTeXMLMath the closure of the set of all LaTeXMLMath subharmonic functions in LaTeXMLMath .
Then , if LaTeXMLMath is a nonnegative function in LaTeXMLMath such that LaTeXMLEquation there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath , where LaTeXMLMath denotes the set of all LaTeXMLMath such that LaTeXMLMath ; in particular , LaTeXMLMath .
A description of the set LaTeXMLMath is given by the following proposition .
Let LaTeXMLMath be a non–compact LaTeXMLMath Riemannian manifold .
Then the following functions are in LaTeXMLMath : 1 ) Any function LaTeXMLMath that is the limit uniformly on compact subsets of LaTeXMLMath of a sequence of functions in LaTeXMLMath , 2 ) LaTeXMLMath subharmonic functions , 3 ) LaTeXMLMath where LaTeXMLMath is a LaTeXMLMath nonnegative subharmonic function and LaTeXMLMath , 4 ) LaTeXMLMath where LaTeXMLMath is a harmonic function and LaTeXMLMath , 5 ) any geodesically convex function .
In general the scalar Laplacian on a Riemannian manifold LaTeXMLMath , acting on LaTeXMLMath functions , is given by LaTeXMLMath , where for a vector field LaTeXMLMath its divergence with respect to an orthonormal frame LaTeXMLMath is given by LaTeXMLEquation .
Here the LaTeXMLMath denote the components of LaTeXMLMath and the LaTeXMLMath the connection forms of the Levi–Civita connection LaTeXMLMath of LaTeXMLMath .
In the following we will show that the above results also apply for the considered hypersurfaces LaTeXMLMath , LaTeXMLMath being arbitrary , obtaining in particular the vanishing of the LaTeXMLMath –kernel of the Laplacian even in case LaTeXMLMath is not complete .
Let us first start considering the function LaTeXMLEquation which is LaTeXMLMath on LaTeXMLMath .
One calculates with respect to the orthonormal frame ( LaTeXMLRef ) LaTeXMLEquation and thus LaTeXMLEquation .
LaTeXMLEquation Because of LaTeXMLMath it follows that LaTeXMLMath is subharmonic and one computes further that LaTeXMLEquation .
We define now the LaTeXMLMath function LaTeXMLMath by LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and put LaTeXMLEquation .
The function LaTeXMLMath is LaTeXMLMath too , monotone , equal to zero for LaTeXMLMath and one for LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLEquation is LaTeXMLMath on LaTeXMLMath and subharmonic on LaTeXMLMath where LaTeXMLEquation .
Note that LaTeXMLMath is compact and that LaTeXMLMath is proper , i. e. , LaTeXMLEquation is compact for all LaTeXMLMath .
We show that LaTeXMLMath is globally Lipschitz .
In order to do so , let us first remark that LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is a constant , since LaTeXMLMath tends asymptotically to LaTeXMLMath on LaTeXMLMath and , as a smooth function , remains bounded on LaTeXMLMath .
Now let LaTeXMLMath and LaTeXMLMath be two points on LaTeXMLMath , and LaTeXMLMath the shortest geodesic between them so that LaTeXMLMath ; we assume that LaTeXMLMath is parametrized by arc length .
Since LaTeXMLEquation one has by Cauchy–Schwarz that LaTeXMLEquation .
LaTeXMLEquation i.e. , LaTeXMLMath is uniformly Lipschitz continuous on LaTeXMLMath .
Summing up we obtain the following proposition .
On the connected non-compact oriented LaTeXMLMath Riemannian manifolds LaTeXMLMath there exists , for every LaTeXMLMath and every curve LaTeXMLMath , a proper continuous function LaTeXMLMath and a compact set LaTeXMLMath such that a ) LaTeXMLMath is LaTeXMLMath , b ) LaTeXMLMath is uniformly Lipschitz , c ) LaTeXMLMath is subharmonic .
In particular the conclusions of Theorem LaTeXMLRef hold .
Note that the above proposition is also true in case LaTeXMLMath , i.e. , for the non–complete LaTeXMLMath Riemannian manifolds LaTeXMLMath .
As a consequence of the proposition we obtain the following vanishing theorem .
Let LaTeXMLMath .
There exist no LaTeXMLMath –harmonic functions , on the hypersurfaces LaTeXMLMath for arbitrary LaTeXMLMath and curves LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be given as in the previous proposition and let LaTeXMLMath be a harmonic function on LaTeXMLMath .
By Proposition LaTeXMLRef one has LaTeXMLMath for all LaTeXMLMath .
Now , by the Aronszajn–Cordes uniqueness theorem for second order differential operators LaTeXMLMath with elliptic metric principal symbol LaTeXMLCite LaTeXMLMath can not vanish identically on LaTeXMLMath unless it vanishes everywhere .
Therefore , for LaTeXMLMath not being trivial , the set LaTeXMLMath is not empty , and by Theorem LaTeXMLRef there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
In particular LaTeXMLMath for all LaTeXMLMath .
∎ In the sequel we will consider the Dirac operator LaTeXMLMath on the hypersurfaces LaTeXMLMath , whose geometry has been studied in the previous sections .
For LaTeXMLMath , the homotopy type of LaTeXMLMath is given by LaTeXMLMath .
If the curve LaTeXMLMath is not closed , LaTeXMLMath can not be complete and admits only one spin structure .
Otherwise LaTeXMLMath has the same homotopy type as the circle LaTeXMLMath and , consequently , admits two spin structures .
The trivial spin structure is characterized by the fact that there exists a global trivialization of the LaTeXMLMath –principal bundle covering an arbitrary orthonormal frame bundle , while the non–trivial spin structure admits a trivialization of this kind only locally .
On the other hand , the unique spin structure of the Eguchi–Hanson space LaTeXMLMath induces a spin structure on the hypersurface LaTeXMLMath by reduction of the former with respect to the normal vector field of LaTeXMLMath .
It turns out that the induced spin structure is the trivial one if and only if the winding number of the closed curve LaTeXMLMath is even .
In the following most of the results will be derived for the induced spin structure , though some of them that follow from purely geometric arguments hold for both spin structures .
First we will try to determine solutions to the Dirac equation that are also solutions to the Einstein equation and we will show that the aforementioned hypersurfaces do not admit such solutions in case LaTeXMLMath .
Nevertheless , it is possible to construct such solutions explicitly by deformation into the singular situation , though these solutions are no longer complete .
In the complete case and if LaTeXMLMath is a minimal surface , one can further show the existence of a spinor field satisfying a generalized Killing equation for spinors .
Let LaTeXMLMath denote the standard basis of the Euclidean space LaTeXMLMath and introduce the complex two-dimensional matrices LaTeXMLEquation .
In case LaTeXMLMath , the spin representation of the LaTeXMLMath –dimensional complex Clifford–algebra LaTeXMLMath is given by the isomorphism LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is equal to LaTeXMLMath and LaTeXMLMath for LaTeXMLMath odd and even , respectively .
For LaTeXMLMath one has the representation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath denote the corresponding representation spaces as well as the representations itself .
The induced representations of LaTeXMLMath will be denoted by the same symbols .
We denote by LaTeXMLMath or simply LaTeXMLMath the spinor bundle considered in each case of LaTeXMLMath , by LaTeXMLMath its hermitean inner product and by LaTeXMLMath the space of smooth sections in LaTeXMLMath .
Further we identify the tangent bundle LaTeXMLMath and the cotangent bundle LaTeXMLMath with the aid of LaTeXMLMath .
The Clifford multiplication LaTeXMLMath of a spinor and a vector can then be extended naturally to a multiplication LaTeXMLMath of a spinor and a form .
The Levi–Civita connection LaTeXMLMath of LaTeXMLMath induces a covariant derivative in LaTeXMLMath , which we will denote by LaTeXMLMath , too .
With respect to an orthonormal frame LaTeXMLMath one has for LaTeXMLMath the local representation LaTeXMLEquation where the LaTeXMLMath are the connection forms of the Levi–Civita connection LaTeXMLMath .
The Dirac operator LaTeXMLMath on LaTeXMLMath is then locally given by LaTeXMLEquation where LaTeXMLMath denotes the Clifford multiplication of a vector field with a spinor ; in the realization of the complex Clifford algebra LaTeXMLMath given above , the vectors LaTeXMLMath are represented by the matrices LaTeXMLMath , respectively .
Note that in the three–dimensional Clifford algebra it hold that LaTeXMLMath , where LaTeXMLMath denotes the totally skew symmetric tensor .
With respect to the global trivialization ( LaTeXMLRef ) the LaTeXMLMath –forms LaTeXMLMath have been computed in Proposition LaTeXMLRef .
Let us now introduce the following definitions .
A non–trivial spinor field LaTeXMLMath on a Riemannian spin manifold LaTeXMLMath with LaTeXMLMath is called a positive resp .
negative Einstein spinor with eigenvalue LaTeXMLMath if it is a solution of the Dirac equation and the Einstein equation LaTeXMLEquation where LaTeXMLMath is the symmetric LaTeXMLMath –tensor field defined by LaTeXMLMath , the energy momentum tensor of LaTeXMLMath .
As shown in LaTeXMLCite , in dimension LaTeXMLMath and in case that the scalar curvature does not vanish , the existence of an Einstein spinor is equivalent to the existence of a so–called WK spinor : Let LaTeXMLMath be a Riemannian spin manifold whose scalar curvature LaTeXMLMath does not vanish anywhere .
A non–trivial spinor field on LaTeXMLMath satisfying the field equation LaTeXMLEquation is called a weak Killing spinor or WK spinor with WK number LaTeXMLMath .
For general LaTeXMLMath each solution LaTeXMLMath of the field equation ( LaTeXMLRef ) with LaTeXMLMath and LaTeXMLMath corresponds to a positive and negative Einstein spinor with eigenvalue LaTeXMLMath , respectively .
For the existence of a WK spinor the following necessary condition is known LaTeXMLCite : Let LaTeXMLMath be a Riemannian spin manifold with non–vanishing scalar curvature and LaTeXMLMath a WK spinor on LaTeXMLMath with WK number LaTeXMLMath .
Then LaTeXMLEquation .
We show in the following that , for LaTeXMLMath , the condition ( LaTeXMLRef ) can not be fulfilled on LaTeXMLMath for any choice of the curve LaTeXMLMath .
For LaTeXMLMath and for any spin structure the hypersurfaces LaTeXMLMath do not admit solutions of the WK equation and , hence , there can be no solution to the Dirac–Einstein system .
Assume that a WK spinor with WK number LaTeXMLMath is given on LaTeXMLMath .
Then , by Proposition LaTeXMLRef LaTeXMLEquation must hold , where LaTeXMLMath has been computed in Theorem LaTeXMLRef Using the relation LaTeXMLMath one computes with respect to the trivialization ( LaTeXMLRef ) LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation thus obtaining for the Laplacian of LaTeXMLMath that LaTeXMLEquation .
LaTeXMLEquation compare Proposition LaTeXMLRef .
One computes further that LaTeXMLEquation .
LaTeXMLEquation as well as LaTeXMLEquation obtaining for LaTeXMLMath the expression LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath , one obtains LaTeXMLEquation for the left–hand side of ( LaTeXMLRef ) and LaTeXMLEquation for the right–hand side , so that the condition ( LaTeXMLRef ) reads LaTeXMLEquation and one sees that in case LaTeXMLMath , it can not be satisfied for any choice of the curve LaTeXMLMath .
Note that since the integrability condition ( LaTeXMLRef ) is purely geometric , the assertion of the proposition holds for any spin structure .
∎ Only for LaTeXMLMath the condition ( LaTeXMLRef ) is fulfilled for arbitrary values of LaTeXMLMath , since then both sides vanish .
In this case the hypersurfaces LaTeXMLMath are no longer complete for any curve , the metric becoming degenerate along the exceptional curve ; one finds that LaTeXMLMath and the Ricci tensor and the scalar curvature are LaTeXMLEquation .
In the following we show that , in this case , solutions of the Dirac–Einstein system can be constructed explicitly on LaTeXMLMath for an arbitrary choice of the curve LaTeXMLMath .
In order to do so let LaTeXMLMath be a non–trivial spinor field on LaTeXMLMath that satisfies the spinor equation ( LaTeXMLRef ) for LaTeXMLMath , LaTeXMLEquation .
Putting LaTeXMLMath , the above equation can be reformulated into an equation for LaTeXMLMath .
Using LaTeXMLMath for a function LaTeXMLMath and the relation LaTeXMLMath in the 3–dimensional Clifford algebra yields LaTeXMLEquation .
As already shown , with respect to the base ( LaTeXMLRef ) only LaTeXMLMath is different from zero and one obtains LaTeXMLEquation .
Further , one has LaTeXMLEquation .
In the realization of the complex Clifford algebra given above one then obtains due to Proposition LaTeXMLRef that LaTeXMLEquation .
LaTeXMLEquation Now , if LaTeXMLMath , LaTeXMLEquation as well as LaTeXMLEquation .
Summing up , ( LaTeXMLRef ) now reads LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation the summands LaTeXMLMath and LaTeXMLMath cancelling out each other .
Since LaTeXMLMath , the system above can be integrated .
Taking into account the equality LaTeXMLMath and the expressions for the LaTeXMLMath one derives the system of partial differential equations LaTeXMLEquation where LaTeXMLEquation are functions in the variables LaTeXMLMath and LaTeXMLMath .
Note that LaTeXMLMath .
Further one has LaTeXMLEquation showing that LaTeXMLEquation is a solution of the system above .
Transforming back to the original WK equation yields the following proposition .
Consider the family of hypersurfaces LaTeXMLMath , where LaTeXMLMath is an arbitrary curve .
Then LaTeXMLEquation is a WK spinor of length LaTeXMLMath and WK number LaTeXMLMath .
Thus , the normalized spinor LaTeXMLEquation is an Einstein spinor on LaTeXMLMath with eigenvalue LaTeXMLMath .
The homotopy type of LaTeXMLMath is given by LaTeXMLMath ; therefore it has at least two spin structures , the one involved here being determined by the global trivialization ( LaTeXMLRef ) .
Recall that LaTeXMLMath is parametrized by the length parameter LaTeXMLMath of the curve LaTeXMLMath and the fiber parameters LaTeXMLMath .
The metric LaTeXMLMath is then given by the formula LaTeXMLEquation and the Ricci tensor has rank two , see equation ( LaTeXMLRef ) .
Similar examples of WK spinors on a LaTeXMLMath –dimensional non–complete Riemannian manifold with negative scalar curvature have been constructed in LaTeXMLCite .
We introduce now the notion of a T–Killing spinor LaTeXMLCite .
Let LaTeXMLMath be a Riemannian spin manifold .
A spinor field LaTeXMLMath without zeros will be called a T–Killing spinor if the trace LaTeXMLMath is constant and LaTeXMLMath is a solution of the field equation LaTeXMLEquation .
Here LaTeXMLMath is the energy momentum tensor of the normalized spinor LaTeXMLMath .
As remarked at the beginning , LaTeXMLMath is endowed with a hyperkähler structure and therefore Ricci–flat and self–dual .
Due to this , there is a parallel spinor on LaTeXMLMath , and the study of its restriction to LaTeXMLMath will enable us to construct a LaTeXMLMath –Killing spinor explicitly .
There we follow a similar construction carried out in LaTeXMLCite , where the restriction of a parallel spinor on the Euclidean space LaTeXMLMath to an isometrically immersed closed LaTeXMLMath –surface of constant mean curvature is considered , yielding examples of LaTeXMLMath –Killing spinors on any surface of constant mean curvature in LaTeXMLMath .
We consider first the restriction of the spinor bundle of LaTeXMLMath to the submanifold LaTeXMLMath ( compare LaTeXMLCite ) .
Note that the Clifford representation LaTeXMLMath can be constructed directly from the Clifford representation LaTeXMLMath by setting LaTeXMLEquation and defining the Clifford multiplication in LaTeXMLMath by means of the Clifford multiplication in LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation The mapping LaTeXMLEquation is an automorphism of the corresponding LaTeXMLMath –representation , and because of LaTeXMLMath it turns out to be an involution .
Thus the spin representation LaTeXMLMath decomposes into the eigensubspaces of LaTeXMLMath , and we denote them by LaTeXMLMath .
Explicitly one has LaTeXMLEquation yielding in particular for LaTeXMLMath the relation LaTeXMLEquation since LaTeXMLMath in the three–dimensional Clifford algebra .
In this way one obtains LaTeXMLEquation i.e. , a spinor in LaTeXMLMath or LaTeXMLMath uniquely defines a spinor in LaTeXMLMath and vice versa .
Thus we have defined two isomorphisms of LaTeXMLMath representations , LaTeXMLEquation .
Since the four–dimensional spin manifold LaTeXMLMath is simply connected , it has only one spin structure , and we denote the corresponding spinor bundle by LaTeXMLMath .
It splits into the subbundles LaTeXMLMath and LaTeXMLMath , according to the above decomposition of LaTeXMLMath , and as a consequence of LaTeXMLMath and ( LaTeXMLRef ) we have the identifications LaTeXMLEquation where LaTeXMLMath is the induced spinor bundle on LaTeXMLMath .
Consider now a spinor field LaTeXMLMath and its restriction LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath is a three–dimensional spinor field .
In particular note that for a field of unit normal vectors on LaTeXMLMath the relation LaTeXMLMath holds , according to the realization of LaTeXMLMath given above .
By using the local formulas for the different covariant derivatives one obtains for the spinorial derivative of LaTeXMLMath on LaTeXMLMath the relation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for every vector field LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath .
Here and until the end of this section LaTeXMLMath denotes an arbitrary section in the frame bundle of LaTeXMLMath .
Since one part of the Weyl tensor of the Eguchi–Hanson space LaTeXMLMath vanishes , we can assume that the parallel spinor on LaTeXMLMath is contained in LaTeXMLMath and given by LaTeXMLMath .
Hence LaTeXMLMath , and with LaTeXMLMath we obtain the equation LaTeXMLEquation for the corresponding three–dimensional spinor LaTeXMLMath .
Further , since LaTeXMLMath is a symmetric bilinear form , LaTeXMLMath is a scalar and one obtains LaTeXMLEquation moreover , LaTeXMLMath has constant length because it is given by the restriction of a parallel spinor .
We summarize these results in the following lemma .
Let LaTeXMLMath denote the induced spinor bundle of LaTeXMLMath .
Then there exists a spinor LaTeXMLMath on LaTeXMLMath with LaTeXMLEquation .
Let now LaTeXMLMath be given as in the previous lemma .
Then LaTeXMLMath , so that LaTeXMLEquation .
Making use of the relation LaTeXMLMath , which holds for an arbitrary vector field LaTeXMLMath and spinor LaTeXMLMath , one computes in the base of the LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation since only the summands with LaTeXMLMath are different from zero .
In particular , one has LaTeXMLEquation and it follows that LaTeXMLMath is constant if LaTeXMLMath is constant .
Since the latter only occurs if LaTeXMLMath vanishes identically , we deduce the following proposition .
Denote by LaTeXMLMath the induced spinor bundle of LaTeXMLMath and let LaTeXMLMath be a minimal surface , i. e. , LaTeXMLMath a great circle in LaTeXMLMath .
Then there exists a T–Killing spinor LaTeXMLMath with LaTeXMLMath satisfying the field equation LaTeXMLEquation .
For any other choice of the curve LaTeXMLMath there are no T–Killing spinors .
In this section we will study some properties of the spectrum LaTeXMLMath of the Dirac operator on the hypersurfaces LaTeXMLMath , LaTeXMLMath being a closed curve , so that LaTeXMLMath is complete .
In general , the Dirac operator LaTeXMLMath on a Riemannian spin manifold LaTeXMLMath is an elliptic formally selfadjoint differential operator of first order and , as a differential operator , closable .
If LaTeXMLMath is complete , LaTeXMLMath is essentially selfadjoint as an unbounded operator in LaTeXMLMath with domain LaTeXMLMath and the kernels of LaTeXMLMath and LaTeXMLMath coincide , see e.g .
LaTeXMLCite .
Here LaTeXMLMath is defined as the completion of LaTeXMLMath , the space of sections in LaTeXMLMath with compact support , with respect to the norm induced by the scalar product LaTeXMLEquation .
One has LaTeXMLMath .
If LaTeXMLMath is complete , LaTeXMLMath is real and consists only of the approximation spectrum since , in this case , LaTeXMLMath has no residual spectrum .
If , additionally , LaTeXMLMath is non–compact , one has to expect point spectrum as well as continuous spectrum ; in particular , we are interested in the essential spectrum of LaTeXMLMath , which is defined by LaTeXMLEquation and represents the continuous spectrum together with the eigenvalues of infinite multiplicity .
The main result of this section will consist in showing that the infimum of LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is a closed curve and LaTeXMLMath , becomes arbitrarily small for arbitrary values of the parameter LaTeXMLMath , and that LaTeXMLMath ; for LaTeXMLMath arising by a Möbius transform from a circle in LaTeXMLMath with center at the origin , we also show that the LaTeXMLMath –kernel of LaTeXMLMath and LaTeXMLMath are trivial , thus obtaining LaTeXMLMath in this case .
As we use the global trivialization ( LaTeXMLRef ) , these results hold for the trivial spin structure .
Let LaTeXMLMath be a closed curve and LaTeXMLMath the closure of the Dirac operator on the hypersurfaces LaTeXMLMath , endowed with the trivial spin structure , where LaTeXMLMath and LaTeXMLMath .
Then , for arbitrary LaTeXMLMath , LaTeXMLEquation and LaTeXMLMath .
We will prove these statements by using the LaTeXMLMath – LaTeXMLMath principle .
For this , we need the following lemmas .
The LaTeXMLMath –kernel of LaTeXMLMath on LaTeXMLMath is non–trivial for arbitrary LaTeXMLMath and LaTeXMLMath .
With respect to the realization of the previous section one has for LaTeXMLMath that LaTeXMLEquation .
LaTeXMLEquation i.e. , LaTeXMLMath for every LaTeXMLMath .
Then LaTeXMLMath implies that the Dirac operator on LaTeXMLMath is given by LaTeXMLEquation since LaTeXMLEquation .
By taking LaTeXMLMath into account one obtains on LaTeXMLMath , for the Dirac operator , the system of partial differential equations LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation Let now LaTeXMLMath and LaTeXMLMath be of the form LaTeXMLMath .
Clearly one has then LaTeXMLEquation as well as LaTeXMLEquation .
Equating these expressions yields the relation LaTeXMLEquation for LaTeXMLMath , so that by integration LaTeXMLEquation .
Putting LaTeXMLMath one sees that all spinors of the form LaTeXMLEquation are harmonic on LaTeXMLMath , where LaTeXMLMath are constants .
Since , further LaTeXMLEquation for an open region LaTeXMLMath and LaTeXMLMath , the harmonic spinors LaTeXMLMath are in LaTeXMLMath .
∎ Let LaTeXMLMath and LaTeXMLMath be arbitrary .
Then there exists a LaTeXMLMath –harmonic spinor LaTeXMLMath on LaTeXMLMath which can be approximated pointwise by spinors LaTeXMLMath depending on a parameter LaTeXMLMath such that LaTeXMLMath .
To begin with , note that LaTeXMLMath converges pointwise to LaTeXMLMath as LaTeXMLMath , and we therefore introduce the function LaTeXMLEquation replacing in LaTeXMLMath the parameter LaTeXMLMath of the Kähler potential by the new parameter LaTeXMLMath .
One computes LaTeXMLEquation .
LaTeXMLEquation as well as LaTeXMLEquation since LaTeXMLMath .
Each other function in LaTeXMLMath and LaTeXMLMath of the functional dependence LaTeXMLMath is also harmonic with respect to LaTeXMLMath and LaTeXMLMath .
We put LaTeXMLEquation .
For LaTeXMLMath one has then LaTeXMLEquation .
As remarked , LaTeXMLMath , so that LaTeXMLEquation .
LaTeXMLEquation Then one computes , since LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , as well as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation i. e. , the LaTeXMLMath are LaTeXMLMath –approximations of LaTeXMLMath –harmonic spinors , LaTeXMLEquation .
LaTeXMLMath being in LaTeXMLMath , too .
∎ In the following we will use the abbreviations LaTeXMLEquation for LaTeXMLMath we then have that LaTeXMLEquation pointwise .
Let LaTeXMLMath be as in ( LaTeXMLRef ) .While LaTeXMLMath becomes unbounded for LaTeXMLMath , it does not follow that LaTeXMLMath for small LaTeXMLMath .
Nevertheless , we will show that for given LaTeXMLMath and LaTeXMLMath small enough , LaTeXMLMath , thus proving Theorem LaTeXMLRef .
For this we have to determine precise estimates for the Rayleigh quotient LaTeXMLMath from above , where the point is to find bounds not depending on LaTeXMLMath .
Let LaTeXMLMath be as in Lemma LaTeXMLRef , equation ( LaTeXMLRef ) .
One has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation therefore LaTeXMLMath is strictly increasing and tends to LaTeXMLMath as LaTeXMLMath , so that it seems natural to estimate LaTeXMLMath from below according to LaTeXMLEquation .
Here LaTeXMLMath is a cutting point to be determined in a convenient manner , such that the resulting lower bound for LaTeXMLMath is as great as possible .
A possible choice would be the turning point LaTeXMLMath of LaTeXMLMath , which can be calculated by the condition LaTeXMLMath by solving the equation of third degree in LaTeXMLMath LaTeXMLEquation .
Since this turns out to be a little bit involved and does not necessarily lead to optimal estimates , we look for a condition for LaTeXMLMath instead such that LaTeXMLEquation i. e. , LaTeXMLEquation .
LaTeXMLEquation This is fulfilled if LaTeXMLEquation where we assumed LaTeXMLMath .
For small LaTeXMLMath and LaTeXMLMath this does not represent a much stronger condition .
Again this is assured if LaTeXMLEquation and we put LaTeXMLEquation .
Then one calculates LaTeXMLEquation the function LaTeXMLMath being given by LaTeXMLEquation .
We remark that , as LaTeXMLMath , the functions LaTeXMLMath and LaTeXMLMath tend to a finite value that is independent of LaTeXMLMath , namely LaTeXMLEquation the cutting point LaTeXMLMath also remaining finite .
We finally obtain an estimate for LaTeXMLMath of the form LaTeXMLEquation .
LaTeXMLEquation Note that LaTeXMLMath tends to LaTeXMLMath if , additionally , LaTeXMLMath so that the value of LaTeXMLMath at the point LaTeXMLMath becomes arbitrarily close to LaTeXMLMath .
This can always be achieved by choosing LaTeXMLMath small enough , though for big LaTeXMLMath the cutting point LaTeXMLMath becomes big , too .
Nevertheless , we will see that this is of no relevance for later arguments .
For small LaTeXMLMath we do not lose too much by the above estimate , since then LaTeXMLMath is also small .
We turn now to estimating LaTeXMLMath .
First , one has LaTeXMLEquation .
LaTeXMLEquation and we set LaTeXMLEquation .
LaTeXMLEquation which yields LaTeXMLMath .
The function LaTeXMLMath vanishes only for LaTeXMLMath .
The zeros of LaTeXMLMath are LaTeXMLMath and the solutions of the equation of fifth degree in LaTeXMLMath , LaTeXMLEquation .
Now LaTeXMLMath becomes zero for LaTeXMLMath and is strictly increasing ; LaTeXMLMath is equal to LaTeXMLMath for LaTeXMLMath and strictly decreasing .
The equation ( LaTeXMLRef ) has therefore exactly one real solution ; it is positive and will be denoted in the following by LaTeXMLMath .
Note that LaTeXMLMath is greater than 0 and bounded from above by LaTeXMLMath .
Since LaTeXMLMath is non–negative and LaTeXMLMath , the numbers LaTeXMLMath and LaTeXMLMath are the only absolute minima of LaTeXMLMath .
The absolute value of LaTeXMLMath can then be estimated according to LaTeXMLEquation .
LaTeXMLEquation The relation LaTeXMLEquation as well as LaTeXMLEquation imply the estimate LaTeXMLEquation .
In a similar way one sees by LaTeXMLEquation that LaTeXMLMath has a maximum at LaTeXMLMath with LaTeXMLEquation and we obtain the estimate LaTeXMLEquation .
Now LaTeXMLMath tends asymptotically to LaTeXMLMath as LaTeXMLMath and one computes LaTeXMLEquation so that for LaTeXMLMath one sees that LaTeXMLMath has a maximum at LaTeXMLMath otherwise it is strictly increasing .
Inserting LaTeXMLMath in LaTeXMLMath we obtain LaTeXMLEquation where LaTeXMLMath , and thus , for LaTeXMLMath , the estimate LaTeXMLEquation .
As LaTeXMLMath , the function LaTeXMLMath tends to LaTeXMLMath .
Summarizing we find that , under the assumption that LaTeXMLMath , LaTeXMLMath can be estimated from above according to LaTeXMLEquation where LaTeXMLMath ; finally we obtain for LaTeXMLMath , assuming LaTeXMLMath to be small , that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Under the assumption that LaTeXMLMath we obtain the estimate LaTeXMLEquation for the Rayleigh quotient .
The expression LaTeXMLEquation tends to zero as LaTeXMLMath , so that the Rayleigh quotient itself becomes arbitrarily small for LaTeXMLMath .
Since for closed curves LaTeXMLMath the hypersurfaces LaTeXMLMath are complete , both LaTeXMLMath and LaTeXMLMath are self–adjoint , and by the LaTeXMLMath – LaTeXMLMath principle , see e. g. LaTeXMLCite , one has LaTeXMLEquation since LaTeXMLMath is bounded from below .
The domain of definition of the closure LaTeXMLMath of the Dirac operator is given by LaTeXMLEquation .
LaTeXMLEquation and in case LaTeXMLMath , one has LaTeXMLMath .
The first assertion of the theorem then follows by noting that the inequalities LaTeXMLMath , LaTeXMLMath and LaTeXMLMath imply that LaTeXMLMath lies in LaTeXMLMath and LaTeXMLMath , respectively , since LaTeXMLMath is assumed to be complete .
To see this , let LaTeXMLMath be fixed and LaTeXMLMath be the function defined in ( LaTeXMLRef ) .
Following LaTeXMLCite we put LaTeXMLEquation .
Then LaTeXMLMath on LaTeXMLMath and LaTeXMLMath .
Further one sees that LaTeXMLMath is Lipschitz–continuous and , hence , almost everywhere differentiable with LaTeXMLMath , where LaTeXMLMath is a constant .
Since LaTeXMLMath is complete , the closed envelopes of the geodesic balls LaTeXMLMath are compact in LaTeXMLMath and therefore LaTeXMLEquation .
Since LaTeXMLMath , one has LaTeXMLMath in LaTeXMLMath .
In the same way LaTeXMLMath implies with the relation LaTeXMLMath that LaTeXMLMath in LaTeXMLMath .
Consequently , one obtains LaTeXMLMath , and in a similar way LaTeXMLMath .Finally , by setting LaTeXMLMath , we obtain a sequence of elements in LaTeXMLMath of unit length for which LaTeXMLMath as LaTeXMLMath , which implies that LaTeXMLMath .
∎ In the following we will study the LaTeXMLMath –kernel of the Dirac operator in case that LaTeXMLMath is a circle in LaTeXMLMath with center at the origin and radius LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
As explained in section LaTeXMLRef , in this case LaTeXMLEquation .
LaTeXMLEquation represent two isometric LaTeXMLMath –actions on LaTeXMLMath .
Putting LaTeXMLEquation one obtains two continuous unitary LaTeXMLMath –representations in LaTeXMLMath , since by the invariance of the volume form under LaTeXMLMath and LaTeXMLMath the equality LaTeXMLEquation and a similar one for LaTeXMLMath hold .
Then , by the theorem of Stone , there exist uniquely determined self–adjoint operators LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath .
They are given by LaTeXMLMath , LaTeXMLMath , while the corresponding eigenfunctions are determined by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are integers and LaTeXMLMath .
Because of LaTeXMLMath , the operator LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath , so that each of the eigensubspaces LaTeXMLMath of LaTeXMLMath and LaTeXMLMath corresponding to the eigenvalue LaTeXMLMath decomposes into the eigensubspaces of the unitary LaTeXMLMath –action according to LaTeXMLEquation in concordance with the spectral decomposition of the operators LaTeXMLMath and LaTeXMLMath ; in particular , one has LaTeXMLMath A general solution of the Dirac equation LaTeXMLMath on LaTeXMLMath can then be written as a product of the form LaTeXMLEquation where LaTeXMLMath is a function of LaTeXMLMath .
Thus , the system of partial differential equations ( LaTeXMLRef ) leads to a system of ordinary differential equations LaTeXMLEquation for the radial function LaTeXMLMath .
Introducing LaTeXMLMath , we put LaTeXMLEquation and make the substitution LaTeXMLEquation so that one obtains for LaTeXMLMath the system of differential equations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation with LaTeXMLMath , LaTeXMLMath .
Note that LaTeXMLEquation .
If LaTeXMLMath or LaTeXMLMath are different from zero , neither LaTeXMLMath nor LaTeXMLMath vanish ; differentiating again gives LaTeXMLEquation .
LaTeXMLEquation and one obtains the differential equations of second order LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
If one puts LaTeXMLMath and LaTeXMLMath , respectively , each solution of ( LaTeXMLRef ) or ( LaTeXMLRef ) corresponds to a solution of the above system of differential equations for LaTeXMLMath , i. e. , solving the latter system of two differential equations of first order is equivalent to finding a solution of the differential equation of second order ( LaTeXMLRef ) or ( LaTeXMLRef ) .
The latter are differential equations of Sturm–Liouville type and our next goal will consist in showing that , for LaTeXMLMath and LaTeXMLMath , they can not have any bounded solutions and , in particular , that they do not lead to LaTeXMLMath –integrable solutions LaTeXMLMath of the Dirac equation .
For this purpose we will make use of the following theorem proved by Hartman LaTeXMLCite .
Let LaTeXMLMath be an interval in LaTeXMLMath and LaTeXMLMath a solution of the differential equation LaTeXMLEquation with continuous complex valued coefficients LaTeXMLMath and LaTeXMLMath .
If LaTeXMLEquation then LaTeXMLMath is concave , i. e. , LaTeXMLMath .
Now , in our case one computes LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and thus LaTeXMLEquation .
LaTeXMLEquation Because of LaTeXMLEquation one recognizes that , for LaTeXMLMath and LaTeXMLMath , the condition ( LaTeXMLRef ) is fulfilled for the differential equations ( LaTeXMLRef ) and ( LaTeXMLRef ) , while for LaTeXMLMath the expression LaTeXMLMath tends asymptotically to LaTeXMLMath for LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath it becomes also negative as LaTeXMLMath .
As a consequence of the preceeding theorem we obtain the following lemma .
Assume that LaTeXMLMath and LaTeXMLMath , and let LaTeXMLMath , LaTeXMLMath be solutions of the differential equations ( LaTeXMLRef ) and ( LaTeXMLRef ) , respectively .
Then LaTeXMLMath and LaTeXMLMath are concave .
We are now in a position to prove the announced theorem .
Let LaTeXMLMath be a circle in LaTeXMLMath with center at the origin and radius LaTeXMLMath , and LaTeXMLMath a spinor on LaTeXMLMath of the form ( LaTeXMLRef ) .
If LaTeXMLMath is a solution of the Dirac equation with respect to the trivial spin structure corresponding to the eigenvalue LaTeXMLMath and if LaTeXMLMath , then LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
By our previous considerations LaTeXMLMath satisfies the differential equation ( LaTeXMLRef ) and we consider its continuation LaTeXMLEquation to the whole complex domain .
For LaTeXMLMath both LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , are meromorphic functions with poles of first and second order at zero , respectively .
The differential equation ( LaTeXMLRef ) is therefore of Fuchssian type and zero is a regular singular point .
Let LaTeXMLMath , LaTeXMLMath form a fundamental system of solutions of ( LaTeXMLRef ) ; they can be expanded around the origin into the uniformly convergent series LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath are constants and LaTeXMLMath , LaTeXMLMath are the roots of the equation LaTeXMLEquation with LaTeXMLEquation see e. g. LaTeXMLCite .
One obtains LaTeXMLMath , LaTeXMLMath , which yields in our case that LaTeXMLMath , and hence LaTeXMLMath , LaTeXMLMath .
Evidently , analogous considerations hold for LaTeXMLMath , too .
Now , LaTeXMLEquation .
In order that the above integral remains bounded it is necessary that LaTeXMLMath and LaTeXMLMath decrease with order greater than one for LaTeXMLMath , since LaTeXMLEquation therefore LaTeXMLMath , LaTeXMLMath , must hold for large LaTeXMLMath .
As , moreover , LaTeXMLMath is smooth , there exists a LaTeXMLMath such that LaTeXMLMath .
However , by Lemma ( LaTeXMLRef ) one has that LaTeXMLMath is monotone increasing so that LaTeXMLEquation must hold .
Consequently , LaTeXMLMath is monotone decreasing and strictly monotone decreasing for LaTeXMLMath .
Let us now assume that LaTeXMLMath without loss of generality .
If LaTeXMLMath is not identically zero , it follows that , in a neighbourhood of the origin , its components LaTeXMLMath and LaTeXMLMath must have the developments LaTeXMLEquation where LaTeXMLMath are constants ; otherwise one would have LaTeXMLMath .
Let now LaTeXMLMath be sufficiently small so that LaTeXMLMath and LaTeXMLMath can be developed as above and , in particular , LaTeXMLEquation .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and hence LaTeXMLMath .
∎ We turn now to the remaining case of LaTeXMLMath .
If LaTeXMLMath is a harmonic spinor on LaTeXMLMath , the components of LaTeXMLMath satisfy the differential equations ( LaTeXMLRef ) and ( LaTeXMLRef ) , respectively , where LaTeXMLEquation i. e. , for LaTeXMLMath and LaTeXMLMath one obtains the differential equations LaTeXMLEquation and these can be integrated explicitly .
Indeed , putting LaTeXMLEquation one verifies that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We continue LaTeXMLMath to a spinor on LaTeXMLMath by setting LaTeXMLMath .
Let now LaTeXMLMath and LaTeXMLMath , so that LaTeXMLMath .
Then one computes LaTeXMLEquation .
LaTeXMLEquation so that LaTeXMLMath .
Nevertheless , LaTeXMLMath is not smooth at LaTeXMLMath , so that LaTeXMLMath .
Thus we have completely determined the LaTeXMLMath –kernel of the Dirac operator in case that LaTeXMLMath is a circle in LaTeXMLMath with center at the origin and obtain the following theorem .
Let LaTeXMLMath be a generalized circle in LaTeXMLMath that arises by a Möbius transform from a circle in LaTeXMLMath with center at the origin , and LaTeXMLMath the Dirac operator on LaTeXMLMath with respect to the trivial spin structure .
Then LaTeXMLEquation while the LaTeXMLMath –kernel of the Dirac operator and its closure are trivial .
In particular , LaTeXMLMath .
Let LaTeXMLMath be a circle in LaTeXMLMath with center around the origin and radius LaTeXMLMath .
Without loss of generality we can assume that LaTeXMLMath .
For LaTeXMLMath and by the previous considerations LaTeXMLEquation .
LaTeXMLEquation are harmonic LaTeXMLMath –spinors on LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath are constants and LaTeXMLMath .
By Theorem LaTeXMLRef , apart from the trivial representation no other representations of the LaTeXMLMath –action LaTeXMLMath can occur in the LaTeXMLMath –kernel of the Dirac operator and we obtain ( LaTeXMLRef ) in case that LaTeXMLMath .
The general statement then follows from the fact that LaTeXMLMath and LaTeXMLMath are isometric for LaTeXMLMath .
If , further , LaTeXMLMath is a harmonic spinor with respect to LaTeXMLMath , then the regularity theorem for solutions of elliptic differential equations implies that LaTeXMLMath .
However , since all LaTeXMLMath –harmonic spinors have to be linear combinations of the LaTeXMLMath , which , nevertheless , are not regular at LaTeXMLMath , the LaTeXMLMath –kernel of the Dirac operator and its closure turn out to be trivial .
Since , by theorem LaTeXMLRef , zero belongs to the spectrum of LaTeXMLMath , it follows that LaTeXMLMath .
∎ In this section we will continue the study of the Laplacian on the hypersurfaces LaTeXMLMath , which we began in Section LaTeXMLRef .
Unlike the Dirac operator , the spectrum of the Laplacian on an open complete manifold is related to the underlying geometry in a much more intrinsic way .
Thus , lower bounds for the Ricci tensor imply upper bounds for its smallest spectral value , and by studying the geodesic flow and the exponential growth of the manifold one obtains statements about the infimum of the essential spectrum of the Laplace operator and vice versa .
Operating on functions , the Hodge–Laplace operator and the Bochner–Laplace operator coincide , and we have LaTeXMLMath on the hypersurfaces LaTeXMLMath ; further , since LaTeXMLMath is complete for a closed curve LaTeXMLMath , LaTeXMLMath is essentially selfadjoint as an operator in LaTeXMLMath with domain LaTeXMLMath , where the domain of LaTeXMLMath is given by the Sobolev space LaTeXMLMath .
One has LaTeXMLMath .
Now , for the smallest spectral value of the Laplacian the following proposition holds in general ( see e.g .
LaTeXMLCite ) .
Let LaTeXMLMath be an open complete Riemannian manifold , the components of the Ricci tensor being bounded from below by LaTeXMLMath , where LaTeXMLMath .
Then the smallest spectral value of the Laplacian LaTeXMLMath satisfies LaTeXMLEquation .
Hence , as an immediate consequence we obtain the following statement .
Let LaTeXMLMath be a closed curve in LaTeXMLMath .
Then the smallest spectral value of the Laplacian on the hypersurfaces LaTeXMLMath satisfies LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
By Theorem LaTeXMLRef , LaTeXMLMath .
Further , since LaTeXMLMath is strictly increasing one has that LaTeXMLMath so that LaTeXMLMath , where LaTeXMLMath .
The assertion then follows from the proposition above .
∎ In the sequel we will proceed to find estimates for the infimum of the spectrum of LaTeXMLMath on the considered hypersurfaces by using again the LaTeXMLMath – LaTeXMLMath principle , and show that it becomes arbitrarily close to zero for any closed curve LaTeXMLMath , so that LaTeXMLMath , where LaTeXMLMath .
Since , by Corollary LaTeXMLRef , this estimate gives also an estimate for the infimum of the essential spectrum , we are in position to compute the exponential growth of LaTeXMLMath for an arbitrary closed curve , thus generalizing the results previously obtained in section LaTeXMLRef , since , as already mentioned , the infimum of the essential spectrum of the Laplacian is closely related to the exponential growth of the underlying manifold .
More precisely the following theorem proved by Brooks LaTeXMLCite holds .
Let LaTeXMLMath be an open complete manifold of infinite volume .
Then LaTeXMLEquation .
Consequently , the exponential growth of the hypersurfaces LaTeXMLMath must be zero for any closed curve LaTeXMLMath .
Let us now prove these assertions .
First note that for LaTeXMLMath , LaTeXMLEquation holds , where LaTeXMLMath denotes the scalar product in LaTeXMLMath and LaTeXMLEquation .
By the LaTeXMLMath – LaTeXMLMath principle we have LaTeXMLEquation .
Now we consider the function LaTeXMLEquation which is derived from the trace LaTeXMLMath of the second fundamental form , and by means of this function we generate estimates for LaTeXMLMath .
Let LaTeXMLMath be a closed curve in LaTeXMLMath and LaTeXMLMath the closure of the scalar Laplacian on LaTeXMLMath , where LaTeXMLMath .
Then , for arbitrary LaTeXMLMath , LaTeXMLEquation .
By Corollary LaTeXMLRef , LaTeXMLMath is LaTeXMLMath –integrable over LaTeXMLMath for LaTeXMLMath .
One computes further that LaTeXMLEquation the derivatives LaTeXMLMath and LaTeXMLMath being zero so that LaTeXMLEquation .
For LaTeXMLMath , and assuming LaTeXMLMath , the monotony of the integral implies LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Similarly , under the assumption that LaTeXMLMath one computes LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation showing that LaTeXMLMath .
Summing up we have LaTeXMLEquation using ( LaTeXMLRef ) one then obtains the stated bound from above for the essential spectrum of the Laplacian since , by Corollary LaTeXMLRef , zero can be no LaTeXMLMath –eigenvalue of LaTeXMLMath and , hence , of LaTeXMLMath .
∎ Let LaTeXMLMath be arbitrary and LaTeXMLMath .
Then for any closed curve LaTeXMLMath in LaTeXMLMath , LaTeXMLMath has subexponential growth .
This is a consequence of the theorems LaTeXMLRef and LaTeXMLRef .
∎
Let LaTeXMLMath be a pseudoconvex domain and let LaTeXMLMath be a locally pluripolar set , LaTeXMLMath .
Put LaTeXMLEquation .
Let LaTeXMLMath be an open connected neighborhood of LaTeXMLMath and let LaTeXMLMath be an analytic subset .
Then there exists an analytic subset LaTeXMLMath of the ‘ envelope of holomorphy ’ LaTeXMLMath of LaTeXMLMath with LaTeXMLMath such that for every function LaTeXMLMath separately holomorphic on LaTeXMLMath there exists an LaTeXMLMath holomorphic on LaTeXMLMath with LaTeXMLMath .
The result generalizes special cases which were studied in LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , and LaTeXMLCite .
1 .
Introduction .
Main Theorem .
Let LaTeXMLMath , LaTeXMLMath , and let LaTeXMLEquation where LaTeXMLMath is a domain , LaTeXMLMath .
We define an LaTeXMLMath –fold cross LaTeXMLEquation .
LaTeXMLEquation Observe that LaTeXMLMath is connected .
Let LaTeXMLMath be an open set and let LaTeXMLMath .
Put LaTeXMLEquation where LaTeXMLMath denotes the set of all functions plurisubharmonic on LaTeXMLMath .
Define LaTeXMLEquation where LaTeXMLMath is a sequence of relatively compact open sets LaTeXMLMath with LaTeXMLMath ( LaTeXMLMath denotes the upper semicontinuous regularization of LaTeXMLMath ) .
Observe that the definition is independent of the chosen exhausting sequence LaTeXMLMath .
Moreover , LaTeXMLMath .
For an LaTeXMLMath –fold cross LaTeXMLMath put LaTeXMLEquation notice that LaTeXMLMath may be empty .
Observe that LaTeXMLMath is pseudoconvex if LaTeXMLMath are pseudoconvex domains .
We say that a subset LaTeXMLMath is locally pluriregular if LaTeXMLMath for any LaTeXMLMath and for any open neighborhood LaTeXMLMath of LaTeXMLMath ( in particular , LaTeXMLMath is non-pluripolar ) .
Note that if LaTeXMLMath are locally pluriregular , then LaTeXMLMath and , moreover , LaTeXMLMath is connected ( Lemma 5 ) .
Let LaTeXMLMath be a connected neighborhood of LaTeXMLMath and let LaTeXMLMath be an analytic subset ( LaTeXMLMath may be empty ) .
We say that a function LaTeXMLMath is separately holomorphic ( LaTeXMLMath ) if for any LaTeXMLMath and LaTeXMLMath the function LaTeXMLMath is holomorphic in the domain LaTeXMLMath .
The main result of our paper is the following extension theorem for separately holomorphic functions .
LaTeXMLMath ” We like to thank Professor Józef Siciak for turning our attention to the problem .
Let LaTeXMLMath be a pseudoconvex domain and let LaTeXMLMath be a locally pluriregular set , LaTeXMLMath .
Let LaTeXMLMath be an analytic subset of an open connected neighborhood LaTeXMLMath of LaTeXMLMath ( LaTeXMLMath may be empty ) .
Then there exists a pure one–codimensional analytic subset LaTeXMLMath such that : LaTeXMLMath LaTeXMLMath for an open neighborhood LaTeXMLMath of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath for every LaTeXMLMath there exists exactly one LaTeXMLMath with LaTeXMLMath .
Moreover , if LaTeXMLMath , then we can take LaTeXMLMath the union of all one–codimensional irreducible components of LaTeXMLMath .
The proof will be given in Sections 3 ( the case LaTeXMLMath ) and 4 ( the general case ) .
Notice that the Main Theorem may be generalized to the case where LaTeXMLMath is Riemann–Stein domains over LaTeXMLMath , LaTeXMLMath .
Observe that in the case LaTeXMLMath , LaTeXMLMath , the Main Theorem is nothing else than the following cross theorem .
Theorem 1 ( cf .
LaTeXMLCite ) .
Let LaTeXMLMath , LaTeXMLMath be pseudoconvex domains and let LaTeXMLMath , LaTeXMLMath be locally pluriregular .
Put LaTeXMLMath .
Then for any LaTeXMLMath there exists exactly one LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
( a ) LaTeXMLMath : There is a long list of papers discussing the case LaTeXMLMath ( under various assumptions ) : LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
The case LaTeXMLMath , LaTeXMLMath can be found in LaTeXMLCite .
The general case LaTeXMLMath , LaTeXMLMath was solved in LaTeXMLCite LaTeXMLMath ” We like to thank Professor Nguyen Thanh Van for calling our attention to that paper.. ( b ) LaTeXMLMath : J. Siciak LaTeXMLCite solved the case : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is a non-zero polynomial of LaTeXMLMath complex variables .
The special subcase LaTeXMLMath , LaTeXMLMath had been studied in LaTeXMLCite .
The general case for LaTeXMLMath , LaTeXMLMath was solved in LaTeXMLCite ; see also LaTeXMLCite for a partial discussion of the case LaTeXMLMath , LaTeXMLMath .
It remains an open problem what happens if the singular set LaTeXMLMath is assumed to be , for instance , a pluripolar relatively closed subset of LaTeXMLMath ( see also LaTeXMLCite ) .
The following particular case will be proved in Section 5 .
Let LaTeXMLMath be locally regular .
Let LaTeXMLMath be a closed pluripolar set and let LaTeXMLMath .
Then there exist a closed pluripolar set LaTeXMLMath and LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , LaTeXMLMath being polar , such that for every LaTeXMLMath there exists exactly one LaTeXMLMath with LaTeXMLMath , where LaTeXMLMath .
Notice that Theorem 2 may be thought as a partial generalization of LaTeXMLCite .
2 .
Auxiliary results .
The following lemma gathers a few standard results , which will be frequently used in the sequel .
Lemma 3 ( cf .
LaTeXMLCite , LaTeXMLCite , § 3.5 ) .
( a ) Let LaTeXMLMath be a bounded open set and let LaTeXMLMath .
Then : LaTeXMLMath If LaTeXMLMath is pluripolar , then LaTeXMLMath .
LaTeXMLMath LaTeXMLMath ( pointwise on LaTeXMLMath ) for any sequence of open sets LaTeXMLMath and any sequence LaTeXMLMath .
LaTeXMLMath LaTeXMLMath .
LaTeXMLMath The following conditions are equivalent : for any connected component LaTeXMLMath of LaTeXMLMath the set LaTeXMLMath is non-pluripolar ; LaTeXMLMath for any LaTeXMLMath .
LaTeXMLMath If LaTeXMLMath is non-pluripolar , LaTeXMLMath , and LaTeXMLMath , then for any connected component LaTeXMLMath of LaTeXMLMath the set LaTeXMLMath is non-pluripolar ( in particular , LaTeXMLMath ) .
LaTeXMLMath If LaTeXMLMath is locally pluriregular , LaTeXMLMath , and LaTeXMLMath , then LaTeXMLMath on LaTeXMLMath .
( b ) Let LaTeXMLMath be an open set and let LaTeXMLMath .
Then : LaTeXMLMath LaTeXMLMath .
LaTeXMLMath If LaTeXMLMath is locally pluriregular , then LaTeXMLMath for any LaTeXMLMath .
LaTeXMLMath If LaTeXMLMath is pluripolar , then LaTeXMLMath .
LaTeXMLMath If LaTeXMLMath is locally pluriregular and LaTeXMLMath is pluripolar , then LaTeXMLMath is locally pluriregular .
Moreover , we get : ( a ) Let LaTeXMLMath be locally pluriregular , LaTeXMLMath , then LaTeXMLMath is locally pluriregular .
( b ) Let LaTeXMLMath , LaTeXMLMath a domain , LaTeXMLMath locally pluriregular , LaTeXMLMath , LaTeXMLMath .
Put LaTeXMLEquation ( observe that LaTeXMLMath ) .
Then LaTeXMLEquation ( a ) is an immediate consequence of the product property for the relatively extremal function LaTeXMLEquation cf .
LaTeXMLCite .
( b ) First observe that LaTeXMLEquation .
To get the opposite inequality we proceed by induction on LaTeXMLMath .
Let LaTeXMLMath : The proof of this step is taken from LaTeXMLCite .
For the reader ’ s convenience we repeat the details .
Put LaTeXMLMath and fix a point LaTeXMLMath .
If LaTeXMLMath ( thus LaTeXMLMath ) , then LaTeXMLMath with LaTeXMLMath and LaTeXMLMath on LaTeXMLMath .
Therefore , LaTeXMLEquation .
In particular , LaTeXMLMath .
Observe that the same argument shows that if LaTeXMLMath , then LaTeXMLMath on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath and therefore LaTeXMLMath .
Put LaTeXMLEquation .
It is clear that LaTeXMLMath .
Put LaTeXMLEquation .
Then LaTeXMLMath and LaTeXMLMath on LaTeXMLMath .
Therefore , by Lemma 3 ( a ) , LaTeXMLEquation .
Consequently , LaTeXMLMath , which finishes the proof for LaTeXMLMath .
Now , assume that the formula is true for LaTeXMLMath .
Put LaTeXMLEquation and fix an arbitrary LaTeXMLMath .
Obviously , LaTeXMLMath .
In virtue of the inductive hypothesis , we conclude that LaTeXMLEquation .
Now we apply the case LaTeXMLMath to the following situation : LaTeXMLEquation .
So LaTeXMLEquation .
Note that LaTeXMLMath .
Hence LaTeXMLEquation .
LaTeXMLMath be an LaTeXMLMath –fold cross as in ( 1 ) .
If LaTeXMLMath are locally pluriregular , then LaTeXMLMath is a domain .
Using exhaustion by bounded domains we may assume that the LaTeXMLMath ’ s are bounded .
We know that LaTeXMLMath .
Let LaTeXMLMath be an arbitrary point .
If LaTeXMLMath , then LaTeXMLMath .
Therefore , LaTeXMLMath can be joined inside LaTeXMLMath with LaTeXMLMath for some LaTeXMLMath .
If LaTeXMLMath , put LaTeXMLEquation .
Then , in virtue of Lemma 3 ( a ) , the connected component LaTeXMLMath of LaTeXMLMath , that contains LaTeXMLMath , intersects LaTeXMLMath .
Therefore , LaTeXMLMath can be joined inside LaTeXMLMath with LaTeXMLMath for some LaTeXMLMath .
Now we repeat the above argument for the second component of the point LaTeXMLMath .
Finally , the point LaTeXMLMath can be joined inside LaTeXMLMath with LaTeXMLMath .
Since LaTeXMLMath is connected , the proof is completed .
LaTeXMLMath be a domain and let LaTeXMLMath be non-pluripolar .
Then any LaTeXMLMath with LaTeXMLMath vanishes identically on LaTeXMLMath .
( b ) Let LaTeXMLMath be domains , let LaTeXMLMath , LaTeXMLMath be locally pluriregular sets , and let LaTeXMLMath .
Let LaTeXMLMath be an analytic subset of an open connected neighborhood LaTeXMLMath of LaTeXMLMath .
Assume that LaTeXMLMath , LaTeXMLMath are such that : LaTeXMLMath LaTeXMLMath and LaTeXMLMath are pluripolar ( in particular , LaTeXMLMath , LaTeXMLMath are also locally pluriregular ) , LaTeXMLMath LaTeXMLMath for any LaTeXMLMath , LaTeXMLMath LaTeXMLMath for any LaTeXMLMath .
Then : ( b LaTeXMLMath ) If LaTeXMLMath and LaTeXMLMath on LaTeXMLMath LaTeXMLMath ” Here and in the sequel to simplify notation we write LaTeXMLMath instead of LaTeXMLMath . , then LaTeXMLMath on LaTeXMLMath .
( b LaTeXMLMath ) If LaTeXMLMath and LaTeXMLMath on LaTeXMLMath , then LaTeXMLMath on LaTeXMLMath .
( a ) is obvious .
( b LaTeXMLMath ) Take a point LaTeXMLMath .
We may assume that LaTeXMLMath .
Since LaTeXMLMath is pluripolar , there exists a sequence LaTeXMLMath such that LaTeXMLMath .
The set LaTeXMLMath is pluripolar .
Consequently , the set LaTeXMLMath is non-pluripolar .
We have LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Hence LaTeXMLMath for any LaTeXMLMath .
Finally , LaTeXMLMath on LaTeXMLMath .
( b LaTeXMLMath ) Take an LaTeXMLMath .
Since LaTeXMLMath , there exists a LaTeXMLMath .
Let LaTeXMLMath ( LaTeXMLMath denotes the polydisc with center LaTeXMLMath and radius LaTeXMLMath ) .
Then LaTeXMLMath on LaTeXMLMath for any LaTeXMLMath .
The set LaTeXMLMath is non-pluripolar .
Hence LaTeXMLMath on LaTeXMLMath for any LaTeXMLMath .
By the same argument for the second variable we get LaTeXMLMath on LaTeXMLMath and , consequently , on LaTeXMLMath .
LaTeXMLMath is uniquely determined ( if exists ) .
To prove the Main Theorem for LaTeXMLMath it suffices to consider only the case where LaTeXMLMath is pure one–codimensional .
Since LaTeXMLMath is pseudoconvex , the arbitrary analytic set LaTeXMLMath can be written as LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath is pure one–codimensional .
Take an LaTeXMLMath .
Observe that LaTeXMLMath .
By the reduction assumption there exists an LaTeXMLMath such that LaTeXMLMath on LaTeXMLMath .
In virtue of Lemma 6 ( a ) , gluing the functions LaTeXMLMath , leads to an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath , LaTeXMLMath .
Therefore , LaTeXMLMath on LaTeXMLMath .
Finally , since codim LaTeXMLMath , the function LaTeXMLMath extends holomorphically across LaTeXMLMath .
LaTeXMLMath is empty or pure one–codimensional and LaTeXMLMath are bounded pseudoconvex .
Let LaTeXMLMath be arbitrary pseudoconvex domains .
Let LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath are pseudoconvex domains with LaTeXMLMath .
Observe that all the LaTeXMLMath ’ s are locally pluriregular .
Put LaTeXMLEquation note that LaTeXMLMath .
Let LaTeXMLMath be given .
By the reduction assumption , for each LaTeXMLMath there exists an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
By Lemma 6 ( a ) , LaTeXMLMath on LaTeXMLMath .
Therefore , gluing the LaTeXMLMath ’ s , we obtain an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
LaTeXMLMath .
In virtue of Theorem 1 , Step 3 finishes the proof of the Main Theorem for LaTeXMLMath .
We proceed by induction on LaTeXMLMath .
Suppose that the Main Theorem is true for LaTeXMLMath .
We have to discuss the case of an LaTeXMLMath –fold cross LaTeXMLMath , where LaTeXMLMath are bounded pseudoconvex .
Let LaTeXMLMath be empty or pure one–codimensional .
Let LaTeXMLMath .
Observe that LaTeXMLEquation where LaTeXMLEquation .
We also mention that for any LaTeXMLMath we have LaTeXMLEquation .
Now fix an LaTeXMLMath such that LaTeXMLEquation in particular , LaTeXMLMath is empty or one–codimensional ( in LaTeXMLMath ) .
Recall that LaTeXMLMath , … , LaTeXMLMath are locally pluriregular .
By inductive assumption there exists an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
To continue define the following LaTeXMLMath –fold cross LaTeXMLEquation .
Notice that LaTeXMLMath satisfies all the properties for the case LaTeXMLMath : LaTeXMLMath are bounded pseudoconvex domains , LaTeXMLMath , LaTeXMLMath are locally pluriregular .
In virtue of Lemma 4 , we have LaTeXMLEquation .
Define LaTeXMLMath , LaTeXMLEquation .
Obviously , LaTeXMLMath is well-defined and therefore LaTeXMLMath .
Using the case LaTeXMLMath , we find another function LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Recall that LaTeXMLMath .
Hence LaTeXMLMath on LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath .
From now on we simplify our notation and we are looking for the following configuration : Let LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath are bounded pseudoconvex domains , LaTeXMLMath are locally pluriregular .
Put , as always , LaTeXMLEquation .
Moreover , let LaTeXMLMath be a pure one–codimensional analytic subset of LaTeXMLMath .
We like to show that any LaTeXMLMath extends holomorphically to LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath be as above .
Let LaTeXMLMath , LaTeXMLMath be sequences of pseudoconvex domains , LaTeXMLMath , LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath .
Moreover , let LaTeXMLMath , LaTeXMLMath be such that LaTeXMLMath , LaTeXMLMath are pluripolar , and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
For each LaTeXMLMath we assume that : for any LaTeXMLMath there exist polydiscs LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , and functions LaTeXMLMath , LaTeXMLMath such that LaTeXMLMath LaTeXMLMath on LaTeXMLMath , LaTeXMLMath LaTeXMLMath on LaTeXMLMath .
Then there exists an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Fix a LaTeXMLMath .
Put LaTeXMLEquation .
Note that LaTeXMLEquation .
We like to glue the functions LaTeXMLMath and LaTeXMLMath to obtain a global holomorphic function LaTeXMLMath on LaTeXMLMath : Let LaTeXMLMath , LaTeXMLMath .
Observe that LaTeXMLEquation .
LaTeXMLEquation Thus LaTeXMLMath on LaTeXMLMath .
Applying Lemma 6 ( a ) , we conclude that LaTeXMLEquation .
Now let LaTeXMLMath be such that LaTeXMLMath .
Fix a LaTeXMLMath .
We know that LaTeXMLMath on LaTeXMLMath .
Hence , by the identity principle , we conclude that LaTeXMLMath on LaTeXMLMath .
The same argument works for LaTeXMLMath .
Consequently , we obtain a function LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be the connected component of LaTeXMLMath with LaTeXMLMath .
Thus we have LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Recall that LaTeXMLMath .
We claim that the envelope of holomorphy of LaTeXMLMath coincides with LaTeXMLMath .
In fact , let LaTeXMLMath , then LaTeXMLMath .
So , in virtue of Theorem 1 , there exists an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Lemma 6 ( b LaTeXMLMath ) implies that LaTeXMLMath on LaTeXMLMath .
Applying the Grauert–Remmert theorem ( cf .
LaTeXMLCite , Th .
3.4.7 ) , we find a function LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
In particular , LaTeXMLMath on LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath are pluripolar , we get LaTeXMLEquation .
LaTeXMLEquation So , in fact , LaTeXMLMath .
Using Lemma 6 ( b LaTeXMLMath ) , we even see that LaTeXMLMath on LaTeXMLMath .
Observe that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Using again Lemma 6 ( a ) , by gluing the LaTeXMLMath ’ s , we get a function LaTeXMLMath with LaTeXMLMath on LaTeXMLMath , which finishes the proof of Step 4 .
LaTeXMLMath .
Put LaTeXMLEquation .
Let LaTeXMLMath be locally pluriregular , LaTeXMLMath , and let LaTeXMLMath be a pure one–codimensional analytic subset of LaTeXMLMath with LaTeXMLMath .
Put LaTeXMLMath , LaTeXMLMath .
Then we say that the condition ( * ) holds if : For any LaTeXMLMath there exists LaTeXMLMath such that for any function LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , there exists an extension LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
If condition ( * ) is satisfied , then the assumptions of Step 4 are fulfilled .
Take LaTeXMLMath , LaTeXMLMath , LaTeXMLMath as is in Step 4 .
Define LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
It clear that LaTeXMLMath , LaTeXMLMath are pluripolar .
Let LaTeXMLMath , LaTeXMLMath be approximation sequences : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
Fix a LaTeXMLMath , LaTeXMLMath and let LaTeXMLMath be the set of all LaTeXMLMath such that there exist a polydisc LaTeXMLMath and a function LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
It is clear that LaTeXMLMath is open .
Observe that LaTeXMLMath .
Indeed , since LaTeXMLMath , we find a point LaTeXMLMath .
Therefore there is a polydisc LaTeXMLMath .
Put LaTeXMLEquation .
By Theorem 1 , we find LaTeXMLMath and LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Consequently , LaTeXMLMath .
Moreover , LaTeXMLMath is relatively closed in LaTeXMLMath .
Indeed , let LaTeXMLMath be an accumulation point of LaTeXMLMath in LaTeXMLMath and let LaTeXMLMath .
Take a point LaTeXMLMath and let LaTeXMLMath , LaTeXMLMath , be such that LaTeXMLMath .
Observe that LaTeXMLMath and LaTeXMLMath for any LaTeXMLMath .
Hence , by ( * ) ( with LaTeXMLMath ) , there exists an extension LaTeXMLMath ( LaTeXMLMath ) such that LaTeXMLMath on LaTeXMLMath .
Take an LaTeXMLMath so small that LaTeXMLMath and put LaTeXMLMath on LaTeXMLMath .
Obviously LaTeXMLMath on LaTeXMLMath .
Hence LaTeXMLMath .
Thus LaTeXMLMath .
There exists a finite set LaTeXMLMath such that LaTeXMLEquation .
Define LaTeXMLMath .
Take LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLMath on LaTeXMLMath .
Consequently , by Lemma 6 ( a ) , LaTeXMLMath on LaTeXMLMath .
In particular , by gluing the functions LaTeXMLMath , we get a function LaTeXMLMath such that LaTeXMLMath on LaTeXMLMath .
Changing the role of LaTeXMLMath and LaTeXMLMath , we get LaTeXMLMath , LaTeXMLMath .
Thus the assumptions of Step 4 are fulfilled .
LaTeXMLMath such that LaTeXMLMath for any LaTeXMLMath with LaTeXMLMath .
Define LaTeXMLEquation .
It suffices to show that LaTeXMLMath .
Suppose that LaTeXMLMath .
Fix LaTeXMLMath and choose LaTeXMLMath as in ( 3 ) with LaTeXMLMath , LaTeXMLMath .
Write LaTeXMLMath .
Put LaTeXMLMath .
Let LaTeXMLMath denote the set of all LaTeXMLMath which satisfy the following condition : LaTeXMLMath There exist LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and holomorphic functions LaTeXMLMath , LaTeXMLMath , such that : LaTeXMLMath LaTeXMLMath , LaTeXMLMath LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath LaTeXMLMath .
For any LaTeXMLMath define a new cross LaTeXMLEquation .
Notice that LaTeXMLMath does not intersect LaTeXMLMath .
In particular , LaTeXMLMath .
Hence , by Theorem 1 , there exists an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Take LaTeXMLMath , and LaTeXMLMath ( LaTeXMLMath ) , such that LaTeXMLMath LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath for LaTeXMLMath , LaTeXMLMath .
Then there exists LaTeXMLMath such that LaTeXMLMath LaTeXMLMath , where LaTeXMLMath .
In particular , LaTeXMLMath .
Fix a LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath for any LaTeXMLMath .
Using the biholomorphic mapping LaTeXMLEquation .
LaTeXMLEquation we see that the function LaTeXMLMath is holomorphic in LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath .
Moreover , LaTeXMLMath for any LaTeXMLMath .
Using Theorem 1 for the cross LaTeXMLEquation immediately shows that LaTeXMLMath extends holomorphically to LaTeXMLMath ( because LaTeXMLMath ) .
Transforming the above information back via LaTeXMLMath for all LaTeXMLMath , we conclude that the function LaTeXMLMath extends holomorphically to LaTeXMLMath for some LaTeXMLMath ; in particular , LaTeXMLMath extends holomorphically to LaTeXMLMath .
Now we prove that LaTeXMLMath is pluripolar .
Write LaTeXMLEquation where LaTeXMLMath is a polydisc and LaTeXMLMath is a defining function for LaTeXMLMath ; cf .
LaTeXMLCite , § 2.9 .
Define LaTeXMLEquation and observe that , by the implicit function theorem , any point from LaTeXMLEquation satisfies ( LaTeXMLMath ) .
It is enough to show that each set LaTeXMLMath is pluripolar .
Fix a LaTeXMLMath .
Let LaTeXMLMath be an irreducible component of LaTeXMLMath .
We have to show that LaTeXMLMath is pluripolar .
If LaTeXMLMath has codimension LaTeXMLMath , then LaTeXMLMath is contained in a countable union of proper analytic sets ( cf .
LaTeXMLCite , § 3.8 ) .
Consequently , LaTeXMLMath is pluripolar .
Thus we may assume that LaTeXMLMath is pure one–codimensional .
The same argument as above shows that LaTeXMLMath is pluripolar .
It remains to prove that LaTeXMLMath is pluripolar ) .
Since LaTeXMLMath is a defining function , for any LaTeXMLMath there exists a LaTeXMLMath such that LaTeXMLMath .
Thus LaTeXMLEquation where LaTeXMLMath .
We only need to prove that each set LaTeXMLMath is pluripolar , LaTeXMLMath .
Fix a LaTeXMLMath .
To simplify notation , assume that LaTeXMLMath .
Observe that , by the implicit function theorem , we can write LaTeXMLEquation where LaTeXMLMath is a polydisc , LaTeXMLMath , and LaTeXMLMath is holomorphic , LaTeXMLMath .
It suffices to prove that the projection of each set LaTeXMLMath is pluripolar .
Fix an LaTeXMLMath .
Since LaTeXMLEquation we conclude that LaTeXMLMath and consequently LaTeXMLMath is independent of LaTeXMLMath .
Thus LaTeXMLMath and therefore the projection is pluripolar .
The proof that LaTeXMLMath is pluripolar is completed .
Using Step 4 , we conclude that LaTeXMLMath extends holomorphically to the domain LaTeXMLMath , where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ( here we have used the product property of the relative extremal function ) .
Since LaTeXMLMath , we find a LaTeXMLMath and a function LaTeXMLMath such that LaTeXMLEquation .
If LaTeXMLMath we get a contradiction ( because LaTeXMLMath ) .
Let LaTeXMLMath .
Repeating the above argument for the coordinates LaTeXMLMath , LaTeXMLMath , we find a LaTeXMLMath and a function LaTeXMLMath holomorphic in LaTeXMLEquation such that LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath denote the envelope of holomorphy of the domain LaTeXMLEquation .
Applying the Grauert–Remmert theorem , we can extend LaTeXMLMath holomorphically to LaTeXMLMath , i.e .
there exists an LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
Observe that LaTeXMLMath .
Recall that LaTeXMLMath ; contradiction .
LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath be as in ( * ) .
Let LaTeXMLMath be a pure one–codimensional analytic subset of LaTeXMLMath ( we do not assume that LaTeXMLMath ) .
Then : For any LaTeXMLMath there exists LaTeXMLMath such that for any function LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , there exists an extension LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
4 .
Proof of the Main Theorem in the general case .
First observe that the function LaTeXMLMath is uniquely determined ( cf .
§ 3 ) .
We proceed by induction on LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath are pseudoconvex domains with LaTeXMLMath .
Put LaTeXMLEquation .
It suffices to show that for each LaTeXMLMath the following condition ( LaTeXMLMath ) holds .
( LaTeXMLMath ) There exists a domain LaTeXMLMath , LaTeXMLMath , such that for any LaTeXMLMath there exists an LaTeXMLMath with LaTeXMLMath .
Indeed , fix a LaTeXMLMath and observe that LaTeXMLMath is the envelope of holomorphy of LaTeXMLMath ( cf .
the proof of Step 4 ) .
Hence , in virtue of the Dloussky theorem ( cf .
LaTeXMLCite , Th .
3.4.8 , see also LaTeXMLCite ) , there exists an analytic subset LaTeXMLMath of LaTeXMLMath , LaTeXMLMath , such that LaTeXMLMath is the envelope of holomorphy of LaTeXMLMath .
In particular , for each LaTeXMLMath there exists an LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath .
It is known that ( cf .
LaTeXMLCite , Prop .
3.4.5 ) there exists a pure one–codimensional analytic subset LaTeXMLMath , LaTeXMLMath , such that any point of LaTeXMLMath is singular with respect to LaTeXMLMath , i.e .
LaTeXMLMath any function LaTeXMLMath extends to a function LaTeXMLMath and LaTeXMLMath for any LaTeXMLMath and an open neighborhood LaTeXMLMath of LaTeXMLMath , LaTeXMLMath , there exists an LaTeXMLMath such that LaTeXMLMath can not be holomorphically extended to the whole LaTeXMLMath .
In particular , LaTeXMLMath .
Consequently , LaTeXMLMath is a pure one–codimensional analytic subset of LaTeXMLMath , LaTeXMLMath , and for each LaTeXMLMath , the function LaTeXMLMath is holomorphic on LaTeXMLMath with LaTeXMLMath .
It remains to prove ( LaTeXMLMath ) .
Fix a LaTeXMLMath .
For any LaTeXMLMath let LaTeXMLMath be such that LaTeXMLMath .
If LaTeXMLMath , then we additionally define LaTeXMLMath –fold crosses LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and we assume that LaTeXMLMath is so small that LaTeXMLEquation .
Since LaTeXMLMath , we may assume that LaTeXMLEquation .
We define LaTeXMLMath –fold crosses LaTeXMLEquation .
Note that LaTeXMLMath .
Since LaTeXMLMath , there exists an LaTeXMLMath , LaTeXMLMath , so small that LaTeXMLEquation .
Put LaTeXMLEquation
ON WICK POWER SERIES CONVERGENT TO NONLOCAL FIELDS A. G. Smirnov , M. A. Soloviev Abstract The infinite series in Wick powers of a generalized free field are considered that are convergent under smearing with analytic test functions and realize a nonlocal extension of the Borchers equivalence classes .
The nonlocal fields to which they converge are proved to be asymptotically commuting , which serves as a natural generalization of the relative locality of the Wick polynomials .
The proposed proof is based on exploiting the analytic properties of the vacuum expectation values in LaTeXMLMath -space and applying the Cauchy–Poincaré theorem .
In this paper we continue the investigation LaTeXMLCite of the infinite series of the form LaTeXMLEquation in Wick powers of a neutral scalar field LaTeXMLMath , whose basic point is the systematic use of the analytic properties of the vacuum expectation values in LaTeXMLMath -space .
The developed approach is primarily aimed at applications to gauge field theory , where the two-point function LaTeXMLMath does not necessarily satisfy the positivity condition LaTeXMLMath ( where LaTeXMLMath is a test function ) , and it not only allows easily finding the test function class on which a given series is convergent , but also enables one to establish the properties of the limiting field LaTeXMLMath .
In the positive metric case , it is customary to use another approach LaTeXMLCite based on estimating the terms of the series representing the vacuum expectation value LaTeXMLMath in momentum space , where they are expressible through positive measures .
It is commonly assumed that this is the only possible way of handling the problem if the sum LaTeXMLMath of a series is a nonlocal field because in this case the analyticity domain of its vacuum expectation values in LaTeXMLMath -space is empty .
Nevertheless , our approach is applicable to such series as well if the analyticity properties of each their particular term are duly taken into account , see LaTeXMLCite .
It is essential that this approach covers the fields LaTeXMLMath of zero mass , which were not considered in LaTeXMLCite , and , moreover , the generalized free fields in a space-time of arbitrary dimension LaTeXMLMath .
Here we shall prove the relative asymptotic commutativity of the nonlocal fields to which the series in Wick powers of a generalized free field converge .
The role of the asymptotic commutativity condition in the theory of nonlocal interactions was analyzed in LaTeXMLCite , where it was proved that this condition ensures the normal connection between spin and statistics and the CPT-invariance .
Within the framework of the traditional approach , the properties of nonlocal Wick series of a free field with nonzero mass were considered earlier in LaTeXMLCite , whereas the relation of the essential locality condition used in LaTeXMLCite to the asymptotic commutativity is discussed in LaTeXMLCite .
The general construction of Wick powers of generalized free fields was considered in LaTeXMLCite .
Other important motivations for a deeper analysis of nonlocal Wick series ( in addition to the fact that they form an extension of the Borchers equivalence class of the field LaTeXMLMath ) are the connection of nonlocal quantum field models exhibiting singular ultraviolet behavior with string theory and M-theory LaTeXMLCite , especially in the context of AdS/CFT correspondence LaTeXMLCite , and the use of nonlocal formfactors for removing ultraviolet divergences in phenomenological models proposed as an alternative to string theory LaTeXMLCite .
In particular , the developed technique may be useful for the treatment of the problem of a possible CPT-invariance breaking in such models , which is discussed in LaTeXMLCite .
From this point on , we shall assume LaTeXMLMath to be a tempered distribution generalized free field LaTeXMLCite .
The LaTeXMLMath -point vacuum expectation values are expressible in terms of the two-point one by the same recurrence relation as in the case of a free field , which makes it possible to define the Wick ordered powers LaTeXMLMath .
The analytic function whose boundary value is the distribution LaTeXMLMath will be denoted by LaTeXMLMath .
As is shown in LaTeXMLCite , if the positivity condition is satisfied , the test function class on which series ( LaTeXMLRef ) is convergent is determined by the behavior of LaTeXMLMath in the imaginary directions , its growth for LaTeXMLMath ( resp .
for LaTeXMLMath ) being essential in the localizable ( resp .
nonlocalizable ) case .
As a characteristic of this behavior one can take the restriction of the function LaTeXMLMath to the semi-axis LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and we shall denote this restriction by LaTeXMLMath .
The formula for the Laplace transformation LaTeXMLEquation , where LaTeXMLMath is a positive polynomially bounded measure supported by the closed upper light cone LaTeXMLMath , shows that LaTeXMLMath is a strictly positive nondecreasing function which majorizes LaTeXMLMath for LaTeXMLMath of the specified form and for any LaTeXMLMath .
Hence , by the Lorentz invariance of LaTeXMLMath , we have LaTeXMLEquation where LaTeXMLMath is the Lorentz square of LaTeXMLMath .
Thus , LaTeXMLMath indeed can serve as an indicator function according to the definition LaTeXMLCite , where it was denoted by LaTeXMLMath in order to distinguish it from the function characterizing the infrared behavior of LaTeXMLMath , which is necessary in theories with an indefinite metric .
Moreover , the function LaTeXMLMath is the least one among all functions satisfying ( LaTeXMLRef ) and , therefore , is the best characteristic of the behavior of LaTeXMLMath .
It should also be noted that LaTeXMLMath is infinitely differentiable and increases indefinitely with decreasing argument .
As usual , we denote by LaTeXMLMath the component of identity of the complex Lorentz group and LaTeXMLMath the extended analyticity domain of LaTeXMLMath which is generated from the primitive domain LaTeXMLMath by applying arbitrary transformations in LaTeXMLMath according to the Bargmann–Hall–Wightman theorem , whose proof for an arbitrary space-time dimension can be found in LaTeXMLCite .
The domain LaTeXMLMath is invariant under the full reflection LaTeXMLMath .
Indeed , for an even LaTeXMLMath the reflection belongs to LaTeXMLMath , and the general case can be treated as follows .
If LaTeXMLMath is in LaTeXMLMath , then there exists a transformation LaTeXMLMath which takes LaTeXMLMath to a point with an imaginary part belonging to the negative LaTeXMLMath -semi-axis .
Now the statement follows if we note that the composition of LaTeXMLMath with the partial reflection LaTeXMLMath , which also belongs to LaTeXMLMath , takes LaTeXMLMath to a point with the same imaginary part .
In particular , the inclusion LaTeXMLMath is valid .
The following simple lemma allows us to estimate the function LaTeXMLMath for real arguments provided we know its behavior in the imaginary directions .
Lemma 1 .
Let LaTeXMLMath , LaTeXMLMath , and let LaTeXMLMath , LaTeXMLMath .
Then there exists LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Proof .
Suppose first that LaTeXMLMath and let LaTeXMLMath be a real Lorentz transformation taking LaTeXMLMath to a vector of the form LaTeXMLMath .
Let LaTeXMLMath be a pure rotation which takes LaTeXMLMath to a vector LaTeXMLMath .
Set LaTeXMLMath , where LaTeXMLMath .
Then LaTeXMLMath belongs to LaTeXMLMath under the proper choice of the sign , and LaTeXMLMath .
Now suppose LaTeXMLMath .
Then there exists a Lorentz transformation LaTeXMLMath taking LaTeXMLMath to a vector LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath , and LaTeXMLMath be defined as before , and let LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath , and the proper choice of the sign in the definition of LaTeXMLMath ensures that LaTeXMLMath .
The lemma is proved .
It is well known that all spacelike vectors belong to LaTeXMLMath , which also follows from Lemma 1 .
For any such vector LaTeXMLMath one can find LaTeXMLMath such that LaTeXMLMath , and hence LaTeXMLMath .
Therefore , by the uniqueness theorem , LaTeXMLEquation .
In particular , at the level of the two-point vacuum expectation values , the locality is a consequence of the other Wightman axioms .
Using the notation LaTeXMLEquation and combining ( LaTeXMLRef ) with Lemma 1 , we obtain the estimate LaTeXMLEquation which holds for LaTeXMLMath .
Let us denote the series ( LaTeXMLRef ) by LaTeXMLMath .
We shall also consider its subordinate series and use the notation LaTeXMLMath which means that for all indices LaTeXMLMath , with the possible exception of a finite subset the inequality LaTeXMLMath holds , where LaTeXMLMath is a positive constant .
It is reasonable to impose the following conditions on the coefficients of series ( LaTeXMLRef ) : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath are constants whose role is explained in LaTeXMLCite ; the subordinate series need not satisfy them .
We use the Gelfand–Shilov spaces LaTeXMLMath as test function spaces , see LaTeXMLCite .
The defining index LaTeXMLMath can be regarded as an indicator function characterizing the momentum space behavior of the test functions .
More precisely , the derivatives of the Fourier transform of LaTeXMLMath satisfy the inequalities LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath positive constants depending on LaTeXMLMath .
For definiteness , we assume the norm LaTeXMLMath in LaTeXMLMath to be uniform .
In the context of QFT , the function LaTeXMLMath characterizes the high energy behavior of the fields defined over LaTeXMLMath .
Let LaTeXMLMath be the fields determined by the series LaTeXMLMath .
Consider the LaTeXMLMath -point vacuum expectation value LaTeXMLMath .
Applying the Wick theorem gives the well-known formal representation LaTeXMLEquation where LaTeXMLMath is an integer-valued multi-index with nonnegative components LaTeXMLMath , LaTeXMLMath , which have the sense of the number of pairings between the terms of the series LaTeXMLMath and LaTeXMLMath , and the designation LaTeXMLEquation is used .
The numerical coefficients LaTeXMLMath are expressible in terms of the coefficients of the series LaTeXMLMath in the following way : LaTeXMLEquation .
If the distribution series on the right-hand side of ( LaTeXMLRef ) absolutely converges on each test function in the space LaTeXMLMath , then it is unconditionally summable with respect to the strong topology of its dual space because the latter is a Montel space .
If this is the case for any set of series subordinate to LaTeXMLMath , then , as is shown in LaTeXMLCite , the fields LaTeXMLMath , LaTeXMLMath are well defined as operator-valued generalized functions over LaTeXMLMath acting in the Hilbert space LaTeXMLMath of the initial field LaTeXMLMath .
In particular , the vector series that define the repeated action of these operators on the vacuum LaTeXMLMath are unconditionally convergent , and the linear span of all vectors of the form LaTeXMLMath , LaTeXMLMath serves as a common dense invariant domain of definition for the family of fields LaTeXMLMath in LaTeXMLMath .
We shall denote this domain by LaTeXMLMath .
The distribution ( LaTeXMLRef ) is the boundary value of the analytic function LaTeXMLEquation from the cone LaTeXMLEquation and the representation ( LaTeXMLRef ) can be rewritten in the following more precise form : LaTeXMLEquation where LaTeXMLMath is the boundary value operator .
It is worth noting that the function ( LaTeXMLRef ) is defined and analytic in the open set LaTeXMLMath and , in particular , in the tube LaTeXMLMath , where LaTeXMLMath .
For what follows , it is essential to know the transformation law for representation ( LaTeXMLRef ) under the rearrangements of the operators LaTeXMLMath entering into the vacuum expectation value .
Lemma 2 .
Suppose the distribution series on the right-hand side of LaTeXMLMath is unconditionally convergent in LaTeXMLMath for any LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be a permutation of the indices LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath .
Proof .
Because of ( LaTeXMLRef ) we have LaTeXMLEquation where LaTeXMLMath is the coefficient corresponding , by ( LaTeXMLRef ) , to the permuted set LaTeXMLMath .
Let LaTeXMLMath be the multi-index whose components LaTeXMLMath are equal to LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
From ( LaTeXMLRef ) , it follows that LaTeXMLMath coincides with the product defining LaTeXMLMath to within the signs of the arguments of some factors , and in view of ( LaTeXMLRef ) we conclude that LaTeXMLMath in LaTeXMLMath .
Passing in this equality to the boundary values from the cone LaTeXMLMath , we obtain LaTeXMLMath , whereas the relation LaTeXMLMath , which follows from ( LaTeXMLRef ) , implies that LaTeXMLMath .
Making the change LaTeXMLMath of the summation indices in the unconditionally convergent series ( LaTeXMLRef ) and applying the above identities , we arrive at ( LaTeXMLRef ) .
The lemma is proved .
If the space LaTeXMLMath on which series ( LaTeXMLRef ) converges contains functions of compact support , i.e. , the field LaTeXMLMath is an ( operator-valued ) ultradistribution , then the fulfilment of the Wightman axioms for this field is easily established by the same arguments as in LaTeXMLCite , where even the more general case of an indefinite metric was considered .
In particular , the locality of LaTeXMLMath and , moreover , the relative locality of the fields LaTeXMLMath , LaTeXMLMath , immediately follow from the relative locality of the Wick monomials LaTeXMLMath .
This property can be also derived from Lemma 2 if LaTeXMLMath is taken to be the transposition LaTeXMLMath of the neighbouring indices LaTeXMLMath and LaTeXMLMath .
The distribution LaTeXMLMath is supported by the closed cone LaTeXMLEquation .
Therefore , the support of the functional LaTeXMLEquation is also contained in this cone , whence LaTeXMLMath for spacelike separated supports of the test functions LaTeXMLMath and for any LaTeXMLMath because LaTeXMLMath and the rest of LaTeXMLMath in ( LaTeXMLRef ) can be taken arbitrary .
The locality condition implies that the vacuum expectation values in momentum space have less than exponential growth , see , e.g. , LaTeXMLCite .
For this reason , the Gelfand–Shilov space determined by the indicator function LaTeXMLMath , which is customarily denoted by LaTeXMLMath , is universal for local fields .
The elements of LaTeXMLMath allow analytic continuation into a complex neighbourhood of the real space and never have compact support .
Nevertheless , the methods of the hyperfunction theory make it possible to give a correct definition of support for the functionals belonging to the dual space LaTeXMLMath .
We refer the reader to LaTeXMLCite for details and confine ourselves to saying that the support of LaTeXMLMath is contained in a closed cone LaTeXMLMath if and only if LaTeXMLMath has a continuous extension to each space LaTeXMLMath , where LaTeXMLMath is an open cone containing LaTeXMLMath and LaTeXMLMath consists of analytic functions on the complex LaTeXMLMath -neighbourhood LaTeXMLMath of the set LaTeXMLMath with the property that the norms LaTeXMLEquation are finite .
The corresponding generalization of local commutativity for the fields LaTeXMLMath , LaTeXMLMath defined over LaTeXMLMath means that the matrix elements LaTeXMLMath have support in the closed cone LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath belonging to the common domain of definition of these fields .
Within the framework of hyperfunction theory , Wick power series of the free field of nonzero mass were studied earlier in the work LaTeXMLCite .
In our notation , the restriction on the series coefficients found there takes the form LaTeXMLEquation .
The fulfilment of the Wightman axioms for the ( hyper ) fields defined by Wick series was established in LaTeXMLCite indirectly , by means of an equivalence theorem of Osterwalder–Schrader type for a properly modified Euclidean field theory and the Minkowski quantum field theory formulated in terms of Fourier-hyperfunctions .
Below we shall obtain , as a by product , a simple direct proof showing that in this case the limiting fields satisfy the generalized locality condition .
If LaTeXMLMath grows faster than linearly , then the elements of LaTeXMLMath are entire functions and the functionals belonging to LaTeXMLMath are nonlocal .
However , under suitable restrictions on LaTeXMLMath , they inherit an important part of the properties of hyperfunctions which has the sense of angular localizability .
In LaTeXMLCite a corresponding theory has been developed for the spaces whose indicator functions are exponentials of order LaTeXMLMath , and in LaTeXMLCite it has been extended to a more general case .
In the work LaTeXMLCite , another scale of spaces LaTeXMLMath was considered which is required for the indefinite metric field theory , but the construction proposed there is applicable to LaTeXMLMath as well .
Namely , let LaTeXMLMath be a nonnegative , convex , differentiable , and indefinitely increasing function on the half-axis LaTeXMLMath and let LaTeXMLMath be an open cone in LaTeXMLMath .
We denote by LaTeXMLMath the distance from the point LaTeXMLMath to the cone LaTeXMLMath and consider the space LaTeXMLMath , where LaTeXMLMath consists of entire analytic functions on LaTeXMLMath such that the norms LaTeXMLEquation are finite for any LaTeXMLMath .
The topology of LaTeXMLMath is defined to be that of the inductive limit of the countably normed spaces LaTeXMLMath with the index LaTeXMLMath .
Lemma 3 .
The space LaTeXMLMath coincides with the space LaTeXMLMath defined by the indicator function LaTeXMLMath , where LaTeXMLMath .
Proof .
Taking into account the elementary inequalities LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is arbitrarily small and LaTeXMLMath is a positive constant , we see that replacing the factor LaTeXMLMath in ( LaTeXMLRef ) by LaTeXMLMath leads to an equivalent definition of LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLEquation .
Making use of the freedom in the choice of the plane of integration , we set LaTeXMLMath and take the infimum with respect to LaTeXMLMath .
As a result , we obtain an estimate of the form ( LaTeXMLRef ) with LaTeXMLMath .
Conversely , if ( LaTeXMLRef ) holds for such an indicator function , then taking the inverse Fourier–Laplace transformation , we find that LaTeXMLEquation and so LaTeXMLMath because the Legendre transformation is involutory .
The lemma is proved .
The spaces LaTeXMLMath of the specified type will be called the Gelfand–Shilov–Gurevich spaces because they also belong to the class of spaces of type LaTeXMLMath introduced by B. L. Gurevich .
Among such spaces , a special role is played by that defined by LaTeXMLMath .
This space customarily denoted by LaTeXMLMath is nothing but the Fourier transformed Schwartz ’ s space LaTeXMLMath .
It is universal for nonlocal fields because in this case LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , i.e. , the test functions have compact support in momentum space , and so fields with an arbitrarily singular ultraviolet behavior can be smeared with them .
Definition 1 .
A closed cone LaTeXMLMath is called a carrier cone of a functional LaTeXMLMath if LaTeXMLMath has a continuous extension to each space LaTeXMLMath , LaTeXMLMath .
For any LaTeXMLMath there exists a unique minimal closed carrier cone .
This has been proved in LaTeXMLCite for the spaces LaTeXMLMath defined by the exponentials of order LaTeXMLMath , and just the properties of them that were used in this proof are included into the definition of LaTeXMLMath .
Definition 2 .
We say that the fields LaTeXMLMath , LaTeXMLMath defined on the test function space LaTeXMLMath asymptotically commute for large spacelike separations of their arguments if the matrix elements LaTeXMLMath are carried by cone LaTeXMLMath for any LaTeXMLMath in the common domain of definition of these fields .
A general convergence criterion for Wick series can be formulated in terms of the above-mentioned characteristic LaTeXMLMath of the two-point function LaTeXMLMath and the indicator function LaTeXMLMath as follows .
The series ( LaTeXMLRef ) is convergent under smearing with test functions in LaTeXMLMath if LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath .
The proof of this criterion is the same as that of Theorem 4 in LaTeXMLCite , where it has been established in the case of the free field of mass LaTeXMLMath and the explicit form of the corresponding function LaTeXMLMath has been used .
For the Gelfand–Shilov–Gurevich spaces , an alternative formulation is possible in terms of the function LaTeXMLMath , which will be useful below .
Its derivation is much simpler than that in the general case considered in LaTeXMLCite , where test functions are not necessarily analytic .
Theorem 1 .
Let LaTeXMLMath be a scalar neutral generalized free field and let LaTeXMLEquation where LaTeXMLMath is the Fourier transform of its two-point function .
Suppose restrictions LaTeXMLMath on the coefficients of the Wick power series LaTeXMLMath hold .
If the function LaTeXMLMath defining the space LaTeXMLMath satisfies the condition LaTeXMLEquation for arbitrarily large LaTeXMLMath , then the field LaTeXMLMath and all fields LaTeXMLMath , LaTeXMLMath , are well defined as operator-valued generalized functions over LaTeXMLMath acting in the Hilbert space LaTeXMLMath of the initial field LaTeXMLMath .
Proof .
According to what has been said above , it is sufficient to show that ( LaTeXMLRef ) implies the absolute convergence of the series on the right-hand side of ( LaTeXMLRef ) on every test function LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath .
Then LaTeXMLEquation .
Making use of ( LaTeXMLRef ) , ( LaTeXMLRef ) , the monotonicity of LaTeXMLMath , and the equality LaTeXMLMath , we obtain the estimate LaTeXMLEquation .
In view of the freedom in the choice of LaTeXMLMath we have LaTeXMLEquation .
Thus , the required convergence of the series ( LaTeXMLRef ) is ensured by the convergence of the number series LaTeXMLEquation with arbitrarily large LaTeXMLMath .
From the conditions ( LaTeXMLRef ) and the properties of the polynomial coefficients , it follows that LaTeXMLMath , see LaTeXMLCite .
Since the number of multi-indices with the norm LaTeXMLMath does not exceed LaTeXMLMath , we conclude that series ( LaTeXMLRef ) is majorized by series ( LaTeXMLRef ) for sufficiently large LaTeXMLMath .
The theorem is thus proved .
Lemma 4 .
Criterion LaTeXMLMath is equivalent to condition LaTeXMLMath for LaTeXMLMath and condition LaTeXMLMath for LaTeXMLMath .
Proof .
Let LaTeXMLMath .
Under condition ( LaTeXMLRef ) the series LaTeXMLMath is convergent everywhere .
Majorizing the infimum on the left-hand side of inequality ( LaTeXMLRef ) by the value of the function at LaTeXMLMath , we see that it is valid with the constant LaTeXMLEquation .
To prove the inverse implication ( LaTeXMLRef ) LaTeXMLMath ( LaTeXMLRef ) , choose LaTeXMLMath such that LaTeXMLEquation .
Then LaTeXMLMath and condition ( LaTeXMLRef ) ensures that LaTeXMLEquation for LaTeXMLMath .
Setting LaTeXMLMath and making use of the arbitrariness of LaTeXMLMath , we obtain ( LaTeXMLRef ) .
Now let LaTeXMLMath .
If LaTeXMLMath grows linearly , then the statement of the lemma is verified immediately .
So we assume that LaTeXMLEquation .
Let us demonstrate that LaTeXMLEquation ( This relation generalizes the conclusion of Lemma 1 in LaTeXMLCite , where LaTeXMLMath . )
Since the Legendre transformation is involutory , the equality ( LaTeXMLRef ) holds for LaTeXMLMath .
Let LaTeXMLMath .
Since LaTeXMLMath , the infimum on the left-hand side occurs at some finite point LaTeXMLMath satisfying the equation LaTeXMLMath .
Set LaTeXMLMath .
Making use of the relation LaTeXMLEquation and taking into account that , for convex functions , every stationary point is the point of absolute minimum , we conclude that the extrema LaTeXMLMath and LaTeXMLMath are attained at LaTeXMLMath .
Estimating from below the supremum with respect to LaTeXMLMath by the value of the function at the point LaTeXMLMath , we obtain LaTeXMLEquation .
Since LaTeXMLMath for any function LaTeXMLMath , the inverse inequality also holds , and so ( LaTeXMLRef ) is proved .
Supposing ( LaTeXMLRef ) is valid and applying ( LaTeXMLRef ) , we have LaTeXMLEquation whence ( LaTeXMLRef ) immediately follows .
Conversely , setting LaTeXMLMath in ( LaTeXMLRef ) , using ( LaTeXMLRef ) and estimating from below the suprema with respect to LaTeXMLMath by the value of the function at a fixed point , we arrive at ( LaTeXMLRef ) .
The lemma is proved .
We proceed to show that the generalizations of local commutativity considered in Section 4 are fulfilled for the fields determined by the Wick series convergent on analytic test functions .
Theorem 2 .
Let LaTeXMLMath be a series in the Wick powers of a generalized free field LaTeXMLMath and let its coefficients satisfy assumption LaTeXMLMath .
Suppose the indicator function LaTeXMLMath of the test function space satisfies condition LaTeXMLMath .
If LaTeXMLMath is a Gelfand–Shilov–Gurevich space , then the fields LaTeXMLMath determined by the series LaTeXMLMath commute asymptotically .
If LaTeXMLMath , then they are relatively local in the sense of hyperfunction theory .
Lemma 5 .
Let the condition LaTeXMLMath be satisfied .
If for any set of series LaTeXMLMath , LaTeXMLMath , functional LaTeXMLMath has a continuous extension to the space LaTeXMLMath , where LaTeXMLMath is the open cone LaTeXMLMath , then all the fields determined by series subordinate to LaTeXMLMath commute asymptotically .
If condition LaTeXMLMath holds and the functional LaTeXMLMath allows a continuous extension to LaTeXMLMath , then they are relatively local in the sense of hyperfunction theory .
Proof .
Let LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath .
Then the value of the functional LaTeXMLMath on a test function LaTeXMLMath coincides with that of the functional of the form LaTeXMLMath with a suitable set of indices on the test function LaTeXMLMath .
If LaTeXMLMath , then this tensor product is the element of LaTeXMLMath which depends continuously on LaTeXMLMath .
Thus , the existence of a continuous extension to LaTeXMLMath for the functionals of the form ( LaTeXMLRef ) ensures that the matrix elements in question have continuous extensions to LaTeXMLMath and all the more to LaTeXMLMath , where LaTeXMLMath .
Analogous statements , with proper changes in notation , are valid for the space LaTeXMLMath .
Since LaTeXMLMath is the linear span of the vectors LaTeXMLMath of the specified form , the lemma is proved .
In the above derivation of the convergence criterion ( LaTeXMLRef ) the key role is played by the variation of the plane of integration in the representation ( LaTeXMLRef ) which enables us to obtain the best estimate for each term of the series on the right-hand side of ( LaTeXMLRef ) .
We shall apply the same idea to prove the asymptotic commutativity of the sums of Wick series .
However , this will require integrating the corresponding analytic functions over surfaces of a more complicated form .
The variation of such surfaces in the analyticity domain is admissible by the Cauchy–Poincaré theorem LaTeXMLCite , but for obtaining concrete estimates it will be convenient to use directly the Stokes theorem which lies at the basis of its derivation .
According to Lemma 2 functional ( LaTeXMLRef ) is represented by the series LaTeXMLEquation .
In view of the barrelledness of LaTeXMLMath and LaTeXMLMath , in order to extend continuously this functional to these spaces it is sufficient to construct a continuous extension of each term of series ( LaTeXMLRef ) and show that it is absolutely convergent on every element of the corresponding space .
We shall consider in detail such a procedure for LaTeXMLMath and then explain how the proof should be modified for the case of hyperfunctions .
For LaTeXMLMath and LaTeXMLMath , we define the regions LaTeXMLEquation .
LaTeXMLEquation The relations ( LaTeXMLRef ) – ( LaTeXMLRef ) show that , for LaTeXMLMath , the set LaTeXMLMath lies in the analyticity domain of LaTeXMLMath , and for LaTeXMLMath , the following inequality holds : LaTeXMLEquation .
Lemma 6 .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
With the notation LaTeXMLEquation .
LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , the identity LaTeXMLEquation holds , where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath is the linear operator taking LaTeXMLMath to LaTeXMLMath , LaTeXMLMath is the unit inward normal to LaTeXMLMath , and LaTeXMLMath is the surface measure on LaTeXMLMath .
Proof .
Note that LaTeXMLMath and LaTeXMLMath and hence LaTeXMLEquation .
We split the integration domain into LaTeXMLMath and LaTeXMLMath , rewrite the right-hand side of the last equality as LaTeXMLMath , where LaTeXMLEquation and shall show that LaTeXMLMath .
Let LaTeXMLMath .
From ( LaTeXMLRef ) , ( LaTeXMLRef ) , it follows that , for any compactum LaTeXMLMath , the estimate LaTeXMLEquation holds if LaTeXMLMath .
Making use of Cauchy ’ s integral formula , one can easily show that analogous inequalities are satisfied for the derivatives as well .
Therefore , the function LaTeXMLEquation is differentiable and , applying the Cauchy–Riemann equations , we have LaTeXMLEquation .
Since LaTeXMLMath for LaTeXMLMath and LaTeXMLEquation we obtain LaTeXMLEquation .
Let us denote LaTeXMLMath and make the change of variables LaTeXMLMath .
Then , observing that LaTeXMLMath , we get the equality LaTeXMLEquation .
From ( LaTeXMLRef ) , ( LaTeXMLRef ) , it follows that LaTeXMLMath as LaTeXMLMath , LaTeXMLMath , and therefore the integration of LaTeXMLMath and LaTeXMLMath yields zero .
Reducing the multiple integral to the iterated one , we obtain LaTeXMLEquation where LaTeXMLMath .
The last integrand can be regarded as the divergence of the vector field LaTeXMLMath .
Applying the Stokes theorem and replacing LaTeXMLMath by LaTeXMLMath , we conclude that LaTeXMLMath .
Lemma 7 .
The functional LaTeXMLMath can be continuously extended to LaTeXMLMath and , for any LaTeXMLMath , LaTeXMLMath there exist LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath Proof .
Let us show that formulas ( LaTeXMLRef ) - ( LaTeXMLRef ) determine the desired extension if LaTeXMLMath is assumed to be an element of the space LaTeXMLMath .
We first estimate LaTeXMLMath supposing LaTeXMLMath .
It is easy to see that LaTeXMLMath for LaTeXMLMath .
Besides , LaTeXMLMath for LaTeXMLMath and , in view of ( LaTeXMLRef ) and the monotonicity of LaTeXMLMath , we have LaTeXMLEquation .
Taking into account that LaTeXMLMath and using relations ( LaTeXMLRef ) , ( LaTeXMLRef ) , the monotonicity of LaTeXMLMath , and the equality LaTeXMLMath , we find that LaTeXMLEquation for LaTeXMLMath .
Thus , we have LaTeXMLEquation .
Now let us estimate LaTeXMLMath .
Observe that if LaTeXMLMath , then LaTeXMLMath .
Furthermore , LaTeXMLMath , where LaTeXMLMath .
Using ( LaTeXMLRef ) and the definition LaTeXMLMath , we obtain LaTeXMLEquation .
Next we apply ( LaTeXMLRef ) , taking into account that LaTeXMLMath for LaTeXMLMath and using the monotonicity of LaTeXMLMath .
As a result , we get LaTeXMLEquation .
Further , there exists a constant LaTeXMLMath independent on LaTeXMLMath and such that LaTeXMLEquation .
Indeed , LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is the area element for the surface of the unit sphere in LaTeXMLMath .
Let us denote LaTeXMLEquation and substitute relations ( LaTeXMLRef ) , ( LaTeXMLRef ) into ( LaTeXMLRef ) , ( LaTeXMLRef ) .
As a result , we obtain LaTeXMLEquation .
The presence of the factor LaTeXMLMath in estimates ( LaTeXMLRef ) , ( LaTeXMLRef ) ensures that formulas ( LaTeXMLRef ) - ( LaTeXMLRef ) define a continuous extension of the distribution LaTeXMLMath to LaTeXMLMath and , by the Cauchy–Poincaré theorem , the extensions corresponding to different LaTeXMLMath coincide with each other .
Since LaTeXMLMath is unbounded from above and convex , we have LaTeXMLMath , where LaTeXMLMath .
Besides , LaTeXMLMath because of the convexity of LaTeXMLMath .
Combining these inequalities with ( LaTeXMLRef ) , ( LaTeXMLRef ) and passing to the infimum with respect to LaTeXMLMath , we obtain ( LaTeXMLRef ) with LaTeXMLMath .
The lemma is proved .
From Lemma 7 , it immediately follows that series ( LaTeXMLRef ) is absolutely convergent on every test function in LaTeXMLMath provided number series ( LaTeXMLRef ) converges for any LaTeXMLMath .
As was established in proving Theorem 1 , this is ensured by condition ( LaTeXMLRef ) which is equivalent to the criterion ( LaTeXMLRef ) by Lemma 4 .
Thus , Theorem 2 is proved for the case of spaces LaTeXMLMath .
The proof of Lemma 6 is extended immediately to the case LaTeXMLMath , if we assume LaTeXMLMath .
Repeating , with appropriate changes , the derivation of Lemma 7 , we make sure that , for all LaTeXMLMath , formulas ( LaTeXMLRef ) - ( LaTeXMLRef ) define the same continuous extension of the functional LaTeXMLMath to LaTeXMLMath and LaTeXMLEquation .
The extensions corresponding to different LaTeXMLMath are obviously compatible and so define a continuous extension to LaTeXMLMath .
Because of ( LaTeXMLRef ) , series ( LaTeXMLRef ) is absolutely convergent on every LaTeXMLMath if LaTeXMLMath for all LaTeXMLMath .
The estimate of the coefficients LaTeXMLMath mentioned in the proof of Theorem 1 shows that the latter is ensured by the condition ( LaTeXMLRef ) , and it remains to apply Lemma 4 to complete the proof .
Together with the results of work LaTeXMLCite Theorem 2 shows that the nonlocal fields determined by series in Wick powers of a generalized free field satisfy all requirements of Wightman ’ s formulation if the locality axiom is replaced by the asymptotic commutativity condition .
It is noteworthy that using the generalized Gelfand–Shilov spaces enables one to consider also the series convergent on quasianalytic test function classes defined by the indicator functions LaTeXMLMath that grow slower than any linear exponential but do not satisfy the strict localizability condition LaTeXMLEquation ensuring that LaTeXMLMath contains functions of compact support .
In fact , Theorem 2 is applicable to this case as well because the generalized functions defined on such spaces have the same supports as their restrictions to LaTeXMLMath , see LaTeXMLCite .
Acknowledgments .
This work was supported in part by the Russian Foundation for Basic Research under Grants No .
99-01-00376 , 99-02-17916 , and 00-15-96566 , and in part by INTAS Grant No .
99-1-590 .
We shall give , in an optimal form , a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface .
We shall moreover prove that , if the normal K3 surface is elliptic and the above fundamental group is not finite , then there is a finite covering which is a complex torus .
Throughout this note , we work in the category of separated complex analytic spaces endowed with the classical topology .
The first aim of this short note is to add some evidence to the following interesting Conjecture posed by De-Qi Zhang ( See [ KT ] , [ SZ ] , [ KZ ] and [ Ca1,2 ] for related work ) : The universal cover of the smooth locus of a normal K3 surface is a big open set of either a normal K3 surface or of LaTeXMLMath .
Furthermore , in the latter case , the universal cover factors through a finite étale cover by a big open set of a torus .
To explain our notation , here and hereafter , a normal K3 surface means a normal surface whose minimal resolution is a K3 surface and a torus is a 2-dimensional complex torus .
By a big open set we mean the complement of a discrete subset .
Note that , by a result of Siu [ Si ] , a K3 surface is always Kähler .
Since the canonical divisor of a smooth K3 surface is trivial , it follows that the singularities of a normal K3 surface can only be the so called Du Val singularities ( also called Rational Double Points ) : these are also the Kleinian singularities obtained as quotients LaTeXMLMath with LaTeXMLMath ) .
The property that the singularities LaTeXMLMath are of the form LaTeXMLMath allows to define the orbifold Euler number of the normal K3 surface .
For each singular point LaTeXMLMath we take a neighbourhood LaTeXMLMath of LaTeXMLMath which is the quotient of a ball in LaTeXMLMath and decree that its orbifold Euler number equals LaTeXMLMath : using a Mayer Vietoris sequence we can then extend the definition to the whole of the normal K3 surface .
One has that the orbifold Euler number is non negative , and indeed R. Kobayashi and A. Todorov [ KT ] showed that the second case in Zhang ’ s conjecture happens if and only if the orbifold Euler number of the normal K3 surface is zero .
Let LaTeXMLMath be a normal K3 surface , let LaTeXMLMath and let LaTeXMLMath the minimal resolution .
Our first result is If the normal K3 surface LaTeXMLMath admits an elliptic fibration then either LaTeXMLMath is finite or there is a finite covering of LaTeXMLMath , ramified only on a finite set , which yields a complex torus .
Our second aim in this note is to establish a sharp sufficient condition for the validity of the first alternative in Zhang ’ s conjecture .
We set LaTeXMLMath to be the reduced exceptional divisor LaTeXMLMath and decompose LaTeXMLMath into irreducible components LaTeXMLMath .
An important invariant is the number LaTeXMLMath of irreducible components of the exceptional divisor .
Clearly , LaTeXMLMath .
Our main observation is as follows : If LaTeXMLMath , then LaTeXMLMath is finite .
In particular , if LaTeXMLMath , then the universal cover of LaTeXMLMath is a big open set of a normal K3 surface .
This easy remark however gives us the best possible uniform bound on LaTeXMLMath in order that LaTeXMLMath be finite , in view of the following facts [ KT ] , [ KZ ] : In our actual proof , the numerical condition LaTeXMLMath will be used twice : in Lemmas 1 and 2 .
These two Lemmas allow us to establish the existence of an elliptic pencil , whence to reduce the proof of theorem B to theorem A .
For the last statement of Theorem B , we recall that the category of Du Val singularities is closed under the operation of taking the normalization of a finite covering which is unramified outside a finite set .
Therefore , by the classification of smooth compact Kähler surfaces due to Castelnuovo , Enriques , Kodaira , the normalization of a finite cover of a normal K3 surface is either a normal K3 surface or a 2-dimensional torus if the covering is unramified outside a finite set .
This fact will be frequently used in our proof , too .
The proof of theorem A is on one side based on an exact sequence for open fibred surfaces , and which relies on the notion of orbifold fundamental group ( cf .
e.g .
[ Ca1 ] ) .
On the other hand , it is based on the existence of finite branched covers of given branching type provided by finite index subgroups of the orbifold fundamental group .
Of course , the most interesting part of the Conjecture concerns the case where the fundamental group is infinite .
As such , the question seems to belong more to the transcendental theory and we hope to return on the question using the current technologies on uniformization problems .
In the last paragraph of this note ( Remark 4 ) we will just comment on a more or less obvious reduction process .
An initial idea of this work was found during the first and the third authors ’ stay at KIAS in Seoul under the financial support by the Institute .
They would like to express their gratitude to Professor Jun-Muk Hwang and KIAS for making their stay enjoyable .
The third author would like to express his thanks to Professors Yujiro Kawamata and Eckart Viehweg for their warm encouragement during his stay at the Institute .
If LaTeXMLMath then LaTeXMLMath does not admit any finite covering by a complex torus which is unramified outside a finite set .
Assuming the contrary , we shall show that LaTeXMLMath .
Let LaTeXMLMath ( LaTeXMLMath be the singular points of LaTeXMLMath and let LaTeXMLMath be the number of the irreducible components of LaTeXMLMath .
Note that the contribution of LaTeXMLMath to the orbifold Euler number of LaTeXMLMath is LaTeXMLMath , where LaTeXMLMath is the order of the local fundamental group around LaTeXMLMath .
Since the orbifold Euler number of LaTeXMLMath equals LaTeXMLMath ( the obvious direction of [ KT ] ) one has LaTeXMLEquation .
Note that LaTeXMLMath if LaTeXMLMath .
Then , one has also LaTeXMLEquation .
Now , substituting the second inequality into the first equality , one obtains LaTeXMLEquation .
This implies LaTeXMLMath .
∎ If LaTeXMLMath then there exist an elliptic K3 surface LaTeXMLMath and an effective divisor LaTeXMLMath supported in fibers of LaTeXMLMath such that LaTeXMLMath is diffeomorphic to LaTeXMLMath and such that LaTeXMLMath and LaTeXMLMath are of the same Dynkin type .
We argue by descending induction on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is algebraic and the orthogonal complement of LaTeXMLMath in LaTeXMLMath is a hyperbolic lattice of rank LaTeXMLMath .
Then , by Meyer ’ s theorem ( cf .
e.g .
[ Se ] , Chapter IV , 3 , Cor .
2 ) , there exists LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and such that LaTeXMLMath .
( However , in general this LaTeXMLMath does not lie in the nef cone of LaTeXMLMath . )
Let LaTeXMLMath be the Kuranishi family of LaTeXMLMath and choose a trivialization LaTeXMLMath over LaTeXMLMath .
Here LaTeXMLMath is an even unimodular lattice of index LaTeXMLMath called the K3 lattice .
Then one can define the period map LaTeXMLMath , where LaTeXMLMath is the period domain .
This LaTeXMLMath is a local isomorphism by the local Torelli Theorem and allows us to identify LaTeXMLMath with a small open set of the period domain LaTeXMLMath ( denoted again by LaTeXMLMath ) .
Let ’ s denote by LaTeXMLMath , LaTeXMLMath the elements in LaTeXMLMath corresponding to LaTeXMLMath and LaTeXMLMath .
Let us take the sublocus LaTeXMLMath defined by the equations LaTeXMLEquation and consider the induced family LaTeXMLMath .
Here LaTeXMLMath is of dimension LaTeXMLMath .
Note that LaTeXMLMath and LaTeXMLMath .
Then , by construction and by the base change Theorem , the smooth rational curves LaTeXMLMath lift uniquely to effective divisors LaTeXMLMath flat over LaTeXMLMath in such a way that the fibers LaTeXMLMath are smooth rational curves ( of the same Dynkin type as LaTeXMLMath ) and LaTeXMLMath is diffeomorphic to LaTeXMLMath for all LaTeXMLMath if LaTeXMLMath is chosen to be sufficiently small .
Furthermore , for generic LaTeXMLMath , the Picard group of LaTeXMLMath is isomorphic to the primitive closure of LaTeXMLMath in LaTeXMLMath ( See for instance [ Og ] ) and is a semi-negative definite lattice .
Thus LaTeXMLMath is of algebraic dimension one .
Now , the algebraic reduction map of this LaTeXMLMath gives an elliptic fibration with the required properties .
Assume that LaTeXMLMath .
As before , we let LaTeXMLMath the Kuranishi family of LaTeXMLMath and choose a trivialization LaTeXMLMath over LaTeXMLMath and identify LaTeXMLMath with a small open set of the period domain LaTeXMLMath .
By LaTeXMLMath ( LaTeXMLMath ) we denote the elements in LaTeXMLMath corresponding to some integral basis of LaTeXMLMath .
Let us take the sublocus LaTeXMLMath defined by the equations LaTeXMLEquation and consider the induced family LaTeXMLMath .
Here LaTeXMLMath is of dimension LaTeXMLMath .
Then , for the same reason as before , the smooth rational curves LaTeXMLMath lift uniquely to effective divisors LaTeXMLMath flat over LaTeXMLMath in such a way that the fibers LaTeXMLMath are smooth rational curves ( of the same Dynkin type as LaTeXMLMath ) and LaTeXMLMath is diffeomorphic to LaTeXMLMath for all LaTeXMLMath if LaTeXMLMath is chosen to be sufficiently small .
Therefore the orbifold Euler numbers are also the same for LaTeXMLMath and LaTeXMLMath .
Here LaTeXMLMath is the normal K3 surface obtained by the contraction of LaTeXMLMath .
Furthermore , by construction , LaTeXMLMath is not isotrivial and the fibers LaTeXMLMath satisfy LaTeXMLMath .
Then by [ Og ] there is LaTeXMLMath such that LaTeXMLMath .
Now , we are done by descending induction on LaTeXMLMath .
∎ By Lemmas 1 and 2 , it is now clear that Theorem A implies Theorem B .
Before we proceed to the proof of theorem A we state a very easy and quite general lemma Let LaTeXMLMath be a fibration of a complete surface onto a complete curve with general fiber LaTeXMLMath , and let LaTeXMLMath be the Zariski open set of LaTeXMLMath which is defined as the complement of a finite number of divisors LaTeXMLMath each contained in a fibre LaTeXMLMath , but with support properly contained in LaTeXMLMath .
Denote by LaTeXMLMath the maximal divisor LaTeXMLMath which has no common component with LaTeXMLMath , and by LaTeXMLMath the greatest common divisor of the multiplicities of the components of LaTeXMLMath .
Set LaTeXMLMath , assume that the points LaTeXMLMath are all distinct , denote by LaTeXMLMath and define the orbifold fundamental group LaTeXMLMath of the open fibration LaTeXMLMath as the quotient of LaTeXMLMath by the normal subgroup generated by the elements LaTeXMLMath , where LaTeXMLMath is a simple loop going around the point LaTeXMLMath .
Then the fundamental group LaTeXMLMath fits into an exact sequence LaTeXMLEquation .
Let LaTeXMLMath be the complement of the given fibres , i.e. , LaTeXMLMath , so that LaTeXMLMath is a fibre bundle with fibre LaTeXMLMath .
Then we have the homotopy exact sequence LaTeXMLEquation .
However , LaTeXMLMath where LaTeXMLMath : therefore the kernel LaTeXMLMath of the surjection LaTeXMLMath is normally generated by simple loops LaTeXMLMath going around the components LaTeXMLMath of the LaTeXMLMath ’ s .
Each LaTeXMLMath maps into LaTeXMLMath to the LaTeXMLMath -th power of a conjugate of LaTeXMLMath , where LaTeXMLMath is the multiplicity of LaTeXMLMath in LaTeXMLMath .
Whence the image LaTeXMLMath in LaTeXMLMath of LaTeXMLMath is normally generated by the elements LaTeXMLMath .
Since LaTeXMLMath we see that LaTeXMLMath is normally generated by the elements LaTeXMLMath , thus we have the desired surjection LaTeXMLMath with kernel generated by LaTeXMLMath .
∎ Let LaTeXMLMath be a normal K3 surface such that LaTeXMLMath admits an elliptic fibration LaTeXMLMath .
Then either Note by the way that all the assumptions and assertions made in the theorem are stable under replacement of LaTeXMLMath by a finite unramified covering .
Let LaTeXMLMath be the elliptic fibration induced by LaTeXMLMath .
We denote by LaTeXMLMath a general fiber of LaTeXMLMath and set as in lemma 3 LaTeXMLMath , LaTeXMLMath , let LaTeXMLMath be the scheme theoretic fiber , and LaTeXMLMath .
Note that LaTeXMLMath is not empty for each LaTeXMLMath because the intersection product on the fibre is not negative definite .
Lemma 3 gives us an exact sequence LaTeXMLEquation .
We subdivide our analysis into two cases : In case ( 1 ) , we take the unramified covering of LaTeXMLMath associated with the epimorphism onto LaTeXMLMath .
We thus get another elliptic normal K3 surface since the minimal resolution has the canonical divisor trivial and the first Betti number LaTeXMLMath ( by the classification theorem , either LaTeXMLMath and we have a torus , or LaTeXMLMath and we have a K3 surface ) , and we are reduced to the case where LaTeXMLMath is trivial .
In this case it is immediate to see that , the fundamental group LaTeXMLMath being abelian , the fundamental group LaTeXMLMath is the quotient of LaTeXMLMath by the subgroup generated by the images of LaTeXMLMath , LaTeXMLMath being the identity matrix and LaTeXMLMath being the local monodromy matrix around the point LaTeXMLMath .
However , this quotient equals precisely the fundamental group of the complete smooth surface LaTeXMLMath , which is a K3 surface .
But then LaTeXMLMath is simply connected and we have proven our assertion .
In case ( 2 ) , we use that LaTeXMLMath is the covering group of a non compact simply connected Riemann surface LaTeXMLMath branched over LaTeXMLMath with branching locus equal to LaTeXMLMath and branching multiplicities LaTeXMLMath ( i.e. , LaTeXMLMath acts on LaTeXMLMath with quotient LaTeXMLMath ) .
If the Riemann surface LaTeXMLMath were the disk then we would get a Fuchsian group and there is a normal subgroup LaTeXMLMath of finite index acting freely on LaTeXMLMath , whence a finite Galois covering LaTeXMLMath where LaTeXMLMath has genus at least LaTeXMLMath .
The epimorphism LaTeXMLMath yields an unramified covering of LaTeXMLMath which compactifies to a smooth surface with trivial canonical bundle .
But such a surface admits no nontrivial holomorphic map to a curve of genus LaTeXMLMath , thus we conclude that LaTeXMLMath is the affine line .
By considering again a normal subgroup LaTeXMLMath of finite index acting freely on LaTeXMLMath , we get an elliptic curve LaTeXMLMath and a Galois covering LaTeXMLMath .
Taking the normalization of the fibre product we obtain again an unramified covering of LaTeXMLMath which compactifies to a surface LaTeXMLMath with trivial canonical bundle .
Since we get a map of LaTeXMLMath to an elliptic curve , LaTeXMLMath is a torus ( observe that LaTeXMLMath is minimal ) , what proves our claims .
∎ If LaTeXMLMath is residually finite for all LaTeXMLMath with LaTeXMLMath , then the Conjecture is affirmative .
The case LaTeXMLMath is covered by [ KT ] .
The remaining case follows from the assumption together with the uniform boundedness of the order of finite automorphism groups of K3 surfaces .
Here the uniform boundedness is a consequence of the global Torelli Theorem together with the Burnside property of finite groups of LaTeXMLMath .
In the algebraic case , one may use the result of Mukai [ Mu ] instead .
∎
Journal of Knot Theory and Its Ramifications ©World Scientific Publishing Company 1 IMPOSSIBILITY OF OBTAINING SPLIT LINKS FROM SPLIT LINKS VIA TWISTINGS MAKOTO OZAWA Department of Mathematics , School of Education , Waseda University , Nishiwaseda 1-6-1 , Shinjuku-ku , Tokyo 169-8050 , Japan ozawa @ musubime.com ABSTRACT We show that if a split link is obtained from a split link LaTeXMLMath in LaTeXMLMath by LaTeXMLMath -Dehn surgery along a trivial knot LaTeXMLMath , then the link LaTeXMLMath is splittable .
That is to say , it is impossible to obtain a split link from a split link via a non-trivial twisting .
As its corollary , we completely determine when a trivial link is obtained from a trivial link via a twisting .
Let LaTeXMLMath be a link in LaTeXMLMath and LaTeXMLMath a trivial knot in LaTeXMLMath missing LaTeXMLMath .
Then we can get a new link LaTeXMLMath in LaTeXMLMath as the image of LaTeXMLMath after doing LaTeXMLMath -Dehn surgery along LaTeXMLMath .
We say that LaTeXMLMath is obtained from LaTeXMLMath by an LaTeXMLMath -twisting along LaTeXMLMath .
In this paper , we consider the following problem .
Problem .
Is it possible that both LaTeXMLMath and LaTeXMLMath are splittable ?
For this problem , it is reasonable to make the following definition .
An LaTeXMLMath -twisting is said to be non-trivial if LaTeXMLMath and the link LaTeXMLMath is non-splittable .
Then our result is stated as follows .
It is impossible to obtain a split link from a split link by a non-trivial twisting .
Next , we consider when a trivial link is obtained from a trivial link by an LaTeXMLMath -twisting .
For a trivial knot , this problem has been solved as follows .
LaTeXMLMath LaTeXMLCite , LaTeXMLCite LaTeXMLMath Suppose that a trivial knot LaTeXMLMath is obtained from a trivial knot LaTeXMLMath by an LaTeXMLMath -twisting along LaTeXMLMath .
Then one of the following conclusions holds .
( 1 ) The link LaTeXMLMath is a trivial link .
( 2 ) The link LaTeXMLMath is a Hopf link .
( 3 ) The link LaTeXMLMath is a torus link of type LaTeXMLMath or LaTeXMLMath , and LaTeXMLMath or LaTeXMLMath respectively .
By Theorems 1 and 2 , we obtain the next corollary .
Suppose that a trivial link LaTeXMLMath is obtained from a trivial link LaTeXMLMath by an LaTeXMLMath -twisting along LaTeXMLMath .
Then one of the following conclusions holds .
( 1 ) The link LaTeXMLMath is a trivial link .
( 2 ) The link LaTeXMLMath is a split union of a Hopf link LaTeXMLMath and a trivial link LaTeXMLMath for some LaTeXMLMath ( 3 ) The link LaTeXMLMath is a split union of a torus link LaTeXMLMath of type LaTeXMLMath or LaTeXMLMath and a trivial link LaTeXMLMath for some LaTeXMLMath , and LaTeXMLMath or LaTeXMLMath respectively .
In this section , we prepare some lemmas for Theorem 1 .
All manifolds are assumed to be compact and orientable , and any srufaces in a 3-manifold are assumed to be properly embedded and in general position .
Let LaTeXMLMath be a 3-manifold , and LaTeXMLMath and LaTeXMLMath two surfaces in LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath be the closed surfaces obtained by capping off LaTeXMLMath , LaTeXMLMath with disks .
Then , for LaTeXMLMath , one defines a graph LaTeXMLMath in LaTeXMLMath , where the edges of LaTeXMLMath correspond to the arc components of LaTeXMLMath , and the vertices to the components of LaTeXMLMath .
Recall that a 1-sided face in a graph is a disk face with exactry one edge in its boundary .
Recall that if LaTeXMLMath is a 3-mainfold with torus boundary and LaTeXMLMath is a slope on LaTeXMLMath , then LaTeXMLMath denotes the closed maifold obtained by attaching a solid torus LaTeXMLMath to LaTeXMLMath so that the boundary of a meridian disk of LaTeXMLMath has slope LaTeXMLMath on LaTeXMLMath .
Recall also that if LaTeXMLMath are two slopes on LaTeXMLMath , then LaTeXMLMath denotes the minimal geometric intersection number of LaTeXMLMath and LaTeXMLMath .
The following lemma will be needed for Theorem 1 .
Let LaTeXMLMath be a 3-manifold with torus boundary and let LaTeXMLMath be planar surfaces in LaTeXMLMath with boundary slopes LaTeXMLMath .
Suppose that the graphs LaTeXMLMath contain no 1-sided faces , and that LaTeXMLMath .
Then either the first homology groups LaTeXMLMath or LaTeXMLMath has a torsion .
Proof .
If LaTeXMLMath , then Lemma 1 follows LaTeXMLCite .
Otherwise , by LaTeXMLCite , LaTeXMLMath contains a Scharlemann cycle or LaTeXMLMath represents all LaTeXMLMath -types .
In the formar case , LaTeXMLMath has a lens space as a connected summand .
In the latter case , by LaTeXMLCite , LaTeXMLMath has a torsion .
This completes the proof of Lemma 2 .
Suppose that a split link LaTeXMLMath is obtained from a split link LaTeXMLMath by a non-trivial LaTeXMLMath -twisting along LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the splitting spheres for LaTeXMLMath and LaTeXMLMath respectively .
Put LaTeXMLMath .
We may assume that LaTeXMLMath intersects LaTeXMLMath and LaTeXMLMath transversely in the 3-spheres LaTeXMLMath and LaTeXMLMath respectively , and assume that LaTeXMLMath and LaTeXMLMath are minimal among all 2-spheres isotopic to LaTeXMLMath and LaTeXMLMath respectively .
Then , since LaTeXMLMath and LaTeXMLMath are non-splittable , LaTeXMLMath and LaTeXMLMath are not equal to zero .
Put LaTeXMLMath and LaTeXMLMath .
Then by the minimality of LaTeXMLMath and LaTeXMLMath and by the irreducibility of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath satisfy the hypothesis of Lemma 2 .
Hence LaTeXMLMath or LaTeXMLMath has a torsion , this is impossible .
Acknowledgement The author would like to thank Prof. Chuichiro Hayashi for his helpful comments .
For a discrete group LaTeXMLMath there are two well-known completions .
The first is the Malcev ( or unipotent ) completion .
This is a prounipotent group LaTeXMLMath , defined over LaTeXMLMath , together with a homomorphism LaTeXMLMath that is universal among maps from LaTeXMLMath into prounipotent LaTeXMLMath -groups .
To construct LaTeXMLMath , it suffices to consider the case where LaTeXMLMath is nilpotent ; the general case is handled by taking the inverse limit of the Malcev completions of the LaTeXMLMath , where LaTeXMLMath denotes the lower central series of LaTeXMLMath .
If LaTeXMLMath is abelian , then LaTeXMLMath .
We review this construction in Section LaTeXMLRef .
The second completion of LaTeXMLMath is the LaTeXMLMath -completion .
For a prime LaTeXMLMath , we set LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath -lower central series of LaTeXMLMath .
If LaTeXMLMath is a finitely generated abelian group , then LaTeXMLMath , where LaTeXMLMath is the ring of LaTeXMLMath -adic integers LaTeXMLCite .
The group LaTeXMLMath is a pro- LaTeXMLMath -group and each LaTeXMLMath is nilpotent provided LaTeXMLMath is finite dimensional .
Both of these completions are instances of a general construction .
Let LaTeXMLMath be a field .
The unipotent LaTeXMLMath -completion of a group LaTeXMLMath is a prounipotent LaTeXMLMath -group LaTeXMLMath together with a homomorphism LaTeXMLMath .
The group LaTeXMLMath is required to satisfy the obvious universal mapping property .
The Malcev completion is the case LaTeXMLMath and the LaTeXMLMath -completion is the case LaTeXMLMath .
This construction for other fields LaTeXMLMath is probably well-known to the experts , but it does not seem to be in the literature .
One reason to study such completions is that they may be used to gain cohomological information about the groups LaTeXMLMath and LaTeXMLMath .
Indeed , the restriction map LaTeXMLMath is injective ( the definition of LaTeXMLMath will be recalled below ) .
This allows one to obtain either a lower bound for LaTeXMLMath or an upper bound for LaTeXMLMath .
Unfortunately , the group LaTeXMLMath may be trivial ( e.g. , LaTeXMLMath perfect , or more generally , if LaTeXMLMath ) .
To circumvent this , Deligne suggested the notion of relative completion .
Suppose that LaTeXMLMath is a representation of LaTeXMLMath in a reductive group LaTeXMLMath defined over LaTeXMLMath .
Assume that the image of LaTeXMLMath is Zariski dense .
The completion of LaTeXMLMath relative to LaTeXMLMath is a proalgebraic group LaTeXMLMath over LaTeXMLMath which is an extension of LaTeXMLMath by a prounipotent group LaTeXMLMath together with a map LaTeXMLMath .
The group LaTeXMLMath should satisfy the obvious universal mapping property .
The basic theory of relative completion in characteristic zero was worked out by R. Hain LaTeXMLCite .
Many of his results remain valid in positive characteristic .
We shall review this in Section LaTeXMLRef below .
For examples of relative completions in characteristic zero beyond those presented here , the reader is referred to Hain ’ s study LaTeXMLCite of the completion of the mapping class group LaTeXMLMath of a surface LaTeXMLMath of genus LaTeXMLMath with LaTeXMLMath marked points relative to its symplectic representation as the group of automorphisms of LaTeXMLMath .
Other examples , due to the author LaTeXMLCite , are the groups LaTeXMLMath and LaTeXMLMath relative to their obvious representations in LaTeXMLMath .
Recent work by Hain and M. Matsumoto LaTeXMLCite tackles the absolute Galois group LaTeXMLMath relative to the cyclotomic character LaTeXMLMath .
In this paper we study the completions of groups such as LaTeXMLMath , where LaTeXMLMath is the local ring of a closed point LaTeXMLMath on a smooth affine curve LaTeXMLMath .
We also study the completion of LaTeXMLMath relative to the reduction map LaTeXMLMath .
We use these completions to compute the second continuous cohomology of special linear groups over complete local rings .
A sample result is the following .
Theorem LaTeXMLRef .
If LaTeXMLMath is a finite field or a number field , then for all LaTeXMLMath , LaTeXMLMath .
As far as the author knows , this is the first calculation of continuous cohomology with coefficients of the same characteristic as LaTeXMLMath .
This paper is organized as follows .
In Section LaTeXMLRef , we review Hain ’ s construction of the completion of LaTeXMLMath .
In Section LaTeXMLRef , we discuss in detail the case where LaTeXMLMath is the trivial group ; this is the unipotent completion mentioned above .
In Section LaTeXMLRef , we present several examples of unipotent completions .
Section LaTeXMLRef deals with the basic theory of relative completion and the computation of examples .
Finally , in Section LaTeXMLRef we carry out some cohomology calculations .
Acknowledgements .
I would like to thank Dick Hain and Chuck Weibel for many useful discussions .
I am also grateful to the referee for insisting that I not omit so many details .
The following construction is due to R. Hain LaTeXMLCite .
Let LaTeXMLMath be a group and suppose that LaTeXMLMath is a Zariski dense representation in a reductive LaTeXMLMath -group LaTeXMLMath .
Consider all commutative diagrams of the form LaTeXMLEquation
Riemann zeta function is an important object of number theory .
It was also used for description of disordered systems in statistical mechanics .
We show that Riemann zeta function is also useful for the description of integrable model .
We study XXX Heisenberg spin 1/2 anti-ferromagnet .
We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit .
We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments .
Quantum Spin Chains and Riemann Zeta Function with Odd Arguments a H.E .
Boos Institute for High Energy Physics Protvino , 142284 , Russia a a V.E .
Korepin C.N .
Yang Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook , NY 11794–3840 , USA Riemann zeta function for LaTeXMLMath can be defined as follows : LaTeXMLEquation .
It also can be represented as a product with respect to all prime numbers LaTeXMLMath LaTeXMLEquation .
It can be analytically continued in the whole complex plane of LaTeXMLMath .
It has only one pole , at LaTeXMLMath and it has ’ trivial ’ zeros at LaTeXMLMath ( LaTeXMLMath is an integer ) .
The famous Riemann hypothesis LaTeXMLCite states that nontrivial zeros belong to the straight line LaTeXMLMath .
Riemann zeta function is useful for study of distribution of prime numbers on the real axis LaTeXMLCite .
The values of Riemann zeta function at special points were studied in LaTeXMLCite , LaTeXMLCite .
At even values of its argument zeta function can be expressed in terms of powers of LaTeXMLMath .
The values of Riemann zeta function at odd arguments provide infinitely many different irrational numbers LaTeXMLCite .
Riemann zeta function plays an important role , not only in pure mathematics but also theoretical physics .
Some Feynman diagrams in quantum field theory can be expressed in terms of LaTeXMLMath , see , for example , LaTeXMLCite .
It appears also in string theory LaTeXMLCite .
In statistical mechanics Riemann zeta function was used for the description of chaotic systems .
This is large field with many publications .
Important contributions to this field were made by Berry , Connes , Julia , Kac , Keating , Knauf , Odlyzko , Pitkanen , Polya , Ruelle , Sarnak and Zagier .
One can find more information and citation on the following web cite http : //www.maths.ex.ac.uk/ mwatkins/ .
We argue that LaTeXMLMath is also important for exactly solvable models .
One of the most famous integrable models is the Heisenberg XXX spin chain .
This model was first suggested by Heisenberg LaTeXMLCite in 1928 and solved by Bethe LaTeXMLCite in 1931 .
Since that time it found multiple applications in solid state physics and statistical mechanics .
Recently the XXX spin chain was used for study of the entanglement in quantum computations LaTeXMLCite .
The Hamiltonian of the XXX spin chain can be written like this LaTeXMLEquation .
Here LaTeXMLMath is the length of the lattice and LaTeXMLMath are Pauli matrices .
We consider thermodynamics limit , when LaTeXMLMath goes to infinity .
The sign in front of the Hamiltonian indicates that we are considering the anti-ferromagnetic case .
We consider periodic boundary conditions .
Notice that this Hamiltonian annihilates the ferromagnetic state [ all spins up ] .
The construction of the anti-ferromagnetic ground state wave function LaTeXMLMath can be credited to Hulthén LaTeXMLCite .
An important correlation function was defined in LaTeXMLCite .
It was called the emptiness formation probability LaTeXMLEquation where LaTeXMLMath is a projector on the state with spin up in LaTeXMLMath th lattice site .
Averaging is over the anti-ferromagnetic ground state .
It describes the probability of formation of a ferromagnetic string of the length LaTeXMLMath in the anti-ferromagnetic background LaTeXMLMath .
In this paper we shall first study short strings ( LaTeXMLMath is small ) , in the end we shall discuss long distance asymptotics ( at finite temperature ) .
The four first values of the emptiness-formation probability look as follows : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Let us comment .
The value of LaTeXMLMath is evident from the symmetry , LaTeXMLMath can be extracted from the explicit expression of the ground state energy LaTeXMLCite .
LaTeXMLMath can be extracted from the results of M.Takahashi LaTeXMLCite on the calculation of the nearest neighbor correlation .
It was confirmed in paper LaTeXMLCite .
One should also mention independent calculation of LaTeXMLMath in LaTeXMLCite .
One can express LaTeXMLMath in terms of next to the nearest neighbor correlation LaTeXMLEquation .
The calculation of LaTeXMLMath and LaTeXMLMath is discussed in this paper .
The expression above for LaTeXMLMath is our main result here .
The plan of the paper is as follows .
In the next section we discuss some main steps of the calculation of LaTeXMLMath and LaTeXMLMath .
The thermodynamics of LaTeXMLMath for the non-zero temperature is briefly discussed in section 3 .
Then we summarize the results in the conclusion .
There are several different approaches to investigate LaTeXMLMath : representation of correlation functions as determinants of Fredholm integral operators described in detail in the book LaTeXMLCite the vertex operator approach developed by the RIMS group LaTeXMLCite One can also mention the application of connection with other correlation functions , for instance , the correlation function LaTeXMLMath .
We shall use the integral representation obtained by Korepin , Izergin , Essler and Uglov LaTeXMLCite in framework of the vertex operator approach at the zero magnetic field : LaTeXMLEquation .
The contour LaTeXMLMath in each integral goes parallel to the real axis with the imaginary part between LaTeXMLMath and LaTeXMLMath .
Recently such formula was generalized by de Gier and Korepin in paper LaTeXMLCite to the case , where averaging is done over arbitrary Bethe state [ with no strings ] instead of anti-ferromagnetic state .
Let us describe in general the strategy we used in order to come to the answers ( LaTeXMLRef ) and ( LaTeXMLRef ) .
The integral formula ( LaTeXMLRef ) can be easily represented as follows : LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
As appeared we can make a lot of simplifications without taking integrals but using some simple observations .
First of all , let us note that the function LaTeXMLMath is antisymmetric in respect to transposition of any pair of integration variables , say , LaTeXMLMath and LaTeXMLMath .
This simple observation turns out to be very useful because LaTeXMLEquation if the function LaTeXMLMath is symmetric for at least one pair of LaTeXMLMath -s. The next observation is also trivial , namely , we can try to reduce the power of denominator in ( LaTeXMLRef ) using simple algebraic relations like LaTeXMLEquation .
Combining these two simple observations one can reduce integration functions for LaTeXMLMath to a sum of terms with denominators of power 2 and for LaTeXMLMath to a more complicated sum of terms with denominators of power not higher than 3 .
In order to calculate the integrals one can close the contours in the complex plane by the infinite semi-circles either in upper half-plane or in the lower half-plane not changing the integrals .
Then it is possible to apply Cauchy theorem using the following formulae LaTeXMLEquation .
LaTeXMLEquation for the cases LaTeXMLMath and LaTeXMLMath respectively where LaTeXMLMath is a small contour surrounding the point LaTeXMLMath with an integer LaTeXMLMath in anti-clockwise direction .
Then the integrals can be expressed in terms of the differential operator acting on some functions .
For instance , for the case LaTeXMLMath LaTeXMLEquation where LaTeXMLMath is the differential operator LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
Here all three contours were closed in the upper half-plane but in real calculations it turns out to be more convenient to close some of them in another direction taking into consideration appearance of an additional sign .
It is not difficult to get generalization of these formulae to the case LaTeXMLMath .
So the problem is reduced to the calculation of sums like ( LaTeXMLRef ) , expanding the result into the series in powers of LaTeXMLMath -s and applying the differential operator LaTeXMLMath .
This procedure is straightforward but can be rather tedious especially for the case LaTeXMLMath .
Proceeding in this way we can come to the results ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Let us note that both of these final answers appeared to be expressed in terms of the logarithmic function and the Riemann zeta function of odd arguments and do not depend on polylogarithms in spite of the fact that polylogarithm LaTeXMLMath appeared in the intermediate stage of calculation .
All coefficients before those functions in ( LaTeXMLRef - LaTeXMLRef ) are rational .
Also they do not contain any powers of LaTeXMLMath which could be considered as Riemann zeta functions of even arguments .
Our conjecture is that the final answer for any LaTeXMLMath will also be expressed in terms of logarithm LaTeXMLMath and Riemann zeta functions LaTeXMLMath with odd integers LaTeXMLMath and with rational coefficients .
If we had the exact answer for LaTeXMLMath for any LaTeXMLMath we could calculate an asymptotics of LaTeXMLMath when LaTeXMLMath tends to infinity .
Unfortunately , for a moment we can not do this because we have LaTeXMLMath only for LaTeXMLMath .
Nevertheless we can discuss a possible behavior of LaTeXMLMath with LaTeXMLMath using some other arguments .
For non-zero temperature one can conclude that the asymptotics of the partition function in thermodynamic limit is as follows LaTeXMLEquation where LaTeXMLMath is the free energy per site and LaTeXMLMath is the length of the chain , it was evaluated in LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
In fact , for LaTeXMLMath the LaTeXMLMath neighboring spins are frozen .
Therefore one has the asymptotics of LaTeXMLMath when LaTeXMLMath tends to infinity LaTeXMLEquation .
For zero temperature we expect Gaussian decay .
We think that our work provide a link between integrable models and chaotic models .
The same mathematical apparatus appears in the description of both kind of models .
Let us repeat that the main result of this paper is the calculation of LaTeXMLMath and LaTeXMLMath ( LaTeXMLRef - LaTeXMLRef ) by means of the multi-integral representation ( LaTeXMLRef ) .
The fact that only the logarithm LaTeXMLMath and Riemann zeta function with odd arguments participate in the answers for LaTeXMLMath and with rational coefficients before these functions allows us to suppose that this is the general property of LaTeXMLMath .
One could compare the calculation of LaTeXMLMath with the many-loop calculation of the self-energy diagrams in the renormalizable quantum field theory which can also be expressed in terms of LaTeXMLMath functions of odd arguments LaTeXMLCite .
Unfortunately , so far we have not got even a conjecture for LaTeXMLMath but we believe that it is not an unsolvable problem .
May be already after calculation of LaTeXMLMath one could guess the right formula for a generic case LaTeXMLMath .
It would give an answer to the question discussed in the previous section , namely , the question about the law of decay of LaTeXMLMath when LaTeXMLMath tends to infinity .
Also it would be interesting to generalize above results to the XXZ spin chain .
Some interesting conjectures were recently invented by Razumov and Stroganov LaTeXMLCite for the special case of the XXZ model with LaTeXMLMath .
These conjectures would be supported if it were possible to get LaTeXMLMath from the general integral representation obtained by the RIMS group LaTeXMLCite .
The authors would like to thank A. Kirillov , B. McCoy , A. Razumov , M. Shiroishi , Yu .
Stroganov , M. Takahashi , L.Takhtajan and V. Tarasov for useful discussions .
This research has been supported by the NSF grant PHY-9988566 and by INTAS Grant no .
01-561 .
We show that representations of the group of spacetime diffeomorphisms and the Dirac algebra both arise in a phase-space histories version of canonical general relativity .
This is the general-relativistic analogue of the novel time structure introduced previously in history theory : namely , the existence in non-relativistic physics of two types of time translation ; and the existence in relativistic field theory of two distinct Poincaré groups .
The ‘ problem of time ’ in quantum gravity takes on a different form according to the approach to quantum gravity that one adopts .
However , in all cases , this question of the status of the notion of time is a fundamental one ; indeed , it is true in all cases that we need a better understanding of conceptual issues concerning the nature of space and time .
In light of the recent developments on the distinction that has been made between time as a causal ordering parameter , and time as an evolution parameter in dynamics LaTeXMLCite , the main goal of the present paper is to show how the application of history ideas to general relativity opens up a novel way of viewing that subject ; and hence ultimately to a completely new way of tackling the quantisation of gravity .
In what follows , we apply the ideas of classical history theory LaTeXMLCite to the general theory of relativity .
A preliminary step in this direction was the application of the history methods to parametrised systems ( often used as simple models for general relativity ) .
Thus , in LaTeXMLCite we studied the quantisation of constrained systems using the continuous-time histories scheme .
In particular , the existence of the two times in a history version of parametrised systems was exploited to show the existence of an intrinsic time that does not disappear when the constraints are enforced , either classically or quantum mechanically .
Hence this provides a solution to the ‘ problem of time ’ for systems of this type .
This work is a natural precursor for dealing with the problem of time as it appears in canonical quantum gravity .
In the context of general relativity , we start by considering a Lorentzian geometry on a spacetime LaTeXMLMath as being equivalent to a history of Riemannian metrics on the three-manifold LaTeXMLMath .
Thus we consider paths LaTeXMLMath of Riemannian metrics , which together with the paths LaTeXMLMath of conjugate momenta , are postulated to form the fundamental classical history algebra ( the history analogue of the normal canonical Poisson brackets ) LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we define LaTeXMLMath , and where LaTeXMLMath is some strictly positive scalar density of weight -1 in the variable LaTeXMLMath .
In the standard approach to canonical general relativity , the relation between the spacetime diffeomorphisms algebra and the Dirac constraint algebra has long been an important matter for discussion LaTeXMLCite .
Therefore , it is of considerable significance that in this new construction the two algebras appear together for the first time in a completely natural way : specifically , as we shall show , the history theory contains a representation of both the spacetime diffeomorphisms group and the Dirac algebra of constraints of the canonical theory .
In particular , we augment the history space of the canonical general relativity treatment , to define appropriate covariant Poisson brackets .
A key result in this respect is the observation that for each vector field LaTeXMLMath on LaTeXMLMath , the ‘ Liouville ’ function LaTeXMLMath that is defined as LaTeXMLEquation satisfies the Lie algebra of the group of spacetime diffeomorphisms LaTeXMLEquation for all spacetime vector fields LaTeXMLMath , and where LaTeXMLMath denotes their commutator .
This is a very significant result since it implies that in this history theory there is a central role for spacetime concepts , whereas the canonical approaches to general relativity are dominated by spatial ideas .
Furthermore , it makes particularly clear how the distinction between the Dirac constraint algebra and the spacetime diffeomorphisms group arises as a facet of the non-trivial temporal structure of the histories description .
In section 2 we present the basic ideas of the histories temporal structure .
Of particular importance is the distinction between the aspects of the concept of time as ( i ) an ordering parameter , and ( ii ) an evolution parameter .
This is realised mathematically with the construction of two distinct generators of time translations .
Furthermore , the definition of the action operator S —which was proved to be the generator of both types of time translations—nicely intertwines the two modes of time LaTeXMLCite .
We briefly present the histories classical non-relativistic physics and relativistic field theory .
We especially emphasise the existence of two Poincaré groups as the analogue of the two types of time translation in non-relativistic physics .
LaTeXMLCite .
The histories theory of parameterised systems is a natural precursor for the study of histories general relativity theory .
We recall how the existence of the two modes of time for such systems has as a result an ‘ intrinsic time ’ that does not disappear when enforcing the constraints LaTeXMLCite .
This provides a solution to the analogue for the parametrised particle of the famous ‘ problem of time ’ in canonical quantum gravity .
In section 3 we first present the structure of the history version of canonical general relativity theory .
We explicitly write the analogue of the Dirac algebra of constraints .
We then show that—after appropriately augmenting the history space of canonical histories—there exists a representation of the group of spacetime diffeomorphisms .
This novel result is a direct analogue of the two Poincaré groups in relativistic field theory : it is therefore grounded in the distinction between the two aspects of time .
Next we explicitly write the extended canonical history algebra and we show that the original history state space is the state space of the standard LaTeXMLMath decomposition .
Finally we apply the history ideas for the treatment of parameterised systems and we show that , of all spacetime diffeomorphisms it is only the generators of time reparameterisations— i.e .
the ones related to the Liouville function LaTeXMLMath —that are still defined on the histories reduced state space .
In recent years , it has become better understood that the problem of quantum gravity—and especially the way in which time might appear in such a theory—suggests the need for a new form of quantum theory : in particular , one where the notion of ‘ time ’ is introduced in some novel way .
One such formalism is the ‘ HPO ’ ( ‘ History Projection Operator ’ ) approach to a quantum history theory .
Although the programme originated from the consistent histories theory , as formulated initially by Griffiths , Omnès , and Gell-Mann and Hartle LaTeXMLCite , it was developed so that the logical structure of the histories theory was recovered ; in particular it introduced a ‘ temporal ’ logic of the theory LaTeXMLCite .
The HPO theory takes a completely different turn in the way the concept of time was introduced in LaTeXMLCite .
The consistent histories formalism was developed to deal with closed systems .
A history LaTeXMLMath is defined to be a collection of projection operators LaTeXMLMath , LaTeXMLMath , each of which represents a property of the system at the single time LaTeXMLMath .
Therefore , the emphasis is placed on histories , rather than properties at a single time , which in turn gives rise to the possibility of generalized histories with novel concepts of time .
The History Projection Operator approach , developed originally by Isham LaTeXMLCite , and Isham and Linden LaTeXMLCite , is an approach to the consistent histories formalism that places emphasis on temporal logic .
This is achieved by representing the history LaTeXMLMath as the operator LaTeXMLMath which is a genuine projection operator on the tensor product LaTeXMLMath of copies of the standard Hilbert space LaTeXMLMath .
Note that to use this construction in any type of field theory requires an extension to a continuous time label , and hence to an appropriate definition of the continuous tensor product LaTeXMLMath .
This has been done successfully for non-relativistic particle physics LaTeXMLCite , and relativistic quantum field theory LaTeXMLCite .
A central feature of the HPO histories theory is the development of the novel temporal structure that was introduced in LaTeXMLCite .
Specifically , it was shown that there exist two distinct types of time transformation , each of which represents a distinct quality , or mode , of the concept of time .
The first such mode corresponds to time considered purely as a kinematical parameter of a physical system , with respect to which a history is defined as a succession of possible events .
It is strongly connected with the temporal-logical structure of the theory and it is related to the view of time as a parameter that determines the ordering of events .
The second mode corresponds to the dynamical evolution generated by the Hamiltonian .
Classically , these two ways of considering time are nicely intertwined through the histories analogue of the action principle which provides the paths that are solutions to the classical equations of motion .
A main result of the theory is that physical quantities appear naturally time-averaged .
Hence these new ideas on the concept of time have as a consequence that observables admit two different time labels : ( i ) a time parameter LaTeXMLMath which corresponds to the ‘ external ’ time that labels events at different moments of time , and with respect to which the time averages are taken ; and ( ii ) a time parameter LaTeXMLMath which corresponds to the ‘ internal ’ time that appears as the evolution parameter for a fixed external time LaTeXMLMath .
In the corresponding quantum theory , the Hamiltonian LaTeXMLMath means the history quantity that is the time-averaged energy of the system .
operator LaTeXMLMath , and the ‘ Liouville ’ operator LaTeXMLMath are the generators of the two types of time transformation LaTeXMLCite .
Specifically , the Hamiltonian LaTeXMLMath is the generator of the unitary time evolution with respect to the ‘ internal ’ time label LaTeXMLMath ; this has no effect on the ‘ external ’ time label LaTeXMLMath .
On the other hand , the Liouville operator LaTeXMLMath —defined in analogy to the kinematical part of the classical action functional—generates time translations along the LaTeXMLMath -time axis without affecting the LaTeXMLMath -label .
The key feature of the ensuing temporal structure , however , is the definition of the action operator LaTeXMLMath as a quantum analogue of the classical action functional : LaTeXMLEquation .
It transpires that the action operator LaTeXMLMath generates both types of time transformation , and in this sense it is the generator of physical time translations in the HPO formalism .
The time transformations generated by the action operator LaTeXMLMath resemble the canonical transformations generated by the Hamilton-Jacobi action functional .
In this sense , there is an interesting relation between the definition of LaTeXMLMath and the well-known work by Dirac on the Lagrangian theory for quantum mechanics LaTeXMLCite .
In particular , motivated by the fact that—contrary to the Hamiltonian method—the Lagrangian method can be expressed relativistically ( on account of the action function being a relativistic invariant ) , Dirac tried to take over the general LaTeXMLMath of the classical Lagrangian theory , albeit not the equations of the Lagrangian theory per se .
Recently , these ideas have been applied in various theories , with some intriguing results .
For example , the temporal structure of HPO histories enables us to treat parameterised systems in such a way that the problem of time does not arise LaTeXMLCite .
Indeed , histories keep their intrinsic temporality after the implementation of the constraint : thus there is no uncertainty about the temporal-ordering properties of the physical system .
In relativistic quantum field theory , the analogue of the two types of time transformation is the existence of two groups of Poincaré transformations LaTeXMLCite .
It transpires that different representations of the theory—that correspond to different choices of foliation—can all be defined on the same Hilbert space , and they are related by transformations generated by the ‘ external ’ Poincaré group .
As we shall see in what follows , the histories description of general relativity blends together the structure of the two systems referred above : namely , the parameterised systems and the relativistic field .
In the histories formalism for Newtonian classical mechanics , the space of classical histories LaTeXMLMath is the set of all smooth paths on the classical state space LaTeXMLMath .
It can be equipped with a natural symplectic structure , which gives rise to the Poisson bracket LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation and similarly for LaTeXMLMath .
The classical analogue of the Liouville operator is defined as LaTeXMLEquation and the Hamiltonian ( i.e .
, time-averaged energy ) function LaTeXMLMath is defined as LaTeXMLEquation where LaTeXMLMath is the Hamiltonian that is associated with the copy LaTeXMLMath of the normal classical state space with the same time label LaTeXMLMath .
The temporal structure leads to the histories analogue of the classical equations of motion LaTeXMLEquation where LaTeXMLMath is any function on LaTeXMLMath , and where the path LaTeXMLMath is a solution of the equations of motion .
A crucial result therefore is that , the history equivalent of the classical equations of motion is given by the following condition that holds for all functions LaTeXMLMath on LaTeXMLMath when LaTeXMLMath is a classical solution : LaTeXMLEquation where LaTeXMLEquation is the classical analogue of the action operator .
This is the history analogue of the least action principle LaTeXMLCite .
A natural precursor to general relativity is the theory of parameterised systems .
Such systems have a vanishing Hamiltonian LaTeXMLMath , when the constraints are imposed .
Classically this implies that two points of the constraint surface LaTeXMLMath correspond to the same physical state ; hence the true degrees of freedom are represented by points in the reduced state space LaTeXMLMath LaTeXMLEquation .
An element of the reduced state space is itself a solution to the classical equations of motion ; on the other hand , a point in state space also corresponds to a possible configuration of the physical system at an instant of time .
Hence the notion of time is unclear : in particular , it is not obvious how to recover the notion of temporal ordering unless we choose to arbitrarily impose a gauge-fixing condition .
In the histories approach to parameterised systems , the history constraint surface LaTeXMLMath is defined as LaTeXMLMath —the set of all smooth paths from the real line to the constraint surface LaTeXMLMath .
The history Hamiltonian constraint is defined by LaTeXMLMath , where LaTeXMLMath is first-class constraint .
For all values of the smearing function LaTeXMLMath the history Hamiltonian constraint LaTeXMLMath generates canonical transformations on the history constraint surface LaTeXMLMath .
The history reduced state space LaTeXMLMath is then defined as LaTeXMLMath —the set of all smooth paths on the canonical reduced state space LaTeXMLMath : it is identical to the space of orbits of LaTeXMLMath on LaTeXMLMath .
The novel result here is that , contrary to what is the case for existing treatments of parameterised systems , the classical equations of motion can be explicitly realised on the reduced state space LaTeXMLMath .
They are given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are respectively the action and Liouville functions projected on LaTeXMLMath .
Both LaTeXMLMath and LaTeXMLMath commute weakly with the Hamiltonian constraint LaTeXMLCite .
Furthermore , the smeared form of the Liouville function LaTeXMLMath generates time reparameterisations on LaTeXMLMath , and it leaves invariant the classical equations of motion .
We write the history version of classical field theory for Minkowski spacetime , foliated with respect to a time-like vector LaTeXMLMath , that is normalised by LaTeXMLMath .
We shall take the signature of the Minkowski metric LaTeXMLMath to be LaTeXMLMath .
In the histories formalism of a scalar field , the space of state-space histories LaTeXMLMath is an appropriate subset of the continuous Cartesian product LaTeXMLMath of copies of the standard state space LaTeXMLMath , each labeled by the time parameter LaTeXMLMath .
The choice of LaTeXMLMath depends on the choice of a foliation vector LaTeXMLMath , hence the space of histories also has an implicit dependence on LaTeXMLMath and should therefore be written as LaTeXMLMath .
Furthermore , for each space-like surface LaTeXMLMath —defined with respect to its normal vector LaTeXMLMath , and labeled by the parameter LaTeXMLMath —we consider the state space LaTeXMLMath that is defined in such a way as to give the basic Poisson algebra relations of the history theory : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are spacetime points .
Note that a spacetime point LaTeXMLMath can be associated with the pair LaTeXMLMath as LaTeXMLMath , where the three-vector LaTeXMLMath has been associated with a corresponding four-vector LaTeXMLMath that is LaTeXMLMath -spatial ( i.e .
, LaTeXMLMath ) ; note that LaTeXMLMath .
We then define the action , Liouville and Hamiltonian functionals for the scalar field as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation respectively .
Here LaTeXMLMath , where LaTeXMLMath is the mass of the free field .
It can be shown that the variation of the action functional LaTeXMLMath leaves invariant the paths LaTeXMLMath that are the classical solutions of the system : LaTeXMLEquation .
LaTeXMLEquation For each copy LaTeXMLMath of the standard state space , there exists a Poincaré group , as one would expect in a canonical treatment of relativistic field theory .
On the other hand , in histories theory the state space LaTeXMLMath is , heuristically , the Cartesian product of such copies .
Hence , for each copy of the standard state space , labeled by a fixed value of LaTeXMLMath , there exists an ‘ internal ’ Poincaré group acting on the copy of standard canonical field theory , that is labeled with the same time label LaTeXMLMath .
However , the physical quantities in histories theory appear naturally time-averaged LaTeXMLCite , and hence a central role is played by a time-averaged form of these internal groups .
Of special interest is the action of the corresponding Hamiltonian LaTeXMLMath , and the boost generator LaTeXMLMath , on the field LaTeXMLMath .
In particular , we can define a classical , history analogue of the Heisenberg picture fields LaTeXMLMath LaTeXMLCite as LaTeXMLEquation or LaTeXMLEquation .
The action of boost transformations is best shown upon objects LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are related by a Lorentz boost .
In addition to these ‘ internal ’ Poincaré groups ( and the time-averaged version ) there exists an ‘ external ’ Poincaré group with the same space translations and rotations generators as those of the internal Poincaré group , but with different time translator and boosts .
In particular , the time-translation generator for the ‘ external ’ Poincaré group is the Liouville functional LaTeXMLMath LaTeXMLCite : LaTeXMLEquation .
The boost generator LaTeXMLMath generates Lorentz transformations LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the element of the Lorentz group parameterised by the boost parameter LaTeXMLMath .
Furthermore under the action of the external Poincaré group , the action functional LaTeXMLMath transforms as LaTeXMLEquation .
It can be shown from Eqs .
( LaTeXMLRef – LaTeXMLRef ) that the two types of boost transformation coincide for the classical solutions LaTeXMLMath LaTeXMLCite LaTeXMLEquation .
LaTeXMLEquation In order to apply the histories theory to general relativity two methods may be followed .
From the spacetime perspective advocated by Hartle LaTeXMLCite , a history is a Lorentzian metric .
On the other hand , from the canonical perspective , a history is a path in the space of Riemannian metrics on a fixed three-manifold LaTeXMLMath .
We shall start by following the latter approach here .
Another interesting way of formulating general relativity histories is a covariant-like treatment—similar to the one developed by Wald LaTeXMLCite —that provides a clarifying spacetime description of the theory .
This will be relevant in future work , where we study the change of foliation in a covariant description of histories theory .
The history space LaTeXMLMath for general relativity is a suitable subset of the Cartesian product LaTeXMLMath of copies of the classical general relativity state space LaTeXMLMath , labeled by a parameter LaTeXMLMath , with LaTeXMLMath .
Here LaTeXMLMath is a fixed three-manifold .
In particular , LaTeXMLMath , where LaTeXMLMath is the space of Riemannian metrics on LaTeXMLMath ; i.e .
, an element of LaTeXMLMath is a pair LaTeXMLMath .
A history is defined to be any smooth map LaTeXMLMath .
The history version of the canonical Poisson brackets is postulated—in accord to the histories ideas LaTeXMLCite , where the entries of the history algebra are defined as histories , i.e. , paths of an appropriate history space—to be LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we have defined LaTeXMLMath and where LaTeXMLMath is some strictly positive scalar density of weight -1 in the variable LaTeXMLMath .
The appearance of LaTeXMLMath on the right-hand side of the canonical Poisson brackets can be justified in the following way .
The quantity LaTeXMLMath is a density in the spatial variable LaTeXMLMath LaTeXMLCite , but a scalar in the parameter LaTeXMLMath .
This means that although it makes sense to put a LaTeXMLMath on the right hand side of the Poisson bracket Eq .
( LaTeXMLRef ) —where the insertion of the comma in the notation LaTeXMLMath indicates that it is a scalar in LaTeXMLMath but a density LaTeXMLMath , the quantity LaTeXMLMath is a tensor density on LaTeXMLMath of the appropriate weight , whereas LaTeXMLMath is just a tensor field .
of weight 1 in LaTeXMLMath —it would not be correct to add the term LaTeXMLMath which is a density in LaTeXMLMath .
Thus , if LaTeXMLMath is a test density in both variables LaTeXMLMath and LaTeXMLMath , and if LaTeXMLMath is a density in LaTeXMLMath but a function in LaTeXMLMath , then the smeared version of Eq .
( LaTeXMLRef ) is LaTeXMLEquation .
We shall discuss next the physical meaning of the quantity LaTeXMLMath .
We note however that it can be regarded as a time-dependent analogue of the dimensioned parameter LaTeXMLMath which should appear on the right hand side of the history version of Poisson brackets , as we have showed in LaTeXMLCite .
We have been long discussing in work so far LaTeXMLCite , the essential difference between the internal and external modes of time , and in particular the way they appear in the histories theory scheme .
Ever since their original construction LaTeXMLCite , the interpretation of the ‘ two types of time ’ has served as the key tool to further the particular histories theory ( ‘ History Projection Operator ’ ) formalism , originally presented by Isham LaTeXMLCite and Isham et al LaTeXMLCite .
In the past , we have attempted to present parts of the conceptual issues involved ; yet this will be the subject of a future work , that it will mainly involve presenting in a detailed way these novel ideas about the concept of time .
However , we can not avoid here some comments on these issues , as it is the first time that the mathematical structure of the theory enables an immediate comparison between the ‘ internal ’ and the ‘ external ’ pictures of the theory .
First , we can think of the function LaTeXMLMath as follows .
In the case of a single particle we can write the canonical symplectic form LaTeXMLMath on the phase space LaTeXMLMath , for each moment of time LaTeXMLMath .
Then the history symplectic form for this system is defined by integrating LaTeXMLEquation where LaTeXMLMath is an arbitrary measure on the real line LaTeXMLMath .
For the case of continuous-time paths , and when time is defined along the whole real axis LaTeXMLMath , we can write LaTeXMLMath , where LaTeXMLMath is a density , that is defined to have dimensions of time , so that the history observables have the same dimension as the canonical ones .
Hence LaTeXMLMath is naturally associated with the notion of ‘ time-averaging ’ .
In particular , the freedom to choose an arbitrary function LaTeXMLMath reflects the freedom of the histories construction , to arbitrarily select the ‘ weight ’ by which each moment of time will contribute to the time-averaging of physical quantities .
The above comment is , however , conceptually distinct from the notion of different time-parameters arising from different foliations : the freedom in the choice of LaTeXMLMath is present even in simple non-relativistic systems .
In all systems we have studied so far LaTeXMLCite , the choice of LaTeXMLMath was practically of no consequence , and we chose to set it equal to a constant .
However , in the context of general relativity , LaTeXMLMath has an additional significance : if the history observables are to treat time and space coordinates in the same footing—the choice we followed through all our work so far and in accord to the ‘ two modes of time ’ interpretation—the introduction of a density LaTeXMLMath is unavoidable .
We believe that this is related to the interplay between canonical formalism and covariant formalism , as they have appeared naturally intertwined in the histories formalism once the introduction of two types of time transformation was made .
Indeed , the definition of the action operator LaTeXMLMath in LaTeXMLCite already establishes an interplay between Lagrangian formalism and Hamiltonian formalism , as LaTeXMLMath is defined in analogy to the classical Hamilton-Jacobi action functional .
In future work we shall argue that , the canonical and covariant descriptions implicitly involve a correspondence to the ‘ external ’ and ‘ internal ’ time distinction , as it has been presented so far in the histories theory .
Hence , the Lagrangian and the Hamiltonian formalism refer to different treatment of the two modes of time , even though the two formalisms coincide at the level of the equations of motion .
In general relativity there exists already an implied distinction at the level of symmetries , as the spacetime diffeomorphism group is different from the group of the canonical constraints .
The construction above leads naturally to a one-parameter family of Dirac super-hamiltonians LaTeXMLMath and super-momenta LaTeXMLMath .
In the standard canonical approach to general relativity LaTeXMLCite , the super-hamiltonian and super-momenta are LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath denotes the spatial covariant derivative .
We note that both these quantities are spatial scalar densities , hence they can be smeared with scalar quantities .
The history analogue of these expressions is LaTeXMLEquation .
LaTeXMLEquation For each choice of the weight function LaTeXMLMath , these quantities on LaTeXMLMath satisfy the history version of the Dirac algebra LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The smeared form of the super-hamiltonian LaTeXMLMath and the super-momentum LaTeXMLMath history quantities are defined using as their smearing functions a scalar function LaTeXMLMath , and a spatial vector field LaTeXMLMath in the following way LaTeXMLEquation .
LaTeXMLEquation Hence we can write the smeared form of the ‘ Hamiltonian ’ as LaTeXMLEquation .
The smeared form of this history version of the Dirac algebra is LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where in Eq .
( LaTeXMLRef ) we have LaTeXMLMath , with LaTeXMLMath .
We note that this smeared form Eqs .
( LaTeXMLRef – LaTeXMLRef ) of the Dirac algebra is the analogue of the internal Poincaré group of the histories quantum field theory , in the sense that it does not affect the external time label LaTeXMLMath .
We shall now see that there is also an analogue of the external Poincaré group—namely the group of spacetime diffeomorphisms .
Since we have considered a one-parameter family of Riemannian metrics LaTeXMLMath on the three-surface LaTeXMLMath , we can now identify LaTeXMLMath as the spacetime LaTeXMLMath .
The critical observation here is that we can write a representation of the spacetime diffeomorphisms group LaTeXMLMath on a suitable extension of the canonical history space LaTeXMLMath , which will also carry the representation of the history version of the Dirac algebra discussed above .
In order to demonstrate this statement we start by postulating the ‘ covariant ’ Poisson brackets , on the extended history space LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is a point in the spacetime LaTeXMLMath , and where LaTeXMLMath is a four-metric that belongs to the space of Lorentzian metrics LaTeXMLMath , and LaTeXMLMath is the conjugate variable .
We have defined LaTeXMLMath .
In previous applications of the histories formalism we have defined the ‘ Liouville ’ function LaTeXMLMath as the generator of time translations with respect to the ‘ external ’ time LaTeXMLMath that appears as a kinematical ordering parameter that distinguishes between past , present and future LaTeXMLCite .
In the present case , in analogy with previous history constructions , we can define the ‘ Liouville ’ function LaTeXMLMath associated with any vector field LaTeXMLMath on LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath denotes the Lie derivative with respect to LaTeXMLMath .
This is the direct analogue of the expression that is used in the normal canonical theory for the representations of spatial diffeomorphisms .
The fundamental result is that these generalised Liouville functions LaTeXMLMath , defined for any vector field LaTeXMLMath as in Eq .
( LaTeXMLRef ) , satisfy the Lie algebra of the spacetime diffeomorphisms group LaTeXMLMath LaTeXMLEquation where LaTeXMLMath is the Lie bracket between vector fields LaTeXMLMath and LaTeXMLMath on the manifold LaTeXMLMath .
Now , the aim is to show how the use of the covariant brackets Eqs .
( LaTeXMLRef – LaTeXMLRef ) leads to an augmented history space LaTeXMLMath , in which we can recover the history Dirac algebra constructed in the previous section .
The first step in recovering the history algebra Eqs .
( LaTeXMLRef – LaTeXMLRef ) is to choose a foliation LaTeXMLMath .
Then we define the spatial parts of the pull-back of LaTeXMLMath to LaTeXMLMath by LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath .
For a fixed LaTeXMLMath , we can choose the foliation to be spacelike LaTeXMLMath , this spacelike character will be maintained for some open neighborhood of the Lorentzian metric LaTeXMLMath .
However , this foliation will fail to be spacelike for certain other Lorentzian metrics on LaTeXMLMath .
This feature is not important at the level of the classical theory we are discussing here ; however it can be expected to be a non-trivial issue in the quantum theory .
in the sense that LaTeXMLMath is a path in the space of Riemannian metrics on LaTeXMLMath .
Next , we need to pull-back the conjugate variable LaTeXMLMath to LaTeXMLMath also .
For this purpose , we lower the indices and define the field LaTeXMLMath .
Hence , using the Poisson brackets Eqs .
( LaTeXMLRef – LaTeXMLRef ) , we get the relations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath .
We must now pull back these equations Eq .
( LaTeXMLRef – LaTeXMLRef ) to LaTeXMLMath using the foliation LaTeXMLMath .
Here , it is important to notice that , since LaTeXMLMath is a density in the variable LaTeXMLMath , we have—in coordinates LaTeXMLMath adapted to the split LaTeXMLMath —the relation LaTeXMLEquation where LaTeXMLMath is an appropriate power of the Jacobian of the diffeomorphism LaTeXMLMath .
However , since LaTeXMLMath is a tensor density on LaTeXMLMath of the same weight as the second variable in LaTeXMLMath , we consider the quantity LaTeXMLMath defined by LaTeXMLEquation .
These new quantities LaTeXMLMath , and LaTeXMLMath satisfy the Poisson brackets LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we have defined LaTeXMLMath .
Finally , we define LaTeXMLEquation where LaTeXMLMath is the inverse of the Riemannian metric LaTeXMLMath on LaTeXMLMath , for each LaTeXMLMath .
Hence we have regained the canonical Poisson brackets Eqs .
( LaTeXMLRef – LaTeXMLRef ) for the histories canonical treatment in section LaTeXMLMath .
In the previous section , although we referred to an augmented version LaTeXMLMath of the history space LaTeXMLMath , in which both a representation of the group of spacetime diffeomorphisms and of the Dirac algebra of constraints exist , there was no need , for the purposes of that section , for a detailed presentation of the extended state space .
However , now we shall present the augmented canonical history algebra , to show that the physical predictions of the previous sections hold , and to further examine possible interesting implications of the new construction for the histories version of general relativity .
When we go from the covariant Poisson brackets , that involve the spacetime metric LaTeXMLMath , to the Poisson brackets that involve the paths of Riemannian metric LaTeXMLMath , we ignore the quantities that in the standard LaTeXMLMath decomposition correspond to the lapse function and the shift vector .
When we do take them into account it amounts into a space of paths on an extended state space LaTeXMLMath .
To this end , we first recall that the choice of a foliation LaTeXMLMath enables us to decompose the spacetime metric LaTeXMLMath in a coordinate system adapted to the foliation LaTeXMLMath , as for instance LaTeXMLEquation .
LaTeXMLEquation where we write LaTeXMLMath .
If the unit , timelike vector field LaTeXMLMath is normal to the foliation , we can write LaTeXMLEquation .
The above equation defines the lapse function LaTeXMLMath and the shift vector LaTeXMLMath , associated with the foliation LaTeXMLMath and the metric LaTeXMLMath .
It is a standard result that the inverse metric LaTeXMLMath can be written in the coordinate system adapted to the foliation as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is the inverse of LaTeXMLMath , as in Eq .
( LaTeXMLRef ) .
Next we write the Poisson brackets in terms of the inverse metric LaTeXMLMath and the field LaTeXMLMath as LaTeXMLEquation .
From this expression it is easy to check that the quantities LaTeXMLMath and LaTeXMLMath defined as LaTeXMLEquation .
LaTeXMLEquation are conjugate momenta to the history objects LaTeXMLMath and LaTeXMLMath The objects LaTeXMLMath and LaTeXMLMath are densities with respect to reparameterisations of the LaTeXMLMath label , hence the association LaTeXMLMath does not correspond to a path in the space of scalar fields on LaTeXMLMath .
This is the reason we prefer to use as history canonical variables the objects LaTeXMLMath and LaTeXMLMath , that do correspond to a path on the space of scalar fields or vector fields on LaTeXMLMath respectively .
respectively , in the sense that they satisfy the Poisson brackets equations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and that all quantities LaTeXMLMath and LaTeXMLMath have vanishing Poisson brackets with LaTeXMLMath and LaTeXMLMath .
Hence , given a foliation LaTeXMLMath , one can write the covariant Poisson brackets in terms of objects that represent paths into an extended phase space having as basic Poisson brackets Eqs .
( LaTeXMLRef – LaTeXMLRef ) and Eqs .
( LaTeXMLRef – LaTeXMLRef ) .
It is important to emphasise here that , because the generators LaTeXMLMath and LaTeXMLMath of the history Dirac algebra Eqs .
( LaTeXMLRef – LaTeXMLRef ) , trivially commute with the additional variables of the extended history algebra , we recover exactly the history version of the Dirac algebra , as it was originally defined in section LaTeXMLMath .
Therefore , on the extended history space LaTeXMLMath we have a representation of the Dirac algebra together with a representation of the spacetime diffeomorphisms group LaTeXMLMath .
In the standard Hamiltonian analysis of the Einstein-Hilbert action , one goes from the extended phase space ( involving the lapse function LaTeXMLMath , shift vector LaTeXMLMath , and their conjugate momenta LaTeXMLMath and LaTeXMLMath ) , to the state space ( containing only three-metrics LaTeXMLMath and their conjugate momenta LaTeXMLMath ) , by imposing as first-class constraints the vanishing of LaTeXMLMath and LaTeXMLMath .
In histories theory , and in analogy to the parameterised systems algorithm we established in LaTeXMLCite , we implement the history analogues of the canonical constraints in the extended history phase space LaTeXMLMath , i.e. , we impose the conditions LaTeXMLEquation .
LaTeXMLEquation Together with the vanishing of the super-hamiltonian LaTeXMLMath and the super-momentum LaTeXMLMath , the above equations form a set of first-class constraints .
We impose the constraints Eqs .
( LaTeXMLRef – LaTeXMLRef ) to vanish on the constraint surface LaTeXMLMath , which is essentially the space of all paths LaTeXMLMath , from the real line LaTeXMLMath to the constraint surface LaTeXMLMath of standard canonical theory LaTeXMLCite .
We then consider the space of orbits LaTeXMLMath with respect to the action of the symplectic transformations generated by the constraints .
The constraint functions LaTeXMLMath and LaTeXMLMath commute with LaTeXMLMath and LaTeXMLMath , hence the symplectic transformations generated by these constraints leaves LaTeXMLMath and LaTeXMLMath invariant .
Hence , we recover the history space LaTeXMLMath —originally defined in section LaTeXMLMath —as being the space of orbits LaTeXMLMath and LaTeXMLMath are imposed early in the discussion hence the ‘ reduced state space ’ —which is defined as the space of orbits of the constraints ’ action on the constraint surface—consists of the three-metrics LaTeXMLMath and their conjugate momenta LaTeXMLMath .
and it is referred to as the state space of general relativity .
corresponding to these constraints .
Furthermore , it is interesting to see how the generators LaTeXMLMath of the diffeomorphisms group can be projected into LaTeXMLMath .
To this end , we first write the above set of constraints with the equivalent covariant expression LaTeXMLEquation where LaTeXMLMath is the vector field corresponding to the parameter LaTeXMLMath of the foliation , and LaTeXMLMath is an arbitrary vector field that serves as a smearing function .
We then impose the constraint LaTeXMLMath on the constraint surface .
Then , the commutator of the generators LaTeXMLMath with the constraint LaTeXMLMath is LaTeXMLEquation .
Note here that LaTeXMLMath is a tensor density .
The first term of the expression Eq .
( LaTeXMLRef ) is equal to LaTeXMLMath , the second to LaTeXMLMath and the third to LaTeXMLMath , and all three terms vanish on the constraint surface .
However , the fourth term vanishes if and only if LaTeXMLEquation for some scalar function LaTeXMLMath .
The above expression implies that the vector field LaTeXMLMath preserves the foliation in the sense that its Lie bracket with the transverse field LaTeXMLMath yields a field in the same direction .
Written in a coordinate system adapted to the foliation this condition implies that the component LaTeXMLMath is a function of LaTeXMLMath only .
Therefore we conclude that , the histories space LaTeXMLMath carries a representation of the sub group of foliation-preserving diffeomorphisms LaTeXMLMath , of the diffeomorphisms group LaTeXMLMath .
The generators of this sub-group can be written as LaTeXMLEquation where LaTeXMLMath denotes vector fields that are horizontal to the foliation .
For the vector field LaTeXMLMath , LaTeXMLMath , we write the Liouville function LaTeXMLMath as LaTeXMLEquation .
Following the histories methods of LaTeXMLCite we further define the histories action functional LaTeXMLMath as LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are appropriate smearing functions .
We notice here that the quantities LaTeXMLMath and LaTeXMLMath are not the original lapse function and shift vector , since all trace of them was lost when passing from the extended state space LaTeXMLMath to LaTeXMLMath .
It is easy to show that the usual dynamical equations for the canonical fields LaTeXMLMath and LaTeXMLMath are equivalent to the history Poisson bracket equations LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is defined in Eq .
( LaTeXMLRef ) .
The path LaTeXMLMath is a solution of the classical equations of motion , and therefore corresponds to a spacetime metric that is a solution of the Einstein equations .
Next we should employ the algorithm we used in LaTeXMLCite to treat parameterised systems .
Again the ‘ history constraint ’ surface LaTeXMLMath is the space of paths from LaTeXMLMath to the canonical constraint surface LaTeXMLMath , which is defined from the requirement that the constraints should vanish for all times LaTeXMLMath .
We then study the action of the constraints by symplectic transformations on LaTeXMLMath .
The reduced space of histories LaTeXMLMath is the space of the orbits that are obtained by the action of the constraints ( i.e .
, equivalence classes of points of LaTeXMLMath that are related by a constraint transformation ) : LaTeXMLEquation .
In fact LaTeXMLMath is isomorphic to the space of continuous paths on LaTeXMLMath .
In order for a function on LaTeXMLMath to be a physical observable ( i.e. , it can be projected into a function on LaTeXMLMath ) , if and only if it commutes with the constraints on the constraint surface .
We shall now discuss the extent to which the generators LaTeXMLMath of the restricted spacetime diffeomorphisms group LaTeXMLMath , that is represented in LaTeXMLMath satisfy this condition .
Indeed , LaTeXMLEquation .
LaTeXMLEquation The generators of the restricted spacetime diffeomorphisms LaTeXMLMath clearly commute with the super-hamiltonian LaTeXMLMath on the constraint surface LaTeXMLMath .
However , LaTeXMLMath only commutes with LaTeXMLMath on LaTeXMLMath if LaTeXMLMath is a spatial vector field , i.e .
, if the diffeomorphisms generated by LaTeXMLMath preserve the spatial nature of LaTeXMLMath .
This is equivalent to the condition that the diffeomorphisms generated by LaTeXMLMath preserve the foliation .
Hence , amongst all spacetime diffeomorphisms , it is only the Liouville function LaTeXMLMath —the time translations generator—that can be non-trivially projected on the reduced phase space ( horizontal diffeomorphisms vanish on LaTeXMLMath ) LaTeXMLEquation for any function LaTeXMLMath .
We note that the function LaTeXMLMath generates time reparametrisations of the parameter LaTeXMLMath on LaTeXMLMath LaTeXMLCite .
Hence , of all spacetime diffeomorphisms , it is only the generators of time reparametrisations that is defined on the reduced phase space LaTeXMLMath .
We have showed how the recent development in introducing the distinction between time as a causal ordering parameter , and as an evolution parameter in dynamics LaTeXMLCite , leads to the construction of a history version of general relativity in which there emerges a new relation between the group structures associated with the normal Lagrangian and Hamiltonian approaches .
In particular , we have showed that in this histories version of canonical general relativity there exists a representation of the spacetime diffeomorphisms group LaTeXMLMath , together with a history analogue of the Dirac algebra of constraints .
However , various important issues arise .
The immediate one to be addressed is that the history algebra Eqs .
( LaTeXMLRef – LaTeXMLRef ) depends on the choice of a Lorentzian foliation .
This leads to two distinct questions .
First , what is the degree to which physical results depend upon this choice ?
The solutions to the equations of motion for each choice allow us to construct different 4-metrics .
If different descriptions are to be equivalent , two distinct 4-metrics should be related by a spacetime diffeomorphism .
We should therefore establish that the action of the spacetime diffeomorphisms group intertwines between constructions corresponding to different choices of the foliation .
This involves considering state space histories corresponding to arbitrary choices of foliation .
Second , and perhaps more important , is to question the notion of a spacelike foliation itself .
Since the spacetime causal structure is a dynamical object , the notion of a foliation being spacelike has meaning only after the solution to the classical equations of motion has been selected .
However , in the histories description we do not just use a single solution of the classical equations of motion ( indeed , many of the possible histories are not solutions at all ) , and in these circumstances the notion of a ‘ spacelike ’ foliation loses its meaning .
The issues mentioned above are fundamental in the treatment of general relativity .
Once they have been resolved , further applications will be technically straightforward : for instance , the appearance of Noether ’ s theorem in the histories formalism .
Of particular significance is the fact that the histories description contains a mixture of Lagrangian and Hamiltonian structures .
An interesting application would be to apply the history ideas to the description of gravity in terms of the Ashtekar variables .
The work presented here is only the beginning of a programme for constructing a history theory of general relativity .
What is necessary next , is to develop the formalism to find a description that focuses on a manifestly covariant treatment of the theory .
Acknowledgements I would like to thank Charis Anastopoulos for a very fruitful interaction and Karel Kuchar for useful discussions .
I would like to especially thank Chris Isham for his help on differential-geometric construction issues .
I gratefully acknowledge support from the L.D .
Rope Third Charitable Settlement and from EPSRC GR/R36572 grant .
It is shown that with finitely many exceptions , the fundamental group obtained by Dehn surgery on a one cusped hyperbolic 3–manifold contains the fundamental group of a closed surface .
57M27 57M50 , 20H10 LaTeXMLMath eometry & LaTeXMLMath opology LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Volume 5 ( 2001 ) 347–367 Email : Abstract AMS Classification numbers Primary : Secondary : Keywords Proposed : Walter Neumann Received : 09 January 2001 A central unresolved question in the theory of closed hyperbolic LaTeXMLMath –manifolds is whether they are covered by manifolds which contain closed embedded incompressible surfaces .
An affirmative resolution of this conjecture would imply in particular that all closed hyperbolic LaTeXMLMath –manifolds contain the fundamental group of a closed surface of genus at least two .
Even the simplest case of this conjecture , namely that of the manifolds obtained by surgery on a hyperbolic manifold with a single cusp has remained open for many years .
In this article we prove the following theorem : Suppose that LaTeXMLMath is a hyperbolic LaTeXMLMath –manifold with a single torus boundary component .
Then all but finitely many surgeries on LaTeXMLMath contain the fundamental group of a closed orientable surface of genus at least two .
Our proof rests upon : Suppose that LaTeXMLMath is an incompressible , LaTeXMLMath –incompressible quasifuchsian surface with boundary slope LaTeXMLMath .
Then there is a LaTeXMLMath so that if LaTeXMLMath is any simple curve on LaTeXMLMath with LaTeXMLMath , the Dehn filled manifold LaTeXMLMath contains the fundamental group of a closed surface of genus at least two .
This result is similar in spirit to the main theorem of LaTeXMLCite ; that result applied only to surfaces of slope zero , however in that context we were able to give an explicit ( and fairly small ) value for LaTeXMLMath .
It follows from LaTeXMLCite that LaTeXMLMath contains at least two distinct strict boundary slopes ; and we shall show using Proposition 1.2.7 loc .
cit .
( see Lemma LaTeXMLRef ) that both these slopes are represented by quasifuchsian surfaces .
Then LaTeXMLRef implies the result .
∎ We now outline the proof .
Using the quasifuchsian surface , we are able to construct a certain complex in the universal covering of LaTeXMLMath .
This in turn gives rise to a map of a manifold with convex boundary LaTeXMLMath .
We are able to prove that this map extends to a map of Dehn filled manifolds , LaTeXMLMath .
The explicit nature of LaTeXMLMath makes it possible to prove that LaTeXMLMath contains a surface group : a geometrical argument using convexity then shows that LaTeXMLMath is LaTeXMLMath –injective , completing the proof .
The intuition for where the surface comes from in this construction is the same as LaTeXMLCite : two surfaces glued together by a very long annulus should remain incompressible in fillings distant from the boundary slope .
It seems worth pointing out that it follows that the surface group which we produce comes from an immersion of a surface into LaTeXMLMath without triple points .
A proof of Theorem LaTeXMLRef by completely different methods has been given by T Li ( see LaTeXMLCite ) .
His proof has the advantage that it gives bounds on the constant LaTeXMLMath .
Throughout this article we fix a hyperbolic manifold LaTeXMLMath with a single torus boundary component .
This gives rise to a discrete , faithful representation LaTeXMLMath which is unique up to conjugacy .
We recall that if LaTeXMLMath is a proper map which is injective at the level of fundamental groups then the surface LaTeXMLMath is said to be quasifuchsian if the limit set of the group LaTeXMLMath is a topological circle .
Many equivalent formulations exist ; for our purpose , it suffices to observe that it is shown in Chapter V , Corollary 9.2 of LaTeXMLCite that a representation of a surface group is quasifuchsian if and only if the representation is geometrically finite and contains no accidental parabolics .
( We recall that a surface is said to have no accidental parabolics if the conjugacy classes of elements representing boundary components in LaTeXMLMath are exactly the conjugacy classes that are parabolic under the representation . )
We begin with a purely topological lemma .
Suppose that LaTeXMLMath is an incompressible , LaTeXMLMath –incompressible surface which represents boundary slope LaTeXMLMath .
Then there is a incompressible , LaTeXMLMath –incompressible surface LaTeXMLMath also with boundary slope LaTeXMLMath which contains no accidental parabolics .
Furthermore , LaTeXMLMath is a subgroup of LaTeXMLMath .
If the original surface LaTeXMLMath contains no accidental parabolics , then we are done .
Otherwise , it is shown in LaTeXMLCite Proposition 8.11.1 , that there are two finite collections of disjoint simple closed curves LaTeXMLMath so that any loop which is an accidental parabolic is freely homotopic to ( a power of ) some leaf of one of these laminations .
Let LaTeXMLMath be a leaf in one of these laminations .
We claim that the other end of the free homotopy in LaTeXMLMath must be a curve which is parallel to a ( power of ) LaTeXMLMath .
The reason is this : The free homotopy defines a map of an annulus LaTeXMLMath , one of whose boundary components LaTeXMLMath lies on LaTeXMLMath and is parallel to LaTeXMLMath and one of whose boundary components lies on LaTeXMLMath .
Put LaTeXMLMath in general position with respect to LaTeXMLMath .
Using a very small move , lift LaTeXMLMath so that it lies slightly away from LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are in general position , we see LaTeXMLMath as a collection of arcs and circles in LaTeXMLMath .
Since LaTeXMLMath is empty , every arc must run from the boundary component of LaTeXMLMath which is mapped to LaTeXMLMath back to this boundary component .
If such an arc of intersection is essential when mapped into LaTeXMLMath , then it give rise to a mapped in boundary compression and hence a compression of LaTeXMLMath , since LaTeXMLMath is not an annulus .
This contradicts our assumption on LaTeXMLMath .
Otherwise , then the arc of intersection can be removed by homotopy .
It follows that we may suppose that there are no arcs of intersection in LaTeXMLMath .
However this implies that the boundary component of LaTeXMLMath which lies on LaTeXMLMath can be homotoped so as to be disjoint from LaTeXMLMath , proving the claim .
Consider an annulus mapped into LaTeXMLMath , one end mapping to LaTeXMLMath and the other end mapped into LaTeXMLMath .
Put this annulus in general position with respect to LaTeXMLMath ; we see easily that there is an essential annulus with one boundary component ( now possibly not embedded ) in LaTeXMLMath and the other boundary component a power of LaTeXMLMath , so that the interior of the annulus does not map into LaTeXMLMath .
Thurston ’ s observation implies that the end which lies on LaTeXMLMath must be a power of some simple loop .
Call this simple loop LaTeXMLMath , say .
Let LaTeXMLMath be LaTeXMLMath , where LaTeXMLMath is some open regular neighbourhood of LaTeXMLMath .
This is an irreducible LaTeXMLMath –manifold with a boundary component of positive genus , so that LaTeXMLMath is Haken .
The observation of the previous paragraph shows that LaTeXMLMath contains an essential annulus .
Following Jaco ( see LaTeXMLCite ) we define a Haken manifold pair LaTeXMLMath to be a Haken manifold LaTeXMLMath together with some incompressible , possibly disconnected LaTeXMLMath –manifold LaTeXMLMath .
In our setting , take LaTeXMLMath to be a neighbourhood of the simple closed curve LaTeXMLMath and LaTeXMLMath to be a neighbourhood of LaTeXMLMath ; the annulus of the above paragraph provides an essential map of pairs LaTeXMLEquation .
Now Theorem VIII.13 of LaTeXMLCite implies that there is an embedding of an annulus with one end in LaTeXMLMath and the other end in LaTeXMLMath .
We now return to LaTeXMLMath and use this embedded annulus to form a new embedded surface LaTeXMLMath by removing a neighbourhood of LaTeXMLMath from LaTeXMLMath and replacing with two copies of LaTeXMLMath .
This surface could be disconnected , in which case choose some component and rename this as LaTeXMLMath .
The new surface continues to be incompressible ( and therefore LaTeXMLMath –incompressible since LaTeXMLMath can not be an annulus ) .
We may repeat this process now with the surface LaTeXMLMath .
However this procedure must eventually terminate in a surface LaTeXMLMath which therefore contains no accidental parabolics .
∎ We recall that a LaTeXMLMath –injective surface is a virtual fibre of LaTeXMLMath , if there is some finite sheeted covering of LaTeXMLMath , LaTeXMLMath , to which LaTeXMLMath lifts and is isotopic to the fibre of a fibration of LaTeXMLMath .
Although we do not actually use this fact in the sequel , we note that we may deduce : Suppose that LaTeXMLMath is a surface of maximal Euler characteristic representing boundary slope LaTeXMLMath .
Then either LaTeXMLMath is a virtual fibre of LaTeXMLMath , or it is quasifuchsian .
Such a surface can not contain an accidental parabolic , for their removal constructs a surface of the same boundary slope of increased Euler characteristic .
If the surface is geometrically finite , then as observed above , this implies that the surface is quasifuchsian .
If the surface is geometrically infinite , this implies that it is a virtual fibre .
This is an argument originally due to Thurston .
The ingredients are contained in LaTeXMLCite Theorem 5.2.18 and LaTeXMLCite .
∎ Remark This , together with Gabai ’ s proof of Property R , gives an alternative proof of a result due to Fenley LaTeXMLCite , who proved that if LaTeXMLMath is the minimal genus Seifert surface of a nonfibred hyperbolic knot in LaTeXMLMath , then LaTeXMLMath is quasifuchsian .
We now recall some of the results and terminology of LaTeXMLCite : Suppose that we are given a curve of representations of the fundamental group of a complete hyperbolic LaTeXMLMath –manifold which contains a faithful representation—for the purposes of this paper it always suffices to take the component containing the complete structure .
Then we can associate to an ideal point of this curve an action of the group on a simplicial tree .
Surfaces are constructed by a transversality argument as subgroups of edge stabilisers .
Such an action or surface we say is associated to an ideal point .
For our purpose , we need only note that LaTeXMLCite Proposition 1.2.7 shows given an action associated to an ideal point , a nontrivial normal subgroup can not fix any point of this simplicial tree .
We show : Suppose that LaTeXMLMath is an incompressible , LaTeXMLMath –incompressible surface which is associated to some ideal point and that LaTeXMLMath represents boundary slope LaTeXMLMath .
Then there is a incompressible , LaTeXMLMath –incompressible quasifuchsian surface LaTeXMLMath also with boundary slope LaTeXMLMath .
By hypothesis , there is a curve of representations which yields the surface LaTeXMLMath and tree LaTeXMLMath via some ideal point .
We note that the modifications in removing annuli in the proof of Lemma LaTeXMLRef yield a incompressible , boundary incompressible surface LaTeXMLMath with the property that LaTeXMLMath , so that LaTeXMLMath also lies in an edge stabiliser of LaTeXMLMath .
We claim that this implies that the surface group LaTeXMLMath can not be geometrically infinite .
For if it were , then as above , we deduce that LaTeXMLMath is a virtual fibre of LaTeXMLMath , so that there is a finite sheeted covering LaTeXMLMath of LaTeXMLMath with the property that LaTeXMLMath is a nontrivial normal subgroup in LaTeXMLMath .
However this is impossible , since restriction gives an action of LaTeXMLMath on the tree LaTeXMLMath and this contradicts Proposition 1.2.7 of LaTeXMLCite .
We deduce that LaTeXMLMath is geometrically finite , it contains no accidental parabolics , so that it is quasifuchsian , as was claimed .
∎ Suppose that LaTeXMLMath is a quasi-fuchsian surface with boundary slope LaTeXMLMath .
Then there is a LaTeXMLMath so that if LaTeXMLMath is any simple closed curve in LaTeXMLMath with LaTeXMLMath , we have LaTeXMLEquation .
Since the image of LaTeXMLMath is quasifuchsian its limit set LaTeXMLMath is a quasi-circle .
Choose basepoints and identify the torus subgroup with LaTeXMLMath so that this stabilises LaTeXMLMath and acts on the complex plane in the upper half space model .
The quotient LaTeXMLMath is a torus and the quasi-circle descends to a closed curve LaTeXMLMath which is a homotopic to a closed geodesic on this torus .
Fix some homotopy which moves LaTeXMLMath to a Euclidean geodesic , this homotopy is covered by a homotopy which carries LaTeXMLMath to a straight line LaTeXMLMath .
Since the homotopy is the image of a compact set , this shows that the set LaTeXMLMath lies within some fixed distance LaTeXMLMath of LaTeXMLMath in the Euclidean metric on LaTeXMLMath .
Choose LaTeXMLMath so large that translation by LaTeXMLMath moves the line LaTeXMLMath a distance of LaTeXMLMath away from itself .
If we choose an element LaTeXMLMath as in the statement of Theorem LaTeXMLRef , ( ie , LaTeXMLMath , where LaTeXMLMath ) , it follows that the limit set of the subgroup LaTeXMLMath and the limit set of the subgroup LaTeXMLMath contain only the point at infinity in common .
It follows that the only possibility for elements in LaTeXMLMath lie in the parabolic subgroup fixing LaTeXMLMath since such elements must stabilise both limit sets ; thus the intersection contains LaTeXMLMath , but can be no larger .
∎ Notice that the proof actually constructs a straight strip in LaTeXMLMath which contains the limit set of LaTeXMLMath .
We also observe that the LaTeXMLMath of the lemma need not yet be the LaTeXMLMath of Theorem LaTeXMLRef , we may need to enlarge it further .
To be specific , let us henceforth suppose that the quasifuchsian surface LaTeXMLMath contains two boundary components , the case of a single boundary component having been dealt with in LaTeXMLCite and the case of more boundary components being entirely analogous .
We now begin our construction of a certain complex LaTeXMLMath .
The surface LaTeXMLMath is finite area and quasifuchsian .
We recall that LaTeXMLMath is the intersection of all the hyperbolic halfspaces which contain LaTeXMLMath ; this is a convex LaTeXMLMath invariant set .
In the degenerate case that the surface is totally geodesic , we adopt the convention that LaTeXMLMath is defined to be small LaTeXMLMath -neighbourhood of the intersection of the hyperbolic halfspaces containing LaTeXMLMath .
The fact that LaTeXMLMath is quasifuchsian means that LaTeXMLMath is a finite volume hyperbolic manifold with cusps homeomorphic to LaTeXMLMath .
We may homotope LaTeXMLMath so that LaTeXMLMath lies inside its hull , for example by homotoping it just inside one of the pleated surface boundaries .
By further homotopy , we arrange that LaTeXMLMath meets the boundary of a cusp in simple closed curves .
From this it follows that we can bound the “ thickness ” of the hull , that is to say , there is some constant , LaTeXMLMath , so that every point on one of the surfaces of the convex hull of LaTeXMLMath is a distance at most LaTeXMLMath from the other surface in the convex hull and in particular , every point in the hull is within distance LaTeXMLMath of some point of LaTeXMLMath .
We may as well assume that LaTeXMLMath is fairly large , at least LaTeXMLMath say .
Fix a horoball neighbourhood in LaTeXMLMath of the cusp , LaTeXMLMath , so small that the smallest distance between two preimages of LaTeXMLMath in the universal covering is very large compared to LaTeXMLMath .
We can assume that LaTeXMLMath was chosen small enough so that only the thin part of LaTeXMLMath coming from the cusps enters LaTeXMLMath .
Now chose a horoball LaTeXMLMath so far inside LaTeXMLMath that the hyperbolic distance between LaTeXMLMath and LaTeXMLMath is greater than LaTeXMLMath .
This now guarantees that in the universal covering , every preimage of LaTeXMLMath either is centred at some limit point of LaTeXMLMath or is distance greater than LaTeXMLMath from LaTeXMLMath .
Using our thickness estimate , we see that every preimage of LaTeXMLMath is either centred at a limit point of LaTeXMLMath or it is at a distance much greater than LaTeXMLMath from the convex hull of LaTeXMLMath .
Define LaTeXMLMath to be the complete hyperbolic manifold LaTeXMLMath with the interior of LaTeXMLMath excised .
This is a compact manifold with a single torus boundary component .
We have arranged the surface LaTeXMLMath so that the surface LaTeXMLMath has two boundary components on this torus .
We will base all fundamental groups at some point LaTeXMLMath on one of these boundary components .
The universal covering of LaTeXMLMath embeds into the universal covering of LaTeXMLMath in the obvious way .
We use the upper halfspace model and let LaTeXMLMath be the horoball preimage of LaTeXMLMath centred at infinity .
Fix some preimage of LaTeXMLMath , LaTeXMLMath which passes through infinity .
As in the proof of Lemma LaTeXMLRef , we may find a pair of vertical planes which define a three dimensional strip which meets LaTeXMLMath in a strip containing the limit set of LaTeXMLMath .
In the upper halfspace model all horoballs not centred at LaTeXMLMath have some uniformly bounded Euclidean size , so by moving these planes apart if necessary , we may suppose that the three dimensional strip contains the intersection of LaTeXMLMath with the horosphere corresponding to LaTeXMLMath as well as all the translates of LaTeXMLMath which meet LaTeXMLMath .
Denote the three dimensional strip this produces by LaTeXMLMath .
The intersection of LaTeXMLMath with the LaTeXMLMath horosphere will be denoted LaTeXMLMath .
Notice that the set LaTeXMLMath is bounded by two totally geodesic planes , so that it is hyperbolically convex , moreover it contains the limit set of LaTeXMLMath by construction , so that it contains the convex hull of this limit set .
The set LaTeXMLMath meets the horosphere LaTeXMLMath in a line which covers one of the boundary components of LaTeXMLMath , let us denote this component by LaTeXMLMath .
This situation is depicted in Figure 1 .
¡-4pt,0pt¿ S1 LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Now we fix a LaTeXMLMath –translate of LaTeXMLMath which passes through infinity and meets LaTeXMLMath in a line which covers the other boundary component LaTeXMLMath .
Denote this translate by LaTeXMLMath .
We then perform the construction of the above paragraph with LaTeXMLMath to form the sets LaTeXMLMath and LaTeXMLMath .
Note that LaTeXMLMath and LaTeXMLMath are parallel strips , so by applying some preliminary covering translation if necessary , we may assume that LaTeXMLMath was chosen so that the distance between LaTeXMLMath and LaTeXMLMath is very large in both the hyperbolic metric and in the Euclidean metric of the horosphere .
( See Figure 2A . )
The proof of LaTeXMLRef together with the fact that the translation coming from LaTeXMLMath stabilises the strips implies that we may choose a LaTeXMLMath so that if LaTeXMLMath is in the stabiliser of LaTeXMLMath and LaTeXMLMath , then the distance between LaTeXMLMath and LaTeXMLMath is very much larger than the distance between LaTeXMLMath and LaTeXMLMath in both the hyperbolic metric and in the Euclidean metric of the horosphere LaTeXMLMath .
( In particular , this distance is very large . )
Since this distance is very large , we may as well assume at the same time that if LaTeXMLMath , then the length of LaTeXMLMath in the Euclidean metric on LaTeXMLMath is much larger than LaTeXMLMath .
Suppose that we wish to do Dehn filling corresponding to some curve LaTeXMLMath which satisfies the requirement that LaTeXMLMath , with the choice of LaTeXMLMath in the above paragraph .
Consider the subset of hyperbolic LaTeXMLMath –space defined as follows .
Begin with the two convex hulls coming from LaTeXMLMath and LaTeXMLMath .
Adjoin all the horoballs which cover LaTeXMLMath and which are incident on LaTeXMLMath or LaTeXMLMath .
Note that this configuration is invariant under the element LaTeXMLMath .
Now add in all LaTeXMLMath translates in the configuration so that it becomes invariant under the subgroup LaTeXMLMath .
Denote this set by LaTeXMLMath .
Suppose we have inductively constructed the complex LaTeXMLMath ; this consists of certain translates of the convex hulls of LaTeXMLMath and LaTeXMLMath and certain translates of horoballs which cover LaTeXMLMath .
Inductively , we assume that the entire complex is invariant under the group LaTeXMLMath .
Some of these horoballs of LaTeXMLMath are centred at points which meet translates of the limit sets of both LaTeXMLMath and LaTeXMLMath ; that is to say , the surface LaTeXMLMath has two ends ( corresponding to the fact that we ’ re assuming that LaTeXMLMath has two boundary components ) and both of these ends appear in this type of horoball .
( By way of example , LaTeXMLMath is the only such horoball in LaTeXMLMath . )
Inductively we may assume that for such horoballs , the collection of surfaces which is incident to the horoball is invariant under the orbit of the conjugate of the subgroup LaTeXMLMath which stabilises the horoball .
However some of the horoballs of LaTeXMLMath only meet translates of the preimage of one end of LaTeXMLMath .
It is along these horoballs that we enlarge the complex : To be specific , we will denote such a surface by LaTeXMLMath , and the horoball by LaTeXMLMath , assuming it is centred at the complex number LaTeXMLMath .
There are now two cases .
( In general , the number of cases is the number of components of LaTeXMLMath . )
The first case is that LaTeXMLMath meets LaTeXMLMath in some closed curve which covers LaTeXMLMath .
In this case we may actually choose an element of LaTeXMLMath which maps LaTeXMLMath to LaTeXMLMath while mapping LaTeXMLMath to LaTeXMLMath .
The ambiguity in such an element is easily seen to come from premultiplication by any element of LaTeXMLMath .
The second case is that LaTeXMLMath meets LaTeXMLMath in some closed curve which covers LaTeXMLMath .
In this case we may choose an element of LaTeXMLMath which maps LaTeXMLMath to LaTeXMLMath while mapping LaTeXMLMath to LaTeXMLMath .
Again , the only ambiguity in such an element comes from premultiplication by an element of LaTeXMLMath .
In either case , denote some choice of such an element by LaTeXMLMath .
We form LaTeXMLMath by adding to LaTeXMLMath all the complexes LaTeXMLMath as we run over all the relevant horoballs .
Since LaTeXMLMath is LaTeXMLMath –invariant , this is independent of the LaTeXMLMath –choices .
One sees easily from the construction that LaTeXMLMath satisfies the inductive hypothesis .
Define LaTeXMLMath to be the union of all the LaTeXMLMath ’ s .
A schematic for LaTeXMLMath is shown in Figure 2A .
To expedite our analysis of LaTeXMLMath , we define a graph LaTeXMLMath by taking one vertex for each preimage of LaTeXMLMath lying in LaTeXMLMath and one vertex for each translate of LaTeXMLMath in LaTeXMLMath .
Edges are defined by the obvious incidence relations .
By construction , LaTeXMLMath has two types of vertex and two types of edge .
( In general , there are two types of vertex and LaTeXMLMath types of edge . )
V1Very large LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath The graph LaTeXMLMath is a tree .
Suppose not and that there is a path in LaTeXMLMath which runs between vertices without backtracking .
This corresponds to some path in LaTeXMLMath which runs from one preimage of the surface LaTeXMLMath to another .
We will replace this path by a piecewise geodesic path which will in fact be a long quasi-geodesic .
From this it will follow that the path runs between different lifts of LaTeXMLMath , so that it is not a loop in LaTeXMLMath .
Here is the construction : Since it does not involve backtracking , such a path consists of pieces of two types .
There are segments which run from horoball to horoball inside a convex hull of some lift of LaTeXMLMath , and segments which run inside the horoball from one lift of LaTeXMLMath to another .
If we see a segment of the first type , we replace it by the common geodesic perpendicular to the pair of horoballs .
Notice that this geodesic runs through the centres of both horoballs , which lie in the limit set of this preimage of LaTeXMLMath , so that the geodesic continues to lie inside the convex hull .
By construction , such a geodesic makes angle LaTeXMLMath with the horosphere that it meets .
If we see a segment of the path lying inside the horoball connecting different lifts of LaTeXMLMath , we replace it by the geodesic in the horoball between endpoints of the relevant common perpendiculars constructed in the previous paragraph .
Note that the distance between such points is enormous , so that this geodesic makes an angle with the horosphere which is very close to LaTeXMLMath .
( See Figure 2B . )
Our choices ensure that all the geodesics we construct this way are all very long , moreover , the angle a horoball geodesic piece makes with a convex hull geodesic piece is very close to LaTeXMLMath .
This makes the path quasi-geodesic , hence it has distinct endpoints and does not correspond to a closed path in the graph LaTeXMLMath as was required .
∎ Remark As a first application of this lemma , we note that it implies that if LaTeXMLMath is the centre of some horoball in LaTeXMLMath , then there are no “ unexpected ” translates of LaTeXMLMath in LaTeXMLMath incident on LaTeXMLMath .
That is to say , in the above notation , if LaTeXMLMath , then the first appearance of LaTeXMLMath in our construction places into LaTeXMLMath the LaTeXMLMath image of the orbit LaTeXMLMath at LaTeXMLMath .
These are the only preimages of LaTeXMLMath in LaTeXMLMath incident at LaTeXMLMath ; the reason being that any preimage added at a later stage in the construction would give rise to a loop in the graph LaTeXMLMath .
Suppose that LaTeXMLMath carries some preimage of LaTeXMLMath lying in LaTeXMLMath to another preimage of LaTeXMLMath which lies in LaTeXMLMath .
Then LaTeXMLMath stabilises LaTeXMLMath .
We begin by noting that LaTeXMLMath has the following property : Suppose that LaTeXMLMath is the centre of some horoball in LaTeXMLMath , LaTeXMLMath say .
This means that there is some translate of LaTeXMLMath , LaTeXMLMath say , in the complex LaTeXMLMath whose limit set contains LaTeXMLMath .
We claim that knowing any translate determines the entire complex LaTeXMLMath .
The reason for the claim is this : The translates of LaTeXMLMath which lie in LaTeXMLMath and have limit point at LaTeXMLMath are constructed as the images of the LaTeXMLMath orbit of LaTeXMLMath under some element LaTeXMLMath , which in particular satisfies LaTeXMLMath .
By the remark following LaTeXMLRef , this is exactly the collection of translates in LaTeXMLMath at LaTeXMLMath .
Since LaTeXMLMath lies in LaTeXMLMath , it follows that we can find an element LaTeXMLMath so that LaTeXMLMath , where LaTeXMLMath denotes one of the reference surfaces , ie , either LaTeXMLMath or LaTeXMLMath .
( Of course , which one is determined by LaTeXMLMath and which component of LaTeXMLMath that this covers . )
Suppose that LaTeXMLMath is any element of LaTeXMLMath which throws LaTeXMLMath to LaTeXMLMath and the relevant reference surface to LaTeXMLMath .
The element LaTeXMLMath stabilises LaTeXMLMath and the reference surface , so it is a power of LaTeXMLMath , that is to say LaTeXMLMath for some integer LaTeXMLMath .
We deduce that we can reconstruct the translates of LaTeXMLMath in LaTeXMLMath at LaTeXMLMath as the LaTeXMLMath image of the orbit LaTeXMLMath .
It follows then that we can unambiguously reconstruct the complex LaTeXMLMath outwards using the new set of horoballs that this created , since each such horoball meets a translate of LaTeXMLMath and we may apply the same argument .
This proves our claim .
The corollary now follows : If LaTeXMLMath is some element of LaTeXMLMath throwing LaTeXMLMath to LaTeXMLMath and LaTeXMLMath to LaTeXMLMath , then in the notation established in this proof , we may choose LaTeXMLMath to construct the surfaces in LaTeXMLMath incident to LaTeXMLMath and LaTeXMLMath to construct all the surfaces in LaTeXMLMath incident to LaTeXMLMath .
From this it follows that LaTeXMLMath maps the surfaces in LaTeXMLMath incident to LaTeXMLMath to the surfaces in LaTeXMLMath incident to LaTeXMLMath .
The result follows by building LaTeXMLMath outwards as described above .
∎ We will need to know that LaTeXMLMath has the following property .
The set LaTeXMLMath has the property that any LaTeXMLMath translate of LaTeXMLMath is either entirely contained inside LaTeXMLMath or is disjoint from it .
The analysis which achieves this is technical and will be deferred to section LaTeXMLRef Define LaTeXMLMath to be the stabiliser in LaTeXMLMath of the complex LaTeXMLMath .
We will show that LaTeXMLMath is isomorphic to the fundamental group of a certain LaTeXMLMath –manifold .
There is a minor difference between the cases that LaTeXMLMath is the boundary of a twisted LaTeXMLMath -bundle or it is not and so we shall assume henceforth : LaTeXMLMath is not the boundary of a twisted LaTeXMLMath –bundle neighbourhood of a nonorientable surface embedded in LaTeXMLMath .
The minor changes which need to be made in this situation are explained in section LaTeXMLRef .
In fact the only case which really needs to be singled out is the case when LaTeXMLMath is the boundary of a twisted LaTeXMLMath –bundle neighbourhood of a nonorientable surface with one boundary component .
Define this LaTeXMLMath –manifold as follows : Take LaTeXMLMath and glue on a manifold homeomorphic to LaTeXMLMath by identifying LaTeXMLMath with two annuli parallel to LaTeXMLMath contained in LaTeXMLMath .
Denote this manifold by LaTeXMLMath .
Of course , a manifold homeomorphic to LaTeXMLMath is already embedded inside LaTeXMLMath , however the group LaTeXMLMath corresponds to an immersion of LaTeXMLMath which intuitively comes by attaching one boundary component of LaTeXMLMath to LaTeXMLMath , then spinning the other boundary component around a very long annulus parallel to LaTeXMLMath before attaching to LaTeXMLMath .
The manifold LaTeXMLMath is easily seen to be irreducible and its boundary contains a component of positive genus , so that LaTeXMLMath is Haken .
We will show that LaTeXMLMath .
To this end we need : The complex LaTeXMLMath is simply connected .
We note that LaTeXMLMath is constructed as horoballs glued to copies of LaTeXMLMath along thin neighbourhoods of copies of the real line which cover boundary components of LaTeXMLMath .
The result then follows from the Seifert–Van Kampen theorem , together with Lemma LaTeXMLRef .
∎ We may now prove : The complex LaTeXMLMath is homeomorphic to LaTeXMLMath , in particular , LaTeXMLMath .
The group LaTeXMLMath acts on LaTeXMLMath and Corollary LaTeXMLRef shows that LaTeXMLMath acts transitively on the copies of LaTeXMLMath lying in LaTeXMLMath .
LaTeXMLMath is a subgroup of LaTeXMLMath .
Taking together LaTeXMLMath and LaTeXMLMath we see that LaTeXMLMath acts transitively on the horoballs of LaTeXMLMath .
Moreover , since we assumed that LaTeXMLMath was not the boundary of an LaTeXMLMath –bundle , it follows from results in Chapter LaTeXMLMath of LaTeXMLCite , that LaTeXMLMath .
Given this we see that LaTeXMLMath is homeomorphic to LaTeXMLMath .
Since the complex LaTeXMLMath is simply connected , the result follows .
∎ Remark The proof shows that this isomorphism identifies the cusp subgroup of LaTeXMLMath with the torus subgroup of LaTeXMLMath in the obvious way .
We begin with a simple lemma ( see also LaTeXMLCite , Proposition 2.1 ) : Suppose that LaTeXMLMath is an orientable surface with LaTeXMLMath boundary components .
Fix some simple closed curve LaTeXMLMath in the boundary of a solid torus LaTeXMLMath which does not bound a disc in LaTeXMLMath .
Form a LaTeXMLMath –manifold LaTeXMLMath by identifying all the annuli LaTeXMLMath with disjoint annuli which are all parallel to neighbourhoods of LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath contains the fundamental group of a closed orientable surface .
By passing to a covering if necessary , it is easy to see that we may assume that LaTeXMLMath meets the disc in the solid torus exactly once transversally .
We form a covering space of LaTeXMLMath in the following way : Take LaTeXMLMath copies of the solid torus and LaTeXMLMath copies of LaTeXMLMath .
Any way that we glue these objects together subject to the obvious restrictions coming from the way LaTeXMLMath is glued on to LaTeXMLMath will be a covering space of LaTeXMLMath .
Fix a copy LaTeXMLMath and glue up in any way ( subject to being compatible with being a covering ) so that each of the boundary components of LaTeXMLMath appears on a different torus .
Now glue on a second copy LaTeXMLMath similarly .
We claim that this LaTeXMLMath –manifold already contains a closed surface group , which will imply the result , since the remaining part of the covering is constructed by forming HNN construction along curves which are nontrivial in both groups .
We see the claim by noting that all we have done abstractly is take two copies of LaTeXMLMath and identified some component of LaTeXMLMath of one copy with some component of LaTeXMLMath in the other .
This manifold has incompressible ( though possibly nonorientable ) boundary .
In particular , it contains a closed surface group , proving the result .
∎ Consider the manifold LaTeXMLMath .
Of course , LaTeXMLMath is convex and therefore contractible , so LaTeXMLMath is a LaTeXMLMath , moreover the isomorphism of LaTeXMLMath with LaTeXMLMath is the obvious isomorphism between the boundary torus of LaTeXMLMath and the cusp group LaTeXMLMath of LaTeXMLMath .
For future reference we note that LaTeXMLMath is a hyperbolic manifold with convex boundary .
The restriction of the covering map LaTeXMLMath is a map LaTeXMLMath which is a local isometry .
By Theorem LaTeXMLRef , every translate of LaTeXMLMath is a horoball in LaTeXMLMath which is either contained in LaTeXMLMath or is disjoint from it .
It follows that we may excise LaTeXMLMath from LaTeXMLMath and LaTeXMLMath from LaTeXMLMath and define a new map LaTeXMLEquation which continues to be a local isometry .
This map induces a covering of boundary tori of degree given by the index of LaTeXMLMath in LaTeXMLMath .
Since the curve LaTeXMLMath is mapped 1–1 by this covering , we may extend to a map of the surgered manifolds LaTeXMLEquation .
Recall that by construction LaTeXMLMath has length more than LaTeXMLMath on LaTeXMLMath , so that using the LaTeXMLMath –theorem LaTeXMLCite , we may put negatively curved metrics on both spaces and arrange that the map LaTeXMLMath continues to be a local isometry .
We now may prove : The map LaTeXMLMath is injective .
We begin by observing that although the manifold LaTeXMLMath is not convex , the manifold LaTeXMLMath with the extended metric is convex , since LaTeXMLMath is convex .
Suppose then that the theorem were false and that we could find some element in the kernel of LaTeXMLMath .
Since LaTeXMLMath is convex , any such element is freely homotopic to a geodesic LaTeXMLMath in LaTeXMLMath .
The loop LaTeXMLMath is a geodesic in LaTeXMLMath , since LaTeXMLMath is a local isometry .
This is a contradiction ; if the loop LaTeXMLMath were nullhomotopic , it would lift to be a nullhomotopic geodesic in LaTeXMLMath , a negatively curved complete simply connected manifold and this contradicts the theorem of Hadamard-Cartan .
( LaTeXMLCite Theorem 3.1 ) .
∎ Theorem LaTeXMLRef now follows : The manifold LaTeXMLMath contains the fundamental group of a closed orientable surface The manifold LaTeXMLMath contains the fundamental group of a closed surface by Lemma LaTeXMLRef and this group injects into LaTeXMLMath by Theorem LaTeXMLRef .
∎ Remarks The reason for using LaTeXMLMath rather than , for example LaTeXMLMath is that there seems to be no a fortiori control over the size of the horoball which embeds into this latter set .
Such control is needed in order to apply the LaTeXMLMath theorem .
It also seems worth clarifying why we do not work with the map LaTeXMLMath directly .
The reason is that in this setting , to define a map into LaTeXMLMath it is necessary to excise the full preimage in LaTeXMLMath of the horoball LaTeXMLMath .
The group LaTeXMLMath stabilises no parabolic fixed points of LaTeXMLMath which lie outside LaTeXMLMath , so there is no way to extend the surgery over those horospheres—and the resulting manifold can not be made to have convex boundary .
A variation on the construction above is necessary in the case that that LaTeXMLMath is the twisted LaTeXMLMath –bundle neighbourhood of a nonorientable surface since the stabiliser of LaTeXMLMath is a LaTeXMLMath extension of the surface group LaTeXMLMath .
It follows that the manifold LaTeXMLMath as described above is not the correct model for the fundamental group .
The above proof is easily modified if the surface core of the LaTeXMLMath -bundle has at least two boundary components .
One performs an analogous construction by gluing a twisted LaTeXMLMath -bundle neighbourhood onto a solid torus .
With this proviso , we continue to have LaTeXMLMath and the manifold LaTeXMLMath continues to contain a surface group .
However , if the nonorientable surface core of the LaTeXMLMath –bundle has only a single boundary component , then LaTeXMLMath is double covered by a handlebody .
We now sketch how to modify the above proof to deal with this case , and in fact the more general case that LaTeXMLMath is the boundary of the twisted LaTeXMLMath –bundle over a nonorientable surface LaTeXMLMath and that LaTeXMLMath has LaTeXMLMath boundary components .
The construction of LaTeXMLMath is a mild variation on the construction described above .
Take LaTeXMLMath “ white ” copies of the universal covering of LaTeXMLMath , which pass through LaTeXMLMath and meet the boundary of a preimage horoball LaTeXMLMath ( chosen to be small as above ) in lines which cover the LaTeXMLMath boundary components of LaTeXMLMath .
As usual arrange that these LaTeXMLMath copies are very far apart .
Similarly , take LaTeXMLMath “ black ” copies and assume that the black and white complexes are very distant from each other .
Now choose a LaTeXMLMath so that if LaTeXMLMath , then LaTeXMLMath maps this complex of LaTeXMLMath surfaces a very long distance away from itself .
The rest of LaTeXMLMath is constructed much as before save only that we do not allow the identification of white and black surfaces .
The group LaTeXMLMath is defined to be the stabiliser of the coloured complex .
The proof that the associated graph is a tree is identical .
However the complex LaTeXMLMath is a little different and for this we need to modify LaTeXMLMath .
The reason for this difference is that the stabiliser of the vertex corresponding to LaTeXMLMath is no longer LaTeXMLMath , but the two-fold extension LaTeXMLMath .
This having been noted , we define the complex LaTeXMLMath to be LaTeXMLMath with two copies of the twisted LaTeXMLMath –bundle over LaTeXMLMath attached in the obvious way .
That this complex contains a surface group now can be proved as in LaTeXMLRef .
The rest of the proof is identical .
Remark Though this seems artificial , we note that in the case of a surface with a single boundary component ( eg a Seifert surface , see LaTeXMLCite ) some sort of operation of this sort is necessary ; one can not tube a surface with a single boundary component to itself .
This section is devoted to the proof of the technical result LaTeXMLRef , which for the reader ’ s convenience we restate here : The set LaTeXMLMath has the property that any LaTeXMLMath translate of LaTeXMLMath is either entirely contained inside LaTeXMLMath or is disjoint from it .
We recall that the thin triangles constant , LaTeXMLMath , for hyperbolic space , is a constant with the property that every point on one side of a geodesic triangle is within a distance LaTeXMLMath of some point on the union of the other two sides .
In fact LaTeXMLMath There is a constant LaTeXMLMath with the following property .
Suppose that LaTeXMLMath and LaTeXMLMath are two points in LaTeXMLMath .
Then every point on the geodesic segment connecting LaTeXMLMath to LaTeXMLMath is within a distance LaTeXMLMath of LaTeXMLMath .
Throughout this proof we will refer to a translate of the chosen horoball LaTeXMLMath simply as a horoball , and to a translate of LaTeXMLMath as a thick surface .
In LaTeXMLRef we constructed a piecewise geodesic , LaTeXMLMath in LaTeXMLMath with endpoints LaTeXMLMath and LaTeXMLMath The geodesic segments in this path are of two types .
The first type is a geodesic segment in a thick surface .
The second type is a geodesic segment in a horoball .
The endpoints of these segments , other than LaTeXMLMath and LaTeXMLMath are on the intersection of some thick surface with some horoball .
A segment contained in a thick surface is orthogonal to the boundary of any horoballs at its endpoints .
The same is not generally true for segments in horoballs .
In LaTeXMLRef every segment in a horoball was long ( length at least LaTeXMLMath ) and therefore almost orthogonal to the boundary of the horoball at its endpoints .
In the present situation , this fails precisely when LaTeXMLMath or LaTeXMLMath is in a horoball .
Thus LaTeXMLMath is a piecewise geodesic , and all the segments , except possibly the first and last , have length at least LaTeXMLMath The angle between two adjacent segments is very close to LaTeXMLMath except possibly for the first and last angles .
Using the above properties of LaTeXMLMath it is well known that the geodesic with endpoints LaTeXMLMath lies within LaTeXMLMath of LaTeXMLMath .
We sketch this : Let LaTeXMLMath be the first geodesic segment and LaTeXMLMath the last segment of LaTeXMLMath .
The subpath LaTeXMLMath of LaTeXMLMath excluding the first and last segment consists of segments all of length at least LaTeXMLMath and with all angles between adjacent segments within the range LaTeXMLMath Let LaTeXMLMath be the geodesic with endpoints LaTeXMLMath and LaTeXMLMath .
Then every point of LaTeXMLMath is within LaTeXMLMath of LaTeXMLMath Now consider the geodesic path LaTeXMLMath Let LaTeXMLMath be the geodesic with the same endpoints as LaTeXMLMath Then LaTeXMLMath is a geodesic quadrilateral .
By dividing this into two geodesic triangles , we see that every point on LaTeXMLMath is within LaTeXMLMath of LaTeXMLMath Now every point of LaTeXMLMath is within LaTeXMLMath of a point of LaTeXMLMath Hence every point of LaTeXMLMath is within LaTeXMLMath of LaTeXMLMath which is contained in LaTeXMLMath ∎ There is a constant LaTeXMLMath such that if LaTeXMLMath is a geodesic tetrahedron with all four vertices in LaTeXMLMath then every point of LaTeXMLMath is within a distance LaTeXMLMath of LaTeXMLMath .
We prove below that every point in the boundary of LaTeXMLMath is within a distance LaTeXMLMath of some point in LaTeXMLMath Given a point LaTeXMLMath in the interior of LaTeXMLMath there is a geodesic triangle with edges contained in the boundary of LaTeXMLMath and which contains LaTeXMLMath Hence LaTeXMLMath is within a distance of LaTeXMLMath of some point in the boundary of LaTeXMLMath Therefore within LaTeXMLMath of some point of LaTeXMLMath Let LaTeXMLMath be a face of LaTeXMLMath Thus LaTeXMLMath is a geodesic hyperbolic triangle with vertices LaTeXMLMath contained in LaTeXMLMath Suppose that LaTeXMLMath is a point in LaTeXMLMath There is a geodesic segment , LaTeXMLMath in LaTeXMLMath containing LaTeXMLMath and with one endpoint LaTeXMLMath and the other endpoint , LaTeXMLMath on LaTeXMLMath Let LaTeXMLMath be a point of LaTeXMLMath closest to LaTeXMLMath By Lemma LaTeXMLRef , LaTeXMLMath Moreover , LaTeXMLMath is within LaTeXMLMath of LaTeXMLMath The points LaTeXMLMath are both in LaTeXMLMath .
With another application of Lemma LaTeXMLRef , we see that every point on LaTeXMLMath is within a distance LaTeXMLMath of LaTeXMLMath Since LaTeXMLMath is in LaTeXMLMath every point on LaTeXMLMath is within LaTeXMLMath of LaTeXMLMath Therefore LaTeXMLMath is within a distance LaTeXMLMath of a point within a distance LaTeXMLMath of LaTeXMLMath Hence LaTeXMLMath ∎ Suppose that LaTeXMLMath is an increasing sequence of compact subsets of LaTeXMLMath The convex hull , LaTeXMLMath of LaTeXMLMath is the closure of LaTeXMLMath It is clear that LaTeXMLMath contains the convex hull of LaTeXMLMath Since LaTeXMLMath is closed , it contains the closure of LaTeXMLMath Moreover , LaTeXMLMath is an increasing sequence of convex sets and it follows that LaTeXMLMath is convex .
Hence the closure is convex .
Thus LaTeXMLMath is contained in this closure.∎ Every point in the convex hull of LaTeXMLMath is within a distance LaTeXMLMath of LaTeXMLMath The convex hull of a compact subset , LaTeXMLMath of LaTeXMLMath is equal to the union of all geodesic tetrahedra having all four vertices in LaTeXMLMath ( we allow degenerate tetrahedra where some of the vertices coincide ) .
Fix a point in LaTeXMLMath in LaTeXMLMath and define LaTeXMLMath to be the compact subset of LaTeXMLMath of points within a distance LaTeXMLMath of LaTeXMLMath Suppose that LaTeXMLMath is a point in LaTeXMLMath By Lemma LaTeXMLRef , LaTeXMLMath Thus LaTeXMLMath Thus LaTeXMLMath Then LaTeXMLRef gives the result .
∎ By our initial careful choices of LaTeXMLMath , any translate of LaTeXMLMath not lying in LaTeXMLMath is very far from LaTeXMLMath .
However , by LaTeXMLRef , LaTeXMLMath is very close to LaTeXMLMath , whence the result .
∎ The authors thank A W Reid for carefully reading an early version of this paper and the referee for several useful comments .
Both authors are supported in part by the NSF .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be of finite order .
We show that the indices of certain twisted Dirac operators vanish if the action of LaTeXMLMath has sufficiently large fixed point codimension .
These indices occur in the Fourier expansion of the elliptic genus of LaTeXMLMath in one of its cusps .
As a by-product we obtain a new proof of a theorem of Hirzebruch and Slodowy on involutions .
Let LaTeXMLMath be a smooth closed connected LaTeXMLMath -manifold with smooth LaTeXMLMath -action and let LaTeXMLMath be the element of order two .
Hirzebruch and Slodowy LaTeXMLCite showed that the elliptic genus of LaTeXMLMath can be computed in terms of the transversal self-intersection of the fixed point manifold LaTeXMLMath and used this property to deduce a vanishing theorem for certain characteristic numbers which occur in the Fourier expansion of the elliptic genus of LaTeXMLMath in one of its cusps .
In this note we extend this vanishing theorem from involutions to cyclic actions of arbitrary order .
Our main result ( see Theorem LaTeXMLRef ) is used in LaTeXMLCite to exhibit obstructions against the existence of positively curved metrics with symmetry on LaTeXMLMath -manifolds .
The proof of Theorem LaTeXMLRef relies on the rigidity theorem for the elliptic genus which we shall recall first .
As a general reference for the theory of elliptic genera we recommend LaTeXMLCite .
The elliptic genus LaTeXMLMath , in the normalization considered in LaTeXMLCite , is a ring homomorphism from the oriented bordism ring to the ring of modular functions ( with LaTeXMLMath -character ) for LaTeXMLMath .
In one of the cusps of LaTeXMLMath ( the signature cusp ) the Fourier expansion of LaTeXMLMath has an interpretation as a series of twisted signatures LaTeXMLEquation .
Here LaTeXMLMath denotes the index of the signature operator twisted with the complexified vector bundle LaTeXMLMath , LaTeXMLMath denotes the tangent bundle and LaTeXMLMath ( resp .
LaTeXMLMath ) denotes the exterior ( resp .
symmetric ) power operation .
Following Witten LaTeXMLCite the series above is best thought of as the “ signature ” of the free loop space LaTeXMLMath of LaTeXMLMath formally localized at the manifold LaTeXMLMath of constant loops .
We denote the series of twisted signatures by LaTeXMLMath .
The main feature of the elliptic genus is its rigidity under LaTeXMLMath -actions .
This phenomenon was first explained by Witten LaTeXMLCite using standard conjectures from quantum field theory and then shown rigorously by Taubes and Bott-Taubes in LaTeXMLCite ( cf .
also LaTeXMLCite ) .
If LaTeXMLMath acts by isometries LaTeXMLMath -action .
on LaTeXMLMath and if LaTeXMLMath is a vector bundle associated to LaTeXMLMath then the signature operator twisted with the complexified vector bundle LaTeXMLMath refines to an LaTeXMLMath -equivariant operator .
Its index is a virtual LaTeXMLMath -representation which we denote by LaTeXMLMath .
In particular , the expansion of the elliptic genus in the signature cusp refines to a series of equivariant twisted signatures LaTeXMLMath .
Let LaTeXMLMath be a closed manifold with LaTeXMLMath -action .
If LaTeXMLMath is LaTeXMLMath then each equivariant twisted signature occurring as coefficient in the series LaTeXMLMath is constant as a character of LaTeXMLMath .
LaTeXMLMath We use the rigidity theorem to study the action of cyclic subgroups of LaTeXMLMath .
Our investigation is inspired by work of Hirzebruch and Slodowy LaTeXMLCite on elliptic genera and involutions .
As a motivation we shall briefly recall relevant aspects of their work .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be of order two .
By the rigidity theorem the expansion of the elliptic genus in the signature cusp is equal to the LaTeXMLMath -equivariant expansion evaluated at LaTeXMLMath , i.e .
LaTeXMLMath .
The latter can be computed via the Lefschetz fixed point formula LaTeXMLCite as a sum of local contributions LaTeXMLMath at the connected components LaTeXMLMath of the fixed point manifold LaTeXMLMath .
Hirzebruch and Slodowy showed that LaTeXMLMath is equal to the expansion of the elliptic genus ( in the signature cusp ) of the transversal self-intersection LaTeXMLMath ( cf .
LaTeXMLCite for details ) : LaTeXMLEquation .
LaTeXMLEquation Note that , by taking constant terms , one obtains the classical formula LaTeXMLMath for the ordinary signature which holds for the larger class of oriented manifolds ( cf .
LaTeXMLCite ) .
Formula ( LaTeXMLRef ) has two immediate consequences .
If the codimension of LaTeXMLMath , LaTeXMLMath , is greater than half of the dimension of LaTeXMLMath then the series LaTeXMLMath vanishes identically .
If the codimension of LaTeXMLMath is equal to half of the dimension of LaTeXMLMath then all the twisted signatures occurring as coefficients of LaTeXMLMath , LaTeXMLMath , in the series LaTeXMLMath vanish , i.e .
LaTeXMLMath .
If the codimension of LaTeXMLMath is less than half of the dimension of LaTeXMLMath then formula ( LaTeXMLRef ) still gives some information on the action of the involution LaTeXMLMath .
Namely it implies that certain twisted Dirac operators have vanishing index provided that the codimension of LaTeXMLMath is sufficiently large .
These indices are related to the elliptic genus in the following way .
Recall that the LaTeXMLMath -series LaTeXMLMath is the expansion of the elliptic genus LaTeXMLMath in one of the cusps of LaTeXMLMath .
In a different cusp ( the LaTeXMLMath -cusp ) the expansion of LaTeXMLMath may be described ( using a suitable change of cusps ) by LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath is a characteristic number of the pair LaTeXMLMath which , in the presence of a LaTeXMLMath -structure , is equal to the index of the Dirac operator twisted with the complexified vector bundle LaTeXMLMath .
We call the series above the expansion of LaTeXMLMath in the LaTeXMLMath -cusp .
Note that LaTeXMLMath and LaTeXMLMath are different expansions of the same modular function LaTeXMLMath and determine each other .
By formula ( LaTeXMLRef ) LaTeXMLMath which implies the following generalization of the Atiyah-Hirzebruch vanishing theorem for the LaTeXMLMath -genus LaTeXMLCite .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be of order two .
If LaTeXMLMath then the expansion of the elliptic genus of LaTeXMLMath in the LaTeXMLMath -cusp has a pole of order less than LaTeXMLMath .
LaTeXMLMath The reasoning indicated above also leads to obstructions against the existence of LaTeXMLMath -actions on highly connected manifolds which might be of independent interest .
Let LaTeXMLMath be a LaTeXMLMath -connected LaTeXMLMath -manifold .
Assume LaTeXMLMath .
If LaTeXMLMath admits a non-trivial LaTeXMLMath -action then the expansion of the elliptic genus of LaTeXMLMath in the LaTeXMLMath -cusp has a pole of order less than LaTeXMLMath .
Note that for LaTeXMLMath the LaTeXMLMath -condition follows from the connectivity assumption .
We remark that the conclusion of Theorem LaTeXMLRef also holds if LaTeXMLMath is a connected LaTeXMLMath -manifold with non-trivial LaTeXMLMath -action and LaTeXMLMath for LaTeXMLMath ( see Section LaTeXMLRef for a proof ) .
The next result extends Theorem LaTeXMLRef to finite cyclic actions of arbitrary order .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action and let LaTeXMLMath be of order LaTeXMLMath .
If LaTeXMLMath then the expansion of the elliptic genus of LaTeXMLMath in the LaTeXMLMath -cusp has a pole of order less than LaTeXMLMath .
The theorem follows from a more general result ( see Theorem LaTeXMLRef and the proof in Section LaTeXMLRef ) .
As indicated above the proof of Theorem LaTeXMLRef given in LaTeXMLCite is specific to actions of order two .
To deal with the general situation we consider the expansion of the equivariant elliptic genus in the LaTeXMLMath -cusp and study the local contributions of the LaTeXMLMath -fixed point components using the rigidity theorem .
We close this section with some consequences of Theorem LaTeXMLRef .
Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action .
Let LaTeXMLMath be of order LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath vanishes .
If LaTeXMLMath then LaTeXMLMath and LaTeXMLMath vanish .
If LaTeXMLMath acts with isolated fixed points then LaTeXMLMath vanishes identically .
Let LaTeXMLMath be of order LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath vanishes .
If LaTeXMLMath then LaTeXMLMath and LaTeXMLMath vanish .
If LaTeXMLMath acts with isolated fixed points then LaTeXMLMath is equal to the signature of LaTeXMLMath .
Let LaTeXMLMath be of order LaTeXMLMath .
If LaTeXMLMath acts with isolated fixed points then LaTeXMLMath and LaTeXMLMath vanish .
LaTeXMLMath In this section we state the main result of this note .
Let LaTeXMLMath be a connected LaTeXMLMath -manifold and let LaTeXMLMath be a natural number .
At a connected component LaTeXMLMath of the fixed point manifold LaTeXMLMath the tangent bundle LaTeXMLMath splits equivariantly as the direct sum of LaTeXMLMath and the normal bundle LaTeXMLMath .
The latter splits ( non-canonically ) as a direct sum LaTeXMLMath corresponding to the irreducible real LaTeXMLMath -dimensional LaTeXMLMath -representations LaTeXMLMath , LaTeXMLMath .
We fix such a decomposition of LaTeXMLMath .
For each LaTeXMLMath choose LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath .
On each vector bundle LaTeXMLMath introduce a complex structure such that LaTeXMLMath acts on LaTeXMLMath by scalar multiplication with LaTeXMLMath .
The LaTeXMLMath ( taken with multiplicities ) are called the rotation numbers of the LaTeXMLMath -action at LaTeXMLMath .
Finally define LaTeXMLEquation where LaTeXMLMath denotes the complex dimension of LaTeXMLMath and LaTeXMLMath runs over the connected components of LaTeXMLMath ( to keep notation light we have suppressed the dependence of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath on LaTeXMLMath ) .
We are now in the position to state Let LaTeXMLMath be a LaTeXMLMath -manifold with LaTeXMLMath -action .
If LaTeXMLMath then the expansion of the elliptic genus of LaTeXMLMath in the LaTeXMLMath -cusp has a pole of order less than LaTeXMLMath .
If LaTeXMLMath has order LaTeXMLMath then LaTeXMLMath and LaTeXMLMath is the codimension of the connected component of LaTeXMLMath which contains LaTeXMLMath .
Thus LaTeXMLMath and one recovers Theorem LaTeXMLRef .
In general if LaTeXMLMath has order LaTeXMLMath then LaTeXMLMath and one obtains Theorem LaTeXMLRef .
Note that without the LaTeXMLMath condition the conclusion of the theorem fails in general , e.g .
for complex projective spaces of even complex dimension ( see however Remark LaTeXMLRef ) .
We may assume that the dimension of LaTeXMLMath is divisible by LaTeXMLMath and that the fixed point manifold LaTeXMLMath is not empty since otherwise LaTeXMLMath is rationally zero bordant by the Lefschetz fixed point formula LaTeXMLCite and LaTeXMLMath vanishes .
We may also assume that the LaTeXMLMath -action lifts to the LaTeXMLMath -structure ( otherwise the action is odd which forces the elliptic genus to vanish , see for example LaTeXMLCite ) .
We fix an LaTeXMLMath -equivariant Riemannian metric on LaTeXMLMath .
The proof is divided into three steps .
Step 1 : We describe the equivariant elliptic genus at LaTeXMLMath .
Consider the expansion of LaTeXMLMath in the LaTeXMLMath -cusp .
Recall that the coefficients are indices of twisted Dirac operators associated to the LaTeXMLMath -structure .
Since the LaTeXMLMath -action lifts to the LaTeXMLMath -structure each index refines to a virtual LaTeXMLMath -representation and the series refines to an element of LaTeXMLMath which we denote by LaTeXMLMath .
Note that LaTeXMLMath and LaTeXMLMath are different expansions of the same function .
Hence the rigidity of LaTeXMLMath ( see Theorem LaTeXMLRef ) is equivalent to the rigidity of LaTeXMLMath , i.e .
each coefficient of the series LaTeXMLMath is constant as a character of LaTeXMLMath .
Let LaTeXMLMath be a fixed topological generator .
By the Lefschetz fixed point formula LaTeXMLCite the series LaTeXMLMath is equal to a sum of local data LaTeXMLEquation where LaTeXMLMath runs over the connected components of LaTeXMLMath .
Recall from Section LaTeXMLRef that we have decomposed the normal bundle LaTeXMLMath of LaTeXMLMath as a direct sum LaTeXMLMath of complex vector bundles .
Fix the orientation for LaTeXMLMath which is compatible with the orientation of LaTeXMLMath and the complex structure of LaTeXMLMath .
Let LaTeXMLMath denote the set of roots of LaTeXMLMath and let LaTeXMLMath denote the set of roots of the complex vector bundle LaTeXMLMath .
The local datum LaTeXMLMath may be described in cohomological terms as ( cf .
LaTeXMLCite , Section 3 ) : LaTeXMLEquation .
Here LaTeXMLMath is equal to LaTeXMLEquation .
LaTeXMLMath , LaTeXMLMath denotes the fundamental cycle of LaTeXMLMath and LaTeXMLMath is the Kronecker pairing .
In general each local datum LaTeXMLMath depends on LaTeXMLMath .
However , the sum LaTeXMLMath is equal to LaTeXMLMath and therefore independent of LaTeXMLMath by the rigidity theorem .
Step 2 : Each local datum is the expansion of a meromorphic function on LaTeXMLMath where LaTeXMLMath denotes the upper half plane .
As in the proof of the rigidity theorem given in LaTeXMLCite ( cf .
also LaTeXMLCite ) modularity properties of these functions will be central for the argument .
In this step we examine some of their properties .
We begin to recall relevant properties of the series LaTeXMLMath ( see for example LaTeXMLCite ) .
For LaTeXMLMath and LaTeXMLMath satisfying LaTeXMLMath the series LaTeXMLMath converges normally to a holomorphic function .
This function extends to a meromorphic function LaTeXMLMath on LaTeXMLMath after the change of variables LaTeXMLMath where LaTeXMLMath is in LaTeXMLMath .
The function LaTeXMLMath is elliptic in LaTeXMLMath for the lattice LaTeXMLMath and satisfies LaTeXMLEquation .
The zeros of LaTeXMLMath are simple and located at LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath be a topological generator of LaTeXMLMath .
In view of formula ( LaTeXMLRef ) and the properties of LaTeXMLMath the local datum LaTeXMLMath converges to a meromorphic function LaTeXMLMath on LaTeXMLMath evaluated at LaTeXMLMath .
We proceed to explain how this function is related to LaTeXMLMath .
For a function LaTeXMLMath in the variables LaTeXMLMath which is smooth in the origin let LaTeXMLMath denote the Taylor expansion of LaTeXMLMath with respect to LaTeXMLMath .
It follows from formula ( LaTeXMLRef ) that LaTeXMLMath is related to LaTeXMLMath by ( see for example LaTeXMLCite ) : LaTeXMLEquation .
The properties of LaTeXMLMath stated above imply corresponding properties for LaTeXMLMath .
In particular , LaTeXMLMath is elliptic for the lattice LaTeXMLMath and satisfies LaTeXMLEquation .
For fixed LaTeXMLMath the poles of LaTeXMLMath are contained in LaTeXMLMath for some LaTeXMLMath depending on the rotation numbers of the LaTeXMLMath -action at LaTeXMLMath ( see for example LaTeXMLCite ) .
In general LaTeXMLMath depends on LaTeXMLMath .
If LaTeXMLMath is a topological generator of LaTeXMLMath , i.e .
if LaTeXMLMath is irrational , then LaTeXMLMath converges to the sum LaTeXMLMath by the Lefschetz fixed point formula and the latter is independent of LaTeXMLMath by the rigidity theorem .
Note that the original data may be recovered from LaTeXMLMath by taking the expansion of LaTeXMLMath with respect to LaTeXMLMath .
Step 3 : In the final step we study the series LaTeXMLMath in terms of the sum LaTeXMLMath where LaTeXMLMath approximates LaTeXMLMath .
We choose LaTeXMLMath in such a way that LaTeXMLMath is periodic with respect to LaTeXMLMath for some LaTeXMLMath ( see below ) .
Note that in general the series LaTeXMLMath does not converge if LaTeXMLMath is close to LaTeXMLMath and the LaTeXMLMath -expansion of LaTeXMLMath , denoted by LaTeXMLMath , is different from the corresponding contribution LaTeXMLMath in the Lefschetz fixed point formula for LaTeXMLMath .
In particular , we can not compare LaTeXMLMath and LaTeXMLMath directly .
However , since the sum LaTeXMLMath is independent of LaTeXMLMath the sum LaTeXMLMath is equal to the elliptic genus in the LaTeXMLMath -cusp ( see last step ) .
Using the properties of LaTeXMLMath described above and the assumption on LaTeXMLMath we will show that LaTeXMLMath has a pole of order less than LaTeXMLMath .
This will complete the proof .
Here are the details .
The discussion in the last step implies that the poles of LaTeXMLMath , LaTeXMLMath , are contained in LaTeXMLMath for some LaTeXMLMath .
Choose LaTeXMLMath , where LaTeXMLMath is a fixed rational positive number LaTeXMLMath .
Hence , LaTeXMLMath is close to LaTeXMLMath and LaTeXMLMath is holomorphic on LaTeXMLMath for every LaTeXMLMath .
Using LaTeXMLMath , LaTeXMLMath , and the transformation property LaTeXMLMath one computes that LaTeXMLMath is ( up to sign ) equal to LaTeXMLMath , where LaTeXMLEquation and LaTeXMLMath .
Note that for some LaTeXMLMath ( depending on LaTeXMLMath and the rotation numbers ) every summand LaTeXMLMath is periodic with respect to LaTeXMLMath .
We claim that its expansion LaTeXMLMath has a pole of order less than LaTeXMLMath .
Since the expansion of LaTeXMLMath ( with respect to LaTeXMLMath ) is equal to LaTeXMLMath the expansion of LaTeXMLEquation can be easily computed in terms of LaTeXMLMath .
The computation shows that the expansion of ( LaTeXMLMath ) has a pole of order LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , LaTeXMLMath are arbitrarily small it follows that LaTeXMLMath has a pole of order less than LaTeXMLMath .
As explained in the beginning of this step the sum LaTeXMLMath is equal to the expansion of the elliptic genus in the LaTeXMLMath -cusp .
Hence , LaTeXMLMath has a pole of order less than LaTeXMLMath .
This completes the proof .
LaTeXMLMath Essentially the same reasoning applies to orientable LaTeXMLMath -manifolds ( not necessarily LaTeXMLMath ) for which the equivariant elliptic genus is rigid .
The rigidity theorem is known to hold for oriented manifolds with finite second homotopy group LaTeXMLCite and for LaTeXMLMath -manifolds with first Chern class a torsion class LaTeXMLCite .
Theorem LaTeXMLRef is also true for these manifolds .
In this section we adapt the arguments of LaTeXMLCite to study the elliptic genus of certain LaTeXMLMath -manifolds including highly connected manifolds .
To begin with we recall the Lefschetz fixed point formula for twisted signatures .
Let LaTeXMLMath be an oriented closed LaTeXMLMath -manifold , LaTeXMLMath an LaTeXMLMath -equivariant vector bundle over LaTeXMLMath and LaTeXMLMath the element of order LaTeXMLMath .
In the following we shall always assume that the fixed point manifold LaTeXMLMath is orientable ( this is the case if LaTeXMLMath is LaTeXMLMath LaTeXMLCite ) .
By the Lefschetz fixed point formula the equivariant twisted signature LaTeXMLMath evaluated at LaTeXMLMath is equal to a sum of local data LaTeXMLMath at the connected components LaTeXMLMath of the fixed point manifold LaTeXMLMath LaTeXMLEquation .
The local contributions are given by ( cf .
LaTeXMLCite ) LaTeXMLEquation where LaTeXMLEquation .
Here LaTeXMLMath ( resp .
LaTeXMLMath ) denote the formal roots of LaTeXMLMath ( resp .
the normal bundle LaTeXMLMath of LaTeXMLMath ) for compatible orientations of LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is the Euler class of LaTeXMLMath and LaTeXMLMath denotes the equivariant Chern character of LaTeXMLMath .
The local datum LaTeXMLMath is obtained by evaluating the cohomology class LaTeXMLMath on the fundamental cycle LaTeXMLMath via the Kronecker pairing LaTeXMLMath .
Note that LaTeXMLMath vanishes if LaTeXMLMath is a torsion class .
Hence , the following lemma is immediate .
Let LaTeXMLMath and LaTeXMLMath be as above and let LaTeXMLMath be of codimension LaTeXMLMath .
If LaTeXMLMath then the local datum LaTeXMLMath vanishes .
LaTeXMLMath For the proof of the next lemma recall that the Euler class of the normal bundle of LaTeXMLMath is equal to LaTeXMLMath , where LaTeXMLMath denotes the push forward in cohomology for the oriented normal bundle LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be as above .
If LaTeXMLMath then LaTeXMLMath vanishes for any connected component LaTeXMLMath of codimension LaTeXMLMath .
LaTeXMLMath We shall now apply these observations to the elliptic genus .
Let LaTeXMLMath be a LaTeXMLMath -manifold .
Assume that LaTeXMLMath for LaTeXMLMath .
If LaTeXMLMath admits a non-trivial LaTeXMLMath -action then the expansion of LaTeXMLMath in the LaTeXMLMath -cusp has a pole of order less than LaTeXMLMath .
Proof : Let LaTeXMLMath denote the element of order two .
Arguing as in the proof of Theorem LaTeXMLRef we may assume that the dimension of LaTeXMLMath and the dimension of each connected component LaTeXMLMath is divisible by LaTeXMLMath .
Consider the expansion LaTeXMLMath of the LaTeXMLMath -equivariant elliptic genus in the signature cusp .
By the rigidity theorem LaTeXMLMath is equal to the non-equivariant expansion LaTeXMLMath .
By the Lefschetz fixed point formula LaTeXMLMath is a sum of local contributions LaTeXMLMath at the connected components LaTeXMLMath of LaTeXMLMath : LaTeXMLEquation .
Note that each coefficient of the LaTeXMLMath -power series LaTeXMLMath is the local contribution in the Lefschetz fixed point formula of an equivariant twisted signature evaluated at LaTeXMLMath .
Since LaTeXMLMath for LaTeXMLMath the contribution LaTeXMLMath vanishes if LaTeXMLMath ( see Lemma LaTeXMLRef ) .
If LaTeXMLMath then LaTeXMLMath is equal to LaTeXMLMath ( see formula ( LaTeXMLRef ) ) .
Hence , LaTeXMLEquation .
This implies that the expansion of LaTeXMLMath in the LaTeXMLMath -cusp has a pole of order less than LaTeXMLMath .
LaTeXMLMath Finally note that Theorem LaTeXMLRef is a direct consequence of the theorem above .
Anand Dessai e-mail : dessai @ math.uni-augsburg.de http : //www.math.uni-augsburg.de/geo/dessai/homepage.html Department of Mathematics , University of Augsburg , D-86159 Augsburg
We construct a versal family of deformations of CR structures in five dimensions , using a differential complex closely related to the differential form complex introduced by Rumin for contact manifolds .
A natural problem in several complex variables is that of classifying the deformations of an isolated singularity in a complex-analytic variety .
The problem is solved by constructing a “ versal family ” of deformations of the singularity , which is , roughly speaking , a minimal family of deformations that includes biholomorphic representatives of all other deformations .
( See Section LaTeXMLRef for a precise definition . )
Versal families for isolated singularities were first constructed from an algebraic point of view in the late 1960s and early 1970s by Tjurina , Grauert , and Donin LaTeXMLCite .
Shortly thereafter , M. Kuranishi LaTeXMLCite outlined a program for relating deformations of an isolated singularity to deformations of the CR structure on a real hypersurface obtained by intersecting the variety with a small sphere surrounding the singular point ( the “ link ” of the singularity ) .
Then Kuranishi ’ s construction was extended and simplified by subsequent work of the first author and others LaTeXMLCite .
A fundamental limitation of all of these results has been a dimensional restriction : Because the deformation complex that was introduced in LaTeXMLCite failed to be subelliptic in low dimensions , these results only applied to CR manifolds of dimension LaTeXMLMath or more ( and therefore to singularities of varieties whose complex dimension is at least LaTeXMLMath ) .
The purpose of this paper is to extend the Kuranishi construction of versal families of CR structures to the case of LaTeXMLMath -dimensional CR manifolds .
The new idea here is a subelliptic estimate and consequent Hodge theory for a certain subcomplex of the standard deformation complex inspired by recent work of M. Rumin on contact manifolds .
Recently , Miyajima LaTeXMLCite introduced an alternative approach to constructing versal families in all dimensions , based on analyzing deformations not only of the CR structure , but of the CR structure together with its embedding into LaTeXMLMath .
The present approach is of independent interest , however , because it represents a completion of the original Kuranishi program of constructing an intrinsically-defined versal family of deformations of the CR structure itself .
There appears to be little hope for extending this intrinsic approach to the case of LaTeXMLMath -dimensional CR manifolds , because the relevant cohomology groups in that case are infinite-dimensional .
Let LaTeXMLMath be a compact strictly pseudoconvex CR manifold of real dimension LaTeXMLMath .
Deformations of the CR structure of LaTeXMLMath can be represented as LaTeXMLMath -valued LaTeXMLMath -forms , where LaTeXMLMath is a LaTeXMLMath -dimensional complex subbundle of LaTeXMLMath transverse to the antiholomorphic tangent bundle LaTeXMLMath ( see Section 2 for precise definitions ) .
The space of such forms fits into a complex LaTeXMLMath , the standard deformation complex LaTeXMLCite .
In earlier work on higher-dimensional CR deformation theory , the first author defined a subcomplex LaTeXMLMath of the standard deformation complex corresponding to deformations of the CR structure that leave the contact structure fixed .
When LaTeXMLMath , there is a subelliptic estimate on LaTeXMLMath , which leads to the construction of a versal family LaTeXMLCite .
But if LaTeXMLMath , there is no such estimate .
In this paper , inspired by the differential-form complex introduced by Rumin LaTeXMLCite for studying de Rham theory on contact manifolds , we extend the LaTeXMLMath complex by defining a new second-order operator LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath is a one-dimensional subbundle of LaTeXMLMath transverse to LaTeXMLMath .
This is closely related to Rumin ’ s complex , in a way we will explain in Section 4 .
A similar complex has also been used in LaTeXMLCite .
Once we have proved an a priori estimate on LaTeXMLMath , it follows that there is a Kodaira-Hodge decomposition theorem on LaTeXMLMath .
Using techniques similar to those in LaTeXMLCite , this leads to a construction of the versal family in the LaTeXMLMath -dimensional case .
We remark that Rumin has recently suggested a simpler proof of an analogous estimate for the complex version of his complex in arbitrary dimensions .
We hope to pursue this further in another paper .
Let LaTeXMLMath be a CR manifold .
By this we mean that LaTeXMLMath is a smooth manifold of dimension LaTeXMLMath and LaTeXMLMath is a complex subbundle of the complexified tangent bundle LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation where by LaTeXMLMath we mean the space of LaTeXMLMath sections of the bundle LaTeXMLMath .
For convenience we will write LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for the real bundle LaTeXMLMath .
We assume that there is a global non-vanishing real one-form LaTeXMLMath that annihilates LaTeXMLMath ; that is , such that LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath is naturally oriented , the existence of such a form is equivalent to LaTeXMLMath being orientable .
We define the Levi form LaTeXMLMath by LaTeXMLEquation .
If this Levi form LaTeXMLMath is positive definite or negative definite , then LaTeXMLMath is called strictly ( or strongly ) pseudoconvex .
( After this section , we will always assume that our CR structure is strictly pseudoconvex . )
Notice that the Levi form gives us a metric on LaTeXMLMath that extends to a Riemannian metric on all of LaTeXMLMath by declaring that LaTeXMLMath is unit length and orthogonal to LaTeXMLMath .
We will call this metric the Webster metric ( see LaTeXMLCite ) .
When LaTeXMLMath is strictly pseudoconvex , we will call a choice of LaTeXMLMath -form LaTeXMLMath a pseudohermitian structure .
Let LaTeXMLMath be the unique real vector field satisfying LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
Notice that this implies that for every point LaTeXMLMath of LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath denote the complex line bundle LaTeXMLMath , and set LaTeXMLMath .
We then get vector bundle decompositions LaTeXMLEquation and LaTeXMLEquation .
Note that these decompositions depend on the choice of LaTeXMLMath ( and thus LaTeXMLMath ) and so are not CR-invariant .
We will often take advantage of these decompositions to project onto various components .
For a vector LaTeXMLMath , let us write LaTeXMLMath for the LaTeXMLMath -component of LaTeXMLMath , LaTeXMLMath for the LaTeXMLMath -component , LaTeXMLMath for the LaTeXMLMath -component , and LaTeXMLMath for the LaTeXMLMath -component , according to these decompositions .
Moreover , since we will often be dealing with vector-valued forms , let us use the same notation for the projection of , say , LaTeXMLMath into component parts LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath via equations ( LaTeXMLRef ) and ( LaTeXMLRef ) .
It is often useful to identify LaTeXMLMath with LaTeXMLMath .
Notice that this identification depends on the choice of LaTeXMLMath , and so is not CR-invariant .
There is a natural bigrading on LaTeXMLMath , so we may make a further identification LaTeXMLEquation .
This allows us to identify , for example , LaTeXMLMath with honest forms on LaTeXMLMath .
Finally , we note that we will use the Einstein summation convention whenever possible .
We will use Roman indices ( LaTeXMLMath , LaTeXMLMath , for example ) to indicate sums from LaTeXMLMath to LaTeXMLMath , and Greek indices ( LaTeXMLMath , LaTeXMLMath , and so on ) for sums from LaTeXMLMath to LaTeXMLMath .
In this section we survey previous work on the deformation theory of CR structures .
This work was initiated by Kuranishi LaTeXMLCite as a CR analogue of his work on complex manifolds .
Most of the work reviewed here was done by the first author LaTeXMLCite Following work of the first author LaTeXMLCite , we introduce a first order differential operator LaTeXMLMath by LaTeXMLEquation .
As in the case of scalar-valued differential forms , this generalizes to operators LaTeXMLMath ( LaTeXMLMath ) given by LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
We then have a differential complex LaTeXMLEquation .
LaTeXMLEquation with LaTeXMLMath ( see LaTeXMLCite ) .
This complex is called the standard deformation complex .
A complex subbundle LaTeXMLMath is an almost CR structure ( and the pair LaTeXMLMath is an almost CR manifold ) if LaTeXMLMath and LaTeXMLMath .
An almost CR structure LaTeXMLMath is at finite distance from LaTeXMLMath if LaTeXMLMath is a bundle isomorphism .
These almost CR structures are characterized by the fact that they are graphs over LaTeXMLMath : there is a bijective correspondence between elements LaTeXMLMath and almost CR structures LaTeXMLEquation at finite distance from LaTeXMLMath ( see , for example , LaTeXMLCite ) .
The almost CR structure LaTeXMLMath is a CR structure exactly when it satisfies the integrability condition , which can be written as the non-linear partial differential equation LaTeXMLEquation where LaTeXMLMath ( LaTeXMLMath ) are the parts of LaTeXMLMath that are degree LaTeXMLMath in LaTeXMLMath .
They are given by LaTeXMLEquation and LaTeXMLEquation .
See LaTeXMLCite and the proof given therein for details .
If we consider only deformations LaTeXMLMath that preserve the contact structure ( that is , for which LaTeXMLMath ) , then we are simply restricting to LaTeXMLMath .
For such LaTeXMLMath , we notice that LaTeXMLMath and that LaTeXMLMath ( so LaTeXMLMath ) .
Thus LaTeXMLMath .
Our integrability condition LaTeXMLMath is thus equivalent in this case to LaTeXMLMath and LaTeXMLMath .
( Compare LaTeXMLCite . )
This in part motivates the definition of the following subspaces of LaTeXMLMath : LaTeXMLEquation .
For LaTeXMLMath , then , the integrability condition becomes LaTeXMLMath .
We remark that contrary to appearances , the definition of LaTeXMLMath is an algebraic condition on LaTeXMLMath , not a differential one .
To see this , apply the one-form LaTeXMLMath to both sides of equation ( LaTeXMLRef ) .
By the definition of LaTeXMLMath , the left-hand side is zero , and so LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath maps into LaTeXMLMath , which is annihilated by LaTeXMLMath , the second sum is a sum of zeros .
Using LaTeXMLMath for LaTeXMLMath , the first sum becomes LaTeXMLEquation .
This is an algebraic condition on LaTeXMLMath .
In fact , the spaces LaTeXMLMath are smooth sections of vector bundles .
There are LaTeXMLCite subbundles LaTeXMLMath such that LaTeXMLMath .
By restricting LaTeXMLMath to LaTeXMLMath , we get a sequence of maps LaTeXMLMath LaTeXMLEquation and LaTeXMLMath is integrable for LaTeXMLMath if and only if LaTeXMLMath .
It turns out that LaTeXMLMath and the resulting complex LaTeXMLEquation is a differential subcomplex of the standard deformation complex ( see LaTeXMLCite ) .
This subcomplex still contains enough information to be useful ; for example , the inclusion map LaTeXMLMath induces a map LaTeXMLEquation that is an isomorphism if LaTeXMLMath and surjective if LaTeXMLMath LaTeXMLCite .
Furthermore , there are a subelliptic estimate for this complex LaTeXMLCite and a Kodaira-Hodge decomposition theorem for LaTeXMLMath LaTeXMLCite , provided LaTeXMLMath .
That is , if we define the Laplacian LaTeXMLMath , then there is a harmonic projector LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and a Neumann operator LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
This construction fails if LaTeXMLMath , as there is no subelliptic estimate for this complex .
In this section , we introduce a new complex as a replacement for the differential subcomplex ( LaTeXMLRef ) of the standard differential complex .
Set LaTeXMLEquation .
We then get a new differential subcomplex of the standard differential complex ( LaTeXMLRef ) : LaTeXMLEquation .
This complex is a generalization of ideas of the first author that is new for use in this setting , but it has been introduced by Buchweitz and Millson LaTeXMLCite based in part on ideas of the third author .
It is straightforward to see that this is a complex : the definition of LaTeXMLMath ensures that LaTeXMLMath and the fact that ( LaTeXMLRef ) is a subcomplex of the standard differential complex ( LaTeXMLRef ) means that , in fact , LaTeXMLMath .
We would like to make a few remarks about LaTeXMLMath .
It is not the space of smooth sections of a vector bundle over LaTeXMLMath ; rather , it is the image of a first order differential operator .
We define this operator LaTeXMLMath as follows : for LaTeXMLMath , we may write LaTeXMLMath for some smooth function LaTeXMLMath ( namely , LaTeXMLMath ) .
We then get an element LaTeXMLMath by requiring that LaTeXMLMath : LaTeXMLMath .
This is equivalent to LaTeXMLMath for all LaTeXMLMath .
Another way to write this is LaTeXMLEquation because LaTeXMLMath and LaTeXMLMath .
Since our CR structure is strictly pseudoconvex , equation ( LaTeXMLRef ) uniquely determines LaTeXMLMath .
Thus LaTeXMLMath is the image of the first order differential operator LaTeXMLMath defined by LaTeXMLMath .
Define a second-order operator LaTeXMLMath as the composition LaTeXMLMath .
We then clearly get a complex LaTeXMLEquation .
It is this complex that we will use to get our subelliptic estimate and therefore our decomposition theorems .
Notice that LaTeXMLMath includes a first derivative of LaTeXMLMath .
Using a local moving frame LaTeXMLMath for LaTeXMLMath satisfying LaTeXMLEquation we set LaTeXMLMath ( note implicit sum ) .
Expanding LaTeXMLMath , we get LaTeXMLEquation .
This simplifies to LaTeXMLMath , so LaTeXMLMath .
Thus LaTeXMLMath is indeed a first order operator , and our composition LaTeXMLMath is a second order operator .
Finally , we would like to relate our operator LaTeXMLMath to that of Rumin LaTeXMLCite .
Define , for LaTeXMLMath , LaTeXMLEquation and set LaTeXMLMath for LaTeXMLMath .
Although the definition ( LaTeXMLRef ) seems to depend on non-invariant decomposition ( LaTeXMLRef ) , we may actually express LaTeXMLMath invariantly as LaTeXMLEquation where LaTeXMLMath is the ideal generated by LaTeXMLMath and LaTeXMLMath .
Since this ideal is CR-invariant , the definition of LaTeXMLMath is as well .
Below the middle dimension , we define a slightly different space .
For LaTeXMLMath , set LaTeXMLEquation and LaTeXMLMath , so LaTeXMLEquation is CR-invariant as well .
Rumin ’ s LaTeXMLMath operator is a map LaTeXMLMath given by LaTeXMLMath , where the representative LaTeXMLMath of LaTeXMLMath is chosen so that LaTeXMLMath will be in LaTeXMLMath .
There is then a complex LaTeXMLEquation which decomposes into subcomplexes LaTeXMLEquation .
We hope to provide more details on these complexes in another paper .
The relation between our complex ( LaTeXMLRef ) and Rumin ’ s complex ( LaTeXMLRef ) occurs when LaTeXMLMath in Rumin ’ s complex , in which case ( LaTeXMLRef ) is LaTeXMLEquation and we note that LaTeXMLMath .
Let LaTeXMLMath denote a nonvanishing closed LaTeXMLMath -form ( that is , an element of LaTeXMLMath ) , if one exists .
For any positive LaTeXMLMath , we get a map LaTeXMLMath by interior multiplying the vector part of LaTeXMLMath into LaTeXMLMath , then wedging the remainder with the form part of LaTeXMLMath .
Let LaTeXMLMath be given by LaTeXMLMath .
The claim is that each LaTeXMLMath is an isomorphism and the following diagram commutes : LaTeXMLEquation .
Since LaTeXMLMath always exists locally , the two complexes are locally isomorphic .
If the canonical line bundle is trivial , then this complex version ( LaTeXMLRef ) of the Rumin complex is isomorphic to our new complex ( LaTeXMLRef ) .
In this section , we state two of our main results .
First , we produce a subelliptic estimate at LaTeXMLMath for our complex ( LaTeXMLRef ) in the LaTeXMLMath -dimensional case .
Using this , we get a Hodge-Kodaira decomposition theorem for elements of LaTeXMLMath .
We begin with some preliminaries .
Our choice of pseudohermitian structure LaTeXMLMath determines the pseudohermitian connection LaTeXMLMath ( see LaTeXMLCite ) : this is the unique connection that is compatible with LaTeXMLMath and its complex structure , for which LaTeXMLMath and LaTeXMLMath are parallel , and satisfying an additional torsion condition .
For any tensor field LaTeXMLMath on LaTeXMLMath , the total covariant derivative LaTeXMLMath can be decomposed as LaTeXMLEquation where LaTeXMLMath involves derivatives only with respect to vector fields in LaTeXMLMath , and LaTeXMLMath only with respect to vector fields in LaTeXMLMath .
Writing LaTeXMLMath , the Folland-Stein norms LaTeXMLMath are defined by LaTeXMLEquation where LaTeXMLMath denotes the LaTeXMLMath norm defined with respect to the Webster metric .
( Note that in LaTeXMLCite , the LaTeXMLMath and LaTeXMLMath norms were called LaTeXMLMath and LaTeXMLMath , respectively . )
We will write LaTeXMLMath for the hermitian inner product that corresponds to the norm LaTeXMLMath , and for any bundle LaTeXMLMath we will let LaTeXMLMath denote the completion of LaTeXMLMath with respect to the LaTeXMLMath norm .
Define a second-order operator LaTeXMLMath .
We then define our Laplacian LaTeXMLMath by LaTeXMLMath , where the adjoints are defined with respect to the complex ( LaTeXMLRef ) .
We use this operator and the norms defined above to express our subelliptic estimate in the following theorem .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of dimension LaTeXMLMath .
Then there exists a constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
The details of the proof of this estimate will be confined to the next section .
We define new norms that are Sobolev extensions of the Folland-Stein norms LaTeXMLMath as follows .
We set LaTeXMLEquation .
The first parameter , LaTeXMLMath , specifies the number of derivatives in the LaTeXMLMath directions , whereas the second parameter , LaTeXMLMath , is the number of unconstrained derivatives .
( We remark that in LaTeXMLCite these norms were written slightly differently : for example , LaTeXMLMath was LaTeXMLMath . )
Then our main estimate , Theorem LaTeXMLRef , together with standard integration-by-parts techniques , gives us the following Sobolev estimate .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of dimension LaTeXMLMath .
For each positive integer LaTeXMLMath , there exists a constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
Let us write LaTeXMLMath for the harmonic elements of LaTeXMLMath , with respect to the Laplacian LaTeXMLMath .
In order to find a useful expression for LaTeXMLMath , we use the following lemma to express the adjoint of LaTeXMLMath in simpler terms .
Let LaTeXMLMath be the completion of LaTeXMLMath under the LaTeXMLMath norm , and LaTeXMLMath is orthogonal projection .
Then we have the following relations : LaTeXMLMath , where LaTeXMLMath is the formal adjoint of LaTeXMLMath LaTeXMLMath The first conclusion follows from the relation between the standard deformation complex ( LaTeXMLRef ) and the complex ( LaTeXMLRef ) involving LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , we may write LaTeXMLMath , from which it follows that LaTeXMLMath on LaTeXMLMath .
That LaTeXMLMath is due to two simple facts : first , that LaTeXMLMath and , second , that LaTeXMLMath is an isomorphism .
∎ This lemma then implies that we may write LaTeXMLMath as LaTeXMLEquation .
The subelliptic estimate in Theorem LaTeXMLRef gives us the following Hodge-Kodaira decomposition theorem .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of dimension LaTeXMLMath .
Then LaTeXMLEquation .
Moreover , there exists a Neumann operator LaTeXMLMath and a harmonic projector LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath .
We will construct the Neumann operator LaTeXMLMath and the harmonic projector LaTeXMLMath by considering the differential equation LaTeXMLEquation .
Let us write LaTeXMLMath for elements of LaTeXMLMath that are orthogonal to LaTeXMLMath with respect to the LaTeXMLMath norm .
We begin with a fairly standard lemma .
There is a constant LaTeXMLMath for which LaTeXMLEquation for all LaTeXMLMath .
We assume the conclusion is false .
That is , for each integer LaTeXMLMath , we assume that there is an element LaTeXMLMath satisfying LaTeXMLMath .
Rescaling these LaTeXMLMath if necessary , we may assume that LaTeXMLMath and therefore LaTeXMLMath .
By our estimate ( LaTeXMLRef ) ( extended by continuity to LaTeXMLMath ) , we have LaTeXMLEquation .
LaTeXMLEquation The sequence LaTeXMLMath is thus bounded with respect to LaTeXMLMath , the Folland-Stein LaTeXMLMath -norm .
Any such set is precompact with respect to LaTeXMLMath ; this means there is a subsequence LaTeXMLMath that converges weakly in LaTeXMLMath and strongly in the Folland-Stein LaTeXMLMath -norm .
Let LaTeXMLMath be its limit .
On the one hand , LaTeXMLMath as each element LaTeXMLMath is .
On the other hand , the closedness of the differential operator LaTeXMLMath implies that LaTeXMLMath and LaTeXMLMath .
Thus LaTeXMLMath , so LaTeXMLMath .
But LaTeXMLMath , so this is a contradiction .
∎ By Lemma LaTeXMLRef and Theorem LaTeXMLRef , the quadratic form LaTeXMLEquation defines a norm that is equivalent to LaTeXMLMath .
We endow LaTeXMLMath with this norm , and let LaTeXMLMath denote the associated symmetric bilinear form .
Note that if LaTeXMLMath and LaTeXMLMath are smooth , then LaTeXMLMath .
By Lemma LaTeXMLRef , the linear functional LaTeXMLMath is bounded on LaTeXMLMath for any LaTeXMLMath .
The Riesz representation theorem then implies that there is a unique LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
Thus we have solved ( LaTeXMLRef ) for LaTeXMLMath .
The Neumann operator is given by LaTeXMLMath , the solution LaTeXMLMath to LaTeXMLMath in the above sense .
This makes sense for LaTeXMLMath , so under the orthogonal decomposition LaTeXMLMath we can extend LaTeXMLMath to all of LaTeXMLMath by declaring that it is identically zero on LaTeXMLMath .
We define the harmonic projector LaTeXMLMath as orthogonal projection onto LaTeXMLMath under this decomposition .
The operators LaTeXMLMath and LaTeXMLMath project onto orthogonal spaces , so LaTeXMLMath .
On the other hand , the decompositions LaTeXMLMath follow immediately from the construction of LaTeXMLMath and LaTeXMLMath .
To see that LaTeXMLMath takes a bit more work .
From LaTeXMLMath it follows directly that LaTeXMLMath , so we need only show that , say , LaTeXMLMath .
This follows easily by considering separately LaTeXMLMath ( on which LaTeXMLMath and LaTeXMLMath are separately zero ) and LaTeXMLMath , in which case LaTeXMLMath is a straightforward computation based on the formulas LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Finally , the isomorphism LaTeXMLMath follows as usual from the existence of the Neumann operator , since the harmonic projector LaTeXMLMath restricts to a map LaTeXMLMath whose kernel is exactly LaTeXMLMath by the arguments above .
∎ In this section , we prove Theorem LaTeXMLRef , our subelliptic estimate .
Since our manifold LaTeXMLMath is assumed to be compact , it will suffice to show that ( LaTeXMLRef ) holds for LaTeXMLMath supported in a neighborhood of each point : assuming this , we can choose a locally finite collection LaTeXMLMath of smooth nonnegative functions satisfying LaTeXMLMath , apply ( LaTeXMLRef ) to LaTeXMLMath , and sum over LaTeXMLMath , yielding ( LaTeXMLRef ) plus some lower-order terms that can be absorbed into the right-hand side .
Let LaTeXMLMath be a local moving frame for LaTeXMLMath satisfying ( LaTeXMLRef ) , from which it follows that LaTeXMLEquation and let LaTeXMLMath be the dual sections of LaTeXMLMath , thought of as one-forms according to the decomposition ( LaTeXMLRef ) .
We may then write LaTeXMLMath in coordinates as LaTeXMLEquation ( Notice the implicit sums over LaTeXMLMath and LaTeXMLMath through LaTeXMLMath . )
Throughout this section , we will assume LaTeXMLMath is supported in the neighborhood on which our moving frame is defined , so that LaTeXMLEquation .
We will often find it useful to look only at the top order derivatives .
In light of the commutation relation ( LaTeXMLRef ) , this unfortunately is not possible .
Instead , we will look at only the top weight derivatives , where we allocate a weight of LaTeXMLMath to vector fields in LaTeXMLMath and a weight of LaTeXMLMath to LaTeXMLMath .
We will then write LaTeXMLMath for equal modulo lower weight terms .
This generalizes to LaTeXMLMath and LaTeXMLMath , meaning greater than or less than , modulo negligible terms .
Our main estimate ( LaTeXMLRef ) can thus be written LaTeXMLEquation for all LaTeXMLMath .
To prove this estimate , we will need a local expression for LaTeXMLMath rather than LaTeXMLMath .
Modulo lower weight terms , this expression is LaTeXMLEquation where LaTeXMLMath .
We begin the actual proof of Theorem LaTeXMLRef by describing LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath in terms of our local moving frame .
( Compare LaTeXMLCite . )
Suppose LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
In our local frame , we may write LaTeXMLMath .
( Since LaTeXMLMath , there are no LaTeXMLMath terms . )
In this case LaTeXMLMath is ( see equation ( LaTeXMLRef ) ) LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where we have discarded all the terms without a derivative of a component of LaTeXMLMath .
This proves the second claim ; the first claim follows from applying the one-form LaTeXMLMath to both sides of ( LaTeXMLRef ) : LaTeXMLEquation where we have simplified using LaTeXMLMath .
Finally , we prove equation ( LaTeXMLRef ) .
To compute this adjoint , we take the inner product of LaTeXMLMath with an element LaTeXMLMath of LaTeXMLMath , and integrate by parts : LaTeXMLEquation .
If we write LaTeXMLMath for LaTeXMLMath ( again , there is no LaTeXMLMath term as LaTeXMLMath ) , then we can compute LaTeXMLMath .
The inside term is not difficult to compute , and we get LaTeXMLMath , so LaTeXMLMath .
Undoing the integration by parts above gives equation ( LaTeXMLRef ) .
∎ The primary tool in our proof of Theorem LaTeXMLRef is the following lemma .
This follows at least in part from the local expressions computed in Lemma LaTeXMLRef For all LaTeXMLMath , LaTeXMLEquation .
We begin by computing LaTeXMLMath .
From ( LaTeXMLRef ) , we have LaTeXMLEquation .
We expand this to get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath by Lemma LaTeXMLRef and LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath , one of the cross terms simplifies : LaTeXMLMath .
Four of the other cross terms combine and ( LaTeXMLRef ) simplifies to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We will deal with the remaining cross terms by adding LaTeXMLMath .
By Lemma LaTeXMLRef , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Moreover , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation as LaTeXMLMath .
Hence LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
To cancel the cross terms in ( LaTeXMLRef ) , we will make use of the fact that LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath modulo lower-weight terms , and therefore integrating by parts yields LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
A similar argument shows that three of the cross terms on the right-hand side of ( LaTeXMLRef ) cancel all the cross terms of ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We now have more cross terms , this time involving LaTeXMLMath .
We will deal with some of these cross terms using integration by parts .
The adjoint of LaTeXMLMath is LaTeXMLMath , and so ( using LaTeXMLMath and other commutation relations ) , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Similarly , LaTeXMLEquation .
LaTeXMLEquation Thus LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Now the three parts grouped in parentheses can be removed by the Schwarz inequality .
This gives us LaTeXMLEquation .
LaTeXMLEquation which is equation ( LaTeXMLRef ) .
This concludes the proof of the Key Estimate .
∎ Now to prove Theorem LaTeXMLRef , we need an estimate LaTeXMLEquation .
In our local frame , the right-hand side of this equation can be written as LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath run from LaTeXMLMath to LaTeXMLMath , and LaTeXMLMath .
We construct each of these estimates individually , and organize them in the following lemma .
There exists a positive constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
What we will show , in fact , is that for each LaTeXMLMath and each LaTeXMLMath there is a constant LaTeXMLMath such that LaTeXMLEquation .
The constant LaTeXMLMath can be chosen to be dominated by all the different constants LaTeXMLMath , so that the sum of the various individual estimates ( LaTeXMLRef ) and ( LaTeXMLRef ) yields the subelliptic estimate ( LaTeXMLRef ) .
We prove this lemma in stages : we produce the estimate ( LaTeXMLRef ) for each of the components LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath in turn .
The LaTeXMLMath case : We begin by noting that we have the estimate ( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath by the Key Estimate , Lemma LaTeXMLRef .
Now consider the part of inequality ( LaTeXMLRef ) that we discarded in the last step of the proof of the Key Estimate : LaTeXMLEquation .
Notice that , since LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for any LaTeXMLMath .
As we have already estimated LaTeXMLMath , this allows us to obtain an estimate LaTeXMLEquation for some LaTeXMLMath .
Similarly , we can obtain an estimate LaTeXMLEquation for some LaTeXMLMath .
From these estimates and inequality ( LaTeXMLRef ) , we obtain estimates for LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
We again return to a term we discarded at the end of the proof of the Key Estimate : we have LaTeXMLEquation .
We may rewrite part of this as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In the same way as above , we can control the inner product on the right .
Since we have an estimate already for LaTeXMLMath , we get estimates for LaTeXMLMath and LaTeXMLMath .
We can integrate by parts to write LaTeXMLEquation .
LaTeXMLEquation The previous estimates for the terms on the the right-hand side of this inequality then establish estimates for LaTeXMLMath .
We use the fact that LaTeXMLMath to get LaTeXMLEquation which gives us estimates for LaTeXMLMath and LaTeXMLMath .
Using integration by parts , we get an equality LaTeXMLEquation .
We thus obtain an estimate on LaTeXMLMath from the estimates on LaTeXMLMath and LaTeXMLMath .
Using this same trick , we have LaTeXMLEquation and we get an estimate on LaTeXMLMath .
Using Lemma LaTeXMLRef for the local expression of LaTeXMLMath , we get LaTeXMLEquation .
LaTeXMLEquation On the other hand , LaTeXMLEquation .
LaTeXMLEquation Since we ’ ve already estimated LaTeXMLMath , this gives us an estimate on LaTeXMLMath and LaTeXMLMath .
Similarly , we may use LaTeXMLEquation and LaTeXMLEquation to obtain estimates on LaTeXMLMath .
This completes the proof of the LaTeXMLMath case of Lemma LaTeXMLRef .
The LaTeXMLMath case : Recall that LaTeXMLMath by Lemma LaTeXMLRef , so this case follows from the LaTeXMLMath case .
The LaTeXMLMath case : We begin by recalling that we have our estimate for LaTeXMLMath and LaTeXMLMath by the Key Estimate , Lemma LaTeXMLRef .
We also remark that we have estimated LaTeXMLMath in the proof the LaTeXMLMath case of Lemma LaTeXMLRef .
In the proof of the Key Estimate , Lemma LaTeXMLRef , we did not use the fact that LaTeXMLEquation .
From this fact we have that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Using the same method as in the proof of the LaTeXMLMath case , and noting that we have estimates for all of the LaTeXMLMath terms , we obtain estimates for LaTeXMLMath and LaTeXMLMath .
Using our integration by parts trick , we see that LaTeXMLEquation .
We have estimates for both terms on the right-hand side , so this gives us estimates for LaTeXMLMath and LaTeXMLMath .
Now we produce an estimate for LaTeXMLMath .
We can write LaTeXMLMath for LaTeXMLMath , so integration by parts yields LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This gives us an estimate on LaTeXMLMath .
Since LaTeXMLMath , we get LaTeXMLMath and an estimate on LaTeXMLMath .
Similarly , LaTeXMLMath and we may estimate LaTeXMLMath .
Finally , integration by parts gives us the equalities LaTeXMLEquation and LaTeXMLEquation which allow us to estimate LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
This is the last of the required LaTeXMLMath estimates , and so this completes the proof of the LaTeXMLMath case of Lemma LaTeXMLRef .
The LaTeXMLMath case : It is simplest to notice the symmetry between the LaTeXMLMath case and the LaTeXMLMath case .
For example , the Key Estimate gives us an estimate on LaTeXMLMath and LaTeXMLMath as well as LaTeXMLMath and LaTeXMLMath .
Making the appropriate changes in the proof of the LaTeXMLMath case will then give us a proof in this case as well .
As this is the final case , we have now completed the proof of Lemma LaTeXMLRef .
∎ In this section , we introduce an explicit family of CR structures parameterized by a finite-dimensional analytic set , and show that it gives a local family of solutions to the deformation problem LaTeXMLEquation .
We begin by saying precisely what we mean by a family of CR structures .
Let LaTeXMLMath be a compact strictly pseudoconvex CR manifold of real dimension LaTeXMLMath .
By a family of deformations of a given CR structure LaTeXMLMath we mean a triple LaTeXMLMath , where LaTeXMLMath is a complex analytic subset containing the origin LaTeXMLMath and LaTeXMLMath is a complex analytic map such that , for each LaTeXMLMath , LaTeXMLMath determines an integrable CR structure LaTeXMLMath on LaTeXMLMath .
Recall that this means that LaTeXMLMath for all LaTeXMLMath , as LaTeXMLMath is the integrability condition for CR structures at finite distance from LaTeXMLMath .
Finally , we require that LaTeXMLMath ; that is , that LaTeXMLMath corresponds to the original CR structure LaTeXMLMath .
Then our main result of this section is the following theorem .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of real dimension LaTeXMLMath , and write LaTeXMLMath for the set of harmonic elements of LaTeXMLMath .
Then there is a complex-analytic map LaTeXMLMath defined in a neighborhood of LaTeXMLMath such that if LaTeXMLEquation then LaTeXMLMath is a family of deformations of LaTeXMLMath .
We will prove this theorem by constructing a locally complex analytic family of solutions to the deformation problem ( LaTeXMLRef ) .
We begin by producing some useful Sobolev estimates .
Our Laplacian LaTeXMLMath is a fourth-order differential operator , and so we can expect that the Neumann operator gains four derivatives in the directions of LaTeXMLMath .
This is the content of the following lemma .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of dimension LaTeXMLMath .
For each integer LaTeXMLMath , there exists a constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
We will show that LaTeXMLEquation whenever LaTeXMLMath .
Because LaTeXMLMath is subelliptic , LaTeXMLMath is smooth whenever LaTeXMLMath is smooth , so the required estimate follows by approximating with smooth sections .
The proof of ( LaTeXMLRef ) is by induction on LaTeXMLMath .
By using a partition of unity we may assume that LaTeXMLMath is supported in the domain of a frame satisfying ( LaTeXMLRef ) .
Observe that Lemma LaTeXMLRef and the Cauchy-Schwartz inequality imply that LaTeXMLMath .
As usual , we will let LaTeXMLMath and LaTeXMLMath denote equality and inequality modulo lower-weight terms , which can be absorbed by using standard interpolation inequalities .
We begin by considering derivatives in the LaTeXMLMath direction .
By Lemma LaTeXMLRef and Theorem LaTeXMLRef , LaTeXMLEquation .
LaTeXMLEquation Because LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath modulo terms of weight LaTeXMLMath , it follows that LaTeXMLMath is an operator of weight at most LaTeXMLMath .
Therefore , after integrating by parts , the second term above can be absorbed to yield LaTeXMLEquation .
Now we can prove ( LaTeXMLRef ) for the case LaTeXMLMath .
Observe that the commutation relations for LaTeXMLMath and LaTeXMLMath imply that LaTeXMLMath is equal to a constant multiple of LaTeXMLMath modulo lower-weight terms .
Therefore , using Lemma LaTeXMLRef and Theorem LaTeXMLRef again , we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is some operator of weight LaTeXMLMath .
Integrating by parts and using ( LaTeXMLRef ) , we find LaTeXMLEquation so LaTeXMLEquation .
LaTeXMLEquation Choosing LaTeXMLMath small enough , we can absorb the LaTeXMLMath term and obtain ( LaTeXMLRef ) when LaTeXMLMath .
Now assume that ( LaTeXMLRef ) holds for some LaTeXMLMath .
By induction , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation If LaTeXMLMath denotes any of the vector fields LaTeXMLMath or LaTeXMLMath , then LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are operators of weight LaTeXMLMath and LaTeXMLMath , respectively .
Thus LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath is a sum of terms of the form and LaTeXMLMath and LaTeXMLMath , this completes the induction .
∎ Recall that , for LaTeXMLMath , the almost CR structure LaTeXMLMath is integrable exactly when LaTeXMLMath .
With this in mind , we state the following proposition ( compare to LaTeXMLCite ) .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of dimension LaTeXMLMath .
Then for each positive integer LaTeXMLMath , there exists a positive constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
The proof of this proposition is simply the fact that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath take derivatives only in the LaTeXMLMath directions ; thus LaTeXMLMath can be written in a local frame for LaTeXMLMath as a homogeneous quadratic polynomial in the coefficients of LaTeXMLMath and their derivatives , in which each monomial has a total of no more than four LaTeXMLMath derivatives .
The assumption that LaTeXMLMath and the Sobolev embedding theorem yield the result .
∎ Thus Proposition LaTeXMLRef combined with Lemma LaTeXMLRef in the case LaTeXMLMath yields the following theorem .
Let LaTeXMLMath be a compact , strictly pseudoconvex CR manifold of dimension LaTeXMLMath .
For each integer LaTeXMLMath , there exists a constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
We now use Theorem LaTeXMLRef to prove the main theorem of this section , Theorem LaTeXMLRef .
We will solve this problem first in a Banach space : complete LaTeXMLMath with respect to the norm LaTeXMLMath for some integer LaTeXMLMath to obtain a Banach space , which we denote by LaTeXMLMath .
Consider the Banach analytic map from LaTeXMLMath to itself given by LaTeXMLEquation .
Theorem LaTeXMLRef implies that LaTeXMLMath is actually mapped to another element of LaTeXMLMath .
This is clearly an analytic local isomorphism .
The Banach inverse mapping theorem then gives us an analytic inverse map ; that is , an analytic function LaTeXMLMath from LaTeXMLMath to itself such that LaTeXMLEquation .
Our family ( LaTeXMLRef ) is locally ( near the origin LaTeXMLMath ) parametrized by the analytic set LaTeXMLMath defined in ( LaTeXMLRef ) .
To see this precisely , notice that equation ( LaTeXMLRef ) implies that for LaTeXMLMath , LaTeXMLEquation ( as LaTeXMLMath on LaTeXMLMath ) .
Combining this with the definition of LaTeXMLMath , we see that LaTeXMLEquation .
Since LaTeXMLMath depends complex analytically on LaTeXMLMath , our LaTeXMLMath is a complex analytic subset of LaTeXMLMath .
∎ In this section we prove that the family of CR structures constructed in Theorem LaTeXMLRef is versal , at least with respect to deformations of complex structure parametrized by smooth complex manifolds .
In order to define the notion of versality , we first make clear our definition of deformations of a complex manifold LaTeXMLMath .
( In practice , LaTeXMLMath will be a complex neighborhood of our CR manifold LaTeXMLMath , which is embedded as a hypersurface in a complex manifold LaTeXMLMath . )
A family of deformations of the complex manifold LaTeXMLMath is a triple LaTeXMLMath , where LaTeXMLMath is a complex analytic subset containing the origin LaTeXMLMath , LaTeXMLMath is a complex analytic space that is differentiably ( but not necessarily complex analytically ) isomorphic to LaTeXMLMath , and LaTeXMLMath is projection onto the second factor .
We remark that a family of deformations LaTeXMLMath of a complex manifold LaTeXMLMath gives rise to a unique LaTeXMLMath -valued 1-form LaTeXMLMath , depending complex analytically on LaTeXMLMath .
Moreover , the complex structure over LaTeXMLMath , defined by LaTeXMLEquation is integrable .
Conversely , by the Newlander-Nirenberg theorem , if such an LaTeXMLMath is given , at least in the case in which LaTeXMLMath is nonsingular , then we can construct a family of deformations LaTeXMLMath of the complex manifold LaTeXMLMath .
Now suppose LaTeXMLMath is a strictly pseudoconvex CR manifold .
A family of deformations LaTeXMLMath of CR structures over M is said to be versal if whenever LaTeXMLMath is embedded as a real hypersurface in an LaTeXMLMath -dimensional complex manifold LaTeXMLMath and LaTeXMLMath is any deformation of the complex structure on a neighborhood LaTeXMLMath of LaTeXMLMath in LaTeXMLMath , we have the following two conditions .
First , there exists a neighborhood of the origin LaTeXMLMath for which there is a holomorphic map LaTeXMLMath and smooth embeddings LaTeXMLMath for all LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath is the identity map .
Second , we note that LaTeXMLMath induces a CR structure over LaTeXMLMath when we consider LaTeXMLMath embedded in LaTeXMLMath via LaTeXMLMath .
Let us denote this CR structure by LaTeXMLMath .
If LaTeXMLMath is sufficiently close to the origin , this defines a unique deformation tensor LaTeXMLMath by LaTeXMLEquation .
Our requirement is that this CR structure be the same as the one induced by LaTeXMLMath at the point LaTeXMLMath : LaTeXMLEquation .
We will only deal with smooth deformations ; that is , deformations in which the analytic space LaTeXMLMath is , in fact , a complex manifold , rather than a variety with singularities .
We now state our main theorem of this section .
Suppose LaTeXMLMath is a compact strictly pseudoconvex CR manifold of real dimension LaTeXMLMath that is embedded as a real hypersurface in a complex manifold LaTeXMLMath of complex dimension LaTeXMLMath .
If the family of CR deformations LaTeXMLMath is a smooth family of deformations , then it is versal with respect to smooth deformations ( that is , with respect to deformations LaTeXMLMath of a neighborhood LaTeXMLMath of LaTeXMLMath in LaTeXMLMath , where the analytic space LaTeXMLMath is a complex manifold ) .
Our proof can be modified to work in the case that LaTeXMLMath has a singularity , so the claim would be that the family of CR deformations is versal .
We leave this claim to another paper .
We must construct LaTeXMLMath and LaTeXMLMath .
Suppose that we are given a family of deformations of a neighborhood LaTeXMLMath of LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be a covering of LaTeXMLMath by coordinate domains , indexed by some finite set .
Let LaTeXMLMath be local holomorphic coordinates on LaTeXMLMath , and let LaTeXMLMath be transition functions : LaTeXMLEquation .
For brevity , we will write this as LaTeXMLEquation .
We can extend this to a local coordinate covering LaTeXMLMath for LaTeXMLMath with transition functions LaTeXMLMath defined on LaTeXMLMath , holomorphic in LaTeXMLMath and smooth in LaTeXMLMath .
We use a similar abbreviation as above : LaTeXMLEquation with the requirement that LaTeXMLMath .
For simplicity , we use local complex coordinates LaTeXMLMath depending complex analytically on the parameter LaTeXMLMath .
That is , each function LaTeXMLMath is a smooth function on LaTeXMLMath and complex analytic on LaTeXMLMath , and the corresponding complex structure on LaTeXMLMath ( as an element of LaTeXMLMath ) is determined by LaTeXMLEquation .
Similarly , the induced CR structure defined in equation ( LaTeXMLRef ) is also determined locally by LaTeXMLEquation where LaTeXMLMath .
This equality also means that the map LaTeXMLMath is a CR embedding from LaTeXMLMath to LaTeXMLMath , with the complex structure LaTeXMLMath .
We have to construct LaTeXMLMath , locally expressed by LaTeXMLMath on LaTeXMLMath , which depends complex analytically on LaTeXMLMath , and a holomorphic map LaTeXMLMath from LaTeXMLMath to LaTeXMLMath , satisfying LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath ( where , if necessary , we may shrink LaTeXMLMath to a smaller neighborhood of LaTeXMLMath ) .
The proof of the existence of such functions is a standard formal power series argument .
Consider the power series expansions LaTeXMLEquation .
We are using multi-index notation , so if LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
In general , if LaTeXMLMath is any vector-bundle-valued function of LaTeXMLMath , we will use the notation LaTeXMLMath to mean the part of the power series for LaTeXMLMath about LaTeXMLMath that is homogeneous of order LaTeXMLMath in LaTeXMLMath .
For such homogeneous polynomials , we will use a subscript LaTeXMLMath to indicate the degree in LaTeXMLMath .
Similarly , a superscript LaTeXMLMath will indicate a ( not usually homogeneous ) polynomial of degree LaTeXMLMath in LaTeXMLMath .
First we formally construct these power series , then prove convergence .
Let LaTeXMLMath and LaTeXMLMath be the LaTeXMLMath th partial sums in the above power series expansions : LaTeXMLEquation .
We construct LaTeXMLMath and LaTeXMLMath formally by induction on LaTeXMLMath .
At any step LaTeXMLMath , we wish to have LaTeXMLMath and LaTeXMLMath satisfy LaTeXMLEquation for LaTeXMLMath near LaTeXMLMath .
At our initial step ( that is , at LaTeXMLMath ) , we define LaTeXMLMath and LaTeXMLMath .
These obviously satisfy our criterion ( LaTeXMLRef ) .
Now we assume that we have already constructed LaTeXMLMath and LaTeXMLMath satisfying ( LaTeXMLRef ) .
To begin our construction of LaTeXMLMath and LaTeXMLMath , we define a polynomial LaTeXMLMath on LaTeXMLMath , homogeneous of degree LaTeXMLMath in LaTeXMLMath , such that LaTeXMLEquation ( In this way , LaTeXMLMath is a rough first approximation of LaTeXMLMath , the homogeneous part of LaTeXMLMath in degree LaTeXMLMath . )
To do this , we construct vector-valued polynomials LaTeXMLMath on LaTeXMLMath , again homogeneous of degree LaTeXMLMath in LaTeXMLMath , by the relation LaTeXMLEquation .
This definition of LaTeXMLMath makes sense as the induction hypothesis ( LaTeXMLRef ) implies that the right-hand side of equation ( LaTeXMLRef ) has only terms of order LaTeXMLMath and higher in LaTeXMLMath .
We use these LaTeXMLMath and a partition of unity LaTeXMLMath subordinate to the covering LaTeXMLMath to define LaTeXMLEquation .
We will show that such LaTeXMLMath satisfy ( LaTeXMLRef ) .
To do this , we need to know how LaTeXMLMath ( or LaTeXMLMath ) transform over different coordinate charts .
We have the following lemma ( compare to LaTeXMLCite ) .
On LaTeXMLMath , LaTeXMLEquation .
By the definition of LaTeXMLMath , LaTeXMLEquation .
We replace LaTeXMLMath with LaTeXMLMath to get LaTeXMLEquation .
We expand the first term on the right-hand side in a power series about the point LaTeXMLMath ; this implies LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ( In the last line we have used the inductive hypothesis ( LaTeXMLRef ) and Taylor ’ s theorem applied to LaTeXMLMath .
Any error term involving LaTeXMLMath multiplied by itself or by LaTeXMLMath can be absorbed into LaTeXMLMath . )
The first and third terms simplify to LaTeXMLMath modulo LaTeXMLMath , and so equation ( LaTeXMLRef ) reduces to equation ( LaTeXMLRef ) .
This proves the lemma .
∎ With LaTeXMLMath defined by ( LaTeXMLRef ) , LaTeXMLMath transforms as in equation ( LaTeXMLRef ) .
From the definition of LaTeXMLMath and ( LaTeXMLRef ) , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By Taylor ’ s theorem , this is equivalent to ( LaTeXMLRef ) .
∎ To define the next term in our formal power series , we will write locally LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the local expression for a homogeneous polynomial LaTeXMLMath of degree LaTeXMLMath in LaTeXMLMath , with values in LaTeXMLMath , and LaTeXMLMath is a homogeneous polynomial of degree LaTeXMLMath with values in LaTeXMLMath .
Since the transformation law for sections of LaTeXMLMath is LaTeXMLEquation it follows that our prospective LaTeXMLMath transforms the correct way : LaTeXMLEquation .
We still must construct LaTeXMLMath and LaTeXMLMath so that LaTeXMLMath and LaTeXMLMath , defined as in equation ( LaTeXMLRef ) , satisfy the inductive hypothesis ( LaTeXMLRef ) .
Note first that , by equation ( LaTeXMLRef ) , the CR structure defined by LaTeXMLMath must satisfy LaTeXMLEquation .
LaTeXMLEquation From this it follows that LaTeXMLEquation .
LaTeXMLEquation On the other hand , from the definition , it is clear that ( see equation ( LaTeXMLRef ) ) the map LaTeXMLMath linearizes to the identity , so LaTeXMLEquation .
Finding solutions to the second equation in ( LaTeXMLRef ) is thus reduced to the following theorem .
There are vector-valued polynomials LaTeXMLMath and LaTeXMLMath , homogeneous in LaTeXMLMath of degree LaTeXMLMath , solving LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath takes values in LaTeXMLMath , and LaTeXMLMath takes values in the finite-dimensional harmonic space LaTeXMLMath .
The proof of this theorem will follow from several lemmas and propositions .
There is a homogeneous polynomial LaTeXMLMath of degree LaTeXMLMath in LaTeXMLMath , with values in LaTeXMLMath , such that LaTeXMLEquation where we have written LaTeXMLMath for LaTeXMLMath .
Since our CR structure is strictly pseudoconvex , the map LaTeXMLEquation .
LaTeXMLEquation is an isomorphism .
Hence there is a LaTeXMLMath -valued polynomial LaTeXMLMath which such that LaTeXMLMath is a polynomial that takes values in LaTeXMLMath .
By the inductive hypothesis , for each LaTeXMLMath , the polynomial LaTeXMLMath already takes values in LaTeXMLMath ; thus we may assume LaTeXMLMath .
Writing LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , we thus have LaTeXMLEquation .
To prove the proposition , it suffices to show LaTeXMLEquation .
In order to show this , we first prove the next lemma .
LaTeXMLEquation and LaTeXMLEquation hold .
In particular , LaTeXMLMath as LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath ( LaTeXMLMath ) are the parts of the deformation equation that are order LaTeXMLMath in LaTeXMLMath .
( Of course , each LaTeXMLMath includes first derivatives of LaTeXMLMath . )
The expressions for LaTeXMLMath are given in equations ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Since LaTeXMLMath is quadratic , and we may replace each LaTeXMLMath with LaTeXMLMath in turn .
On the one hand , LaTeXMLMath by the induction hypothesis ( LaTeXMLRef ) .
On the other hand , LaTeXMLMath itself satisfies LaTeXMLMath .
Together , these facts imply that LaTeXMLMath .
The proof for LaTeXMLMath is similar .
∎ Continuing the proof of Proposition LaTeXMLRef , we remark that LaTeXMLMath is , for each LaTeXMLMath , an integrable complex structure .
Since LaTeXMLMath is a CR embedding for each LaTeXMLMath , modulo terms of order LaTeXMLMath and higher , the CR structure induced by LaTeXMLMath is also integrable : LaTeXMLEquation .
Obviously , we may remove the terms of order LaTeXMLMath and higher to see that LaTeXMLEquation .
From the previous lemma , LaTeXMLMath , and so LaTeXMLEquation .
A similar computation shows that LaTeXMLEquation ( and the zero follows from Lemma LaTeXMLRef ) .
The integrability condition is thus LaTeXMLEquation .
LaTeXMLEquation Because LaTeXMLMath takes its values in LaTeXMLMath , we have LaTeXMLMath .
Hence LaTeXMLEquation .
This is equivalent to equation ( LaTeXMLRef ) , so this proves Proposition LaTeXMLRef .
∎ LaTeXMLEquation .
We recall that LaTeXMLEquation ( The map defined on each LaTeXMLMath by LaTeXMLMath makes sense globally modulo LaTeXMLMath . )
Thus LaTeXMLEquation .
LaTeXMLEquation We apply the operator LaTeXMLMath to this equality .
By Proposition LaTeXMLRef , the left-hand side is the image of an element of LaTeXMLMath under LaTeXMLMath , so this makes sense .
The decomposition of Theorem LaTeXMLRef implies that LaTeXMLMath , and from this Lemma LaTeXMLRef follows easily .
∎ LaTeXMLEquation .
The first term on the left-hand side satisfies LaTeXMLEquation .
LaTeXMLEquation as we have seen in the proof of Proposition LaTeXMLRef .
By the construction of LaTeXMLMath ( equation ( LaTeXMLRef ) ) , we have LaTeXMLEquation .
Taking the difference of the last two equations implies LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The proposition then follows from Lemmas LaTeXMLRef and LaTeXMLRef .
∎ We wish to solve equation ( LaTeXMLRef ) , which can be written as LaTeXMLEquation .
LaTeXMLEquation We begin by solving LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
By Proposition LaTeXMLRef , the left-hand side of this equation is in the kernel of LaTeXMLMath , modulo LaTeXMLMath .
The decomposition of Theorem LaTeXMLRef implies that LaTeXMLMath and LaTeXMLMath , defined as follows , satisfy equation ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath , and LaTeXMLMath for elements of LaTeXMLMath , we may define LaTeXMLMath locally by LaTeXMLEquation .
LaTeXMLEquation To solve equation ( LaTeXMLRef ) , and thus equation ( LaTeXMLRef ) , we simply set LaTeXMLMath .
This LaTeXMLMath and LaTeXMLMath solve equation ( LaTeXMLRef ) , so we have proved Theorem LaTeXMLRef .
∎ Continuing the proof of Theorem LaTeXMLRef , we turn to the proof of convergence of the formal series .
This part of the proof uses the standard method of Kodaira and Spencer ( see LaTeXMLCite ) .
We define a Sobolev LaTeXMLMath norm on a power series by setting LaTeXMLEquation .
Consider the power series LaTeXMLEquation this converges for any positive LaTeXMLMath .
Moreover , for positive LaTeXMLMath , we have LaTeXMLMath , where LaTeXMLMath means every coefficient of the left-hand side is less than the corresponding coefficient of the right-hand side .
This implies that LaTeXMLMath for all integers LaTeXMLMath .
By choosing suitable LaTeXMLMath and LaTeXMLMath ( see LaTeXMLCite or LaTeXMLCite ) , we wish to show , for any integer LaTeXMLMath , that LaTeXMLEquation ( The reason for subtracting LaTeXMLMath in ( LaTeXMLRef ) is because LaTeXMLMath has no LaTeXMLMath term . )
By the Sobolev embedding theorem , this would give us all the convergence and regularity claimed in Theorem LaTeXMLRef .
Proof of the convergence ( LaTeXMLRef ) is done by induction on the partial sums .
That is , we assume that we have LaTeXMLEquation then establish the same inequality for LaTeXMLMath .
The special properties of LaTeXMLMath are used here : we bound the LaTeXMLMath st degree terms with lower degree terms that we have previously bounded .
As LaTeXMLMath and LaTeXMLMath are chosen properly , we can bound sums of powers of LaTeXMLMath by LaTeXMLMath itself .
The LaTeXMLMath term is well-behaved : for any LaTeXMLMath there is a constant LaTeXMLMath such that the harmonic projector LaTeXMLMath satisfies the estimate LaTeXMLEquation for any LaTeXMLMath .
However , we may have to correct LaTeXMLMath to ensure convergence , because our construction of of LaTeXMLMath involved first derivatives of LaTeXMLMath .
Recall from Theorem LaTeXMLRef that LaTeXMLMath is a solution to equation ( LaTeXMLRef ) , which can be viewed as a linear LaTeXMLMath equation for the standard deformation complex ( LaTeXMLRef ) .
Because LaTeXMLMath is a holomorphic vector bundle , by the results of LaTeXMLCite there is a Neumann operator LaTeXMLMath satisfying LaTeXMLMath for all LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath is the projection onto LaTeXMLMath .
Arguing as in LaTeXMLCite , we let LaTeXMLEquation and LaTeXMLEquation .
It is true that there is a first derivative of LaTeXMLMath in LaTeXMLMath , but only in the LaTeXMLMath direction .
In fact , we recall that LaTeXMLMath is defined on LaTeXMLMath by LaTeXMLEquation ( The CR structure defined on LaTeXMLMath by LaTeXMLMath makes sense globally , modulo LaTeXMLMath . )
Thus LaTeXMLEquation .
LaTeXMLEquation By the inductive hypothesis , we have LaTeXMLEquation .
LaTeXMLEquation while LaTeXMLMath takes its values in LaTeXMLMath .
Since the composition LaTeXMLMath of the adjoint operator and the standard Neumann operator gains 1 in this direction , there is no problem in the convergence of our formal solution .
This finishes the proof of Theorem LaTeXMLRef .
∎
A bstract The LaTeXMLMath -equivalence is an equivalence relation generated by LaTeXMLMath -moves defined by Habiro .
Habiro showed that the set of LaTeXMLMath -equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order LaTeXMLMath .
We see that the set of LaTeXMLMath -equivalence classes of the spatial LaTeXMLMath -curves forms a group under the vertex connected sum and that if the group is abelian , then it can be classified by the additive Vassiliev invariant of order LaTeXMLMath .
However the group is not necessarily abelian .
In fact , we show that it is nonabelian for LaTeXMLMath .
As an easy consequence , we have the set of LaTeXMLMath -equivalence classes of LaTeXMLMath -string links , which forms a group under the composition , is nonabelian for LaTeXMLMath and LaTeXMLMath .
1 .
LaTeXMLMath -moves and Vassiliev invariants of spatial LaTeXMLMath -curves A tangle LaTeXMLMath is a disjoint union of properly embedded arcs in the unit LaTeXMLMath -ball LaTeXMLMath .
A local move is a pair of tangles LaTeXMLMath with LaTeXMLMath such that for each component LaTeXMLMath of LaTeXMLMath there exists a component LaTeXMLMath of LaTeXMLMath with LaTeXMLMath .
Two local moves LaTeXMLMath and LaTeXMLMath are equivalent if there is an orientation preserving self-homeomorphism LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are ambient isotopic in LaTeXMLMath relative to LaTeXMLMath for LaTeXMLMath .
Here LaTeXMLMath and LaTeXMLMath are ambient isotopic in LaTeXMLMath relative to LaTeXMLMath if LaTeXMLMath is deformed to LaTeXMLMath by an ambient isotopy of LaTeXMLMath that is pointwisely fixed on LaTeXMLMath .
Let LaTeXMLMath be a local move , LaTeXMLMath a component of LaTeXMLMath and LaTeXMLMath a component of LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be regular neighbourhoods of LaTeXMLMath and LaTeXMLMath in LaTeXMLMath and LaTeXMLMath respectively such that LaTeXMLMath .
Let LaTeXMLMath be a disjoint union of properly embedded arcs in LaTeXMLMath as illustrated in Fig .
1.1 .
Let LaTeXMLMath be a homeomorphism with LaTeXMLMath for LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are ambient isotopic in LaTeXMLMath relative to LaTeXMLMath .
Then we say that a local move LaTeXMLMath is a double of LaTeXMLMath with respect to the components LaTeXMLMath and LaTeXMLMath .
Fig .
1.1 A LaTeXMLMath -move is a local move LaTeXMLMath as illustrated in Fig .
1.2 .
A double of a LaTeXMLMath -move is called a LaTeXMLMath -move .
Note that , for each natural number LaTeXMLMath , there are only finitely many LaTeXMLMath -moves up to equivalence .
It is easy to see that if LaTeXMLMath is a LaTeXMLMath -move , then LaTeXMLMath is equivalent to a LaTeXMLMath -move ( but possibly not equivalent to itself ) .
The definition of LaTeXMLMath -move follows that in LaTeXMLCite , and is defferent from the one in LaTeXMLCite .
However by an easy induction on LaTeXMLMath it is shown that these two definitions are essentially same .
In LaTeXMLCite , a LaTeXMLMath -move is called a simple LaTeXMLMath -move , and a LaTeXMLMath -move means a parallel of a LaTeXMLMath -move .
The definition of parallel of a local move appears in Section 3 .
Fig .
1.2 Let LaTeXMLMath be a graph with labeled vertices and edges .
Let LaTeXMLMath be an embedding of LaTeXMLMath into the oriented three sphere LaTeXMLMath .
The embedding LaTeXMLMath is called a spatial graph .
Let LaTeXMLMath and LaTeXMLMath be spatial graphs .
We say that LaTeXMLMath is obtained from LaTeXMLMath by a local move LaTeXMLMath if there is an orientation preserving embedding LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath together with the labels of vertices and edges .
Two spatial graphs LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent if LaTeXMLMath is obtained from LaTeXMLMath by a finite sequence of LaTeXMLMath -moves and ambient isotopies .
We note that the relation is an equivalence relation on spatial graphs .
For a spatial graph LaTeXMLMath , let LaTeXMLMath denote the LaTeXMLMath -equivalence class that contains LaTeXMLMath .
It is known that LaTeXMLMath -equivalence implies LaTeXMLMath -equivalence LaTeXMLCite , LaTeXMLCite .
Let LaTeXMLMath be a positive integer and LaTeXMLMath positive integers .
Suppose that for each LaTeXMLMath , an spatial graph LaTeXMLMath in LaTeXMLMath is assigned .
Suppose that there are mutually disjoint , orientation preserving embeddings LaTeXMLMath LaTeXMLMath such that ( 1 ) LaTeXMLMath together with the labels for any subset LaTeXMLMath , ( 2 ) LaTeXMLMath is a LaTeXMLMath -move LaTeXMLMath , and ( 3 ) LaTeXMLMath Then we call the set LaTeXMLMath a singular spatial graph of type LaTeXMLMath .
The LaTeXMLMath -curve is a graph LaTeXMLMath with two vertices LaTeXMLMath and three edges LaTeXMLMath each of which joins LaTeXMLMath and LaTeXMLMath .
When LaTeXMLMath , a spatial graph and singular spatial graph are called spatial LaTeXMLMath -curve and singular spatial LaTeXMLMath -curve respsctively .
A spatial LaTeXMLMath -curve is trivial if its image is contained in a 2-sphere in LaTeXMLMath .
In the remainder of this sction , we consider only the case that a graph is the LaTeXMLMath -curve .
Let LaTeXMLMath be the set of all spatial LaTeXMLMath -curve types in LaTeXMLMath and LaTeXMLMath the free abelian group generated by LaTeXMLMath .
For a singular spatial LaTeXMLMath -curve LaTeXMLMath of type LaTeXMLMath , we define an element LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation .
Let LaTeXMLMath be the subgroup of LaTeXMLMath generated by all LaTeXMLMath where LaTeXMLMath varies over all singular spatial LaTeXMLMath -curves of type LaTeXMLMath .
For two spatial LaTeXMLMath -curves LaTeXMLMath and LaTeXMLMath , remove small balls centred at LaTeXMLMath and LaTeXMLMath from LaTeXMLMath , then identify the boundaries so that the images of LaTeXMLMath -th edge are joined for each LaTeXMLMath .
Then we obtain a new spatial LaTeXMLMath -curve .
We call this embedding the vertex connected sum of LaTeXMLMath and LaTeXMLMath , and denote by LaTeXMLMath .
The vertex connected sum is well-defined up to ambient isotopy LaTeXMLCite .
Let LaTeXMLMath be the vertex connected sum of two spatial LaTeXMLMath -curves LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath is called a composite relator .
Let LaTeXMLMath be the subgroup of LaTeXMLMath generated by all composite relators .
Let LaTeXMLMath be the natural inclusion map .
Let LaTeXMLMath and LaTeXMLMath be the quotient homomorphisms .
Then the composite maps LaTeXMLMath and LaTeXMLMath are called the universal Vassiliev invariant of type LaTeXMLMath and universal additive Vassiliev invariant of type LaTeXMLMath respectively .
We denote them by LaTeXMLMath and LaTeXMLMath respectively .
In the case of knots , these are same invariants as defined by K. Taniyama and the author LaTeXMLCite .
Similarly , we can also define LaTeXMLMath and the universal Vassiliev invariant LaTeXMLMath for the embeddings of any graph .
Since a LaTeXMLMath -move is a crossing change we see that a singular spatial graph of type LaTeXMLMath is essentially the same as a singular saptial graph with LaTeXMLMath crossing vertices in the sense of T. Stanford LaTeXMLCite .
Therefore we see that LaTeXMLMath is the universal Vassiliev invariant of order LaTeXMLMath .
Note that LaTeXMLMath if and only if LaTeXMLMath for any Vassiliev invariant LaTeXMLMath of order LaTeXMLMath .
In the case of links , LaTeXMLMath is the same as that defined in LaTeXMLCite , LaTeXMLCite .
In LaTeXMLCite Taniyama and the author defined finite type invariants of order LaTeXMLMath for the embeddings of a graph , which are essentially same as LaTeXMLMath .
By the arguments similar to that in Proofs of Theorems 1.1 and 1.2 in LaTeXMLCite and that in Proof of Theorem 1.4 in LaTeXMLCite , we have the following two theorems .
Theorem 1.1 .
Let LaTeXMLMath be positive integers and LaTeXMLMath .
Then the followings hold .
( 1 ) LaTeXMLMath .
( 2 ) LaTeXMLMath .
LaTeXMLMath Remark .
Theorem 1.1 ( 1 ) holds for the spatial embeddings of any graph .
Theorem 1.2 .
The LaTeXMLMath -equivalence classes of the spatial LaTeXMLMath -curves forms a group with the unit element LaTeXMLMath under the vertex connected sum , where LaTeXMLMath is a trivial LaTeXMLMath -curve .
LaTeXMLMath We denote by LaTeXMLMath this group .
Let LaTeXMLMath be a map induced by the inclusion LaTeXMLMath .
By Theorem 1.2 , LaTeXMLMath is a well-defined , epimorphism .
( In fact , LaTeXMLMath is an isomorphism , see the remark after Corollary 1.4 . )
Since LaTeXMLMath , by Theorem 1.1 ( 2 ) , we have the following theorem .
Theorem 1.3 .
Let LaTeXMLMath be positive integers and LaTeXMLMath .
Then LaTeXMLMath is isomorphic to LaTeXMLMath .
LaTeXMLMath Let LaTeXMLMath and LaTeXMLMath be spatial LaTeXMLMath -curves and LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent , then by Theorem 1.1 ( 1 ) , LaTeXMLMath .
Therefore we have LaTeXMLMath .
On the other hand , if LaTeXMLMath , then LaTeXMLMath .
Hence we have LaTeXMLMath .
If LaTeXMLMath is abelian group , i.e. , LaTeXMLMath , then by Theorem 1.3 , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
So we have the following corollary .
Corollary 1.4 .
Let LaTeXMLMath be positive integers and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be spatial LaTeXMLMath -curves .
If LaTeXMLMath is an abelian group , then the following conditions are mutually equivalent .
( 1 ) LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent , ( 2 ) LaTeXMLMath , ( 3 ) LaTeXMLMath .
LaTeXMLMath Remark .
Let LaTeXMLMath and LaTeXMLMath be spatial graphs ( not necessarily LaTeXMLMath -curve ) .
If LaTeXMLMath , then there are singular spatial graphs LaTeXMLMath ’ s of type LaTeXMLMath and integers LaTeXMLMath ’ s such that LaTeXMLMath .
By induction on LaTeXMLMath , we see that LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
Since LaTeXMLMath if LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent , we have the following : Two spatial graphs LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent if and only if LaTeXMLMath .
Theorem 1.5 .
Let LaTeXMLMath be a trivial spatial LaTeXMLMath -curve .
Then the followings hold .
LaTeXMLMath For each LaTeXMLMath , LaTeXMLMath belongs to the center of LaTeXMLMath .
LaTeXMLMath If LaTeXMLMath , then the set LaTeXMLMath is an abelian subgroup of LaTeXMLMath .
By LaTeXMLCite and LaTeXMLCite , we have LaTeXMLMath .
Hence , by Theorem 1.5 ( 2 ) , LaTeXMLMath is abelian .
By Corollary 1.4 , we have Corollary 1.6 .
Let LaTeXMLMath LaTeXMLMath be positive integers and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be spatial LaTeXMLMath -curves .
Then the following conditions are mutually equivalent .
( 1 ) LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent , ( 2 ) LaTeXMLMath , ( 3 ) LaTeXMLMath .
LaTeXMLMath Remark .
As a special case of Corollary 1.6 , we see that for LaTeXMLMath two spatial LaTeXMLMath -curves are LaTeXMLMath -equivalent if and only if the universal ( additive ) Vassiliev invariant of order LaTeXMLMath are equal .
Meanwhile , a basis for the space of Vassiliev invariants of order LaTeXMLMath is known LaTeXMLCite , LaTeXMLCite .
Theorem 1.7 .
Let LaTeXMLMath be a trivial spatial LaTeXMLMath -curve , and let LaTeXMLMath and LaTeXMLMath be in LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent if and only if LaTeXMLMath .
As we saw before , LaTeXMLMath is abelian for LaTeXMLMath .
However LaTeXMLMath is not necessarily abelian .
In fact , we have the following theorem .
Theorem 1.8 .
The group LaTeXMLMath is nonabelian for any LaTeXMLMath .
Remarks ( 1 ) If LaTeXMLMath is abelian , then so is LaTeXMLMath for any LaTeXMLMath .
( 2 ) In the proof of Theorem 1.8 , we see that there are two spatial LaTeXMLMath -curves LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath is in LaTeXMLMath and is not in LaTeXMLMath for LaTeXMLMath .
Hene LaTeXMLMath for LaTeXMLMath by Theorem 1.1 ( 1 ) , while LaTeXMLMath for any LaTeXMLMath .
In contrast , for any knots LaTeXMLMath and LaTeXMLMath , LaTeXMLMath if and only if LaTeXMLMath LaTeXMLCite .
( 3 ) Habiro showed the set of LaTeXMLMath -equivalence classes LaTeXMLMath of LaTeXMLMath -string links forms a group under the composition LaTeXMLCite .
By considering the complement of a regular neighbourhood of one of edges , we have that there is a surjection from the LaTeXMLMath -string links to the spatial LaTeXMLMath -curves .
Since the surjection induces an epimorphism from LaTeXMLMath to LaTeXMLMath and since there is an epimorphism from LaTeXMLMath to LaTeXMLMath , LaTeXMLMath is nonabelian for any LaTeXMLMath and LaTeXMLMath .
The following is still open .
Problem .
Find the minimum number LaTeXMLMath such that LaTeXMLMath is nonabelian .
2 .
Band description of spatial graphs A LaTeXMLMath -link model is a pair LaTeXMLMath where LaTeXMLMath is a disjoint union of properly embedded arcs in LaTeXMLMath and LaTeXMLMath is a disjoint union of arcs on LaTeXMLMath with LaTeXMLMath as illustrated in Fig .
2.1 .
Suppose that a LaTeXMLMath -link model LaTeXMLMath is defined where LaTeXMLMath is a disjoint union of LaTeXMLMath properly embedded arcs in LaTeXMLMath and LaTeXMLMath is a disjoint union of LaTeXMLMath arcs on LaTeXMLMath with LaTeXMLMath such that LaTeXMLMath is a disjoint union of LaTeXMLMath circles .
Let LaTeXMLMath be a component of LaTeXMLMath and LaTeXMLMath a regular neighbourhood of LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be an oriented solid torus , LaTeXMLMath a disk in LaTeXMLMath , LaTeXMLMath properly embedded arcs in LaTeXMLMath and LaTeXMLMath arcs on LaTeXMLMath as illustrated in Fig .
2.2 .
Let LaTeXMLMath be an orientation preserving homeomorphism such that LaTeXMLMath and LaTeXMLMath bounds disjoint disks in LaTeXMLMath .
Then we call the pair LaTeXMLMath a LaTeXMLMath -link model .
A link model is a LaTeXMLMath -link model for some LaTeXMLMath .
It is known that , for a LaTeXMLMath -link model LaTeXMLMath , the local move LaTeXMLMath is equivalent to a LaTeXMLMath -move LaTeXMLCite , where LaTeXMLMath is a slight push in of LaTeXMLMath .
Let LaTeXMLMath be a spatial LaTeXMLMath -curve , and let LaTeXMLMath be link models .
Let LaTeXMLMath LaTeXMLMath be mutually disjoint , orientation preserving embeddings , and let LaTeXMLMath be mutually disjoint disks embedded in LaTeXMLMath .
Suppose that they satisfy the following conditions ; ( 1 ) LaTeXMLMath for each LaTeXMLMath , ( 2 ) LaTeXMLMath is an arc for each LaTeXMLMath , ( 3 ) LaTeXMLMath is a component of LaTeXMLMath for each LaTeXMLMath , ( 4 ) ( LaTeXMLMath for each LaTeXMLMath .
Let LaTeXMLMath be a spatial LaTeXMLMath -curve defined by LaTeXMLEquation where the labels of LaTeXMLMath coincides that of LaTeXMLMath on LaTeXMLMath .
When LaTeXMLMath is a LaTeXMLMath -link model , we call LaTeXMLMath a LaTeXMLMath -link ball .
We set LaTeXMLMath and call LaTeXMLMath a LaTeXMLMath -chord when LaTeXMLMath is a LaTeXMLMath -link model .
We denote LaTeXMLMath by LaTeXMLMath and call it a band description of LaTeXMLMath .
We also say LaTeXMLMath is a band sum of LaTeXMLMath and link models LaTeXMLMath .
By the arguments similar to that in Proof of Lemma 3.6 LaTeXMLCite , we have Lemma 2.1 .
Two spatial LaTeXMLMath -curves LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalence if and only if there are spatial LaTeXMLMath -curves LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath is ambient isotopic to LaTeXMLMath LaTeXMLMath and LaTeXMLMath is a band sum of LaTeXMLMath and some LaTeXMLMath -link models .
LaTeXMLMath In the following lemma , the former assertion follows directly from Lemma 3.9 in LaTeXMLCite and the latter can be shown by the similar arguments as in proof of Lemma 3.9 in LaTeXMLCite .
Lemma 2.2 .
A local move as illustrated in Fig .
2.3 LaTeXMLMath resp .
Fig .
2.4 LaTeXMLMath is realized by a LaTeXMLMath -move LaTeXMLMath resp .
LaTeXMLMath -move LaTeXMLMath .
LaTeXMLMath Fig .
2.3 Fig .
2.4 Proof of Theorem 1.5 .
( 1 ) Suppose LaTeXMLMath , then by Lemma 2.1 , we may assume that LaTeXMLMath is a band sum of LaTeXMLMath and some LaTeXMLMath -link models .
Let LaTeXMLMath be a spatial LaTeXMLMath -curve .
Since LaTeXMLMath , we may suppose LaTeXMLMath is a band sum of LaTeXMLMath and some LaTeXMLMath -link models .
It is not hard to see that LaTeXMLMath and LaTeXMLMath are transposed each other by the moves as in Figs .
2.3 and 2.4 , where we consider the case LaTeXMLMath , and ambient isotopies .
Thus by Lemma 2.2 we have LaTeXMLMath is LaTeXMLMath -equivalent to LaTeXMLMath .
Hence we have LaTeXMLMath .
( 2 ) Suppose that both LaTeXMLMath and LaTeXMLMath belong to LaTeXMLMath .
Then we note that LaTeXMLMath .
If LaTeXMLMath belongs to LaTeXMLMath , then by Lemma 2.1 , we may assume that LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath -chords LaTeXMLMath .
By using Sublemma 3.5 in LaTeXMLCite repeartedly , there are LaTeXMLMath -chords LaTeXMLMath such that LaTeXMLMath is ambient isotopic to LaTeXMLMath .
By Lemma 2.2 , we can deform LaTeXMLMath into LaTeXMLMath by LaTeXMLMath -moves and ambient isotopies , i.e. , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
Since LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
This implies that LaTeXMLMath .
Therefore LaTeXMLMath is a subgroup of LaTeXMLMath .
By the arguments similar to that in ( 1 ) , we see that LaTeXMLMath is abelian .
LaTeXMLMath Let LaTeXMLMath be an invariant of the embeddings of a graph that takes values in an abelian group .
We call LaTeXMLMath a Vassiliev invariant of type LaTeXMLMath if , for any singular spatial graph LaTeXMLMath of type LaTeXMLMath , LaTeXMLEquation .
Proof of Theorem 1.7 .
The ‘ only if ’ part follows from Theorem 1.1 ( 1 ) .
We shall show ‘ if ’ part .
Let LaTeXMLMath be a map defined as follows : LaTeXMLEquation .
Clearly , LaTeXMLMath is an invarinat .
Now we will show that LaTeXMLMath is a Vassiliev invariant of type LaTeXMLMath .
Let LaTeXMLMath be a singular spatial LaTeXMLMath -curve of type LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLMath if LaTeXMLMath .
So we may suppose that LaTeXMLMath .
Then we have LaTeXMLEquation .
Since LaTeXMLMath , by Lemma 2.1 , we may assume that LaTeXMLMath for some LaTeXMLMath -chords LaTeXMLMath .
By Sublemma 3.1 in LaTeXMLCite ( or Lemma 3.7 in the next section ) , there are LaTeXMLMath -chords LaTeXMLMath such that LaTeXMLMath is ambient isotopic to LaTeXMLMath .
By the arguments similar to that in the proof of Sublemma 3.5 in LaTeXMLCite , we see that there are LaTeXMLMath -chords LaTeXMLMath such that LaTeXMLMath is ambient isotopic to LaTeXMLMath .
Since LaTeXMLMath are LaTeXMLMath -chords , by Lemma 2.2 , we have LaTeXMLEquation and LaTeXMLEquation .
Hence we have LaTeXMLEquation .
Therefore LaTeXMLMath is a Vassiliev invariant of type LaTeXMLMath .
This and the assumption LaTeXMLMath imply LaTeXMLMath .
By the definition of LaTeXMLMath , we have LaTeXMLMath .
LaTeXMLMath 3 .
Disk/band surfaces and Vassiliev invariants of spatial graphs A graph LaTeXMLMath is trivalent if the valence of any vertex of LaTeXMLMath is equal to LaTeXMLMath .
A graph LaTeXMLMath is planar if there exists an embedding LaTeXMLMath .
A connected , planar graph LaTeXMLMath is said to be prime if , for any embedding LaTeXMLMath , there exist no simple closed curves LaTeXMLMath in LaTeXMLMath satisfying either the following ( 1 ) or ( 2 ) ( cf .
LaTeXMLCite , LaTeXMLCite ) , where LaTeXMLMath , LaTeXMLMath are the two components of LaTeXMLMath .
( 1 ) LaTeXMLMath meets LaTeXMLMath in a single point such that both LaTeXMLMath and LaTeXMLMath are non-empty .
( 2 ) LaTeXMLMath meets LaTeXMLMath in two points such that both LaTeXMLMath , LaTeXMLMath are neither empty nor single open arcs .
For any connected , planar graph LaTeXMLMath , we fix a planar embedding LaTeXMLMath arbitrarily .
The image LaTeXMLMath has complementary domains LaTeXMLMath that are bounded and one unbounded LaTeXMLMath .
The preimage LaTeXMLMath is a 1-complex which can be viewed as a 1-cycle in LaTeXMLMath .
We call LaTeXMLMath LaTeXMLMath , LaTeXMLMath respectively a boundary cycle and the outermost cycle in LaTeXMLMath with respect to LaTeXMLMath .
For a spatial embedding LaTeXMLMath of a graph LaTeXMLMath , a disk/band surface LaTeXMLMath of LaTeXMLMath is a compact , orientable surface in LaTeXMLMath such that LaTeXMLMath is a deformation retract of LaTeXMLMath contained in LaTeXMLMath LaTeXMLCite .
In LaTeXMLCite , T. Soma , H. Sugai and the author showed the following theorem .
Theorem 3.1 .
( LaTeXMLCite ) Suppose that LaTeXMLMath is a connected , planar , prime and trivalent graph , and LaTeXMLMath is an embedding .
Then , for any embedding LaTeXMLMath , there exists the unique disk/band surface LaTeXMLMath of LaTeXMLMath up to ambient isotopy of which the Seifert pairings satisfying the following equation .
LaTeXMLEquation where LaTeXMLMath are boundary cycles and LaTeXMLMath is the outermost cycle with respect to LaTeXMLMath .
LaTeXMLMath We call the disk/band surface above the canonical disk/band surface for LaTeXMLMath .
Note that the Seifert linking form of the canonical disk/band surface depends only on the linking numbers of pairs of disjoint cycles .
If LaTeXMLMath is the LaTeXMLMath -curve or the complete graph with LaTeXMLMath vertices , then the cannonical disk/band surface is same as the disk/band surface with zero Seifert linking form that is defined in LaTeXMLCite .
By the proof of Theorem 1 in LaTeXMLCite , we note that the canonical disk/band surface is given as the image of an embedding of the regular neighborhood LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
Thus by fixing orientation and label of LaTeXMLMath , we have an ordered , oriented link as the image of an embedding of LaTeXMLMath .
From now on , we always assume that , for each graph LaTeXMLMath , LaTeXMLMath has fixed orientation and label .
Let LaTeXMLMath be a local move , LaTeXMLMath the components of LaTeXMLMath and LaTeXMLMath the components of LaTeXMLMath with LaTeXMLMath LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be regular neighbourhoods of LaTeXMLMath and LaTeXMLMath in LaTeXMLMath respectively such that LaTeXMLMath LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Let LaTeXMLMath LaTeXMLMath be disjoint union of properly embedded LaTeXMLMath arcs in LaTeXMLMath as illustrated in Fig .
3.1 .
Let LaTeXMLMath be a homeomorphism with LaTeXMLMath for LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are ambient isotopic in LaTeXMLMath relative to LaTeXMLMath .
Then we say that a local move LaTeXMLMath is a parallel of LaTeXMLMath with weight LaTeXMLMath .
Fig .
3.1 Proposition 3.2 .
Let LaTeXMLMath be a connected , planar , prime and trivalent graph .
Let LaTeXMLMath LaTeXMLMath be embeddings and LaTeXMLMath the canonical disk/band surface for LaTeXMLMath LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent , then LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
Let LaTeXMLMath and LaTeXMLMath be tangles .
We say that LaTeXMLMath is obtained from LaTeXMLMath by a local move LaTeXMLMath if there is an orientation preserving embedding LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Two tangles LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent if LaTeXMLMath is obtained from LaTeXMLMath by a finite sequence of LaTeXMLMath -moves and ambient isotopies relative LaTeXMLMath .
Lemma 3.3 .
( cf .
LaTeXMLCite ) Let LaTeXMLMath be a parallel of a LaTeXMLMath -move .
Then LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
Proof .
Let LaTeXMLMath be a LaTeXMLMath -move , LaTeXMLMath and LaTeXMLMath the components of LaTeXMLMath and LaTeXMLMath respectively .
Suppose that LaTeXMLMath is a parallel of LaTeXMLMath with weight LaTeXMLMath .
We give a proof by induction on LaTeXMLMath .
In the case that LaTeXMLMath , it obviously holds .
Suppose LaTeXMLMath .
We may suppose LaTeXMLMath .
By Lemma 2.1 in LaTeXMLCite , we may assume that the LaTeXMLMath -move LaTeXMLMath is as illustrated in Fig .
3.2 , i.e. , the arcs except for LaTeXMLMath are contained in the shaded part in LaTeXMLMath LaTeXMLMath .
It is not hard to see that LaTeXMLMath is obtained from LaTeXMLMath by LaTeXMLMath local moves that are paralells of LaTeXMLMath with weight LaTeXMLMath .
This completes the proof .
LaTeXMLMath Fig .
3.2 Proof of Proposition 3.2 .
In the case LaTeXMLMath , it clearly holds .
We consider the case LaTeXMLMath .
It is sufficient to consider the case that LaTeXMLMath is obtained from LaTeXMLMath by a single LaTeXMLMath -move .
Suppose that LaTeXMLMath is obtained from LaTeXMLMath by a LaTeXMLMath -move .
Then there is an embedding LaTeXMLMath such that LaTeXMLMath is a LaTeXMLMath -move and LaTeXMLMath together with the labels .
We may suppose that LaTeXMLMath is a parallel of the LaTeXMLMath -move LaTeXMLMath with weight LaTeXMLMath .
Then we have a new disk/band surface LaTeXMLMath for LaTeXMLMath from LaTeXMLMath by the local move LaTeXMLMath .
Since LaTeXMLMath -move ( LaTeXMLMath ) does not change the linking number , by Theorem 3.1 , we have LaTeXMLMath , and since the local move LaTeXMLMath does not change the Seifert linking form of a disk/band surface , we have LaTeXMLMath .
So we have LaTeXMLMath .
By Theorem 3.1 , LaTeXMLMath is ambient isotopic to LaTeXMLMath .
Thus LaTeXMLMath is obtained from LaTeXMLMath by a parallel of a LaTeXMLMath -move .
Lemma 3.3 completes the proof .
LaTeXMLMath Let LaTeXMLMath be a connected , planar , prime and trivalent graph and LaTeXMLMath the set of spatial graph types .
Let LaTeXMLMath be an invariant of ordered , oriented links that takes values in an abelian group LaTeXMLMath .
Then we define a map LaTeXMLMath as LaTeXMLMath , where LaTeXMLMath is the canonical disk/band surface for LaTeXMLMath .
By Theorem 3.1 , LaTeXMLMath is an invariant of LaTeXMLMath .
We call LaTeXMLMath the invariant induced from LaTeXMLMath .
Theorem 3.4 .
Let LaTeXMLMath be a connected , planar , prime and trivalent graph and LaTeXMLMath the set of spatial graph types .
Let LaTeXMLMath be a Vassiliev invariant of type LaTeXMLMath for ordered , oriented links .
Then the invariant for LaTeXMLMath induced from LaTeXMLMath is a Vassiliev invariant of type LaTeXMLMath .
In Theorem 3.4 , the case of that a graph is the LaTeXMLMath -curve and LaTeXMLMath is given by Stanford LaTeXMLCite .
By the arguments similar to that in the proof of Lemma 1.4 in LaTeXMLCite , we have Lemma 3.5 .
Let LaTeXMLMath be a Vassiliev invariant of type LaTeXMLMath for ordered , oriented links and LaTeXMLMath the invariant induced from LaTeXMLMath .
Let LaTeXMLMath be a singular spatial graph of type LaTeXMLMath .
Let LaTeXMLMath be the ordered , oriented link that is the boundary of the canonical disk/band surface for LaTeXMLMath LaTeXMLMath .
Suppose there are mutually disjoint embeddings LaTeXMLMath LaTeXMLMath such that LaTeXMLMath LaTeXMLMath together with orientations and labels of the components for any subset LaTeXMLMath , LaTeXMLMath LaTeXMLMath is a LaTeXMLMath -move LaTeXMLMath , and LaTeXMLMath LaTeXMLMath Then we have LaTeXMLEquation .
The following lemma follows directly from the proof of Theorem 1 in LaTeXMLCite .
Lemma 3.6 .
Let LaTeXMLMath be a connected , planar , prime and trivalent graph , LaTeXMLMath an embedding , and LaTeXMLMath the regular neighborhood of LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be a disk/band surface for an embedding LaTeXMLMath .
Suppose that LaTeXMLMath is the image of an embedding of LaTeXMLMath that is an extension of LaTeXMLMath .
Then the canonical disk/band surface for LaTeXMLMath is obtained from LaTeXMLMath by a finite sequence of the moves as illustrated in Fig .
LaTeXMLMath .
LaTeXMLMath Fig .
3.3 In the definition of band sum in Section 2 , by replacing LaTeXMLMath with LaTeXMLMath LaTeXMLMath , we can define that LaTeXMLMath is a band sum of LaTeXMLMath and link models LaTeXMLMath .
By the arguments similar to that in Proof of Lemma 3.6 LaTeXMLCite , we have the following lemma .
Lemma 3.7 .
Two tangles LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent if and only if there are tangless LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath is ambient isotopic to LaTeXMLMath LaTeXMLMath relative LaTeXMLMath and LaTeXMLMath is a band sum of LaTeXMLMath and some LaTeXMLMath -link models .
LaTeXMLMath Lemma 3.8 .
Let LaTeXMLMath and LaTeXMLMath be tangles .
If LaTeXMLMath is obtained from LaTeXMLMath by a parallel of a LaTeXMLMath -move , then there are tangles LaTeXMLMath and mutually disjoint , orientation preserving embeddings LaTeXMLMath LaTeXMLMath such that LaTeXMLMath LaTeXMLMath is ambient isotopic to LaTeXMLMath LaTeXMLMath relative LaTeXMLMath , LaTeXMLMath LaTeXMLMath , and LaTeXMLMath LaTeXMLMath is a LaTeXMLMath -move LaTeXMLMath .
Proof .
By Lemma 3.3 , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
By Lemma 3.7 , there are tangles LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath is ambient isotopic to LaTeXMLMath LaTeXMLMath relative LaTeXMLMath and that LaTeXMLMath is a band sum of LaTeXMLMath and some LaTeXMLMath -link models LaTeXMLMath .
Since LaTeXMLMath LaTeXMLMath are LaTeXMLMath -moves , we have the conclusion .
LaTeXMLMath Proof of Theorem 3.4 .
Since LaTeXMLMath is a singular spatial graph of type LaTeXMLMath , by the definition , there are mutually disjoint , orientation preserving embeddings LaTeXMLMath LaTeXMLMath such that ( 1 ) LaTeXMLMath together with the labels for any subset LaTeXMLMath , ( 2 ) LaTeXMLMath is a LaTeXMLMath -move LaTeXMLMath , and ( 3 ) LaTeXMLMath Let LaTeXMLMath be the canonical disk/band surface for LaTeXMLMath .
By considering the intersections LaTeXMLMath LaTeXMLMath , we find disk/band surfaces LaTeXMLMath for LaTeXMLMath LaTeXMLMath such that ( 1 ) LaTeXMLMath ( 2 ) LaTeXMLMath is a parallel of a LaTeXMLMath -move LaTeXMLMath , and ( 3 ) LaTeXMLMath By the proof of Proposition 3.2 , if LaTeXMLMath for any LaTeXMLMath , then LaTeXMLMath is the canonical disk/band surface .
Set LaTeXMLMath LaTeXMLMath .
By Lemma 3.6 , there are mutually disjoint , orientation preserving embeddings LaTeXMLMath LaTeXMLMath , where LaTeXMLMath if LaTeXMLMath , and the canonical disk/band surfaces LaTeXMLMath for LaTeXMLMath such that ( 1 ) LaTeXMLMath , ( 2 ) LaTeXMLMath is a paralell of LaTeXMLMath -move LaTeXMLMath , and ( 3 ) LaTeXMLMath By combining this , Lemmas 3.8 and 3.5 , we have the conclusion .
LaTeXMLMath Let LaTeXMLMath be a connected , planar , prime and trivalent graph and LaTeXMLMath the set of edges of LaTeXMLMath .
Let LaTeXMLMath be the cannonical disk/band surface for a spatial embedding LaTeXMLMath of LaTeXMLMath , and let LaTeXMLMath LaTeXMLMath be a surface obtained from LaTeXMLMath as illustrated in Fig .
3.4 .
We note that LaTeXMLMath depends only on LaTeXMLMath and the integers LaTeXMLMath .
This means LaTeXMLMath is the unique surface for LaTeXMLMath .
Let LaTeXMLMath be an invariant of ordered , oriented links that takes values in an abelian group LaTeXMLMath .
Then we can define an invariant LaTeXMLMath as LaTeXMLMath .
We call LaTeXMLMath the invariant induced from LaTeXMLMath with respect to LaTeXMLMath .
Fig .
3.4 By the arguments similar to that in the proofs of Proposition 3.2 and Theorem 3.4 , we have the following theorem .
Theorem 3.9 Let LaTeXMLMath be a connected , planar , prime and trivalent graph and LaTeXMLMath the set of edges of LaTeXMLMath .
Then the followings hold .
( 1 ) Let LaTeXMLMath and LaTeXMLMath be spatial graphs and LaTeXMLMath the surface obtained from the canonical disk/band surface for LaTeXMLMath LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent , then LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
( 2 ) Let LaTeXMLMath be a Vassiliev invariant of type LaTeXMLMath for ordered , oriented links .
Then the invariant LaTeXMLMath for spatial embeddings of LaTeXMLMath induced from LaTeXMLMath is a Vassiliev invariant of type LaTeXMLMath .
LaTeXMLMath Proof of Theorem 1.8 .
Suppose that LaTeXMLMath is abelian .
Let LaTeXMLMath and LaTeXMLMath be spatial LaTeXMLMath -curves as illustrated in Fig .
3.5 .
Since LaTeXMLMath is abelian , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -equivalent .
Then , by Theorem 1.1 ( 1 ) , LaTeXMLMath .
This means that LaTeXMLMath and LaTeXMLMath can not be distinguished by any Vassiliev invariant of order LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the surfaces obtained from the cannonical disk/band surfaces for LaTeXMLMath and LaTeXMLMath respectively .
We note that LaTeXMLMath and LaTeXMLMath contain pretzel knots LaTeXMLMath and LaTeXMLMath respectively , see Fig .
3.6 .
Let LaTeXMLMath be a Vassiliev invariant for oriented link of order LaTeXMLMath .
By combining Theorem 3.9 ( 1 ) and the fact that a LaTeXMLMath -moves preserves Vassiliev invariants of order LaTeXMLMath LaTeXMLCite ( or simply by Theorem 3.9 ( 2 ) ) , we have LaTeXMLMath .
Hence LaTeXMLMath .
Let LaTeXMLMath be the quantum invariant of a knot LaTeXMLMath corresponding to the representation of the partition LaTeXMLMath of the quantum enveloping algebra LaTeXMLMath .
Then , using the computer software ‘ K2K ’ by M. Ochiai and N. Imafuji LaTeXMLCite , we have LaTeXMLEquation where LaTeXMLMath is a trivial knot .
Since this is divisible by LaTeXMLMath and is not divisible by LaTeXMLMath , these pretzel knots can be distinguished by a Vassiliev invariant of order LaTeXMLMath LaTeXMLCite .
Hence we have LaTeXMLMath .
This completes the proof .
LaTeXMLMath Fig .
3.5 Fig .
3.6 Acknowledgement The author would like to thank Professor Jun Murakami , Professor Józef Przytycki and Professor Paweł Traczyk for their valuable advice on distingushing two mutant knots by Vassiliev invariants .
Their advice was useful for proving Theorem 1.8
In 1986 , Matveev defined the notion of Borromean surgery for closed oriented LaTeXMLMath -manifolds and showed that the equivalence relation generated by this move is characterized by the pair ( first betti number , linking form up to isomorphism ) .
We explain how this extends for LaTeXMLMath -manifolds with spin structure if we replace the linking form by the quadratic form defined by the spin structure .
We then show that the equivalence relation among closed spin LaTeXMLMath -manifolds generated by spin Borromean surgeries is characterized by the triple ( first betti number , linking form up to isomorphism , Rochlin invariant modulo LaTeXMLMath ) .
+¿ The notion of Borromean surgery was introduced by Matveev in LaTeXMLCite as an example of what he called a LaTeXMLMath -surgery .
Since then , this transformation has become the elementary move of Goussarov-Habiro finite type invariants theory for oriented LaTeXMLMath -manifolds ( LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
Matveev showed that the equivalence relation , among closed oriented LaTeXMLMath -manifolds , generated by Borromean surgery is characterized by the pair : LaTeXMLEquation where LaTeXMLMath is the first Betti number of a LaTeXMLMath -manifold LaTeXMLMath and LaTeXMLEquation is its torsion linking form .
This result gives a characterization of degree LaTeXMLMath invariants in Goussarov-Habiro theory for closed oriented LaTeXMLMath -manifolds .
As mentioned by Habiro and Goussarov , their finite type invariants theory ( in short : “ FTI theory ” ) makes sense also for LaTeXMLMath -manifolds with spin structure because Borromean surgeries work well with spin structures ( see § LaTeXMLRef ) .
So , the question is : what is the “ spin ” analogue of Matveev ’ s theorem ?
For each closed spin LaTeXMLMath -manifold LaTeXMLMath , a quadratic form LaTeXMLEquation can be defined by many ways ( see LaTeXMLCite , LaTeXMLCite , and also LaTeXMLCite , LaTeXMLCite ) .
The bilinear form associated to LaTeXMLMath is LaTeXMLMath .
Its Gauss-Brown invariant is equal to LaTeXMLMath modulo LaTeXMLMath , where LaTeXMLEquation is the Rochlin function of LaTeXMLMath , sending a spin structure LaTeXMLMath of LaTeXMLMath to the modulo LaTeXMLMath signature of a spin LaTeXMLMath -manifold which spin-bounds LaTeXMLMath .
The main result of this paper is the following refinement of Matveev ’ s theorem : Let LaTeXMLMath and LaTeXMLMath be connected closed spin LaTeXMLMath -manifolds .
Then , the following assertions are equivalent : LaTeXMLMath and LaTeXMLMath can be obtained from one another by spin Borromean surgeries , there exists a homology isomorphism LaTeXMLMath such that : LaTeXMLEquation .
LaTeXMLMath modulo LaTeXMLMath and there exists a homology isomorphism LaTeXMLMath such that : LaTeXMLEquation .
The equivalence between assertions 2 and 3 will be the topological statement of an algebraic fact : a nondegenerate quadratic form on a finite Abelian group is determined , up to isomorphism , by its associated bilinear form and its Gauss-Brown invariant .
In § LaTeXMLRef we recall Matveev ’ s notion of LaTeXMLMath -surgery .
With this background , we then recall the definition of Borromean surgery , and give equivalent descriptions of other authors .
In § LaTeXMLRef , we clarify how all of these notions have to be understood in the spin case : in particular , spin Borromean surgeries are introduced .
As a motivation to Theorem LaTeXMLRef , FTI for spin LaTeXMLMath -manifolds , in the sense of Habiro and Goussarov , are then defined : the Rochlin invariant is shown to be a finite type degree LaTeXMLMath invariant .
It should be mentioned that Cochran and Melvin have proposed a different FTI theory in LaTeXMLCite , and have also refined their theory to the case of spin manifolds .
§ LaTeXMLRef is of an algebraic nature .
We recall some definitions and results about quadratic forms on finite Abelian groups .
We also prove the above mentioned algebraic fact : the proof makes use of Kawauchi-Kojima classification of linking pairings .
§ LaTeXMLRef is the topological cousin of the former : we review the quadratic form LaTeXMLMath .
Starting from Turaev LaTeXMLMath -dimensional definition in LaTeXMLCite , we then give an intrinsic definition for LaTeXMLMath ( no reference to dimension LaTeXMLMath ) .
§ LaTeXMLRef is devoted to the proof of Theorem LaTeXMLRef .
It goes as a refinement of the original proof by Matveev for the “ unspun ” case .
Last section will give some of its applications .
The author wants to thank his advisor Pr .
Christian Blanchet who supervised this work , and Florian Deloup for conversations regarding quadratic forms .
He is also indebted to Félicie Pastore for correcting his written english .
First of all , we want to recall the unifying idea of LaTeXMLMath -surgery by Matveev in LaTeXMLCite .
This will allow us to have a more conceptual view of Borromean surgeries .
We begin with some general definitions .
A Matveev triple is a triple of oriented LaTeXMLMath -manifolds : LaTeXMLEquation where LaTeXMLMath is closed and is the union of LaTeXMLMath and LaTeXMLMath along their common boundary LaTeXMLMath , as depicted in Figure LaTeXMLRef .
The triple LaTeXMLMath is called the inverse of LaTeXMLMath and is denoted by LaTeXMLMath .
Let now LaTeXMLMath be a closed oriented LaTeXMLMath -manifold and let LaTeXMLMath be an orientation-preserving embedding .
Form the following closed oriented LaTeXMLMath -manifold : LaTeXMLEquation .
With the above notations , LaTeXMLMath is said to be obtained from LaTeXMLMath by LaTeXMLMath -surgery along LaTeXMLMath .
Note that if LaTeXMLMath denotes the embedding of LaTeXMLMath in LaTeXMLMath , then LaTeXMLMath is obtained from LaTeXMLMath by LaTeXMLMath -surgery along LaTeXMLMath .
Two Matveev triples LaTeXMLMath and LaTeXMLMath are said to be equivalent if there exists an orientation-preserving diffeomorphism from LaTeXMLMath to LaTeXMLMath sending LaTeXMLMath to LaTeXMLMath , and LaTeXMLMath to LaTeXMLMath .
Note that , if the triples LaTeXMLMath and LaTeXMLMath are equivalent , then they have the same surgery effect .
Let LaTeXMLMath denote the genus LaTeXMLMath closed oriented surface and let LaTeXMLMath be the genus LaTeXMLMath oriented handlebody .
Then , each orientation-preserving diffeomorphism LaTeXMLMath leads to a triple : LaTeXMLEquation .
A LaTeXMLMath -surgery amounts to “ twist ” an embedded genus LaTeXMLMath handlebody by LaTeXMLMath .
For instance , from the standard genus one Heegaard decomposition of LaTeXMLMath , integral Dehn surgery is recovered .
The original Matveev ’ s point of view was : A Borromean surgery is a LaTeXMLMath -surgery with : LaTeXMLEquation where the “ halves ” LaTeXMLMath and LaTeXMLMath are obtained from the genus LaTeXMLMath handlebody by surgery on three-component framed links as shown in Figure LaTeXMLRef .
We now recall Goussarov ’ s notion of LaTeXMLMath -surgery in LaTeXMLCite .
This move is equivalent to the LaTeXMLMath -move of Habiro in LaTeXMLCite .
A LaTeXMLMath -graph LaTeXMLMath in a closed oriented LaTeXMLMath -manifold LaTeXMLMath is an ( unoriented ) embedding of the surface drawn in Figure LaTeXMLRef , together with its decomposition between leaves , edges and node .
The closed oriented LaTeXMLMath -manifold obtained from LaTeXMLMath by LaTeXMLMath -surgery along LaTeXMLMath is : LaTeXMLEquation where LaTeXMLMath is a regular neighbourhood of LaTeXMLMath in LaTeXMLMath , and LaTeXMLMath is the surgered handlebody on the six-component link LaTeXMLMath drawn on Figure LaTeXMLRef .
We call LaTeXMLMath -equivalence the equivalence relation among closed oriented LaTeXMLMath -manifolds generated by orientation-preserving diffeomorphisms and LaTeXMLMath -surgeries .
Note that a LaTeXMLMath -surgery is a LaTeXMLMath -surgery if we call LaTeXMLMath the triple : LaTeXMLEquation the corresponding LaTeXMLMath -graph gives the place where the LaTeXMLMath -surgery must be performed .
The Matveev triples LaTeXMLMath and LaTeXMLMath are equivalent .
Thus , Borromean surgery is equivalent to LaTeXMLMath -surgery .
We will show that both of the triples LaTeXMLMath and LaTeXMLMath are equivalent to a triple LaTeXMLMath , defined by an orientation-preserving diffeomorphism LaTeXMLMath .
We start with the “ half ” LaTeXMLMath of Figure LaTeXMLRef : handle-sliding of each node-component over the corresponding leaf-component , followed by some isotopies of framed links gives Figure LaTeXMLRef , where only part of the link is drawn .
Up to a LaTeXMLMath -framing correction , the three depicted components LaTeXMLMath can be normally pushed off at once towards the boundary : we obtain three disjoint curves LaTeXMLMath on LaTeXMLMath .
Note that during this push-off , none of the three components LaTeXMLMath is intersected .
Then , the components LaTeXMLMath can also be pushed off so that the framing correction is now LaTeXMLMath : the result is a family of three disjoint curves LaTeXMLMath .
After a convenient isotopy of the handles , the curves LaTeXMLMath and LaTeXMLMath can be depicted as on Figure LaTeXMLRef .
We define LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are the following composites of ( commuting ) Dehn twists : LaTeXMLEquation .
According to the Lickorish trick LaTeXMLCite , a LaTeXMLMath -surgery is therefore equivalent to a LaTeXMLMath -surgery .
On the other hand , from Figure LaTeXMLRef , we deduce that a LaTeXMLMath -surgery is equivalent to a LaTeXMLMath -surgery where LaTeXMLMath .
Let LaTeXMLMath denote the meridian of the LaTeXMLMath handle of LaTeXMLMath , for LaTeXMLMath .
Then , the equation : LaTeXMLEquation holds for both LaTeXMLMath and LaTeXMLMath so that LaTeXMLMath .
∎ Note that , in the proof of Lemma LaTeXMLRef , the curve LaTeXMLMath is homologous in the surface LaTeXMLMath to the corresponding curve LaTeXMLMath ( look at Figure LaTeXMLRef ) .
As a consequence , the diffeomorphism LaTeXMLMath belongs to the Torelli group .
In the sequel , we will call LaTeXMLMath the Borromean diffeomorphism .
In order to have a complete understanding of these equivalent triples LaTeXMLMath , LaTeXMLMath or LaTeXMLMath , it remains to recognize their underlying closed LaTeXMLMath -manifolds .
The closed LaTeXMLMath -manifolds LaTeXMLMath and LaTeXMLMath , respectively defined by the triples LaTeXMLMath and LaTeXMLMath , are both homeomorphic to the LaTeXMLMath -torus LaTeXMLMath .
According to Lemma LaTeXMLRef , LaTeXMLMath and LaTeXMLMath are diffeomorphic .
Let us identify LaTeXMLMath .
Recall that LaTeXMLMath was defined as : LaTeXMLEquation .
Write LaTeXMLMath as LaTeXMLMath where LaTeXMLMath ( resp .
LaTeXMLMath ) is the sublink containing the leaf ( resp .
node ) -components .
Note that LaTeXMLMath can be isotoped in LaTeXMLMath to some Borromean rings contained in a LaTeXMLMath -ball disjoint from LaTeXMLMath : make LaTeXMLMath leave the handlebody where it was lying , towards the handlebody with the minus sign in ( LaTeXMLRef ) .
So LaTeXMLMath is obtained from LaTeXMLMath by surgery on some Borromean rings contained in a little LaTeXMLMath -ball .
The lemma then follows from the fact that the latter is nothing but LaTeXMLMath .
∎ We now go into the world of spin LaTeXMLMath -manifolds .
We refer to LaTeXMLCite for an introduction to spin structures .
As a warming up , we recall a few facts in the next subsection .
Let LaTeXMLMath , and let LaTeXMLMath be a compact smooth oriented LaTeXMLMath -manifold endowed with a Riemannian metric .
Its bundle of oriented orthonormal frames will be denoted by LaTeXMLMath : it is a principal LaTeXMLMath -bundle with total space LaTeXMLMath and with projection LaTeXMLMath .
Recall that if LaTeXMLMath is spinnable , LaTeXMLMath can be thought of as : LaTeXMLEquation and is essentially independant of the metric .
The set LaTeXMLMath is then an affine space over LaTeXMLMath , the action being defined by : LaTeXMLEquation .
For LaTeXMLMath , let LaTeXMLMath be a compact smooth oriented LaTeXMLMath -manifold and let LaTeXMLMath be a submanifold of LaTeXMLMath with orientation induced by LaTeXMLMath .
Let also LaTeXMLMath be an orientation-reversing diffeomorphism and let LaTeXMLMath .
Assume that LaTeXMLMath and LaTeXMLMath are spinnable , that LaTeXMLMath is connected , and that the set : LaTeXMLEquation is not empty .
Then , LaTeXMLMath is spinnable and the restriction map : LaTeXMLEquation is injective with LaTeXMLMath as image .
For LaTeXMLMath or LaTeXMLMath , let LaTeXMLMath be the principal LaTeXMLMath -bundle derived from LaTeXMLMath and the inclusion of groups : LaTeXMLMath the canonical map from LaTeXMLMath to LaTeXMLMath .
Then , ( E ( F^+S_i ) ; Z _2 ) & ( E ( FS_i ) ; Z _2 ) is an isomorphism .
The bundle LaTeXMLMath can be identified with LaTeXMLMath .
In particular , there is an inclusion map LaTeXMLMath .
Moreover , the diffeomorphism LaTeXMLMath induces a further identification : ( F^+S_2 ) & ≃ ^f & E ( F^+S_1 ) , such that the total space LaTeXMLMath is homeomorphic to the glueing : LaTeXMLEquation .
We denote by LaTeXMLMath the corresponding inclusion of LaTeXMLMath into LaTeXMLMath .
But now , by the Mayer-Vietoris sequence , we have : ( E ( FM ) ; Z _2 ) & ( j_1^* , j_2^* ) & H^1 ( E ( FM_1 ) ; Z _2 ) ⊕ H^1 ( E ( FM_2 ) ; Z _2 ) & & ∘ f ) ^*-k_2^* & & H^1 ( E ( F^+S_2 ) ; Z _2 ) .
Note that LaTeXMLMath sends each LaTeXMLMath to LaTeXMLMath , while LaTeXMLMath sends each LaTeXMLMath to LaTeXMLMath .
Note also that , since LaTeXMLMath is connected , the map LaTeXMLMath is injective .
The whole lemma then follows from these two remarks and from the exactness of the Mayer-Vietoris sequence .
∎ With the notations and hypothesis of Lemma LaTeXMLRef , for each LaTeXMLMath , the unique spin structure of LaTeXMLMath sent by LaTeXMLMath to LaTeXMLMath is called the glueing of the spin structures LaTeXMLMath and LaTeXMLMath , and is denoted by LaTeXMLMath .
In some cases , a Matveev LaTeXMLMath -surgery , whose definition has been recalled in § LaTeXMLRef , makes sense for spin LaTeXMLMath -manifolds .
A Matveev triple LaTeXMLMath is said to be spin-admissible , if LaTeXMLMath is connected , and if the maps : LaTeXMLEquation induced by inclusions , are isomorphisms .
Suppose now that LaTeXMLMath is a spin-admissible triple .
Note that the restriction maps : Let also LaTeXMLMath be a closed oriented LaTeXMLMath -manifold and let LaTeXMLMath be an orientation-preserving embedding .
As in § LaTeXMLRef , we denote by LaTeXMLMath the result of the LaTeXMLMath -surgery along LaTeXMLMath , and we want to define a canonical bijection : Θ _j , V & Spin ( M ’ ) .
First , the embedding LaTeXMLMath allows us to define the following map : LaTeXMLEquation .
From formula ( LaTeXMLRef ) and from Definition LaTeXMLRef , we can define LaTeXMLMath as the following glueing : LaTeXMLEquation .
The inverse of LaTeXMLMath is LaTeXMLMath , where LaTeXMLMath denotes the embedding of LaTeXMLMath in LaTeXMLMath .
With the above notations , the spin manifold LaTeXMLMath is said to be obtained from LaTeXMLMath by spin LaTeXMLMath -surgery along LaTeXMLMath .
Let LaTeXMLMath be an orientation-preserving diffeomorphism .
Denote by LaTeXMLMath the lagrangian subspace of LaTeXMLMath span by the meridians .
Then , as can be easily verified , the triple LaTeXMLMath of Example LaTeXMLRef is spin-admissible if and only if LaTeXMLMath .
For instance , this condition is satisfied when LaTeXMLMath belongs to the Torelli modulo LaTeXMLMath group .
In the particular case of Example LaTeXMLRef , that is when LaTeXMLMath with LaTeXMLMath , then for each LaTeXMLMath , LaTeXMLMath is the unique spin structure of LaTeXMLMath extending LaTeXMLMath .
In that case , LaTeXMLMath is a handlebody and so LaTeXMLMath is zero .
Therefore , the restriction map LaTeXMLMath is injective .
∎ From Lemma LaTeXMLRef and from Remark LaTeXMLRef above , we have learnt that both of the triples LaTeXMLMath and LaTeXMLMath are equivalent to the triple LaTeXMLMath where LaTeXMLMath is the Borromean diffeomorphism which belongs to the Torelli group .
So , by Example LaTeXMLRef , they are spin-admissible and the following definition makes sense : A LaTeXMLMath -surgery , or equivalently a spin Borromean surgery , is the surgery move among closed spin LaTeXMLMath -manifolds defined equivalently by the triples LaTeXMLMath or LaTeXMLMath .
We call LaTeXMLMath -equivalence the equivalence relation among them generated by spin diffeomorphisms and LaTeXMLMath -surgeries .
Let LaTeXMLMath be a closed spin LaTeXMLMath -manifold and let LaTeXMLMath be a LaTeXMLMath -graph in LaTeXMLMath .
The LaTeXMLMath -surgery along LaTeXMLMath gives a new spin manifold which will be denoted by : LaTeXMLEquation .
Let LaTeXMLMath be an embedding of the genus 3 handlebody onto a regular neighbourhood of LaTeXMLMath in LaTeXMLMath .
Then , LaTeXMLEquation and according to Lemma LaTeXMLRef , LaTeXMLMath is the unique spin structure of LaTeXMLMath extending LaTeXMLMath .
Let LaTeXMLMath be a closed spin LaTeXMLMath -manifold and let LaTeXMLMath and LaTeXMLMath be disjoint LaTeXMLMath -graphs in LaTeXMLMath .
Then , up to diffeomorphism of manifolds with spin structure , LaTeXMLEquation .
The equality LaTeXMLMath is obvious .
By construction , both of LaTeXMLMath and LaTeXMLMath are extensions of LaTeXMLMath .
The lemma then follows from the fact that the restriction map : ( ( M_G ) _H ) & ( M ∖ ( N ( G ) ∪ N ( H ) ) ) is injective since the relative cohomology group LaTeXMLMath is zero .
∎ Let LaTeXMLMath be a family of disjoint LaTeXMLMath -graphs in a closed LaTeXMLMath -manifold LaTeXMLMath with spin structure LaTeXMLMath .
Lemma LaTeXMLRef says that LaTeXMLMath -surgery along the family LaTeXMLMath is well-defined .
We denote the result by LaTeXMLMath .
Following Habiro and Goussarov definition of a finite type invariant ( LaTeXMLCite , LaTeXMLCite ) , we can now define : Let LaTeXMLMath be an Abelian group and let LaTeXMLMath be an LaTeXMLMath -valued invariant of LaTeXMLMath -manifolds with spin structure .
Then , LaTeXMLMath is an invariant of degree at most LaTeXMLMath if for any closed spin LaTeXMLMath -manifold LaTeXMLMath and any family LaTeXMLMath of at least LaTeXMLMath LaTeXMLMath -graphs in LaTeXMLMath , the following identity holds : LaTeXMLEquation where the sum is taken over all subfamilies LaTeXMLMath of LaTeXMLMath .
Moreover , LaTeXMLMath is of degree LaTeXMLMath if it is of degree at most LaTeXMLMath , but is not of degree at most LaTeXMLMath .
Note that the degree LaTeXMLMath invariants are precisely those invariants which are constant on each LaTeXMLMath -equivalence class .
So , the refined Matveev theorem will quantify how powerful they can be .
The next subsection will provide us some examples of invariants which are finite type in the sense of Definition LaTeXMLRef .
Let LaTeXMLMath be a closed spin LaTeXMLMath -manifold , and let LaTeXMLMath be a LaTeXMLMath -graph in LaTeXMLMath .
Then , the following formula holds : LaTeXMLEquation where the map LaTeXMLMath , induced by the LaTeXMLMath -surgery along LaTeXMLMath , has been defined in § LaTeXMLRef .
According to Lemma LaTeXMLRef , we can think of the LaTeXMLMath -torus as : LaTeXMLEquation .
The surgered manifold LaTeXMLMath will be thought of concretely as in ( LaTeXMLRef ) .
Pick a spin LaTeXMLMath -manifold LaTeXMLMath spin-bounded by LaTeXMLMath , and a spin LaTeXMLMath -manifold LaTeXMLMath spin-bounded by the LaTeXMLMath -torus with LaTeXMLMath as a spin structure .
Glue the “ generalized ” handle LaTeXMLMath to LaTeXMLMath along the first handlebody of the LaTeXMLMath -torus in decomposition ( LaTeXMLRef ) , using LaTeXMLMath as glue .
We obtain a LaTeXMLMath -manifold LaTeXMLMath .
Orient LaTeXMLMath coherently with LaTeXMLMath and LaTeXMLMath , and then give to LaTeXMLMath the spin structure obtained by glueing those of LaTeXMLMath and LaTeXMLMath ( see Definition LaTeXMLRef ) .
It follows from definitions that the spin-boundary of LaTeXMLMath is LaTeXMLMath .
According to Wall theorem on non-additivity of the signature ( see LaTeXMLCite ) , we have : LaTeXMLEquation .
The involved correcting term is the signature of a real bilinear symmetric form explicitely described by Wall .
It is defined by means of the intersection form in LaTeXMLMath , with domain : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are subspaces of LaTeXMLMath defined to be respectively the kernels of : Σ _3 ; R ) & ( j—_ ∂ ) _* & H_1 ( ∂ N ( G ) ; R ) & ∖ int ( N ( G ) ) ; R ) , & H_1 ( Σ _3 ; R ) & & R ) , and : & H_1 ( Σ _3 ; R ) & Σ _3 ; R ) & R ) .
No matter who is LaTeXMLMath , since the Borromean diffeomorphism LaTeXMLMath lies in the Torelli group , we certainly have LaTeXMLMath .
The space LaTeXMLMath then vanishes and so does the correcting term .
The announced equality then follows by taking equation ( LaTeXMLRef ) modulo LaTeXMLMath .
∎ The Rochlin invariant is a degree LaTeXMLMath invariant of closed spin LaTeXMLMath -manifolds for Goussarov-Habiro theory , and its modulo LaTeXMLMath reduction is of degree LaTeXMLMath .
In Cochran-Melvin theory , the Rochlin invariant is a degree LaTeXMLMath finite type invariant ( see LaTeXMLCite ) .
Last statement is clear from Proposition LaTeXMLRef and from the fact that the Rochlin function of the LaTeXMLMath -torus takes values in LaTeXMLMath .
Let us show that the Rochlin invariant is at most of degree LaTeXMLMath .
Take a closed spin LaTeXMLMath -manifold LaTeXMLMath and two disjoint LaTeXMLMath -graphs LaTeXMLMath and LaTeXMLMath in LaTeXMLMath .
According to Proposition LaTeXMLRef , in order to show that : LaTeXMLEquation it suffices to show that : LaTeXMLEquation where the left LaTeXMLMath is defined by LaTeXMLMath and the right LaTeXMLMath is defined by LaTeXMLMath .
But this follows from definition of the maps LaTeXMLMath and from the fact that LaTeXMLMath extends LaTeXMLMath .
It remains to show that the Rochlin invariant is not of degree LaTeXMLMath ( and so it will be “ exactly ” of degree LaTeXMLMath ) .
For instance , all of the spin structures of the LaTeXMLMath -torus are related one to another by LaTeXMLMath -surgeries ( Cf Example LaTeXMLRef below ) and so are not distinguished one to another by degree LaTeXMLMath invariants .
But Rochlin distinguishes one of them from the others .
∎ We call LaTeXMLMath the LaTeXMLMath -manifold obtained from LaTeXMLMath by surgery along a ordered oriented framed link LaTeXMLMath of length LaTeXMLMath , and LaTeXMLMath the corresponding LaTeXMLMath -manifold obtained from LaTeXMLMath by attaching LaTeXMLMath -handles and sometimes called the trace of the surgery .
Let also LaTeXMLMath be the linking matrix of LaTeXMLMath .
Recall that LaTeXMLMath is free Abelian of rank LaTeXMLMath .
For each LaTeXMLMath , choose a Seifert surface of LaTeXMLMath in LaTeXMLMath and push it off into the interior of LaTeXMLMath : denote the result by LaTeXMLMath .
Then , glue LaTeXMLMath to the core of the LaTeXMLMath LaTeXMLMath -handle to obtain a closed surface LaTeXMLMath .
A basis of LaTeXMLMath is then given by LaTeXMLMath .
This is by means of the so-called “ characteristic solutions of LaTeXMLMath ” or , equivalently , “ characteristic sublinks of LaTeXMLMath ” .
A vector LaTeXMLMath or , equivalently , the sublink of LaTeXMLMath containing the components LaTeXMLMath such that LaTeXMLMath , are said to be characteristic if the following equation is satisfied : LaTeXMLEquation .
We denote by LaTeXMLMath the subset of LaTeXMLMath comprising the characteristic solutions of LaTeXMLMath .
There is a bijection : ≃ & S _L which is defined by the following composition : Z _2 ) & ≃ & H_2 ( W_L ; Z _2 ) & ≃ & ( Z _2 ) ^l where LaTeXMLMath sends any LaTeXMLMath to the obstruction to extend LaTeXMLMath to the whole of LaTeXMLMath , LaTeXMLMath is the Poincaré duality isomorphism and the last map is defined by the basis LaTeXMLMath .
With this combinatorial description , Kirby theorem can be refined to closed spin LaTeXMLMath -manifolds ( see LaTeXMLCite ) .
The following lemma , more general than needed , will allow us to enunciate in those terms the effect of a LaTeXMLMath -surgery .
Let LaTeXMLMath be the ordered union of two ordered oriented framed links in LaTeXMLMath and let LaTeXMLMath be an embedded handlebody such that LaTeXMLMath is contained in the interior of LaTeXMLMath , LaTeXMLMath is disjoint from LaTeXMLMath , and LaTeXMLMath is a LaTeXMLMath -homology handlebody .
Suppose that LaTeXMLMath is represented by a characteristic solution LaTeXMLMath of LaTeXMLMath satisfying the following two properties : LaTeXMLMath is a characteristic solution of LaTeXMLMath , for all LaTeXMLMath such that LaTeXMLMath , the component LaTeXMLMath bounds a Seifert surface within LaTeXMLMath .
Then , the restricted spin structure : LaTeXMLEquation extends to the spin structure of LaTeXMLMath represented by LaTeXMLMath .
In the following , all ( co ) homology groups are assumed to be with coefficients in LaTeXMLMath .
We use the above fixed notations .
Let us consider the map : ( V_L ∖ H ) & ( W_L , V_L ∖ H ) , where LaTeXMLMath is the obstruction to extend any LaTeXMLMath to the whole of LaTeXMLMath .
Let also LaTeXMLMath denote the connecting homomorphism for the pair LaTeXMLMath .
Note that the following equation holds : LaTeXMLEquation .
Since LaTeXMLMath is injective , it follows that LaTeXMLMath is injective .
The same map LaTeXMLMath can be defined for LaTeXMLMath relatively to LaTeXMLMath , and for LaTeXMLMath and LaTeXMLMath relatively to LaTeXMLMath .
We have thus the following commutative diagram : ∪ K ) & ∪ K , V_L ∪ K ) & ≃ _P & H_2 ( W_L ∪ K ) & Spin ( V_L ∪ K ∖ H_K ) & ∪ K , V_L ∪ K ∖ H_K ) & ≃ _P & H_2 ( W_L ∪ K , H_K ) & & r Spin ( V_L ∖ H ) & ∖ H ) & ≃ _P & H_2 ( W_L , H ) & Spin ( V_L ) & ≃ _P & H_2 ( W_L ) where the letter LaTeXMLMath stands for a Poincaré duality isomomorphism , the vertical arrows are induced by inclusions and the map LaTeXMLMath is defined by planar commutativity .
From intersection theory , we deduce that : LaTeXMLEquation .
Let now LaTeXMLMath be a spin structure of LaTeXMLMath such that the corresponding characteristic solution LaTeXMLMath of LaTeXMLMath satisfies the conditions 1 and 2 of the lemma .
We define LaTeXMLMath .
By hypothesis 1 , there exists a unique spin structure LaTeXMLMath of LaTeXMLMath with LaTeXMLMath as associated characteristic solution of LaTeXMLMath .
We want to show that LaTeXMLMath .
Diagram chasing shows that proving LaTeXMLMath should suffice .
This follows from hypothesis 2 , formulas ( LaTeXMLRef ) and from the fact that LaTeXMLMath and LaTeXMLMath .
∎ Let LaTeXMLMath be a closed oriented LaTeXMLMath -manifold and let LaTeXMLMath be obtained from LaTeXMLMath by surgery along a LaTeXMLMath -graph LaTeXMLMath .
According to § LaTeXMLRef , LaTeXMLMath -surgery along LaTeXMLMath induces a bijective map : Θ _G & Spin ( M_G ) .
With the notations of § LaTeXMLRef , each LaTeXMLMath is sent by LaTeXMLMath to LaTeXMLMath .
Suppose now that we are given a surgery presentation LaTeXMLMath of LaTeXMLMath .
Isotope the graph LaTeXMLMath in LaTeXMLMath to make it disjoint from the dual of LaTeXMLMath , so that LaTeXMLMath is in LaTeXMLMath .
Let LaTeXMLMath be a regular neighbourhood of LaTeXMLMath .
A few Kirby calculi , inside LaTeXMLMath , show that surgery along this LaTeXMLMath -graph is equivalent to surgery on the two-component link LaTeXMLMath of Figure LaTeXMLRef .
We prefer this unsymmetric link to Figure LaTeXMLRef because of the fewer components .
We then have LaTeXMLMath .
The linking matrix of LaTeXMLMath , when LaTeXMLMath is appropriately oriented , looks like : LaTeXMLEquation .
Writing the characteristic condition ( LaTeXMLRef ) , we find that each characteristic solution of LaTeXMLMath is of the form : LaTeXMLEquation where LaTeXMLMath must be characteristic for LaTeXMLMath .
Equation ( LaTeXMLRef ) then defines a combinatorial bijection : S _L & ≃ & S _L ∪ K. With the above notations , the map LaTeXMLMath is a combinatorial version of the map LaTeXMLMath in terms of characteristic solutions for surgery presentations on LaTeXMLMath .
More precisely , the following diagram is commutative : S _L & S _L ∪ K ≃ & & ≃ Spin ( M ) & Θ _G & Spin ( M_G ) This follows from the definitions and from Lemma LaTeXMLRef : note that LaTeXMLMath is nul-homologous in LaTeXMLMath , and that here LaTeXMLMath is merely a handlebody .
∎ Let LaTeXMLMath be a surgery presentation of a LaTeXMLMath -manifold LaTeXMLMath on LaTeXMLMath , and let LaTeXMLMath be a LaTeXMLMath -graph in LaTeXMLMath .
Then , LaTeXMLMath is said to be simple ( with respect to this surgery presentation ) , if LaTeXMLMath can be isotoped in LaTeXMLMath so that , in LaTeXMLMath , its leaves bound disjoint discs , each intersecting LaTeXMLMath in exactly one point .
For a LaTeXMLMath -surgery along a simple LaTeXMLMath -graph , the spin-diffeomorphism of Figure LaTeXMLRef holds .
Replace in the lhs of Figure LaTeXMLRef , this simple LaTeXMLMath -graph by the 2-component link of Figure LaTeXMLRef such that LaTeXMLMath links the LaTeXMLMath component of LaTeXMLMath and use equation ( LaTeXMLRef ) to obtain the intermediate link of Figure LaTeXMLRef .
Perform then some spin Kirby moves to obtain the rhs of Figure LaTeXMLRef .
∎ As a consequence of Corollary LaTeXMLRef , the Lie spin structure of the LaTeXMLMath -torus is LaTeXMLMath -equivalent to the seven other ones .
The two spin structures of LaTeXMLMath are equivalent , so are the eight ones of LaTeXMLMath .
Furthermore , LaTeXMLMath can be obtained from LaTeXMLMath by surgery along a trivial LaTeXMLMath -framed three-component link .
Surgery on the LaTeXMLMath -framed Borromean rings gives rise to the LaTeXMLMath -torus LaTeXMLMath , and this link can be obtained from the trivial link by a simple LaTeXMLMath -surgery .
∎ We recall here standard algebraic constructions : notations are that of Deloup in LaTeXMLCite , where a brief review of the subject can be found .
A linking pairing on a finite Abelian group LaTeXMLMath is a nondegenerate symmetric bilinear map LaTeXMLMath .
A quadratic form on LaTeXMLMath is a map LaTeXMLMath such that the map LaTeXMLMath defined by LaTeXMLMath is bilinear , and such that LaTeXMLMath satisfies : LaTeXMLMath .
LaTeXMLMath is said to be nondegenerate when the associated bilinear form LaTeXMLMath is a linking pairing .
Let now LaTeXMLMath be a symmetric bilinear form on a free finitely generated Abelian group LaTeXMLMath .
We denote by LaTeXMLMath the adjoint map , and by LaTeXMLMath its rational extension .
Form : LaTeXMLEquation .
Note that LaTeXMLMath , the torsion subgroup of LaTeXMLMath .
We now define a linking pairing : LaTeXMLEquation by the formula : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath is such that LaTeXMLMath , and LaTeXMLMath is the rational extension of LaTeXMLMath .
LaTeXMLMath is said to be a presentation of the linking pairing LaTeXMLMath .
Suppose now that the form LaTeXMLMath comes equipped with a Wu class , that is an element LaTeXMLMath such that LaTeXMLMath .
We can then define a quadratic form over LaTeXMLMath , denoted by : LaTeXMLEquation and defined by : LaTeXMLEquation .
LaTeXMLMath is determined by the modulo LaTeXMLMath class of LaTeXMLMath .
The triple LaTeXMLMath is said to be a presentation of the quadratic form LaTeXMLMath .
Any linking pairing and any nondegenerate quadratic form admit such presentations with LaTeXMLMath nondegenerate ( see LaTeXMLCite ) .
Given an arbitrary quadratic form on a finite Abelian group LaTeXMLMath , we can calculate its Gauss sum : LaTeXMLEquation .
This complex number is a LaTeXMLMath -root of unity or is LaTeXMLMath ( only if LaTeXMLMath is degenerate ) .
We then define the corresponding Gauss-Brown invariant LaTeXMLMath by the formula : LaTeXMLEquation using the convention LaTeXMLMath .
If LaTeXMLMath admits a triple LaTeXMLMath as a presentation , a useful formula of Van der Blij states that LaTeXMLCite , in case when LaTeXMLMath is nondegenerate .
Milnor and Husemoller have included a detailed proof in LaTeXMLCite , for LaTeXMLMath nondegenerate and LaTeXMLMath .
The general case can be reduced to this special case .
: LaTeXMLEquation .
For LaTeXMLMath a linking pairing , denote by LaTeXMLMath the set of quadratic forms with LaTeXMLMath as associated linking pairing , and denote by LaTeXMLMath the subgroup of elements of LaTeXMLMath of order at most LaTeXMLMath .
Using the nondegenerativity of LaTeXMLMath , we easily obtain : The set LaTeXMLMath is an affine space over LaTeXMLMath , with action defined by : LaTeXMLEquation .
The following lemma says how the Gauss-Brown invariant behaves under this action .
Let LaTeXMLMath and let LaTeXMLMath .
Then , LaTeXMLEquation .
Since LaTeXMLMath , we have : LaTeXMLEquation ∎ We now want to prove the following result : Let LaTeXMLMath and LaTeXMLMath be two nondegenerate quadratic forms on finite Abelian groups .
We denote by LaTeXMLMath and LaTeXMLMath the linking pairings going respectively with them .
The following two assertions are equivalent : LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath denotes the Gauss-Brown invariant of quadratic forms .
Let LaTeXMLMath ( respectively LaTeXMLMath ) be a nondegenerate quadratic form over the linking pairing LaTeXMLMath ( respectively LaTeXMLMath ) .
Recall that the relation between LaTeXMLMath and LaTeXMLMath is : LaTeXMLEquation .
From this formula and the definition of LaTeXMLMath , “ LaTeXMLMath ” of Theorem LaTeXMLRef is obvious .
Suppose momentarily that LaTeXMLMath is a LaTeXMLMath -group , with LaTeXMLMath an odd prime .
Then , equation LaTeXMLMath makes LaTeXMLMath determine LaTeXMLMath , for if LaTeXMLMath is another quadratic form over LaTeXMLMath , then LaTeXMLMath is an order at most LaTeXMLMath element of LaTeXMLMath , and so vanishes .
So , if LaTeXMLMath and LaTeXMLMath are both LaTeXMLMath -groups , then LaTeXMLMath implies LaTeXMLMath .
Come back now to the general case and suppose that condition 2 is satisfied .
LaTeXMLMath splits along its LaTeXMLMath -primary components LaTeXMLMath : LaTeXMLEquation and , according to formula ( LaTeXMLRef ) , the same holds for LaTeXMLMath .
The given isomorphism between LaTeXMLMath and LaTeXMLMath , induces then for each prime LaTeXMLMath an isomorphism between LaTeXMLMath and LaTeXMLMath .
From the above lines , we deduce that , for LaTeXMLMath odd , LaTeXMLMath and LaTeXMLMath are isomorphic .
In particular , LaTeXMLMath for LaTeXMLMath odd , and so , by additivity of the Gauss-Brown invariant , this is also true for LaTeXMLMath .
Consequently , it is enough to prove Theorem LaTeXMLRef when LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -groups .
In the sequel , we recall a construction due to Wall ( see LaTeXMLCite ) , establishing a one to one correspondence ( up to isomorphism ) between nondegenerate quadratic forms on LaTeXMLMath -groups and linking pairings on LaTeXMLMath -groups without direct summand of order two .
Next , we give a brief review of Kawauchi and Kojima classification of linking pairings on LaTeXMLMath -groups .
We will finally end the proof of Theorem LaTeXMLRef .
A linking pairing LaTeXMLMath will be said here to be special if LaTeXMLMath is a finite LaTeXMLMath -group without direct summand of order two .
Let give us a special linking pairing LaTeXMLMath .
Set LaTeXMLMath where LaTeXMLMath is the subgroup of elements of order at most LaTeXMLMath , and denote by LaTeXMLMath the canonical projection LaTeXMLMath .
Note that LaTeXMLMath can be any LaTeXMLMath -group .
Define now LaTeXMLMath by : LaTeXMLEquation .
The quantity LaTeXMLMath is well-defined because of the special feature of the group LaTeXMLMath .
Then , LaTeXMLMath is easily seen to be quadratic and nondegenerate , with associated linking pairing LaTeXMLMath defined by : LaTeXMLEquation .
Let us denote by LaTeXMLMath the construction LaTeXMLMath .
Wall showed this to be surjective onto the set of nondegenerate quadratic forms on LaTeXMLMath -groups .
He also proved that if LaTeXMLMath and LaTeXMLMath give rise to the same LaTeXMLMath by LaTeXMLMath , then they have to be isomorphic ( see LaTeXMLCite ) .
As a consequence , the classification , up to isomorphism , of nondegenerate quadratic forms on LaTeXMLMath -groups is reduced to that of special linking pairings .
Let LaTeXMLMath be a linking pairing on a finite LaTeXMLMath -group .
Find a cyclic decomposition of LaTeXMLMath : LaTeXMLEquation .
The natural numbers LaTeXMLMath are group invariants of LaTeXMLMath .
The very next construction is due to Wall ( see LaTeXMLCite ) .
Denote by LaTeXMLMath the subgroup of LaTeXMLMath of elements of order at most LaTeXMLMath for LaTeXMLMath , and set : LaTeXMLEquation .
The group LaTeXMLMath is clearly a LaTeXMLMath -vector space of rank LaTeXMLMath .
Let also LaTeXMLMath be defined by : LaTeXMLEquation .
The form LaTeXMLMath was shown by Wall to be nondegenerate ( see LaTeXMLCite ) .
Consider now the map LaTeXMLMath sending LaTeXMLMath to LaTeXMLMath .
It is additive , and so we can define an element LaTeXMLMath in LaTeXMLMath by the equation : LaTeXMLEquation .
When LaTeXMLMath , the following map can be defined : LaTeXMLEquation .
The form LaTeXMLMath can be verified to be quadratic and nondegenerate .
In particular , its Gauss-Brown invariant LaTeXMLMath is not equal to LaTeXMLMath .
Kawauchi and Kojima defined LaTeXMLMath by : LaTeXMLEquation and showed the following theorem ( see LaTeXMLCite ) .
If LaTeXMLMath is a linking pairing on a finite LaTeXMLMath -group , its isomorphism class is determined by the invariant family LaTeXMLMath .
Let LaTeXMLMath be a quadratic form on a finite LaTeXMLMath -group LaTeXMLMath , going with a linking pairing LaTeXMLMath .
Let also LaTeXMLMath be a special linking pairing , giving rise to LaTeXMLMath by Wall construction LaTeXMLMath .
We want to compare the invariants LaTeXMLMath and LaTeXMLMath of LaTeXMLMath and LaTeXMLMath , in order to quantify how much LaTeXMLMath determines LaTeXMLMath , and so LaTeXMLMath , up to isomorphism .
Recall that LaTeXMLMath .
Denote by LaTeXMLMath the canonical projection .
The map LaTeXMLMath induces a morphism from LaTeXMLMath onto LaTeXMLMath with kernel LaTeXMLMath .
So , LaTeXMLMath induces a natural isomorphism : ~ G ’ _k+1 & ~ π _k_ ≃ & ~ G _k .
In particular , the LaTeXMLMath -vector spaces LaTeXMLMath and LaTeXMLMath have the same rank .
So , LaTeXMLEquation .
Besides , since LaTeXMLMath is special , we have : LaTeXMLEquation .
The isomorphism LaTeXMLMath makes LaTeXMLMath and LaTeXMLMath commute because of equation ( LaTeXMLRef ) .
As a consequence , LaTeXMLMath sends LaTeXMLMath to LaTeXMLMath .
Furthermore , when these ( simultaneously ) vanish , the natural isomorphism between LaTeXMLMath and LaTeXMLMath induced by LaTeXMLMath , make LaTeXMLMath and LaTeXMLMath commute ( because of equation ( LaTeXMLRef ) ) .
As a consequence , these two quadratic forms will have the same Gauss-Brown invariant .
So , to sum up , LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath vanishes .
It remains to be noticed that LaTeXMLMath is nothing but LaTeXMLMath .
Thus , LaTeXMLEquation .
Now , from equations ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) and Kawauchi-Kojima theorem , we see that LaTeXMLMath together with LaTeXMLMath determine LaTeXMLMath up to isomorphism .
What has been remaining to be proved for Theorem LaTeXMLRef , then follows .
∎ We now give a result of Durfee ( see LaTeXMLCite ) as a corollary of Theorem LaTeXMLRef .
Let LaTeXMLMath be a linking pairing on a finite Abelian group LaTeXMLMath without cyclic direct summand of order LaTeXMLMath or LaTeXMLMath .
Then , LaTeXMLMath .
Take some quadratic forms LaTeXMLMath and LaTeXMLMath over LaTeXMLMath , and let LaTeXMLMath be such that LaTeXMLMath ( see Lemma LaTeXMLRef ) .
By the hypothesis on LaTeXMLMath , there exits some LaTeXMLMath such that LaTeXMLMath , and so , by homogeneity of LaTeXMLMath , we have : LaTeXMLEquation .
But since LaTeXMLMath is then of order at most LaTeXMLMath , LaTeXMLMath has to be of order at most LaTeXMLMath ( see LaTeXMLCite ) .
It follows that LaTeXMLMath and so , by Lemma LaTeXMLRef , we obtain that LaTeXMLMath .
Theorem LaTeXMLRef allows us to conclude .
∎ In this section , when not specified , integer coefficients are assumed .
Let LaTeXMLMath be a connected closed oriented LaTeXMLMath -manifold , and let LaTeXMLMath be a surgery presentation on LaTeXMLMath given by an ordered oriented framed link LaTeXMLMath ( see the beginning of § LaTeXMLRef ) .
We use notations and apply constructions of § LaTeXMLRef to LaTeXMLMath , taking for LaTeXMLMath the intersection form of LaTeXMLMath .
Recall that the matrix of LaTeXMLMath relative to the preferred basis of LaTeXMLMath is LaTeXMLMath , the linking matrix of LaTeXMLMath .
The composite : Z ) & ≃ & H^2 ( W_L ) & ≃ & H_2 ( W_L , ∂ W_L ) is equal to LaTeXMLMath , induced by inclusion .
Since LaTeXMLMath , we obtain the following isomorphism LaTeXMLMath : ≃ & T ( Coker ( i_* ) ) & ≃ ^ ψ _* & TH_1 ( M ) In fact , it is well-known that the above LaTeXMLMath is via LaTeXMLMath a presentation of the torsion linking form LaTeXMLMath of LaTeXMLMath , the definition of which we now recall : Let LaTeXMLMath be respectively realized by oriented disjoint knots LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be such that LaTeXMLMath .
Pick a LaTeXMLMath -times connected sum of LaTeXMLMath .
We obtain a null-homologous knot in LaTeXMLMath for which we can thus find a Seifert surface LaTeXMLMath in general position with LaTeXMLMath .
Then : LaTeXMLEquation where LaTeXMLMath is the intersection form of LaTeXMLMath .
Now to each LaTeXMLMath is associated a characteristic solution of LaTeXMLMath or , alternatively , a Wu class ( modulo LaTeXMLMath ) of LaTeXMLMath , denoted by LaTeXMLMath .
Then , Turaev defined : The quadratic form of the spin LaTeXMLMath -manifold LaTeXMLMath : ϕ _M , σ & Q / Z is defined to be LaTeXMLMath .
We still have to verify that LaTeXMLMath does not depend on the choice of the surgery presentation .
Let LaTeXMLMath be another one .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) be the characteristic solution of LaTeXMLMath ( resp .
LaTeXMLMath ) corresponding to the spin structure LaTeXMLMath of LaTeXMLMath ( resp .
LaTeXMLMath of LaTeXMLMath ) .
According to the refined Kirby theorem ( see LaTeXMLCite ) , there exists a sequence of spin Kirby moves from LaTeXMLMath to LaTeXMLMath , inducing a spin-diffeomorphism from LaTeXMLMath to LaTeXMLMath isotopic to LaTeXMLMath .
These Kirby moves induce a path LaTeXMLMath whose elementary steps are : LaTeXMLEquation .
As a consequence , this path induces an isomorphism LaTeXMLMath from LaTeXMLMath to LaTeXMLMath making the following diagram commutative : & & & TH_1 ( M ) The well-definition of LaTeXMLMath then follows .
Let LaTeXMLMath be a closed spin LaTeXMLMath -manifold and let LaTeXMLMath be a smooth oriented knot in LaTeXMLMath .
Each parallel LaTeXMLMath of LaTeXMLMath defines a trivialization of the normal bundle of LaTeXMLMath in LaTeXMLMath , and so allows us to restrict LaTeXMLMath to a spin structure on LaTeXMLMath .
We define : LaTeXMLEquation .
LaTeXMLMath is a LaTeXMLMath -valued invariant of framed knots in LaTeXMLMath .
Let LaTeXMLMath be a closed spin LaTeXMLMath -manifold .
Then , for each oriented smooth knot LaTeXMLMath in LaTeXMLMath with LaTeXMLMath as a parallel and meridian LaTeXMLMath , LaTeXMLEquation .
Let LaTeXMLMath denote the boundary of a regular neighbourhood of LaTeXMLMath in LaTeXMLMath .
The normal bundle of LaTeXMLMath in LaTeXMLMath is naturally trivialized , so LaTeXMLMath inherits from LaTeXMLMath a spin structure .
Let LaTeXMLMath be the quadratic form associated to the spin smooth surface LaTeXMLMath as defined by Johnson in LaTeXMLCite .
The following identity then holds for each parallel LaTeXMLMath : LaTeXMLEquation when LaTeXMLMath is thought of as a curve on LaTeXMLMath .
Since LaTeXMLMath is quadratic with respect to the modulo LaTeXMLMath intersection form LaTeXMLMath on LaTeXMLMath , we have : LaTeXMLEquation ∎ We now recall the definition of the framing number LaTeXMLMath of a rationally nulhomologous oriented framed knot LaTeXMLMath in a closed oriented LaTeXMLMath -manifold LaTeXMLMath : Choose LaTeXMLMath such that LaTeXMLMath .
Pick a LaTeXMLMath -times connected sum of LaTeXMLMath .
We obtain a null-homologous knot in LaTeXMLMath for which we can thus pick a Seifert surface LaTeXMLMath in general position with the knot LaTeXMLMath .
Then : LaTeXMLEquation .
Let LaTeXMLMath be a closed spin LaTeXMLMath -manifold and LaTeXMLMath .
Choose a smooth oriented knot LaTeXMLMath in LaTeXMLMath representative for LaTeXMLMath , and pick a parallel LaTeXMLMath for LaTeXMLMath satisfying LaTeXMLMath .
Then , LaTeXMLEquation .
Note that , according to Lemma LaTeXMLRef , the rhs of ( LaTeXMLRef ) is an invariant of the oriented knot LaTeXMLMath ( it does not depend on the choice of LaTeXMLMath satisfying the above condition ) .
This lemma claims that it only depends on the homology class LaTeXMLMath of LaTeXMLMath , and gives a LaTeXMLMath -dimensional definition for the quadratic form LaTeXMLMath .
From this lemma , we can see that LaTeXMLMath coincides with the quadratic form defined by Lannes and Latour in LaTeXMLCite when specialized to our case ( see also LaTeXMLCite ) .
Consider the LaTeXMLMath -manifold LaTeXMLMath obtained from LaTeXMLMath by attaching a LaTeXMLMath -handle to LaTeXMLMath along LaTeXMLMath .
Identify LaTeXMLMath with LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath extends in a unique way to a spin structure LaTeXMLMath of LaTeXMLMath .
LaTeXMLMath is then a spin cobordism between LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is the closed oriented LaTeXMLMath -manifold obtained from LaTeXMLMath by the corresponding surgery , and where LaTeXMLMath is the restriction of LaTeXMLMath to LaTeXMLMath .
Note also that the core of the LaTeXMLMath -handle is a LaTeXMLMath -disc LaTeXMLMath in LaTeXMLMath with boundary LaTeXMLMath in LaTeXMLMath , and whose normal bundle can be trivialized in accordance with the trivialization of the normal bundle of LaTeXMLMath in LaTeXMLMath given by LaTeXMLMath .
The framed knot LaTeXMLMath will briefly be said to have property LaTeXMLMath in LaTeXMLMath .
According to Kaplan Theorem ( see LaTeXMLCite ) , the spin LaTeXMLMath -manifold LaTeXMLMath admits an even surgery presentation in LaTeXMLMath ( i.e .
the linking matrix is even and its characteristic solution corresponding to LaTeXMLMath is the trivial one ) .
Denote by LaTeXMLMath the trace of the surgery and by LaTeXMLMath the unique extension of LaTeXMLMath to the whole of LaTeXMLMath .
By glueing LaTeXMLMath to LaTeXMLMath along LaTeXMLMath , we obtain a spin LaTeXMLMath -manifold LaTeXMLMath with boundary LaTeXMLMath .
The LaTeXMLMath -handle from LaTeXMLMath to LaTeXMLMath can be reversed .
After a rearrangement , the LaTeXMLMath -manifold LaTeXMLMath appears as LaTeXMLMath to which have been simultaneously attached some LaTeXMLMath -handles ( one more than LaTeXMLMath ) , with boundary LaTeXMLMath , and to which LaTeXMLMath can be extended .
So , LaTeXMLMath is the trace of an even surgery presentation .
As a summary , we have found so for an even surgery presentation LaTeXMLMath for LaTeXMLMath such that LaTeXMLMath has property ( LaTeXMLMath ) in LaTeXMLMath .
Let us work with this surgery presentation of LaTeXMLMath .
Notations of § LaTeXMLRef will be used : LaTeXMLMath , LaTeXMLMath stands for the intersection form of LaTeXMLMath and so on .
Let the LaTeXMLMath -disc LaTeXMLMath give an element of LaTeXMLMath .
The latter is identified with an element LaTeXMLMath of LaTeXMLMath .
Recall from the definition of LaTeXMLMath that , in this even case , LaTeXMLEquation where LaTeXMLMath is the rational extension of LaTeXMLMath and where LaTeXMLMath is such that LaTeXMLMath .
Let LaTeXMLMath such that LaTeXMLMath .
Then , there exists LaTeXMLMath such that LaTeXMLMath .
So LaTeXMLMath works as a LaTeXMLMath .
Equation ( LaTeXMLRef ) can be rewritten as : LaTeXMLEquation .
When LaTeXMLMath is seen as belonging to LaTeXMLMath , the integer LaTeXMLMath is equal to LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -cycles representatives for LaTeXMLMath in transverse position in LaTeXMLMath .
By means of a “ collar ” trick appearing in LaTeXMLCite , we will be able to give examples of such LaTeXMLMath and LaTeXMLMath .
We add a collar LaTeXMLMath to LaTeXMLMath such that LaTeXMLMath is identified with LaTeXMLMath .
Let LaTeXMLMath be a Seifert surface for LaTeXMLMath in LaTeXMLMath in transverse position with LaTeXMLMath , and LaTeXMLMath be a Seifert surface for LaTeXMLMath in LaTeXMLMath .
Because of the property LaTeXMLMath , LaTeXMLMath can be pushed off to a disc LaTeXMLMath in such a way that LaTeXMLMath and LaTeXMLMath .
Figure LaTeXMLRef is a good summary .
We define LaTeXMLMath and LaTeXMLMath .
Then : LaTeXMLEquation .
The last LaTeXMLMath in ( LaTeXMLRef ) is intersection in LaTeXMLMath .
The lemma follows from ( LaTeXMLRef ) , ( LaTeXMLRef ) and the definition of a framing number .
∎ The algebraic results of § LaTeXMLRef have a topological meaning .
First , it has been shown by Turaev in LaTeXMLCite : For each LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is the Rochlin function of the LaTeXMLMath -manifold LaTeXMLMath .
Find an even surgery presentation of the spin LaTeXMLMath -manifold LaTeXMLMath .
So we are led to apply formula ( LaTeXMLRef ) with LaTeXMLMath .
∎ So , in view of Lemma LaTeXMLRef , the topological translations of Theorem LaTeXMLRef and its Corollary LaTeXMLRef are respectively : Let LaTeXMLMath and LaTeXMLMath be connected closed spin LaTeXMLMath -manifolds .
The following two assertions are equivalent : their quadratic forms LaTeXMLMath and LaTeXMLMath are isomorphic , their linking forms LaTeXMLMath and LaTeXMLMath are isomorphic , and LaTeXMLMath modulo LaTeXMLMath .
Let LaTeXMLMath be a connected closed oriented LaTeXMLMath -manifold , such that LaTeXMLMath does not admit LaTeXMLMath nor LaTeXMLMath as a direct summand .
Then , all of its quadratic forms are isomorphic one to another .
Part of the work has already been done in previous sections .
First , “ LaTeXMLMath ” follows from Corollary LaTeXMLRef and from the easy part of ( unspun ) Matveev theorem : a LaTeXMLMath -surgery preserves homology and torsion linking forms .
This can be verified seeing LaTeXMLMath -surgery as a LaTeXMLMath -surgery ( where LaTeXMLMath is the Borromean diffeomorphism of Remark LaTeXMLRef ) , using a Mayer-Vietoris argument and the fact that LaTeXMLMath belongs to the Torelli group .
Second , “ LaTeXMLMath ” follows from Proposition LaTeXMLRef .
What remains to be proved is then “ LaTeXMLMath ” .
We start by recalling an algebraic result of Durfee ( see LaTeXMLCite ) about even symmetric bilinear forms on finitely generated free Abelian groups .
Let LaTeXMLMath and LaTeXMLMath be the unimodular even forms whose matrices are respectively : LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath be two symmetric even bilinear forms on finitely generated free Abelian groups .
They are said to be stably equivalent if they become isomorphic after some stabilizations with unimodular symmetric even bilinear forms .
Note that a unimodular even form becomes indefinite after taking direct sum with LaTeXMLMath .
Recall that every even unimodular indefinite form splits as a direct sum of LaTeXMLMath and LaTeXMLMath ( see for example LaTeXMLCite ) .
Thus , in Definition LaTeXMLRef , LaTeXMLMath and LaTeXMLMath suffice as unimodular even forms to stabilize .
For LaTeXMLMath an even symmetric bilinear form on a finitely generated free Abelian group , we will shortly denote by LaTeXMLMath the quadratic form LaTeXMLMath corresponding to the zero Wu class .
Let LaTeXMLMath and LaTeXMLMath be two symmetric even bilinear forms on finitely generated free Abelian groups .
Then , the following two assertions are equivalent : LaTeXMLMath and LaTeXMLMath are stably equivalent , LaTeXMLMath and LaTeXMLMath .
Implication “ 1 LaTeXMLMath 2 ” is obvious since LaTeXMLMath and LaTeXMLMath are both unimodular .
Now suppose that condition 2 is satisfied .
For each LaTeXMLMath , there exists a nondegenerate symmetric bilinear form LaTeXMLMath such that : LaTeXMLEquation where LaTeXMLMath ( note that it is free ) , LaTeXMLMath , and LaTeXMLMath is the zero form .
The form LaTeXMLMath is still even and LaTeXMLMath and LaTeXMLMath are still isomorphic .
Consequently , without loss of generality , we can assume both LaTeXMLMath to be nondegenerate .
But this case was treated by Durfee in LaTeXMLCite LaTeXMLCite .
.
∎ Let LaTeXMLMath and LaTeXMLMath be connected closed spin LaTeXMLMath -manifolds such that LaTeXMLMath and LaTeXMLMath .
Suppose given for them some even surgery presentations LaTeXMLMath and LaTeXMLMath with respective linking matrices LaTeXMLMath and LaTeXMLMath .
According to Proposition LaTeXMLRef , there exists a unimodular integer matrix LaTeXMLMath satisfying for some stabilizations : LaTeXMLEquation .
We have the following geometric realizations of algebraic operations : stabilizations by LaTeXMLMath correspond to connected sums with LaTeXMLMath surgery presented on the zero-framed Hopf link , a stabilization by LaTeXMLMath is concrete when thought of as a connected sum with the Poincaré sphere surgery presented on an appropriate height-component link as in LaTeXMLCite , congruence by LaTeXMLMath can be realized by some spin Kirby moves ( handle-slidings and changes of orientation of components of LaTeXMLMath ) .
The Poincaré sphere can also be obtained by surgery along a LaTeXMLMath -framed trefoil knot ( LaTeXMLCite ) , which can be obtained from the LaTeXMLMath -framed unknot by a simple LaTeXMLMath -surgery ( see Definition LaTeXMLRef ) .
As a consequence , the Poincaré sphere and the sphere LaTeXMLMath , equipped with their unique spin structures , are LaTeXMLMath -equivalent .
Since LaTeXMLMath -equivalence is compatible with connected sums , we can assume that LaTeXMLMath .
A theorem of Murakami and Nakanishi ( LaTeXMLCite LaTeXMLCite is not detailed . )
states that two ordered oriented links have identical linking matrices if and only if they are LaTeXMLMath -equivalent .
A LaTeXMLMath -move is a certain unknotting operation , which is equivalent to surgery along a simple LaTeXMLMath -graph .
Finally , from Corollary LaTeXMLRef , we see that a simple LaTeXMLMath -surgery between even surgery presentations leaves the trivial characteristic solution fixed .
We conclude that LaTeXMLMath is LaTeXMLMath -equivalent to LaTeXMLMath , which completes the proof .
According to Theorem LaTeXMLRef , two connected closed spin LaTeXMLMath -manifolds are LaTeXMLMath -equivalent if and only if they are LaTeXMLMath -equivalent as plain LaTeXMLMath -manifolds and their Rochlin invariants are identical modulo LaTeXMLMath .
In other words , while studying the degree LaTeXMLMath part of Goussarov-Habiro theory , the spin problem can be “ factored out ” .
Now , given a closed connected oriented LaTeXMLMath -manifold , one can wonder whether all of its spin structures are LaTeXMLMath -equivalent one to another .
This has been verified to be true in the case of LaTeXMLMath by a direct calculation ( Example LaTeXMLRef ) .
More generally we have : Let LaTeXMLMath be a connected oriented closed LaTeXMLMath -manifold such that LaTeXMLMath has no cyclic direct summand of order LaTeXMLMath or LaTeXMLMath .
Then , all spin structures of LaTeXMLMath are LaTeXMLMath -equivalent one to another .
This follows directly from Theorem LaTeXMLRef and Corollary LaTeXMLRef .
∎ On the contrary , we have : The two spin structures of LaTeXMLMath are not LaTeXMLMath -equivalent , for the Rochlin function of LaTeXMLMath takes LaTeXMLMath and LaTeXMLMath as values .
Commutative diagrams were drawn with Paul Taylor ’ s package .
Abstract A number is perfect if it is the sum of its proper divisors ; here we call a finite group ‘ perfect ’ if its order is the sum of the orders of its proper normal subgroups .
( This conflicts with standard terminology but confusion should not arise . )
The notion of perfect group generalizes that of perfect number , since a cyclic group is perfect just when its order is perfect .
We show that , in fact , the only abelian perfect groups are the cyclic ones , and exhibit some non-abelian perfect groups of even order .
This article was originally composed in 1996 for Eureka , the journal of the Cambridge student mathematical society ( but has yet to appear , as no issue has been published since ) .
It is therefore written to be comprehensible to an undergraduate readership , and contains many reminders of basic facts .
Contents LaTeXMLRef Perfect numbers LaTeXMLRef LaTeXMLRef Definition and first examples of perfect groups LaTeXMLRef LaTeXMLRef Multiplicativity LaTeXMLRef LaTeXMLRef The abelian quotient theorem : proof by counting LaTeXMLRef LaTeXMLRef The abelian quotient theorem : proof by prime-index subgroups LaTeXMLRef Perfect numbers are an ancient object of study .
A number is called perfect if it is the sum of its proper divisors—for instance , 6 is perfect , since LaTeXMLMath .
It is straightforward to classify the even perfect numbers , but it is a long-standing question as to whether there are any odd perfect numbers at all .
This article generalizes the notion of ‘ perfection ’ from numbers to groups .
We define what it means for a group to be perfect , explain in what sense this is a generalization of the notion for numbers , and go on to give some theory and examples of perfect groups .
Signposts are provided for the reader not well versed in group theory , so that at least the rough shape of the ideas should be discernible .
The first properties of perfect numbers are summarized in Section LaTeXMLRef .
In Section LaTeXMLRef we give the definition of a perfect group and look at some examples .
Section LaTeXMLRef is devoted to ‘ multiplicativity ’ .
This shows some close parallels with the world of numbers , including the results of Section LaTeXMLRef , and the new theory also enables us to give some more interesting examples of perfect groups than was possible previously .
The climax of the article , such as it is , is a theorem concerning the abelian quotients of perfect groups , a corollary of which classifies the perfect abelian groups .
Two rather different proofs of this result are offered , one in each of Sections LaTeXMLRef and LaTeXMLRef .
I would like to thank Robin Bhattacharyya for his careful reading-through of an early version of this document , and Vin de Silva for reading a later version .
Alan Bain made some useful suggestions on adapting it to an undergraduate readership , and Simon Norton made some further helpful remarks .
I must also make two apologies .
First of all , I crave the reader ’ s indulgence for the use of the term ‘ perfect group ’ when it is firmly established to mean something else .
Faced with a group-theoretic concept generalizing that of perfect number , any other name seemed unnatural .
My second apology is for the lack of pointers to the literature : some of the results included here are surely widely known , but I am not well enough educated to provide references .
Here we go over the basic properties of perfect numbers .
For any number LaTeXMLMath , define LaTeXMLMath , the sum of the divisors of LaTeXMLMath , and call LaTeXMLMath perfect if LaTeXMLMath .
By a ‘ number ’ I mean a positive integer .
The function LaTeXMLMath is multiplicative : that is to say , if LaTeXMLMath and LaTeXMLMath are coprime ( have no common divisors other than LaTeXMLMath ) then LaTeXMLMath .
To see this , first observe that any divisor LaTeXMLMath of LaTeXMLMath can be written uniquely as LaTeXMLMath where LaTeXMLMath is a divisor of LaTeXMLMath ( LaTeXMLMath ) ; conversely , if LaTeXMLMath is a divisor of LaTeXMLMath ( LaTeXMLMath ) then LaTeXMLMath is a divisor of LaTeXMLMath .
Hence LaTeXMLEquation as required .
It is easy to classify the even perfect numbers : they are precisely those numbers LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is prime .
( Of course , computing which values of LaTeXMLMath make LaTeXMLMath prime is itself a hard problem . )
The first three perfect numbers are LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
In one direction , suppose that LaTeXMLMath and LaTeXMLMath is prime : then LaTeXMLEquation so LaTeXMLMath is an even perfect number .
In the other direction , suppose that LaTeXMLMath is an even perfect number .
Write LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is odd : then LaTeXMLMath being perfect says that LaTeXMLEquation i.e .
LaTeXMLEquation i.e .
LaTeXMLEquation .
Hence LaTeXMLMath is a divisor of LaTeXMLMath , and since LaTeXMLEquation it is a proper divisor of LaTeXMLMath .
But LaTeXMLMath is by definition the sum of the proper divisors of LaTeXMLMath , so LaTeXMLMath is the unique proper divisor of LaTeXMLMath .
Thus LaTeXMLMath is prime and LaTeXMLMath , and by ( LaTeXMLRef ) , the latter means that LaTeXMLMath .
So LaTeXMLMath with LaTeXMLMath and LaTeXMLMath prime , as required .
In this section we define the notion of a perfect group , and search for examples among some of the well-known families of groups ( symmetric , alternating , … ) .
In fact , the only examples of perfect groups we will find are cyclic , although by Section LaTeXMLRef we will have developed enough theory to be able to exhibit some more interesting examples .
Of the examples below , only the cyclic groups ( LaTeXMLRef ) and the symmetric and alternating groups ( LaTeXMLRef ) will be needed later on .
The reader is reminded that a normal subgroup of a group LaTeXMLMath is a subset of LaTeXMLMath which is the kernel of some homomorphism from LaTeXMLMath to some other group ; equivalently , it is a subgroup LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
We write LaTeXMLMath to mean that LaTeXMLMath is a normal subgroup of LaTeXMLMath .
From here on , ‘ group ’ will mean ‘ finite group ’ .
If LaTeXMLMath is a group , define LaTeXMLMath , the sum of the orders of the normal subgroups of LaTeXMLMath , and say that LaTeXMLMath is perfect if LaTeXMLMath .
Let LaTeXMLMath be the cyclic group of order LaTeXMLMath .
Then LaTeXMLMath has one normal subgroup of order LaTeXMLMath for each divisor LaTeXMLMath of LaTeXMLMath , and no others , so LaTeXMLMath and LaTeXMLMath is perfect just when LaTeXMLMath is perfect .
Thus perfect groups provide a generalization of the concept of perfect numbers , and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are all perfect groups .
None of the symmetric groups LaTeXMLMath or alternating groups LaTeXMLMath is perfect .
If LaTeXMLMath then LaTeXMLMath is simple and the only normal subgroups of LaTeXMLMath are 1 , LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath and LaTeXMLMath are too small .
For LaTeXMLMath , we have LaTeXMLEquation .
A ( finite ) LaTeXMLMath -group is a group of order LaTeXMLMath , where LaTeXMLMath is prime and LaTeXMLMath .
Lagrange ’ s Theorem says that the order of any subgroup of a group divides the order of the group , so if LaTeXMLMath is a LaTeXMLMath -group then LaTeXMLMath .
Hence no LaTeXMLMath -group is perfect .
Let LaTeXMLMath be the dihedral group of order LaTeXMLMath : that is , the group of all isometries of a regular LaTeXMLMath -sided polygon .
Of the LaTeXMLMath isometries , LaTeXMLMath are rotations ( forming a cyclic subgroup of order LaTeXMLMath ) and LaTeXMLMath are reflections .
We examine the cases of LaTeXMLMath odd and LaTeXMLMath even separately .
LaTeXMLMath odd : All reflections are in an axis passing through a vertex and the midpoint of the opposite side , and any reflection is conjugate to any other by a suitable rotation .
Thus if LaTeXMLMath and LaTeXMLMath contains a reflection , then LaTeXMLMath contains all reflections ; but LaTeXMLMath too , so LaTeXMLMath , so LaTeXMLMath .
So any proper normal subgroup is inside the rotation group LaTeXMLMath ; conversely , any ( normal ) subgroup of LaTeXMLMath is normal in LaTeXMLMath .
Thus LaTeXMLEquation and LaTeXMLMath is perfect if and only if LaTeXMLMath is a perfect number .
LaTeXMLMath even : The reflections split into two conjugacy classes , LaTeXMLMath and LaTeXMLMath , each of size LaTeXMLMath : those in an axis through two opposite vertices , and those in an axis through the midpoints of two opposite sides .
Write LaTeXMLMath for the group of rotations by LaTeXMLMath or LaTeXMLMath or … or LaTeXMLMath vertices , a subgroup of LaTeXMLMath which is cyclic of order LaTeXMLMath .
Then we can show that the smallest subgroup of LaTeXMLMath containing LaTeXMLMath is LaTeXMLMath , for LaTeXMLMath and 2 .
Moreover , LaTeXMLMath is of order LaTeXMLMath , i.e .
index 2 , therefore normal in LaTeXMLMath .
So we have two different normal subgroups , LaTeXMLMath and LaTeXMLMath , of order LaTeXMLMath .
We also have the normal subgroups LaTeXMLMath and LaTeXMLMath , hence LaTeXMLEquation and LaTeXMLMath is not perfect .
In summary , the perfect dihedral groups are in one-to-one correspondence with the odd perfect numbers—so it is an open question as to whether there are any .
We proved in LaTeXMLRef that the function LaTeXMLMath , on numbers LaTeXMLMath , was multiplicative .
The aim of this section is to prove an analogous result for groups , and then to give some examples of nonabelian perfect groups by using this result .
Some difficulties are present for the reader not acquainted with composition series and the Jordan-Hölder Theorem .
However , it is still possible for him or her to understand an example ( LaTeXMLRef ) of a nonabelian perfect group , provided that the following fact is taken on trust : if LaTeXMLMath and LaTeXMLMath are groups whose orders are coprime , and LaTeXMLMath their direct product , then LaTeXMLMath .
This done , the reader may proceed to LaTeXMLRef straight away .
The Jordan-Hölder Theorem states that any two composition series for a group LaTeXMLMath have the same set-with-multiplicities of factors , up to isomorphism of the factors .
I shall write this set-with-multiplicities as LaTeXMLMath , and use LaTeXMLMath to denote the disjoint union ( or ‘ union counting multiplicities ’ ) of two sets-with-multiplicities .
Thus if LaTeXMLEquation then LaTeXMLEquation .
We will use the fundamental fact that if LaTeXMLMath then LaTeXMLEquation .
A pair of groups will be called coprime if they have no composition factor in common ; alternatively , we will say that one group is prime to the other .
( In particular , if two groups have coprime orders then they are coprime . )
We will prove that LaTeXMLMath is multiplicative : that is , if LaTeXMLMath and LaTeXMLMath are coprime then LaTeXMLMath .
First of all we establish the group-theoretic analogue of a number-theoretic result from Section LaTeXMLRef —namely , the second sentence of LaTeXMLRef .
Let LaTeXMLMath and LaTeXMLMath be coprime groups .
Then the normal subgroups of LaTeXMLMath are exactly the subgroups of the form LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath then LaTeXMLMath ; conversely , suppose LaTeXMLMath .
Write LaTeXMLMath ( LaTeXMLMath , 2 ) for the projections , and regard LaTeXMLMath as a normal subgroup of LaTeXMLMath by identifying it with LaTeXMLMath , and similarly LaTeXMLMath .
We have LaTeXMLEquation so by the ‘ fundamental fact ’ above , LaTeXMLEquation and therefore by symmetry LaTeXMLEquation .
But LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are coprime , so LaTeXMLMath and LaTeXMLMath have no element in common ; similarly LaTeXMLMath , so LaTeXMLMath and LaTeXMLMath have no element in common .
Hence LaTeXMLMath .
We also know that LaTeXMLMath determines the order of a group LaTeXMLMath and that LaTeXMLMath , so in fact LaTeXMLMath .
Thus LaTeXMLEquation and as always LaTeXMLEquation so LaTeXMLMath , with LaTeXMLMath .
LaTeXMLMath LaTeXMLMath is multiplicative .
LaTeXMLMath This is a direct analogue of LaTeXMLRef .
For by LaTeXMLRef , if LaTeXMLMath and LaTeXMLMath are coprime then LaTeXMLEquation .
We can now exhibit three nonabelian perfect groups .
The group LaTeXMLMath , of order 30 , is perfect .
For LaTeXMLMath and LaTeXMLMath have coprime orders ( 6 and 5 ) , so are coprime , so LaTeXMLEquation .
We present this example ( of order LaTeXMLMath ) along with the method by which it was found .
Firstly , LaTeXMLMath is a simple group of order LaTeXMLMath .
Now , let us try to find a perfect group LaTeXMLMath of the form LaTeXMLMath where LaTeXMLMath is some group prime to LaTeXMLMath .
Since LaTeXMLEquation we need to find a LaTeXMLMath such that LaTeXMLEquation .
Let us look for such a group LaTeXMLMath amongst those of the form LaTeXMLMath , where LaTeXMLMath is prime to LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLEquation we need to find a LaTeXMLMath such that LaTeXMLEquation .
In turn , let us look for such a group LaTeXMLMath amongst those of the form LaTeXMLMath , where LaTeXMLMath is prime to LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLEquation we need to find a LaTeXMLMath such that LaTeXMLEquation .
This is satisfied by LaTeXMLMath , and the groups LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are pairwise coprime .
Thus if LaTeXMLEquation then LaTeXMLMath is perfect .
By the same technique we get this next example , of order LaTeXMLMath .
This time , we start with the simple group LaTeXMLMath of order LaTeXMLMath , and the sequence of groups LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ‘ works ’ in the sense of the previous example .
The details are left to the reader ; note that LaTeXMLMath and that 127 is prime .
In each of the next two sections we present a separate proof of our main classification result , the abelian quotient theorem .
The two proofs have rather different flavours , and each produces its own insights , which is why both are included .
We start with the more elementary of the two .
An abelian quotient of a group LaTeXMLMath is just a quotient of LaTeXMLMath which is abelian .
That is , it ’ s an abelian group LaTeXMLMath for which there exists a surjective homomorphism LaTeXMLMath ; alternatively , it ’ s an abelian group isomorphic to LaTeXMLMath for some normal subgroup LaTeXMLMath of LaTeXMLMath .
We will prove : If LaTeXMLMath is a group with LaTeXMLMath then any abelian quotient of LaTeXMLMath is cyclic .
This result has the following corollaries , the second of which says that abelian perfect groups ‘ are ’ just perfect numbers : If LaTeXMLMath is a perfect group then any abelian quotient of LaTeXMLMath is cyclic .
The perfect abelian groups are precisely the cyclic groups LaTeXMLMath of order LaTeXMLMath with LaTeXMLMath perfect .
Part ( LaTeXMLRef ) is immediate .
For ( LaTeXMLRef ) , if LaTeXMLMath is perfect abelian then LaTeXMLMath is an abelian quotient of the perfect group LaTeXMLMath , hence LaTeXMLMath is cyclic .
But we have already seen ( LaTeXMLRef ) that the perfect cyclic groups correspond exactly to the perfect numbers .
LaTeXMLMath ( Those who know about such things will recognize that the theorem could be stated more compactly in this way : if LaTeXMLMath is a group with LaTeXMLMath then LaTeXMLMath is cyclic .
Here LaTeXMLMath is the abelianization of LaTeXMLMath : it is an abelian quotient of LaTeXMLMath with the property that any abelian quotient of LaTeXMLMath is also a quotient of LaTeXMLMath .
In particular , if LaTeXMLMath is abelian then LaTeXMLMath , which is how we would deduce Corollary LaTeXMLRef ( LaTeXMLRef ) from this formulation . )
The proof of the abelian quotient theorem given in this section uses two ingredients .
The first is a new way of evaluating LaTeXMLMath : For any group LaTeXMLMath , LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLMath The second ingredient is the ‘ standard ’ fact that the inverse image ( under a homomorphism ) of a normal subgroup is a normal subgroup .
For let LaTeXMLMath be a homomorphism of groups , and let LaTeXMLMath .
Then LaTeXMLMath is the kernel of the natural homomorphism LaTeXMLMath , in other words , LaTeXMLMath .
So LaTeXMLEquation i.e .
LaTeXMLMath is the kernel of the homomorphism LaTeXMLMath .
Thus LaTeXMLMath is a normal subgroup of LaTeXMLMath .
We are now ready to assemble these ingredients into the following proposition , from which the abelian quotient theorem follows immediately .
Two pieces of terminology will be used .
An element LaTeXMLMath of LaTeXMLMath is called a normal generator of LaTeXMLMath if the only normal subgroup of LaTeXMLMath containing LaTeXMLMath is LaTeXMLMath itself .
A group is called simple if it has precisely two normal subgroups—inevitably , the whole group and the one-element subgroup .
Let LaTeXMLMath be a group .
If LaTeXMLMath then LaTeXMLMath has a normal generator .
If LaTeXMLMath has a normal generator then any abelian quotient of LaTeXMLMath is cyclic .
By Lemma LaTeXMLRef , LaTeXMLMath if and only if the mean over all LaTeXMLMath of LaTeXMLEquation is LaTeXMLMath .
If LaTeXMLMath is not simple or trivial then LaTeXMLMath ( where LaTeXMLMath is the identity element of LaTeXMLMath ) ; so for the mean to be LaTeXMLMath , there must be some LaTeXMLMath for which LaTeXMLMath —and this says exactly that LaTeXMLMath is a normal generator of LaTeXMLMath .
On the other hand , if LaTeXMLMath is simple then any nonidentity element of LaTeXMLMath is a normal generator , and if LaTeXMLMath is trivial then LaTeXMLMath is a normal generator .
So ( LaTeXMLRef ) is proved in all cases .
Let LaTeXMLMath be an abelian quotient of LaTeXMLMath , with LaTeXMLMath a surjective homomorphism , and let LaTeXMLMath be a normal generator of LaTeXMLMath .
Then LaTeXMLMath is a normal generator of LaTeXMLMath : for if LaTeXMLMath and LaTeXMLMath then LaTeXMLMath is a normal subgroup of LaTeXMLMath containing LaTeXMLMath , so LaTeXMLMath ; and since LaTeXMLMath is surjective , this means that LaTeXMLMath .
But LaTeXMLMath is abelian , so all subgroups are normal , so the fact that LaTeXMLMath is a normal generator of LaTeXMLMath says that the only subgroup of LaTeXMLMath containing LaTeXMLMath is LaTeXMLMath itself .
And this in turn says exactly that the cyclic subgroup generated by LaTeXMLMath is LaTeXMLMath itself .
LaTeXMLMath This last section is devoted to a second proof of the abelian quotient theorem , LaTeXMLRef .
This time , the proof reveals something about the normal subgroup structure of a perfect group LaTeXMLMath : namely , that LaTeXMLMath has at most one normal subgroup of each prime index ( LaTeXMLRef ( LaTeXMLRef ) ) .
It is a corollary of this that any abelian quotient of LaTeXMLMath is cyclic .
This section assumes some more sophisticated group theory than the last .
Let LaTeXMLMath be a group and LaTeXMLMath a prime : then the number of normal subgroups of LaTeXMLMath with index LaTeXMLMath is LaTeXMLEquation for some LaTeXMLMath .
‘ Usually ’ LaTeXMLMath , in which case both sides of the equation evaluate to LaTeXMLMath .
For this proof we write the cyclic group of order LaTeXMLMath additively , as LaTeXMLMath .
We also write LaTeXMLMath for the set of all homomorphisms LaTeXMLMath , and LaTeXMLMath for the set of all automorphisms of the group LaTeXMLMath ( that is , invertible homomorphisms LaTeXMLMath ) .
The key observation is that a normal subgroup of LaTeXMLMath of index LaTeXMLMath is just the kernel of a surjection from LaTeXMLMath to LaTeXMLMath .
All but one element of LaTeXMLMath is surjective , and the remaining one is trivial .
Two surjections LaTeXMLMath have the same kernel if and only if LaTeXMLMath for some LaTeXMLMath ; moreover , if such an LaTeXMLMath exists for LaTeXMLMath and LaTeXMLMath then it is unique .
So the nontrivial elements of LaTeXMLMath have LaTeXMLEquation different kernels between them .
In other words , there are this many index- LaTeXMLMath normal subgroups of LaTeXMLMath .
We now just have to evaluate LaTeXMLMath and LaTeXMLMath .
Firstly , LaTeXMLMath is cyclic with LaTeXMLMath generators , so LaTeXMLMath .
Secondly , LaTeXMLMath is abelian , so LaTeXMLMath forms an abelian group under pointwise addition .
Each element has order 1 or LaTeXMLMath , so LaTeXMLMath can be given scalar multiplication over the field LaTeXMLMath , and thus becomes a finite vector space over LaTeXMLMath .
This vector space has a dimension LaTeXMLMath , and then LaTeXMLMath .
( Alternatively , Cauchy ’ s Theorem gives this result . )
The lemma is now proved .
LaTeXMLMath Let us temporarily call a group LaTeXMLMath tight if for each prime LaTeXMLMath , LaTeXMLMath has at most one normal subgroup of index LaTeXMLMath .
Putting together the three parts of the following proposition gives us our second proof of the abelian quotient theorem .
A group LaTeXMLMath with LaTeXMLMath is tight .
A quotient of a tight group is tight .
A tight abelian group is cyclic .
For each prime LaTeXMLMath , we have LaTeXMLEquation where LaTeXMLMath is as in Lemma LaTeXMLRef .
If LaTeXMLMath then LaTeXMLEquation giving a contradiction .
Thus LaTeXMLMath is 0 or 1 , and so LaTeXMLMath is 0 or 1 .
Let LaTeXMLMath be a surjective homomorphism .
If LaTeXMLMath and LaTeXMLMath are distinct normal subgroups of LaTeXMLMath with index LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath are distinct normal subgroups of LaTeXMLMath with index LaTeXMLMath .
For this we invoke the classification theorem for finite abelian groups , which tells us that for any abelian group LaTeXMLMath there exist primes LaTeXMLMath and numbers LaTeXMLMath such that LaTeXMLEquation .
Suppose that LaTeXMLMath ( LaTeXMLMath , say ) for some LaTeXMLMath .
Then , since LaTeXMLMath , LaTeXMLMath has a ( normal ) subgroup LaTeXMLMath of index LaTeXMLMath ; and similarly LaTeXMLMath .
Hence LaTeXMLMath and LaTeXMLMath are distinct index- LaTeXMLMath subgroups of LaTeXMLMath , and LaTeXMLMath is not tight .
Since LaTeXMLMath is a quotient of LaTeXMLMath , part ( LaTeXMLRef ) implies that LaTeXMLMath is not tight either .
Thus if LaTeXMLMath is tight then all the LaTeXMLMath ’ s are distinct , so that LaTeXMLEquation .
LaTeXMLMath There are still other lines of proof for the abelian quotient theorem .
In part ( LaTeXMLRef ) of the Proposition , the fact that LaTeXMLMath was prime was quite irrelevant , and in just the same manner we can prove that LaTeXMLEquation whenever LaTeXMLMath is a quotient of LaTeXMLMath .
( If LaTeXMLMath is the quotient map , with kernel of order LaTeXMLMath , then a normal subgroup LaTeXMLMath of LaTeXMLMath gives rise to a normal subgroup LaTeXMLMath of LaTeXMLMath of order LaTeXMLMath . )
Thus if LaTeXMLMath is a group with LaTeXMLMath and LaTeXMLMath is an abelian quotient of LaTeXMLMath then LaTeXMLMath .
So we have reduced the abelian quotient theorem to the abelian case : if LaTeXMLMath is abelian and LaTeXMLMath then LaTeXMLMath is cyclic .
Certainly this is provable by methods derived from one of the two proofs of the general case , but other approaches exist ; I leave that for the reader .
We finish with some general speculative thoughts , roughly in order of the material above .
The chosen definition of the function LaTeXMLMath , and therefore of perfect group , is one amongst many candidates .
We defined LaTeXMLMath to be the sum of the orders of the normal subgroups , but we could change ‘ normal subgroups ’ to ‘ subgroups ’ , ‘ characteristic subgroups ’ , ‘ subnormal subgroups ’ , … , or we could define LaTeXMLMath to be the sum of the indices of the normal subgroups , etc .
In all cases we preserve the identity LaTeXMLMath , but only in some of them does LaTeXMLMath remain multiplicative ( a feature we probably like ) .
More abstractly , this article was about lifting the classical function LaTeXMLMath to a function LaTeXMLMath .
We might consider it natural to go the whole hog and create a function assigning not just a number , but some kind of algebraic structure , to each group LaTeXMLMath .
I do not know of any very useful way to do this .
In number theory there is a whole body of work on multiplicative functions of integers , which include the number-of-divisors function , the sum-of-divisors function , the Euler function LaTeXMLMath , and the Möbius function LaTeXMLMath .
In the world of groups we have at least the beginning of an analogue .
For let LaTeXMLMath be a multiplicative function from groups to numbers : then just as in Corollary 3.2 , the function LaTeXMLMath is multiplicative .
For instance , if LaTeXMLMath is the function with constant value LaTeXMLMath then LaTeXMLMath gives the number of normal subgroups of a group , and is multiplicative .
The abelian quotient theorem says that if LaTeXMLMath then LaTeXMLMath has some special property expressible in standard group-theoretic terms .
We can prove this in at least two ways , but it seems rather more challenging to prove something in the other direction : that if LaTeXMLMath is ‘ too big ’ then LaTeXMLMath has a certain form .
Finally , we can make various conjectures on perfect groups , based on the skimpy evidence above : for instance , ‘ there are no odd-order perfect groups ’ , or ‘ there are infinitely many nonabelian perfect groups ’ .
Example LaTeXMLRef , on the dihedral groups , tells us that classifying the even-order perfect groups is at least as hard as determining whether there are any odd perfect numbers .
Clearly such problems are unlikely to be easy to solve .
Clifford Index of ACM Curves in LaTeXMLMath robin hartshorne Department of Mathematics University of California Berkeley , California 94720–3840 In this paper we review the notions of gonality and Clifford index of an abstract curve .
For a curve embedded in a projective space , we investigate the connection between the Clifford index of the curve and the geometrical properties of its embedding .
In particular if LaTeXMLMath is a curve of degree LaTeXMLMath in LaTeXMLMath , and if LaTeXMLMath is a multisecant of maximum order LaTeXMLMath , then the pencil of planes through LaTeXMLMath cuts out a LaTeXMLMath on LaTeXMLMath .
If the gonality of LaTeXMLMath is equal to LaTeXMLMath we say the gonality of LaTeXMLMath can be computed by multisecants .
We discuss the question whether the gonality of every smooth ACM curve in LaTeXMLMath can be computed by multisecants , and we show the answer is yes in some special cases .
Let LaTeXMLMath be a nonsingular projective curve over an algebraically closed field LaTeXMLMath .
A linear system of degree LaTeXMLMath and ( projective ) dimension LaTeXMLMath will be denoted by LaTeXMLMath .
The least integer LaTeXMLMath for which there exists a complete linear system LaTeXMLMath without base points is called the gonality of LaTeXMLMath .
Thus a curve is rational if and only if its gonality is 1 .
Curves of genus 1 and 2 have gonality 2 .
For curves of genus LaTeXMLMath , the curve is hyperelliptic if and only if the gonality is 2 .
It is well known that for curves of genus LaTeXMLMath the gonality LaTeXMLMath lies between 2 and LaTeXMLMath ; there exist curves having each possible gonality in this range ; and a curves of genus LaTeXMLMath of general moduli has gonality LaTeXMLMath .
See LaTeXMLCite for references to proofs of these results .
Thus the gonality of a curve provides a stratification of the variety of moduli LaTeXMLMath of curves of genus LaTeXMLMath , with the hyperelliptic curves at one end , and the curves of general moduli at the other end .
To illustrate this principle , let us describe some different types of curves of genus LaTeXMLMath for small values of LaTeXMLMath .
For LaTeXMLMath there is just one curve , LaTeXMLMath , having gonality 1 and a unique LaTeXMLMath .
For LaTeXMLMath there is a one-parameter family of non-isomorphic curves .
They all have gonality 2 , and each one has infinitely many LaTeXMLMath ’ s .
For LaTeXMLMath , the curves are hyperelliptic , each having a unique LaTeXMLMath .
For LaTeXMLMath , there are hyperelliptic curves , with a unique LaTeXMLMath , and there are non-hyperelliptic curves , each having infinitely many LaTeXMLMath ’ s .
These are called trigonal ( meaning a curve with gonality 3 ) .
The canonical embedding of a trigonal curve of genus 3 is a nonsingular plane quartic curve .
The LaTeXMLMath ’ s on the curve are cut out by the pencils of lines through a point on the curve .
For LaTeXMLMath , there are again two types , hyperelliptic and trigonal .
The canonical embedding of a trigonal curves of genus 4 is a complete intersection of an irreducible quadric surface LaTeXMLMath and a cubic surface LaTeXMLMath in LaTeXMLMath .
If LaTeXMLMath is nonsingular ( the general case ) , the the curve LaTeXMLMath has exactly two LaTeXMLMath ’ s cut out by the two families of lines on LaTeXMLMath .
If LaTeXMLMath is a cone , then LaTeXMLMath has a unique LaTeXMLMath .
For LaTeXMLMath there are three types of curves : the hyperelliptic curves , the trigonal curves , and the general curves .
The canonical embedding of a non-hyperelliptic curve of 5 is a curve of degree 8 in LaTeXMLMath .
If the curve is trigonal , then it lies on a rational ruled cubic surface in LaTeXMLMath , and the unique LaTeXMLMath is cut out by the rulings of that surface .
In the general case , the curve is a complete intersection of three quadric hypersurfaces in LaTeXMLMath , and has infinitely many LaTeXMLMath ’ s .
For LaTeXMLMath the situation becomes more complicated and more interesting .
We can distinguish ( at least ) five different types of curves : a ) The hyperelliptic curves , having a unique LaTeXMLMath .
b ) The trigonal curves , having a unique LaTeXMLMath .
c ) Plane quartic curves , having a unique LaTeXMLMath .
These curves have infinitely many LaTeXMLMath ’ s , cut out by the pencils of lines through a point of the curve .
d ) Double cover of an elliptic curve , having infinitely many LaTeXMLMath ’ s , but no LaTeXMLMath .
e ) Curves having only finitely many LaTeXMLMath ’ s .
The general curve has exactly five LaTeXMLMath ’ s ; some others may have fewer .
For references , see e.g .
LaTeXMLCite .
From these few examples it is already clear that the gonality does not tell the whole story in distinguishing different types of curves of genus LaTeXMLMath .
More generally , we should take into account all possible special linear systems LaTeXMLMath that might exist on the curve .
Here special means LaTeXMLMath , or equivalently LaTeXMLMath , where LaTeXMLMath is a divisor in the linear system .
In this connection we consider the Brill-Noether number LaTeXMLEquation .
Then one knows , for given LaTeXMLMath that if LaTeXMLMath , every curves of genus LaTeXMLMath has a LaTeXMLMath , and if LaTeXMLMath , then a general curves of genus LaTeXMLMath does not have a LaTeXMLMath LaTeXMLCite .
So to distinguish among curves of genus LaTeXMLMath , we will be interested in the existence of linear systems LaTeXMLMath for which LaTeXMLMath .
Now we can introduce the Clifford index of a curve LaTeXMLMath of genus LaTeXMLMath ( first defined by H. Martens LaTeXMLCite ) .
For a particular linear system LaTeXMLMath we define its Clifford index to be LaTeXMLMath .
Then the Clifford index of the curve is the minimum of Clifford indices of certain special linear series , namely Cliff LaTeXMLMath is the minimum of LaTeXMLMath , taken over all linear systems LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Equivalently , one can take the minimum of LaTeXMLMath over all LaTeXMLMath containing a divisor LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
( The equivalence of these two criteria is easy using the Riemann-Roch theorem , and replacing LaTeXMLMath by LaTeXMLMath if LaTeXMLMath . )
Recall that Clifford ’ s theorem tells us that LaTeXMLMath for a special linear system LaTeXMLMath on a curve , with equality if and only if the corresponding divisor LaTeXMLMath is 0 or the canonical divisor LaTeXMLMath , or the curve is hyperelliptic .
Since the inequalities in the definition rule out the possibility LaTeXMLMath or LaTeXMLMath , we see that the Clifford index is always LaTeXMLMath , with equality if and only if the curve is hyperelliptic .
On the other hand , the Brill-Noether theory tells us that Cliff LaTeXMLMath , and is equal to this value for a curve of general moduli .
In most cases , the Clifford index can be computed by a pencil , that is , there exists a LaTeXMLMath with Cliff LaTeXMLMath .
In this case Cliff LaTeXMLMath , where LaTeXMLMath denotes the gonality .
This suggests the definition of the Clifford dimension of the curve , which is the least LaTeXMLMath for which there exists a LaTeXMLMath with Cliff LaTeXMLMath , i.e. , the LaTeXMLMath computes the Clifford index .
Then LaTeXMLMath is the normal situation .
Curves with Clifford dimension LaTeXMLMath are rare .
The first example is the curve of genus 6 with a LaTeXMLMath mentioned above .
In this case Cliff LaTeXMLMath while LaTeXMLMath , and so the Clifford dimension of the LaTeXMLMath is 2 .
The nonsingular plane curves of degree LaTeXMLMath all have Clifford dimension 2 , and these are the only such .
Any curve of Clifford dimension 3 must be a complete intersection of two cubic surfaces in LaTeXMLMath , having degree 9 and genus 10 LaTeXMLCite .
There exist curves of every possible Clifford dimension LaTeXMLMath , and for LaTeXMLMath conjecturally only one possible degree-genus pair in LaTeXMLMath LaTeXMLCite .
We now consider a nonsingular curve LaTeXMLMath embedded in a projective space LaTeXMLMath , and we ask , how are the gonality and the Clifford index of LaTeXMLMath related to the geometry of the embedding ?
The prototype for this kind of question is the following well-known theorem about plane curves .
Let LaTeXMLMath be a nonsingular plane curve of degree LaTeXMLMath .
Then ( a ) There is no LaTeXMLMath on LaTeXMLMath , but there are LaTeXMLMath ’ s on LaTeXMLMath , so the gonality is LaTeXMLMath .
( b ) Every LaTeXMLMath on LaTeXMLMath is cut out by the pencil of lines in LaTeXMLMath through some fixed point of LaTeXMLMath .
Proof .
This result was known to M.Noether and has received a number of modern proofs LaTeXMLCite , LaTeXMLCite , … We will give an elementary proof to illustrate the ideas involved .
( a ) Suppose LaTeXMLMath is a divisor of degree LaTeXMLMath on LaTeXMLMath with LaTeXMLMath .
Since LaTeXMLMath moves in a pencil , we may assume that LaTeXMLMath consists of LaTeXMLMath distinct points LaTeXMLMath .
By the Riemann-Roch theorem LaTeXMLMath .
This means that the LaTeXMLMath points LaTeXMLMath do not impose independent conditions on the canonical divisors LaTeXMLMath containing them .
Now the canonical divisor LaTeXMLMath on LaTeXMLMath is cut out by curves of degree LaTeXMLMath in LaTeXMLMath .
Any LaTeXMLMath distinct points impose independent conditions on these curves , a contradiction .
So no such LaTeXMLMath exists .
On the other hand , for any LaTeXMLMath , the lines through LaTeXMLMath cut out a LaTeXMLMath , so these do exist .
( b ) Now let LaTeXMLMath be any divisor of degree LaTeXMLMath with LaTeXMLMath .
The argument above shows that the LaTeXMLMath points of LaTeXMLMath impose dependent conditions on plane curves of degree LaTeXMLMath , and this can only happen if these points lie on a line in LaTeXMLMath .
This line LaTeXMLMath will meet LaTeXMLMath in one further point LaTeXMLMath , and then it is clear that the LaTeXMLMath is equal to the one cut out by lines through LaTeXMLMath .
This result has been generalized to irreducible plane curves LaTeXMLMath of degree LaTeXMLMath with LaTeXMLMath nodes and cusps , when LaTeXMLMath is not too large in relation to LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
In those cases , the desired result would be that the gonality of the normalization LaTeXMLMath is LaTeXMLMath , and is given by the linear systems cut out by lines through one of the double points .
We can not expect such a result to hold for arbitrary plane curves with nodes , however , as the following example shows .
Example 2.2 .
Let LaTeXMLMath be a smooth curve in LaTeXMLMath , of degree 6 and genus 3 , not lying on a quadric surface .
Such a curve arises of type LaTeXMLMath on a nonsingular cubic surface LaTeXMLMath in LaTeXMLMath , for example .
This is the proper transform of a plane curve of degree 4 passing through the 6 points blown up to get LaTeXMLMath .
It has gonality 3 by LaTeXMLMath .
On the other hand , its general projection to LaTeXMLMath is a plane curve of degree 6 with 7 nodes .
The pencil of lines through one of the nodes cuts out a LaTeXMLMath on the normalization , which does not give us the correct gonality .
Passing now to curves in higher dimensional projective spaces , let us first consider a nonsingular curve LaTeXMLMath of degree LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be a multisecant of maximum order LaTeXMLMath ( that is , the scheme-theoretic intersection of LaTeXMLMath and LaTeXMLMath has length LaTeXMLMath ) .
Then the pencil of planes through LaTeXMLMath cuts out a LaTeXMLMath on LaTeXMLMath .
If the gonality of LaTeXMLMath is equal to LaTeXMLMath , we say the gonality of LaTeXMLMath can be computed by multisecants .
We can also ask the stronger question , whether every LaTeXMLMath on LaTeXMLMath arises in this way .
For a curve LaTeXMLMath of degree LaTeXMLMath in LaTeXMLMath , with LaTeXMLMath , the corresponding situation would be to look for a multisecant linear space LaTeXMLMath of codimension 2 in LaTeXMLMath , meeting LaTeXMLMath in a scheme of length LaTeXMLMath .
The hyperplanes through LaTeXMLMath will cut out a LaTeXMLMath on LaTeXMLMath , and if this gives the gonality , we say again that the gonality can be computed by multisecants .
Example 2.3 .
Let LaTeXMLMath be the canonical embedding of a nonhyperelliptic curve of genus LaTeXMLMath in LaTeXMLMath , with LaTeXMLMath .
Let LaTeXMLMath be a special complete linear system without base points , and let LaTeXMLMath be any divisor of the LaTeXMLMath .
Then LaTeXMLMath .
Let LaTeXMLMath , where LaTeXMLMath is the canonical divisor .
Then LaTeXMLMath .
Since LaTeXMLMath is cut out by hyperplanes in LaTeXMLMath , this means that the divisor LaTeXMLMath is contained in two distinct hyperplanes .
Let them meet in the linear space LaTeXMLMath of codimension 2 .
Then the pencil of hyperplanes through LaTeXMLMath , after removing the fixed points LaTeXMLMath , cuts out the original linear system LaTeXMLMath on LaTeXMLMath .
In particular , the gonality can be computed by multisecants .
If LaTeXMLMath , we have a plane curve of degree 4 , and recover the result of LaTeXMLMath in this case .
If LaTeXMLMath , the curve LaTeXMLMath is the complete intersection of a quadric surface LaTeXMLMath with a cubic surface LaTeXMLMath .
The pencil of planes through a line of LaTeXMLMath cuts out the other family of lines on LaTeXMLMath , if LaTeXMLMath is nonsingular , or the only family of lines , if LaTeXMLMath is a cone .
Thus the LaTeXMLMath ’ s are computed by multisecants .
If LaTeXMLMath , there are two cases .
When LaTeXMLMath is trigonal , it lies on a rational ruled cubic surface LaTeXMLMath in LaTeXMLMath .
This surface contains conics meeting the curve in 5 points .
The plane of the conic is therefore a 5-secant plane , and the hyperplane sections of LaTeXMLMath containing this conic cut out the rulings of LaTeXMLMath , which in turn cut out the unique LaTeXMLMath on LaTeXMLMath .
If LaTeXMLMath is not trigonal , then our result tells us that the curve LaTeXMLMath has 4-secant planes , so that the pencil of hypersurface through them cut out the LaTeXMLMath ’ s on LaTeXMLMath .
Example 2.4 .
Basili LaTeXMLCite shows that if LaTeXMLMath is a smooth complete intersection curve in LaTeXMLMath , not contained in a plane , then the gonality can be computed by multisecant lines .
Furthermore every LaTeXMLMath with LaTeXMLMath arises in this manner .
One can also ask what are the possible orders of multisecant lines , and hence what are the possible gonalities of these complete intersection curves .
Let LaTeXMLMath be the complete intersection LaTeXMLMath of surfaces of degrees LaTeXMLMath .
Nollet LaTeXMLCite has shown that the maximum order LaTeXMLMath of a multisecant is either LaTeXMLMath or LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is a curve of bidegree LaTeXMLMath on the quadric surface , so LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , again LaTeXMLMath since there are lines on the cubic surface , so LaTeXMLMath .
If LaTeXMLMath , then Ellia and Franco LaTeXMLCite have shown that every value of LaTeXMLMath satisfying LaTeXMLMath or LaTeXMLMath can occur .
In particular , the general complete intersection curve with LaTeXMLMath has at most 4-secants , and gonality LaTeXMLMath .
Example 2.5 .
If LaTeXMLMath is a smooth curve of bidegree LaTeXMLMath on a nonsingular quadric surface , with LaTeXMLMath , then the maximum order of a multisecant is LaTeXMLMath .
In this case G. Martens LaTeXMLCite and Ballico LaTeXMLCite have shown that the gonality of LaTeXMLMath is LaTeXMLMath , and thus is computed by multisecants .
Example 2.6 .
The case of complete intersection curves in LaTeXMLMath LaTeXMLMath has been generalized by Ellia and Franco LaTeXMLCite to curves arising as the zero locus of a section of a rank 2 vector bundle LaTeXMLMath on LaTeXMLMath , twisted sufficiently : LaTeXMLMath with LaTeXMLMath .
This includes “ most ” subcanonical curves in LaTeXMLMath , but it is hard to get specific results for small degree curves .
They show the gonality of these curves can also be computed by multisecants .
Example 2.7 .
Farkas LaTeXMLCite using the method of Mori , shows the existence of curves LaTeXMLMath on a nonsingular quartic surface LaTeXMLMath in LaTeXMLMath , for certain values of degree LaTeXMLMath and genus LaTeXMLMath satisfying complicated conditions , for which the gonality is LaTeXMLMath and can be computed by 4-secants to the curve .
Example 2.8 .
Eisenbud et al LaTeXMLCite , also studying curves on LaTeXMLMath surfaces , show the existence , for every LaTeXMLMath of a smooth curve in LaTeXMLMath having LaTeXMLMath , LaTeXMLMath , whose gonality is computed by multisecants , and having Clifford dimension LaTeXMLMath .
The gonality is LaTeXMLMath and the Clifford index is LaTeXMLMath .
Lest the reader begin to think that all these examples are evidence for supposing that the gonality of any space curve is computed by multisecants , let us give a few counterexamples .
Example 2.9 .
We consider rational curves LaTeXMLMath of degree LaTeXMLMath in LaTeXMLMath .
There exist such curves having a LaTeXMLMath -secant , for example , curves of bidegree LaTeXMLMath on a quadric surface .
In this case the gonality is computed by multisecants .
However , for LaTeXMLMath , Ellia and Franco LaTeXMLCite have shown that there exist smooth rational curves whose maximum order of a multisecant LaTeXMLMath can take any value LaTeXMLMath .
In particular , the general such curve has only 4-secants , and if LaTeXMLMath these do not give the correct gonality .
Example 2.10 .
For an example of curves of higher genus , take a plane curve of degree LaTeXMLMath , blow up 6 points not on the curve , and let LaTeXMLMath be the image curve in the nonsingular cubic surface LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath has degree LaTeXMLMath .
It has multisecants of order LaTeXMLMath and LaTeXMLMath on LaTeXMLMath .
The pencil of planes through one of the latter cut out a LaTeXMLMath on LaTeXMLMath .
But the gonality is LaTeXMLMath , so the gonality is not computed by multisecants .
From all this evidence , it seems reasonable to pose the following question .
Question 2.11 ( Peskine ) If LaTeXMLMath is a smooth ACM ( i.e .
projectively normal ) curve in LaTeXMLMath , is its gonality computable by multisecants ?
We will discuss this question in the following sections .
We consider a smooth surface LaTeXMLMath , together with a proper morphism LaTeXMLMath , where LaTeXMLMath is a nonsingular curve .
Then the fibers of LaTeXMLMath from a flat family of curves on LaTeXMLMath .
We assume that the general fiber LaTeXMLMath is irreducible and nonsingular for LaTeXMLMath , LaTeXMLMath , and that the special fiber LaTeXMLMath , a union of two smooth irreducible curves LaTeXMLMath , LaTeXMLMath , meeting transversally at LaTeXMLMath distinct points .
In a flat family as above , whose general curve LaTeXMLMath is smooth , and whose special curve LaTeXMLMath is a union of two smooth curves meeting transversally at LaTeXMLMath points , we have LaTeXMLEquation for all sufficiently general LaTeXMLMath .
Proof .
Suppose that the general curves LaTeXMLMath in the family all have a LaTeXMLMath with LaTeXMLMath .
Then we can find an open set LaTeXMLMath and an invertible sheaf LaTeXMLMath on LaTeXMLMath inducing a LaTeXMLMath on each fiber .
This invertible sheaf extends to an invertible sheaf LaTeXMLMath on LaTeXMLMath , but the extension is not unique , because we can replace LaTeXMLMath by LaTeXMLMath for any LaTeXMLMath and still get the same LaTeXMLMath on restricting to LaTeXMLMath .
Let us compute some intersection numbers .
Since LaTeXMLMath , we have LaTeXMLMath , so LaTeXMLMath .
Similarly LaTeXMLMath .
Let us denote by LaTeXMLMath the restriction of LaTeXMLMath to LaTeXMLMath , and by LaTeXMLMath , the restrictions to LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath .
If we replace LaTeXMLMath by LaTeXMLMath then LaTeXMLMath becomes LaTeXMLMath while LaTeXMLMath becomes LaTeXMLMath .
Thus by choosing LaTeXMLMath appropriately , we may assume that LaTeXMLMath and consequently LaTeXMLMath .
Since LaTeXMLMath cuts out a LaTeXMLMath on the general curve LaTeXMLMath , we have LaTeXMLMath for general LaTeXMLMath .
Hence by semicontinuity , LaTeXMLMath .
We consider the exact sequence LaTeXMLEquation where LaTeXMLMath is the set of LaTeXMLMath points LaTeXMLMath .
On cohomology this gives LaTeXMLEquation we consider three cases , depending on the dimensions of LaTeXMLMath , LaTeXMLMath .
Case 1 .
If one of LaTeXMLMath , LaTeXMLMath is zero , then the other must be LaTeXMLMath .
So we have a LaTeXMLMath on one of the curves , say LaTeXMLMath , and since the section of LaTeXMLMath giving this LaTeXMLMath is identically zero on LaTeXMLMath , the divisor of the LaTeXMLMath must contain LaTeXMLMath as a fixed component .
But then LaTeXMLMath , contrary to our assumptions .
Case 2 .
If one of the LaTeXMLMath is LaTeXMLMath , say LaTeXMLMath , then we can find a section of LaTeXMLMath inducing LaTeXMLMath on LaTeXMLMath and a nonzero section of LaTeXMLMath on LaTeXMLMath .
In this case LaTeXMLMath , and LaTeXMLMath since LaTeXMLMath , so LaTeXMLMath and we have LaTeXMLMath .
Case 3 .
If both of LaTeXMLMath are LaTeXMLMath , then we have a LaTeXMLMath as LaTeXMLMath and a LaTeXMLMath on LaTeXMLMath , so LaTeXMLMath , as required .
In the statement of the theorem , if LaTeXMLMath , then there exist morphisms LaTeXMLMath and LaTeXMLMath of degrees equal to the gonality , such that LaTeXMLMath and LaTeXMLMath agree on the LaTeXMLMath points LaTeXMLMath .
Proof .
Indeed , if LaTeXMLMath , then we must be in Case 3 of the proof above , and the LaTeXMLMath on LaTeXMLMath and LaTeXMLMath on LaTeXMLMath are induced by LaTeXMLMath , so must agree on LaTeXMLMath .
Note : A special case of this kind of argument appears in a paper of Ballico LaTeXMLCite .
Example 3.3 .
We can use the theorem to give another proof of a weak form of LaTeXMLMath , namely , a general curve LaTeXMLMath of degree LaTeXMLMath in LaTeXMLMath has gonality LaTeXMLMath .
The pencil of lines through a point on LaTeXMLMath cuts out a LaTeXMLMath , so we always have LaTeXMLMath .
To prove the reverse inequality , we use induction on LaTeXMLMath .
For LaTeXMLMath , the conic is isomorphic to LaTeXMLMath , so has gonality LaTeXMLMath .
For LaTeXMLMath , consider a family of smooth curves LaTeXMLMath of degree LaTeXMLMath degenerating to the union of a general smooth curve LaTeXMLMath of degree LaTeXMLMath and a transversal line LaTeXMLMath .
Then LaTeXMLMath is LaTeXMLMath points , and LaTeXMLMath , LaTeXMLMath , so by the theorem we find LaTeXMLMath .
Note that LaTeXMLMath in this proof , so we do not get any additional information from the Corollary .
Let LaTeXMLMath be a nonsingular quadric surface in LaTeXMLMath , and let LaTeXMLMath be a general smooth curve of bidegree LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLMath .
Proof .
By projection onto one of the factors of LaTeXMLMath we know that LaTeXMLMath .
For LaTeXMLMath , the curve is rational , so has gonality LaTeXMLMath .
For LaTeXMLMath , we let LaTeXMLMath move in a family specializing to the union of a general curve LaTeXMLMath of bidegree LaTeXMLMath and a conic LaTeXMLMath of bidegree LaTeXMLMath .
Then LaTeXMLMath .
Also by induction LaTeXMLMath and LaTeXMLMath .
So by LaTeXMLMath , LaTeXMLMath , as required .
We recover a slightly weaker version of LaTeXMLMath , since our method of proof applies only to the general curve in a family .
For curves on a cubic surface , we have seen LaTeXMLMath that not every smooth curve on a smooth cubic surface has its gonality determined by multisecants .
However , we can obtain a result for sufficiently general ACM curves on a cubic surface .
Let LaTeXMLMath be a smooth ACM curve on a nonsingular cubic surface LaTeXMLMath in LaTeXMLMath .
If LaTeXMLMath is sufficiently general in its linear system on LaTeXMLMath , then LaTeXMLMath has a multisecant LaTeXMLMath such that the pencil of planes through LaTeXMLMath cuts out a pencil on LaTeXMLMath computing the gonality of LaTeXMLMath .
Proof .
First we must identify the smooth ACM curves on LaTeXMLMath .
Using the postulation character LaTeXMLMath of LaTeXMLCite we see that if LaTeXMLMath is not contained in any quadric surface , then its LaTeXMLMath -character has LaTeXMLMath , and is positive , and connected .
This means it must have one of the following four types , where LaTeXMLMath : a ) LaTeXMLMath b ) LaTeXMLMath c ) LaTeXMLMath d ) LaTeXMLMath .
These can all be obtained by ascending biliaison on LaTeXMLMath from one of the following curves on a quadric surface : a ) LaTeXMLMath LaTeXMLMath , LaTeXMLMath b ) LaTeXMLMath LaTeXMLMath , LaTeXMLMath c ) LaTeXMLMath LaTeXMLMath , LaTeXMLMath d ) LaTeXMLMath LaTeXMLMath , LaTeXMLMath .
Now , for suitable choice of the basis of LaTeXMLMath , we can represent these curves by the following divisor classes on LaTeXMLMath : a ) LaTeXMLMath LaTeXMLMath , line LaTeXMLMath meets LaTeXMLMath in LaTeXMLMath points b ) LaTeXMLMath LaTeXMLMath , line LaTeXMLMath meets LaTeXMLMath in LaTeXMLMath points c ) LaTeXMLMath LaTeXMLMath , line LaTeXMLMath meets LaTeXMLMath in LaTeXMLMath points d ) LaTeXMLMath LaTeXMLMath , has a trisecant not on LaTeXMLMath .
Thus for each of these curves the gonality is computed by a multisecant LaTeXMLMath , and in the first three cases , we can choose LaTeXMLMath to be a line lying on LaTeXMLMath .
In th fourth case , we can not find a trisecant lying on LaTeXMLMath , so we make one biliaison ( i.e. , replace LaTeXMLMath by LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is the plane section ) and obtain d LaTeXMLMath ) LaTeXMLMath .
This last curve LaTeXMLMath in case d LaTeXMLMath ) is a complete intersection of two cubic surfaces .
This is the exceptional case of Clifford dimension LaTeXMLMath studied by Martens LaTeXMLCite .
He shows it has gonality LaTeXMLMath , but Clifford index LaTeXMLMath given by the linear system LaTeXMLMath giving the embedding in LaTeXMLMath .
Now this curve does have a trisecant LaTeXMLMath on LaTeXMLMath , say LaTeXMLMath , and this line computes the gonality .
To prove our result , we use the fact that every smooth ACM curve on the cubic surface LaTeXMLMath is obtained from one of a ) , b ) , c ) , d ) by biliaison on the surface LaTeXMLMath .
These curves all have gonality computed by multisecants .
We prove our result then , by induction on the degree .
Our induction statement is the stronger claim that if LaTeXMLMath is sufficiently general , then there is a multisecant LaTeXMLMath on LaTeXMLMath of order LaTeXMLMath , such that the gonality of LaTeXMLMath is LaTeXMLMath .
We begin the induction with cases a ) , b ) , c ) , d LaTeXMLMath ) .
For the induction step , suppose that LaTeXMLMath on LaTeXMLMath has degree LaTeXMLMath , a multisecant LaTeXMLMath on LaTeXMLMath of order LaTeXMLMath , and gonality LaTeXMLMath .
We take LaTeXMLMath a general member of the linear system LaTeXMLMath , and let it specialize to LaTeXMLMath union LaTeXMLMath .
Then LaTeXMLMath , and LaTeXMLMath , since LaTeXMLMath is a plane cubic curve .
Then by LaTeXMLMath , LaTeXMLMath .
Since LaTeXMLMath in all our starting cases , we conclude LaTeXMLMath .
On the other hand , the degree of LaTeXMLMath is LaTeXMLMath , and LaTeXMLMath .
So the pencil of planes through LaTeXMLMath cuts out a LaTeXMLMath and we find LaTeXMLMath as required .
In particular , for the complete intersection curves on LaTeXMLMath , we recover a weak form of Basili ’ s result LaTeXMLCite , since our proof works only for sufficiently general curves .
For curves on quartic surfaces , Farkas LaTeXMLCite has shown that the gonality of some special classes of curves is computed by LaTeXMLMath -secants .
His method does not cover all ACM curves on quartic surfaces , because he always assumes the surface contains no rational and no elliptic curves .
If we apply the methods of this paper to ACM curves lying on surfaces of degree four and higher , we obtain only an inequality for the gonality , not an exact figure , and so we are unable to answer Question LaTeXMLMath in general .
If the surface contains a line , and the curve is either in the biliaison class of the line , or residual to the line , then the line becomes a multisecant of high order that computes the gonality .
This case was also observed by Paoletti LaTeXMLCite .
To make further progress on Question LaTeXMLMath will require some other technique .
Connes ’ distance formula is applied to endow linear metric to three 1D lattices of different topology , with a generalization of lattice Dirac operator written down by Dimakis et al to contain a non-unitary link-variable .
Geometric interpretation of this link-variable is lattice spacing and parallel transport .
PACS : 02.40.Gh , 11.15.Ha Key words : Connes ’ distance , one-dimensional lattice , Dirac operator , link-variable , lattice spacing , parallel transport Lattice as a universal regulator for the non-perturbative definition of a quantum field theory works well for bosonic fields LaTeXMLCite .
However , when fermionic fields are involved , lattice formalism encounters two well-known seemingly insurmountable problems : implementation of grassmann number in simulations and No-Go theorem for chiral fermion on lattices LaTeXMLCite .
On the other hand , lattice provides one simplest model of noncommutative geometry ( NCG ) LaTeXMLCite ; NCG in Connes ’ formulation has an intimate relation with fermion through a Hilbert space and a generalized Dirac operator LaTeXMLCite .
Therefore , to explore lattice field theory in NCG context is significant for to understand those old puzzles .
As the first step , because NCG endows a metric , hence a geometry , onto a space through Dirac operator , to consider this ( Dirac-operator ) induced metric on lattices exhibits new relation between lattice fermions and lattice geometry .
In fact , the first striking nontrivial result along this line is that this distance is non-Euclidean , providing Naïve or Wilson-Dirac operator is adopted LaTeXMLCite LaTeXMLCite .
On the contrary , Dimakis and Müller-Hoissen ( DM ) proposed a new free Dirac operator which induces correct linear distance on a 1D lattice LaTeXMLCite .
In this paper , we generalize DM ’ s result in case that a link-variable field is presented on this 1D lattice .
We will show that the amplitude of this field modify the induced distance in the sense that its inverse provides a localized lattice spacing and that the phase of this field can play the role of a LaTeXMLMath -parallel transport , hence a gauge potential .
This paper is organized as following .
Connes ’ distance is introduced in Sect .
LaTeXMLRef , and is calculated for three types of 1D lattices in Sect .
LaTeXMLRef after generalized DM ’ s lattice Dirac operator is defined .
Geometric interpretation is given in Sect .
LaTeXMLRef .
A spectral geometry in Connes ’ sense , commutative or not , is defined to be a triple LaTeXMLMath in which LaTeXMLMath is a pre- LaTeXMLMath algebra being represented faithfully on Hilbert space LaTeXMLMath and LaTeXMLMath is a self-adjoint operator on LaTeXMLMath playing the role of Dirac operator in Classical spinor geometry LaTeXMLCite .
In this paper , LaTeXMLMath is taken to be the algebra of complex functions on a lattice , LaTeXMLMath is the Hilbert space of fermionic fields which are not considered as grassmann-valued sections , and LaTeXMLMath is lattice Dirac operator to be specified .
Connes ’ distance is introduced by the formula LaTeXMLEquation for all points LaTeXMLMath of this lattice , where LaTeXMLMath is operator norm on LaTeXMLMath .
Note that we do not distinguish LaTeXMLMath from its imagine represented on LaTeXMLMath due to the faithfulness .
To obtain a manipulable algorithm for Eq .
( LaTeXMLRef ) , we define a LaTeXMLMath -Hamiltonian , LaTeXMLMath .
Then it is easy to verify that LaTeXMLMath .
Consequently , Eq .
( LaTeXMLRef ) can be expressed as LaTeXMLEquation .
We specify the term one-dimensional lattice by a discrete set LaTeXMLMath together with a isomorphism LaTeXMLMath acting on LaTeXMLMath .
LaTeXMLMath is denoted for algebra of complex functions on LaTeXMLMath and Hilbert space is chosen to be LaTeXMLMath which is a free module over LaTeXMLMath of rank 2 .
LaTeXMLMath induces an isomorphism of LaTeXMLMath and an isometry of LaTeXMLMath to which we still write as LaTeXMLMath .
DM ’ s free lattice Dirac operator can be written as LaTeXMLEquation where LaTeXMLMath are defined using Pauli matrices LaTeXMLMath .
We generalize it to be LaTeXMLEquation where LaTeXMLMath .
Below we consider three types of LaTeXMLMath corresponding to three topologies in continuum limit .
In this case , LaTeXMLMath is coordinatized by LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath , which we refer as LaTeXMLMath .
Notice Eq .
( LaTeXMLRef ) , LaTeXMLMath , where LaTeXMLMath .
One can check that LaTeXMLMath -Hamiltonian LaTeXMLMath where LaTeXMLMath .
Therefore , LaTeXMLMath in which LaTeXMLMath is sup -norm of LaTeXMLMath .
According Eq .
( LaTeXMLRef ) , LaTeXMLEquation for all LaTeXMLMath .
If we assume LaTeXMLMath is non-singular , i.e .
LaTeXMLMath for all LaTeXMLMath , then LaTeXMLMath possesses an upper bound LaTeXMLEquation in which LaTeXMLMath is supposed to be larger than LaTeXMLMath .
Define LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath saturates the upper bound in Eq .
( LaTeXMLRef ) .
Subsequently , ( LaTeXMLRef ) becomes an equality , especially it holds that LaTeXMLMath , to which a clear interpretation is that the inverse of amplitude of LaTeXMLMath is the lattice spacing between LaTeXMLMath and LaTeXMLMath .
Note that it is obvious that the value of LaTeXMLMath at LaTeXMLMath makes no sense in this case .
Here LaTeXMLMath is labeled by LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath , so addition of the argument of LaTeXMLMath makes LaTeXMLMath a finite group LaTeXMLMath .
If we define LaTeXMLMath , then the deduction is exactly the same as that in Subsect .
LaTeXMLRef and LaTeXMLEquation .
With the non-singular assumption on LaTeXMLMath and cyclic addition on LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath .
Now we design a function to saturate this upper bound .
Without losing generality , let LaTeXMLMath and define LaTeXMLMath .
It is easy to check that LaTeXMLMath and that LaTeXMLMath saturates the upper bound in Eq .
( LaTeXMLRef ) .
If LaTeXMLMath satisfies triangle-inequalities LaTeXMLMath , then LaTeXMLMath is able to be interpreted as lattice spacing between LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath is parametrized by integer LaTeXMLMath in this case and LaTeXMLMath .
However to guarantee convergency , we must consider LaTeXMLMath and LaTeXMLMath here .
Still define LaTeXMLMath , then deduction is the same as that in Subsect .
LaTeXMLRef , and it follows that LaTeXMLEquation .
With non-singular LaTeXMLMath and that LaTeXMLMath , LaTeXMLEquation .
Let LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath saturates the upper bound in Eq .
( LaTeXMLRef ) .
Since LaTeXMLMath , LaTeXMLMath is the lattice spacing between LaTeXMLMath and LaTeXMLMath .
Notice that non-singular LaTeXMLMath can be polarized as LaTeXMLMath with two real functions LaTeXMLMath , we conclude that LaTeXMLMath is determined entirely by lattice spacing function LaTeXMLMath and that LaTeXMLMath is still linear distance in the sense of additivity .
We claim that LaTeXMLMath in the above decomposition plays the role of unitary link-variable in lattice gauge theory , or equivalently parallel transport in mathematical literature .
In fact , a local LaTeXMLMath -gauge transformation on LaTeXMLMath is defined to be LaTeXMLMath where LaTeXMLMath is a unitary in LaTeXMLMath and a LaTeXMLMath -parallel transport LaTeXMLMath on LaTeXMLMath is a link-variable satisfying LaTeXMLMath , LaTeXMLMath in which LaTeXMLMath is hermitian-structure on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is a parallel transport and LaTeXMLMath is gauge-invariant where LaTeXMLMath is inner product of LaTeXMLMath .
Therefore geometric interpretation of LaTeXMLMath is clear : LaTeXMLMath is a link-variable not necessarily unitary , whose amplitude provides a vierbein and phase is the usual integrated LaTeXMLMath -connection .
Non-unitary link-variable has been noticed in the work of Majid and Raineri LaTeXMLCite discussing field theory on permutation group LaTeXMLMath and ours LaTeXMLCite .
Nevertheless , its geometric picture is the clearest on 1D lattices .
Acknowledgements This work was supported by Climb-Up ( Pan Deng ) Project of Department of Science and Technology in China , Chinese National Science Foundation and Doctoral Programme Foundation of Institution of Higher Education in China .
We are grateful to Prof. S. Majid for introducing his work to us .
Hexagonal circle patterns are introduced , and a subclass thereof is studied in detail .
It is characterized by the following property : For every circle the multi-ratio of its six intersection points with neighboring circles is equal to LaTeXMLMath .
The relation of such patterns with an integrable system on the regular triangular lattice is established .
A kind of a Bäcklund transformation for circle patterns is studied .
Further , a class of isomonodromic solutions of the aforementioned integrable system is introduced , including circle patterns analogons to the analytic functions LaTeXMLMath and LaTeXMLMath .
The theory of circle packings and , more generally , of circle patterns enjoys in recent years a fast development and a growing interest of specialists in complex analysis .
The origin of this interest was connected with the Thurston ’ s idea about approximating the Riemann mapping by circle packings , see LaTeXMLCite , LaTeXMLCite .
Since then the theory bifurcated to several subareas .
One of them concentrates around the uniformization theorem of Koebe–Andreev–Thurston , and is dealing with circle packing realizations of cell complexes of a prescribed combinatorics , rigidity properties , constructing hyperbolic 3-manifolds , etc LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Another one is mainly dealing with approximation problems , and in this context it is advantageous to stick from the beginning with fixed regular combinatorics .
The most popular are hexagonal packings , for which the LaTeXMLMath convergence to the Riemann mapping was established by He and Schramm LaTeXMLCite .
Similar results are available also for circle patterns with the combinatorics of the square grid introduced by Schramm LaTeXMLCite .
It is also the context of regular patterns ( more precisely , the two just mentioned classes thereof ) where some progress was achieved in constructing discrete analogs of analytic functions ( Doyle ’ s spiralling hexagon packings LaTeXMLCite and their generalizations including the discrete analog of a quotient of Airy functions LaTeXMLCite , discrete analogs of LaTeXMLMath and LaTeXMLMath for the square grid circle patterns LaTeXMLCite , discrete versions of LaTeXMLMath and LaTeXMLMath for the same class of circle patterns LaTeXMLCite , LaTeXMLCite ) .
And it is again the context of regular patterns where the theory comes into interplay with the theory of integrable systems .
Strictly speaking , only one instance of such an interplay is well–established up to now : namely , Schramm ’ s equation describing the square grid circle packings in terms of Möbius invariants turns out to coincide with the stationary Hirota ’ s equation , known to be integrable , see LaTeXMLCite , LaTeXMLCite .
It should be said that , generally , the subject of discrete integrable systems on lattices different from LaTeXMLMath is underdeveloped at present .
The list of relevant publications is almost exhausted by LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
The present paper contributes to several of the above mentioned issues : we introduce a new interesting class of circle patterns , and relate them to integrable systems .
Besides , for this class we construct , in parallel to LaTeXMLCite , LaTeXMLCite , the analogs of the analytic functions LaTeXMLMath , LaTeXMLMath .
This class is constituted by hexagonal circle patterns , or , in other words , by circle patterns with the combinatorics of the regular hexagonal lattice ( the honeycomb lattice ) .
This means that each elementary hexagon of the honeycomb lattice corresponds to a circle , and each common vertex of two hexagons corresponds to an intersection point of the corresponding circles .
In particular , each circle carries six intersection points with six neighboring circles .
Since at each vertex of the honeycomb lattice there meet three elementary hexagons , there follows that at each intersection point there meet three circles .
This class of hexagonal circle patterns is still too wide to be manageable , but it includes several very interesting subclasses , leading to integrable systems .
For example , one can prescribe intersection angles of the circles .
This situation will be considered in a subsequent publication .
In the present one we consider the following requirement : the six intersection points on each circle have the multi-ratio equal to LaTeXMLMath , where the multi–ratio is a natural generalization of the notion of a cross-ratio of four points on a plane .
We show that , adding to the intersection points of the circles their centers , one embeds hexagonal circle patterns with the multi-ratio property into an integrable system on the regular triangular lattice .
Each solution of this latter system describes a peculiar geometrical construction : it consists of three triangulations of the plane , such that the corresponding elementary triangles in all three tilings are similar .
Moreover , given one such tiling , one can reconstruct the other two almost uniquely ( up to an affine transformation ) .
If one of the tilings comes from the hexagonal circle pattern , so do the other two .
This results are contained in Sect .
LaTeXMLRef , LaTeXMLRef .
In the intermediate Sect .
LaTeXMLRef we discuss a general notion of integrable systems on graphs as flat connections with the values in loop groups .
It should be noticed that closely related integrable equations ( albeit on the standard grid LaTeXMLMath ) were previously introduced by Nijhoff LaTeXMLCite in a totally different context ( discrete Bussinesq equation ) , see also similar results in LaTeXMLCite .
However , these results did not go beyond writing down the equations : geometrical structures behind the equations were not discussed in these papers .
Having included hexagonal circle patterns with the multi-ratio property into the framework of the theory of integrable systems , we get an opportunity of applying the immense machinery of the latter to studying the properties of the former .
This is illustrated in Sect .
LaTeXMLRef , LaTeXMLRef , where we introduce and study some isomonodromic solutions of our integrable system on the triangular lattice , as well as the corresponding circle patterns .
Finally , in Sect .
LaTeXMLRef we define a subclass of these “ isomonodromic circle patterns ” which are natural discrete versions of the analytic functions LaTeXMLMath , LaTeXMLMath .
The results of Sect .
LaTeXMLRef – LaTeXMLRef constitute an extension to the present , somewhat more intricate , situation of the similar constructions for Schramm ’ s circle patterns with the combinatorics of the square grid LaTeXMLCite .
First of all we define the regular triangular lattice LaTeXMLMath as the cell complex whose vertices are LaTeXMLEquation whose edges are all non–ordered pairs LaTeXMLEquation and whose 2-cells are all regular triangles with the vertices in LaTeXMLMath and the edges in LaTeXMLMath .
We shall use triples LaTeXMLMath as coordinates of the vertices of the regular triangular lattice , identifying two such triples iff they differ by the vector LaTeXMLMath with LaTeXMLMath .
We call two points LaTeXMLMath neighbors in LaTeXMLMath , iff LaTeXMLMath .
To the complex LaTeXMLMath there correspond three regular hexagonal sublattices LaTeXMLMath , LaTeXMLMath .
Each LaTeXMLMath is the cell complex whose vertices are LaTeXMLEquation whose edges are LaTeXMLEquation and whose 2-cells are all regular hexagons with the vertices in LaTeXMLMath and the edges in LaTeXMLMath .
Again , we call two points LaTeXMLMath neighbors in LaTeXMLMath , iff LaTeXMLMath .
Obviously , every point in LaTeXMLMath has three neighbors in LaTeXMLMath , as well as three neighbors in LaTeXMLMath which do not belong to LaTeXMLMath .
The centers of 2-cells of LaTeXMLMath are exactly the points of LaTeXMLMath , i.e .
the points LaTeXMLMath with LaTeXMLMath .
In the following definition we consider only LaTeXMLMath , since , clearly , LaTeXMLMath and LaTeXMLMath are obtained from LaTeXMLMath via shifting all the corresponding objects by LaTeXMLMath , resp .
by LaTeXMLMath .
We say that a map LaTeXMLMath defines a hexagonal circle pattern , if the following condition is satisfied : Let LaTeXMLEquation be the vertices of any elementary hexagon in LaTeXMLMath with the center LaTeXMLMath .
Then the points LaTeXMLMath lie on a circle , and their circular order is just the listed one .
We denote the circle through the points LaTeXMLMath by LaTeXMLMath , thus putting it into a correspondence with the center LaTeXMLMath of the elementary hexagon above .
As a consequence of this condition , we see that if two elementary hexagons of LaTeXMLMath with the centers in LaTeXMLMath have a common edge LaTeXMLMath , then the circles LaTeXMLMath and LaTeXMLMath intersect in the points LaTeXMLMath , LaTeXMLMath .
Similarly , if three elementary hexagons of LaTeXMLMath with the centers in LaTeXMLMath meet in one point LaTeXMLMath , then the circles LaTeXMLMath , LaTeXMLMath and LaTeXMLMath also have a common intersection point LaTeXMLMath .
( Note that in every point LaTeXMLMath there meet three distinct elementary hexagons of LaTeXMLMath ) .
Remark .
Sometimes it will be convenient to consider circle patterns defined not on the whole of LaTeXMLMath , but rather on some connected subgraph of the regular hexagonal lattice .
We shall study in this paper a subclass of hexagonal circle patterns satisfying an additional condition .
We need the following generalization of the notion of cross-ratio .
Given a LaTeXMLMath -tuple LaTeXMLMath of complex numbers , their multi-ratio is the following number : LaTeXMLEquation where it is agreed that LaTeXMLMath .
In particular , LaTeXMLEquation is the usual cross-ratio , while in the present paper we shall be mainly dealing with LaTeXMLEquation .
The following two obvious properties of the multi-ratio will be important for us : The multi-ratio LaTeXMLMath is invariant with respect to the action of an arbitrary Möbius transformation LaTeXMLMath on all of its arguments .
The multi-ratio LaTeXMLMath is a Möbius transformation with respect to each one of its arguments .
We shall need also the following , slightly less obvious , property : If the points LaTeXMLMath lie on a circle LaTeXMLMath , and the multi-ratio LaTeXMLMath is real , then also LaTeXMLMath .
We say that a map LaTeXMLMath defines a hexagonal circle pattern with LaTeXMLMath , if in addition to the condition of Definition LaTeXMLRef the following one is satisfied : For any elementary hexagon in LaTeXMLMath with the vertices LaTeXMLMath ( listed counterclockwise ) , the multi-ratio LaTeXMLEquation where LaTeXMLMath .
Geometrically the condition ( LaTeXMLRef ) means that , first , the lengths of the sides of the hexagon with the vertices LaTeXMLMath satisfy the condition LaTeXMLEquation and , second , that the sum of the angles of the hexagon at the vertices LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is equal to LaTeXMLMath , as well as the sum of the angles at the vertices LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Notice that if a hexagon is inscribed in a circle and satisfies ( LaTeXMLRef ) , then it is conformally symmetric , i.e .
there exists a Möbius transformation mapping it onto a centrally symmetric hexagon .
Notice also that the regular hexagons satisfy this condition .
To demonstrate quickly the existence of hexagonal circle patterns with LaTeXMLMath we give their construction via solving a suitable Cauchy problem .
Consider a row of elementary hexagons of LaTeXMLMath running from the north–west to the south-east , with the centers in the points LaTeXMLMath .
Let the map LaTeXMLMath be defined in five vertices of each hexagon – in all except LaTeXMLMath .
Suppose that the five points LaTeXMLMath , LaTeXMLMath , lie on the circles LaTeXMLMath .
These data determine uniquely a map LaTeXMLMath yielding a hexagonal circle pattern with LaTeXMLMath on the whole lattice .
Proof .
Equation ( LaTeXMLRef ) determines the points LaTeXMLMath , which , according to the property above , lie also on LaTeXMLMath .
Now for every hexagon of the parallel row next to north–east , with the centers in the points LaTeXMLMath , we know the value of the map LaTeXMLMath in three vertices , namely in LaTeXMLEquation .
This uniquely defines the circle LaTeXMLMath , as the only circle through three points LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
The intersection points of these circles of the second row give us the values of the map LaTeXMLMath in the points LaTeXMLMath and LaTeXMLMath .
Namely , LaTeXMLMath is the intersection point of LaTeXMLMath with LaTeXMLMath , different from LaTeXMLMath , and LaTeXMLMath is the intersection point of LaTeXMLMath with LaTeXMLMath , different from LaTeXMLMath .
Therefore we get the values of the map LaTeXMLMath in five vertices of each hexagon of the next parallel row – in all except LaTeXMLMath .
The induction allows to continue the construction ad infinitum .
Now we show that , adding the centers of the circles of a hexagonal pattern with LaTeXMLMath to their intersection points , we come to a new interesting notion .
Let the map LaTeXMLMath define a hexagonal circle pattern with LaTeXMLMath .
Extend LaTeXMLMath to the points of LaTeXMLMath by the following rule .
Fix some point LaTeXMLMath .
Let LaTeXMLMath be a center of an elementary hexagon of LaTeXMLMath .
Set LaTeXMLMath to be the reflection of the point LaTeXMLMath in the circle LaTeXMLMath .
Then the condition ( LaTeXMLRef ) holds also for LaTeXMLMath in the case when the points LaTeXMLMath are the vertices of any elementary hexagon of the two complementary hexagonal sublattices LaTeXMLMath and LaTeXMLMath .
Proof .
Consider the situation corresponding to an elementary hexagon of the sublattice LaTeXMLMath or LaTeXMLMath ( see Fig .
LaTeXMLRef ) .
The point LaTeXMLMath is the intersection point of the three circles LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , the points LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are obtained by reflection of LaTeXMLMath in the corresponding circles , and the points LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are the pairwise intersection points of these circles different from LaTeXMLMath .
To simplify the geometry behind this situation , perform a Möbius transformation sending LaTeXMLMath to infinity .
Then the circles LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath become straight lines , and the points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are the reflections of LaTeXMLMath in these lines ( see Fig .
LaTeXMLRef ; for definiteness we suppose here that the Möbius image of LaTeXMLMath lies in the interior of the triangle formed by these straight lines ) .
By construction , one gets : LaTeXMLEquation the angles by the vertices LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are equal to LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , respectively , so that their sum is equal to LaTeXMLEquation the angles by the vertices LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are equal to LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , respectively , so that their sum is equal to LaTeXMLEquation .
This proves that the hexagon under consideration satisfies ( LaTeXMLRef ) .
A particular case of the construction of Theorem LaTeXMLRef is when LaTeXMLMath , so that the map LaTeXMLMath is extended by the centers of the corresponding circles .
In any case , this theorem suggests to consider the class of maps described in the following definition .
We say that the map LaTeXMLMath defines a triangular lattice with LaTeXMLMath , if the equation ( LaTeXMLRef ) holds for LaTeXMLMath , whenever the points LaTeXMLMath are the vertices ( listed counterclockwise ) of any elementary hexagon of any of the sublattices LaTeXMLMath LaTeXMLMath .
In the next section we shall discuss an integrable system on the regular triangular lattice , each solution of which delivers , in a single construction , three different triangular lattices with LaTeXMLMath .
However , these three lattices are not independent : given such a lattice , the two associated ones can be constructed almost uniquely ( up to an affine transformation LaTeXMLMath ) .
It will turn out that if the original lattice comes from a hexagonal circle pattern with LaTeXMLMath , then the two associated ones do likewise .
Let us describe a general construction of “ integrable systems ” on graphs which does not hang on the specific features of the regular triangular lattice .
This notion includes the following ingredients : An oriented graph LaTeXMLMath ; the set of its vertices will be denoted LaTeXMLMath , the set of its edges will be denoted LaTeXMLMath .
A loop group LaTeXMLMath , whose elements are functions from LaTeXMLMath into some group LaTeXMLMath .
The complex argument LaTeXMLMath of these functions is known in the theory of integrable systems as the “ spectral parameter ” .
A “ wave function ” LaTeXMLMath , defined on the vertices of LaTeXMLMath .
A collection of “ transition matrices ” LaTeXMLMath defined on the edges of LaTeXMLMath .
It is supposed that for any oriented edge LaTeXMLMath the values of the wave functions in its ends are connected via LaTeXMLEquation .
Therefore the following discrete zero curvature condition is supposed to be satisfied .
Consider any closed contour consisting of a finite number of edges of LaTeXMLMath : LaTeXMLEquation .
Then LaTeXMLEquation .
In particular , for any edge LaTeXMLMath , if LaTeXMLMath , then LaTeXMLEquation .
Actually , in applications the matrices LaTeXMLMath depend also on a point of some set LaTeXMLMath ( the “ phase space ” of an integrable system ) , so that some elements LaTeXMLMath are attached to the edges LaTeXMLMath of LaTeXMLMath .
In this case the discrete zero curvature condition ( LaTeXMLRef ) becomes equivalent to the collection of equations relating the fields LaTeXMLMath , LaTeXMLMath , LaTeXMLMath attached to the edges of each closed contour .
We say that this collection of equations admits a zero curvature representation .
For an arbitrary graph , the analytical consequences of the zero curvature representation for a given collection of equations are not clear .
However , in case of regular lattices , like LaTeXMLMath , such representation may be used to determine conserved quantities for suitably defined Cauchy problems , as well as to apply powerful analytical methods for finding concrete solutions .
Remark .
The above construction of integrable systems on graphs is not the only possible one .
For example , in the construction by Adler LaTeXMLCite the fields are defined on the vertices of a planar graph , and the equations relate the fields on stars consisting of the edges incident to each single vertex , rather than the fields on closed contours .
Examples are given by discrete time systems of the relativistic Toda type .
In the corresponding zero curvature representation the wave functions LaTeXMLMath naturally live on 2-cells rather than on vertices .
The transition matrices live on edges : the matrix LaTeXMLMath corresponds to the transition across LaTeXMLMath and depends on the fields sitting on two ends of LaTeXMLMath .
We now introduce an orientation of the edges of the regular triangular lattice LaTeXMLMath .
Namely , we declare as positively oriented all edges of the types LaTeXMLEquation .
Correspondingly , all edges of the types LaTeXMLEquation are negatively oriented .
Thus all elementary triangles become oriented .
There are two types of elementary triangles : those “ pointing upwards ” LaTeXMLMath are oriented counterclockwise , while those “ pointing downwards ” LaTeXMLMath are oriented clockwise .
The group LaTeXMLMath we use in our construction is the twisted loop group over LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath .
The elements of LaTeXMLMath we attach to every positively oriented edge of LaTeXMLMath are of the form LaTeXMLEquation .
Hence , to each positively oriented edge we assign a triple of complex numbers LaTeXMLMath satisfying an additional condition LaTeXMLMath .
In other words , choosing LaTeXMLMath ( say ) as the basic variables , we can assume that the “ phase space ” LaTeXMLMath mentioned in the previous section , is LaTeXMLMath .
The scalar factor LaTeXMLMath is not very essential and assures merely that LaTeXMLMath .
It is obvious that the zero curvature condition ( LaTeXMLRef ) is fulfilled for every closed contour in LaTeXMLMath , if and only if it holds for all elementary triangles .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath be the consecutive positively oriented edges of an elementary triangle of LaTeXMLMath .
Then the zero curvature condition LaTeXMLEquation is equivalent to the following set of equations : LaTeXMLEquation and LaTeXMLEquation with the understanding that LaTeXMLMath , LaTeXMLMath .
Proof .
An easy calculation shows that the matrix equation LaTeXMLMath consists of the following nine scalar equations : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It remains to isolate the independent ones among these nine equations .
First of all , equations ( LaTeXMLRef ) are equivalent to ( LaTeXMLRef ) , provided ( LaTeXMLRef ) and LaTeXMLMath hold .
For example : LaTeXMLEquation .
Next , the conditions LaTeXMLMath allow us to rewrite ( LaTeXMLRef ) as LaTeXMLEquation .
Further , all equations in ( LaTeXMLRef ) are equivalent provided ( LaTeXMLRef ) holds .
For example : LaTeXMLEquation .
Finally , LaTeXMLMath follows from ( LaTeXMLRef ) , ( LaTeXMLRef ) .
Indeed , LaTeXMLEquation .
The theorem is proved .
For want of a better name we shall call the system of equations ( LaTeXMLRef ) , ( LaTeXMLRef ) the fgh–system .
The equations ( LaTeXMLRef ) may be interpreted in the following way : there exist functions LaTeXMLMath such that for any positively oriented edge LaTeXMLMath there holds : LaTeXMLEquation .
The function LaTeXMLMath is determined by LaTeXMLMath uniquely , up to an additive constant , and similarly for the functions LaTeXMLMath , LaTeXMLMath .
Having introduced functions LaTeXMLMath sitting in the vertices of LaTeXMLMath , we may reformulate the remaining equations ( LaTeXMLRef ) as follows : let LaTeXMLMath be the consecutive vertices of a positively oriented elementary triangle , then LaTeXMLEquation .
The equations arising by cyclic permutations of indices LaTeXMLMath are equivalent to this one due to ( LaTeXMLRef ) .
So , we have one equation pro elementary triangle LaTeXMLMath .
Its geometrical meaning is the following : the triangle LaTeXMLMath is similar to the triangle LaTeXMLMath ( where the corresponding vertices are listed on the corresponding places ) .
Of course , these two triangles are also similar to the third one , LaTeXMLMath .
We discuss now the Cauchy data which allow one to determine a solution of the LaTeXMLMath –system .
The key observation is the following .
Given the values of two fields , say LaTeXMLMath and LaTeXMLMath , in three points LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , the equations of the LaTeXMLMath –system determine uniquely the values of LaTeXMLMath and LaTeXMLMath in the point LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Proof .
The formula ( LaTeXMLRef ) follows by eliminating LaTeXMLMath from LaTeXMLEquation .
These equations yield then ( LaTeXMLRef ) .
This immediately yields the following statement .
The values of the fields LaTeXMLMath and LaTeXMLMath in the vertices of the zig–zag line running from the north–west to the south–east , LaTeXMLEquation uniquely determine the functions LaTeXMLMath on the whole lattice .
The values of the fields LaTeXMLMath and LaTeXMLMath on the two positive semi-axes , LaTeXMLEquation uniquely determine the functions LaTeXMLMath on the whole sector LaTeXMLEquation .
Proof follows by induction with the help of the formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) .
There holds the following result having many analogs in the differential geometry described by integrable systems ( “ Sym formula ” , see , e.g. , LaTeXMLCite ) .
Let LaTeXMLMath be the solution of ( LaTeXMLRef ) with the initial condition LaTeXMLMath for some LaTeXMLMath .
Then the fields LaTeXMLMath may be found as LaTeXMLEquation .
Proof .
Note , first of all , that from LaTeXMLMath and LaTeXMLMath there follows that LaTeXMLMath for all LaTeXMLMath .
Consider an arbitrary positively oriented edge LaTeXMLMath .
From ( LaTeXMLRef ) there follows : LaTeXMLEquation .
At LaTeXMLMath we find : LaTeXMLEquation .
This proves the Proposition .
Next terms of the power series expansion of the wave function LaTeXMLMath around LaTeXMLMath also deliver interesting and important results .
Let LaTeXMLMath be the solution of ( LaTeXMLRef ) with the initial condition LaTeXMLMath for some LaTeXMLMath .
Then LaTeXMLEquation where the function LaTeXMLMath satisfies the difference equation LaTeXMLEquation and similar equations hold for the functions LaTeXMLMath ( with the cyclic permutation LaTeXMLMath ) .
Proof .
Proceeding as in the proof of Proposition LaTeXMLRef , we have : LaTeXMLEquation .
Taking into account that LaTeXMLMath , we find at LaTeXMLMath : LaTeXMLEquation .
This implies the statement of the proposition .
Notice that it is à priori not obvious that the equation ( LaTeXMLRef ) admits a well–defined solution on LaTeXMLMath , or , in other words , that its right–hand side defines a closed form on LaTeXMLMath .
This fact might be proved by a direct calculation , based upon the equations of the LaTeXMLMath –system , but the above argument gives a more conceptual and a much shorter proof .
Under the conditions of Propositions LaTeXMLRef , LaTeXMLRef , we have : LaTeXMLEquation where the function LaTeXMLMath satisfies the difference equation LaTeXMLEquation and similar equations hold for the functions LaTeXMLMath ( with the cyclic permutation LaTeXMLMath ) .
Further examples of such exact forms may be obtained from the values of higher derivatives of the wave function LaTeXMLMath at LaTeXMLMath .
We discuss now the equations satisfied by the field LaTeXMLMath alone , as well as by the field LaTeXMLMath alone .
In this point we make contact with the geometric considerations of Sect .
LaTeXMLRef .
Both maps LaTeXMLMath define triangular lattices with LaTeXMLMath .
In other words , if LaTeXMLMath are the vertices ( listed counterclockwise ) of any elementary hexagon of any of the hexagonal sublattices LaTeXMLMath LaTeXMLMath , and if LaTeXMLMath and LaTeXMLMath , then there hold both the equations LaTeXMLEquation and LaTeXMLEquation .
Given a triangular lattice LaTeXMLMath with LaTeXMLMath , there exists a unique , up to an affine transformation LaTeXMLMath , function LaTeXMLMath such that ( LaTeXMLRef ) are satisfied everywhere .
This function also defines a triangular lattice with LaTeXMLMath .
Given a pair of complex–valued functions LaTeXMLMath defined on LaTeXMLMath and satisfying the equation ( LaTeXMLRef ) everywhere , there exists a unique , up to an affine transformation , function LaTeXMLMath such that the pairs LaTeXMLMath and LaTeXMLMath satisfy the same equation .
The function LaTeXMLMath also defines a triangular lattice with LaTeXMLMath .
Proof .
1 .
To prove the first statement , we proceed as follows .
Let LaTeXMLMath , and let the vertices of an elementary hexagonal with the center in LaTeXMLMath be enumerated as LaTeXMLMath , LaTeXMLMath .
Then the following elementary triangles are positively oriented : LaTeXMLMath and LaTeXMLMath for LaTeXMLMath ( with the agreement that LaTeXMLMath ) .
According to ( LaTeXMLRef ) , we have : LaTeXMLEquation .
Dividing the first equation by the second one and taking the product over LaTeXMLMath , we find : LaTeXMLEquation which is nothing but ( LaTeXMLRef ) .
The proof of ( LaTeXMLRef ) is similar .
2 .
As for the second statement , suppose we are given a function LaTeXMLMath on the whole of LaTeXMLMath .
For an arbitrary elementary triangle , if the values of LaTeXMLMath in two vertices are known , the equation ( LaTeXMLRef ) allows us to calculate the value of LaTeXMLMath in the third vertex .
Therefore , choosing arbitrarily the values of LaTeXMLMath in two neighboring vertices , we can extend this function on the whole of LaTeXMLMath , provided this procedure is consistent .
It is easy to understand that it is enough to verify the consistency in running once around a vertex .
But this is assured exactly by the equation ( LaTeXMLRef ) .
3 .
To prove the third statement , notice that the proof of Theorem LaTeXMLRef shows that the formula LaTeXMLEquation valid for every edge LaTeXMLMath of LaTeXMLMath , correctly defines the third field LaTeXMLMath of the LaTeXMLMath –system .
All affine transformations of the field LaTeXMLMath thus obtained , and only they , lead to pairs LaTeXMLMath and LaTeXMLMath satisfying ( LaTeXMLRef ) .
Remark .
Notice that the above results remain valid in the more general context , when the fields LaTeXMLMath do not commute anymore , e.g .
when they take values in LaTeXMLMath , the field of quaternions .
The formulation and the proof of Theorem LaTeXMLRef hold in this case literally , while the formula ( LaTeXMLRef ) reads then as LaTeXMLEquation and similarly for LaTeXMLMath .
Recall that hexagonal circle patterns with LaTeXMLMath lead to a subclass of triangular lattices with LaTeXMLMath , namely those where the points of one of the three hexagonal sublattices lie on circles .
We now prove a remarkable statement , assuring that this subclass is stable with respect to the transformation LaTeXMLMath described in Theorem LaTeXMLRef .
Let LaTeXMLMath define a hexagonal circle pattern with LaTeXMLMath .
Extend it with the centers of the circles to LaTeXMLMath , a triangular lattice with LaTeXMLMath .
Let LaTeXMLMath be the triangular lattice with LaTeXMLMath related to LaTeXMLMath via ( LaTeXMLRef ) .
Then the restriction of the map LaTeXMLMath to the sublattice LaTeXMLMath also defines a hexagonal circle pattern with LaTeXMLMath , while the points LaTeXMLMath corresponding to LaTeXMLMath are the centers of the corresponding circles .
Proof starts as the proof of Theorem LaTeXMLRef .
Let LaTeXMLMath be a center of an arbitrary elementary hexagon of the sublattice LaTeXMLMath , i.e .
LaTeXMLMath with LaTeXMLMath .
Denote by LaTeXMLMath , LaTeXMLMath the vertices of the hexagon .
As before , considering the positively oriented triangles LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , surrounding the point LaTeXMLMath , we come to the relations LaTeXMLEquation .
But , obviously , LaTeXMLMath LaTeXMLMath are centers of elementary hexagons of the sublattice LaTeXMLMath .
By condition , the points LaTeXMLMath , LaTeXMLMath and LaTeXMLMath lie on a circle with the center in LaTeXMLMath .
Therefore , LaTeXMLEquation .
So , the absolute values of the left–hand sides of all equations in ( LaTeXMLRef ) are equal to 1 .
It follows that all six points LaTeXMLMath lie on a circle with the center in LaTeXMLMath .
Recall that we use triples LaTeXMLMath as coordinates of the vertices LaTeXMLMath , and that two such triples are identified iff they differ by the vector LaTeXMLMath with LaTeXMLMath .
By the LaTeXMLMath –axis we call the straight line LaTeXMLMath , resp .
by the LaTeXMLMath –axis the straight line LaTeXMLMath , and by the LaTeXMLMath –axis the straight line LaTeXMLMath .
It will be sometimes convenient to use the symbols LaTeXMLMath , LaTeXMLMath and LaTeXMLMath to denote the shifts of various objects in the positive direction of the axes LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , respectively , and the symbols LaTeXMLMath LaTeXMLMath , LaTeXMLMath LaTeXMLMath , LaTeXMLMath to denote the shifts in the negative directions .
This will apply to vertices , edges and elementary triangles of LaTeXMLMath , as well as to various objects assigned to them .
For example , if LaTeXMLMath , then LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLEquation .
Similarly , if LaTeXMLMath , then LaTeXMLEquation .
A fundamental role in the subsequent presentation will be played by a non-autonomous constraint for the solutions of the LaTeXMLMath –system .
This constraint consists of a pair of equations which are formulated for every vertex LaTeXMLMath and include the values of the fields on the edges incident to LaTeXMLMath , i.e .
on the star of this vertex .
It will be convenient to fix a numeration of these edges as follows : LaTeXMLEquation .
LaTeXMLEquation The notations LaTeXMLMath will refer to the values of the field LaTeXMLMath on these edges : LaTeXMLEquation .
LaTeXMLEquation and similarly for the fields LaTeXMLMath , LaTeXMLMath , see Fig .
LaTeXMLRef .
The constraint looks as follows : LaTeXMLEquation .
LaTeXMLEquation These are supposed to be the equations for the vertex LaTeXMLMath , and we use the notations LaTeXMLMath , LaTeXMLMath .
Since the fields LaTeXMLMath , LaTeXMLMath are defined only up to an affine transformation , one should replace the left–hand sides of the above equations by LaTeXMLMath , LaTeXMLMath , respectively , with arbitrary constants LaTeXMLMath , LaTeXMLMath .
In the form we have choosen it is imposed that the fields LaTeXMLMath , LaTeXMLMath are normalized to vanish in the origin .
The equations ( LaTeXMLRef ) , ( LaTeXMLRef ) are well defined equations for the point LaTeXMLMath , i.e .
they are invariant under the shift LaTeXMLMath , provided the equations ( LaTeXMLRef ) hold .
Proof is technical and is given in the Appendix LaTeXMLRef .
We mention an important consequence of this proposition .
Apparently , the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) relates the values of the fields LaTeXMLMath , LaTeXMLMath in seven points shown on Fig .
LaTeXMLRef .
However , we are free to choose any representative LaTeXMLMath for LaTeXMLMath .
In particular , we can let vanish any one of the coordinates LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
In the corresponding representation the constraint relates the values of the fields LaTeXMLMath , LaTeXMLMath in five points , belonging to any one of the three possible four–leg crosses through LaTeXMLMath .
An essential algebraic property of the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) is given by the following statement .
If the equations ( LaTeXMLRef ) hold , then the constraints ( LaTeXMLRef ) , ( LaTeXMLRef ) imply a similar equation for the field LaTeXMLMath ( vanishing at LaTeXMLMath ) : LaTeXMLEquation where LaTeXMLMath .
Proof is again based on calculations and is relegated to the Appendix LaTeXMLRef .
Remark .
We notice that restoring the fields LaTeXMLMath allows us to rewrite the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) as LaTeXMLEquation .
LaTeXMLEquation which coincides with ( LaTeXMLRef ) via a cyclic permutation of fields LaTeXMLMath performed once or twice , respectively , and accompanied by changing LaTeXMLMath to LaTeXMLMath , LaTeXMLMath , respectively .
Another similar remark : as it follows from the formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) used in the proof of Proposition LaTeXMLRef ( and their analogs for the fields LaTeXMLMath , LaTeXMLMath ) , the constraints ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) may be rewritten as equations for the single field LaTeXMLMath , resp .
LaTeXMLMath , LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
However , in this form , unlike the previous one , the terms attached to the variable LaTeXMLMath ( say ) , contain not only the fields on two edges LaTeXMLMath , LaTeXMLMath parallel to the LaTeXMLMath –axis .
This form is therefore less suited for the solution of the Cauchy problem for the constrained LaTeXMLMath –system , which we discuss now .
For arbitrary LaTeXMLMath the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) is compatible with the equations ( LaTeXMLRef ) .
Proof .
To prove this statement , one has to demonstrate the solvability of a reasonably posed Cauchy problem for the LaTeXMLMath –system constrained by ( LaTeXMLRef ) , ( LaTeXMLRef ) .
In this context , it is unnatural to assume that the fields LaTeXMLMath , LaTeXMLMath vanish at the origin , so that we replace ( only in this proof ) the left–hand sides of ( LaTeXMLRef ) , ( LaTeXMLRef ) by LaTeXMLMath , LaTeXMLMath , with arbitrary LaTeXMLMath .
We show that reasonable Cauchy data are given by the values of two fields LaTeXMLMath , LaTeXMLMath ( say ) in three points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is arbitrary .
According to Lemma LaTeXMLRef , these data yield via the equations of the LaTeXMLMath –system the values of LaTeXMLMath , LaTeXMLMath in LaTeXMLMath .
Further , these data together with the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) determine uniquely the values of LaTeXMLMath , LaTeXMLMath in LaTeXMLMath .
Indeed , assign LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath are two arbitrary complex numbers .
The constraint uniquely defines the values of LaTeXMLMath , LaTeXMLMath in the point LaTeXMLMath .
The requirement that these values agree with the ones obtained via Lemma LaTeXMLRef from the points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , gives us two equations for LaTeXMLMath , LaTeXMLMath .
It is shown by a direct computation that these equations have a unique solution , which is expressed via rational functions of the data at LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
It is also shown that the same solution is obtained , if we work with LaTeXMLMath instead of LaTeXMLMath .
Having found the fields LaTeXMLMath , LaTeXMLMath at LaTeXMLMath , we determine simultaneously LaTeXMLMath , LaTeXMLMath at LaTeXMLMath , LaTeXMLMath .
Now a similar procedure allows us to determine LaTeXMLMath , LaTeXMLMath at LaTeXMLMath and LaTeXMLMath , using the constraint at the points LaTeXMLMath and LaTeXMLMath , respectively .
Simultaneously the values of LaTeXMLMath , LaTeXMLMath are found at LaTeXMLMath and LaTeXMLMath .
A continuation of this procedure delivers the values of LaTeXMLMath , LaTeXMLMath on the both semiaxes LaTeXMLEquation using the condition that the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) is fulfilled on these semiaxes .
As we know from Proposition LaTeXMLRef , these data are enough to determine the solution of the LaTeXMLMath –system in the whole sector LaTeXMLEquation .
It remains to prove that this solution fulfills also the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) in the whole sector .
This follows by induction from the following statement : If the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) is satisfied in LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , then it is satisfied also in LaTeXMLMath .
The constraint at LaTeXMLMath includes the data at five points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
As we have seen , the data at LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are certain ( complicated ) functions of the data at LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Therefore , to check the constraint at LaTeXMLMath , one has to check that two ( complicated ) equations for the values of LaTeXMLMath , LaTeXMLMath at LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are satisfied identically .
This has been done with the help of the Mathematica computer algebra system .
Now we show how the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) appears in the context of isomonodromic solutions of integrable systems .
In this context , the results look better with a different gauge of the transition matrices for the LaTeXMLMath –system .
Namely , we conjugate them with the matrix LaTeXMLMath , and then multiply by LaTeXMLMath in order to get rid of the normalization of the determinant .
Writing then LaTeXMLMath for LaTeXMLMath , we end up with the matrices LaTeXMLEquation .
The zero curvature condition turns into LaTeXMLEquation .
LaTeXMLMath , LaTeXMLMath , LaTeXMLMath being the consecutive positively oriented edges of an elementary triangle of LaTeXMLMath .
This implies some slight modifications also for the notion of the wave function .
Namely , the previous formula does not allow to define the function LaTeXMLMath on LaTeXMLMath such that LaTeXMLEquation holds , whenever LaTeXMLMath .
The way around this difficulty is the following .
We define the wave function LaTeXMLMath on a covering of LaTeXMLMath .
Namely , over each point LaTeXMLMath now sits a sequence LaTeXMLEquation .
The values of these functions in neighboring vertices are related by natural formulas LaTeXMLEquation .
We call a solution LaTeXMLMath of the equations ( LaTeXMLRef ) isomonodromic ( cf .
LaTeXMLCite ) , if there exists the wave function LaTeXMLMath satisfying ( LaTeXMLRef ) and some linear differential equation in LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath are LaTeXMLMath matrices , meromorphic in LaTeXMLMath , with the poles whose position and order do not depend on LaTeXMLMath .
Obviously , due to ( LaTeXMLRef ) , the matrix LaTeXMLMath has to fulfill the condition LaTeXMLEquation .
Solutions of the equations ( LaTeXMLRef ) satisfying the constraints ( LaTeXMLRef ) , ( LaTeXMLRef ) are isomonodromic .
The corresponding matrix LaTeXMLMath is given by the following formulas : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are LaTeXMLMath –independent matrices : LaTeXMLEquation .
LaTeXMLMath are rank 1 matrices LaTeXMLEquation and the matrix LaTeXMLMath is well defined on LaTeXMLMath and not only on its covering LaTeXMLMath : LaTeXMLEquation where the functions LaTeXMLMath are solutions of the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) .
Proof can be found in the Appendix LaTeXMLRef .
We now consider isomonodromic solutions of the LaTeXMLMath –system satisfying the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) , which are special in two respects : First , the constants LaTeXMLMath and LaTeXMLMath in the constraint equations are not arbitrary , but are equal : LaTeXMLMath , so that LaTeXMLMath .
Second , the initial conditions will be choosen in a special way .
We will show that the resulting solutions lead to hexagonal circle patterns .
First of all , we discuss the Cauchy data which allow one to determine a solution of the LaTeXMLMath –system augmented by the constraints ( LaTeXMLRef ) , ( LaTeXMLRef ) .
Of course , the fields LaTeXMLMath , LaTeXMLMath , LaTeXMLMath have to vanish in the origin LaTeXMLMath .
Next , one sees easily that , given LaTeXMLMath and LaTeXMLMath in one of the points neighboring to LaTeXMLMath , the constraint allows to calculate one after another the values of LaTeXMLMath and LaTeXMLMath in all points of the corresponding axis .
For instance , fixing some values of LaTeXMLMath and LaTeXMLMath , we can calculate all LaTeXMLMath and LaTeXMLMath from the relations LaTeXMLEquation .
LaTeXMLEquation where we have set LaTeXMLEquation .
Indeed , we start with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and continue via the recurrent formulas , which are easily seen to be equivalent to ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
So , given the values of the fields LaTeXMLMath and LaTeXMLMath ( and hence of LaTeXMLMath ) in the points LaTeXMLMath and LaTeXMLMath , we get their values in all points LaTeXMLMath and LaTeXMLMath of the positive LaTeXMLMath - and LaTeXMLMath -semiaxes .
It is easy to see that LaTeXMLMath and LaTeXMLMath do not depend on LaTeXMLMath and LaTeXMLMath , respectively , so that all points LaTeXMLMath lie on a straight line , and so do all points LaTeXMLMath .
Similar statements hold also for all points LaTeXMLMath and for all points LaTeXMLMath .
And , of course , the third field LaTeXMLMath behaves analogously .
So , we get the values of LaTeXMLMath and LaTeXMLMath in all points on the border of the sector LaTeXMLEquation .
Proposition LaTeXMLRef assures that these data determine the values of LaTeXMLMath and LaTeXMLMath in all points of LaTeXMLMath .
By Theorem LaTeXMLRef ( more precisely , by Lemma LaTeXMLRef ) the solution thus obtained will satisfy the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) in the whole sector LaTeXMLMath .
Now we are in a position to specify the above mentioned isomonodromic solutions .
Let LaTeXMLMath .
Let LaTeXMLMath be the solutions of the LaTeXMLMath –system with the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) , with the initial conditions LaTeXMLEquation where LaTeXMLMath .
Then all three maps LaTeXMLMath define hexagonal circle patterns with LaTeXMLMath in the sector LaTeXMLMath .
More precisely , if LaTeXMLMath , LaTeXMLMath are the vertices of an elementary hexagon in this sector , then : LaTeXMLMath lie on a circle with the center in LaTeXMLMath whenever LaTeXMLMath , LaTeXMLMath lie on a circle with the center in LaTeXMLMath whenever LaTeXMLMath , LaTeXMLMath lie on a circle with the center in LaTeXMLMath whenever LaTeXMLMath .
Proof follows from the above inductive construction with the help of two lemmas .
The first one shows that if LaTeXMLMath then the constraint yields a very special property of the sequences of the values of the fields LaTeXMLMath , LaTeXMLMath , LaTeXMLMath in the points of the LaTeXMLMath - and LaTeXMLMath -axes .
If LaTeXMLMath , then for LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation The second one allows to extend inductively these special properties to the whole sector ( LaTeXMLRef ) .
Consider two elementary triangles with the vertices LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Suppose that LaTeXMLMath ; LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath .
Then LaTeXMLEquation and hence LaTeXMLEquation and LaTeXMLEquation .
The assertion of this lemma is illustrated on Fig .
LaTeXMLRef .
First of all , we show how do these lemmas work towards the proof of Theorem LaTeXMLRef .
The initial conditions ( LaTeXMLRef ) imply : LaTeXMLEquation .
Therefore , the conditions of Lemma LaTeXMLRef are fulfilled in the point LaTeXMLMath with the fields LaTeXMLMath instead of LaTeXMLMath .
From this Lemma it follows that The points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath ; The points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath ; The points LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath ; The points LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath .
Since , by Lemma LaTeXMLRef , we have LaTeXMLMath , there follows from LaTeXMLMath that LaTeXMLMath .
Finally , from Lemma LaTeXMLRef there follows that ( see Fig .
LaTeXMLRef ) LaTeXMLEquation .
Therefore the conditions of Lemma LaTeXMLRef are fulfilled in the point LaTeXMLMath with the fields LaTeXMLMath .
We deduce that The points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath ; The points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath ; The points LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath , which adds the point LaTeXMLMath to the list of equidistant neighbors of LaTeXMLMath from the conclusion LaTeXMLMath above ; and The points LaTeXMLMath , LaTeXMLMath are equidistant from LaTeXMLMath .
By Lemma LaTeXMLRef , we have LaTeXMLMath , and there follows from LaTeXMLMath that LaTeXMLMath .
Finally , from Lemma LaTeXMLRef there follows that ( see Fig .
LaTeXMLRef ) LaTeXMLEquation .
Hence , the conditions of Lemma LaTeXMLRef are again fulfilled in the point LaTeXMLMath with the fields LaTeXMLMath .
These arguments may be continued by induction along the LaTeXMLMath -axis , and , by symmetry , along the LaTeXMLMath -axis .
This delivers all the necessary relations which involve the points LaTeXMLMath with LaTeXMLMath or LaTeXMLMath .
We call them the relations of the level 1 .
The arguments of the level 2 start with the pair of fields LaTeXMLMath at the point LaTeXMLMath .
We have the level 1 relation LaTeXMLEquation .
For the angles , we have from the level 1 ( see Fig .
LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation So , the conditions of Lemma LaTeXMLRef are again satisfied in the point LaTeXMLMath for the fields LaTeXMLMath .
Continuing this sort of arguments , we prove all the necessary relations which involve the points LaTeXMLMath with LaTeXMLMath or LaTeXMLMath , and which will be called the relations of the level 2 .
The induction with respect to the level finishes the proof .
It remains to prove Lemmas LaTeXMLRef and LaTeXMLRef above .
What concerns the key Lemma LaTeXMLRef , it might be instructive to give two proofs for it , an analytic and a geometric ones .
The first one is shorter , but the second one seems to provide more insight into the geometry .
Analytic proof of Lemma LaTeXMLRef .
We rewrite the assumptions of the lemma as LaTeXMLEquation and LaTeXMLEquation .
Geometric proof of Lemma LaTeXMLRef .
The equations of the LaTeXMLMath –system imply that the triangles LaTeXMLMath and LaTeXMLMath are similar , and the triangles LaTeXMLMath and LaTeXMLMath are similar .
Therefore , LaTeXMLEquation .
From LaTeXMLMath there follows now LaTeXMLEquation .
Denoting the angles as on Fig .
LaTeXMLRef , we have : LaTeXMLEquation hence LaTeXMLEquation .
In other words , LaTeXMLEquation .
The relations ( LaTeXMLRef ) , ( LaTeXMLRef ) yield that the triangles LaTeXMLMath and LaTeXMLMath are similar .
But they have a common edge LaTeXMLMath , therefore they are congruent ( symmetric with respect to this edge ) .
This implies that the triangles LaTeXMLMath and LaTeXMLMath are isosceles , so that LaTeXMLMath and LaTeXMLMath , and LaTeXMLEquation .
Therefore LaTeXMLEquation .
Lemma is proved .
As for Lemma LaTeXMLRef , its statement is a small part of the following theorem and its corollary .
If LaTeXMLMath , then the recurrent relations ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) with LaTeXMLMath can be solved for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath in a closed form : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
Proof .
Elementary calculations show that the expressions above satisfy the recurrent relations ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) with LaTeXMLMath , as well as the initial conditions .
The uniqueness of the solution yields the statement .
We remark that similar formulas can be found also in the general case LaTeXMLMath , however , the property formulated in Lemma LaTeXMLRef fails to hold in general .
If LaTeXMLMath , and LaTeXMLMath , then for the third field LaTeXMLMath , LaTeXMLMath LaTeXMLMath we have : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
Proof .
The formulas for LaTeXMLMath follow from ( LaTeXMLRef ) , ( LaTeXMLRef ) .
The formulas for LaTeXMLMath with LaTeXMLMath follow by induction .
Although the construction of the previous section always delivers hexagonal circle patterns with LaTeXMLMath , these do not always behave regularly .
As a rule , they are not embedded ( i.e .
some elementary triangles overlap ) , and even not immersed ( i.e .
some neighboring triangles overlap ) , cf .
Fig .
LaTeXMLRef ) .
However , there exists a choice of the initial values ( i.e .
of LaTeXMLMath in Theorem LaTeXMLRef ) which assures that this is not the case .
Let LaTeXMLMath , so that LaTeXMLMath .
Set LaTeXMLMath .
Then the hexagonal circle patterns of Theorem LaTeXMLRef are called : the hexagonal LaTeXMLMath with an intersection point at the origin ; the hexagonal LaTeXMLMath with a circle at the origin .
In other words , for the hexagonal LaTeXMLMath the opening angle of the image of the sector ( LaTeXMLRef ) is equal to LaTeXMLMath , exactly as for the analytic function LaTeXMLMath .
For LaTeXMLMath the hexagonal circle patterns LaTeXMLMath with an intersection point at the origin and LaTeXMLMath with a circle at the origin are embedded .
For the proof of a similar statement for LaTeXMLMath circle patterns with the combinatorics of the square grid see LaTeXMLCite , where it is proven that they are immersed .
Remark .
Actually , the LaTeXMLMath and LaTeXMLMath versions of the hexagonal LaTeXMLMath with an intersection point at the origin are not essentially different .
Indeed , it is not difficult to see that the half–sector of the LaTeXMLMath pattern , corresponding to LaTeXMLMath , being rotated by LaTeXMLMath , coincides with the half–sector of the LaTeXMLMath pattern , corresponding to LaTeXMLMath , and vice versa .
For the LaTeXMLMath pattern , both sectors are identical ( up to the rotation by LaTeXMLMath ) .
So , for every LaTeXMLMath we have two essentially different hexagonal pattrens LaTeXMLMath .
It is important to notice the peculiarity of the case when LaTeXMLMath with LaTeXMLMath .
Then one can attach to the LaTeXMLMath –images of the sector LaTeXMLMath its LaTeXMLMath copies , rotated each time by the angle LaTeXMLMath .
The resulting object will satisfy the conditions for the hexagonal circle pattern everywhere except the origin LaTeXMLMath , which will be an intersection point of LaTeXMLMath circles .
Similarly , if LaTeXMLMath , and we attach to the LaTeXMLMath –image of the sector LaTeXMLMath its LaTeXMLMath copies , rotated each time by the angle LaTeXMLMath , then the origin LaTeXMLMath will be the center of a circle intersecting with LaTeXMLMath neighboring circles .
See Fig .
LaTeXMLRef for the examples of the LaTeXMLMath –pattern with LaTeXMLMath and the LaTeXMLMath –pattern with LaTeXMLMath .
Now we turn our attention to the limiting cases LaTeXMLMath and LaTeXMLMath .
It is easy to see that the quantities LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , become singular as LaTeXMLMath ( see ( LaTeXMLRef ) and ( LaTeXMLRef ) ) .
As a compensation , the quantities LaTeXMLMath , LaTeXMLMath , vanish with LaTeXMLMath , so that LaTeXMLMath for all LaTeXMLMath .
Similar effects hold for the LaTeXMLMath –axis , where LaTeXMLMath , LaTeXMLMath , become singular , and LaTeXMLMath for all LaTeXMLMath .
( Recall that for the LaTeXMLMath pattern we have : LaTeXMLMath ) .
These observations suggest the following rescaling : LaTeXMLEquation .
In order to be able to go to the limit LaTeXMLMath , we have to calculate the values of our fields in several lattice points next to LaTeXMLMath .
Applying formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) , we find : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
For the rescaled variables LaTeXMLMath , LaTeXMLMath , LaTeXMLMath in the limit LaTeXMLMath we find : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
These initial values have to be supplemented by the values in all further points of the LaTeXMLMath – and LaTeXMLMath –axes .
From the formulas of Theorem LaTeXMLRef there follows : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation which have to be augmented by LaTeXMLMath , LaTeXMLMath .
From Corollary LaTeXMLRef there follow the formulas for the edges of the LaTeXMLMath lattice : LaTeXMLEquation .
LaTeXMLEquation The hexagonal circle patterns corresponding to the solutions of the LaTeXMLMath –system in the sector ( LaTeXMLRef ) defined by the boundary values ( LaTeXMLRef ) – ( LaTeXMLRef ) are called : the hexagonal LaTeXMLMath with an intersection point at the origin ; the symmetric hexagonal LaTeXMLMath .
Alternatively , one could define the lattices LaTeXMLMath , LaTeXMLMath , LaTeXMLMath as the solutions of the LaTeXMLMath –system with the initial values ( LaTeXMLRef ) – ( LaTeXMLRef ) , satisfying the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) with LaTeXMLMath .
In this appoach the values ( LaTeXMLRef ) – ( LaTeXMLRef ) would be derived from the constraint .
Notice also that the formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) in this case turns into LaTeXMLEquation .
LaTeXMLEquation Considerations similar to those of the previous subsection show that , as LaTeXMLMath , the quantities LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , become singular ( see ( LaTeXMLRef ) and ( LaTeXMLRef ) ) .
As a compensation , the quantities LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , vanish with LaTeXMLMath , so that LaTeXMLMath for all LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath .
Similar effects hold for the LaTeXMLMath –axis .
These observations suggest the following rescaling : LaTeXMLEquation .
It turns out that in this case we need to calculate the values of these functions in a larger number of lattice points in the vicinity of LaTeXMLMath .
To this end , we add to ( LaTeXMLRef ) – ( LaTeXMLRef ) the following values , which are obtained by a direct calculation : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation From ( LaTeXMLRef ) – ( LaTeXMLRef ) and ( LaTeXMLRef ) – ( LaTeXMLRef ) we obtain in the limit LaTeXMLMath under the rescaling ( LaTeXMLRef ) the following initial values : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation These initial values have to be supplemented by the values in all further points of the LaTeXMLMath – and LaTeXMLMath –axes .
From the formulas of Theorem LaTeXMLRef there follow the expressions for the edges of the lattices LaTeXMLMath , LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation The formulas of Corollary LaTeXMLRef yield the results for the lattice LaTeXMLMath : LaTeXMLEquation so that LaTeXMLEquation .
Of course , one has also LaTeXMLMath .
The hexagonal circle patterns corresponding to the solutions of the LaTeXMLMath –system in the sector ( LaTeXMLRef ) defined by the boundary values ( LaTeXMLRef ) – ( LaTeXMLRef ) are called : the asymmetric hexagonal LaTeXMLMath ; the hexagonal LaTeXMLMath with a ( degenerate ) circle at the origin .
It is meant that the LaTeXMLMath –image of the half-sector LaTeXMLMath is not symmetric with respect to the line LaTeXMLMath ( the image of LaTeXMLMath , and the same for LaTeXMLMath .
Instead , this symmetry interchanges the LaTeXMLMath pattern and the LaTeXMLMath pattern , see Fig .
LaTeXMLRef .
Alternatively , one can define these lattices as the solutions of the LaTeXMLMath –system with the initial values ( LaTeXMLRef ) – ( LaTeXMLRef ) , satisfying the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) , which in the present situation degenerates into LaTeXMLEquation .
LaTeXMLEquation Just as in the non–degenerate case , these formulas allow one to calculate inductively the values of LaTeXMLMath , LaTeXMLMath on the LaTeXMLMath – and LaTeXMLMath –axes .
The formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) hold literally with LaTeXMLMath .
In this paper we introduced the notion of hexagonal circle patterns , and studied in some detail a subclass consisting of circle patterns with the property that six intersection points on each circle have the multi-ratio LaTeXMLMath .
We established the connection of this subclass with integrable systems on the regular triangular lattice , and used this connection to describe some Bäcklund–like transformations of hexagonal circle patterns ( transformation LaTeXMLMath , see Theorems LaTeXMLRef , LaTeXMLRef ) , and to find discrete analogs of the functions LaTeXMLMath , LaTeXMLMath .
Of course , this is only the beginning of the story of hexagonal circle patterns .
In a subsequent publication we shall demonstrate that there exists another subclass related to integrable systems , namely the patterns with fixed intersection angles .
The intersection of both subclasses constitute conformally symmetric patterns , including analogs of Doyle ’ s spirals ( cf .
LaTeXMLCite ) .
A very interesting question is , what part of the theory of integrable circle patterns can be applied to hexagonal circle packings .
This also will be a subject of our investigation .
This research was financially supported by DFG ( Sonderforschungsbereich 288 “ Differential Geometry and Quantum Physics ” ) .
Dropping all edges of LaTeXMLMath parallel to the LaTeXMLMath –axis , we end up with the cell complex isomorphic to the regular square lattice : its vertices LaTeXMLMath may be identified with LaTeXMLMath , its edges are then identified with those pairs LaTeXMLMath for which LaTeXMLMath , and its 2-cells ( parallelograms ) are identified with the elementary squares of the square lattice .
Hence , flat connections on LaTeXMLMath form a subclass of flat connections on the square lattice .
A natural question is , whether this inclusion is strict , i.e .
whether there exist flat connections on the square lattice which can not be extended to flat connections on LaTeXMLMath .
At least for the LaTeXMLMath –system , the answer is negative : denote by LaTeXMLMath the set of matrices ( LaTeXMLRef ) , then flat connections on the regular square grid with values in LaTeXMLMath are essentially in a one-to-one correspondence with flat connections on LaTeXMLMath with values in LaTeXMLMath , i.e .
with solutions of the LaTeXMLMath –system .
This is a consequence of the following statement dealing with an elementary square of the regular square lattice : a flat connection on such an elementary square with values in LaTeXMLMath can be extended by an element of LaTeXMLMath sitting on its diagonal without violating the flatness property .
More precisely : Let LaTeXMLEquation and let the off–diagonal parts of LaTeXMLMath , LaTeXMLMath be componentwise distinct from the off–diagonal parts of LaTeXMLMath , LaTeXMLMath , respectively .
Then there exists LaTeXMLMath such that LaTeXMLEquation .
Proof .
We have to prove that LaTeXMLMath .
It is easy to see that it is necessary and sufficient to prove that the entries 13 , 21 , 32 of this matrix vanish , i.e .
that there holds LaTeXMLEquation as well as two similar equations resulting by two successive permutations LaTeXMLMath .
We are given the relations LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation In order to prove ( LaTeXMLRef ) , we start with the third equation in ( LaTeXMLRef ) : LaTeXMLEquation .
Using LaTeXMLMath and ( LaTeXMLRef ) , we find : LaTeXMLEquation .
Plugging this into ( LaTeXMLRef ) , we get : LaTeXMLEquation .
Now , due to the second equation in ( LaTeXMLRef ) , we find : LaTeXMLEquation .
Substituting this into ( LaTeXMLRef ) , we come to the equation : LaTeXMLEquation .
Since , by condition , LaTeXMLMath , we obtain LaTeXMLMath , which is the equation ( LaTeXMLRef ) .
This result shows that the LaTeXMLMath –system could be alternatively studied in a more common framework of integrable systems on a square lattice .
However , such an approach would hide a rich and interesting geometric structures immanently connected with the triangular lattice .
It should be said at this point that the one–field equation ( LaTeXMLRef ) was first found , under the name of the “ Schwarzian lattice Bussinesq equation ” by Nijhoff in LaTeXMLCite using a ( different ) Lax representation on the square lattice .
The same holds for the one–field form of the constraint ( LaTeXMLRef ) .
Proof of Proposition LaTeXMLRef .
The arguments are similar for both equations ( LaTeXMLRef ) , ( LaTeXMLRef ) .
For instance , for the first one we have to demonstrate that LaTeXMLEquation .
LaTeXMLEquation To eliminate the fields LaTeXMLMath from this equation , consider six elementary triangles surrounding the vertex LaTeXMLMath .
The equations ( LaTeXMLRef ) imply : LaTeXMLEquation .
LaTeXMLEquation Therefore , LaTeXMLEquation .
LaTeXMLEquation By the way , this again yields the property LaTeXMLMath of the lattice LaTeXMLMath , which can be written now as LaTeXMLEquation .
Using ( LaTeXMLRef ) , an analogous expression along the LaTeXMLMath –axis , and an expression analogous to ( LaTeXMLRef ) along the LaTeXMLMath –axis , we rewrite ( LaTeXMLRef ) as LaTeXMLEquation .
Clearing denominators , we put it in the equivalent form LaTeXMLEquation .
LaTeXMLEquation But the polynomial on the left–hand side of the last formula is equal to LaTeXMLEquation and hence vanishes in virtue of ( LaTeXMLRef ) .
Proof of Proposition LaTeXMLRef .
Denote the right–hand sides of ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) through LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , respectively .
In order to prove ( LaTeXMLRef ) , i.e .
LaTeXMLMath , it is necessary and sufficient to demonstrate that LaTeXMLEquation ( or , actually , any two of these three equations ) .
We perform the proof for the first one only , since for the other two everything is similar .
In dealing with our constraints we are free to choose any representative LaTeXMLMath for LaTeXMLMath .
In order to keep things shorter , we always assume in this proof that LaTeXMLMath .
Writing the formula LaTeXMLEquation in long hand , we have to prove that LaTeXMLEquation .
Assuming that ( LaTeXMLRef ) and ( LaTeXMLRef ) hold , we have : LaTeXMLEquation .
Taking into account that LaTeXMLMath , LaTeXMLMath , we find : LaTeXMLEquation or , equivalently , LaTeXMLEquation .
The first two terms on the right–hand side already have the required form , since LaTeXMLMath .
So , it remains to prove that LaTeXMLEquation .
The most direct and unambiguous way to do this is to notice that everything here may be expressed with the help of the LaTeXMLMath –equations in terms of a single field LaTeXMLMath .
After straightforward calculations one obtains : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Taking into account that LaTeXMLMath , we see that ( LaTeXMLRef ) and Proposition LaTeXMLRef are proved .
Proof of Theorem LaTeXMLRef .
In order for the isomonodromy property to hold , the following compatibility conditions of ( LaTeXMLRef ) with ( LaTeXMLRef ) are necessary and sufficient : ( LaTeXMLRef ) and LaTeXMLEquation .
Substituting the ansatz ( LaTeXMLRef ) and calculating the residues at LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we see that the above system is equivalent to the following nine matrix equations : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLEquation .
We do not aim at solving these equations completely , but rather at finding a certain solution leading to the constraint ( LaTeXMLRef ) , ( LaTeXMLRef ) .
The subsequent reasoning will be divided into several steps .
Step 1 .
Consistency of the ansatz for LaTeXMLMath .
First of all , we have to convince ourselves that the ansatz ( LaTeXMLRef ) , ( LaTeXMLRef ) does not violate the necessary condition ( LaTeXMLRef ) , i.e .
that LaTeXMLEquation .
Notice that the entries 12 and 23 of this matrix equation are nothing but the content of Proposition LaTeXMLRef .
Upon the cyclic permutation of the fields LaTeXMLMath this gives also the entry 31 .
To check the entry 21 , we proceed as in the proof of Proposition LaTeXMLRef .
We have to prove that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Clearing denominators , we put it in the equivalent form LaTeXMLEquation .
But the polynomial on the left–hand side is equal to LaTeXMLEquation and vanishes due to ( LaTeXMLRef ) .
Via the cyclic permutation of fields this proves also the entries 32 and 13 of the matrix identity ( LaTeXMLRef ) .
Finally , turning to the diagonal entries , we consider , for the sake of definiteness , the entry 22 .
We have to prove that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation or LaTeXMLEquation .
LaTeXMLEquation Again , the polynomial on the left–hand side is equal to LaTeXMLEquation and vanishes due to ( LaTeXMLRef ) .
The formula ( LaTeXMLRef ) is proved .
Step 2 .
Checking the equations for the matrix LaTeXMLMath .
Next , we have to show that the ansatz ( LaTeXMLRef ) , ( LaTeXMLRef ) verifies ( LaTeXMLRef ) – ( LaTeXMLRef ) .
Notice that the matrices LaTeXMLEquation are degenerate , and that LaTeXMLEquation are the right null–vector and the left null–vector of LaTeXMLMath , respectively .
In terms of these vectors one can write the projectors LaTeXMLMath as LaTeXMLEquation .
Therefore we have : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In order to demonstrate ( LaTeXMLRef ) – ( LaTeXMLRef ) it is sufficient to prove that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
All these equations are verified in a similar manner , therefore we restrict ourselves to the first one .
LaTeXMLEquation or , in long hand , LaTeXMLEquation .
To prove this we have , first , to check that these two rank one matrices are proportional , and then to check that their entries 31 ( say ) coincide .
The second of these claims reads : LaTeXMLEquation and follows from ( LaTeXMLRef ) , ( LaTeXMLRef ) .
The first claim above is equivalent to : LaTeXMLEquation which , in turn , is equivalent to : LaTeXMLEquation and LaTeXMLEquation .
All these relations easily follow from the equations of the LaTeXMLMath –system .
For instance , to check the first equation in ( LaTeXMLRef ) , one has to consider the two elementary positively oriented triangles LaTeXMLMath and LaTeXMLMath .
Denoting the edge LaTeXMLMath , we have : LaTeXMLEquation .
Eliminating LaTeXMLMath from these two equations , we end up with the desired one .
This finishes the proof of ( LaTeXMLRef ) – ( LaTeXMLRef ) .
Step 3 .
Checking the equations for the matrix LaTeXMLMath .
Notice that the matrices LaTeXMLEquation are upper triangular .
We require that the matrices LaTeXMLMath are also upper triangular : LaTeXMLEquation .
It is immediately seen that the diagonal entries are constants .
By multiplying the wave function LaTeXMLMath from the right by a constant ( LaTeXMLMath –dependent ) matrix one can arrange that the matrices LaTeXMLMath are traceless .
Hence the diagonal part of LaTeXMLMath is parameterized by two arbitrary numbers .
It will be convenient to choose this parametrization as LaTeXMLEquation .
Equating the entries 12 and 23 in ( LaTeXMLRef ) – ( LaTeXMLRef ) , we find for an arbitrary positively oriented edge LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Obviously , a solution ( unique up to an additive constant ) is given by LaTeXMLEquation .
Finally , equating in ( LaTeXMLRef ) – ( LaTeXMLRef ) the entries 13 , we find : LaTeXMLEquation .
LaTeXMLEquation Comparing this with ( LaTeXMLRef ) , ( LaTeXMLRef ) , we see that ( LaTeXMLRef ) is proved .
Step 4 .
Equations relating the matrices LaTeXMLMath and LaTeXMLMath .
It remains to consider the equations ( LaTeXMLRef ) – ( LaTeXMLRef ) .
Denoting entries of the matrix LaTeXMLMath by LaTeXMLMath , we see that these matrix equations are equivalent to the following scalar ones : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ( In the last three equations we took into account that LaTeXMLMath , LaTeXMLMath are constants . )
It is easy to see that the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) are nothing but the constraint equations ( LaTeXMLRef ) , ( LaTeXMLRef ) , respectively .
We show now that the remaining equations ( LaTeXMLRef ) – ( LaTeXMLRef ) are not independent , but rather follow from the equations of the LaTeXMLMath –system and the constraints ( LaTeXMLRef ) , ( LaTeXMLRef ) .
We start with the last three equations , and prove the claim for ( LaTeXMLRef ) , since for other two everything is similar .
As in the proof of Proposition LaTeXMLRef , we write the formulas here with LaTeXMLMath .
Writing ( LaTeXMLRef ) in long hand , using the ansätze ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , we see that it is equivalent to LaTeXMLEquation .
But this follows immediately from ( LaTeXMLRef ) , ( LaTeXMLRef ) .
Finally , we turn to ( LaTeXMLRef ) .
Actually , since the entry 13 of the matrix LaTeXMLMath is defined only up to an additive constant , this equation is equivalent to the system of the following three ones : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
As usual , we restrict ourselves to the first one .
Upon using the equation ( LaTeXMLRef ) and the constraints ( LaTeXMLRef ) , ( LaTeXMLRef ) , we see that it is equivalent to LaTeXMLEquation .
Writing in long hand , in the representation with LaTeXMLMath , we see that the terms proportional to LaTeXMLMath and LaTeXMLMath vanish identically , while the vanishing of the terms proportional to LaTeXMLMath is equivalent to : LaTeXMLEquation .
But this follows immediately from ( LaTeXMLRef ) and the formulas LaTeXMLEquation which are similar to ( and follow from ) the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) .
This finishes the proof of Theorem LaTeXMLRef .
The notion of gluing of abelian categories was introduced in LaTeXMLCite and studied in LaTeXMLCite .
We observe that this notion is a particular case of a general categorical construction .
We then apply this general notion to the study of the ring of global differential operators LaTeXMLMath on the basic affine space LaTeXMLMath ( here LaTeXMLMath is a semi-simple simply connected algebraic group over LaTeXMLMath and LaTeXMLMath is a maximal unipotent subgroup ) .
We show that the category of LaTeXMLMath -modules is glued from LaTeXMLMath copies of the category of LaTeXMLMath -modules on LaTeXMLMath where LaTeXMLMath is the Weyl group , and the Fourier transform is used to define the gluing data .
As an application we prove that the algebra LaTeXMLMath is noetherian , and get some information on its homological properties .
Let LaTeXMLMath be an affine algebraic variety over LaTeXMLMath .
Let LaTeXMLMath be the ring of regular functions on LaTeXMLMath , and LaTeXMLMath be the algebra of differential operators on LaTeXMLMath in the sense of Grothendieck ( thus LaTeXMLMath , where for LaTeXMLMath we have LaTeXMLMath for all LaTeXMLMath ; and LaTeXMLMath ) .
If LaTeXMLMath is normal , then LaTeXMLMath is identified with global sections of the sheaf of differential operators with polynomial coefficients on the smooth part LaTeXMLMath .
If LaTeXMLMath is smooth , then it is well-known that algebra LaTeXMLMath possesses the following properties : a ) LaTeXMLMath is noetherian b ) The homological dimension of LaTeXMLMath is equal to LaTeXMLMath .
On the other hand , it is known after the work of J. Bernstein , I. Gelfand and S. Gelfand ( cf .
LaTeXMLCite ) that a ) and b ) fail if we do not assume that LaTeXMLMath is non-singular .
In fact in LaTeXMLCite the authors show that the algebra LaTeXMLMath does not ” behave nicely ” already when LaTeXMLMath is the cubic cone in a three-dimensional space .
In this paper we exhibit some examples of singular affine varieties LaTeXMLMath for which the algebra LaTeXMLMath of global differential operators behaves somewhat similarly to the algebra of differential operators on a non-singular variety .
These examples come from semi-simple groups .
Namely , let LaTeXMLMath be a semi-simple , simply connected algebraic group over LaTeXMLMath and let LaTeXMLMath be a maximal unipotent subgroup of LaTeXMLMath .
Consider the variety LaTeXMLMath .
Then it is known that LaTeXMLMath is a quasi-affine variety .
This means that the algebra LaTeXMLMath of global regular functions on LaTeXMLMath is finitely-generated and separates points of LaTeXMLMath .
Let LaTeXMLMath .
Then it is easy to see that LaTeXMLMath is singular unless LaTeXMLMath .
Let LaTeXMLMath be the algebra of global differential operators on LaTeXMLMath .
Since LaTeXMLMath is normal , LaTeXMLMath is equal to the algebra of differential operators on LaTeXMLMath .
The following result is proven in Section LaTeXMLRef .
The algebra LaTeXMLMath is ( left and right ) noetherian .
Theorem LaTeXMLRef is a generalization of a ) for LaTeXMLMath .
Let us now turn to the generalization of b ) .
The homological dimension of the algebra LaTeXMLMath is probably infinite , cf .
LaTeXMLCite .
However in Section LaTeXMLRef we prove the following The injective dimension of LaTeXMLMath as a left LaTeXMLMath -module is equal to LaTeXMLMath .
Note that LaTeXMLMath is a projective generator of the category of left LaTeXMLMath -modules ; and for a category of finite homological dimension the homological dimension of the category equals the injective dimension of a projective generator ( thus if the homological dimension of this category were finite then it would be equal to LaTeXMLMath ) .
Also Theorem LaTeXMLRef implies that one can define an analogue of Verdier duality for LaTeXMLMath -modules .
Denote by LaTeXMLMath the sheaf of differential operators on LaTeXMLMath .
Let us now explain the relation between the category LaTeXMLMath of left LaTeXMLMath -modules and the category LaTeXMLMath of LaTeXMLMath -modules on LaTeXMLMath ( i.e .
quasi-coherent sheaves of modules over the sheaf of algebras LaTeXMLMath ) .
In LaTeXMLCite D. Kazhdan and G. Laumon considered the category LaTeXMLMath of LaTeXMLMath -adic perverse sheaves on the variety LaTeXMLMath over a finite field LaTeXMLMath .
In particular they introduced for every element LaTeXMLMath of the Weyl group LaTeXMLMath of LaTeXMLMath a functor LaTeXMLMath .
The functors LaTeXMLMath are generalizations of the Fourier-Deligne transform .
Using the functors LaTeXMLMath they defined a new glued category LaTeXMLMath .
For example , when LaTeXMLMath then LaTeXMLMath and the Kazhdan-Laumon category LaTeXMLMath is equivalent to the category of perverse sheaves on LaTeXMLMath .
However , for general LaTeXMLMath the category LaTeXMLMath is not equivalent to the category of perverse sheaves on any variety .
This construction was studied by A. Polishchuk in LaTeXMLCite .
The main result of this paper asserts that the LaTeXMLMath -module counterpart of the Kazhdan-Laumon construction produces a category equivalent to LaTeXMLMath ; thus it says that LaTeXMLMath is glued from LaTeXMLMath copies of LaTeXMLMath .
This result implies both Theorem LaTeXMLRef and Theorem LaTeXMLRef .
The procedure of Kazhdan-Laumon gluing is in fact a particular case of a very general categorical construction called the category of coalgebras over a comonad ( alternatively , it can also be obtained as a category of algebras over a monad ) .
For a pair of adjoint functors between two categories LaTeXMLMath , LaTeXMLMath satisfying some additional assumptions it allows to describe the category LaTeXMLMath in terms of objects in LaTeXMLMath endowed with an additional structure .
These assumptions hold in particular if the categories are abelian , and one of the two functors is exact and faithful .
The same notion of gluing was used by A. Rosenberg in his paper about noncommutative schemes LaTeXMLCite .
Kontsevich and Rosenberg LaTeXMLCite used comonads of a more general nature for construction of noncommutative stacks .
The proof of our main result reduces to the following .
The algebra LaTeXMLMath carries an action of the Weyl group LaTeXMLMath , with the action of a simple reflection defined by means of Fourier transform ( this statement is in fact due to Gelfand and Graev , LaTeXMLCite ) .
Let LaTeXMLMath denote the localization functor sending every LaTeXMLMath to LaTeXMLMath .
Then our gluing theorem is equivalent to the fact that for every non-zero LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath .
This fact is proved by a direct calculation , similar to some standard computations in the theory of semi-simple Lie algebras ( e.g .
the computation of the determinant of the Shapovalov form on a Verma module ) .
This paper ows its very existence to D. Kazhdan who communicated to us its main results as a conjecture .
The second author is also grateful to J. Bernstein and A. Polishchuk for very useful discussions on the subject .
The third author is grateful to D. Kaledin , who first told him about monads many years ago .
The proof of Theorem LaTeXMLRef was obtained by the first two authors in 1995 , when they were graduate students at the University of Tel-Aviv .
They take the pleasure of acknowledging their gratitude to alma mater .
Likewise , the third author did his part of this research in 1996 , when he was a graduate student at Harvard University .
He is glad to use the opportunity to express his gratitude to this institution .
In this section we recall some general categorical constructions , most of which can be found e.g .
in chapter VI of the book LaTeXMLCite ( one has to reverse the direction of various arrows , i.e. , to replace all categories by the opposite ones , to pass from our setting to the one of loc .
cit . )
Examples are provided in the end of this section and in Section LaTeXMLRef .
Let LaTeXMLMath be a category .
The category of functors from LaTeXMLMath to itself is a monoidal category with respect to the operation of composition .
Therefore , one can consider monoid or comonoid objects in this category .
In LaTeXMLCite they are called “ monads ” and “ comonads ” in LaTeXMLMath .
Let us spell out the resulting definition .
A comonad in LaTeXMLMath is a functor LaTeXMLMath together with morphisms of functors LaTeXMLMath ( counit ) , LaTeXMLMath ( comultiplication ) such that the two compositions LaTeXMLEquation coincide ; and each of the two compositions LaTeXMLEquation equals identity .
In the next definition we stick to the terminology from LaTeXMLCite .
In our situation ( when LaTeXMLMath is abelian and LaTeXMLMath is additive ) it would be more natural to call such objects “ comodules ” rather then “ coalgebras ” .
A coalgebra over a comonad LaTeXMLMath is an object LaTeXMLMath of LaTeXMLMath together with a morphism LaTeXMLMath ( comultiplication ) such that LaTeXMLEquation .
Coalgebras over a comonad LaTeXMLMath form a category , which we denote by LaTeXMLMath .
Let LaTeXMLMath be an abelian category , and LaTeXMLMath be a comonad on LaTeXMLMath .
Assume that LaTeXMLMath is ( additive and ) left exact ; then LaTeXMLMath is an abelian category , and the forgetful functor LaTeXMLMath is exact .
For LaTeXMLMath and a morphism LaTeXMLMath we need to check that kernel and cokernel of LaTeXMLMath exist , and that LaTeXMLEquation ( the other axioms are clear ) .
Let LaTeXMLMath denote LaTeXMLMath considered as an element of LaTeXMLMath .
Then there are unique morphism LaTeXMLMath and LaTeXMLMath which make the obvious diagrams commutative ; moreover , LaTeXMLMath , LaTeXMLMath are easily seen to be a kernel , and a cokernel of LaTeXMLMath ( here the diagrams involving LaTeXMLMath are commutative because they inject into the corresponding diagrams for LaTeXMLMath ; while in the ones involving LaTeXMLMath one only needs to verify equalities of elements in LaTeXMLMath for various objects LaTeXMLMath , thus it is enough to verify the equality of their compositions with a surjective arrow LaTeXMLMath ) .
This shows existence of LaTeXMLMath , LaTeXMLMath ; since an arrow in LaTeXMLMath is an isomorphism if and only if the forgetful functor to LaTeXMLMath sends it into an isomorphism , ( LaTeXMLRef ) follows .
Exactness of the forgetful functor is clear from the explicit description of LaTeXMLMath , LaTeXMLMath .
∎ For every LaTeXMLMath the object LaTeXMLMath is naturally equipped with a structure of a coalgebra over LaTeXMLMath ; thus we get a functor LaTeXMLMath ( the “ cofree coalgebra ” construction ) .
This functor is right adjoint to the forgetful functor LaTeXMLMath , i.e .
we have a natural isomorphism LaTeXMLEquation where we omit the forgetful functor LaTeXMLMath from our notation ( see LaTeXMLCite §VI.2 , Theorem 1 ) .
Let LaTeXMLMath be a functor , and LaTeXMLMath be a right adjoint functor .
Then the composition LaTeXMLMath is equipped with natural transformations LaTeXMLMath and LaTeXMLMath , which together form a comonad .
The functor LaTeXMLMath factors canonically ( cf .
LaTeXMLCite , VI.3 , Theorem 1 ) through a functor LaTeXMLEquation .
The next statement is a particular case of the general “ triality Theorem ” of Barr and Beck ( see LaTeXMLCite , §VI.7 , Theorem 1 ( and Exercise 6 ) ) which gives an explicit criterion for LaTeXMLMath to be an equivalence ; we include the proof since it is a bit shorter in our particular case .
Let LaTeXMLMath , LaTeXMLMath be abelian categories , LaTeXMLMath be an additive functor , and LaTeXMLMath be a right adjoint functor .
Assume that LaTeXMLMath is exact and faithful .
Then ( LaTeXMLRef ) provides an equivalence LaTeXMLMath .
LaTeXMLMath is faithful and exact , because such is its composition with the faithful exact forgetful functor LaTeXMLMath .
Let us check that LaTeXMLMath is a full imbedding , i.e .
LaTeXMLEquation .
First we claim that ( LaTeXMLRef ) holds if LaTeXMLMath ; moreover , this is true for any pair of adjoint functors LaTeXMLMath , LaTeXMLMath ( not necessarily between additive categories ) .
Indeed , using ( LaTeXMLRef ) we get LaTeXMLEquation and it is immediate to see that the resulting isomorphism coincides with the map induced by LaTeXMLMath .
Thus to check ( LaTeXMLRef ) it suffices to see that any LaTeXMLMath is a kernel of an arrow LaTeXMLMath for some LaTeXMLMath .
It is enough to find an injection LaTeXMLMath ( then apply the same construction to its cokernel ) ; but LaTeXMLMath being faithful implies that the adjunction arrow LaTeXMLMath is an injection .
To see that LaTeXMLMath is surjective on isomorphism classes of objects it sufficies to prove that any object LaTeXMLMath is a subobject in LaTeXMLMath for some LaTeXMLMath ( then it is also a kernel of a morphism LaTeXMLMath for some LaTeXMLMath ; since we know already that LaTeXMLMath for some LaTeXMLMath we conclude that LaTeXMLMath by exactness of LaTeXMLMath ) .
Now for LaTeXMLMath consider the adjunction arrow LaTeXMLMath ( coming from ( LaTeXMLRef ) ) .
We claim it is injective ; indeed , since the forgetful functor is exact and faithful , it is enough to see that the corresponding arrow LaTeXMLMath in LaTeXMLMath is injective ; it is in fact a split injection , because LaTeXMLMath by the definition of a coalgebra over a comonad .
∎ The following definitions appear ( in a somewhat less economical notation ) in LaTeXMLCite and LaTeXMLCite .
( In LaTeXMLCite left adjoint functors are used instead of right adjoint ones ; in LaTeXMLCite it is assumed that both left and right adjoint functors exist . )
Let LaTeXMLMath be a finite set and let LaTeXMLMath be a collection of abelian categories , and let LaTeXMLMath be their product .
For a functor LaTeXMLMath we will write LaTeXMLMath , where LaTeXMLMath .
A right gluing data for LaTeXMLMath is a comonad LaTeXMLMath on LaTeXMLMath such that 1 ) LaTeXMLMath is ( additive and ) left-exact ; 2 ) for each LaTeXMLMath the morphism LaTeXMLMath induced by LaTeXMLMath is an isomorphism .
For a gluing data LaTeXMLMath we call the category LaTeXMLMath the category glued from LaTeXMLMath .
A right localization data for LaTeXMLMath is an abelian category LaTeXMLMath together with a collection of exact functors LaTeXMLMath , such that LaTeXMLMath has a right adjoint LaTeXMLMath , and the adjunction arrow LaTeXMLMath is an isomorphism .
A localization data is called faithful if the sum LaTeXMLMath is faithful .
Lemma LaTeXMLRef , Theorem LaTeXMLRef imply the following If LaTeXMLMath is a localization data , then LaTeXMLMath , LaTeXMLMath is a gluing data for LaTeXMLMath .
We have a canonical exact functor from LaTeXMLMath to the glued category LaTeXMLMath ; it is an equivalence if and only if the localization data is faithful .
LaTeXMLMath Conversely , if LaTeXMLMath is a gluing data , then the glued category LaTeXMLMath with the forgetful functor LaTeXMLMath is a faithful localization data for LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath be a pair of adjoint functors between abelian categories LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is exact .
It is not difficult to show that the adjunction arrow LaTeXMLMath is an isomorphism if and only if LaTeXMLMath induces an equivalence LaTeXMLMath , where LaTeXMLMath is a Serre subcategory , and LaTeXMLMath is the Serre quotient .
Thus localization data amounts to the data of an abelian category LaTeXMLMath , and a finite collection of Serre subcategories LaTeXMLMath , such that the projection LaTeXMLMath admits a right adjoint .
The localization data is faithful if the intersection LaTeXMLMath is the zero category ( recall that a Serre subcategory is strictly full by definition ) .
Since any gluing data admits a unique ( up to equivalence ) faithful localization data , we see that a gluing data amounts to the data of an abelian category LaTeXMLMath , and a finite collection of Serre subcategories LaTeXMLMath with zero intersection , and such that the projection LaTeXMLMath has a right adjoint .
Assume that an abelian category LaTeXMLMath admits arbitrary direct sums , and the functors of direct sums in LaTeXMLMath are exact .
Assume also that LaTeXMLMath admits a set of generators .
Then it follows from LaTeXMLCite , 5.51–5.53 that for a Serre subcategory LaTeXMLMath the projection LaTeXMLMath admits a right adjoint if and only if LaTeXMLMath is closed under arbitrary direct sums .
Let LaTeXMLMath be a topological space and let LaTeXMLMath be a collection of open subsets of LaTeXMLMath .
Set LaTeXMLMath to be the category of sheaves of abelian groups on LaTeXMLMath , LaTeXMLMath be the category of sheaves of abelian groups on LaTeXMLMath ; let LaTeXMLMath be the restriction functor and LaTeXMLMath be the functor of direct image of sheaves ( this example explains our terminology ) .
Let LaTeXMLMath be a nonsingular quasi-affine complex algebraic variety .
Denote by LaTeXMLMath the sheaf of differential operators on LaTeXMLMath and set LaTeXMLMath to be the algebra of global sections of LaTeXMLMath .
Let LaTeXMLMath be any finite set of automorphisms of LaTeXMLMath and set LaTeXMLMath to be the category of LaTeXMLMath -modules .
Then every LaTeXMLMath defines a functor LaTeXMLMath ( twisting of the action by LaTeXMLMath ) .
Set LaTeXMLMath to be the category of quasi-coherent sheaves of LaTeXMLMath -modules for any LaTeXMLMath LaTeXMLMath for every LaTeXMLMath LaTeXMLMath for any LaTeXMLMath and LaTeXMLMath ( here LaTeXMLMath denotes the functor of global sections for LaTeXMLMath -modules ) .
This is another example of a localization data whose particular case is considered in Section LaTeXMLRef below .
Let LaTeXMLMath be a ring and LaTeXMLMath be the category of left LaTeXMLMath -modules .
For any idempotent element LaTeXMLMath consider the full subcategory LaTeXMLMath whose objects are all LaTeXMLMath -modules LaTeXMLMath such that LaTeXMLMath .
It is easy to see that LaTeXMLMath is a Serre subcategory .
Consider the subring LaTeXMLMath of the ring LaTeXMLMath .
It does not contain the unit element of LaTeXMLMath , but it has its own unit LaTeXMLMath .
We claim that the quotient category LaTeXMLMath can be identified with the category of left LaTeXMLMath -modules , and the projection functor sends an LaTeXMLMath -module LaTeXMLMath to the LaTeXMLMath -module LaTeXMLMath .
Indeed , let us set LaTeXMLMath and denote by LaTeXMLMath the category of left LaTeXMLMath -modules .
Let LaTeXMLMath be the functor LaTeXMLMath .
We have LaTeXMLEquation where LaTeXMLMath is considered as a LaTeXMLMath - LaTeXMLMath -bimodule and LaTeXMLMath is an LaTeXMLMath - LaTeXMLMath -bimodule .
Therefore , both left and right adjoint functors to LaTeXMLMath exist and they can be computed as LaTeXMLEquation .
It is easy to check that LaTeXMLMath .
Now suppose that we are given a finite set of idempotents LaTeXMLMath .
Let LaTeXMLMath denote the categories of left modules over the rings LaTeXMLMath and LaTeXMLMath , LaTeXMLMath be the above-defined functors .
This is obviously a localization data .
It is faithful whenever LaTeXMLMath .
The following examples show how badly can the homological dimension behave with respect to our gluing .
Let LaTeXMLMath be the associative algebra ( over a field LaTeXMLMath ) generated by elements LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath with the following relations : LaTeXMLMath are orthogonal idempotents , LaTeXMLMath , and LaTeXMLMath .
Consider the quotient algebras LaTeXMLMath and LaTeXMLMath .
The last two algebras are finite-dimensional : LaTeXMLMath and LaTeXMLMath .
It is easy to see that the algebra LaTeXMLMath has infinite homological dimension .
On the other hand , we have LaTeXMLMath and LaTeXMLMath .
So an abelian category of infinite homological dimension is glued out of two copies of the category of vector spaces .
The algebra LaTeXMLMath has homological dimension 2 .
On the other hand , we have LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath .
Thus a category of homological dimension 2 is glued out of the category of vector spaces and a category of infinite homological dimension .
In this section we will deal with algebraic varieties over LaTeXMLMath .
For any such variety LaTeXMLMath , the symbol LaTeXMLMath will denote the algebra of global differential operators on LaTeXMLMath .
Let LaTeXMLMath be a semisimple simply-connected algebraic group over LaTeXMLMath ; let LaTeXMLMath be a maximal unipotent subgroup , and let LaTeXMLMath denote the homogeneous space LaTeXMLMath .
Then LaTeXMLMath is the unipotent radical of a Borel subgroup LaTeXMLMath , and LaTeXMLMath is known as the basic affine space of LaTeXMLMath .
The variety LaTeXMLMath is a quasi-affine .
Let LaTeXMLMath denote the algebra of global differential operators on LaTeXMLMath , and let LaTeXMLMath denote the algebra of regular functions on LaTeXMLMath .
For a simple root LaTeXMLMath of LaTeXMLMath let LaTeXMLMath be the minimal parabolic subgroup of type LaTeXMLMath containing LaTeXMLMath .
Let LaTeXMLMath be the commutator subgroup of LaTeXMLMath , and set LaTeXMLMath .
We have the obvious projection of homogeneous spaces LaTeXMLMath .
It is a fibration with the fiber LaTeXMLMath ( here LaTeXMLMath denotes the affine plane ) .
Let LaTeXMLMath be the relative affine completion of the morphism LaTeXMLMath .
( So LaTeXMLMath is the affine morphism corresponding to the sheaf of algebras LaTeXMLMath on LaTeXMLMath . )
Then LaTeXMLMath has the structure of a 2-dimensional vector bundle ; LaTeXMLMath is identified with the complement to the zero-section in LaTeXMLMath .
The LaTeXMLMath -action on LaTeXMLMath obviously extends to LaTeXMLMath ; moreover , it is easy to see that the determinant of the vector bundle LaTeXMLMath admits a unique ( up to a constant ) LaTeXMLMath -invariant trivialization , i.e .
LaTeXMLMath admits unique up to a constant LaTeXMLMath -invariant fiberwise symplectic form LaTeXMLMath ( cf .
LaTeXMLCite for more details ) .
In what follows we will fix these forms for every simple root LaTeXMLMath .
Recall that for any symplectic vector bundle LaTeXMLMath we have a canonical automorphism LaTeXMLMath of the sheaf of algebras LaTeXMLMath and , in particular , of the ring LaTeXMLMath of global differential operators on LaTeXMLMath , called Fourier transform .
In particular , we get a canonical automorphism LaTeXMLMath of the ring LaTeXMLMath ( the first equality follows from the fact that LaTeXMLMath has codimension 2 in LaTeXMLMath ) .
( cf .
LaTeXMLCite ) Let LaTeXMLMath be the Weyl group , LaTeXMLMath be the simple reflection of type LaTeXMLMath .
Then the assignment LaTeXMLMath extends to a homomorphism LaTeXMLMath of the Weyl group to the group of automorphisms of LaTeXMLMath .
Let LaTeXMLMath denote the group LaTeXMLMath -points of a split real form of LaTeXMLMath .
Let also LaTeXMLMath be the group of points of a maximal unipotent subgroup of LaTeXMLMath defined over LaTeXMLMath .
Then the manifold LaTeXMLMath admits a unique up to a constant LaTeXMLMath -invariant measure which has unique smooth extension to every LaTeXMLMath .
Let LaTeXMLMath denote the space of LaTeXMLMath -functions on LaTeXMLMath with respect to this measure .
Since for every LaTeXMLMath as above we have LaTeXMLMath it follows that we have well defined unitary operators LaTeXMLMath acting on LaTeXMLMath ( Fourier transform along the fibers of LaTeXMLMath ) .
We claim now that these operators LaTeXMLMath define an action of LaTeXMLMath on LaTeXMLMath .
Indeed , this is proved in LaTeXMLCite when LaTeXMLMath is replaced by a non-archimedian local field and in the archimedian case is essentially a word-by-word repetition .
For every LaTeXMLMath we denote by LaTeXMLMath the corresponding unitary automorphism of LaTeXMLMath .
The operators LaTeXMLMath commute with the natural action of LaTeXMLMath on LaTeXMLMath .
Set now LaTeXMLEquation ( here by LaTeXMLMath we mean the space of complex valued LaTeXMLMath -functions ) .
It is clear that LaTeXMLMath is a dense subspace of LaTeXMLMath ( since it contains the dense subspace of LaTeXMLMath -functions with compact support ) .
Let us show that LaTeXMLMath is invariant with respect to the operators LaTeXMLMath .
Indeed , every LaTeXMLMath is a LaTeXMLMath -vector in the LaTeXMLMath representation LaTeXMLMath .
Therefore , since every LaTeXMLMath commutes with LaTeXMLMath it follows that LaTeXMLMath is again a LaTeXMLMath -vector with respect to LaTeXMLMath .
Hence for every LaTeXMLMath the function LaTeXMLMath makes sense .
Moreover , it is easy to see that for every LaTeXMLMath and for every simple root LaTeXMLMath of LaTeXMLMath we have LaTeXMLEquation .
Hence LaTeXMLMath which implies that LaTeXMLMath .
Hence LaTeXMLMath is invariant with respect to LaTeXMLMath ’ s and therefore it is also invariant with respect to all LaTeXMLMath .
It is clear that LaTeXMLMath is a faithful module over LaTeXMLMath .
Moreover ( LaTeXMLRef ) implies that in the space LaTeXMLMath we have the equality LaTeXMLEquation .
Clearly , this implies our claim .
∎ One can also give an algebraic proof of Proposition LaTeXMLRef ( the braid relations can be verified using the analogue of the Radon transform associated to any LaTeXMLMath ) .
We do not present details in this paper .
Let LaTeXMLMath be the Cartan group of LaTeXMLMath and let LaTeXMLMath .
Then LaTeXMLMath carries a natural action of LaTeXMLMath , commuting with the LaTeXMLMath -action ( it comes from the action of LaTeXMLMath on LaTeXMLMath by right translations ) .
So LaTeXMLMath acts in a locally finite way on the rings LaTeXMLMath , i.e .
these rings are LaTeXMLMath -graded , where LaTeXMLMath is the weight lattice of LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) denote the graded component of LaTeXMLMath ( resp .
of LaTeXMLMath ) of degree LaTeXMLMath .
Let also LaTeXMLMath be the set of dominant weights .
We denote by LaTeXMLMath the half-sum of the positive roots .
Note that every element LaTeXMLMath defines a LaTeXMLMath -invariant vector field on LaTeXMLMath .
This defines an embedding of algebras LaTeXMLMath , where LaTeXMLMath is the universal enveloping algebra of LaTeXMLMath .
The following Lemma is an immediate consequence of the definitions and Proposition LaTeXMLRef .
The operators LaTeXMLMath commute with the LaTeXMLMath -action on LaTeXMLMath .
For every LaTeXMLMath we have LaTeXMLEquation ( Here LaTeXMLMath is the natural pairing between LaTeXMLMath and LaTeXMLMath ) .
In particular we have LaTeXMLMath Here we formulate our main computational result ( it is proved in Section LaTeXMLRef below ) .
Let us denote by LaTeXMLMath be the category of modules over the ring LaTeXMLMath ; and let as before LaTeXMLMath be the category of LaTeXMLMath -modules on LaTeXMLMath .
We have the functor of global sections LaTeXMLMath , and the left-adjoint functor LaTeXMLMath .
For an affine open LaTeXMLMath and LaTeXMLMath we have : LaTeXMLEquation .
It is easy to see that LaTeXMLMath is an exact functor , and that LaTeXMLMath .
For any LaTeXMLMath we will use the same notation LaTeXMLMath for an automorphism of an associative ring LaTeXMLMath and the corresponding auto-equivalence of the category LaTeXMLMath of modules over LaTeXMLMath .
Set LaTeXMLEquation .
For any LaTeXMLMath , LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath .
The statements formulated in the Introduction are immediate consequences of Theorem LaTeXMLRef Proof of Theorem LaTeXMLRef .
If LaTeXMLMath is a chain of left ideals in LaTeXMLMath then LaTeXMLMath is a chain of sub LaTeXMLMath -modules in the free LaTeXMLMath -module LaTeXMLMath .
Since the category of LaTeXMLMath -modules on a smooth variety is Noetherian , this chain stabilizes for all LaTeXMLMath ; thus LaTeXMLMath for large LaTeXMLMath , hence LaTeXMLMath by Theorem LaTeXMLRef .
We checked that LaTeXMLMath is left Noetherian ; since LaTeXMLMath , because LaTeXMLMath has a non-vanishing volume form , we see that LaTeXMLMath is also right Noetherian .
LaTeXMLMath Proof of Theorem LaTeXMLRef .
It is enough to show that LaTeXMLMath for LaTeXMLMath and any finitely generated LaTeXMLMath ; and that LaTeXMLMath for some LaTeXMLMath .
Now , LaTeXMLMath may be considered as a right LaTeXMLMath -module by means of the right action of LaTeXMLMath on itself .
Therefore , it is enough to show that for any LaTeXMLMath one has LaTeXMLMath ( Theorem LaTeXMLRef is valid , of course , also for right LaTeXMLMath -modules ) .
Suppose that this is not so , i.e .
that there exists LaTeXMLMath such that LaTeXMLMath .
We may assume , without loss of generality , that LaTeXMLMath .
For a finitely generated projective object LaTeXMLMath we have a canonical isomorphism LaTeXMLMath , where LaTeXMLMath is the sheaf of Hom ’ s ; hence for any finitely generated LaTeXMLMath we have LaTeXMLMath where LaTeXMLMath is the corresponding sheaf of Ext ’ s .
This means that for any affine open subset LaTeXMLMath of LaTeXMLMath one has LaTeXMLEquation .
But the right hand side of this equality vanishes when LaTeXMLMath , since for a non-singular affine variety LaTeXMLMath the algebra LaTeXMLMath has homological dimension equal to LaTeXMLMath .
Also , for LaTeXMLMath it is non-zero , for example if LaTeXMLMath , so that LaTeXMLMath .
LaTeXMLMath copies of the category LaTeXMLMath ( indexed by LaTeXMLMath ) together with functors LaTeXMLMath and natural tranformations LaTeXMLMath , LaTeXMLMath arising from adjointness of LaTeXMLMath and LaTeXMLMath form a gluing data .
The glued category is naturally equivalent to LaTeXMLMath .
Theorem LaTeXMLRef shows that categories LaTeXMLMath , LaTeXMLMath for LaTeXMLMath ; and functors LaTeXMLMath ; LaTeXMLMath form a faithful localization data .
Thus the statement follows from Corollary LaTeXMLRef .
∎ Let LaTeXMLMath be the affine completion of LaTeXMLMath .
We have the obvious open embedding LaTeXMLMath .
Let LaTeXMLMath be the ideal of functions vanishing on LaTeXMLMath .
Then for LaTeXMLMath we have LaTeXMLMath if and only if LaTeXMLMath acts on LaTeXMLMath locally nilpotently .
The following result is well-known .
( Bott-Borel-Weil ) LaTeXMLMath is the irreducible LaTeXMLMath -module of highest weight LaTeXMLMath when LaTeXMLMath and LaTeXMLMath otherwise .
The ideal LaTeXMLMath is generated by LaTeXMLMath .
We now proceed to the proof of Theorem LaTeXMLRef .
Assume that LaTeXMLMath is such that LaTeXMLMath for all LaTeXMLMath .
Then LaTeXMLMath acts on LaTeXMLMath locally nilpotently for all LaTeXMLMath .
Fix LaTeXMLMath , LaTeXMLMath .
Then from the second statement of Lemma LaTeXMLRef we see that for some LaTeXMLMath we have LaTeXMLMath for all LaTeXMLMath .
So Theorem LaTeXMLRef follows from the following result .
Let LaTeXMLMath denote the longest element .
For any LaTeXMLMath we set LaTeXMLMath .
For any dominant weight LaTeXMLMath the left ideal in LaTeXMLMath generated by the space LaTeXMLMath contains 1 .
For a LaTeXMLMath -module LaTeXMLMath , let LaTeXMLMath denote the isotypic part of LaTeXMLMath corresponding to the irreducible LaTeXMLMath -module of highest weight LaTeXMLMath .
We start the proof of Proposition LaTeXMLRef with the following ( cf .
LaTeXMLCite for a different proof ) .
We have : LaTeXMLMath .
It is obvious that LaTeXMLMath .
Let us prove the inverse inclusion .
Let LaTeXMLMath .
Since LaTeXMLMath is irreducible for any LaTeXMLMath , we have LaTeXMLMath for some LaTeXMLMath .
Let us prove that the function LaTeXMLMath , LaTeXMLMath is polynomial .
For any LaTeXMLMath consider the operator LaTeXMLMath acting on the functions on LaTeXMLMath which is defined by : LaTeXMLMath .
Then for any set of functions LaTeXMLMath , where LaTeXMLMath , we have : LaTeXMLEquation .
In particular , if LaTeXMLMath is a differential operator of order LaTeXMLMath , then LaTeXMLMath for any LaTeXMLMath .
But it is well known that the latter property implies that the function LaTeXMLMath is polynomial .
It is clear that for every polynomial function LaTeXMLMath there exists an element LaTeXMLMath such that LaTeXMLMath .
Let us take LaTeXMLMath .
Then LaTeXMLMath for any LaTeXMLMath .
Hence LaTeXMLMath which finishes the proof .
∎ The plan of the proof of Proposition LaTeXMLRef is as follows .
Let LaTeXMLMath as before be the longest element .
Consider the map LaTeXMLMath , defined by LaTeXMLMath .
Then LaTeXMLMath lands to LaTeXMLMath .
Let LaTeXMLMath be the unique ( up to a constant ) LaTeXMLMath -invariant element .
( Recall that LaTeXMLMath . )
Then LaTeXMLMath by Lemma LaTeXMLRef .
Let LaTeXMLMath denote LaTeXMLMath .
It is clear that for any LaTeXMLMath the element LaTeXMLMath lies in the left ideal generated by LaTeXMLMath .
On the other hand we will prove the following The element LaTeXMLMath is of the form LaTeXMLEquation where LaTeXMLMath are positive coroots , LaTeXMLMath are positive integers and LaTeXMLMath is a non-zero constant .
Proposition LaTeXMLRef implies the following The ideal in LaTeXMLMath generated by LaTeXMLMath for all LaTeXMLMath contains 1 .
Proof of the Corollary .
By Hilbert Nullstellensatz it suffices to prove that for any point LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath .
( Here we identified LaTeXMLMath with the algebra of polynomial functions on LaTeXMLMath . )
It is enough to take LaTeXMLMath such that LaTeXMLMath is an antidominant weight .
LaTeXMLMath Corollary LaTeXMLRef obviously implies Proposition LaTeXMLRef .
Let us now prove Proposition LaTeXMLRef .
We will prove a more precise We have : LaTeXMLEquation where LaTeXMLMath is the set of positive coroots and LaTeXMLMath .
The first step towards the proof is the following We have LaTeXMLEquation .
Here order denotes the standard order of a differential operator .
Let LaTeXMLMath as before be a symplectic vector bundle .
Let LaTeXMLMath .
Suppose that LaTeXMLMath transforms by a character LaTeXMLMath under the natural action of the group LaTeXMLMath on LaTeXMLMath ( coming from the action of LaTeXMLMath on LaTeXMLMath by dilatations ) .
Then it is easy to check that LaTeXMLEquation .
For LaTeXMLMath we know that LaTeXMLMath .
So dilatations in the fibers of the vector bundle LaTeXMLMath act on LaTeXMLMath by the character LaTeXMLMath .
By induction on the length of LaTeXMLMath we deduce that LaTeXMLMath .
In particular , for any LaTeXMLMath and LaTeXMLMath we have : LaTeXMLMath .
The lemma follows .
∎ For LaTeXMLMath and an integer LaTeXMLMath such that LaTeXMLMath the element LaTeXMLMath divides LaTeXMLMath .
Let us choose a simple coroot LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
By Lemma LaTeXMLRef the statement of Lemma LaTeXMLRef is equivalent to saying that LaTeXMLMath divides LaTeXMLMath .
We are going to show that for LaTeXMLMath such that LaTeXMLMath we have LaTeXMLMath .
( Recall that we have identified LaTeXMLMath with the algebra of polynomial functions on LaTeXMLMath ) .
This is enough , since the set LaTeXMLMath is a Zariski dense subset of the hyperplane LaTeXMLMath .
The latter statement is equivalent to saying that LaTeXMLMath for LaTeXMLMath .
Let us prove this .
We have LaTeXMLMath where LaTeXMLMath , LaTeXMLMath are the dual bases of LaTeXMLMath and LaTeXMLMath respectively .
So LaTeXMLMath .
We claim that LaTeXMLMath for LaTeXMLMath .
Indeed , LaTeXMLMath , hence LaTeXMLMath since LaTeXMLMath is not dominant .
( Recall that LaTeXMLMath by the assumption ) .
This proves the lemma .
∎ Now we are ready to prove Proposition LaTeXMLRef .
Let RHS denote the right hand side of equality ( LaTeXMLRef ) .
From Lemma LaTeXMLRef it follows that RHS divides LaTeXMLMath .
From Lemma LaTeXMLRef we see that LaTeXMLMath .
Since both LaTeXMLMath and RHS are LaTeXMLMath -invariant and since LaTeXMLMath the equality will follow provided we know that LaTeXMLMath .
To check this take the dual bases LaTeXMLMath , LaTeXMLMath of LaTeXMLMath and LaTeXMLMath respectively , compatible with the weight decomposition .
Assume that LaTeXMLMath is a highest weight vector , and LaTeXMLMath is a lowest weight vector .
Then LaTeXMLMath which implies that there exists LaTeXMLMath and a highest weight vector LaTeXMLMath , such that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Hence LaTeXMLMath .
The proof is finished .
LaTeXMLMath
We present the construction of an associative , commutative algebra LaTeXMLMath of generalized functions on a manifold LaTeXMLMath satisfying the following optimal set of permanence properties : LaTeXMLMath is linearly embedded into LaTeXMLMath , LaTeXMLMath is the unity in LaTeXMLMath .
For every smooth vector field LaTeXMLMath on LaTeXMLMath there exists a derivation operator LaTeXMLMath which is linear and satisfies the Leibniz rule .
LaTeXMLMath is the usual Lie derivative .
LaTeXMLMath is the pointwise product of functions .
Moreover , the basic building blocks of LaTeXMLMath are defined in purely intrinsic terms of the manifold LaTeXMLMath .
Key words .
Algebras of generalized functions , Colombeau algebras , generalized functions on manifolds .
Mathematics Subject Classification ( 2000 ) .
Primary 46F30 ; Secondary 46T30 .
Recent applications of Colombeau ’ s theory of algebras of generalized functions to problems of a primarily geometric nature ( LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) have very clearly indicated the need for a global intrinsic version of the construction on differentiable manifolds .
The development of local diffeomorphism invariant Colombeau algebras on open subsets of LaTeXMLMath , initiated in LaTeXMLCite , has only recently been completed in LaTeXMLCite .
Based on LaTeXMLCite as well as Parts I and II of this series ( also in this volume ) in the present article we present an intrinsic global version of Colombeau ’ s theory on differentiable manifolds adapted to the needs of applications in mathematical physics .
In what follows we shall use freely notation and terminology from LaTeXMLCite .
Additionally , we will use the following conventions .
LaTeXMLMath will denote an orientable , paracompact , LaTeXMLMath -dimensional smooth manifold .
with atlas LaTeXMLMath .
LaTeXMLMath is the space of compactly supported ( smooth ) LaTeXMLMath -forms on LaTeXMLMath .
For coordinates LaTeXMLMath on LaTeXMLMath , elements of LaTeXMLMath will be written as LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath and if LaTeXMLMath resp .
LaTeXMLMath then LaTeXMLMath .
Concerning distributions on manifolds we follow the terminology of LaTeXMLCite and LaTeXMLCite .
The space of distributions on LaTeXMLMath is defined as LaTeXMLMath ( the dual of the space of compactly supported LaTeXMLMath -forms ) and operations on distributions are defined as ( sequentially ) continuous extensions of classical operations on smooth functions .
For example , for LaTeXMLMath ( the space of smooth vector fields on LaTeXMLMath ) and LaTeXMLMath the Lie derivative of LaTeXMLMath with respect to LaTeXMLMath is given by LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath then the local representation of LaTeXMLMath on LaTeXMLMath is the element LaTeXMLMath defined by LaTeXMLEquation .
An intrinsic formulation on differentiable manifolds of the diffeomorphism invariant Colombeau algebra LaTeXMLMath introduced in LaTeXMLCite faces a number of serious obstacles due to the following indispensible technical ingredients of the construction of LaTeXMLMath : Translation ( convolution ) used for the embedding of LaTeXMLMath into LaTeXMLMath , leading to terms of the form LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath , LaTeXMLMath open ) .
Scaling operations of the form LaTeXMLMath .
Moment integrals of the form LaTeXMLMath .
Obviously , none of these operations allows a direct generalization to the manifold setting .
Our task therefore consists in unfolding the diffeomorphism invariant ‘ essence ’ underlying the local testing procedures used for determining relationship in the spaces of moderate resp .
negligible mappings ( cf .
LaTeXMLCite , ch .
3 ) .
Definition LaTeXMLRef below introduces test objects on LaTeXMLMath ( so called smoothing kernels ) that display precisely those properties of local test objects corresponding to regularization via convolution and linear scaling on LaTeXMLMath ( part ( i ) ) resp .
to the interplay between LaTeXMLMath - and LaTeXMLMath -differentiation in the local context .
Definition .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath is our basic space , both moderate and negligible maps will be elements of LaTeXMLMath .
Localization of elements of LaTeXMLMath is effected by the map LaTeXMLEquation .
Suppose that LaTeXMLMath is smooth such that for each LaTeXMLMath , LaTeXMLMath is contained in LaTeXMLMath .
Then for any LaTeXMLMath , we introduce the following two notions of Lie derivatives of LaTeXMLMath with respect to LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLEquation After these preparations we finally turn to the definition of smoothing kernels , the global analogue on LaTeXMLMath of the translated scaled test objects LaTeXMLMath in the local theory : Definition .
LaTeXMLMath is called a smoothing kernel if LaTeXMLMath LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath : LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLEquation .
The space of smoothing kernels on LaTeXMLMath is denoted by LaTeXMLMath .
Here LaTeXMLMath denotes the ball of radius LaTeXMLMath , measured with respect to the distance function LaTeXMLMath induced on LaTeXMLMath by some Riemannian metric LaTeXMLMath on LaTeXMLMath and LaTeXMLMath denotes the norm induced by LaTeXMLMath on LaTeXMLMath .
Both notions are independent of the chosen metric LaTeXMLMath ( cf .
LaTeXMLCite , Lemma 3.4 ) .
The grading on LaTeXMLMath into the subspaces LaTeXMLMath consisting of those test functions of unit integral whose moments up to order LaTeXMLMath vanish is replaced by the following subspaces of LaTeXMLMath in the global case .
Definition .
Let LaTeXMLMath ( LaTeXMLMath ) be the set of all LaTeXMLMath satisfying LaTeXMLEquation .
The technical motivation for the exact form of this definition is that it provides precisely what is needed for proving LaTeXMLMath later on .
Moreover , in the local case the above requirement in essence reproduces the original spaces LaTeXMLMath .
It is shown in LaTeXMLCite , Lemma 3.7 that the spaces LaTeXMLMath are in fact nontrivial for all LaTeXMLMath .
Definition .
For any LaTeXMLMath and any LaTeXMLMath we define the Lie derivative of LaTeXMLMath with respect to LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath denotes the derivative of LaTeXMLMath ( cf .
LaTeXMLCite ) .
In order to derive the local form of this Lie derivative we let LaTeXMLMath , set LaTeXMLMath and calculate as follows LaTeXMLEquation .
Now setting LaTeXMLMath ( LaTeXMLMath ) we obtain the local algebra derivative LaTeXMLMath in the J-formalism ( cf .
LaTeXMLCite , Ch .
5 and LaTeXMLCite ) .
We begin by introducing the subspaces of moderate and negligible maps of LaTeXMLMath .
Definition .
LaTeXMLMath is called moderate if the following condition is satisfied : LaTeXMLEquation .
LaTeXMLEquation The subset of moderate elements of LaTeXMLMath is denoted by LaTeXMLMath .
Definition .
LaTeXMLMath is called negligible if the following condition is satisfied : LaTeXMLEquation .
LaTeXMLEquation The subset of negligible elements of LaTeXMLMath will be denoted by LaTeXMLMath .
We shall see in LaTeXMLRef below that in fact Lie derivatives can be omitted completely in the definition of LaTeXMLMath if we additionally suppose that LaTeXMLMath .
This fact is another instance of a very general result ( Th .
13.1 of LaTeXMLCite ) stating a similar characterization of the Colombeau ideal as a subspace of the space of moderate functions without resorting to derivatives for practically all types of Colombeau algebras .
We immediately obtain Theorem .
LaTeXMLMath is a subalgebra of LaTeXMLMath .
LaTeXMLMath is an ideal in LaTeXMLMath .
In order to connect the development of LaTeXMLMath already at this early stage to that of LaTeXMLMath we need a localization procedure allowing to transport smoothing kernels on the manifold to local test objects on open subsets of LaTeXMLMath and vice versa .
However , a direct translation is not feasible since localizations of smoothing kernels display rather poor properties concerning domain of definition .
To precisely formulate the translation process we introduce the following spaces of functions .
We denote by LaTeXMLMath the space of smooth maps LaTeXMLMath such that for each LaTeXMLMath ( i.e. , LaTeXMLMath a compact subset of LaTeXMLMath ) and any LaTeXMLMath , the set LaTeXMLMath is bounded in LaTeXMLMath .
For any LaTeXMLMath we set LaTeXMLEquation .
LaTeXMLEquation Elements of LaTeXMLMath are said to have asymptotically vanishing moments of order LaTeXMLMath ( more precisely , in the terminology LaTeXMLCite elements of LaTeXMLMath are of type LaTeXMLMath , the abbreviation standing for asymptotic vanishing of moments globally , i.e. , on each LaTeXMLMath ) .
It is easily seen that LaTeXMLMath .
Moreover , by LaTeXMLMath we denote the space of all LaTeXMLMath where LaTeXMLMath is some subset ( depending on LaTeXMLMath ) of LaTeXMLMath such that for LaTeXMLMath the following holds : For each LaTeXMLMath there exists LaTeXMLMath and a subset LaTeXMLMath of LaTeXMLMath which is open in LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation Here the subscript LaTeXMLMath indicates the weaker requirements on the domain of definition of LaTeXMLMath .
The subspace of LaTeXMLMath consisting of those LaTeXMLMath whose moments up to order LaTeXMLMath vanish asymptotically on each compact subset of LaTeXMLMath will be written as LaTeXMLMath .
LaTeXMLMath is defined analogously .
With this terminology at hand we can now precisely state the transport properties of smoothing kernels .
Lemma .
Denote by LaTeXMLMath a chart in LaTeXMLMath .
( A ) Transforming smoothing kernels to local test objects .
Let LaTeXMLMath be a smoothing kernel .
Then the map LaTeXMLMath defined by LaTeXMLEquation .
LaTeXMLMath is an element of LaTeXMLMath .
If , in addition , LaTeXMLMath for some LaTeXMLMath then LaTeXMLMath , i.e. , LaTeXMLEquation uniformly on compact sets .
In particular , if LaTeXMLMath then LaTeXMLMath .
( B ) Transporting local test objects onto the manifold .
Let LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath with LaTeXMLMath on an open neighborhood of LaTeXMLMath and LaTeXMLMath on an open neighborhood of LaTeXMLMath .
Then LaTeXMLEquation is a smoothing kernel ( with LaTeXMLMath a smooth cut-off function ) .
If , in addition , LaTeXMLMath and LaTeXMLMath then LaTeXMLMath .
In particular , if LaTeXMLMath and LaTeXMLMath then LaTeXMLMath .
For a proof of this result we refer to LaTeXMLCite .
Using the above result we may now derive local characterizations of moderateness and negligibility , thereby establishing the link to LaTeXMLMath promised above .
Theorem .
Let LaTeXMLMath .
Then for all LaTeXMLMath , LaTeXMLEquation .
Proof .
( LaTeXMLMath ) Let LaTeXMLMath , LaTeXMLMath and set LaTeXMLMath .
Then for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath LaTeXMLMath LaTeXMLMath .
we have to show that LaTeXMLEquation for some LaTeXMLMath .
To this end we fix some LaTeXMLMath and define LaTeXMLMath by ( LaTeXMLRef ) .
Let LaTeXMLMath and for LaTeXMLMath choose LaTeXMLMath such that the local expression of LaTeXMLMath coincides with LaTeXMLMath on a neighborhood of LaTeXMLMath .
Then for LaTeXMLMath sufficiently small ( LaTeXMLRef ) equals LaTeXMLEquation ( LaTeXMLMath ) Let LaTeXMLMath and LaTeXMLMath for some chart LaTeXMLMath .
Let LaTeXMLMath and define LaTeXMLMath by ( LaTeXMLRef ) .
Since LaTeXMLMath belongs to LaTeXMLMath it follows from LaTeXMLCite , Th .
10.5 that given LaTeXMLMath there exists some LaTeXMLMath and some LaTeXMLMath such that for LaTeXMLMath and LaTeXMLMath we have LaTeXMLMath and LaTeXMLMath .
From this and ( LaTeXMLRef ) the result follows .
LaTeXMLMath Theorem .
Let LaTeXMLMath .
Then for all LaTeXMLMath , LaTeXMLEquation .
Proof .
( LaTeXMLMath ) Let LaTeXMLMath , LaTeXMLMath and set LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Set LaTeXMLMath and choose LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath and all LaTeXMLMath .
Then by LaTeXMLRef , for LaTeXMLMath , LaTeXMLMath as in ( LaTeXMLRef ) is an element of LaTeXMLMath .
Hence ( with LaTeXMLMath as in the proof of LaTeXMLRef ) we obtain LaTeXMLEquation so the claim follows from the characterization results in LaTeXMLCite ( Ch .
7 and 10 ) .
( LaTeXMLMath ) Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
There exists LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath and all LaTeXMLMath .
Now set LaTeXMLMath and let LaTeXMLMath .
Then LaTeXMLMath defined by ( LaTeXMLRef ) is in LaTeXMLMath by LaTeXMLRef ( A ) ( ii ) .
Hence inserting local representations of LaTeXMLMath , ( LaTeXMLRef ) implies ( LaTeXMLRef ) .
LaTeXMLMath From this and LaTeXMLCite , Th .
13.1 we conclude Corollary .
Let LaTeXMLMath .
Then LaTeXMLEquation .
As a final ingredient in the construction of LaTeXMLMath , in the following result we establish stability of LaTeXMLMath and LaTeXMLMath under Lie derivatives Theorem .
Let LaTeXMLMath .
Then LaTeXMLMath .
LaTeXMLMath .
Proof .
Let LaTeXMLMath , LaTeXMLMath .
By LaTeXMLRef for any chart LaTeXMLMath we have LaTeXMLMath .
Thus by the stability of the space of moderate functions on LaTeXMLMath ( LaTeXMLCite , Th .
7.10 ) also LaTeXMLMath LaTeXMLMath LaTeXMLMath ( where LaTeXMLMath denotes the local representation of LaTeXMLMath ) , which , again by LaTeXMLRef gives the result .
The claim for LaTeXMLMath follows analogously from the stability of the space of negligible functions under differentiation ( LaTeXMLCite , Th .
7.11 ) .
LaTeXMLMath Having collected all the necessary properties of LaTeXMLMath and LaTeXMLMath ( cf .
the general scheme of construction for Colombeau algebras given in LaTeXMLCite , Ch .
3 ) we turn to the actual definition of LaTeXMLMath .
Definition .
The full Colombeau algebra on LaTeXMLMath is defined as LaTeXMLEquation .
LaTeXMLMath is a differential algebra with respect to the Lie derivative LaTeXMLMath induced by ( LaTeXMLRef ) , cf .
LaTeXMLRef .
For any LaTeXMLMath , its class in LaTeXMLMath will be denoted by LaTeXMLMath .
In order to complete the list of properties of LaTeXMLMath given in the abstract of this paper we still need to embed LaTeXMLMath and LaTeXMLMath into LaTeXMLMath .
Definition .
Let LaTeXMLMath , LaTeXMLMath ; LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath and LaTeXMLMath are linear embeddings , LaTeXMLMath respecting multiplication and unit of LaTeXMLMath .
Next we show that LaTeXMLMath and LaTeXMLMath commute with arbitrary Lie derivatives .
Concerning LaTeXMLMath , we obtain from ( LaTeXMLRef ) LaTeXMLEquation .
Similarly , for LaTeXMLMath the first term LaTeXMLMath of LaTeXMLMath vanishes , so LaTeXMLMath .
The most important properties of LaTeXMLMath and LaTeXMLMath are collected in the following result .
Theorem .
LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath .
Proof .
To show ( i ) , let LaTeXMLMath ; then LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath it follows from the local theory that indeed LaTeXMLMath .
( ii ) is immediate and ( iii ) follows from LaTeXMLRef and the corresponding local result .
Finally , to establish ( iv ) suppose that LaTeXMLMath .
By the same reasoning as above LaTeXMLMath for each LaTeXMLMath .
Thus again by the respective local result LaTeXMLMath for each LaTeXMLMath , i.e .
LaTeXMLMath .
LaTeXMLMath Corollary .
LaTeXMLMath is a linear embedding that commutes with Lie derivatives and coincides with LaTeXMLMath on LaTeXMLMath .
Thus LaTeXMLMath renders LaTeXMLMath a linear subspace and LaTeXMLMath a faithful subalgebra of LaTeXMLMath .
LaTeXMLRef completes the construction of an intrinsic Colombeau algebra on LaTeXMLMath preserving all the distinguishing properties of the local algebra LaTeXMLMath .
In this paper we study the integral representation in the space LaTeXMLMath of special functions with bounded deformation of some LaTeXMLMath -norm lower semicontinuous functionals invariant with respect to rigid motions .
Keywords : functions with bounded deformation , integral representation , homogenization , symmetric quasiconvexity 1991 Mathematics Subject Classification : 35J50 , 49J45 , 49Q20 , 73E99 .
2000 Mathematics Subject Classification : 35J50 , 49J45 , 49Q20 , 74C15 , 74G65 .
F. Ebobisse ] ebobisse @ sissa.it R. Toader ] toader @ sissa.it Several phenomena in phase transition , fracture mechanics , liquid crystals , can be modelled as energy minimization problems where the natural energy has both volume and surface terms .
In many cases the energy functional is obtained as a limit of approximating functionals and some of its properties can be deduced from the approximation process .
A basic step is then to obtain , starting from these properties , an integral representation of the energy .
We consider here this problem for local functionals LaTeXMLMath defined on the space LaTeXMLMath of functions with bounded deformation , which are lower semicontinuous with respect to the LaTeXMLMath -topology , satisfy linear growth and coercivity conditions , as set functions are ( restrictions of ) Radon measures , and are invariant with respect to rigid motions .
In order to identify the volume and the surface densities we follow the global method for relaxation introduced by Bouchitté , Fonseca and Mascarenhas in LaTeXMLCite for functionals defined on the space LaTeXMLMath of functions with bounded variation , which is characterized by the identification of both bulk and surface densities from a local Dirichlet problem .
This kind of approach has already been used in some other contexts , as , for instance , homogenization , where the homogenized density is obtained from a Dirichlet problem in the cell .
An example of functional in the class we consider is given by the relaxed functional LaTeXMLMath of the bulk energy LaTeXMLEquation with respect to the LaTeXMLMath -norm topology , where LaTeXMLMath is a bounded open subset of LaTeXMLMath and LaTeXMLMath is a Borel function satisfying standard linear growth assumptions .
The integral representation of LaTeXMLMath on LaTeXMLMath was studied by Barroso , Fonseca and Toader in LaTeXMLCite , where the global method was applied in order to derive the surface density , while the volume density was obtained by a direct proof using the explicit form of the functional LaTeXMLMath .
In this paper , the bulk density is deduced from the global method and the approximate differentiability of LaTeXMLMath functions proved by Ambrosio , Coscia and Dal Maso in LaTeXMLCite , while the surface density is obtained exactly as in LaTeXMLCite .
Note that both our result and the one in LaTeXMLCite are valid for functions in LaTeXMLMath , i.e .
integrable functions LaTeXMLMath for which the Cantor part LaTeXMLMath of the measure LaTeXMLMath vanishes .
An integral representation in all the space LaTeXMLMath would require more information on LaTeXMLMath , since the only property that LaTeXMLMath vanishes on LaTeXMLMath - LaTeXMLMath finite Borel subsets , proved in LaTeXMLCite , is not sufficient .
We recall in Section 2 some useful properties of LaTeXMLMath functions .
In Section 3 we prove the integral representation theorem ( Theorem LaTeXMLRef ) and give an example showing why we assume the invariance with respect to rigid motions .
In the last section we apply Theorem LaTeXMLRef to obtain the integral representation in LaTeXMLMath of some LaTeXMLMath -limits arising in the homogenization of multi-dimensional structures recently studied in the context of linear elasticity and perfect plasticity by Ansini and Ebobisse in LaTeXMLCite , following the measure-theoretic approach introduced by Ansini , Braides and Chiadò Piat in LaTeXMLCite .
Let LaTeXMLMath be an integer .
We denote by LaTeXMLMath the space of LaTeXMLMath matrices and by LaTeXMLMath the subspace of symmetric matrices in LaTeXMLMath .
For any LaTeXMLMath , LaTeXMLMath is the transposition of LaTeXMLMath .
Given LaTeXMLMath , LaTeXMLMath and LaTeXMLMath denote the tensor and symmetric products of LaTeXMLMath and LaTeXMLMath , respectively .
We use the standard notation , LaTeXMLMath and LaTeXMLMath to denote respectively the Lebesgue and LaTeXMLMath -dimensional Hausdorff measures .
Let LaTeXMLMath be a bounded open subset of LaTeXMLMath ; we denote by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath the family of Borel , open and open subsets of LaTeXMLMath with Lipschitz boundary , respectively .
For any LaTeXMLMath and LaTeXMLMath , we denote by LaTeXMLMath the open ball of LaTeXMLMath centered at LaTeXMLMath with radius LaTeXMLMath , by LaTeXMLMath the cube of centre LaTeXMLMath and sidelength LaTeXMLMath , while LaTeXMLMath is the cube with two its faces perpendicular to the unit vector LaTeXMLMath .
When LaTeXMLMath and LaTeXMLMath we simply write LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is a Radon measure , we denote by LaTeXMLMath its total variation .
A function LaTeXMLMath is with bounded deformation in LaTeXMLMath if LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is the distributional gradient of LaTeXMLMath and LaTeXMLMath is the space of LaTeXMLMath -valued Radon measures with finite total variation in LaTeXMLMath .
The space LaTeXMLMath of functions with bounded deformation in LaTeXMLMath , introduced in LaTeXMLCite , has been widely studied , for instance by Anzellotti-Giaquinta LaTeXMLCite , Kohn LaTeXMLCite , Suquet LaTeXMLCite , and Temam LaTeXMLCite .
It is a Banach space when equipped with the norm LaTeXMLEquation .
It is sometimes convenient to consider also the distance between two functions LaTeXMLMath , LaTeXMLMath given by LaTeXMLEquation .
The topology induced by this distance in LaTeXMLMath is called intermediate topology .
We denote by LaTeXMLMath the convergence with respect to this topology .
It is well known ( see Temam LaTeXMLCite ) that the trace operator LaTeXMLMath is continuous when LaTeXMLMath is equipped with the intermediate topology .
Whenever the open set LaTeXMLMath is assumed to be connected , the kernel of the operator LaTeXMLMath is the class of rigid motions denoted here by LaTeXMLMath , and composed of affine maps of the form LaTeXMLMath , where LaTeXMLMath is a skew-symmetric LaTeXMLMath matrix and LaTeXMLMath .
Therefore LaTeXMLMath is closed and finite-dimensional .
Fine properties of LaTeXMLMath functions were studied , for instance , in LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
We recall that if LaTeXMLMath , then the jump set LaTeXMLMath of LaTeXMLMath is a countably LaTeXMLMath -rectifiable Borel set and the following decomposition of the measure LaTeXMLMath holds LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are the one-sided Lebesgue limits of LaTeXMLMath with respect to the measure theoretic normal LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is the density of the absolutely continuous part of LaTeXMLMath with respect to LaTeXMLMath , LaTeXMLMath is the singular part , and LaTeXMLMath is the Cantor part and vanishes on the Borel sets that are LaTeXMLMath -finite with respect to LaTeXMLMath ( see LaTeXMLCite ) .
Moreover , the following theorem on the approximate differentiability of LaTeXMLMath functions was proved in LaTeXMLCite .
Let LaTeXMLMath be a bounded open set in LaTeXMLMath with Lipschitz boundary .
Let LaTeXMLMath .
Then for LaTeXMLMath almost every LaTeXMLMath there exists an LaTeXMLMath matrix LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation for LaTeXMLMath almost every LaTeXMLMath .
In particular , by ( LaTeXMLRef ) LaTeXMLMath is approximately differentiable LaTeXMLMath almost everywhere in LaTeXMLMath and the function LaTeXMLMath satisfies the weak LaTeXMLMath estimate LaTeXMLEquation where LaTeXMLMath is a positive constant depending only on LaTeXMLMath and LaTeXMLMath .
From ( LaTeXMLRef ) and ( LaTeXMLRef ) one can easily see that LaTeXMLEquation .
Analogously to the space LaTeXMLMath introduced by De Giorgi and Ambrosio in LaTeXMLCite , the space LaTeXMLMath was defined in LaTeXMLCite .
The space LaTeXMLMath of special functions with bounded deformation , is the space of functions LaTeXMLMath such that the measure LaTeXMLMath in ( LaTeXMLRef ) is zero .
Let LaTeXMLMath be a functional satisfying the properties mentioned in the introduction , more precisely , LaTeXMLMath is LaTeXMLMath lower semicontinuous ; for every LaTeXMLMath , LaTeXMLEquation .
LaTeXMLMath is the restriction to LaTeXMLMath of a Radon measure ; LaTeXMLMath for every LaTeXMLMath and every rigid motion LaTeXMLMath .
Since the properties ( 2 ) and ( 3 ) give the absolute continuity of LaTeXMLMath with respect to the measure LaTeXMLMath , in order to obtain the integral representation of LaTeXMLMath , we need only to identify the volume and the surface densities whenever LaTeXMLMath .
To do this we define , as in LaTeXMLCite , see also LaTeXMLCite , for every LaTeXMLMath and every LaTeXMLMath LaTeXMLEquation .
The basic idea of the global method in LaTeXMLCite consists in comparing the asymptotic behaviours of LaTeXMLMath and LaTeXMLMath with respect to LaTeXMLMath as LaTeXMLMath , and to show via a blow-up argument that , the volume and surface densities are obtained from a local Dirichlet problem ( see Lemma LaTeXMLRef ) .
We shall use the following lemmas , similar to Lemmas 3.1 and 3.5 in LaTeXMLCite for LaTeXMLMath -functions , proved in the case of LaTeXMLMath -functions in LaTeXMLCite .
( LaTeXMLCite ) There exists a positive constant LaTeXMLMath such that for any LaTeXMLMath , LaTeXMLMath and any LaTeXMLMath we have LaTeXMLEquation ( LaTeXMLCite ) If LaTeXMLMath satisfies conditions ( 1 ) - ( 3 ) then LaTeXMLEquation .
We prove now the integral representation result .
Let LaTeXMLMath be a functional satisfying properties ( 1 ) - ( 4 ) .
Then for every LaTeXMLMath and LaTeXMLMath we have LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and where LaTeXMLEquation .
We use the same notation for LaTeXMLMath and its extension to the Borel subsets of LaTeXMLMath .
( i ) The volume part .
Let LaTeXMLMath and choose LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Let , for every LaTeXMLMath , LaTeXMLEquation .
By ( LaTeXMLRef ) the functions LaTeXMLMath converge to LaTeXMLMath in LaTeXMLMath .
Moreover , LaTeXMLEquation .
Indeed , by definition LaTeXMLEquation where LaTeXMLMath .
Then from ( LaTeXMLRef ) we get LaTeXMLMath , where we used also the formula ( LaTeXMLRef ) .
This shows that LaTeXMLMath in LaTeXMLMath .
On the other hand from the continuity of the trace with respect to the intermediate topology it follows that LaTeXMLEquation .
LaTeXMLEquation Then by ( LaTeXMLRef ) , Lemmas LaTeXMLRef and LaTeXMLRef we obtain LaTeXMLEquation .
Now condition ( 4 ) with LaTeXMLMath implies that LaTeXMLEquation ( ii ) The surface part .
As in LaTeXMLCite , it can be proved that LaTeXMLEquation where LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and where LaTeXMLEquation .
Using again condition ( 4 ) we obtain LaTeXMLEquation hence ( LaTeXMLRef ) , concluding thus the proof .
∎ As a particular case the result in LaTeXMLCite is recovered , i.e .
if LaTeXMLMath is the localized lower semicontinuous envelope of the functional LaTeXMLMath given by ( LaTeXMLRef ) , then LaTeXMLEquation for every LaTeXMLMath and every LaTeXMLMath , where LaTeXMLMath is the symmetric quasiconvex envelope of LaTeXMLMath introduced by Ebobisse in LaTeXMLCite , and characterized by LaTeXMLEquation for every LaTeXMLMath and for every bounded open subset LaTeXMLMath of LaTeXMLMath , and LaTeXMLMath is the recession function of LaTeXMLMath .
Note that hypothesis ( 4 ) is not a consequence of hypotheses ( 1 ) - ( 3 ) .
In fact , without condition ( 4 ) of Theorem LaTeXMLRef we would obtain that LaTeXMLEquation for every LaTeXMLMath .
In particular , for every LaTeXMLMath , LaTeXMLEquation which , under some continuity assumption on LaTeXMLMath with respect to LaTeXMLMath , for instance , assuming that there exists a modulus of continuity LaTeXMLMath satisfying LaTeXMLEquation for all LaTeXMLMath , such that LaTeXMLMath , implies that for LaTeXMLMath -almost every LaTeXMLMath and every LaTeXMLMath , the function LaTeXMLMath is quasiconvex .
By ( 2 ) , LaTeXMLEquation .
The following example shows that there exists a rank-one convex function LaTeXMLMath which satisfies LaTeXMLEquation and which depends also on the antisymmetric part of the matrix LaTeXMLMath .
Let LaTeXMLMath .
It is enough to define such a function on the matrix LaTeXMLMath and then to add the quantity LaTeXMLMath .
Since LaTeXMLMath we look for a function LaTeXMLMath which is separately convex , satisfies the linear growth condition LaTeXMLEquation and which depends also on LaTeXMLMath .
An example of such a function is the following : LaTeXMLEquation .
Therefore , the function LaTeXMLMath given by LaTeXMLEquation is rank-one convex , has the linear growth ( LaTeXMLRef ) , and does not depend only on the symmetric part of the matrix LaTeXMLMath .
In LaTeXMLCite , the authors studied the asymptotic behaviour of functionals of the form LaTeXMLEquation defined on a particular class of functions with bounded deformation , denoted by LaTeXMLMath , and given by the functions LaTeXMLMath whose deformation tensor LaTeXMLMath is an absolutely continuous measure with respect to LaTeXMLMath , with LaTeXMLMath -summable density LaTeXMLMath , where LaTeXMLMath is defined by LaTeXMLMath , with LaTeXMLMath a fixed LaTeXMLMath -periodic Radon measure and LaTeXMLMath is a Borel function LaTeXMLMath -periodic in the first variable .
Assuming the standard LaTeXMLMath -growth condition on LaTeXMLMath and that the measure LaTeXMLMath is ’ LaTeXMLMath -homogenizable ’ ( see LaTeXMLCite ) , the authors proved a homogenization theorem ( Theorem 5.1 ) .
Precisely , they proved the existence of the LaTeXMLMath -limit LaTeXMLMath of the functionals LaTeXMLMath with respect to LaTeXMLMath -convergence in the Sobolev space LaTeXMLMath , and with respect to LaTeXMLMath -convergence in LaTeXMLMath .
They showed that the LaTeXMLMath -limit admits the integral representation LaTeXMLEquation in LaTeXMLMath ; moreover , if LaTeXMLMath is convex and LaTeXMLMath then LaTeXMLEquation in LaTeXMLMath , where LaTeXMLMath is described by an asymptotic formula .
However , in the second case , the question about the integral representation of the LaTeXMLMath -limit without the convexity assumption on LaTeXMLMath remained open .
Notice that such an assumption is too strong in the vectorial calculus of variations .
As shown in LaTeXMLCite , ( see the proof of Theorem 5.1 ) , the LaTeXMLMath -limit verifies the properties ( 1 ) - ( 3 ) and the invariance with respect to rigid motions follows from the fact that the approximating functionals LaTeXMLMath have this property .
So we can apply Theorem LaTeXMLRef to obtain that LaTeXMLEquation for every LaTeXMLMath and every LaTeXMLMath .
Now , from the integral representation ( LaTeXMLRef ) in LaTeXMLMath and the relaxation theorem 3.5 in LaTeXMLCite , one can easily see that LaTeXMLEquation for every LaTeXMLMath , and for every LaTeXMLMath Notice that , since LaTeXMLMath is symmetric quasiconvex , that is LaTeXMLEquation for every LaTeXMLMath , LaTeXMLMath and for every bounded open subset LaTeXMLMath of LaTeXMLMath , then LaTeXMLMath is well defined .
Acknowledgements .
The authors wish to thank G. Dal Maso for many useful discussions concerning the subject of this paper , in particular for suggesting the example in Remark LaTeXMLRef .
The work of Rodica Toader is part of the European Research Training Network “ Homogenization and Multiple Scales ” under contract HPRN-2000-00109 .
Semigroups in finite von Neumann algebras Gilles Cassier Dedicated to the memory of Professor Béla Szőkefalvi-Nagy Abstract .
Let LaTeXMLMath be a finite von Neumann algebra .
In the first part , we give asymptotic results about LaTeXMLMath -stable sequences of weak*-continuous mappings which are related with operators belonging to LaTeXMLMath .
In the second part , we extend , by a shorter way , similarity results given in [ CaFa2 ] to unbounded semigroups of operators contained in a finite von Neumann algebra .
I .
Introduction and preliminaries Let LaTeXMLMath be a separable complex Hilbert space and let LaTeXMLMath be the algebra of bounded linear operators acting on LaTeXMLMath .
The ultra-weak topology of LaTeXMLMath is the weak* topology ( in the sequel we will shorten weak* to w* ) that comes from the well known duality LaTeXMLMath , where LaTeXMLMath is the Banach space of trace class operators on LaTeXMLMath endowed with the trace norm ( see [ Dix ] ) .
A von Neumann algebra acting on LaTeXMLMath is by definition an ultra-weakly closed *-subalgebra of LaTeXMLMath .
Such a von Neumann algebra LaTeXMLMath is finite if it admits a faithful normal trace LaTeXMLMath , which means that LaTeXMLMath is an ultra-weakly continuous linear functional on LaTeXMLMath satisfying : 1 ) LaTeXMLMath for any LaTeXMLMath ; 2 ) for any positive element LaTeXMLMath in LaTeXMLMath , we have LaTeXMLMath and LaTeXMLMath .
We denote by LaTeXMLMath the set of all faithful normal traces acting on LaTeXMLMath .
A good example of a finite von Neumann algebra is the w*-algebra generated by the left regular representation of a countable discrete group .
We will denote by LaTeXMLMath the predual of LaTeXMLMath .
For any subset LaTeXMLMath of LaTeXMLMath , we shall denote by LaTeXMLMath the family of operators commuting with every element of LaTeXMLMath .
Let LaTeXMLMath denote the algebra of bounded linear operators acting on LaTeXMLMath , and let LaTeXMLMath stand for the algebra of operators LaTeXMLMath wich are weak*-continuous .
Recall that LaTeXMLMath if and only if LaTeXMLMath is the adjoint of a bounded linear operator acting on the Banach space LaTeXMLMath ( see for instance [ BCP ] ) .
For any LaTeXMLMath , let LaTeXMLMath denote the uniquely determined operator whose ( Banach space ) adjoint is LaTeXMLMath , that is LaTeXMLMath .
For more details on von Neumann algebras , we refer the reader to [ Dix ] and [ Sak ] .
As usual LaTeXMLMath denotes the LaTeXMLMath matrix which acts on the orthogonal sum of LaTeXMLMath copies of LaTeXMLMath ; its entries are operators acting on LaTeXMLMath .
We remind the reader that LaTeXMLMath inherits a unique structure of von Neumann algebra .
Let LaTeXMLMath be a linear mapping from LaTeXMLMath into itself , we define LaTeXMLMath by LaTeXMLMath .
We call LaTeXMLMath LaTeXMLMath -positive if LaTeXMLMath is positive ( that is positive operators are transformed into positive ones ) and we call LaTeXMLMath completely positive if LaTeXMLMath is LaTeXMLMath -positive for all LaTeXMLMath .
We proved in [ CaFa2 ] that a power bounded operator LaTeXMLMath in a finite von Neumann algebra LaTeXMLMath is similar to a unitary element of LaTeXMLMath if and only if LaTeXMLMath for any LaTeXMLMath ( LaTeXMLMath is said to be of class LaTeXMLMath in the Sz.-Nagy–Foias terminology ) .
We will extend this result into two directions .
On the one hand , we will consider general semigroups .
On the other hand , we will work with operators which are not necessarily power bounded .
To achieve this , we have to find a proper framework , which will allow short and well adapted methods .
In similarity problems , the idea of using limits in the sense of Banach comes from B. Sz.-Nagy [ Nag ] .
In the sequel , we frequently use this idea .
Recall that a Banach limit is a state , that is a linear functional LaTeXMLMath with LaTeXMLMath , acting on the classical space LaTeXMLMath of all complex bounded sequences and satisfying LaTeXMLMath .
A bounded sequence LaTeXMLMath is said to be almost convergent to a complex number LaTeXMLMath if LaTeXMLEquation .
Lorentz proved in [ Lor ] that LaTeXMLMath is almost convergent to LaTeXMLMath if and only if for every Banach limit LaTeXMLMath we have LaTeXMLMath .
A sequence LaTeXMLMath is said to be strongly almost convergent to LaTeXMLMath if the sequence LaTeXMLMath is almost convergent to LaTeXMLMath .
We will say that a sequence LaTeXMLMath of operators in LaTeXMLMath is weakly almost convergent to LaTeXMLMath if LaTeXMLMath almost converges to LaTeXMLMath for any LaTeXMLMath .
Definition 1.1 .
A mapping LaTeXMLMath is called a gauge if there exists LaTeXMLMath such that the sequence LaTeXMLMath is strongly almost convergent to LaTeXMLMath .
Moreover , if in addition the sequence LaTeXMLMath strongly almost converges to LaTeXMLMath , then we say that LaTeXMLMath is a regular gauge .
We will say that a sequence LaTeXMLMath of operators , acting on a Banach space , is dominated by a gauge LaTeXMLMath if LaTeXMLMath holds for every positive integer LaTeXMLMath .
We follow [ Ker ] in saying that LaTeXMLMath is compatible with a gauge LaTeXMLMath if in addition the sequence LaTeXMLMath does not almost converge to LaTeXMLMath .
An operator LaTeXMLMath is dominated by ( compatible with ) LaTeXMLMath if the sequence LaTeXMLMath is dominated by ( resp .
compatible with ) LaTeXMLMath .
Finally , a family LaTeXMLMath of operators is called dominated by ( compatible with ) LaTeXMLMath if each operator in LaTeXMLMath is dominated by ( resp .
compatible with ) LaTeXMLMath .
For some recent contributions in this area , we refer the reader to [ Ker ] , [ Ker1 ] , [ Ker2 ] , [ Ker3 ] , [ Ker4 ] , [ Ker5 ] and [ KeMü ] .
Assume that LaTeXMLMath is a gauge and LaTeXMLMath is dominated by LaTeXMLMath .
Given a Banach limit LaTeXMLMath , let us introduce the ( bounded , linear ) operator LaTeXMLMath , acting on LaTeXMLMath , by setting LaTeXMLEquation for any LaTeXMLMath .
The following proposition summarizes some useful properties of the operator LaTeXMLMath .
Proposition 1.2 .
Let LaTeXMLMath be an element in a von Neumann algebra LaTeXMLMath acting on a separable Hilbert space LaTeXMLMath .
Assume that LaTeXMLMath is dominated by a gauge LaTeXMLMath .
Then , for any Banach limit LaTeXMLMath , we have ( i ) LaTeXMLMath is a completely positive mapping ; ( ii ) LaTeXMLMath for any LaTeXMLMath ; ( iii ) if LaTeXMLMath commute with LaTeXMLMath , then we have LaTeXMLMath for any LaTeXMLMath ; ( iv ) LaTeXMLMath for any LaTeXMLMath ; ( v ) there exists LaTeXMLMath such that LaTeXMLMath ; ( vi ) moreover , if LaTeXMLMath is a finite von Neumann algebra , then the mapping LaTeXMLMath belongs to LaTeXMLMath .
Remark 1.3 .
If LaTeXMLMath is compatible with a gauge LaTeXMLMath , then the spectral radius LaTeXMLMath satisfies LaTeXMLMath ( see [ Ker ) ] ) .
Proof .
( i ) Let LaTeXMLMath be a positive LaTeXMLMath matrix whose entries are operators in LaTeXMLMath and let LaTeXMLMath be vectors in LaTeXMLMath .
For any LaTeXMLMath , we define the linear functional LaTeXMLMath acting on LaTeXMLMath by setting LaTeXMLMath .
It is obvious that LaTeXMLMath , hence LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The positivity of the last term follows from the positivity of the matrix LaTeXMLMath and the positivity of the state LaTeXMLMath .
( ii ) Given any LaTeXMLMath in LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath is a gauge , we see that the sequence LaTeXMLMath is almost convergent to LaTeXMLMath .
It follows that LaTeXMLMath also almost converges to LaTeXMLMath .
By Lemma 1 from [ Ker ] , we get LaTeXMLEquation .
LaTeXMLEquation and ( ii ) follows .
( iii ) Let LaTeXMLMath , LaTeXMLMath be two operators in LaTeXMLMath commuting with LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation This establishes the formula .
( iv ) follows immediately from ( ii ) and ( iii ) .
( v ) Let LaTeXMLMath ; using ( iv ) , we get LaTeXMLEquation .
LaTeXMLEquation by setting LaTeXMLMath .
From the formula LaTeXMLMath ( see [ Ker ] Proposition 1 ) , we immediately deduce that LaTeXMLMath .
( vi ) It suffices to show that the linear functional LaTeXMLMath is ultra-weakly continuous for any LaTeXMLMath .
Let LaTeXMLMath be in LaTeXMLMath , for clarity we will denote by LaTeXMLMath the element in LaTeXMLMath given by LaTeXMLMath for any LaTeXMLMath .
Given LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation hence LaTeXMLEquation .
Let LaTeXMLMath be in LaTeXMLMath , we deduce from the last equation that LaTeXMLMath is ultra-weakly continuous .
Since the linear functionals LaTeXMLMath with LaTeXMLMath are dense in LaTeXMLMath , it follows that LaTeXMLMath is ultra-weakly continuous .
This completes the proof .
Q.E.D .
II .
Convergence of LaTeXMLMath - LaTeXMLMath -stable maps Given any LaTeXMLMath , the linear functional LaTeXMLMath ( LaTeXMLMath ) is weak*-continuous , and so LaTeXMLMath .
The mapping LaTeXMLEquation is a bounded linear quasiaffinity ; the linear manifold LaTeXMLMath is dense in LaTeXMLMath .
Let us consider the set LaTeXMLEquation of LaTeXMLMath - LaTeXMLMath -stable weak*-continuous operators .
For any LaTeXMLMath , we can introduce the linear mapping LaTeXMLEquation .
For any LaTeXMLMath we have LaTeXMLEquation .
An application of the Closed Graph Theorem yields that LaTeXMLMath is bounded .
In fact , we see that LaTeXMLMath is also in LaTeXMLMath and we have LaTeXMLMath .
We say that LaTeXMLMath is LaTeXMLMath -stable if it is LaTeXMLMath - LaTeXMLMath -stable for every LaTeXMLMath .
We will consider the set LaTeXMLEquation of all LaTeXMLMath -stable operators .
Remarks 2.1 .
1 .
Denote by LaTeXMLMath the algebra of square matrices of order LaTeXMLMath , and consider the finite von Neumann algebra LaTeXMLMath acting on the Hilbert space LaTeXMLMath in an obvious sense .
We consider the faithful normal traces LaTeXMLMath and LaTeXMLMath defined by setting LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the usual trace acting on LaTeXMLMath and LaTeXMLMath is given by LaTeXMLEquation .
Let us consider the mapping LaTeXMLMath defined by LaTeXMLEquation where LaTeXMLMath is given in an obvious sense by LaTeXMLEquation .
Then , we can check that LaTeXMLMath but LaTeXMLMath .
2 .
If LaTeXMLMath is a factor ( LaTeXMLMath ) , then all faithful normal traces are proportional ( see [ Dix , p. 249 ] ) .
Consequently we have LaTeXMLMath for every LaTeXMLMath .
3 .
Let LaTeXMLMath be a finite von Neumann algebra and LaTeXMLMath , then the mapping LaTeXMLMath belongs to LaTeXMLMath .
Let LaTeXMLMath be a finite von Neumann algebra .
Recall that LaTeXMLMath is also a finite von Neumann algebra with the faithful normal trace LaTeXMLMath defined by setting LaTeXMLEquation .
We begin with some useful properties of the operators LaTeXMLMath when LaTeXMLMath is a LaTeXMLMath - LaTeXMLMath -stable mapping .
Proposition 2.2 .
Let LaTeXMLMath be a finite von Neumann algebra , LaTeXMLMath a faithful normal trace on LaTeXMLMath and LaTeXMLMath .
Then ( i ) the mapping LaTeXMLMath belongs to LaTeXMLMath if and only if there exists LaTeXMLMath such that LaTeXMLMath for every LaTeXMLMath ; and then LaTeXMLMath ; ( ii ) the set LaTeXMLMath is an algebra ; the mapping LaTeXMLMath is linear , involutive and LaTeXMLMath ; ( iii ) if LaTeXMLMath , then LaTeXMLMath and we have LaTeXMLMath ; ( iv ) if LaTeXMLMath is LaTeXMLMath -positive LaTeXMLMath , then LaTeXMLMath is also LaTeXMLMath -positive ; ( v ) if LaTeXMLMath is completely positive , then LaTeXMLMath is completely positive ; ( vi ) assume that LaTeXMLMath is LaTeXMLMath -positive , then the mappings LaTeXMLMath and LaTeXMLMath extend uniquely to bounded operators from LaTeXMLMath into itself ; moreover , we have LaTeXMLEquation .
Proof .
( i ) If LaTeXMLMath , it suffices to set LaTeXMLMath .
Conversely , assume that there exists LaTeXMLMath such that LaTeXMLMath for every LaTeXMLMath .
We immediately deduce that the linear functional LaTeXMLMath is ultra-weakly continuous for each LaTeXMLMath .
Since LaTeXMLMath is dense in LaTeXMLMath , we see that LaTeXMLMath .
Moreover , we have LaTeXMLEquation for any LaTeXMLMath , thus we have LaTeXMLMath .
This gives LaTeXMLMath .
( ii ) This statement follows clearly from the characterization of elements of LaTeXMLMath given in ( i ) .
( iii ) Assume that LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be two elements in LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It follows easily by ( i ) that LaTeXMLMath and we have LaTeXMLMath .
( iv ) Assume that LaTeXMLMath is positive .
Let LaTeXMLMath and LaTeXMLMath be two positive elements in LaTeXMLMath , we have LaTeXMLEquation .
Hence , we derive easily the positivity of LaTeXMLMath from the previous calculation .
If LaTeXMLMath is LaTeXMLMath -positive , the map LaTeXMLMath is positive , thus LaTeXMLMath is positive and the formula LaTeXMLMath implies that LaTeXMLMath is LaTeXMLMath -positive .
( v ) It is clear from ( iv ) that LaTeXMLMath is completely positive if LaTeXMLMath is completely positive .
( vi ) Let LaTeXMLMath be LaTeXMLMath -positive and LaTeXMLMath , then the matrix LaTeXMLEquation is positive , a fact which implies that LaTeXMLEquation .
Given a pair LaTeXMLMath of elements of LaTeXMLMath , we deduce from the previous inequality that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation It follows that LaTeXMLMath .
Using the density of LaTeXMLMath in LaTeXMLMath , we see that the map LaTeXMLMath extends uniquely to a bounded operator from LaTeXMLMath into itself .
We also get LaTeXMLEquation .
Observe that the adjoint of LaTeXMLMath in LaTeXMLMath is given by LaTeXMLMath for any LaTeXMLMath .
The rest of the proof follows immediately .
Q.E.D .
Remark 2.3 .
It follows immediately from ( ii ) that LaTeXMLMath is also an algebra .
Let LaTeXMLMath be a sequence in LaTeXMLMath dominated by a gauge LaTeXMLMath .
Given a Banach limit LaTeXMLMath , let us consider the limit operator LaTeXMLMath , defined by LaTeXMLEquation for any LaTeXMLMath .
Note that the previous formulas actually define LaTeXMLMath as an element of LaTeXMLMath .
We write LaTeXMLMath .
The following theorem seems to be of independent interest .
It presents some fine properties of abelian sequences included in LaTeXMLMath which are compatible with a gauge LaTeXMLMath , where LaTeXMLMath is a finite von Neumann algebra and LaTeXMLMath is a faithful normal trace on LaTeXMLMath .
Theorem 2.4 .
Let LaTeXMLMath be a finite von Neumann algebra , LaTeXMLMath a faithful normal trace on LaTeXMLMath and LaTeXMLMath a sequence in LaTeXMLMath dominated by a gauge LaTeXMLMath and such that LaTeXMLMath is also dominated by LaTeXMLMath .
( i ) The operator LaTeXMLMath belongs to LaTeXMLMath and we have LaTeXMLMath for any Banach limit LaTeXMLMath .
( ii ) Suppose LaTeXMLMath is abelian , then the operators LaTeXMLMath and LaTeXMLMath commute for any pair LaTeXMLMath of Banach limits .
( iii ) Assume that LaTeXMLMath and that the sequences LaTeXMLMath and LaTeXMLMath are dominated by the gauge LaTeXMLMath .
Then we have LaTeXMLEquation for any Banach limits LaTeXMLMath and LaTeXMLMath .
In particular , if LaTeXMLMath almost converges to a nonzero limit , then LaTeXMLMath weakly almost converges to an operator LaTeXMLMath belonging to LaTeXMLMath .
Proof .
( i ) Given a pair LaTeXMLMath of elements of LaTeXMLMath , we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for every Banach limit LaTeXMLMath .
Now Proposition 2.2 .
( i ) implies the statement .
( ii ) Assume that the sequence LaTeXMLMath is abelian .
We have LaTeXMLEquation for any pair LaTeXMLMath of positive integers and any pair LaTeXMLMath .
By taking LaTeXMLMath -limit with respect to LaTeXMLMath we get LaTeXMLEquation .
LaTeXMLEquation Now , taking LaTeXMLMath -limit with respect to LaTeXMLMath and using ( i ) , we obtain that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We thus have LaTeXMLMath .
( iii ) Let LaTeXMLMath and LaTeXMLMath be a pair of Banach limits .
For any LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation The sequence LaTeXMLMath is strongly almost convergent to LaTeXMLMath , when LaTeXMLMath goes to infinity .
By taking LaTeXMLMath -limit with respect to LaTeXMLMath , we get LaTeXMLEquation .
LaTeXMLEquation After taking LaTeXMLMath -limit with respect to LaTeXMLMath , it follows that LaTeXMLEquation .
LaTeXMLEquation whence LaTeXMLMath .
We deduce from ( ii ) that LaTeXMLMath and LaTeXMLMath commute .
Interchanging the role of LaTeXMLMath and LaTeXMLMath we conclude that LaTeXMLEquation .
When the sequence LaTeXMLMath almost converges to a nonzero number , then we deduce that the limit of LaTeXMLMath is independent of LaTeXMLMath .
Applying Lorentz ’ s result , we obtain that LaTeXMLMath is weakly almost convergent to an operator in LaTeXMLMath .
The proof is now complete .
Q.E.D .
Remarks 2.5 .
1 .
If LaTeXMLMath is almost convergent to a nonzero limit , then we see that the Cesaro means LaTeXMLMath weakly converge to an operator in LaTeXMLMath ( which is obviously ultra-weakly continuous ) .
2 .
The assumption that LaTeXMLMath is a sequence in LaTeXMLMath dominated by a gauge LaTeXMLMath does not imply that LaTeXMLMath is also dominated by LaTeXMLMath .
Denote by LaTeXMLMath the algebra of square matrices of order LaTeXMLMath , and consider the finite von Neumann algebra LaTeXMLMath acting on the Hilbert space LaTeXMLMath in an obvious sense .
We consider the faithful normal trace LaTeXMLMath defined by LaTeXMLEquation where LaTeXMLMath is the usual trace on LaTeXMLMath .
Fix a unit vector LaTeXMLMath in LaTeXMLMath and write LaTeXMLMath for the orthogonal projection onto LaTeXMLMath .
For any LaTeXMLMath , set LaTeXMLMath .
We can easily see that LaTeXMLMath is a sequence in LaTeXMLMath such that LaTeXMLMath for every LaTeXMLMath .
Hence LaTeXMLMath is dominated by the constant gauge LaTeXMLMath equal to LaTeXMLMath , but LaTeXMLMath is not dominated by LaTeXMLMath , actually LaTeXMLMath .
Let LaTeXMLMath be an abelian set included in LaTeXMLMath .
We consider the ( abelian ) semigroup LaTeXMLMath induced by LaTeXMLMath , that is LaTeXMLEquation .
We define a partial ordering on LaTeXMLMath by setting LaTeXMLMath if there exists LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath .
( It is clear that LaTeXMLMath is a directed set with this partial ordering , and it can be considered as a net ( generalized sequence ) indexed by itself . )
Proposition 2.6 .
Let LaTeXMLMath be a finite von Neumann algebra , LaTeXMLMath a faithful normal trace on LaTeXMLMath , and let LaTeXMLMath be an abelian set of LaTeXMLMath -positive projections belonging to LaTeXMLMath ( LaTeXMLMath ) .
Assume that LaTeXMLMath and LaTeXMLMath are bounded in LaTeXMLMath .
Then the net LaTeXMLMath weakly converges to an LaTeXMLMath -positive projection LaTeXMLMath .
Proof .
Let us introduce the classical Hilbert space LaTeXMLMath equipped with the inner product LaTeXMLMath .
Since every LaTeXMLMath is LaTeXMLMath -positive , Proposition 2.2 .
( vi ) shows that LaTeXMLMath extends uniquely to a bounded projection ( still denoted by LaTeXMLMath ) from LaTeXMLMath into itself .
Using again Proposition 2.2 .
( vi ) , we see that the set LaTeXMLMath is bounded in LaTeXMLMath , thus it is weakly relatively compact .
Choose two cofinal subnets LaTeXMLMath and LaTeXMLMath in LaTeXMLMath , which converge respectively to LaTeXMLMath and LaTeXMLMath in the weak operator topology of LaTeXMLMath .
Fix LaTeXMLMath and consider the set LaTeXMLMath .
Then we have LaTeXMLEquation .
LaTeXMLEquation for any LaTeXMLMath and any pair LaTeXMLMath .
Thus , taking limit with respect to the set LaTeXMLMath , we obtain LaTeXMLEquation .
Now , taking limit with respect to the directed set LaTeXMLMath , we get LaTeXMLEquation whence LaTeXMLMath follows .
Interchanging the role of LaTeXMLMath and LaTeXMLMath , we see that LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are limit points of elements belonging to the commutative set LaTeXMLMath , they commute .
Hence LaTeXMLMath , in particular LaTeXMLMath .
We deduce that LaTeXMLMath is weakly convergent in LaTeXMLMath to a projection LaTeXMLMath .
Now , we want to show that LaTeXMLMath and LaTeXMLMath .
To this order let LaTeXMLMath be arbitrary , and let us consider the linear functional LaTeXMLMath .
Choosing a cofinal subnet LaTeXMLMath in LaTeXMLMath , we have LaTeXMLMath .
Thus LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath ( see [ Dix ] Section I.6.10 ) .
We deduce that there exists unique LaTeXMLMath such that LaTeXMLEquation holds for every LaTeXMLMath .
Hence LaTeXMLMath and LaTeXMLMath .
It is clear that LaTeXMLEquation is true for every LaTeXMLMath .
Since LaTeXMLMath is dense in LaTeXMLMath , and LaTeXMLMath is bounded , it follows that LaTeXMLMath weakly converges to LaTeXMLMath .
We can prove in the same manner that LaTeXMLMath converges weakly to an operator LaTeXMLMath .
Taking into account that LaTeXMLMath , we obtain by passing to the limit that LaTeXMLEquation holds for every LaTeXMLMath .
It follows by Proposition 2.2 .
( i ) that LaTeXMLMath belongs to LaTeXMLMath .
It remains to prove that LaTeXMLMath is LaTeXMLMath -positive .
It is clear that every operator in LaTeXMLMath is LaTeXMLMath -positive .
Let LaTeXMLMath be a positive operator .
Given any vector LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation and so LaTeXMLMath is an LaTeXMLMath -positive projection .
Q.E.D .
Let LaTeXMLMath be a finite von Neumann algebra and let LaTeXMLMath be an operator in LaTeXMLMath dominated by the regular gauge LaTeXMLMath .
We know by Theorem 2.4 .
( iii ) that the sequence of LaTeXMLMath -stable mappings LaTeXMLMath defined by LaTeXMLMath is weakly almost convergent , we will denote its limit by LaTeXMLMath .
Now Proposition 1.2 shows that LaTeXMLMath is a completely positive projection .
Notice also that LaTeXMLMath is an ultra-weakly continuous LaTeXMLMath -stable operator with LaTeXMLMath and that LaTeXMLMath .
Let LaTeXMLMath be an abelian subset of LaTeXMLMath , which is dominated by the regular gauge LaTeXMLMath .
We consider the abelian semigroup LaTeXMLMath induced by the ( abelian ) set LaTeXMLMath .
Corollary 2.7 .
Let LaTeXMLMath be a finite von Neumann algebra and let LaTeXMLMath be an abelian subset of LaTeXMLMath .
Assume that LaTeXMLMath is dominated by a regular gauge LaTeXMLMath .
Then the net LaTeXMLMath weakly converges to a completely positive LaTeXMLMath -stable projection LaTeXMLMath satisfying the following properties : ( i ) LaTeXMLMath for any LaTeXMLMath and LaTeXMLMath ; ( ii ) LaTeXMLMath for any pair LaTeXMLMath and LaTeXMLMath .
Proof .
We apply Proposition 2.6 to the net LaTeXMLMath .
We deduce that LaTeXMLMath converges weakly to LaTeXMLMath , for any LaTeXMLMath .
Since properties ( i ) and ( ii ) are true for the operators LaTeXMLMath by Proposition 1.2 , we see that the same properties hold for LaTeXMLMath .
Q.E.D .
Proposition 2.8 .
Let LaTeXMLMath be a finite von Neuman algebra , LaTeXMLMath a faithful normal trace on LaTeXMLMath , and let LaTeXMLMath be an abelian subset of LaTeXMLMath which is dominated by a regular gauge LaTeXMLMath .
Let us assume that the limit projection LaTeXMLMath of the net LaTeXMLMath is such that LaTeXMLMath is injective .
Then there exists an abelian set LaTeXMLMath of unitaries belonging to LaTeXMLMath such that for any LaTeXMLMath there exists LaTeXMLMath satisfying LaTeXMLMath .
Moreover , if LaTeXMLMath denotes the limit of the net LaTeXMLMath , then we have the following properties : ( i ) LaTeXMLMath for any LaTeXMLMath ; ( ii ) LaTeXMLMath for any LaTeXMLMath .
Proof .
( i ) First of all , observe that the equation LaTeXMLMath and the injectivity of LaTeXMLMath imply that LaTeXMLMath is also injective .
Taking the polar decomposition of LaTeXMLMath , we see that there exists a unique isometry LaTeXMLMath such that LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath is finite , it follows that LaTeXMLMath is unitary .
The previous intertwining relations readily imply that the set LaTeXMLMath is abelian .
Given LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation for every positive integer LaTeXMLMath .
Taking a Banach limit we get the relation LaTeXMLEquation .
Now taking limits in the nets LaTeXMLMath and LaTeXMLMath we get ( i ) .
( ii ) Since LaTeXMLMath and LaTeXMLMath are LaTeXMLMath - LaTeXMLMath -stable , we can now get ( ii ) by the following computation .
For any LaTeXMLMath we have LaTeXMLEquation .
LaTeXMLEquation Hence , we have LaTeXMLMath for any LaTeXMLMath .
This completes the proof .
Q.E.D .
III .
Similarity We say that an operator LaTeXMLMath is asymptotically controlled by a gauge LaTeXMLMath if LaTeXMLMath is compatible with LaTeXMLMath and satisfies the condition that LaTeXMLMath , for every nonzero vector LaTeXMLMath , where LaTeXMLEquation for any bounded real sequence LaTeXMLMath ( see [ Ker ] for the role of this functional in the study of Banach limits ) .
For any real sequence LaTeXMLMath , let LaTeXMLMath , where LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath , where LaTeXMLMath .
A set LaTeXMLMath of operators , acting on the Hilbert space LaTeXMLMath , is called asymptotically controlled by a gauge LaTeXMLMath , if every operator in LaTeXMLMath is compatible with LaTeXMLMath , and if for every nonzero vector LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLEquation is true for every LaTeXMLMath and LaTeXMLMath .
Remark 3.1 .
Let LaTeXMLMath be an operator compatible with a gauge LaTeXMLMath .
Assume that LaTeXMLMath satisfies LaTeXMLEquation for any nonzero LaTeXMLMath in LaTeXMLMath ; then LaTeXMLMath is asymptotically controlled by LaTeXMLMath .
In particular , power bounded operators of class LaTeXMLMath ( in the Sz.-Nagy–Foias terminology ) are exactly operators which are asymptotically controlled by constant gauges .
Theorem 3.2 .
Let LaTeXMLMath be an abelian set of operators which is contained in a finite von Neumann algebra .
Assume that LaTeXMLMath is asymptotically controlled by a regular gauge LaTeXMLMath .
Then , there exists an invertible operator LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath is a unitary operator for any LaTeXMLMath .
Proof .
Let LaTeXMLMath be a faithful normal trace acting on LaTeXMLMath .
Let LaTeXMLMath be the completely positive limit projection provided by Corollary 2.7 .
Since LaTeXMLMath is asymptotically controlled by the gauge LaTeXMLMath , we can infer by a short computation that , given any nonzero vector LaTeXMLMath , LaTeXMLEquation is true for every choice of LaTeXMLMath with a LaTeXMLMath , whence LaTeXMLEquation .
Thus , the positive operator LaTeXMLMath is injective .
Let us consider the associated set LaTeXMLMath and the corresponding limit operator LaTeXMLMath occurring in Proposition 2.8 .
Set LaTeXMLMath , LaTeXMLMath and consider the positive operator LaTeXMLMath .
Note that LaTeXMLMath commutes with LaTeXMLMath .
Let LaTeXMLMath be a projection associated with the spectral decomposition of LaTeXMLMath ( which still commutes with LaTeXMLMath ) .
By the Cauchy–Schwarz Inequality , we get LaTeXMLEquation .
Applying the properties of LaTeXMLMath and LaTeXMLMath described in Corollary 2.7 and Proposition 2.8 , we infer that LaTeXMLEquation .
LaTeXMLEquation Now , note that the operator LaTeXMLMath commutes with LaTeXMLMath , because we have LaTeXMLEquation .
LaTeXMLEquation Using again Corollary 2.7 and Proposition 2.8 we obtain LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Substituting these results into ( 1 ) , we get LaTeXMLEquation whence LaTeXMLEquation .
Let LaTeXMLMath be a compact set contained in the interval LaTeXMLMath , and denote by LaTeXMLMath the spectral projection associated to LaTeXMLMath by the functional calculus of LaTeXMLMath .
We thus have LaTeXMLEquation .
Combining ( 2 ) with ( 3 ) yields LaTeXMLEquation .
The operator LaTeXMLMath is positive , so it is necessarily equal to LaTeXMLMath .
Therefore LaTeXMLMath is an orthogonal projection .
But LaTeXMLMath is compact and contained in LaTeXMLMath , thus there exists LaTeXMLMath such that LaTeXMLMath .
Consequently , we have LaTeXMLMath .
It follows that LaTeXMLEquation .
The last step is devoted to show that LaTeXMLMath .
Let us denote by LaTeXMLMath the spectral projection associated to LaTeXMLMath .
We have LaTeXMLMath , thus LaTeXMLEquation .
It follows that LaTeXMLMath .
Since LaTeXMLMath is injective , we deduce that LaTeXMLMath .
Finally , we see that LaTeXMLMath is invertible ( actually , LaTeXMLMath ) , therefore LaTeXMLMath is also invertible .
By Proposition 2.8 we know that LaTeXMLMath , see [ Ker ] ) .
It follows that , for any LaTeXMLMath , LaTeXMLMath is unitary , where LaTeXMLMath is invertible .
Q.E.D .
Acknowledgments .
The author wishes to thank the organizers of the “ Memorial Conference for Béla Szőkefalvi-Nagy ” , the organizers of “ Journées d ’ Analyse Fonctionnelle de Lens ” and the University of Besançon where these results were presented .
The author wishes to express his gratitude to László Kérchy for some stimulating questions .
References [ Bea ] B. Beauzamy , Introduction to Operator Theory and Invariant Subspaces , North-Hol-land , Amsterdam ( 1988 ) .
[ BCP ] S. Brown , B. Chevreau and C. Pearcy , Contractions with rich spectrum have invariant subspaces , J .
Operator Theory , 1 ( 1979 ) , 123–136 .
[ CaFa ] G. Cassier and T. Fack , Contractions in von Neumann algebras , J. Funct .
Anal .
, 135 ( 1996 ) , 297–338 .
[ CaFa2 ] G. Cassier and T. Fack , On power-bounded operators in finite von Neumann algebras , J. Funct .
Anal .
, 141 ( 1996 ) , 133–158 .
[ Dix ] J. Dixmier , Les algèbres d ’ opérateurs dans l ’ espace Hilbertien ( Algèbre de von Neumann ) , Gauthier-Villards , Paris ( 1969 ) .
[ Dix2 ] J. Dixmier , Les moyennes invariantes dans les semi-groupes et leurs applications , Acta Sci .
Math .
( Szeged ) , 12 ( 1950 ) , 213–227 .
[ Lor ] G. G. Lorentz , A contribution to the theory of divergent sequences , Acta .
Math .
, 80 ( 1948 ) , 167–190 .
[ Ker ] L. Kérchy , Operators with regular norm sequences , Acta Sci .
Math .
( Szeged ) , 63 ( 1997 ) , 571–605 .
[ Ker2 ] L. Kérchy , Criteria of regularity for norm sequences , Integral Equations Operator Theory , 34 ( 1999 ) , 458-477 .
[ Ker3 ] L. Kérchy , Representations with regular norm-behavior of discrete abelian semigroups , Acta Sci .
Math .
( Szeged ) , 65 ( 1999 ) , 702–726 .
[ Ker4 ] L. Kérchy , Hyperinvariant subspaces of operators with non-vanishing orbits , Proc .
Amer .
Math .
Soc .
, 127 ( 1999 ) , 1363–1370 .
[ Ker5 ] L. Kérchy , Unbounded representations of discrete abelian semigroups , Progress in Nonlinear Differential Equations and Their Applications , 42 ( 2000 ) , 141–150 .
[ KeMü ] L. Kérchy and V. Müller , Criteria of regularity for norm sequences .
II , Acta Sci .
Math .
( Szeged ) , 65 ( 1999 ) , 131–138 .
[ Sak ] S. Sakai , LaTeXMLMath algebras and LaTeXMLMath algebras , Ergebnisse der Mathematik und ihrer Grenzgebiete , 60 ( 1971 ) , Springer-Verlag , Berlin–New York .
[ Nag ] B. Sz.-Nagy , On uniformly bounded linear transformations in Hilbert space , Acta Sci .
Math .
( Szeged ) , 11 ( 1947 ) , 152–157 .
Institut Girard Desargues UPRES-A 5028 Mathématiques Université Claude Bernard Lyon I 69622 Villeurbanne Cedex , France e-mail : cassier @ desargues.univ-lyon1.fr
In this paper we first apply the chain level Floer theory to the study of Hofer ’ s geometry of Hamiltonian diffeomorphism group in the cases without quantum contribution : we prove that any quasi-autonomous Hamiltonian path on weakly exact symplectic manifolds or any autonomous Hamiltonian path on arbitrary symplectic manifolds is length minimizing in its homotopy class with fixed ends , as long as it has a fixed maximum and a fixed minimum which are not over-twisted and all of its contractible periodic orbits of period less than one are sufficiently LaTeXMLMath -small .
Next we give a construction of new invariant norm of Viterbo ’ s type on the Hamiltonian diffeomorphism group of arbitrary compact symplectic manifolds .
Contents §1 .
Introduction §2 .
Normalization of Hamiltonians and the action spectrum §3 .
Floer homology with real filtrations 3.1 .
Behavior of filtration under the chain homotopy 3.2 .
Adiabatic homotopy and adiabatic chain map §4 .
LaTeXMLMath -small Hamiltonians and local Floer complex 4.1 .
Local Floer homology 4.2 .
LaTeXMLMath versus LaTeXMLMath : comparison of two Floer homology §5 .
Calculation §6 .
Handle sliding lemma §7 .
Non-pushing down lemma and existence §8 .
Construction of spectral invariants Appendix In [ H ] , Hofer introduced an invariant pseudo-norm on the group LaTeXMLMath of compactly supported Hamiltonian diffeomorphisms of the symplectic manifold LaTeXMLMath by putting LaTeXMLEquation where LaTeXMLMath means that LaTeXMLMath is the time-one map of Hamilton ’ s equation LaTeXMLEquation and LaTeXMLMath is the function defined by LaTeXMLEquation .
He also proved that ( 1.1 ) is non-degenerate for the case LaTeXMLMath with respect to the standard symplectic structure .
Subsequently , Polterovich [ Po1 ] and Lalonde-McDuff [ LM ] proved the non-degeneracy for the case of rational symplectic manifolds and in complete generality , respectively .
We also refer to [ Ch ] for the proof in the case of tame symplectic manifolds based on the Floer homology theory of Lagrangian intersections and its simplification to [ Oh4 ] .
The invariant norm ( 1.1 ) induces a bi-invariant distance on LaTeXMLMath by LaTeXMLEquation which is the Finsler distance induced by the invariant Finsler norm LaTeXMLEquation on the Lie algebra LaTeXMLMath of the group LaTeXMLMath .
A natural problem of current interest in the literature is the study of geodesics in this Finsler manifold .
Hofer proved that the path of any autonomous Hamiltonian on LaTeXMLMath is length minimizing as long as the corresponding Hamilton ’ s equation has no non-constant time-one periodic orbit .
This result was generalized in [ MS ] on general symplectic manifolds for the case of slow autonomous Hamiltonians among the paths homotopic with fixed ends : According to [ En ] , [ MS ] and [ Mc ] , an autonomous Hamiltonian is called slow if it has no non-constant contractible periodic orbit of period less than 1 and the linearized flow at each fixed point is not over-twisted i.e. , has no closed trajectory of period less than one We call two Hamiltonians LaTeXMLMath and LaTeXMLMath are called equivalent if there exists a family LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
We denote LaTeXMLMath in that case and say that two Hamiltonian paths LaTeXMLMath and LaTeXMLMath are homotopic to each other with fixed ends , or just homotopic to each other when there is no danger of confusion .
In the present paper , we study length minimizing property of the quasi-autonomous Hamiltonian path : Such a Hamiltonian path was proven to be geodesics in the sense of Finsler geometry [ LM ] ( up to time reparametrization ) .
We refer to [ Po2 ] for the precise variational definition of geodesics from the first principle and an elegant proof of this latter fact .
We will just borrow theorems from [ LM ] or [ Po2 ] for a concrete description of geodesics in terms of quasi-autonomous Hamiltonians .
Definition 1.1 A Hamiltonian LaTeXMLMath is called quasi-autonomous if there exists two points LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
It has been proven in [ BP ] , [ LM ] , [ Po2 ] that a path LaTeXMLMath is a geodesic in the variational sense iff the corresponding Hamiltonian LaTeXMLMath is locally quasi-autonomous .
Based on this theorem , we just say that a geodesic is the Hamiltonian path generated by a locally quasi-autonomous Hamiltonian .
We now recall Lalonde-McDuff ’ s necessary condition on the stability of geodesics .
In [ Corollary 4.11 , LM ] , Lalonde-McDuff proved that for a generic LaTeXMLMath in the sense that all its fixed points are isolated , any stable geodesic LaTeXMLMath from the identity to LaTeXMLMath must have at least two fixed points at which the linearized isotopy has no non-constant closed trajectory in time less than 1 in the sense of Definition 1.2 below .
Definition 1.2 .
Let LaTeXMLMath be a Hamiltonian which is not necessarily time-periodic and LaTeXMLMath be its Hamiltonian flow .
The following is the main result of the present paper .
Suppose that LaTeXMLMath is a quasi-autonomous Hamiltonian such that ( i ) all contractible periodic orbit of period less than one are sufficiently LaTeXMLMath -small , ( ii ) it has a fixed minimum and a fixed maximum which are not over-twisted .
Then its Hamiltonian path LaTeXMLMath is length minimizing in its homotopy class with fixed ends for LaTeXMLMath , in cases The case ( 1 ) extends the result by Siburg [ Si ] on LaTeXMLMath , and ( 2 ) extends Entov ’ s [ En ] and Lalonde-McDuff-Slimowitz ’ s result [ MS ] for the slow autonomous case in several ways : first , it removes the slowness assumption in the case of autonomous Hamiltonian .
Secondly it allows both time-dependent Hamiltonians and appearance of non-constant periodic orbits .
Whether Theorem I holds in general cases , when there exists quantum contribution , is still to be seen .
Our proof of Theorem I will be based on the Floer homology theory on general symplectic manifolds , which has been established by now in the general context [ FOn ] , [ LT ] , [ Ru ] .
The idea of studying length minimizing property using the Floer theory was introduced by Polterovich [ Po2 ] for the case of small autonomous Hamiltonians when the action functional is single valued as in the case of exact symplectic manifolds .
We generalize his scheme to the case of quasi-autonomous Hamiltonian paths when the action functional is not single valued .
We first summarize Polterovich ’ s scheme of the proof for the case of small autonomous Hamiltonian when the symplectic form LaTeXMLMath is exact , say LaTeXMLMath .
A crucial idea behind his scheme is to relate the norm LaTeXMLMath with two homologically essential critical values of the action functional LaTeXMLEquation corresponding to the maximum and minimum points LaTeXMLMath and LaTeXMLMath of the function LaTeXMLMath , which is precisely LaTeXMLMath and LaTeXMLMath respectively .
This is carried out first by proving some existence result for the Floer ’ s continuity equation LaTeXMLEquation where LaTeXMLMath is the linear homotopy LaTeXMLEquation for small autonomous Hamiltonian LaTeXMLMath and arbitrary Hamiltonian LaTeXMLMath with LaTeXMLMath , and then by making some calculations involving the action functional and the solution of ( 1.4 ) .
( Similar calculations of this sort were previously employed by Chekanov [ Ch ] and by the present author [ Oh3,5 ] . )
For the existence result , Polterovich exploits the fact that when LaTeXMLMath is sufficiently small , then the Floer complex is diffeomorphic to the Morse complex of LaTeXMLMath and so the maximum point on the compact manifold LaTeXMLMath is homologically essential , which in turn is translated into the existence of a solution of ( 1.4 ) , via the fact that the Floer complexes of LaTeXMLMath and LaTeXMLMath are conjugate to each other ( see Proposition 5.3 ) , when LaTeXMLMath .
When we try to use Floer homology theory in the study of quasi-autonomous Hamiltonian paths , the first obvious point we need to take care of is that the Hamiltonian may not be one-periodic .
This can be taken care of using canonical modification of Hamiltonians into time periodic ones without changing their time-one maps and quasi-autonomous property ( see Lemma 5.2 for the precise statements ) .
There are many difficulties to overcome for the non-autonomous Hamiltonians especially when the action functional is not single-valued .
However using the full power of Floer homology theory developed by now ( in the level of chain , though ) and an idea of mini-max theory via the Floer homology developed by the author in [ Oh3,5 ] , we again reduce the proof of Theorem I to a similar existence result ( Proposition 5.3 ) for ( 1.4 ) where LaTeXMLMath is replaced by a quasi-autonomous Hamiltonian .
Unlike the small autonomous case , such an existence result is highly non-trivial ( even in the autonomous case ) for large Hamiltonians .
In fact , the method we employ to prove the existence theorem heavily relies on the extensive chain level Floer theory .
The latter turns out to carry applicability much wider than as we use in the present paper and leads us to the construction of spectral invariants on arbitrary compact symplectic manifolds ( See §8 and [ Oh7 ] ) .
The proof of Theorem I will then be carried out by a continuation argument over the homotopy LaTeXMLEquation combined with a delicate mini-max argument via Floer homology over the adiabatic homotopy .
One important point that we are exploiting in the first step is that when the Hamiltonian is LaTeXMLMath -small as in the case of LaTeXMLMath for LaTeXMLMath sufficiently small , the Floer boundary operator is decomposed into LaTeXMLEquation where LaTeXMLMath is the classical contribution and LaTeXMLMath is the quantum contribution ( see §5 , and [ Oh2 ] in the context of Lagrangian intersections ) .
This enables us to define the concept of local Floer homology which is invariant under local continuation ( see [ Oh1 ] in the context of Lagrangian submanifolds ) .
In general LaTeXMLMath is not zero , but is so either when LaTeXMLMath is weakly exact , or when the Hamiltonian is LaTeXMLMath -small and autonomous which is due to the extra LaTeXMLMath symmetry ( see [ Fl2 ] , [ FHS ] , [ FOn ] , [ LT ] ) .
This is one place where we used the hypotheses in Theorem I .
The second ingredient we use in this paper is several versions of the Non pushing down lemma culminating in Proposition 7.14 .
In fact this kind of non-pushing down lemma is the heart of the matter in the chain level Floer theory ( see [ Oh7 ] for more such arguments in general ) .
The proofs of these Non-pushing down lemmas use the above hypothesis in a more serious way and also use the concept of adiabatic homotopy and adiabadic chain map .
The third ingredient is a Floer theoretic version of the Handle sliding lemma ( Proposition 6.3 ) .
These tools enable us to develop a mini-max theory of the action functional in the non-exact case .
In the much simpler setting of the ( weakly- ) exact case where the action functional is single valued , similar mini-max idea was previously developed by the author in [ Oh3,5 ] for the Lagrangian submanifolds on the cotangent bundle , and subsequently by Schwarz [ Sc ] for the Hamiltonian diffeomorphisms on symplectically aspherical symplectic manifolds .
As an application of this mini-max theory , we prove the following construction of spectral invariants For each cohomology class LaTeXMLMath and Hamiltonian LaTeXMLMath , there exists an invariant LaTeXMLMath such that LaTeXMLMath and the assignment LaTeXMLMath is LaTeXMLMath -continuous .
In a sequel [ Oh7 ] to the present paper , we have further developed the techniques used here and applied them to extend the definition of these spectral invariants to the arbitrary quantum cohomology classes LaTeXMLMath .
These are then applied to give a construction of invariant norm and to obtain a new lower bound for the Hofer norm and to the study of length minimizing property of Hofer ’ s geodesics .
We would like to thank L. Polterovich for introducing us to the idea of studying length minimizing property of geodesics in terms of the Floer theory during his visit of KIAS Seoul , Korea , in April 2000 and giving us a copy of his book [ Po2 ] before its publication .
We also thank D. McDuff for sending us the preprints [ MS ] and [ Mc ] and informing us that the proof in [ LM ] already proves local length minimizing property of geodesics once construction of Gromov-Witten invariants on general symplectic manifolds is established .
We would also like to thank her for several helpful e-mail communications .
Let LaTeXMLMath be the set of contractible loops and LaTeXMLMath be its standard covering space in the Floer theory .
We recall the definition of this covering space from [ HS ] here .
Note that the universal covering space of LaTeXMLMath can be described as the set of equivalence classes of the pair LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is a map from the unit disc LaTeXMLMath to LaTeXMLMath such that LaTeXMLMath : the equivalence relation to be used is that LaTeXMLMath is zero in LaTeXMLMath .
Following Seidel [ Se ] , we say that LaTeXMLMath is LaTeXMLMath -equivalent to LaTeXMLMath iff LaTeXMLEquation where LaTeXMLMath is the map with opposite orientation on the domain and LaTeXMLMath is the obvious glued sphere .
And LaTeXMLMath denotes the first Chern class of LaTeXMLMath .
We denote by LaTeXMLMath the LaTeXMLMath -equivalence class of LaTeXMLMath and by LaTeXMLMath the canonical projection .
We also call LaTeXMLMath the LaTeXMLMath -covering space of LaTeXMLMath .
The action functional LaTeXMLMath is defined by LaTeXMLEquation .
Two LaTeXMLMath -equivalent pairs LaTeXMLMath and LaTeXMLMath have the same action and so the action is well-defined on LaTeXMLMath .
When a periodic Hamiltonian LaTeXMLMath is given , we consider the functional LaTeXMLMath by LaTeXMLEquation .
Here the sign convention is chosen to be consistent with that of [ Oh3,5 ] , LaTeXMLEquation where LaTeXMLMath for the canonical one form LaTeXMLMath on the cotangent bundle which in turn is precisely the classical mechanics Lagrangian on the cotangent bundle .
We would like to note that under this convention the maximum and minimum are reversed when we compare the action functional LaTeXMLMath and the ( quasi-autonomous ) Hamiltonian LaTeXMLMath .
We denote by LaTeXMLMath the set of periodic orbits of LaTeXMLMath .
Definition 2.1 [ Action Spectrum ] .
We define the action spectrum of LaTeXMLMath , denoted as LaTeXMLMath , by LaTeXMLEquation i.e. , the set of critical values of LaTeXMLMath .
For each given LaTeXMLMath , we denote LaTeXMLEquation .
Note that LaTeXMLMath is a principal homogeneous space modeled by the period group of LaTeXMLMath LaTeXMLEquation and LaTeXMLEquation .
Recall that LaTeXMLMath is either a discrete or a countable dense subset of LaTeXMLMath .
Lemma 2.2 .
LaTeXMLMath is a measure zero subset of LaTeXMLMath .
We first note that LaTeXMLMath is a countable subset of LaTeXMLMath for each LaTeXMLMath .
We consider the Poincaré return map in a tubular neighborhood of each LaTeXMLMath .
More precisely , we choose a small neighborhood LaTeXMLMath of LaTeXMLMath .
We identify LaTeXMLMath with LaTeXMLMath -ball LaTeXMLMath with the point LaTeXMLMath identified with the center of the ball .
Choose another ball neighborhood LaTeXMLMath with LaTeXMLMath such that the ( first ) Poincaré return map denoted by LaTeXMLEquation is well-defined .
We now define a continuous map from LaTeXMLMath to the space of piecewise smooth maps from LaTeXMLMath on LaTeXMLMath as follows : for each LaTeXMLMath , we first follow the flow of LaTeXMLMath and then follow from LaTeXMLMath to LaTeXMLMath by the straight line under the identification of LaTeXMLMath with LaTeXMLMath .
We reparameterize the domain of the loop by re-scaling it to be LaTeXMLMath .
We denote by LaTeXMLMath the loop corresponding to LaTeXMLMath constructed as above , and by LaTeXMLMath the image of the assignment LaTeXMLMath .
Obviously LaTeXMLMath is homotopic to LaTeXMLMath and so any given disc LaTeXMLMath bounding LaTeXMLMath can be naturally continued to bound the loop LaTeXMLMath .
We denote by LaTeXMLMath the disc continued from LaTeXMLMath and corresponding to LaTeXMLMath .
It can be easily checked that the function LaTeXMLEquation defines a smooth function on LaTeXMLMath and its critical values comprise those of LaTeXMLMath near LaTeXMLMath .
This can be proven by writing LaTeXMLMath explicitly and by a simple local calculation .
Noting that LaTeXMLMath is a finite dimensional ( in fact , LaTeXMLMath dimensional ) manifold , Sard ’ s theorem implies that the set of critical values is a measure zero subset in LaTeXMLMath .
Since a finite number of such tubular neighborhoods together with their complement cover LaTeXMLMath , LaTeXMLMath is a finite union of measure zero subset of LaTeXMLMath and so itself has measure zero .
∎ For given LaTeXMLMath , we denote by LaTeXMLMath if LaTeXMLMath , and denote LaTeXMLEquation .
We say that two Hamiltonians LaTeXMLMath and LaTeXMLMath are equivalent if they are connected by one parameter family of Hamiltonians LaTeXMLMath such that LaTeXMLMath i.e. , LaTeXMLEquation for all LaTeXMLMath .
We denote by LaTeXMLMath the equivalence class of LaTeXMLMath .
Then the universal covering space LaTeXMLMath of LaTeXMLMath is realized by the set of such equivalence classes .
Let LaTeXMLMath and denote LaTeXMLEquation .
Note that LaTeXMLMath defines a loop based at the identity .
Suppose LaTeXMLMath so there exists a family LaTeXMLMath with LaTeXMLMath and LaTeXMLMath and satisfying ( 2.3 ) .
In particular LaTeXMLMath defines a contractible loop .
If we denote LaTeXMLMath , this family provides a natural contraction of the loop LaTeXMLMath to the identity through LaTeXMLEquation which in turn provides a natural lifting of the action of the loop LaTeXMLMath on LaTeXMLMath to LaTeXMLMath which we define LaTeXMLEquation where LaTeXMLMath is the natural map from LaTeXMLMath obtained from identifying LaTeXMLMath as a map from LaTeXMLMath .
Even when LaTeXMLMath and so LaTeXMLMath is not contractible , note that the ( based ) loop group LaTeXMLMath naturally acts on the loop space LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
An interesting consequence of Arnold ’ s conjecture is that this action maps the particular component LaTeXMLMath to itself ( see e.g. , [ Lemma 2.2 , Se ] ) .
Seidel [ Lemma 2.4 , Se ] proves that this action ( by a based loop ) can be lifted to LaTeXMLMath .
In this paper , we will consider only the action by contractible loops in LaTeXMLMath .
We now study behavior of the action spectrum LaTeXMLMath when LaTeXMLMath varies .
In particular , we would like to study continuity property of certain critical values which are relevant to the uniform minimum point of the given quasi-autonomous Hamiltonian .
For this purpose , we need to normalize the spectrum LaTeXMLMath .
We will achieve this by restricting ourselves to LaTeXMLMath the set of normalized Hamiltonians with LaTeXMLMath by LaTeXMLMath as in [ Sc ] .
The following is proved in [ Oh6 ] ( see [ Sc ] for the symplectically aspherical case where the action fuctional is single-valued .
In this case Schwarz [ Sc ] proved that the normalization works on LaTeXMLMath not just on LaTeXMLMath as long as LaTeXMLMath without assuming LaTeXMLMath ) .
Let LaTeXMLMath and LaTeXMLMath be a path in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Denote LaTeXMLMath and LaTeXMLMath for a LaTeXMLMath .
Then the function LaTeXMLMath defined by LaTeXMLEquation is constant .
In particular , we have LaTeXMLEquation .
From now on , we will always assume that the Hamiltonian functions are normalized so that LaTeXMLEquation 1 .
Behavior of filtration under the chain map For each given generic LaTeXMLMath , we consider the free LaTeXMLMath vector space over LaTeXMLEquation .
To be able to define the Floer boundary operator correctly , we need to complete this vector space downward with respect to the real filtration provided by the action LaTeXMLMath of the element LaTeXMLMath of ( 3.1 ) .
More precisely , Definition 3.1 .
We call the formal sum LaTeXMLEquation a Novikov chain if there are only finitely many non-zero terms in the expression ( 3.2 ) above any given level of the action .
We denote by LaTeXMLMath the set of Novikov chains .
Here , we put ‘ tilde ’ over LaTeXMLMath to distinguish this LaTeXMLMath vector space with more standard Floer complex module over the Novikov ring in the literature .
Note that this is an infinite dimensional LaTeXMLMath -vector space in general , unless LaTeXMLMath .
It appears that for the purpose of studying Hofer ’ s geometry this set-up of Floer homology with real filtration on the LaTeXMLMath -covering space LaTeXMLMath suits better than the more standard Floer homology on LaTeXMLMath with the Novikov ring as its coefficient , although they provide equivalent descriptions .
Since , for the study of action changes under the chain maps , we will frequently use the chain level property of various operators in the Floer theory , we briefly review construction of basic operators in the Floer homology theory [ Fl2 ] .
Let LaTeXMLMath be a periodic family of compatible almost complex structure on LaTeXMLMath .
For each given pair LaTeXMLMath , we define the boundary operator LaTeXMLEquation considering the perturbed Cauchy-Riemann equation LaTeXMLEquation .
This equation , when lifted to LaTeXMLMath , defines nothing but the negative gradient flow of LaTeXMLMath with respect to the LaTeXMLMath -metric on LaTeXMLMath induced by the metrics LaTeXMLMath .
For each given LaTeXMLMath and LaTeXMLMath , we define the moduli space LaTeXMLMath of solutions LaTeXMLMath of ( 3.3 ) satisfying LaTeXMLEquation .
LaTeXMLMath has degree LaTeXMLMath and satisfies LaTeXMLMath .
When we are given a family LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , the chain homomorphism LaTeXMLEquation is defined by the non-autonomous equation LaTeXMLEquation where LaTeXMLMath is functions of the type LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation for some LaTeXMLMath .
LaTeXMLMath has degree 0 and satisfies LaTeXMLEquation .
Finally when we are given a homotopy LaTeXMLMath of homotopies with LaTeXMLMath , LaTeXMLMath , consideration of the parameterized version of ( 3.5 ) for LaTeXMLMath defines the chain homotopy map LaTeXMLEquation which has degree LaTeXMLMath and satisfies LaTeXMLEquation .
By now , construction of these maps using these moduli spaces has been completed with rational coefficients ( See [ FOn ] , [ LT ] and [ Ru ] ) .
We will freely use this advanced machinery throughout the paper .
However the main stream of the proof can be read independently of these papers once it is understood that the bubbling of spheres is a codimension two phenomenon , which is exactly what the advanced machinery establishes .
Therefore we do not explicitly mention these technicalities in this paper , unless it is absolutely necessary .
The following upper estimate of the action change can be proven by the same argument as that of [ Oh3 ] .
Because this will be a crucial ingredient in our proof , we include its proof here for reader ’ s convenience .
Proposition 3.2 [ Theorem 7.2 , Oh3 ] .
When there are two Hamiltonians LaTeXMLMath and LaTeXMLMath , the canonical chain map LaTeXMLEquation provided by the linear homotopy LaTeXMLMath respects the filtration LaTeXMLEquation and so induces the homomorphism LaTeXMLEquation .
We fix LaTeXMLMath here .
Let LaTeXMLMath and LaTeXMLMath be given .
As argued in [ Oh3 ] , for any given solution LaTeXMLMath of ( 3.5 ) and ( 3.4 ) , we compute LaTeXMLEquation .
Here we have LaTeXMLEquation .
However since LaTeXMLMath satisfies ( 3.5 ) , we have LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation Combining these and using that LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By definition of the chain map LaTeXMLMath , this finishes the proof .
∎ Proposition 3.3 [ Lemma 4.3 , Oh3 ] .
For a fixed LaTeXMLMath and for a given one parameter family LaTeXMLMath , the natural chain map LaTeXMLEquation respects the filtration .
A similar computation , this time using ( 3.2 ) and ( 3.3 ) with LaTeXMLMath fixed , leads to LaTeXMLEquation .
We refer to the proof of [ Lemma 4.3 , Oh3 ] for complete details .
∎ We would like to remark that there is also some upper estimate for chain maps over general homotopy or for the chain homotopy maps .
This general upper estimate is used in our construction of spectral invariants in [ Oh7 ] .
3.2 .
Adiabatic homotopy and adiabatic chain map For our purpose of using the Floer theory in the study of Hofer ’ s geometry , we also need to consider a family version of the Floer homology to keep track of the behavior of the action spectrum over one parameter family of Hamiltonians as in §2 .
Let LaTeXMLMath and LaTeXMLMath be a path in LaTeXMLMath .
We normalize LaTeXMLMath so that ( 2.5 ) ( and so Proposition 2.3 ) holds .
With this normalization , if LaTeXMLMath were isomorphic to LaTeXMLMath or LaTeXMLMath like the case where LaTeXMLMath or more generally where LaTeXMLMath is integral , the “ adiabatic ” homotopy LaTeXMLEquation as defined in [ MO1,2 ] will induce an isomorphism LaTeXMLEquation for any LaTeXMLMath .
Since we will use this adiabatic homotopy in an essential way later , we carefully explain how it is constructed following the exposition from [ MO1,2 ] .
Suppose that there is a ‘ gap ’ in the spectrum LaTeXMLMath , i.e , that there is a positive number LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
Since LaTeXMLMath is a smooth path , there exists some LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath with LaTeXMLMath .
We consider the partition LaTeXMLEquation so that LaTeXMLEquation .
By Proposition 3.2 , the chain map LaTeXMLEquation over the linear path LaTeXMLEquation restricts to LaTeXMLEquation for any LaTeXMLMath with LaTeXMLMath .
Similarly , we have LaTeXMLEquation for any LaTeXMLMath .
Combining these two , we have the composition LaTeXMLEquation .
By the condition ( 3.5 ) and the gap condition , all of these three maps in fact restrict to the same levels and induces homomorphisms LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation provided LaTeXMLMath is chosen sufficiently close to LaTeXMLMath .
However , if we choose LaTeXMLMath sufficiently small , we can also prove the identity LaTeXMLEquation which implies that ( 3.8 ) is an isomorphism for all LaTeXMLMath with LaTeXMLMath .
By repeating the above to LaTeXMLMath for LaTeXMLMath , we conclude that the composition LaTeXMLEquation restricts to LaTeXMLEquation for all LaTeXMLMath , and so induces the composition LaTeXMLEquation which becomes an isomorphism .
In particular , we have the isomorphism LaTeXMLEquation .
Definition 3.4 .
Let LaTeXMLMath be a partition .
We define its mesh , denoted as LaTeXMLMath , by LaTeXMLEquation .
We call the associated piecewise continuous linear path LaTeXMLMath and the chain map ( 3.9 ) the adiabatic homotopy , denoted as LaTeXMLMath , and the adiabatic chain map over the path LaTeXMLMath .
We denote LaTeXMLEquation .
We define the mesh LaTeXMLMath of the adiabatic homotopy LaTeXMLMath along the path LaTeXMLMath to be LaTeXMLEquation .
We simply denote by LaTeXMLMath , LaTeXMLMath when we do not specify the partition LaTeXMLMath .
Note that the mesh of the adiabatic homotopy can be made arbitrarily small by making LaTeXMLMath small .
This adiabatic construction of homotopy in the chain level will be used in a crucial way to study the global case of length minimizing property of geodesics , where the action spectrum is not necessarily fixed and does not have a ‘ gap ’ in general .
4.1 .
Local Floer complex In this section , we consider LaTeXMLMath -small Hamiltonians LaTeXMLMath .
We consider the subset LaTeXMLMath of loops LaTeXMLMath with LaTeXMLMath is contained in a fixed Darboux neighborhood LaTeXMLMath of the diagonal LaTeXMLMath for all LaTeXMLMath .
In particular , any periodic orbit LaTeXMLMath of LaTeXMLMath contained in LaTeXMLMath has a canonical isotopy class of contraction LaTeXMLMath .
We will always use this convention LaTeXMLMath whenever there is a canonical contraction of LaTeXMLMath like in this case of small loops .
This provides a canonical embedding of LaTeXMLMath defined by LaTeXMLEquation .
We denote by LaTeXMLMath the set LaTeXMLEquation .
Imitating the construction from [ Fl2 ] and [ Oh1 ] , we define Definition 4.1 .
For any LaTeXMLMath and for the given Darboux neighborhood LaTeXMLMath of the diagonal LaTeXMLMath such that LaTeXMLEquation we define LaTeXMLEquation .
Consider the evaluation map LaTeXMLEquation .
For each open subset LaTeXMLMath with LaTeXMLMath , we define the local Floer complex in LaTeXMLMath by LaTeXMLEquation .
We say LaTeXMLMath is isolated in LaTeXMLMath if its closure is contained in the interior of LaTeXMLMath .
The following can be proved by the same method as that of [ Fl2 ] ( See Proposition 3.2 [ Oh1 ] ) , to which we refer readers for its proof .
Proposition 4.2 .
If LaTeXMLMath is isolated in LaTeXMLMath , then for all LaTeXMLMath LaTeXMLMath -close enough to LaTeXMLMath in the LaTeXMLMath -topology , LaTeXMLMath is also isolated in LaTeXMLMath .
Using this proposition , we can define the local Floer homology , denoted by LaTeXMLMath .
Furthermore , the restriction of the action functional to the image of the embedding LaTeXMLMath provides a filtration on the local Floer complex .
Proof of the following proposition is standard combining existing methods in the Floer theory ( see [ §3 , Oh1 ] ) .
Proposition 4.3 .
Let LaTeXMLMath be as above and LaTeXMLMath .
Assume that LaTeXMLMath so small that ( 4.2 ) holds for LaTeXMLMath .
Then there exists a canonical isomorphism , we have LaTeXMLEquation whose matrix elements are given by the number of solutions of ( 4.4 ) below whose images are contained in LaTeXMLMath : LaTeXMLEquation .
Following [ Oh1 ] , we call thin trajectories the solutions of the Cauchy-Riemann equations defining the boundary map or the chain map whose images are contained in LaTeXMLMath .
4.2 .
LaTeXMLMath versus LaTeXMLMath : comparison of two Floer homology The main goal of this sub-section is to prove that when LaTeXMLMath is LaTeXMLMath -small quasi-autonomous Hamiltonian , the minimum point LaTeXMLMath , which corresponds to a ( local ) maximum point of LaTeXMLMath in the local Floer complex , is homologically essential in the local Floer complex .
There does not seem to be a direct way of proving this in the context of Floer theory of Hamiltonian diffeomorphisms .
We will need to use the intersection theoretic version of the Floer theory of Lagrangian submanifolds between LaTeXMLMath and LaTeXMLMath in the product LaTeXMLMath .
This kind of comparison argument has been around among the experts in the Floer theory but never been rigorously carried out before .
As we will see below , contrary to the conventional wisdom in the literature , this comparison does not work in the chain level but works only in the homology level .
We now compare the local Floer homology LaTeXMLMath of LaTeXMLMath -small Hamiltonian LaTeXMLMath and two versions of its intersection counterparts , one LaTeXMLMath and the other LaTeXMLMath .
We will be especially keen to keep track of filtration changes .
First we note that the two Floer complexes LaTeXMLMath and LaTeXMLMath are canonically isomorphic by the assignment LaTeXMLEquation and so the two Lagrangian intersection Floer homology are canonically isomorphic : Here the above two moduli spaces are the solutions sets of the following Cauchy-Riemann equations LaTeXMLEquation and LaTeXMLEquation respectively , where LaTeXMLMath .
The relevant action functionals for these cases are given by LaTeXMLEquation on LaTeXMLMath and LaTeXMLEquation on LaTeXMLMath where we denote LaTeXMLEquation and similarly for LaTeXMLMath .
Again the ‘ tilde ’ means the covering space which can be represented by the set of pairs LaTeXMLMath in a similar way ( see [ §2 , FOOO ] for the complete discussion on this set-up for the Lagrangian intersection Floer homology theory ) .
The relations between the action functionals ( 4.5 ) , ( 4.6 ) and ( 2.1 ) are evident and respects the filtration .
Next we will attempt to compare LaTeXMLMath and LaTeXMLMath .
Without loss of generality , we will concern Hamiltonians LaTeXMLMath such that LaTeXMLMath near LaTeXMLMath , which one can always achieve by perturbing LaTeXMLMath without changing its time-one map ( See Lemma 5.2 ) .
It turns out that there is no direct way of identifying the corresponding Floer complexes between the two .
As an intermediate case , we consider the Hamiltonian LaTeXMLMath defined by LaTeXMLEquation and the assignment LaTeXMLEquation with LaTeXMLMath .
Here the map LaTeXMLMath is the map defined by LaTeXMLEquation is well-defined and continuous because LaTeXMLEquation .
LaTeXMLEquation Furthermore near LaTeXMLMath , this is smooth ( and so holomorphic ) by the elliptic regularity since LaTeXMLMath is smooth ( Recall that we assume that LaTeXMLMath near LaTeXMLMath .
Conversely , any element LaTeXMLMath can be written as the form of LaTeXMLMath which is uniquely determined by LaTeXMLMath .
This proves that ( 4.7 ) is a diffeomorphism from LaTeXMLMath to LaTeXMLMath which induces a filtration-preserving isomorphism between LaTeXMLMath and LaTeXMLMath Finally , we need to relate LaTeXMLMath and LaTeXMLMath .
For this we note that LaTeXMLMath and LaTeXMLMath can be connected by a one-parameter family LaTeXMLMath with LaTeXMLEquation .
And we have LaTeXMLEquation .
Noting that there are only finite number of periodic trajectories in LaTeXMLMath , the “ adiabatic argument ” explained in §3 indeed proves that the adiabatic homomorphism LaTeXMLEquation respects the filtration and so the induced homomorphism in its homology LaTeXMLEquation becomes a filtration-preserving isomorphism .
We note that LaTeXMLMath is isolated in LaTeXMLMath .
Therefore if follows from Proposition 3.1 that if LaTeXMLMath and LaTeXMLMath are sufficiently small , both LaTeXMLMath and LaTeXMLMath are also isolated in LaTeXMLMath .
We now apply the above discussion to the LaTeXMLMath -small quasi-autonomous Hamiltonian LaTeXMLMath to prove the following homological essentialness of the minimum points of LaTeXMLMath in the local Floer homology .
Recall from the remark in the beginning of §2 that the minimum of LaTeXMLMath corresponds to the maximum of the action functional and vice versa .
We refer to [ Definition 13.2.F , Po2 ] for a formulation of the homological essentialness of the critical point and its consequence on the existence result [ Corollary 13.2.H , Po2 ] .
Suppose that LaTeXMLMath with LaTeXMLMath so small that LaTeXMLMath lies in the given Darboux neighborhood of LaTeXMLMath .
Suppose that LaTeXMLMath is quasi-autonomous with the unique maximum point LaTeXMLMath and minimum point LaTeXMLMath .
Then the critical point LaTeXMLMath is homologically essential in LaTeXMLMath .
In the above discussion , we have shown that LaTeXMLMath is diffeomorphic to LaTeXMLMath .
We will first show that the intersection point LaTeXMLMath is homologically essential in the latter Floer complex , which in turn will imply the homological essentialness of LaTeXMLMath in LaTeXMLMath .
Identifying LaTeXMLMath with a neighborhood of the zero section of the cotangent bundle LaTeXMLMath , we denote by LaTeXMLMath the canonical almost complex structure on LaTeXMLMath associated to the Levi-Civita connection of a given Riemannian metric on LaTeXMLMath .
Since the image of LaTeXMLMath is isolated in LaTeXMLMath , we may perturb LaTeXMLMath to LaTeXMLMath in LaTeXMLMath so that LaTeXMLMath near the boundary of LaTeXMLMath and also that LaTeXMLEquation .
We connect LaTeXMLMath and LaTeXMLMath by a path LaTeXMLMath on LaTeXMLMath so that LaTeXMLMath near the boundary for all LaTeXMLMath .
Noting that LaTeXMLMath is pseudo-convex with respect to LaTeXMLMath , the two local Floer complexes LaTeXMLMath and LaTeXMLMath can be connected by an isolated continuation in LaTeXMLMath .
Recall from Proposition 3.2 that this continuation preserves the filtration of Floer homology .
On the other hand LaTeXMLMath is diffeomorphic to LaTeXMLMath for a generating function of the Lagrangian submanifold LaTeXMLMath , if LaTeXMLMath is LaTeXMLMath -small .
Moreover LaTeXMLMath corresponds to LaTeXMLMath which is the minimum point of the generating function LaTeXMLMath .
Since LaTeXMLMath ( and so LaTeXMLMath ) is assumed to be compact , LaTeXMLMath is homologically essential in the Morse homology of LaTeXMLMath .
On the other hand , LaTeXMLMath is diffeomorphic to LaTeXMLMath ( see [ FOh1 ] for its proof ) , where LaTeXMLMath is the almost complex structure on LaTeXMLMath that is associated to the Levi-Civita connection of a chosen metric LaTeXMLMath on LaTeXMLMath .
Therefore LaTeXMLMath is homologically essential in LaTeXMLMath .
Combining all these , we derive that the constant solution LaTeXMLMath is homologically essential in the local Floer complex LaTeXMLMath .
By the uniqueness of the minimum points , under the chain isomorphism ( 4.8 ) , the image LaTeXMLMath must involve LaTeXMLMath in its expression and so LaTeXMLMath is also homologically essential in LaTeXMLMath .
We refer readers to the proof of this kind of result in a more difficult context in §7 .
∎ In this section , we start with the proof of Theorem I in the introduction .
We consider the rescaled Hamiltonians LaTeXMLEquation and choose LaTeXMLMath so small that it has no non-constant contractible periodic orbit for all LaTeXMLMath .
We first prove the following simple lemma .
Lemma 5.1 .
Let LaTeXMLMath be a sequence of smooth Hamiltonians such that LaTeXMLMath in LaTeXMLMath -topology and LaTeXMLMath in LaTeXMLMath -topology .
If all LaTeXMLMath are length minimizing over LaTeXMLMath , then so is LaTeXMLMath .
Suppose the contrary that there exists LaTeXMLMath such that LaTeXMLMath , but LaTeXMLMath .
We choose LaTeXMLMath with LaTeXMLEquation .
Therefore LaTeXMLEquation for sufficiently large LaTeXMLMath .
We consider the Hamiltonian LaTeXMLMath defined by LaTeXMLEquation .
LaTeXMLEquation This generates the flow LaTeXMLMath and so LaTeXMLMath .
This implies , by the hypothesis that LaTeXMLMath are length minimizing over LaTeXMLMath , we have LaTeXMLEquation and so LaTeXMLEquation for all sufficiently large LaTeXMLMath .
However since LaTeXMLMath , LaTeXMLMath by the hypotheses ( and also so LaTeXMLMath ) in LaTeXMLMath -topology , we have LaTeXMLMath in LaTeXMLMath -topology .
Therefore we have LaTeXMLEquation which gives rise to a contradiction to ( 5.3 ) .
∎ Now , using the Floer homology theory , we would like to show LaTeXMLEquation for any LaTeXMLMath when the quasi-autonomous Hamiltonian LaTeXMLMath satisfies the hypothesis that there is no non-constant contractible periodic orbits .
However we need to take care of a problem before applying the Floer theory , that is , LaTeXMLMath not being time-periodic .
The following lemma will be important in this respect .
Lemma 5.2 .
Let LaTeXMLMath be a given Hamiltonian LaTeXMLMath and LaTeXMLMath be its time-one map .
Then we can perturb LaTeXMLMath so that the perturbed Hamiltonian LaTeXMLMath has the properties Furthermore , this modification is canonical with the “ smallness ” in ( 3 ) can be chosen uniformly over LaTeXMLMath depending only on the LaTeXMLMath -norm of LaTeXMLMath .
We first reparameterize LaTeXMLMath in the following way : We choose a smooth function LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation and consider the isotopy LaTeXMLEquation .
It is easy to check that the Hamiltonian generating the isotopy LaTeXMLMath is LaTeXMLMath with LaTeXMLMath .
By definition , it follows that LaTeXMLMath satisfies ( 1 ) and ( 2 ) .
For ( 3 ) , we compute LaTeXMLEquation .
LaTeXMLEquation For the first term , LaTeXMLEquation .
LaTeXMLEquation which can be made arbitrarily small by choosing LaTeXMLMath so that LaTeXMLMath become sufficiently small .
For the second term , LaTeXMLEquation .
LaTeXMLEquation Again by appropriately choosing LaTeXMLMath , we can make LaTeXMLEquation as small as we want .
Combining these two , we have verified LaTeXMLMath can be made as small as we want .
Similar consideration applies to LaTeXMLMath and hence we have finished the proof of ( 3 ) .
The property ( 4 ) and naturality of this modification are evident from the construction .
( 5 ) follows from simple comparison of corresponding actions of periodic orbits .
∎ We will always perform this canonical modification in the rest of the paper whenever we would like to consider the Cauchy-Riemann equation associated to the Hamiltonian LaTeXMLMath , when LaTeXMLMath is not a one-periodic Hamiltonian .
Let LaTeXMLMath be an arbitrary Hamiltonian with LaTeXMLMath .
We want to prove LaTeXMLMath .
Applying Lemma 5.2 to LaTeXMLMath and LaTeXMLMath , we may assume that LaTeXMLMath and LaTeXMLMath are time one periodic , allowing small errors and then getting rid of them by taking the limit .
We will postpone the proof of the following crucial existence result to the next sections .
From now on , we will always denote by LaTeXMLMath the constant disc LaTeXMLMath for each given constant periodic orbit LaTeXMLMath .
Suppose LaTeXMLMath for sufficiently small LaTeXMLMath as before .
Let LaTeXMLMath be a Morse function on LaTeXMLMath and consider the linear homotopy LaTeXMLEquation .
Then there exists LaTeXMLMath such that for any LaTeXMLMath , the continuation equation LaTeXMLEquation has a solution for some LaTeXMLMath and for some LaTeXMLMath with LaTeXMLEquation where we recall LaTeXMLMath .
Assuming this proposition for the moment , we proceed with the proof of Theorem I .
The following calculation is a slight modification used by Polterovich [ Po2 ] in our context which will lead to the proof of Theorem I once we prove Proposition 5.3 .
We compute LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Therefore by integrating this over LaTeXMLMath from LaTeXMLMath to LaTeXMLMath , we have LaTeXMLEquation .
On the other hand , we derive LaTeXMLEquation from the normalization condition ( 2.5 ) , ( 5.6 ) and from the fact that LaTeXMLMath is the fixed minimum point over LaTeXMLMath .
Therefore we have LaTeXMLEquation .
By letting LaTeXMLMath , we have proven LaTeXMLEquation .
By considering LaTeXMLMath and LaTeXMLMath which generate LaTeXMLMath and LaTeXMLMath respectively , we also prove LaTeXMLEquation which is equivalent to LaTeXMLEquation .
Combining ( 5.7 ) and ( 5.8 ) , we have proved LaTeXMLEquation .
This will finish the proof of Theorem I up to the proof of Proposition 5.3 .
∎ In this section , we study an important ingredient in our proof , the Floer theoretic version of the ‘ handle sliding ’ lemma .
Let LaTeXMLMath be any time periodic Hamiltonian and consider the Cauchy Riemann equation LaTeXMLEquation for generic LaTeXMLMath .
We call a solution LaTeXMLMath trivial if it is LaTeXMLMath -independent , i.e. , stationary .
We define LaTeXMLEquation .
LaTeXMLEquation The positivity of LaTeXMLMath is an easy consequence of Gromov compactness type theorem , whose proof we omit .
We will need a family version of LaTeXMLMath .
When there does not occur bifurcation of periodic orbits , one can define this to be LaTeXMLEquation .
However when there does occur bifurcation of periodic orbits , LaTeXMLMath could be zero , which forces us to look at another positive constant the definition of which should be given more subtly to make it suitable for our purpose .
In introducing this constant , we exploit the fact that in the definition of Floer ’ s chain homotopy map , only index zero solutions of Floer ’ s continuity equation ( 3.5 ) or ( 6.9 ) below enter .
We first recall that for a generic one parameter family LaTeXMLMath , there are only finite number of points LaTeXMLMath where there occur either birth-death or death-birth type of bifurcation of periodic orbits ( see [ Lee ] for a detailed proof of this ) .
Furthermore at each such LaTeXMLMath , there is exactly one bifurcation orbit LaTeXMLMath of LaTeXMLMath for which there is a continuous family of the pair LaTeXMLMath of periodic orbits of LaTeXMLMath for LaTeXMLMath , LaTeXMLMath sufficiently small such that ( 1 ) LaTeXMLMath as LaTeXMLMath , ( 2 ) the Conley-Zehnder indices satisfy LaTeXMLEquation where LaTeXMLMath for LaTeXMLMath a canonical ‘ short ’ cylinder between LaTeXMLMath and LaTeXMLMath .
This latter condition makes sense because LaTeXMLMath and LaTeXMLMath are close when LaTeXMLMath is sufficiently small .
We now prove the following important lemma Let LaTeXMLMath be a generic one parameter family as above .
For each LaTeXMLMath , we define LaTeXMLEquation and LaTeXMLEquation .
Then LaTeXMLMath is strictly positive .
Suppose the contrary that LaTeXMLMath , i.e. , that there exists a sequence LaTeXMLMath with LaTeXMLMath and LaTeXMLMath solutions of ( 6.1 ) for LaTeXMLMath such that LaTeXMLEquation .
Then we must have , by choosing a subsequence if necessary , LaTeXMLEquation and a bifurcation orbit LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath uniformly and so LaTeXMLEquation .
Since LaTeXMLMath are solutions of LaTeXMLMath , they must be the pair described in ( 1 ) above ( 6.4 ) and so LaTeXMLEquation .
But this contradicts to the index condition in ( 6.6 ) which finishes the proof .
∎ Again for a generic choice of LaTeXMLMath , we may assume that there are only finitely many points LaTeXMLMath with LaTeXMLMath at which ( 6.1 ) has exactly one non-trivial solution LaTeXMLMath that has Fredholm index 0 .
( See [ Fl1 ] for this kind of generic argument . )
We denote LaTeXMLEquation .
Next we define LaTeXMLEquation which is again positive by Gromov type compactness theorem .
Now we have the following crucial definition of a family version of the constant LaTeXMLMath suitable for our purpose .
We define LaTeXMLEquation .
The following proposition is an important ingredient of our proof .
Proposition 6.3 .
( Handle sliding lemma ) Let LaTeXMLMath be a ( two parameter ) family of almost complex structures and LaTeXMLMath be a generic family of Hamiltonians .
Let LaTeXMLMath be the constant defined in Definition 6.3 and let LaTeXMLMath .
Then there exists a LaTeXMLMath such that if LaTeXMLMath , any finite energy solution LaTeXMLMath with LaTeXMLEquation of LaTeXMLEquation must either satisfy LaTeXMLEquation or LaTeXMLEquation where for LaTeXMLMath as LaTeXMLMath , provided LaTeXMLMath .
Here LaTeXMLMath is the linear path LaTeXMLMath and LaTeXMLMath is the standard function as before .
We call a solution LaTeXMLMath of ( 6.9 ) very short if it satisfies ( 6.10 ) , and long if it satisfies ( 6.11 ) .
We can phrase the content of this proposition as “ Any short path is indeed very short ” .
We prove this by contradiction .
Suppose the contrary that there exists some LaTeXMLMath , LaTeXMLMath and LaTeXMLMath with LaTeXMLMath as LaTeXMLMath , and solutions LaTeXMLMath that satisfy ( 6.8 ) and LaTeXMLEquation but LaTeXMLEquation .
In particular , the right half of ( 6.13 ) implies the uniform bound on the energy of LaTeXMLMath .
As LaTeXMLMath , the equation ( 6.12 ) converges to ( 6.1 ) with LaTeXMLMath .
By Gromov type compactness theorem , we have a cusp curve LaTeXMLEquation in the limit of a subsequence where each LaTeXMLMath is a solution of ( 6.1 ) with LaTeXMLMath .
We also have LaTeXMLEquation .
On the other hand the left half of ( 6.13 ) implies that at least one of LaTeXMLMath is not trivial .
Now we consider three cases separately : the first is the one where LaTeXMLMath and the second where LaTeXMLMath and the rest where LaTeXMLMath .
When LaTeXMLMath , we must have LaTeXMLEquation which gives rise to a contradiction to ( 6.13 ) if LaTeXMLMath is sufficiently large .
On the other hand , if LaTeXMLMath , the cusp curve must contain one component LaTeXMLMath that has Index 0 and is non-constant .
Again the right hand side of ( 6.13 ) prevents this from happening .
Finally when LaTeXMLMath , the index condition LaTeXMLMath and the transversality condition implies that all components LaTeXMLMath must be constant which again contradicts to LHS of ( 6.13 ) if LaTeXMLMath is sufficiently large .
This finishes the proof of proposition .
∎ An immediate corollary of this is the following estimate on the action .
Corollary 6.4 .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath as in Proposition 6.3 .
Suppose LaTeXMLMath .
If LaTeXMLMath is very short , then we have the lower estimate LaTeXMLEquation and so combined with the upper estimate ( 3.6 ) , we have LaTeXMLEquation .
LaTeXMLEquation If LaTeXMLMath is not very short and so must be long , then we have the improved upper estimate LaTeXMLEquation .
A straightforward computation leads to the following general identity LaTeXMLEquation .
LaTeXMLEquation Corollary 6.4 immediately follows from this and Proposition 6.3 .
∎ We will apply the above handle sliding lemma and its corollary to the adiabatic paths in the next section .
In this section , we will assume the main hypothesis .
This is the only section where we use the hypothesis .
All the materials in other sections are valid in arbitrary compact symplectic manifolds .
Hypothesis .
Assume one of the following two cases : In the beginning , we will approach both cases in the general setting of quasi-autonomous cases on arbitrary LaTeXMLMath and then explain how non-existence of quantum contributions enter our proof of the Non-pushing down lemma .
Definition 7.1 .
Let LaTeXMLMath be a Hamiltonian which is not necessarily time-periodic and LaTeXMLMath be its Hamiltonian flow .
The remaining section will be occupied by the proof of the following result ( Theorem I in the introduction ) .
Theorem 7.2 .
We assume one of the two cases in the Hypothesis .
Suppose that the quasi-autonomous Hamiltonian LaTeXMLMath satisfies ( i ) LaTeXMLMath has no non-constant contractible periodic orbit of period less than one , ( ii ) it has at least one fixed minimum and one fixed maximum which are not over-twisted .
Then the Hamiltonian path LaTeXMLMath is length minimizing in its homotopy class with fixed ends .
Remark 7.3 .
( 1 ) Note that the hypotheses ( i ) is slightly different from Theorem I .
However from our proof , it will be clear that the proof for Theorem 7.2 is stable under LaTeXMLMath -small perturbation of the Hamiltonian and so allow sufficiently LaTeXMLMath -small non-constant contractible periodic orbits .
This will prove Theorem I .
It is rather awkward to state how small the perturbation can be .
One might want to consider Theorem I as a stability result of the case in Theorem 7.2 .
( 2 ) Considering LaTeXMLMath with LaTeXMLMath but arbitrarily close to 1 and applying Lemma 5.2 , we may assume stronger assumption “ period less than equal to 1 ” instead of “ period less than 1 ” in both ( 1 ) and ( 2 ) in the hypotheses in the theorem .
We will assume this stronger assumption in the proof .
We consider the reparameterized Hamiltonians LaTeXMLMath .
The assumption ( i ) implies that there is no appearance of non-constant contractible periodic orbit as LaTeXMLMath moves from LaTeXMLMath to 1 .
The only possible bifurcation is by that of critical points of LaTeXMLMath .
This proves Lemma 7.4 .
Suppose LaTeXMLMath satisfies the above .
Then for each LaTeXMLMath , there is one-one correspondence between the set of contractible solutions and the set of points LaTeXMLMath such that LaTeXMLEquation .
We call a point LaTeXMLMath LaTeXMLMath -critical point of LaTeXMLMath if LaTeXMLMath satisfies ( 7.1 ) .
We denote by LaTeXMLEquation the set of LaTeXMLMath -critical points of LaTeXMLMath .
It follows from Lemma 7.4 that for any LaTeXMLMath there is a canonical injection LaTeXMLEquation and that there is a canonical one-one correspondence between the set of LaTeXMLMath -critical points of LaTeXMLMath and that of critical points of LaTeXMLMath which are of the type LaTeXMLMath .
From this description of LaTeXMLMath , it follows that there does not emerge any new critical points of LaTeXMLMath as LaTeXMLMath moves from LaTeXMLMath to 1 .
For any LaTeXMLMath -critical point LaTeXMLMath of LaTeXMLMath , we have LaTeXMLEquation .
We denote LaTeXMLEquation and LaTeXMLEquation .
Using Lemma 5.1 and 5.2 and the conditions ( i ) and ( ii ) in the statement of Theorem 7.2 , by adding a small bump function around LaTeXMLMath , we may assume , without loss of generality , that LaTeXMLMath is the unique minimum point of LaTeXMLMath for each LaTeXMLMath and that there is a ‘ gap ’ between LaTeXMLMath and LaTeXMLMath LaTeXMLEquation for all LaTeXMLMath for any LaTeXMLMath .
Similar statement holds for the maximum point LaTeXMLMath .
We will fix LaTeXMLMath later in ( 7.15 ) .
This implies that for any LaTeXMLMath we have LaTeXMLEquation for any LaTeXMLMath -critical point LaTeXMLMath of LaTeXMLMath .
For the proof of Theorem 7.2 , it will be enough to prove Proposition 5.3 .
The rest of this section will be occupied by its proof .
We recall that we considered the linear homotopy LaTeXMLMath , LaTeXMLEquation and then studied the continuation equation LaTeXMLEquation .
Using Lemma 5.2 , after preliminary perturbation of LaTeXMLMath , we may assume that there are only finitely many constant periodic solutions of LaTeXMLMath .
We will construct a solution of the equation ( 7.6 ) in four steps : First by considering the linear homotopy LaTeXMLEquation we construct a cycle LaTeXMLMath with its Floer homology class LaTeXMLMath being non-zero , and which is a linear combination of the form LaTeXMLEquation where LaTeXMLMath ’ s are the uniform critical points of LaTeXMLMath over LaTeXMLMath .
This is an immediate consequence of homological essentialness ( Proposition 4.4 ) of LaTeXMLMath in the local Floer complex LaTeXMLMath and from the Hypothesis above , which implies that there is no quantum contribution for the Floer boundary operator for the LaTeXMLMath -small Hamiltonians in either case .
( See Proposition 7.6 below ) .
Secondly we consider the homotopy LaTeXMLEquation from LaTeXMLMath to LaTeXMLMath .
This step proves that the Novikov cycle LaTeXMLMath of LaTeXMLMath transferred from LaTeXMLMath via the adiabatic homotopy along LaTeXMLMath satisfies the Non-pushing down lemma , i.e , can not be pushed down by the Cauchy-Riemann flow of LaTeXMLMath .
The proof heavily relies on the Hypothesis .
Thirdly we consider the homotopy LaTeXMLEquation from LaTeXMLMath to LaTeXMLMath which is provided by the definition LaTeXMLMath .
Again this step proves that the Novikov cycle of LaTeXMLMath transferred from LaTeXMLMath via the adiabatic homotopy along LaTeXMLMath can not be pushed down by the Cauchy-Riemann flow of LaTeXMLMath .
However its proof do not use the Hypothesis but the fact LaTeXMLMath and the arguments hold in general .
Finally , we glue the homotopies LaTeXMLMath and LaTeXMLMath and deform the glued homotopy LaTeXMLMath to the linear homotopy LaTeXMLEquation .
The arguments in this step are independent of the Hypothesis .
In the rest of this section , we will carry out these steps .
Step I ; from LaTeXMLMath To carry out the first step , it is essential to further analyze the general structure of the boundary operator for the LaTeXMLMath -small Hamiltonians ( not necessarily quasi-autonomous ) like LaTeXMLMath of LaTeXMLMath sufficiently small .
This will be carried out following the argument used in [ §3 , Oh1 ] .
For each time independent LaTeXMLMath , we consider the quantity LaTeXMLEquation .
We choose LaTeXMLMath so small and in particular so that LaTeXMLMath .
We now state the following proposition , which is the analog of [ Proposition 4.1 , Oh1 ] to which we refer its proof ( see also [ Oh7 ] for its complete proof ) .
Let LaTeXMLMath be the Darboux neighborhood of LaTeXMLMath in LaTeXMLMath chosen as before .
Then , for any given LaTeXMLMath and for any fixed time-independent LaTeXMLMath , there exists a constant LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
In particular , such a path has trivial homotopy class and so LaTeXMLEquation .
Moreover , all the other LaTeXMLMath which are not contained in LaTeXMLMath satisfy LaTeXMLEquation for sufficiently small LaTeXMLMath which is independent of LaTeXMLMath .
By the argument similar to [ §8 , Oh1 ] , we deduce that for LaTeXMLMath chosen as above , the boundary map LaTeXMLEquation is decomposed into LaTeXMLEquation such that LaTeXMLMath maps LaTeXMLMath .
Here the part LaTeXMLMath is derived from the ‘ thin ’ trajectories LaTeXMLMath and LaTeXMLMath from ‘ thick ’ trajectories ( or from quantum contributions ) .
In this LaTeXMLMath -small case where the only time-one periodic orbits are the constant ones , this ‘ thin ’ and ‘ thick ’ decomposition coincides with that of homotopically trivial and nontrivial trajectories .
The essential point of imposing the Hypothesis is that under the Hypothesis , LaTeXMLMath and so LaTeXMLEquation .
Now for each given LaTeXMLMath , we define the chain map LaTeXMLEquation along the linear path LaTeXMLEquation by considering the equation LaTeXMLEquation for given LaTeXMLMath and LaTeXMLMath .
The induced homomorphisms LaTeXMLEquation and its local version LaTeXMLEquation induces an isomorphism in homology with its inverse induced by LaTeXMLMath and LaTeXMLMath respectively .
Now we consider a Novikov cycle LaTeXMLEquation .
The following definition which be crucial for the minimax argument we carry out later .
Definition 7.7 .
Let LaTeXMLMath be a Novikov cycle in LaTeXMLMath .
We define the level of the cycle LaTeXMLMath and denote by LaTeXMLEquation if LaTeXMLMath , and just put LaTeXMLMath as usual .
As in ( 7.7 ) , we can choose a cycle LaTeXMLMath for LaTeXMLMath LaTeXMLEquation with LaTeXMLEquation for all LaTeXMLMath , its Floer homology class satisfying LaTeXMLMath .
By considering the local Floer complexes LaTeXMLMath and LaTeXMLMath and their continuation and using the homological essentialness of the maximum point LaTeXMLMath of LaTeXMLMath , we can write LaTeXMLEquation for some LaTeXMLMath for each given LaTeXMLMath so that LaTeXMLMath is a finite union LaTeXMLEquation where LaTeXMLMath ’ s are critical points of LaTeXMLMath .
Lemma 7.8 .
Assume the conditions in Theorem 7.2 .
Let LaTeXMLMath be as above .
Then for any Novikov cycle LaTeXMLMath homologous to LaTeXMLMath , i.e. , satisfying LaTeXMLEquation for some Novikov chain LaTeXMLMath , we have LaTeXMLEquation .
Note that under the main Hypothesis , we have LaTeXMLEquation for sufficiently small LaTeXMLMath .
In other words , all the contributions to the boundary LaTeXMLMath come from ‘ thin ’ trajectories .
Since LaTeXMLMath is the maximum point of LaTeXMLMath , there can not be any such thin trajectory landing at LaTeXMLMath .
Therefore LaTeXMLMath must have contribution from LaTeXMLMath by ( 7.14 ) since LaTeXMLMath does have contribution from LaTeXMLMath .
Hence we must have ( 7.14 ) by definition of the level function LaTeXMLMath .
This finishes proof of the lemma .
∎ Step II : from LaTeXMLMath to LaTeXMLMath In this step we consider the homotopy LaTeXMLEquation .
We perturb this to a generic path LaTeXMLEquation so that it satisfies the genericity condition as in the Handle sliding lemma ( See the paragraph above ( 6.4 ) ) .
By the gap condition and the non over-twisting condition in ( ii ) in Theorem 7.2 , we can continue the fixed extremum points LaTeXMLMath to isolated fixed extremum points of the perturbed path LaTeXMLMath without having small periodic points bifurcated from them .
In particular the perturbed path LaTeXMLMath itself becomes quasi-autonomous .
Without loss of generality , we may assume that these fixed extrema are the same points LaTeXMLMath .
Other contractible periodic orbits of LaTeXMLMath will be bifurcated from the constant periodic orbits of LaTeXMLMath .
More precisely , we have the following lemma .
Lemma 7.9 .
For any given LaTeXMLMath , there exists a generic path LaTeXMLMath in the above sense such that for each LaTeXMLMath , for any contractible periodic orbit LaTeXMLMath of LaTeXMLMath of period one there exists LaTeXMLMath such that The point of Remark 7.3 ( 1 ) is that the length minimizing property holds for the Hamiltonian path LaTeXMLMath which is perturbed from LaTeXMLMath and this Hamiltonian satisfies the property assumed in Theorem I ( i ) .
Indeed the proof below proves that this path is length minimizing .
Using Lemma 5.1 , we then derive the length minimizing property of the LaTeXMLMath itself .
As in §3 , we consider the partition LaTeXMLEquation and denote its mesh of LaTeXMLMath by LaTeXMLEquation .
We also consider the associated piecewise linear homotopy LaTeXMLEquation where LaTeXMLMath is the linear homotopy LaTeXMLEquation .
We call the above piecewise linear homotopy LaTeXMLMath the adiabatic homotopy associated to LaTeXMLMath and the partition LaTeXMLMath .
We also denote the associated chain map LaTeXMLEquation the adiabatic chain map associated to LaTeXMLMath and LaTeXMLMath .
We will just denote LaTeXMLMath and LaTeXMLMath respectively for the adiabatic homotopy and the adiabatic chain map associated to LaTeXMLMath when we do not specify the partition LaTeXMLMath .
Now we choose LaTeXMLMath with LaTeXMLMath so small that LaTeXMLEquation .
LaTeXMLEquation We recall the Handle sliding lemma , Proposition 6.3 , applied to our perturbed family LaTeXMLMath .
It is easy to see from definition that we have LaTeXMLEquation if LaTeXMLMath is sufficiently LaTeXMLMath -close to LaTeXMLMath , where the constants LaTeXMLMath , LaTeXMLMath are defined as in ( 6.3 ) and ( 6.5 ) .
Because there does not occur bifurcation of contractible periodic orbits along the family LaTeXMLMath , a Gromov compactness type argument proves LaTeXMLMath .
We now state a version of Handle sliding lemma that we need in our proof .
Proposition 7.10 .
Let LaTeXMLMath and LaTeXMLMath be as above and LaTeXMLMath be a smooth periodic ( two parameter ) family of compatible almost complex structures .
Let LaTeXMLMath .
Then for any fixed LaTeXMLMath and for any LaTeXMLMath , there exists a constant LaTeXMLMath such that if LaTeXMLMath , any finite energy solution of LaTeXMLEquation must be either satisfies LaTeXMLEquation or LaTeXMLEquation .
Here LaTeXMLMath is the linear path LaTeXMLMath and LaTeXMLMath is the standard function as before .
By choosing LaTeXMLMath and then LaTeXMLMath sufficiently small , we will also make the constant LaTeXMLMath , satisfy LaTeXMLEquation which is possible because LaTeXMLMath depends only on LaTeXMLMath and LaTeXMLMath but independent of LaTeXMLMath .
Next we consider the cycle LaTeXMLEquation and prove the following proposition , where the condition of no quantum contribution enters .
Proposition 7.11 .
( Non-pushing down lemma II ) Let LaTeXMLMath and LaTeXMLMath as in Theorem 7.2 .
Then the cycle LaTeXMLMath has the properties ( 1 ) LaTeXMLMath ( 2 ) Non pushing-down lemma for LaTeXMLMath holds , i.e. , for any Novikov cycle LaTeXMLMath homologous to LaTeXMLMath , we have LaTeXMLEquation .
We consider the family of cycles LaTeXMLEquation for LaTeXMLMath .
We will prove the following properties of the cycle LaTeXMLMath by induction on LaTeXMLMath : ( P1.j ) LaTeXMLMath gets non-trivial contribution from LaTeXMLMath , ( P2.j ) its level satisfies LaTeXMLEquation ( P3.j ) Non pushing down lemma for LaTeXMLMath holds , i.e. , for any Novikov cycle LaTeXMLMath homologous to LaTeXMLMath , we have LaTeXMLEquation .
Once we prove this , Proposition 7.11 will follow by putting LaTeXMLMath .
For LaTeXMLMath , ( P1 ) , ( P2 ) follow from the definition of LaTeXMLMath and ( P3 ) follows from Lemma 7.8 .
Now suppose ( P1-3.j ) hold for LaTeXMLMath and we will prove them ( P1-3.j+1 ) .
We first prove ( P1.j+1 ) and ( P2.j+1 ) .
We note that LaTeXMLEquation is homologous to LaTeXMLMath and so by ( P3.j ) , we have LaTeXMLEquation .
Therefore ( 7.19 ) and ( P2.j ) together with the upper estimate imply LaTeXMLEquation and so LaTeXMLEquation .
This together with Proposition 7.10 and by ( 7.4 ) , also implies that any trajectory starting from the cycle LaTeXMLMath that lands at the critical point realizing the level LaTeXMLMath must be very short : for not very short path LaTeXMLMath staring from LaTeXMLMath a generator of LaTeXMLMath , it follows from ( 7.24 ) LaTeXMLEquation and so LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Here the last inequality follows from ( 7.19 ) , ( 7.20 ) and ( P2.j ) .
Therefore it follows from ( 7.28 ) that such trajectory LaTeXMLMath can not land at a critical point that realizes the level of LaTeXMLMath since LaTeXMLEquation .
Because of ( 7.18 ) and the upper estimate , it follows that any generator LaTeXMLMath with LaTeXMLMath can not land at the critical point of LaTeXMLMath that realizes the level of LaTeXMLMath .
This proves that the only possible path realizing the level of LaTeXMLMath is a very short path LaTeXMLMath such that LaTeXMLEquation .
This prove ( P1.j+1 ) and ( P2.j+1 ) .
Now it remains to prove ( P3.j+1 ) .
We prove this by contradiction .
Suppose that there is a Novikov cycle LaTeXMLMath homologous to LaTeXMLMath i.e. , LaTeXMLEquation but LaTeXMLEquation .
We study the two cases separately : In the case where LaTeXMLMath is weakly exact , ( 7.31 ) indeed implies LaTeXMLEquation by ( 7.18 ) because action depends only on LaTeXMLMath not on the choice of LaTeXMLMath .
Then the upper estimate and ( 7.19 ) and ( 7.20 ) imply LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and hence LaTeXMLEquation .
However ( 7.33 ) is a contradiction to ( P3.j ) since the cycle LaTeXMLMath is homologous to LaTeXMLEquation which is in turn homologous to LaTeXMLMath .
This finishes proof of ( P3.j+1 ) for this case ( 1 ) .
When LaTeXMLMath is autonomous , we use a generic family of LaTeXMLMath of autonomous Hamiltonians LaTeXMLMath which are Morse except at a finite set of LaTeXMLMath ’ s , and of LaTeXMLMath where each LaTeXMLMath is LaTeXMLMath -independent .
Since LaTeXMLMath is the minimum point of LaTeXMLMath , there is no LaTeXMLMath -independent trajectory of LaTeXMLMath landing at LaTeXMLMath .
Therefore any Floer trajectory landing at LaTeXMLMath must be LaTeXMLMath -dependent .
Let the trajectory start at LaTeXMLMath , LaTeXMLMath with LaTeXMLEquation and denote by LaTeXMLMath the corresponding Floer moduli space of connecting trajectories .
The general index formula shows LaTeXMLEquation .
We consider two cases separately : the cases of LaTeXMLMath or LaTeXMLMath .
If LaTeXMLMath , we derive from ( 7.34 ) , ( 7.35 ) that LaTeXMLMath .
This implies that any such trajectory must come with ( locally ) free LaTeXMLMath -action , i.e. , the moduli space LaTeXMLEquation and its stable map compactification have a locally free LaTeXMLMath -action without fixed points .
Therefore after LaTeXMLMath -invariant perturbation LaTeXMLMath via considering the quotient Kuranishi structure [ FOn ] on the quotient space LaTeXMLMath , the corresponding perturbed moduli space LaTeXMLMath becomes empty .
This is because the quotient Kuranishi structure has virtual dimension -1 by the assumption ( 7.34 ) .
We refer to [ FOn ] or [ LT ] for more explanation on this LaTeXMLMath -invariant regularization process .
Now consider the case LaTeXMLMath .
First note that ( 7.34 ) and ( 7.35 ) imply that LaTeXMLMath .
On the other hand , if LaTeXMLMath , the same argument as above shows the perturbed moduli space becomes empty .
It now follows that there is no trajectory of index 1 that land at LaTeXMLMath after the LaTeXMLMath -invariant regularization .
This together with ( 7.31 ) gives rise to a contradiction to ( 7.30 ) as in Lemma 7.8 and finishes the proof of ( P3.j+1 ) for the second case ( 2 ) .
Hence the proof of Proposition 7.11 .
∎ ( 1 ) We would like to note that a ( Morse ) gradient trajectory of the Morse function LaTeXMLMath is not necessarily regular as a Floer gradient trajectory i.e. , as a solution of the perturbed Cauchy-Riemann equation , unless the LaTeXMLMath -norm of LaTeXMLMath is sufficiently small .
The “ slowness ” condition introduced in [ En ] , [ MS ] is related to this problem .
( 2 ) A careful look of the above proofs shows that the only obstacle to extending them to arbitrary quasi-autonomous Hamiltonians on general symplectic manifolds is that Non pushing-down lemma will not be available for the cycle LaTeXMLEquation defined in ( 7.26 ) in case quantum contribution exists for the Floer boundary operator .
This will prevent us from using the deformation argument used in the end of §7 to produce a solution for the continuity equation along the linear path LaTeXMLMath .
Some simpleness condition as in [ BP ] enables us to prove Non-pushing down lemma , which we will investigate further elsewhere .
Step III ; from LaTeXMLMath to LaTeXMLMath Now we consider the homotopy LaTeXMLMath LaTeXMLEquation .
We take a partition LaTeXMLEquation and its associated adiabatic homotopy LaTeXMLMath .
We first recall from Proposition 2.3 that LaTeXMLEquation which is of measure zero subset LaTeXMLMath .
We consider the family of cycles LaTeXMLEquation and its level function LaTeXMLEquation .
Here LaTeXMLMath is the path LaTeXMLMath .
We will provide the proof of the following proposition in the appendix .
The function LaTeXMLMath is continuous and so constant .
In particular , the cycle LaTeXMLEquation has the level LaTeXMLEquation .
With this proposition at out disposal , we prove Proposition 7.14 ( Non pushing-down lemma III ) .
Let LaTeXMLMath be as above .
If a Novikov cycle LaTeXMLMath is homologous to LaTeXMLMath in LaTeXMLMath , i.e. , satisfies LaTeXMLEquation then we must have LaTeXMLEquation .
Suppose the contrary that there exists LaTeXMLMath and LaTeXMLMath with ( 7.37 ) and LaTeXMLEquation satisfied .
We apply the homotopy LaTeXMLMath to ( 7.37 ) .
Composing this with LaTeXMLMath , we get the identity LaTeXMLEquation for the obvious Floer chain homotopy LaTeXMLMath in a standard way .
We apply ( 7.40 ) to the cycle LaTeXMLMath to get LaTeXMLEquation from the definition of LaTeXMLMath in ( 7.41 ) .
Inserting ( 7.41 ) into ( 7.40 ) and using the chain property of LaTeXMLMath , we get LaTeXMLEquation .
Lemma 7.8 implies that LaTeXMLEquation .
On the other hand , using ( 7.39 ) , ( 7.43 ) and the Handle sliding lemma , and applying the proof of Proposition 7.13 in Appendix to LaTeXMLMath backwards , LaTeXMLMath , we prove that the function LaTeXMLMath is continuous and so constant .
In particular , we have LaTeXMLEquation .
Therefore we have proven LaTeXMLEquation .
Now ( 7.43 ) and ( 7.44 ) give rise to a contradiction .
This finishes the proof .
∎ Step IV : from the LaTeXMLMath to LaTeXMLMath Finally we consider the linear homotopy LaTeXMLMath from LaTeXMLMath to LaTeXMLMath LaTeXMLEquation and the associated chain map LaTeXMLEquation ( by connecting LaTeXMLMath and LaTeXMLMath by a generic path LaTeXMLMath ) .
We connect the glued homotopy LaTeXMLMath and LaTeXMLMath by any generic homotopy ( of homotopies ) LaTeXMLMath and consider the parameterized equation LaTeXMLEquation for LaTeXMLMath .
Again this parameterized equation induces the identity LaTeXMLEquation for the corresponding chain homotopy LaTeXMLMath .
Applying this identity to LaTeXMLMath above , we have LaTeXMLEquation .
Since standard gluing theorem in the Floer theory implies LaTeXMLEquation for sufficiently large LaTeXMLMath , we have LaTeXMLEquation .
Obviously LaTeXMLMath is a Novikov cycle in LaTeXMLMath .
Therefore Proposition 7.14 implies that LaTeXMLEquation .
By definition of the chain map LaTeXMLMath and the cycle LaTeXMLMath in ( 7.13 ) , this then implies existence of LaTeXMLMath and LaTeXMLMath for which there exists a solution of the following Cauchy-Riemann equation : LaTeXMLEquation with LaTeXMLEquation .
This is exactly what we wanted to prove for Proposition 5.3 .
This finally finishes the proof of Proposition 5.3 and hence the proof of Theorem I .
In this section , we outline a construction of spectral invariants of Viterbo ’ s type [ V ] ( more precisely , the type the author constructed in [ Oh3,5 ] ) on arbitrary compact symplectic manifolds .
As a consequence , we also define a new invariant norm on the Hamiltonian diffeomorphism group of arbitrary compact symplectic manifolds .
We just illustrate the main idea of the construction in the present paper with minimal possible sophistication in the presentation and refer readers [ Oh7 ] for precise details of the construction .
The starting point of our construction of the invariants will then be the fact that for any fixed generic autonomous Hamiltonian LaTeXMLMath on LaTeXMLMath we have the isomorphism LaTeXMLEquation as a chain complex when LaTeXMLMath is sufficiently small , and the canonical isomorphism LaTeXMLEquation for any Hamiltonian LaTeXMLMath over the Novikov ring LaTeXMLMath .
A natural isomorphism ( 8.2 ) is induced by the chain map LaTeXMLEquation over the linear path LaTeXMLMath .
Here we also note that ( 8.1 ) also induces a canonical isomorphism LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath denote the Morse chain complex and its associated homology of LaTeXMLMath with LaTeXMLMath -coefficients .
By letting LaTeXMLMath , we will have the corresponding limit isomorphism LaTeXMLEquation by identifying the singular cohomology LaTeXMLMath with LaTeXMLMath by realizing its Poincaré dual by a Morse cycle of LaTeXMLMath and then composing with the map ( 8.1 ) .
Definition 8.1 .
Let LaTeXMLMath be a given generic Hamiltonian .
For each LaTeXMLMath , we denote by LaTeXMLMath its Poincaré dual to LaTeXMLMath .
We consider the Floer homology class LaTeXMLMath .
We define the level of the Floer homology class LaTeXMLMath by LaTeXMLEquation .
Of course , a crucial task in this definition is to show that this is well-defined , i.e , the numbers are finite , independent of the choice of the Morse function LaTeXMLMath and behave continuously over LaTeXMLMath ( in LaTeXMLMath -norm ) .
The following theorem is the analog to [ Theorem II , Oh5 ] which can be proved in a similar way .
However we exploit the isomorphism ( 8.1 ) in a crucial way here .
Theorem 8.2 .
Let LaTeXMLMath be a given Hamiltonian .
For each LaTeXMLMath , the number LaTeXMLMath is finite and the assignment LaTeXMLMath can be extended to LaTeXMLMath as a continuous function with respect to LaTeXMLMath -topology of LaTeXMLMath .
The proof will be the same as [ Oh5 ] once we prove finiteness of the value LaTeXMLMath .
To be more precise , we choose a Morse function LaTeXMLMath on LaTeXMLMath and use the chain map ( 8.3 ) .
The homology class LaTeXMLMath considered as a Morse homology class of LaTeXMLMath defines a Floer homology class of LaTeXMLMath which is non-zero by the fact that the Floer boundary operator LaTeXMLMath .
Therefore we have LaTeXMLEquation since LaTeXMLMath .
In fact , by the same calculation as in Proposition 3.2 , we can prove LaTeXMLEquation .
To prove LaTeXMLMath , we first prove the following lemma .
Lemma 8.3 .
We have LaTeXMLEquation .
Let LaTeXMLMath be a Novikov cycle with LaTeXMLMath .
We write LaTeXMLEquation where LaTeXMLMath is the sum of the terms with trivial homotopy class i.e. , those of the type with LaTeXMLMath and LaTeXMLMath are the ones LaTeXMLMath with non-trivial homotopy class with LaTeXMLMath .
Since LaTeXMLMath preserves this decomposition ( no quantum contribution ! )
and since any LaTeXMLMath can be represented by LaTeXMLMath , both LaTeXMLMath and LaTeXMLMath are closed and have LaTeXMLEquation .
By setting LaTeXMLMath in the Morse homology of LaTeXMLMath , we have LaTeXMLMath .
An easy fact from the ( finite dimensional ) Morse homology theory implies LaTeXMLEquation .
Obviously we have LaTeXMLMath which finishes the proof by ( 8.9 ) .
∎ Now we go back to the proof of Theorem 8.2 .
Let LaTeXMLMath with its Floer homology class LaTeXMLMath .
Note that by the same calculation as in Proposition 3.2 along the linear path from LaTeXMLMath to LaTeXMLMath , we have LaTeXMLEquation where we know LaTeXMLMath because LaTeXMLMath since LaTeXMLMath and LaTeXMLMath induces an isomorphism in homology .
On the other hand , let LaTeXMLMath be a representative as in Lemma 8.3 with LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLEquation .
It follows from Lemma 8.3 that LaTeXMLEquation .
From ( 8.10 ) , we derive LaTeXMLEquation .
LaTeXMLEquation Letting LaTeXMLMath , we have proved LaTeXMLEquation and then taking the infimum over LaTeXMLMath with LaTeXMLMath , we derive LaTeXMLEquation which in particular proves LaTeXMLMath .
To prove continuity of LaTeXMLMath in LaTeXMLMath -topology , we imitate the above argument by replacing LaTeXMLMath by another generic Hamiltonian LaTeXMLMath .
As in ( 8.10 ) , we have LaTeXMLEquation .
We have LaTeXMLMath in LaTeXMLMath because LaTeXMLMath in LaTeXMLMath .
From ( 8.15 ) and the definition of LaTeXMLMath , we have LaTeXMLEquation .
This proves LaTeXMLEquation by taking the infimum of LaTeXMLMath over LaTeXMLMath with LaTeXMLMath .
Equivalently we have LaTeXMLEquation .
Next we want to prove LaTeXMLEquation .
We apply ( 8.15 ) with LaTeXMLMath and LaTeXMLMath switched and LaTeXMLMath with LaTeXMLMath and get LaTeXMLEquation or LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLMath is chain homotopic to LaTeXMLMath , we also have LaTeXMLEquation .
Therefore we derive ( 8.17 ) from this by the same argument as that of ( 8.16 ) .
Combining ( 8.16 ) and ( 8.17 ) , we have proved LaTeXMLEquation .
Now it follows from ( 8.18 ) that the function LaTeXMLMath can be extended to LaTeXMLMath as a continuous function in LaTeXMLMath -topology .
This finishes the proof .
∎ These numbers LaTeXMLMath will satisfy the same kind of properties as the invariants constructed by the author in [ Oh5 ] .
We refer to [ Oh3,5 ] for the statements and proofs of the properties of LaTeXMLMath in the context of Lagrangian submanifolds on the cotangent bundle leaving complete details to [ Oh7 ] for the present case .
We now focus on the special cases where the corresponding class LaTeXMLMath is the class 1 in LaTeXMLMath .
Definition LaTeXMLMath Theorem 8.4 [ Oh7 ] .
Let LaTeXMLMath be the identity class of LaTeXMLMath .
For each given Hamiltonian LaTeXMLMath , we define LaTeXMLEquation .
Then we have LaTeXMLMath , and LaTeXMLEquation as long as LaTeXMLMath .
This makes LaTeXMLMath depends only on the equivalence class LaTeXMLMath , i.e , defines a well-defined function on the covering space LaTeXMLMath .
Now for a given Hamiltonian diffeomorphism LaTeXMLMath , we define LaTeXMLEquation for any Hamiltonian diffeomorphism LaTeXMLMath .
The following theorem has been proven in [ Oh7 ] to which we refer the readers .
Theorem 8.5 [ Oh7 ] .
The above function LaTeXMLMath satisfies the following properties : This norm reduces to the norm Schwarz constructed in [ Sc ] for the symplectically aspherical case where the norm LaTeXMLMath is defined by LaTeXMLEquation where LaTeXMLMath is the volume class in LaTeXMLMath , following [ V ] and [ Oh5 ] .
The reason why the two ( 8.19 ) and ( 8.21 ) coincide is that in the aspherical case , we have the additional identity LaTeXMLEquation .
But Polterovich observed [ Po3 ] that this latter identity fails in the non-exact case due to the quantum contribution .
In fact in the non-exact case , even positivity of ( 8.21 ) seems to fail .
It turns out that our definition ( 8.19 ) in Definition 8.4 is the right one to take , which satisfies all the expected properties .
We refer readers to [ Oh7 ] for the proof of Theorem 8.5 and for further consequences of the spectral invariants in the study of length minimizing property of Hofer ’ s geodesics and new lower bounds for the Hofer norm of Hamiltonian diffeomorphisms .
In this appendix , we prove Proposition 7.13 .
Since this proposition is a general fact for arbitrary pairs LaTeXMLMath of Hamiltonians with LaTeXMLMath , we gather the facts from the main part of the paper that are needed and make this appendix self-contained .
We first recall the Handle sliding lemma .
Let LaTeXMLMath be any Hamiltonian and consider the Cauchy-Riemann equation LaTeXMLEquation .
We call a solution LaTeXMLMath trivial if it is LaTeXMLMath -independent , i.e. , stationary .
We define LaTeXMLEquation .
Let LaTeXMLMath and the family LaTeXMLMath be given .
We define LaTeXMLEquation .
In general , this number could be zero .
When it becomes positive , we have the following result .
This is an easy version of Proposition 6.3 Proposition A.1 .
Let LaTeXMLMath be a smooth family of Hamiltonians and LaTeXMLMath be a smooth periodic ( two parameter ) family of compatible almost complex structures .
Suppose that LaTeXMLMath is positive .
Let LaTeXMLMath .
Then for any fixed LaTeXMLMath and for any LaTeXMLMath , there exists a constant LaTeXMLMath such that if LaTeXMLMath , any finite energy solution of LaTeXMLEquation must be either satisfies LaTeXMLEquation or LaTeXMLEquation .
Here LaTeXMLMath is the linear path LaTeXMLMath and LaTeXMLMath is the standard function as before .
As in Proposition 6.3 , we call a solution LaTeXMLMath of ( A.4 ) very short if it satisfies ( A.4 ) and long if it satisfies ( A.6 ) .
Corollary A.2 [ Corollary 6.4 , §6 ] .
Let LaTeXMLMath be any given number .
Then there exists LaTeXMLMath such that for any LaTeXMLMath with LaTeXMLMath , the following holds : if LaTeXMLMath is very short , then LaTeXMLEquation .
LaTeXMLEquation If LaTeXMLMath is not very short , then we have LaTeXMLEquation .
We would like to apply these results to the path LaTeXMLMath .
We first prove Lemma A.3 .
Let LaTeXMLMath be the family of almost complex structures defined by LaTeXMLEquation .
Then we have LaTeXMLEquation .
In particular , we have LaTeXMLEquation .
We first note that the map LaTeXMLEquation and ( 2.4 ) give one-one correspondence between LaTeXMLMath and LaTeXMLMath and between LaTeXMLMath and LaTeXMLMath respectively .
Furthermore ( A.10 ) also provides one-one correspondence between solution sets of the corresponding Cauchy-Riemann equations by LaTeXMLEquation .
And a straightforward calculation shows the identity LaTeXMLEquation which finishes the proof .
∎ We are now ready to provide the proof of Proposition 7.13 .
We choose the partition LaTeXMLEquation so that its mesh LaTeXMLEquation where LaTeXMLMath is defined by LaTeXMLEquation .
We will prove the proposition in 3 steps : finiteness , upper estimates and lower estimates .
Step 1 : finiteness The finiteness of this function follows from the assumption LaTeXMLMath and from construction of the chain map .
More specifically , the chain map LaTeXMLEquation maps Novikov cycles to Novikov cycles and induces an isomorphism in homology over the Novikov rings .
Since we have chosen LaTeXMLMath so that LaTeXMLMath , we have LaTeXMLMath and in particular LaTeXMLMath for all LaTeXMLMath .
Hence comes the finiteness of the level of LaTeXMLMath , i.e , the value of LaTeXMLMath .
Step 2 : upper estimates In this step , we will prove LaTeXMLEquation for LaTeXMLMath with LaTeXMLMath for sufficiently small LaTeXMLMath .
This upper estimates can be proved without help of the Handle sliding lemma .
We recall that the chain map LaTeXMLMath is defined as the composition of chain maps LaTeXMLMath over the linear homotopy for the partition LaTeXMLMath .
We first consider the first segment LaTeXMLMath .
In this segment , we have LaTeXMLEquation over the linear path LaTeXMLMath .
We consider the chain map LaTeXMLMath which is induced by the assignment LaTeXMLEquation for each LaTeXMLMath .
Here LaTeXMLMath denotes the moduli space of trajectories of the Cauchy-Riemann equation LaTeXMLEquation and LaTeXMLMath denotes its ( rational ) Euler number ( see [ FOn ] , [ LT ] , [ Ru ] for the precise meaning ) .
In the case relevant to the chain map the moduli space is zero-dimensional .
In particular , if this number is not zero , then ( A.12 ) has a solution .
Assuming existence of such pair LaTeXMLMath and LaTeXMLMath for the moment , we proceed with the proof .
Then to every pair LaTeXMLMath and LaTeXMLMath for which LaTeXMLMath is non-zero , we have LaTeXMLEquation .
Taking the maximum over LaTeXMLMath among the generators of LaTeXMLMath , we get LaTeXMLEquation .
Since this holds for any generator LaTeXMLMath of LaTeXMLMath , ( A.14 ) ) proves ( A.11 ) by definition of LaTeXMLMath .
Now it remains to prove existence of a pair LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation and LaTeXMLMath contributes LaTeXMLMath and LaTeXMLMath contributes LaTeXMLMath .
We recall that LaTeXMLEquation where LaTeXMLMath is defined by considering parameterized equation induced by the homotopy ( of homotopies ) LaTeXMLMath connecting the linear homotopy between LaTeXMLMath and LaTeXMLMath and the glued homotopy via LaTeXMLMath .
However if LaTeXMLMath is close to LaTeXMLMath and the Cauchy-Riemann equation for LaTeXMLMath is regular , then those corresponding to LaTeXMLMath are all regular for LaTeXMLMath .
Since LaTeXMLMath is defined by counting generic non-regular solutions on LaTeXMLMath , this proves that LaTeXMLMath if LaTeXMLMath is very close to LaTeXMLMath .
Therefore we have LaTeXMLEquation if LaTeXMLMath for sufficiently small LaTeXMLMath .
By definition of the chain map LaTeXMLMath , there must be such a pair of LaTeXMLMath and LaTeXMLMath for which ( A.15 ) holds .
This finishes the proof of ( A.11 ) .
Step 3 : lower estimate This is the place where the Handle sliding lemma plays a crucial role .
We apply LaTeXMLMath to ( A.16 ) to get LaTeXMLEquation .
Therefore LaTeXMLMath is homologous to LaTeXMLMath in LaTeXMLMath because LaTeXMLMath is so .
By Non pushing-down lemma , Proposition 7.14 , we have LaTeXMLEquation .
This gives rise to LaTeXMLEquation .
LaTeXMLEquation Now we choose LaTeXMLMath so small in ( 7.4 ) that we have LaTeXMLEquation .
Then the trajectory constructed in Step 2 that satisfies ( A.15 ) must be very short .
On the other hand for very short trajectories , the lower estimate ( A.7 ) holds .
Combining Step 1-3 , we have proved that the function LaTeXMLMath is continuous and so must be constant on LaTeXMLMath .
Then this also implies Non pushing-down lemma for LaTeXMLMath from which we can repeat the above argument to the segment LaTeXMLMath .
We repeat this to all LaTeXMLMath which finishes the proof of Proposition 7.13 .
PROJECTIVE RANKS OF COMPACT HERMITIAN SYMMETRIC SPACES Amassa Fauntleroy Department of Mathematics North Carolina State University Raleigh , NC 27695 U.S.A. Abstract Let LaTeXMLMath be a compact irreducible Hermitian symmetric space and write LaTeXMLMath , with LaTeXMLMath the group of holomorphic isometries of LaTeXMLMath and LaTeXMLMath the stability group of the point of LaTeXMLMath .
We determine the maximal dimension of a complex projective space embedded in LaTeXMLMath as a totally geodesic submanifold .
AMS Subject Classification : 14L35 , 22F30 , 20G05 Introduction Let LaTeXMLMath be a simply connected compact complex manifold carrying an Hermitian metric of everywhere nonnegative holomorphic bisectional curvature .
In [ 11 ] Mok proved that if the second Betti number LaTeXMLMath , then LaTeXMLMath is biholomorphic to an irreducible compact Hermitan symmetric space .
It was earlier proved by Siu and Yau [ 14 ] and Mori [ 12 ] that when the above curvature is everywhere positive LaTeXMLMath is biholomorphic to a complex projective space .
In each of the three papers cited above the existence of a minimally embedded projective line in LaTeXMLMath plays a crucial role .
It turns out that such minimally embedded projective lines are totally geodesic in LaTeXMLMath .
In this paper we study the maximal totally geodesic complex submanifolds of LaTeXMLMath which are biholomorphic to a complex projective space .
We call the dimension of such a submanifold the projective rank of LaTeXMLMath .
We calculate the projective ranks of each of the irreducible compact Hermitian symmetric spaces .
The results are given in Section 5 .
In Sections 1 through 4 we develop the techniques used to make these calculations and in the last section we discuss the degrees of the holomorphic totally geodesic maps LaTeXMLMath where LaTeXMLMath projective rank of LaTeXMLMath .
We also discuss the conjugacy of these maximal totally geodesic complex projective spaces in LaTeXMLMath under the action of the group of isometries of LaTeXMLMath .
We shall make use of the work of Chen and Nagano [ 5 ] on totally geodesic submanifolds of symmetric spaces .
Several authors have studied the question of minimal or energy minimizing maps from LaTeXMLMath to a compact symmetric space .
The interested reader may consult [ 2 ] and [ 3 ] for connections with the present paper .
It is the recent work of Robert Bryant [ 18 ] which motivates the author to revive these results which were summarized without proofs in the article [ 19 ] .
Unlike Bryant , the approach taken here is very much an algebraic one with only an occasional nod to the topological and analytic methods which underlie many of the foundational results .
We thank the referee of an earlier version of this paper for pointing out an error in our original discussion of the case of the quadrics .
A complex manifold LaTeXMLMath ( noncompact ) with Hermitian metric LaTeXMLMath is an Hermitian symmetric space or H.S.S .
if each point of LaTeXMLMath is an isolated fixed point of an involutive holomorphic isometry of LaTeXMLMath .
Let LaTeXMLMath denote the connected component of the group of holomorphic isometries of LaTeXMLMath .
Then LaTeXMLMath is a connected Lie group which acts transitively on LaTeXMLMath .
Fix LaTeXMLMath and let LaTeXMLMath be the isotropy group of LaTeXMLMath so LaTeXMLMath .
We introduce the following notations : LaTeXMLMath : Lie algebra of LaTeXMLMath LaTeXMLMath : involutive isometry having LaTeXMLMath as isolated fixed point LaTeXMLMath : Lie algebra of LaTeXMLMath LaTeXMLMath : decomposition of LaTeXMLMath into the LaTeXMLMath and LaTeXMLMath eigenspaces if LaTeXMLMath LaTeXMLMath : complexification of LaTeXMLMath LaTeXMLMath : complexification of LaTeXMLMath LaTeXMLMath : compact real form of LaTeXMLMath where LaTeXMLMath ( cf. , [ 7 ; V 2.1 ] ) .
Let LaTeXMLMath be a Cartan subalgebra of LaTeXMLMath .
Then LaTeXMLMath is a Cartan subalgebra of LaTeXMLMath and LaTeXMLMath and LaTeXMLMath is a Cartan subalgebra of LaTeXMLMath .
In fact , LaTeXMLMath is a connected centerless semi-simple Lie group .
The center LaTeXMLMath of LaTeXMLMath is a torus whose dimension is the number of simple direct factors of LaTeXMLMath and LaTeXMLMath is the centralizer in LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath -root system of LaTeXMLMath so LaTeXMLMath LaTeXMLMath : compact roots ; i.e. , LaTeXMLMath -root system of LaTeXMLMath LaTeXMLMath : noncompact roots , i.e. , LaTeXMLMath LaTeXMLMath : central element of LaTeXMLMath such that LaTeXMLMath =ad ( LaTeXMLMath ) induces the complex structure of LaTeXMLMath LaTeXMLMath : LaTeXMLMath -eigenspace of LaTeXMLMath on LaTeXMLMath LaTeXMLMath , parabolic subspace of LaTeXMLMath that is the sum of nonnegative eigenspaces of ad ( LaTeXMLMath ) : LaTeXMLMath LaTeXMLMath : parabolic subgroup of LaTeXMLMath ; i.e. , the complex analytic subgroup corresponding to LaTeXMLMath .
The following theorem summarizes the major classical results which we will need .
Details and proofs may be found in [ 7 ] , [ 17 ] .
Let the notation be as above .
Then ( 1 ) If LaTeXMLMath then LaTeXMLMath is irreducible , LaTeXMLMath is a compact centerless simple Lie group and LaTeXMLMath is a maximal compact proper subgroup of LaTeXMLMath and LaTeXMLMath and has a 1-dimensional center .
( 2 ) LaTeXMLMath is a maximal parabolic subgroup of LaTeXMLMath acts transitively on LaTeXMLMath and LaTeXMLMath .
( 3 ) LaTeXMLMath embeds holomorphically in LaTeXMLMath as an open LaTeXMLMath -orbit , and LaTeXMLMath is a compact H.S.S .
( 4 ) Assume that LaTeXMLMath .
Let LaTeXMLMath be a base for the root system LaTeXMLMath .
Then there is an LaTeXMLMath such that LaTeXMLEquation .
Moreover , the homogeneous line bundle LaTeXMLMath on LaTeXMLMath corresponding to LaTeXMLMath is ample .
Thus LaTeXMLMath is projective algebraic .
Let LaTeXMLMath be a submanifold containing LaTeXMLMath .
Then LaTeXMLMath is totally geodesic in LaTeXMLMath if and only if LaTeXMLMath satisfies LaTeXMLMath – i.e. , LaTeXMLMath is a Lie triple system .
In this case LaTeXMLMath is a Lie subalgebra of LaTeXMLMath .
If LaTeXMLMath is the corresponding analytic subgroup of LaTeXMLMath then LaTeXMLMath acts transitively on LaTeXMLMath and LaTeXMLMath is again a symmetric space .
Note : LaTeXMLMath need not be Hermitian symmetric .
Let LaTeXMLMath be a simply connected symmetric space and LaTeXMLMath a totally geodesic simply connected submanifold of LaTeXMLMath .
Assume that the connected group of isometries of LaTeXMLMath , is semisimple .
Then every isometry of LaTeXMLMath extends to an isometry of LaTeXMLMath .
Proof .
Fix LaTeXMLMath and let LaTeXMLMath be the corresponding Lie triple system .
Let LaTeXMLMath be the corresponding subalgebra of LaTeXMLMath – the Lie algebra of LaTeXMLMath where LaTeXMLMath is the connected component of the isometry group of LaTeXMLMath .
Let LaTeXMLMath be the analytic subgroup corresponding to LaTeXMLMath and LaTeXMLMath the stability group in LaTeXMLMath of LaTeXMLMath .
Then by [ 7 , p. 225 ] LaTeXMLMath .
In particular , there is a natural mapping LaTeXMLMath and by [ 7 , Remark 2 , p. 211 ] the image is a closed analytic subgroup of LaTeXMLMath .
Let LaTeXMLMath be the stability group of LaTeXMLMath in LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath , then since LaTeXMLMath acts transitively on LaTeXMLMath , there is an LaTeXMLMath which LaTeXMLMath .
Thus LaTeXMLMath .
Since LaTeXMLMath is semisimple , the Lie algebra LaTeXMLMath of LaTeXMLMath is spanned by the set LaTeXMLEquation by [ 9 ; Vol .
II , XI 3.2 ( 5 ) ] .
But for LaTeXMLMath in LaTeXMLMath we can write LaTeXMLMath with LaTeXMLMath in LaTeXMLMath .
Since on LaTeXMLMath and LaTeXMLMath is totally geodesic , we have LaTeXMLEquation .
If follows that LaTeXMLMath .
Hence LaTeXMLEquation is surjective .
Since the image of LaTeXMLMath is closed , LaTeXMLMath .
Now LaTeXMLMath is simply connected to there exists a subgroup LaTeXMLMath such that LaTeXMLMath is an isomorphism and LaTeXMLMath .
This proves the lemma .
Let LaTeXMLMath be a totally geodesic Hermitian symmetric subspace of the H.S.S .
M. Assume that LaTeXMLMath and LaTeXMLMath are compact and that LaTeXMLMath is irreducible .
Let LaTeXMLMath and LaTeXMLMath denote the respective linear groups of holomorphic transformations of LaTeXMLMath and LaTeXMLMath .
Then there exists an isogeny LaTeXMLMath and a homomorphism LaTeXMLMath such that LaTeXMLEquation is the inclusion map .
Proof .
Since the connected groups of holomorphic isometries LaTeXMLMath and LaTeXMLMath are dense in the connected components of LaTeXMLMath and LaTeXMLMath ( cf. , [ 7 , p. 211 ] ) respectively , the corollary follows readily from the lemma .
Remark .
Note that if LaTeXMLMath is irreducible and LaTeXMLMath is the Cartan decomposition of the Lie algebra of LaTeXMLMath then LaTeXMLMath generates LaTeXMLMath , i.e. , LaTeXMLMath .
We give now an alternative description of Hermitian symmetric spaces in terms of LaTeXMLMath -spaces .
Let LaTeXMLMath be a connected complex simply connected Lie group with simple Lie algebra LaTeXMLMath of rank LaTeXMLMath .
Let LaTeXMLMath be a Cartan subalgebra of LaTeXMLMath and let LaTeXMLMath denote the set of roots of LaTeXMLMath so LaTeXMLEquation .
Fix a base LaTeXMLMath for LaTeXMLMath and let LaTeXMLMath denote the set of positive roots .
Fix a simple root LaTeXMLMath and put LaTeXMLEquation .
Define subalgebras of LaTeXMLMath by LaTeXMLEquation .
Then LaTeXMLMath is reductive , LaTeXMLMath is nilpotent and LaTeXMLMath .
Let LaTeXMLMath be a compact real form of LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be the complex analytic subgroup of LaTeXMLMath such that Lie LaTeXMLMath and LaTeXMLMath the real analytic subgroup with Lie LaTeXMLMath .
Put LaTeXMLMath .
Then LaTeXMLMath acts transitively on LaTeXMLMath and induces the structure of a compact complex manifold on the homogenous space LaTeXMLMath .
The irreducible compact Hermitian symmetric spaces are given as follows : ( A III ) Grassmannians : LaTeXMLMath ( BDI ) Quadric Hypersurfaces : LaTeXMLEquation ( CI ) LaTeXMLMath D III LaTeXMLMath E III LaTeXMLMath E VII LaTeXMLMath Note that the group LaTeXMLMath defined above is a maximal parabolic subgroup – the standard parabolic defined by the subset LaTeXMLMath of LaTeXMLMath .
Moreover , LaTeXMLMath is a maximal compact subgroup of LaTeXMLMath and the Levi-factor LaTeXMLMath of LaTeXMLMath is the centralizer in LaTeXMLMath of LaTeXMLMath .
The Lie algebra of LaTeXMLMath is LaTeXMLMath .
A somewhat more concrete description of the classical compact H.S.S .
is as follows : ( cf. , [ 17 , p. 321 ] .
( A III ) Grassmannians : LaTeXMLMath is the space of LaTeXMLMath -dimensional subspaces of LaTeXMLMath ( BDI ) The nonsingular quadratic hypersurface in LaTeXMLMath defined by LaTeXMLEquation .
LaTeXMLMath ( CI ) LaTeXMLMath The space consists of LaTeXMLMath -dimensional linear subspaces of LaTeXMLMath annihilated by a nondegenerate skew-symmetric bilinear form .
( D III ) The subvariety LaTeXMLMath of the same Grassmannian LaTeXMLMath consisting of LaTeXMLMath dimensional subspaces annihilated by a nondegenerate symmetric bilinear form .
In the sequel we sometimes want to distinguish between LaTeXMLMath thought of as LaTeXMLMath -dimensional linear subspaces of LaTeXMLMath and as LaTeXMLMath dimensional subspaces of LaTeXMLMath .
In those instances we use the notation LaTeXMLMath or LaTeXMLMath for the Grassmannian thought of as LaTeXMLMath -dimensional subspaces of the LaTeXMLMath -dimensional vector space V. Let LaTeXMLMath be an LaTeXMLMath dimensional complex vector space and LaTeXMLMath a positive integer with LaTeXMLMath .
Let LaTeXMLMath denote the Grassmann manifold of LaTeXMLMath dimensional subspaces of LaTeXMLMath or equivalently LaTeXMLMath -dimensional linear subspaces of LaTeXMLMath .
The manifold LaTeXMLMath is Hermitian symmetric of type A III .
If LaTeXMLMath is a compact complex submanifold of LaTeXMLMath which has everywhere positive holomorphic bisectional curvature then by [ 12 ] or [ 14 ] LaTeXMLMath is biholomorphic to a complex projective space .
If LaTeXMLMath is also totally geodesic in LaTeXMLMath then LaTeXMLMath is a symmetric subspace of LaTeXMLMath .
In this section we study such submanifolds LaTeXMLMath which have minimal degree in LaTeXMLMath .
Let LaTeXMLMath denote the universal subbundle over LaTeXMLMath and LaTeXMLMath the universal quotient bundle .
Then LaTeXMLEquation is exact and the fiber over LaTeXMLMath of LaTeXMLMath is the subspace of LaTeXMLMath represented by the point LaTeXMLMath .
For a holomorphic embedding LaTeXMLMath we define the degree of LaTeXMLMath to be the degree of the line bundle LaTeXMLMath where LaTeXMLMath is the dual of LaTeXMLMath .
As usual for a sequence of integers LaTeXMLMath we define the Schubert variety LaTeXMLMath as follows : Fix a sequence of subspaces LaTeXMLEquation with LaTeXMLMath .
Then LaTeXMLEquation .
The classes of these cycles in LaTeXMLMath generate the homology .
Suppose LaTeXMLMath for LaTeXMLMath .
Then the Schubert variety LaTeXMLMath is just the set of subspaces LaTeXMLMath which are contained in LaTeXMLMath .
Hence LaTeXMLMath .
The Plücker embedding of LaTeXMLMath is determined by the line bundle LaTeXMLMath .
For any Schubert variety LaTeXMLMath of dimension LaTeXMLMath we have ( cf. , [ 6 ; p. 274 ] ) LaTeXMLEquation where LaTeXMLMath .
In particular for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Now let LaTeXMLMath be a holomorphic embedding .
We say that LaTeXMLMath is minimal if LaTeXMLMath and that LaTeXMLMath is linear if LaTeXMLMath factors as LaTeXMLMath where LaTeXMLMath is a linear embedding of LaTeXMLMath as a subspace of LaTeXMLMath and LaTeXMLMath is an isomorphism LaTeXMLMath with a Schubert variety LaTeXMLMath in LaTeXMLMath .
The embedding LaTeXMLMath is totally geodesic if LaTeXMLMath is a totally geodesic submanifold of LaTeXMLMath .
Let LaTeXMLMath be an embedding .
Then LaTeXMLMath is minimal if and only if it is linear .
Proof .
A linear embedding is clearly minimal .
Conversely , if LaTeXMLMath is minimal then LaTeXMLMath is a rank LaTeXMLMath vector bundle generated by its global sections .
By [ 13 ; p. 22 ] , LaTeXMLEquation with LaTeXMLMath .
Since LaTeXMLMath we have LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Now consider the pull back of the universal sequence LaTeXMLEquation to LaTeXMLMath .
We have LaTeXMLEquation .
Let LaTeXMLMath be a global frame for the trivial bundle LaTeXMLMath such that LaTeXMLMath span the trivial LaTeXMLMath of LaTeXMLMath over LaTeXMLMath .
For any point LaTeXMLMath the corresponding subspace LaTeXMLMath must lie in the subspace LaTeXMLMath spanned by LaTeXMLMath .
Thus LaTeXMLMath hence LaTeXMLMath is linear .
LaTeXMLMath For a line LaTeXMLMath in LaTeXMLMath we have a simple description of LaTeXMLMath as a Schubert variety .
Let LaTeXMLMath .
There is a LaTeXMLMath -dimensional subspace LaTeXMLMath of LaTeXMLMath such that LaTeXMLEquation .
In particular , if LaTeXMLMath are in LaTeXMLMath then LaTeXMLEquation .
So if LaTeXMLMath Let LaTeXMLMath be a minimal embedding .
Then LaTeXMLMath is linear .
Proof .
Let LaTeXMLMath denote two distinct lines in LaTeXMLMath .
Then there exists subspaces LaTeXMLMath and LaTeXMLMath of LaTeXMLMath of dimension LaTeXMLMath and subspaces LaTeXMLMath of dimension LaTeXMLMath such that LaTeXMLEquation .
I claim LaTeXMLMath .
Suppose not .
Then since LaTeXMLMath there is a common subspace LaTeXMLMath of LaTeXMLMath and LaTeXMLMath of dimension LaTeXMLMath .
Thus LaTeXMLMath and LaTeXMLMath , LaTeXMLMath both lie in LaTeXMLMath where LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are both subspaces of LaTeXMLMath we can find a basis LaTeXMLMath of LaTeXMLMath such that the following conditions are fulfilled LaTeXMLEquation .
Then the subspace LaTeXMLMath is in LaTeXMLMath and LaTeXMLMath is in LaTeXMLMath .
Hence LaTeXMLEquation .
But LaTeXMLMath and LaTeXMLMath correspond to points in LaTeXMLMath so lie in a minimally embedded line LaTeXMLMath .
By Lemma 2.1 LaTeXMLMath is linearly embedded so by the above remark LaTeXMLEquation .
This contradiction leads to the desired conclusion LaTeXMLMath and hence LaTeXMLMath so is linearly embedded .
LaTeXMLMath Let us fix a basis LaTeXMLMath of LaTeXMLMath establishing a fixed isomorphism with LaTeXMLMath .
Let LaTeXMLMath be the subspace spanned by LaTeXMLMath .
Then without loss of generality one may assume that a linearly embedded LaTeXMLMath has image LaTeXMLMath .
Restricting the universal sequence to LaTeXMLMath gives LaTeXMLEquation .
If LaTeXMLMath then the corresponding subspace LaTeXMLMath lies in LaTeXMLMath so the LaTeXMLMath vectors LaTeXMLMath thought of as global sections of LaTeXMLMath map onto global sections of LaTeXMLMath free from relations .
This implies that LaTeXMLEquation where LaTeXMLMath is a line bundle .
Thus since LaTeXMLMath .
Conversely , if LaTeXMLMath we can find LaTeXMLMath global sections of LaTeXMLMath which map onto generators of the trivial factor of LaTeXMLMath .
Extending these to a basis LaTeXMLMath for LaTeXMLMath yields LaTeXMLMath for each LaTeXMLMath .
Let LaTeXMLMath be an embedding with LaTeXMLMath .
Then LaTeXMLMath is minimal if and only if it is linear .
Proof .
It suffices by the above remark to show that LaTeXMLMath .
Now for LaTeXMLMath or LaTeXMLMath the Lemmas 2.1 , 2.2 yield the result .
If LaTeXMLMath we have for any line LaTeXMLMath , LaTeXMLMath is minimal so LaTeXMLMath .
It follows that LaTeXMLMath is uniform - i.e. , for each LaTeXMLMath with LaTeXMLMath constant .
By Lemma 2.2 , LaTeXMLMath splits on a LaTeXMLMath also .
Then by a theorem of Horrocks [ 13 , p. 42 ] LaTeXMLMath splits over LaTeXMLMath so LaTeXMLMath is linear .
LaTeXMLMath The above arguments actually yield a more general result .
Let LaTeXMLMath be a vector bundle on LaTeXMLMath of rank LaTeXMLMath and generated by LaTeXMLMath global sections .
If LaTeXMLMath is uniform of type LaTeXMLMath then LaTeXMLMath splits on LaTeXMLMath , i.e. , LaTeXMLMath .
Proof .
By hypothesis there exist a surjection LaTeXMLEquation .
Let ( * ) : LaTeXMLMath be the corresponding exact sequence of locally free sheaves .
Over LaTeXMLMath we obtain LaTeXMLEquation .
It follows that the morphism LaTeXMLMath defined by ( * ) restricts to a linear embedding on each LaTeXMLMath .
The proof of Lemma 2.2 shows that LaTeXMLMath is then linear on each LaTeXMLMath .
By the remarks preceding Theorem 2.3 LaTeXMLMath splits on LaTeXMLMath into LaTeXMLMath .
By the theorem of Horrocks [ 13 , p. 42 ] LaTeXMLMath splits over LaTeXMLMath .
LaTeXMLMath In addition , we can obtain the following restriction on the existence of maps from projective spaces into Grassmannians .
If LaTeXMLMath then every holomorphic map LaTeXMLMath from LaTeXMLMath to LaTeXMLMath with LaTeXMLMath is constant .
Proof .
Suppose LaTeXMLMath and let LaTeXMLMath be a linear subspace of dimension LaTeXMLMath .
If LaTeXMLMath is holomorphic map from LaTeXMLMath to LaTeXMLMath with LaTeXMLMath then for every hyperplane LaTeXMLMath , the restriction of LaTeXMLMath to LaTeXMLMath is a minimal hence linear embedding .
Suppose LaTeXMLMath and LaTeXMLMath are two distinct hyperplanes in LaTeXMLMath and let LaTeXMLMath be a line in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
We can find LaTeXMLMath such that LaTeXMLEquation and LaTeXMLMath of dimension LaTeXMLMath such that LaTeXMLEquation .
Just as in the proof of Lemma 2.2 we can conclude that LaTeXMLMath .
It follows that LaTeXMLMath for every hyperplane LaTeXMLMath so LaTeXMLMath .
But then LaTeXMLMath and the only morphism from LaTeXMLMath to LaTeXMLMath are known to be constant .
LaTeXMLMath The interested reader may compare the above result with the results of Tango in [ 15 ] and [ 16 ] .
Let LaTeXMLMath be a compact H.S.S .
and LaTeXMLMath a totally geodesic compact Hermitian symmetric subspace of LaTeXMLMath .
If LaTeXMLMath is isomorphic to LaTeXMLMath as H.S.S. , then we have seen in Corollary 1.3 that there is a homomorphism LaTeXMLEquation where LaTeXMLMath is the group of analytic automorphisms of LaTeXMLMath .
If LaTeXMLMath is a Grassmannian , say LaTeXMLMath , then LaTeXMLMath yields a homomorphism which we again call LaTeXMLMath LaTeXMLEquation so LaTeXMLMath is a representation of LaTeXMLMath on LaTeXMLMath .
We want to describe these representations when LaTeXMLMath .
Let LaTeXMLMath be the fundamental dominant weights for LaTeXMLMath .
We fix the maximal torus LaTeXMLMath to be the diagonal subgroup and use the standard description of the roots as in [ 8 , p.64 ] .
In particular , if LaTeXMLMath the base for the root system ( B = upper triangular matrices ) is given by LaTeXMLEquation and LaTeXMLEquation for LaTeXMLMath .
A straightforward computation gives LaTeXMLEquation .
If LaTeXMLMath are the standard basis elements for LaTeXMLMath then LaTeXMLMath is the highest weight vector corresponding to LaTeXMLMath .
Hence LaTeXMLEquation .
Recall that for a dominant weight LaTeXMLMath Weyl ’ s character formula ( cf .
[ 7 , VII , 10.2 , 11.2 ] ) gives LaTeXMLEquation .
Following Humphries [ ibid ] we can use the co-root LaTeXMLMath instead of LaTeXMLMath .
Then for LaTeXMLMath height LaTeXMLMath LaTeXMLEquation if LaTeXMLMath is a root of height LaTeXMLMath and LaTeXMLMath Let LaTeXMLMath with LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -module of dimension LaTeXMLMath .
Assume that LaTeXMLMath .
Then LaTeXMLMath contains a trivial LaTeXMLMath -module of dimension LaTeXMLMath as a direct summand .
If LaTeXMLMath and LaTeXMLMath then LaTeXMLMath contains a trivial summand of dimension LaTeXMLMath .
Proof .
Let LaTeXMLMath be a nontrivial irreducible submodule of LaTeXMLMath with highest weight LaTeXMLMath .
I claim that LaTeXMLMath .
Since LaTeXMLMath if LaTeXMLMath it follows that if LaTeXMLMath in the lexicographic ordering then LaTeXMLMath .
If LaTeXMLMath with LaTeXMLMath for some index LaTeXMLMath with LaTeXMLMath then LaTeXMLMath .
But LaTeXMLMath for LaTeXMLMath in this range and LaTeXMLMath .
It follows that LaTeXMLMath if LaTeXMLMath .
Consider now LaTeXMLMath .
Again LaTeXMLMath .
If LaTeXMLMath then LaTeXMLEquation .
Similarly , if LaTeXMLMath then LaTeXMLMath .
Thus the hypothesis LaTeXMLMath implies LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath then LaTeXMLEquation .
Hence we get a contradiction unless LaTeXMLMath is LaTeXMLMath or LaTeXMLMath .
The proposition now follows .
LaTeXMLMath Proposition 3.1 will be used in Section 5 to determine the projective rank of the Grassmann manifolds .
In [ 5 ] B. Y. Chen and T. Nagano give a method of classifying compact symmetric spaces in terms of certain totally geodesic submanifolds .
In this section we summarize their method and in the next section apply it to the problem of determining the projective ranks of the compact irreducible Hermitian symmetric spaces .
Let LaTeXMLMath be a symmetric space , LaTeXMLMath the origin and LaTeXMLMath the geodesic symmetry of LaTeXMLMath fixing the point LaTeXMLMath .
A smooth closed geodesic through LaTeXMLMath , or circle LaTeXMLMath for short , has an antipodal point LaTeXMLMath with LaTeXMLMath .
Write LaTeXMLMath where LaTeXMLMath is the group of isometries of LaTeXMLMath ( the closure of the group generated by all the symmetries LaTeXMLMath and LaTeXMLMath is the isotropy group of LaTeXMLMath .
Let LaTeXMLMath denote the orbit LaTeXMLMath .
Then LaTeXMLMath is a totally geodesic submanifold of LaTeXMLMath [ 5 2.1 ] .
Let LaTeXMLMath be the involution of LaTeXMLMath given by LaTeXMLMath .
If LaTeXMLMath is the Cartan decomposition of the Lie algebra LaTeXMLMath Lie LaTeXMLMath with respect to LaTeXMLMath then LaTeXMLMath leaves LaTeXMLMath stable ( LaTeXMLMath and LaTeXMLMath commute ) and induces the decomposition LaTeXMLEquation of LaTeXMLMath into the LaTeXMLMath and LaTeXMLMath eigenspaces of LaTeXMLMath .
Now LaTeXMLMath so LaTeXMLMath forms a Lie triple system .
Consider the connected subgroup LaTeXMLMath of LaTeXMLMath corresponding to the Lie subalgebra LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLMath is totally geodesic in LaTeXMLMath and the tangent space to LaTeXMLMath at LaTeXMLMath is the normal space to LaTeXMLMath at LaTeXMLMath in LaTeXMLMath ( see [ 5 , 2.2 ] ) .
Given a pair of antipodal points LaTeXMLMath on a circle LaTeXMLMath in LaTeXMLMath we have the system LaTeXMLMath .
The group LaTeXMLMath acts naturally on the set of all such systems .
Denote by LaTeXMLMath the orbit set .
If LaTeXMLMath is an isometric totally geodesic embedding then there arises a mapping LaTeXMLMath induced by the mapping carrying LaTeXMLMath into LaTeXMLMath .
Moreover , LaTeXMLMath and LaTeXMLMath .
We say LaTeXMLMath is a pairwise totally geodesic immersion .
The following theorem summarizes the results of Chen-Nagano which we shall use .
[ 5 , Section 5 ] Let LaTeXMLMath be a point fixed by LaTeXMLMath in the compact symmetric space LaTeXMLMath .
For LaTeXMLMath let LaTeXMLMath denote the fixed point set of LaTeXMLMath .
Then ( 1 ) LaTeXMLMath is the set of antipodal points on circles through LaTeXMLMath .
( 2 ) For each LaTeXMLMath there exists an involutive automorphism LaTeXMLMath such that LaTeXMLEquation ( 3 ) The rank of LaTeXMLMath equals the rank LaTeXMLMath and if LaTeXMLMath is connected , the rank LaTeXMLMath equals the rank of LaTeXMLMath .
( 4 ) If a totally geodesic submanifold LaTeXMLMath of LaTeXMLMath has the same rank as LaTeXMLMath then LaTeXMLEquation ( 5 ) LaTeXMLMath is globally determined by LaTeXMLMath ; i.e. , The set of isomorphism classes of compact irreducible symmetric spaces is in one-to-one correspondence with the set of the corresponding LaTeXMLMath .
The following table gives the pairs LaTeXMLMath and LaTeXMLMath for the compact irreducible Hermitian symmetric spaces .
In this table LaTeXMLMath , and LaTeXMLMath denote respectively the circle , the unoriented real Grassmann manifold LaTeXMLMath , the complex Grassmann manifold LaTeXMLMath , and the quaternion Grassmann manifold LaTeXMLMath .
A determination is outlined in [ 5 , p. 409 ] .
Table 4.2 4.3 An Example Let LaTeXMLMath .
Then LaTeXMLMath and rank LaTeXMLMath .
All geodesics in LaTeXMLMath are circles , have the same length and are permuted transitively by LaTeXMLMath [ 7 , VII , 10.2 , 11.2 ] .
To find the possible geodesic pairs in LaTeXMLMath it suffices to consider just one circle .
It is well-known that any projective line LaTeXMLMath ( embedded linearly ; i.e. , of degree 1 ) is totally geodesic in LaTeXMLMath so we may take the circle LaTeXMLMath to lie in LaTeXMLMath .
Viewing LaTeXMLMath as the set of LaTeXMLMath -dimensional subspaces of LaTeXMLMath , LaTeXMLMath is the Schubert variety LaTeXMLEquation .
Let LaTeXMLMath be an orthonormal basis for LaTeXMLMath such that LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath is an element of LaTeXMLMath then LaTeXMLMath with LaTeXMLMath .
The assignment LaTeXMLMath where LaTeXMLMath are homogeneous point coordinates in LaTeXMLMath defines an isomorphism of LaTeXMLMath with LaTeXMLMath .
We take our circle LaTeXMLMath to be defined by LaTeXMLEquation .
Let LaTeXMLMath .
Then since LaTeXMLMath so LaTeXMLMath .
Hence we have a natural surjective map LaTeXMLEquation where LaTeXMLMath .
Since LaTeXMLMath is a connected totally geodesic subspace and is a proper subspace of LaTeXMLMath and since LaTeXMLMath is simply connected this map is an isomorphism and LaTeXMLMath .
It follows that LaTeXMLMath is the unique totally geodesic pair for LaTeXMLMath .
We determine the projective ranks of the irreducible compact Hermitian symmetric spaces in this section .
We use the classification of Cartan as given in Table 4.2 .
We will show by induction on LaTeXMLMath that LaTeXMLMath embedded linearly in LaTeXMLMath is a maximal totally geodesic complex projective submanifold of LaTeXMLMath .
If LaTeXMLMath there is nothing to prove since LaTeXMLMath .
Assume the result is known for all LaTeXMLMath with LaTeXMLMath where LaTeXMLMath as usual .
Let LaTeXMLMath be a maximal totally geodesic complex projective submanifold in LaTeXMLMath .
Since LaTeXMLMath is totally geodesic in LaTeXMLMath when embedded linearly , LaTeXMLMath .
Suppose LaTeXMLMath .
Using Corollary 1.3 we can find a nontrivial homomorphism LaTeXMLEquation .
Since LaTeXMLMath .
Thus LaTeXMLMath must split as an LaTeXMLMath -module by Proposition 3.1 .
Say LaTeXMLMath .
Let LaTeXMLMath correspond to the subspace LaTeXMLMath .
Since LaTeXMLMath has codimension at least LaTeXMLMath in LaTeXMLMath .
Hence if LaTeXMLMath is the stability group of LaTeXMLMath in LaTeXMLMath and LaTeXMLMath then LaTeXMLMath is not a complex projective space because LaTeXMLMath .
This contradiction implies LaTeXMLMath .
Thus the projective rank of LaTeXMLMath is LaTeXMLMath .
Note that the maximal complex projective spaces LaTeXMLMath in LaTeXMLMath are linearly embedded , are permuted transitively by LaTeXMLMath and are parametrized by the LaTeXMLMath -dimensional subspaces of LaTeXMLMath , i.e. , by LaTeXMLMath .
The Hermitian symmetric space LaTeXMLEquation is biholomorphic to the complex quadric in LaTeXMLMath LaTeXMLEquation .
We have the following result from Chen-Nagano [ 4 ] : Let LaTeXMLMath be a totally geodesic complete connected Riemannian submanifold of LaTeXMLMath .
Then 1 .
The embedding of LaTeXMLMath in LaTeXMLMath is unique up to an isometry of LaTeXMLMath .
2 .
If LaTeXMLMath is maximal in LaTeXMLMath then LaTeXMLMath is one of the following : ( i ) LaTeXMLMath ( ii ) a local Riemannian product of two spheres LaTeXMLMath ( iii ) If LaTeXMLMath as Riemannian manifold 3 .
If LaTeXMLMath is not maximal then either LaTeXMLMath or LaTeXMLMath – the real projective space of dimension LaTeXMLMath if LaTeXMLMath .
LaTeXMLMath The canonical decomposition of the Lie algebra LaTeXMLMath is cf .
[ 9 , Vol .
II , p. 278 ] LaTeXMLMath where LaTeXMLEquation and LaTeXMLEquation .
The complex structure of LaTeXMLMath is given by LaTeXMLMath .
An inner product on LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath denotes the standard Euclidean product of LaTeXMLMath and LaTeXMLMath is a positive scalar .
Both LaTeXMLMath and LaTeXMLMath are invariant under the adjoint action of LaTeXMLMath represented as LaTeXMLEquation .
If LaTeXMLMath is a typical element of LaTeXMLMath as above and LaTeXMLMath then LaTeXMLEquation where the right hand side is a matrix product .
The complex structure is the natural one so that LaTeXMLEquation .
Thus the action of LaTeXMLMath is the natural action of LaTeXMLMath viewed as a real subgroup of LaTeXMLMath .
The expression for LaTeXMLMath may be read as LaTeXMLEquation where LaTeXMLMath is the complex vector with entries LaTeXMLMath .
Let LaTeXMLMath be a Lie triple system which is LaTeXMLMath -invariant .
Then LaTeXMLMath where the orthogonal splitting is relative to the standard Hermitian metric on LaTeXMLMath .
Since LaTeXMLMath is the real part of this metric ( up to scalar factor LaTeXMLMath ) we have an exclusion of LaTeXMLMath , into LaTeXMLMath and a Lie algebra homomorphism LaTeXMLMath compatible with adjoint actions on LaTeXMLMath .
This includes an inclusion of symmetric pairs LaTeXMLEquation and it follows readily that the totally geodesic submanifold determined by LaTeXMLMath is the complex quadric LaTeXMLMath imbedded in LaTeXMLMath in some position .
The only totally geodesic complex submanifolds of LaTeXMLMath are the complex quadrics LaTeXMLMath and the complex projective spaces LaTeXMLMath .
Proof .
Note that the assertion 2 ( iii ) of 4.2 says that for LaTeXMLMath is a totally geodesic submanifold .
When LaTeXMLMath , let LaTeXMLMath have a complex structure LaTeXMLMath .
Relative to this complex structure LaTeXMLMath embeds in LaTeXMLMath .
Similarly for LaTeXMLMath .
Any LaTeXMLMath -dimensional complex subspace has a natural induced orientation as a LaTeXMLMath -dimensional real space .
Thus LaTeXMLMath can be mapped to LaTeXMLMath as follows : Let LaTeXMLMath be the standard orthonormal base for LaTeXMLMath and for LaTeXMLMath an LaTeXMLMath -plane write LaTeXMLMath as orthogonal direct sum .
Orient LaTeXMLMath so that LaTeXMLMath and LaTeXMLMath have the same orientation – LaTeXMLMath and LaTeXMLMath are in the same orbit under LaTeXMLMath .
Then orient LaTeXMLMath so that LaTeXMLMath has the same orientation as LaTeXMLMath .
Define the image of LaTeXMLMath to be the oriented LaTeXMLMath -plane LaTeXMLMath in LaTeXMLMath .
The Lie algebra of LaTeXMLMath embeds in LaTeXMLMath by LaTeXMLEquation from this one can check that LaTeXMLMath is a complex embedding .
Let LaTeXMLMath be a H.S.S .
of type CI .
Then LaTeXMLMath is the subset of LaTeXMLMath consisting of LaTeXMLMath -dimensional subspaces of LaTeXMLMath which are totally isotropic with respect to a skew symmetric bilinear form [ 17 , p. 232 ] .
We take the matrix of this form to be LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath -identity matrix .
Let LaTeXMLMath be the subspace defined by LaTeXMLMath where LaTeXMLMath are the usual coordinates of LaTeXMLMath .
Let LaTeXMLMath be the subspace defined by the equations LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath represent points in LaTeXMLMath and LaTeXMLMath .
If we take the base point to be LaTeXMLMath then LaTeXMLMath where LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
It follows from this description that LaTeXMLMath is a totally geodesic submanifold of LaTeXMLMath .
According to Table 4.2 we expect that the projective rank of LaTeXMLMath is LaTeXMLMath .
To see this consider LaTeXMLMath .
An easy calculation shows that LaTeXMLMath and that LaTeXMLMath is totally isotropic in LaTeXMLMath .
This induces a morphism LaTeXMLMath exhibiting a map LaTeXMLMath .
In fact , since LaTeXMLMath the above map is just the assignment LaTeXMLMath for LaTeXMLMath an LaTeXMLMath plane .
Let LaTeXMLMath be the element such that LaTeXMLMath .
Then LaTeXMLMath acts in LaTeXMLMath by LaTeXMLMath .
Under this action LaTeXMLMath acts transitively on LaTeXMLMath .
Since LaTeXMLMath is LaTeXMLMath and LaTeXMLMath equivariant , LaTeXMLMath is totally geodesic in LaTeXMLMath .
Thus the projective rank of LaTeXMLMath is at least LaTeXMLMath .
Since LaTeXMLMath is totally geodesic in LaTeXMLMath the maximal possible projective rank for LaTeXMLMath is LaTeXMLMath .
But if LaTeXMLMath is an LaTeXMLMath -dimensional subspace then LaTeXMLMath can not be contained in LaTeXMLMath .
Indeed , let LaTeXMLMath be completely singular of codimension one and suppose LaTeXMLMath .
We have LaTeXMLMath ( this is easily seen for LaTeXMLMath and since LaTeXMLMath acts transitively the same holds for LaTeXMLMath ) choose codimension LaTeXMLMath subspaces LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath and put LaTeXMLMath .
If each LaTeXMLMath is completely singular then LaTeXMLMath a contradiction .
Thus no LaTeXMLMath lies in LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the H.S.S .
of type DIII ( LaTeXMLMath ) .
Then LaTeXMLMath may be identified with the submanifold if LaTeXMLMath consisting of completely singular LaTeXMLMath -dimensional subspaces with respect to the form LaTeXMLMath .
Let LaTeXMLMath be the standard basis for LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath are completely singular .
For LaTeXMLMath a codimension one subspace , LaTeXMLMath and LaTeXMLMath is completely singular .
Thus we have a map LaTeXMLMath which one checks as in 5.3 is totally geodesic .
Just as in 5.3 , no LaTeXMLMath can lie in LaTeXMLMath which is itself totally geodesic in LaTeXMLMath .
Hence the projective rank of LaTeXMLMath is also LaTeXMLMath .
The exception space is LaTeXMLMath .
According to Table 4.2 , LaTeXMLMath contains the unique symmetric pair LaTeXMLMath .
We have seen that for any LaTeXMLMath the symmetric pair for LaTeXMLMath is LaTeXMLMath .
As the projective rank of DIII ( 5 ) is 4 , by [ 5 , p. 409 ] the pair LaTeXMLMath is totally geodesically embeddable in LaTeXMLMath .
It follows that the maximal possible value of LaTeXMLMath is 5 .
From the Dynkin diagrams We see that there exists a nontrivial homomorphism LaTeXMLMath into LaTeXMLMath .
From [ 7 , p. 507 ] the Cartan involution for LaTeXMLMath is induced by the permutation ( 1,6 ) ( 3,5 ) of the vertices of the Dynkin diagram for LaTeXMLMath .
Thus LaTeXMLMath is stable under the Cartan involution .
In particular , there exists a Lie triple system LaTeXMLMath such that LaTeXMLMath .
This together with the fact from Table 4.2 that LaTeXMLMath sits in LaTeXMLMath tells us that indeed the projective rank of EIII is 5 .
Let LaTeXMLMath be the H.S.S .
of type E VII .
The unique maximal symmetric pair for LaTeXMLMath given by Table 4.2 is LaTeXMLEquation .
Since LaTeXMLMath we have a symmetric pair LaTeXMLEquation contained in LaTeXMLMath which suggest LaTeXMLMath .
Again by considering the Dynkin diagram for LaTeXMLMath we see that LaTeXMLMath is in fact a subalgebra so again we have LaTeXMLMath .
Since LaTeXMLMath with LaTeXMLMath we have LaTeXMLMath .
Thus the corresponding totally geodesic subspace is of type AIII and is LaTeXMLMath .
Hence the projective rank of LaTeXMLMath is LaTeXMLMath .
Summarizing we have the following result .
The projective ranks of the irreducible compact Hermitian symmetric spaces are as follows : ( i ) pr [ AIII ( n , d ) ] = d , n/2 LaTeXMLMath ( ii ) pr [ BDI ( m ) ] = [ LaTeXMLMath ] ( iii ) pr [ CI ( n ) ] = LaTeXMLMath ( iv ) pr [ DIII ( n ) ] = LaTeXMLMath ( v ) pr [ EIII ] = 5 ( vi ) pr [ EVII ] = 6 .
Let LaTeXMLMath be an irreducible Hermitian symmetric space with complex algebraic group of automorphisms LaTeXMLMath and stability group of LaTeXMLMath the group LaTeXMLMath .
As a LaTeXMLMath -space LaTeXMLMath has a canonical ample line bundle LaTeXMLMath defined as follows : If LaTeXMLMath is a Levi decomposition of LaTeXMLMath with LaTeXMLMath the unipotent radical of LaTeXMLMath , then LaTeXMLMath is the connected centralizer in LaTeXMLMath of a torus LaTeXMLMath .
Let LaTeXMLMath be a maximal torus in LaTeXMLMath containing LaTeXMLMath such that the root system LaTeXMLMath is relative to LaTeXMLMath .
If LaTeXMLMath is the root which defines the LaTeXMLMath -space , LaTeXMLMath and hence LaTeXMLMath induces a character on LaTeXMLMath – the central character – which will again be called LaTeXMLMath .
Let LaTeXMLMath be the corresponding homogeneous line bundle on LaTeXMLMath .
Then LaTeXMLMath is the sheaf of sections of LaTeXMLMath .
It is known that LaTeXMLMath is very ample and we measure the degree of a subvariety LaTeXMLMath relative to the projective embedding defined by this line bundle .
Suppose LaTeXMLMath is a totally geodesic holomorphic embedding with image LaTeXMLMath .
To find the degree of LaTeXMLMath , or LaTeXMLMath , it suffices to find the degree of the restriction of LaTeXMLMath to any projective line LaTeXMLMath linearly embedded in LaTeXMLMath .
With a view toward the conjugacy problem in mind we begin with the following well-known result : There exists totally geodesic projective lines in LaTeXMLMath of degree one .
Proof .
We treat each type separately .
If LaTeXMLMath is of type AIII the result follows from the fact that LaTeXMLMath has degree one in LaTeXMLMath .
Suppose LaTeXMLMath is of type BDI .
Viewing LaTeXMLMath as the complex quadric LaTeXMLMath in LaTeXMLMath we have a natural totally geodesic embedding of LaTeXMLMath in LaTeXMLMath as LaTeXMLEquation .
Consider the holomorphic isomorphism LaTeXMLEquation given by LaTeXMLEquation .
The image of the line LaTeXMLMath is LaTeXMLEquation which clearly has degree one in the ambient projective space .
By Mok [ 11 , Section 1 ] , LaTeXMLMath is totally geodesic in LaTeXMLMath hence the proposition holds for LaTeXMLMath in general .
Now suppose LaTeXMLMath is of type LaTeXMLMath say LaTeXMLMath .
As in Section 5 we view LaTeXMLMath as LaTeXMLMath -dimensional completely singular subspaces of LaTeXMLMath .
With LaTeXMLMath and LaTeXMLMath as in the last section we consider the mapping LaTeXMLEquation given by LaTeXMLMath .
It is easily verified that the image lies in LaTeXMLMath .
Since the image also lies in LaTeXMLMath where LaTeXMLMath it readily follows that the degree of the image is one and , by Mok [ i bid ] again , the image is totally geodesic in LaTeXMLMath .
When LaTeXMLMath is of type DIII a completely analogous argument leads to the desired conclusion .
Finally , the exceptional cases EIII and EVII follow from the descriptions of the maximal totally geodesic complex projective spaces in these manifolds corresponding to the embeddings of LaTeXMLMath and LaTeXMLMath and the fact that the central characters in these cases restrict appropriately to LaTeXMLMath projective rank of EIII ( respectively EVIII ) .
LaTeXMLMath One might hope that the calculation of the degree of LaTeXMLMath could be accomplished using the above proposition .
Unfortunately this is not the case .
Moreover , the degree of a totally geodesic embedding LaTeXMLMath is not a constant function of LaTeXMLMath .
The point is made with the following : Example 6.2 There exists a totally geodesic holomorphic embedding of LaTeXMLMath into LaTeXMLMath of degree 2 .
Consider the subalgebra LaTeXMLMath of LaTeXMLMath consisting of matrices LaTeXMLEquation .
Then LaTeXMLMath is stable under the cartan involution LaTeXMLMath of LaTeXMLMath .
If LaTeXMLMath is the LaTeXMLMath -eigenspace then LaTeXMLEquation so LaTeXMLMath is also LaTeXMLMath -stable .
The corresponding totally geodesic submanifold for this Lie triple system is therefore a complex submanifold .
Consider the two elements LaTeXMLMath and LaTeXMLMath of LaTeXMLMath ; LaTeXMLEquation with LaTeXMLMath as base point in LaTeXMLMath The geodesics given by LaTeXMLMath and LaTeXMLMath are LaTeXMLEquation .
These curves lie on the algebraic curve LaTeXMLEquation in LaTeXMLMath which is the image ( up to linear automorphism in LaTeXMLMath ) of the veronese map of degree 2 mapping LaTeXMLMath into the plane LaTeXMLMath in LaTeXMLMath .
In particular , we have a totally geodesic complex projective line embedded in LaTeXMLMath having degree 2 and not 1 .
LaTeXMLMath We turn now to the spaces of type CI .
Let LaTeXMLMath and LaTeXMLMath be the subspaces of LaTeXMLMath given in Section 5.3 .
We have a morphism ( 6.3 ) LaTeXMLMath given by LaTeXMLMath .
From the definition of the universal subbundle on LaTeXMLMath is follows easily that LaTeXMLMath where LaTeXMLMath denotes projection onto the LaTeXMLMath -th factor .
Consider the morphism LaTeXMLMath given by LaTeXMLMath .
Let LaTeXMLMath be the graph of this mapping in LaTeXMLMath .
Then LaTeXMLMath is precisely the totally geodesic submanifold of LaTeXMLMath given in 5.3 .
We want to compute the degree of the map LaTeXMLMath where LaTeXMLMath .
More precisely we compute the determinant of the bundle LaTeXMLMath .
Since LaTeXMLMath it follows that LaTeXMLMath .
Evidently , LaTeXMLMath .
Now LaTeXMLMath is the line bundle corresponding to LaTeXMLMath on LaTeXMLMath and hence is generated by LaTeXMLMath -global sections .
Thus LaTeXMLMath is a line bundle on LaTeXMLMath generated by LaTeXMLMath -global sections so LaTeXMLMath .
It follows that LaTeXMLMath .
Hence LaTeXMLEquation .
An entirely analogous argument shows that if LaTeXMLMath is of type DIII , then the degree of the embedding given in 5.4 is also 2 .
Summarizing we have Proposition 6.4 .
Let LaTeXMLMath be an irreducible compact Hermitian symmetric space of type CI or DIII .
Let LaTeXMLMath be a totally geodesic complex submanifold biholomorphic to LaTeXMLMath .
Then LaTeXMLMath .
Proof .
We have only to show that the case LaTeXMLMath can not occur .
If LaTeXMLMath then by Theorem 2.3 there is an LaTeXMLMath -dimensional subspace LaTeXMLMath such that LaTeXMLMath is a hyperplane in LaTeXMLMath .
Such a hyperplane consists of all LaTeXMLMath -dimensional subspaces of LaTeXMLMath containing a fixed line , say LaTeXMLMath .
But LaTeXMLMath must be isotropic in this case .
As we have noted in 5.3 , LaTeXMLMath itself can not be totally singular .
Thus there exists LaTeXMLMath with LaTeXMLMath .
The subspace spanned by LaTeXMLMath and LaTeXMLMath is contained in an LaTeXMLMath -dimensional subspace of LaTeXMLMath so there exists at least one such LaTeXMLMath -plane which is not totally isotropic .
It follows that no hyperplane in LaTeXMLMath can be contained entirely in LaTeXMLMath and hence LaTeXMLMath can not be one .
LaTeXMLMath Our next goal is to show that in fact the degree of LaTeXMLMath is precisely 2 when LaTeXMLMath is of type LaTeXMLMath or DIII .
We will need a closer examination of totally geodesic complex projective subspaces of LaTeXMLMath of dimension LaTeXMLMath .
Let LaTeXMLMath be such a submanifold .
Then by 3.1 there exists a homomorphism LaTeXMLMath and a corresponding map LaTeXMLMath .
Let LaTeXMLMath be a fixed LaTeXMLMath -dimension subspace with orbit under LaTeXMLMath isomorphic to LaTeXMLMath .
Let LaTeXMLMath denote the isotropy group of LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath .
Lemma 6.5 .
Let LaTeXMLMath be a compact subgroup of LaTeXMLMath and suppose LaTeXMLMath is biholomorphic to LaTeXMLMath .
Then LaTeXMLMath is conjugate to LaTeXMLMath in LaTeXMLMath .
Proof .
Identify LaTeXMLMath with LaTeXMLMath so that LaTeXMLMath is the stability group of a point .
Then LaTeXMLMath .
By [ 35 , Theorem 6.1ii ) ] R is a maximal connected proper subgroup of LaTeXMLMath so the desired equality follows .
LaTeXMLMath The LaTeXMLMath -module LaTeXMLMath decomposes as a direct sum LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is reductive each LaTeXMLMath decomposes as a sum of irreducible LaTeXMLMath -modules : LaTeXMLEquation .
Thus LaTeXMLMath decomposes as LaTeXMLMath -module LaTeXMLEquation .
Now the submodule LaTeXMLMath is LaTeXMLMath -stable and not fixed by LaTeXMLMath so LaTeXMLMath as LaTeXMLMath -modules .
More precisely we have an element LaTeXMLMath with image LaTeXMLMath .
Suppose for definiteness that LaTeXMLMath .
Then LaTeXMLEquation .
By Schur ’ s lemma LaTeXMLMath must be of the form LaTeXMLMath with LaTeXMLMath .
Thus LaTeXMLMath as LaTeXMLMath -module .
From this last equality we see that an element LaTeXMLMath in LaTeXMLMath maps LaTeXMLMath into LaTeXMLMath .
It now follows that the orbit map LaTeXMLMath of LaTeXMLMath into LaTeXMLMath is the same as the map described in 6.3 and hence LaTeXMLMath as desired .
LaTeXMLMath We put the above calculations together and summarize the findings in the following results .
Theorem 6.6 .
Let LaTeXMLMath be an irreducible Hermitian symmetric space with connected isometry group LaTeXMLMath .
Let LaTeXMLMath be a totally geodesic complex projective subspace of LaTeXMLMath , with LaTeXMLMath .
Let LaTeXMLMath be the degree of LaTeXMLMath in LaTeXMLMath .
Then ( i ) LaTeXMLMath if LaTeXMLMath is of type AIII , EIII , or EVII .
( ii ) LaTeXMLMath if LaTeXMLMath is of CI or DIII .
( iii ) LaTeXMLMath or LaTeXMLMath if LaTeXMLMath is of type BDI .
Moreover , all such submanifolds LaTeXMLMath of LaTeXMLMath of minimal degree are conjugate under LaTeXMLMath .
Proof .
The assertions in ( i ) and ( ii ) have already been established .
As to ( iii ) , by the discussion preceding 5.3 we see that each totally geodesic LaTeXMLMath is contained in a complex quadric LaTeXMLMath .
Now LaTeXMLMath as the hypersurface LaTeXMLMath .
Since LaTeXMLMath is a complex submanifold or LaTeXMLMath , its embedding in LaTeXMLMath must be by a complete linear system LaTeXMLMath .
If LaTeXMLMath , then the forms of degree LaTeXMLMath can not satisfy the single quadratic relation above and yield an embedding so LaTeXMLMath is at most 2 and ( iii ) follows .
As for the conjugacy assertion , again in the case of type AIII this follows immediately .
If LaTeXMLMath is of type BDI , then we need only consider LaTeXMLMath .
Then the result follows from the fact that LaTeXMLMath permutes the two-dimensional LaTeXMLMath -invariant Lie triple systems in LaTeXMLMath and clearly preserves the degree .
To treat the types CI and DIII we make use of the following : Lemma 6.7 .
Let LaTeXMLMath be a compact irreducible Hermitian symmetric space of type AIII , CI or DIII and LaTeXMLMath a totally geodesic holomorphic isometric embedding of degree one .
Then every geodesic circle in LaTeXMLMath has minimal length .
Proof .
Since CI and DIII are totally geodesic in AIII it suffices to prove the result for LaTeXMLMath .
For LaTeXMLMath the image of LaTeXMLMath is contained in LaTeXMLMath and a geodesic circle is given by ( see example # 3 ) LaTeXMLEquation .
LaTeXMLMath .
One checks easily that this has minimal length in LaTeXMLMath and the lemma follows .
LaTeXMLMath By [ 7 , VII , 11.2 ] the closed geodesics of minimal length in LaTeXMLMath are all conjugate .
Now in the case of CI ( LaTeXMLMath ) or DIII ( LaTeXMLMath ) we can consider the geodesic ( in the notation of 5.3 ) LaTeXMLEquation .
The mid-point corresponds to the subspace LaTeXMLMath and the orbit under LaTeXMLMath is precisely LaTeXMLEquation i.e. , the image of the map in 6.3 .
Since the closed geodesics in LaTeXMLMath are conjugate so are the submanifolds which arise from 6.3 , 6.4 .
Finally , we discuss the exceptional types EIII and EVII .
In each case according to Table 4.1 there exists a unique Hermitian pair LaTeXMLMath .
Considering first the case EIII , LaTeXMLEquation .
If LaTeXMLMath are the images of two totally geodesic embeddings of LaTeXMLMath in LaTeXMLMath , then we may first assume LaTeXMLMath by the result for DIII .
Then LaTeXMLMath and LaTeXMLMath meet along a hyperplane and have the same normal space in EIII so coincide .
If LaTeXMLMath , then LaTeXMLEquation .
Again for LaTeXMLMath images of totally geodesic maps from LaTeXMLMath into LaTeXMLMath we may assume LaTeXMLMath so LaTeXMLMath and LaTeXMLMath meet along a LaTeXMLMath and have the same normal space in EVII so coincide .
This completes the proof .
Solving inverse scattering problem for a discrete Sturm–Liouville operator with a rapidly decreasing potential one gets reflection coefficients LaTeXMLMath and invertible operators LaTeXMLMath , where LaTeXMLMath is the Hankel operator related to the symbol LaTeXMLMath .
The Marchenko–Faddeev theorem ( in the continuous case ) [ 6 ] and the Guseinov theorem ( in the discrete case ) [ 4 ] , guarantees the uniqueness of solution of the inverse scattering problem .
In this article we ask the following natural question — can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators LaTeXMLMath are invertible ?
Can one claim that uniqueness implies invertibility or vise versa ?
Moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials .
So we merge here two mostly developed cases of inverse problem for Sturm–Liouville operators : the inverse problem with ( almost ) periodic potential and the inverse problem with the fast decreasing potential .
Asymptotics of polynomials orthogonal on a homogeneous set , we described earlier [ 8 ] , indicated strongly that there should be a scattering theory for Jacobi matrix with almost periodic background like it exists in the classical case of a constant background .
Note that here left and right asymptotics are not necessary the same almost periodic coefficient sequences , but they are of the same spectral class .
In this work we present all important ingredients of such theory : reflection/transmission coefficients , Gelfand–Levitan–Marchenko transformation operators , a Riemann–Hilbert problem related to the inverse scattering problem .
At last now we can say that the reflectionless Jacobi matrices with homogeneous spectrum are those whose reflection coefficient is zero .
Moreover , we extend theory in depth and show that a reflection coefficient determine uniquely a Jacobi matrix of the Szegö class and both transformation operators are invertible if and only if the spectral density satisfies matrix LaTeXMLMath condition [ 10 ] .
Concerning LaTeXMLMath condition in the inverse scattering we have to mention , at least as indirect references , the book [ 7 , Chapter 2 , Sect .
4 ] and the paper [ 1 ] .
Generally references to stationary scattering and inverse scattering problems in connections with spatial asymptotics can be found in [ 3 ] , where explicit expressions of transmission and reflection coefficients in terms of Weyl functions and phases asymptotic wave functions was given .
Let LaTeXMLMath be a Jacobi matrix defining a bounded self–adjoint operator on LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath is the standard basis in LaTeXMLMath , LaTeXMLMath .
The resolvent matrix–function is defined by the relation LaTeXMLEquation where LaTeXMLMath in such a way that LaTeXMLEquation .
This matrix–function possesses an integral representation LaTeXMLEquation with a LaTeXMLMath matrix–measure having a compact support on LaTeXMLMath .
LaTeXMLMath is unitary equivalent to the operator multiplication by independent variable on LaTeXMLEquation .
The spectrum of LaTeXMLMath is called absolutely continuous if the measure LaTeXMLMath is absolutely continuous with respect to the Lebesgue measure on the real axis , LaTeXMLEquation .
Let LaTeXMLMath be a Jacobi matrix with constant coefficients , LaTeXMLMath ( so called Chebyshev matrix ) .
It has the following functional representation , besides the general one mentioned above .
Note that the resolvent set of LaTeXMLMath is the domain LaTeXMLMath .
Let LaTeXMLMath be a uniformization of this domain , LaTeXMLMath .
With respect to the standard basis LaTeXMLMath in LaTeXMLEquation the matrix of the operator of multiplication by LaTeXMLMath , is the Jacobi matrix LaTeXMLMath , since LaTeXMLMath .
The famous Bernstein–Szeqö theorem implies the following proposition .
Let LaTeXMLMath be a Jacobi matrix whose spectrum is an interval LaTeXMLMath .
Assume that the spectrum is absolutely continuous and the density of the spectral measure satisfies the condition LaTeXMLEquation .
Then LaTeXMLEquation .
Moreover , there exist generalized eigenvectors LaTeXMLEquation .
LaTeXMLEquation such that the following asymptotics hold true LaTeXMLEquation .
LaTeXMLEquation in LaTeXMLMath .
To clarify the meaning of the words ” generalized eigenvectors ” we need some definitions and notation .
The matrix LaTeXMLEquation is called the scattering matrix–function .
It is a unitary–valued matrix–function with the following symmetry property : LaTeXMLEquation and the analytic property : LaTeXMLEquation .
We still denote by LaTeXMLMath , the values of the function inside the disk , and in the sequel , we assume that LaTeXMLMath meets the normalization condition LaTeXMLMath .
In fact , these mean that each of the entries LaTeXMLMath ( so called reflection coefficient ) determines the matrix LaTeXMLMath in unique way .
Indeed , since LaTeXMLEquation using ( 0.11 ) , we have LaTeXMLEquation .
Then , we can solve for LaTeXMLMath the relation LaTeXMLEquation .
With the function LaTeXMLMath we associate the metric LaTeXMLEquation .
LaTeXMLEquation Note that the conditions ( 0.11 ) , ( 0.12 ) guarantee that LaTeXMLMath implies LaTeXMLMath .
We denote by LaTeXMLMath or LaTeXMLMath ( for shortness ) the closer of LaTeXMLMath with respect to this new metric .
The following relation sets a unitary map from LaTeXMLMath to LaTeXMLMath : LaTeXMLEquation moreover , in this case , LaTeXMLEquation and the inverse map is of the form LaTeXMLEquation .
We say that a Jacobi matrix LaTeXMLMath with the spectrum LaTeXMLMath is of Szegö class if its spectral measure LaTeXMLMath satisfies ( 0.4 ) , ( 0.5 ) .
Let LaTeXMLMath be a Jacobi matrix of Szegö class with the spectrum LaTeXMLMath .
Then there exists a unique unitary–valued matrix–function LaTeXMLMath of the form ( 0.9 ) possessing the properties ( 0.10 ) , ( 0.11 ) , and a unique pair of Fourier transforms LaTeXMLEquation determining each other by the relations LaTeXMLEquation and possessing the following analytic properties LaTeXMLEquation and the asymptotic properties LaTeXMLEquation where LaTeXMLEquation ( As before , LaTeXMLMath is the standard basis in LaTeXMLMath ) .
Show that ( 0.17 ) is equivalent to ( 0.8 ) .
Due to LaTeXMLEquation ( 0.17 ) is equivalent to ( LaTeXMLMath ) LaTeXMLEquation .
LaTeXMLEquation Using ( 0.15 ) , we rewrite the second relation into the form LaTeXMLEquation .
Substituting LaTeXMLMath , we get the second relation of ( 0.8 ) .
A fundamental question is how to recover the Jacobi matrix from the scattering matrix , in fact , from the reflection coefficient LaTeXMLMath ( or LaTeXMLMath ) ?
When can this be done ?
Do we have a uniqueness theorem ?
We show that for an arbitrary function LaTeXMLMath satisfying LaTeXMLEquation there exists a Jacobi matrix LaTeXMLMath of Szegö class such that LaTeXMLMath is its reflection coefficient .
But we can construct a matrix with this property , at least , in two different ways .
First , consider the space LaTeXMLEquation and introduce the Hankel operator LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath is the Riesz projection from LaTeXMLMath onto LaTeXMLMath .
This operator determines the metric in LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Under the assumptions ( 0.19 ) , the space LaTeXMLMath is a space of holomorphic functions with a reproducing kernel .
Moreover , LaTeXMLMath for any LaTeXMLMath , and the reproducing vector LaTeXMLMath : LaTeXMLEquation is of the form LaTeXMLEquation .
Put LaTeXMLMath .
Let LaTeXMLMath satisfy ( 0.19 ) .
Then the system of functions LaTeXMLMath forms an orthonormal basis in LaTeXMLMath .
With respect to this basis , operator multiplication by LaTeXMLMath is a Jacobi matrix LaTeXMLMath of Szegö class .
Moreover , the initial function LaTeXMLMath is the reflection coefficient of the scattering matrix–function LaTeXMLMath , associated to LaTeXMLMath by Theorem 0.1 , and LaTeXMLEquation .
From the other hand , the system of functions LaTeXMLMath forms an orthonormal basis in LaTeXMLMath , and we are able to define a Jacobi matrix LaTeXMLMath by the relation LaTeXMLEquation where LaTeXMLMath is the dual system to the system LaTeXMLMath ( see ( 0.15 ) ) , i.e .
: LaTeXMLEquation .
Even the invertibility condition for the operators LaTeXMLMath does not guarantee that operators LaTeXMLMath and LaTeXMLMath are the same .
But if LaTeXMLMath , then the uniqueness theorem takes place .
Let LaTeXMLMath satisfy ( 0.19 ) .
Then the reflection coefficient LaTeXMLMath determines a Jacobi matrix LaTeXMLMath of Szegö class in a unique way if and only if the following relations take place LaTeXMLEquation .
Let LaTeXMLMath be a Jacobi matrix of Szegö class with the spectrum LaTeXMLMath and let LaTeXMLMath be the density of its spectral measure .
If LaTeXMLEquation then there is no other Jacobi matrix of Szegö class with the same scattering matrix–function LaTeXMLMath .
It is important to know , when the operators LaTeXMLMath , playing a central role in the inverse scattering problem , are invertible in the proper sense of words .
Let LaTeXMLMath be a Jacobi matrix of Szeqö class with the spectrum LaTeXMLMath .
Let LaTeXMLMath be the density of its spectral measure and let LaTeXMLMath be the reflection coefficient of its scattering matrixÑ-function .
Then the following statements are equivalent .
1 .
The spectral density LaTeXMLMath satisfies condition LaTeXMLMath .
2 .
The reflection coefficient LaTeXMLMath determines a Jacobi matrix of Szeqö class uniquely and both operators LaTeXMLMath are invertible .
To extend these results to the case when a spectrum LaTeXMLMath is a finite system of intervals or a Cantor set of positive measure , we need only to introduce a counterpart of Hardy space .
Let LaTeXMLMath be a uniformization of the domain LaTeXMLMath .
Thus there exists a discrete subgroup LaTeXMLMath of the group LaTeXMLMath consisting of elements of the form LaTeXMLEquation such that LaTeXMLMath is automorphic with respect to LaTeXMLMath , i.e. , LaTeXMLMath , and any two preimages of LaTeXMLMath are LaTeXMLMath –equivalent , i.e. , LaTeXMLEquation .
We normalize LaTeXMLMath by the conditions LaTeXMLMath , LaTeXMLMath .
A character of LaTeXMLMath is a complex–valued function LaTeXMLMath , satisfying LaTeXMLEquation .
The characters form an Abelian compact group denoted by LaTeXMLMath .
For a given character LaTeXMLMath , as usual let us define LaTeXMLEquation .
Generally , a group LaTeXMLMath is said to be of Widom type if for any LaTeXMLMath the space LaTeXMLMath is not trivial ( contains a non–constant function ) .
A group of Widom type acts dissipatively on LaTeXMLMath with respect to LaTeXMLMath , that is there exists a measurable ( fundamental ) set LaTeXMLMath , which does not contain any two LaTeXMLMath –equivalent points , and the union LaTeXMLMath is a set of full measure .
We can choose LaTeXMLMath possessing the symmetry property : LaTeXMLMath .
For the space of square summable functions on LaTeXMLMath ( with respect to the Lebesgue measure ) , we use the notation LaTeXMLMath .
Let LaTeXMLMath be an analytic function in LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then we put LaTeXMLEquation .
Notice that LaTeXMLMath , means that the form LaTeXMLMath is invariant with respect to the substitutions LaTeXMLMath ( LaTeXMLMath is an Abelian integral on LaTeXMLMath ) .
Analogically , LaTeXMLMath , means that the form LaTeXMLMath is invariant with respect to these substitutions .
We recall , that a function LaTeXMLMath is of Smirnov class , if it can be represented as a ratio of two functions from LaTeXMLMath with an outer denominator .
Let LaTeXMLMath be a group of Widom type .
The space LaTeXMLMath ( LaTeXMLMath ) is formed by functions LaTeXMLMath , which are analytic on LaTeXMLMath and satisfy the following three conditions LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLMath is a Hilbert space with the reproducing kernel LaTeXMLMath , moreover LaTeXMLEquation .
Put LaTeXMLEquation .
We need one more special function .
The Blaschke product LaTeXMLEquation is called the Green ’ s function of LaTeXMLMath with respect to the origin .
It is character–automorphic function , i.e. , there exists LaTeXMLMath such that LaTeXMLMath .
Note , if LaTeXMLMath denotes the Green ’ s function of the domain LaTeXMLMath , then LaTeXMLEquation .
Let LaTeXMLMath be a group of Widom type .
The following statements are equivalent : 1 ) The function LaTeXMLMath is continuous on LaTeXMLMath .
2 ) LaTeXMLMath , LaTeXMLMath .
3 ) The Direct Cauchy Theorem holds : LaTeXMLEquation 4 ) Let LaTeXMLMath .
Then LaTeXMLEquation 5 ) Every invariant subspace LaTeXMLMath ( i.e .
LaTeXMLMath ) is of the form LaTeXMLEquation for some character–automorphic inner function LaTeXMLMath .
A measurable set LaTeXMLMath is homogeneous if there is an LaTeXMLMath such that LaTeXMLEquation .
A Cantor set of positive length is an example of a homogeneous set .
Let LaTeXMLMath be a homogeneous set , then the domain LaTeXMLMath ( respectively the group LaTeXMLMath ) is of Widom type and the Direct Cauchy Theorem holds .
Recall , that a sequence of real numbers LaTeXMLMath is called uniformly almost periodic if the set of sequences LaTeXMLMath is a precompact in LaTeXMLMath .
The general way to produce a sequence of this type looks as follows : let LaTeXMLMath be a compact Abelian group , and let LaTeXMLMath be a continuous function on LaTeXMLMath , then LaTeXMLEquation is an almost periodic sequence .
A Jacobi matrix is almost periodic if the coefficient sequences are almost periodic .
We denote by LaTeXMLMath the class of almost periodic Jacobi matrices with absolutely continuous homogeneous spectrum LaTeXMLMath .
In what follows the class LaTeXMLMath plays a role of Chebyshev matrix .
In fact , if LaTeXMLMath then LaTeXMLMath .
Let LaTeXMLMath be a homogeneous set .
Let LaTeXMLMath be a uniformizing mapping .
Then the systems of functions LaTeXMLMath and LaTeXMLMath form an orthonormal basis in LaTeXMLMath and in LaTeXMLMath , respectively , for any LaTeXMLMath .
With respect to this basis , the operator multiplication by LaTeXMLMath is a three–diagonal almost periodic Jacobi matrix LaTeXMLMath .
Moreover , LaTeXMLEquation and LaTeXMLMath is a continuous function on LaTeXMLMath .
We say that a Jacobi matrix LaTeXMLMath with the spectrum LaTeXMLMath is of Szegö class if its spectral measure is absolutely continuous , LaTeXMLMath , and LaTeXMLMath satisfies ( 0.5 ) .
Let LaTeXMLMath be a Jacobi matrix of Szegö class with a homogeneous spectrum LaTeXMLMath .
Then there exists a unique unitary–valued matrix–function LaTeXMLMath of the form ( 0.9 ) possessing the properties ( 0.10 ) , ( 0.11 ) , and a unique pair of Fourier transforms LaTeXMLEquation determining each other by the relations LaTeXMLEquation and possessing the following analytic properties LaTeXMLEquation and the asymptotic properties LaTeXMLEquation where LaTeXMLEquation and LaTeXMLMath is the closer of the functions from LaTeXMLMath with respect to the metric LaTeXMLEquation .
Theorems 0.2–0.4 also have their closely parallel counterparts in the case when the spectrum is a homogeneous set .
Let LaTeXMLMath be a homogeneous set .
Let LaTeXMLMath be a uniformization and LaTeXMLMath be the Green ’ s function .
Throughout the paper we assume that LaTeXMLMath .
Let LaTeXMLMath be a symmetric fundamental set ( LaTeXMLMath ) .
With a function LaTeXMLMath such that LaTeXMLEquation we associate the metric LaTeXMLEquation .
LaTeXMLEquation Condition ( 1.1 ) guarantee that LaTeXMLMath implies LaTeXMLMath .
We denote by LaTeXMLMath or LaTeXMLMath ( for shortness ) the closer of LaTeXMLMath with respect to this metric .
The operator multiplication by LaTeXMLMath in LaTeXMLMath is unitary equivalent to the operator multiplication by LaTeXMLMath in LaTeXMLMath Let us put LaTeXMLEquation .
In this case LaTeXMLMath .
The system of identities LaTeXMLEquation .
LaTeXMLEquation finishes the proof.∎ Let LaTeXMLMath .
In what further , we assume that LaTeXMLMath and LaTeXMLEquation .
We define an outer function LaTeXMLMath , LaTeXMLMath , by the relation LaTeXMLEquation .
It is a character–automorphic function such that LaTeXMLMath .
It is convenient to denote its character by LaTeXMLMath , i.e. , LaTeXMLMath .
Let us discuss some properties of the space LaTeXMLEquation .
First of all , we define ” a Hankel operator ” LaTeXMLMath , LaTeXMLEquation .
Note , that this operator , indeed , does not depend on ” an analytical part ” of its symbol , more precisely , LaTeXMLEquation .
Besides , in the classical case LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , with a function LaTeXMLEquation is associated the operator LaTeXMLMath having the representation LaTeXMLEquation with respect to the standard basis LaTeXMLMath in LaTeXMLMath .
The operator LaTeXMLMath determines the metric in LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Under the assumptions ( 1.2 ) , the space LaTeXMLMath is a space of holomorphic functions with a reproducing kernel .
Moreover , LaTeXMLMath for any LaTeXMLMath , and the reproducing vector LaTeXMLMath : LaTeXMLEquation is of the form LaTeXMLEquation .
From the inequality LaTeXMLEquation it follows that LaTeXMLEquation .
Thus , if a sequence LaTeXMLMath , LaTeXMLMath , converges in LaTeXMLMath , then the sequence LaTeXMLMath converges in LaTeXMLMath .
In the same way we have boundedness of the functional LaTeXMLMath , LaTeXMLEquation .
Let us prove ( 1.3 ) .
Let LaTeXMLMath , then for the norm of the difference we have an estimate LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , LaTeXMLEquation .
Besides , ( 1.5 ) implies that ( 1.3 ) follows from the relation LaTeXMLEquation .
Let us prove ( 1.7 ) .
Since the function LaTeXMLEquation decreases with LaTeXMLMath and it is bounded by ( 1.6 ) , there exists a limit LaTeXMLEquation .
From the other hand , for any LaTeXMLMath and LaTeXMLMath the following inequalities hold LaTeXMLEquation .
LaTeXMLEquation that is LaTeXMLEquation .
Putting LaTeXMLMath , we have LaTeXMLEquation .
Comparing this inequality with ( 1.8 ) , we get ( 1.7 ) , thus ( 1.3 ) is proved.∎ We define LaTeXMLMath by LaTeXMLEquation .
In this case LaTeXMLEquation is a unitary–valued matrix function possessing properties ( 0.10 ) , ( 0.11 ) .
The following relation sets a unitary map from LaTeXMLMath to LaTeXMLMath : LaTeXMLEquation .
Moreover , in this case , LaTeXMLEquation and the inverse map is of the form LaTeXMLEquation .
These follow from the identities LaTeXMLEquation and ( 0.18 ) .∎ Let LaTeXMLMath .
The system of functions LaTeXMLMath forms an orthonormal basis in LaTeXMLMath when LaTeXMLMath and in LaTeXMLMath when LaTeXMLMath .
With respect to this basis the operator multiplication by LaTeXMLMath is a Jacobi matrix .
First , we note that LaTeXMLEquation .
Therefore , LaTeXMLEquation .
Iterating this relation , we get that LaTeXMLMath is an orthonormal basis in LaTeXMLMath , since LaTeXMLMath .
Then , we note that an arbitrary function LaTeXMLMath can be approximated with the given accuracy by a function LaTeXMLMath from LaTeXMLMath .
This function , in its turn , can be approximated by a function LaTeXMLMath with a suitable LaTeXMLMath .
Therefore , linear combinations of functions from LaTeXMLMath are dense in LaTeXMLMath .
Since this system of functions is orthonormal , it forms a basis in LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLEquation .
For this reason , in the basis LaTeXMLMath , the matrix of the operator multiplication by LaTeXMLMath has only one non–zero entry over diagonal in each column .
But the operator is self–adjoint , therefore , the matrix is a three–diagonal Jacobi matrix.∎ Let LaTeXMLMath , LaTeXMLMath .
Define LaTeXMLEquation .
Then LaTeXMLMath is an orthonormal basis in LaTeXMLMath , LaTeXMLEquation and LaTeXMLEquation .
Lemma 1.3 and Lemma 1.4 imply immediately that LaTeXMLMath is an orthonormal basis in LaTeXMLMath .
Moreover , LaTeXMLMath .
To prove ( 1.9 ) consider a scalar product ( LaTeXMLMath ) LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
To prove ( 1.10 ) , we write LaTeXMLEquation .
Due to the Direct Cauchy Theorem , the reproducing kernel LaTeXMLMath possesses the following property : LaTeXMLEquation .
Substituting ( 1.12 ) in ( 1.11 ) , we obtain LaTeXMLEquation .
LaTeXMLEquation Using ( 1.3 ) , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since the limit ( 1.7 ) exists , finally , we get LaTeXMLEquation .
The lemma is proved.∎ Let LaTeXMLMath .
Then LaTeXMLEquation .
Note , that operators LaTeXMLMath are invertible .
We use the notation of Lemma 1.5 .
As we know , LaTeXMLMath .
But , in the case under consideration , LaTeXMLMath .
Hence , the function LaTeXMLMath itself belongs to LaTeXMLMath .
Therefore , we can project each term onto LaTeXMLMath in the relation LaTeXMLEquation .
On the right hand side we get LaTeXMLEquation .
To evaluate the left hand side , using ( 1.10 ) , we write LaTeXMLEquation .
LaTeXMLEquation Using ( 1.12 ) , we get LaTeXMLEquation .
Thus , LaTeXMLEquation .
In particular , LaTeXMLMath , and ( 1.10 ) becomes the statement of the lemma.∎ Assume that for some Jacobi matrix LaTeXMLMath there exists a pair of unitary transforms LaTeXMLEquation determining each other by the relations LaTeXMLEquation such that LaTeXMLEquation .
As before , we put LaTeXMLEquation .
Then LaTeXMLMath has at the origin zero ( poles ) of multiplicity LaTeXMLMath ( LaTeXMLMath ) .
Furthermore , LaTeXMLMath , and , hence , LaTeXMLEquation .
The equality in ( 1.15 ) takes place if and only if LaTeXMLMath .
Let us show that the annihilator of the linear space LaTeXMLMath contains LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By ( 1.13 ) and ( DCT ) , the last scalar product equals zero .
Therefore , LaTeXMLEquation .
Now , from the three-term recurrent relation LaTeXMLEquation and ( 1.13 ) it follows that LaTeXMLMath , LaTeXMLMath , has in the origin zero , at least of multiplicity LaTeXMLMath .
Since LaTeXMLMath , it possesses the decomposition LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath , LaTeXMLEquation in this decomposition .
But , LaTeXMLEquation .
Thus , ( 1.15 ) and the lemma are proved .
∎ Let LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath is the orthogonal projection from LaTeXMLMath onto LaTeXMLMath .
Let us denote by LaTeXMLMath an extremal function of the problem LaTeXMLEquation .
Using properties 1 ) , 2 ) of a group of Widom type with ( DCT ) , Theorem [ 5 ] , and compactness of LaTeXMLMath , for any LaTeXMLMath , we can find a finite covering of LaTeXMLMath LaTeXMLEquation such that LaTeXMLEquation .
It means that LaTeXMLEquation .
For fixed LaTeXMLMath one can find LaTeXMLMath such that LaTeXMLEquation .
Therefore , there exists LaTeXMLMath such that LaTeXMLEquation .
Now , let LaTeXMLMath and let LaTeXMLMath LaTeXMLMath .
For LaTeXMLMath , we write LaTeXMLEquation .
Then LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation Therefore , LaTeXMLEquation .
Putting LaTeXMLMath , we get LaTeXMLEquation .
The lemma is proved.∎ Assume that for some Jacobi matrix LaTeXMLMath there exists a pair of unitary transforms LaTeXMLEquation determining each other by the relations LaTeXMLEquation such that ( 1.13 ) holds .
Then the following relations are equivalent : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is defined by ( 1.14 ) .
( 1.18 ) LaTeXMLMath ( 1.19 ) .
It follows from two remarks .
First , the form on the left in ( 1.19 ) does not depend on LaTeXMLMath ( it is the Wronskian of recurrence relation ( 0.7 ) ) .
Second , the identity LaTeXMLEquation holds for any LaTeXMLMath .
( 1.19 ) LaTeXMLMath ( 1.20 ) .
Let us introduce the matrix LaTeXMLEquation .
Then ( 1.17 ) implies LaTeXMLEquation .
In particular , with a help of ( 1.19 ) , we get LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath are holomorphic functions ( in fact , of Smirnov class ) LaTeXMLEquation .
Now , we only have to mention that LaTeXMLMath .
( 1.20 ) LaTeXMLMath ( 1.18 ) .
This is non–trivial part of the proposition .
The main step is to prove that LaTeXMLEquation .
By Lemma ( 1.7 ) we have an estimate from below LaTeXMLEquation .
LaTeXMLEquation To get an estimate from above we use ( 1.20 ) .
Let us note that due to the recurrence relation , the form LaTeXMLEquation also does not depend on LaTeXMLMath .
Thus , a relation like ( 1.20 ) holds for all LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
With the function LaTeXMLMath , let us associate the functions LaTeXMLMath , LaTeXMLMath and the character LaTeXMLMath ( note , that LaTeXMLMath is not LaTeXMLMath , but LaTeXMLMath ) .
It is important that LaTeXMLMath and LaTeXMLMath depend continuously on LaTeXMLMath .
By Lemma 1.6 LaTeXMLEquation .
Substituting ( 1.26 ) in ( 1.25 ) , and combining the result with ( 1.24 ) , we obtain LaTeXMLEquation .
Lemma 1.8 implies that for any LaTeXMLMath with LaTeXMLMath we have LaTeXMLEquation .
Indeed , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation as LaTeXMLMath .
Also , since LaTeXMLMath depends continuously on LaTeXMLMath and LaTeXMLMath is continuous on a compact group LaTeXMLMath , for any LaTeXMLMath we can choose LaTeXMLMath so small that LaTeXMLEquation .
Thus , returning to ( 1.27 ) , we obtain LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLMath are arbitrary small , ( 1.23 ) is proved .
Now we are in a position to prove ( 1.18 ) .
Consider the norm of the difference LaTeXMLEquation .
Since LaTeXMLEquation using Lemma 1.8 , we conclude that LaTeXMLEquation .
Let us evaluate the scalar product LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation The proposition is proved.∎ The following theorem shows that an arbitrary function LaTeXMLMath , possessing ( 1.1 ) , ( 1.2 ) , is the reflection coefficient of a Jacobi matrix of Szegö class .
Let a function LaTeXMLMath , LaTeXMLMath , be such that that LaTeXMLMath .
Let an outer function LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath be associated to LaTeXMLMath by the relations LaTeXMLEquation .
Then the system of functions LaTeXMLEquation forms an orthonormal basis in LaTeXMLMath .
The dual system , defined by LaTeXMLEquation forms an orthonormal basis in LaTeXMLMath .
The subspaces of LaTeXMLMath , that formed by functions with vanishing negative Fourier coefficients with respect to these basses , are spaces of holomorphic character–automorphic forms , moreover , LaTeXMLEquation .
Further , LaTeXMLEquation and with respect to these basses the operator multiplication by LaTeXMLMath is a Jacobi matrix J of Szegö class .
All statements , besides the last one , only summaries results of Lemmas 1.4 , 1.5 and Proposition 1.1 .
To prove that LaTeXMLMath is of Szegö class we evaluate its spectral density LaTeXMLMath .
Using the definition of the resolvent matrix–function , we get LaTeXMLEquation .
Note , that if LaTeXMLMath are related by LaTeXMLMath then LaTeXMLEquation .
Therefore , using Lemma 1.3 , we have LaTeXMLEquation and , substituting LaTeXMLMath from ( 1.22 ) , we obtain LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
Thus , LaTeXMLEquation and LaTeXMLEquation .
The theorem is proved.∎ Let us note , by the way , that LaTeXMLMath ( see ( 1.21 ) ) and LaTeXMLMath are related by LaTeXMLMath and , besides ( 1.28 ) , LaTeXMLEquation .
We start this section with the remark that the spectral measure LaTeXMLMath determines a Jacobi matrix uniquely , but it is not an arbitrary LaTeXMLMath matrix–measure , or , say , a real-valued ( all entries are real ) LaTeXMLMath matrix–measure .
Indeed , one can represent LaTeXMLMath as a two dimensional perturbation of an orthogonal sum of a pair of one–sided Jacobi matrices , i.e .
: LaTeXMLEquation where LaTeXMLMath .
This formula implies that LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation Thus , the real–valued matrix–measure LaTeXMLMath is determined by two scalar measures LaTeXMLMath ( with the normalization LaTeXMLMath ) and a constant LaTeXMLMath .
In what follows LaTeXMLMath denotes the image of LaTeXMLMath in the spectral representation .
Recall that LaTeXMLEquation and LaTeXMLEquation .
Let LaTeXMLMath be the orthonormal polynomials with respect to the ( scalar ) measure LaTeXMLMath and LaTeXMLEquation ( so–called polynomials of the second kind ) .
In these terms LaTeXMLEquation .
LaTeXMLEquation Now , we prove Theorem 0.5 .
The function LaTeXMLMath is LaTeXMLMath –automorphic , thus it defines a meromorphic function in LaTeXMLMath , LaTeXMLEquation .
The recurrence relations implies that LaTeXMLMath possesses the same decomposition into a continued fraction as LaTeXMLMath .
Therefore , LaTeXMLEquation .
By Proposition 1.1 the asymptotic ( 1.18 ) implies the identity ( 1.19 ) .
Using this identity , we get ( LaTeXMLMath ) LaTeXMLEquation .
This means that an outer part of the function LaTeXMLMath is determined uniquely .
But then ( 2.3 ) means that an outer part of LaTeXMLMath is determined uniquely , and since LaTeXMLMath and LaTeXMLMath are of Smirnov class , these functions are determined up to a common inner factor LaTeXMLMath , i.e. , LaTeXMLEquation where the inner parts of LaTeXMLMath , LaTeXMLMath are relatively prime .
To show that LaTeXMLMath we use ( 0.23 ) , ( 0.24 ) .
Since LaTeXMLEquation .
LaTeXMLEquation we have LaTeXMLEquation .
Substituting ( 2.4 ) and using the symmetry LaTeXMLEquation we obtain LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since the first expression here is a function of Smirnov class and LaTeXMLMath is an outer function we conclude that LaTeXMLMath is a constant .
Since LaTeXMLEquation with LaTeXMLMath defined by ( 1.21 ) , LaTeXMLMath is also determined in a unique way .
At last , by the recurrence relations we get the same conclusion with respect to all functions LaTeXMLMath , not only for LaTeXMLMath .
∎ The key instrument is the following theorem [ 9 ] : if LaTeXMLMath is a meromorphic function in LaTeXMLMath such that LaTeXMLMath and poles of LaTeXMLMath satisfies the Blaschke condition , then LaTeXMLMath is a function of bounded characteristic in LaTeXMLMath without a singular component in the multiplicative representation .
Let us show that poles of LaTeXMLMath satisfies the Blaschke condition .
Diagonal entries LaTeXMLMath and LaTeXMLMath of the resolvent matrix–function LaTeXMLMath are holomorphic in LaTeXMLMath .
By the theorem mentioned above they are functions of bounded characteristic .
In force of ( 2.1 ) , LaTeXMLEquation .
LaTeXMLEquation This means that poles of LaTeXMLMath are subsets of poles of LaTeXMLMath and LaTeXMLMath .
Thus LaTeXMLMath are functions of bounded characteristic .
Now , let us use the Szegö condition LaTeXMLMath .
Since LaTeXMLEquation using again ( 2.1 ) , we have LaTeXMLEquation .
Therefore , each of the functions LaTeXMLMath belongs to LaTeXMLMath .
Thus we can represent LaTeXMLMath ( uniquely ) in the form LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are functions of Smirnov class with coprime inner parts ( in fact , they are Blaschke products ) such that LaTeXMLEquation and LaTeXMLMath , LaTeXMLMath .
Note that LaTeXMLEquation .
As soon as the functions LaTeXMLMath and LaTeXMLMath have been constructed we are able to introduce LaTeXMLMath and LaTeXMLMath in their terms .
First , let us write down an expression for the resolvent matrix–function : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are as in ( 1.21 ) and ( 1.28 ) respectively , and LaTeXMLEquation .
Therefore , LaTeXMLEquation since ( see ( 2.7 ) ) LaTeXMLEquation .
From ( 2.9 ) and LaTeXMLMath we get immediately that the matrix–function LaTeXMLMath defined by ( 2.6 ) is unitary–valued .
Let us show that its element LaTeXMLMath is an outer function .
In fact , we have to show that the function LaTeXMLMath is an outer function ( see ( 1.22 ) ) .
To this end let us use the representation for the diagonal entries of LaTeXMLMath ( see ( 2.8 ) ) LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be an inner part of LaTeXMLMath .
Since LaTeXMLMath is of Smirnov class , LaTeXMLMath is a divisor of LaTeXMLMath .
If LaTeXMLMath is not trivial , then it has a non–trivial divisor LaTeXMLMath that is a divisor of one of these functions , say , LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are coprime ( and LaTeXMLMath is a divisor of LaTeXMLMath ) , the LaTeXMLMath is a divisor of LaTeXMLMath , and , therefore , it is not a divisor of LaTeXMLMath .
Thus , LaTeXMLMath is not a divisor of the product LaTeXMLMath .
But this means that LaTeXMLMath is not of Smirnov class .
We arrive to a contradiction , hence LaTeXMLMath is a constant .
We define LaTeXMLMath by the formulas LaTeXMLEquation .
LaTeXMLEquation Evidently , LaTeXMLMath and by ( 2.6 ) , ( 0.23 ) are fulfilled .
Using the formula for the spectral density LaTeXMLMath and ( 2.9 ) , we have LaTeXMLEquation .
Since LaTeXMLEquation we obtain LaTeXMLEquation .
LaTeXMLEquation Thus LaTeXMLMath is an isometry , and since this map is invertible , LaTeXMLEquation where LaTeXMLMath , it is a unitary map .
Further , using ( 2.2 ) , for LaTeXMLMath we have LaTeXMLEquation .
Due to the well known properties of orthogonal polynomials these functions have no singularity at the origin and hence they are functions of Smirnov class .
This easily implies ( 0.24 ) .
At last , our maps possess properties ( 1.19 ) ( or ( 1.20 ) ) , in force of Proposition 1.1 , ( 0.25 ) holds .
The theorem is proved.∎ Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , satisfy LaTeXMLMath .
Then the reflection coefficient LaTeXMLMath determines a Jacobi matrix of Szegö class in a unique way if and only if LaTeXMLEquation .
Assume on the contrary that LaTeXMLEquation .
We construct two Jacobi matrices .
First , we consider the basis LaTeXMLEquation and by LaTeXMLMath we denote the operator multiplication by LaTeXMLMath in LaTeXMLMath with respect to this basis ( Lemma 1.4 ) .
Then , starting with the basis LaTeXMLMath in LaTeXMLMath , we introduce the basis LaTeXMLEquation .
By LaTeXMLMath we denote the operator multiplication by LaTeXMLMath in LaTeXMLMath with respect to LaTeXMLMath .
By Lemma 1.5 , LaTeXMLEquation .
Thus ( see ( 2.12 ) ) , LaTeXMLMath .
Due to the uniqueness part of Theorem 0.5 , LaTeXMLMath .
The ” only if ” part is proved .
Now , let ( 2.11 ) holds , and let LaTeXMLMath be a Jacobi matrix of Szegö class and LaTeXMLMath be its representations in LaTeXMLMath .
By Lemma 1.7 , LaTeXMLEquation .
Then ( 2.11 ) implies that , in fact , LaTeXMLMath and LaTeXMLMath , thus , due to a conclusion of Lemma 1.7 , LaTeXMLEquation .
Recall that these functions determine the functions LaTeXMLMath and the coefficient LaTeXMLMath ( see ( 2.3 ) ) , and they , in their turn , determine LaTeXMLMath .
The theorem is proved .
Let LaTeXMLMath be a Jacobi matrix of Szegö class with a homogeneous spectrum LaTeXMLMath .
Let LaTeXMLMath be the density of its spectral measure and LaTeXMLMath be its scattering matrix–function .
If LaTeXMLEquation then there is no other Jacobi matrix of Szegö class with the same scattering matrix–function LaTeXMLMath .
By virtue of ( 1.29 ) , ( 2.15 ) is equivalent to LaTeXMLEquation that is LaTeXMLMath and LaTeXMLMath belong to LaTeXMLMath .
Then words by words repetition of arguments in the proof of Lemma 1.6 gives us LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLMath and LaTeXMLMath .
Since , generally , LaTeXMLEquation ( 2.11 ) holds , the corollary is proved .
To finish this section we give an example of a scattering matrix–function , which does not determine a Jacobi matrix of Szegö class .
Moreover , in this example , the associated operators LaTeXMLMath are invertible .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Define outer functions LaTeXMLMath , LaTeXMLMath , by LaTeXMLEquation .
Then , we put LaTeXMLEquation .
At last , LaTeXMLEquation where LaTeXMLEquation and LaTeXMLMath is an inner function from LaTeXMLMath , LaTeXMLMath .
In this case LaTeXMLMath , since their symbols differ by functions from LaTeXMLMath , and therefore LaTeXMLMath are invertible .
From the other hand , the coefficient LaTeXMLMath is of the form LaTeXMLEquation and because of the factor LaTeXMLMath , LaTeXMLMath does not belong to LaTeXMLMath .
The simplest choice of parameters : LaTeXMLEquation .
LaTeXMLMath is a Blaschke product , LaTeXMLMath , gives us an example where LaTeXMLMath , defined by ( 2.14 ) , does not belong to LaTeXMLMath , at the same time LaTeXMLMath , defined by ( 2.13 ) , belongs to LaTeXMLMath .
By LaTeXMLMath we denote the transform LaTeXMLEquation primarily defined on integrable 2D vector–functions .
Let LaTeXMLMath be of Szegö class and LaTeXMLMath give its representations as the operator multiplication by LaTeXMLMath in the model spaces LaTeXMLMath .
Then LaTeXMLEquation for any finite vector LaTeXMLMath .
Let LaTeXMLMath denote the LaTeXMLMath –th matrix orthonormal polynomial with respect the spectral measure LaTeXMLMath .
Recall , that LaTeXMLEquation and , analogically to the scalar case , LaTeXMLEquation .
Based on ( 3.3 ) , we have LaTeXMLEquation .
LaTeXMLEquation Using ( 2.8 ) and definition ( 2.10 ) , we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In fact , this finishes the proof.∎ Let LaTeXMLMath be the spectral density of a Jacobi matrix LaTeXMLMath of Szegö class and LaTeXMLMath be the reflection coefficient .
Then the following statements are equivalent : 1 .
There exist LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation 2 .
LaTeXMLMath determines LaTeXMLMath and the operators LaTeXMLMath are invertible .
LaTeXMLMath .
Since ( see ( 1.30 ) ) LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation we get LaTeXMLMath .
Thus , LaTeXMLMath .
By Lemma 1.7 and Theorem 2.1 we come to the conclusion that LaTeXMLMath determines LaTeXMLMath .
Further , by ( 3.5 ) LaTeXMLEquation .
LaTeXMLEquation Using again LaTeXMLMath , we can represent the last norms in the form LaTeXMLEquation .
This proves the second statement in 2 .
LaTeXMLMath .
Recall that LaTeXMLMath , but in the case under consideration , the norm in LaTeXMLMath is equivalent to the norm in LaTeXMLMath , i.e .
: LaTeXMLEquation .
Further , since LaTeXMLMath determines LaTeXMLMath , by Lemma 1.7 , we have LaTeXMLMath .
So , starting with ( 3.6 ) we obtain ( 3.4 ) .∎ Let us define LaTeXMLMath as the square root of an outer function such that LaTeXMLMath .
Put LaTeXMLEquation .
In this case , LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath .
Thus , LaTeXMLEquation or LaTeXMLEquation .
Besides , LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
But , in fact , the group LaTeXMLMath is defined up to a choice of a half–period LaTeXMLMath .
So , we may assume that LaTeXMLEquation .
Let LaTeXMLMath be a homogeneous set .
Then LaTeXMLEquation is a holomorphic matrix function in LaTeXMLMath satisfying the following RHP LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , with the normalization at infinity : LaTeXMLEquation ( 4.4 ) follows from ( 4.3 ) and ( 4.5 ) follows from ( 4.2 ) and ( 2.6 ) .
To prove ( 4.6 ) , we represent LaTeXMLMath in the form LaTeXMLEquation .
Then , we note that LaTeXMLEquation and LaTeXMLEquation .
Solving together ( 4.7 ) , ( 4.8 ) , we get ( 4.6 ) .∎ Let LaTeXMLMath be a homogeneous set .
Throughout this section LaTeXMLMath denotes orthoprojector from the vector–valued LaTeXMLMath onto LaTeXMLMath in the upper halfplane .
We are interested in the boundedness of the weighted transform LaTeXMLEquation where LaTeXMLMath is a weight on LaTeXMLMath and LaTeXMLMath is the characteristic function of the set LaTeXMLMath .
Here is an analog of the matrix LaTeXMLMath condition LaTeXMLEquation where LaTeXMLMath and LaTeXMLEquation .
This supremum will be called LaTeXMLMath .
The operator ( 5.1 ) is bounded if and only if LaTeXMLMath .
With an arbitrary LaTeXMLMath we associate a subspace LaTeXMLMath of the Hardy space , LaTeXMLMath .
It is well known , that LaTeXMLEquation and LaTeXMLEquation .
Because of the first of these relations we have LaTeXMLEquation .
Now , using the second one we get LaTeXMLEquation .
Let us substitute LaTeXMLEquation in ( 5.3 ) .
This give us LaTeXMLEquation where LaTeXMLMath denotes an average with the Poisson kernel , LaTeXMLEquation .
Thus we proved an inequality with the Poisson ’ s averages LaTeXMLEquation .
At last let us note that LaTeXMLEquation with an absolute and positive constant LaTeXMLMath .
Therefore ( 5.4 ) implies ( 5.2 ) .
If LaTeXMLMath is a centered at LaTeXMLMath interval and LaTeXMLMath is the center of the square built on LaTeXMLMath , then LaTeXMLEquation .
First we note , that for LaTeXMLMath LaTeXMLEquation and therefore LaTeXMLMath .
Let us show that LaTeXMLEquation .
Integrating the inequality LaTeXMLEquation over LaTeXMLMath we get LaTeXMLEquation .
Therefore LaTeXMLEquation or LaTeXMLEquation .
Using ( 5.2 ) we obtain LaTeXMLEquation .
To prove ( 5.5 ) , using LaTeXMLEquation we write the following chain of inequalities : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We want to prove that ( 5.2 ) suffices for LaTeXMLMath in ( 5.1 ) to be bounded .
Fix LaTeXMLMath .
We need to show LaTeXMLEquation .
In other words , introducing a Stolz cone LaTeXMLMath and LaTeXMLEquation one needs to prove that LaTeXMLEquation .
We follow closely the lines of the proof in [ 10 ] .
Let us consider a nonnegative function LaTeXMLMath and LaTeXMLEquation where LaTeXMLEquation .
Let us note that LaTeXMLEquation if the function LaTeXMLMath has the following property : LaTeXMLEquation .
Let us choose LaTeXMLMath to be maximal such that LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath will be chosen a bit later and LaTeXMLMath denotes the maximal function LaTeXMLEquation .
If this LaTeXMLMath satisfies ( 5.10 ) , then ( 5.9 ) and ( 5.8 ) imply what we need .
To choose LaTeXMLMath and to prove that LaTeXMLMath satisfies ( 5.9 ) we follow the algorithm below .
Let LaTeXMLMath be an arbitrary interval on the real axis .
We will consider two cases : LaTeXMLMath and LaTeXMLMath .
In the first case we fix an interval LaTeXMLMath centered at LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
Denote LaTeXMLMath .
Consider LaTeXMLEquation .
LaTeXMLEquation We will fix later LaTeXMLMath .
Now , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Here LaTeXMLMath .
Notice that for every vector LaTeXMLMath the scalar function LaTeXMLMath is uniformly in scalar LaTeXMLMath .
In particular , there exists such an LaTeXMLMath that we have the inverse Hölder inequality for all such functions uniformly : LaTeXMLEquation .
Let us choose LaTeXMLMath LaTeXMLMath , LaTeXMLMath , then we have LaTeXMLEquation .
LaTeXMLEquation We use ( 5.13 ) and inverse Hölder inequality ( 5.12 ) in ( 5.11 ) to rewrite it as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
We used the doubling property of LaTeXMLMath : LaTeXMLMath , the inequality which can be proved in the same way as ( 5.6 ) .
The last inequality ensures that for any LaTeXMLMath , LaTeXMLMath , using Kolmogorov-type inequalities we can find a subset LaTeXMLMath , LaTeXMLMath such that LaTeXMLEquation .
Similarly , for every LaTeXMLMath there exists a set LaTeXMLMath , LaTeXMLMath such that LaTeXMLEquation .
Here we use the same calculations and the fact that for any LaTeXMLMath centered at LaTeXMLMath LaTeXMLEquation .
Now let us work with LaTeXMLMath .
Let LaTeXMLMath be the center of the square built on LaTeXMLMath .
Using the representation LaTeXMLEquation clearly , we obtain for every LaTeXMLMath LaTeXMLEquation .
Therefore , using the inverse Hölder inequality ( 5.12 ) , we have again LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Here LaTeXMLMath is close to 2 ( LaTeXMLMath ) .
Finally , using Lemma 5.1 we estimate the last sum by a constant : LaTeXMLEquation .
That is LaTeXMLEquation .
The same for LaTeXMLMath .
Combining all ( * ) inequalities we obtain that with a suitable LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation at least on a quarter of LaTeXMLMath .
Of course , LaTeXMLMath .
In the case LaTeXMLMath we fix an interval LaTeXMLMath centered at LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the center of the square built on LaTeXMLMath .
We can use again a representation of the form ( 5.15 ) : LaTeXMLEquation to obtain an analog of ( 5.16 ) LaTeXMLEquation for all LaTeXMLMath .
Continue in this way we get LaTeXMLEquation .
The same for LaTeXMLMath .
Thus LaTeXMLEquation everywhere on LaTeXMLMath .
Let LaTeXMLMath be the largest constant in ( 5.18 ) , ( 5.19 ) .
We have already chosen LaTeXMLMath .
Now we introduce the following function LaTeXMLMath : LaTeXMLEquation .
What we proved can be summarized in : LaTeXMLEquation .
LaTeXMLEquation In any case , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation because LaTeXMLMath , and we can use the Hardy–Littlewood maximal theorem .
The theorem is proved .
In this article I propose a new method for reducing a co-oriented contact manifold LaTeXMLMath equipped with an action of a Lie group LaTeXMLMath by contact transformations .
With a certain regularity and integrality assumption the contact quotient LaTeXMLMath at LaTeXMLMath is a naturally a co-oriented contact orbifold which is independent of the contact form used to represent the given contact structure .
Removing the regularity and integrality assumptions and replacing them with one concerning the existence of a slice , which is satisfied for compact symmetry groups , results in a contact stratified space ; i.e , a stratified space equipped with a line bundle which , when restricted to each stratum , defines a co-oriented contact structure .
This extends the previous work of the author and E. Lerman ( LaTeXMLCite ) .
In this article I propose a new method for reducing a co-oriented contact manifold LaTeXMLMath under the action of a Lie group LaTeXMLMath by contact transformations .
With a certain regularity and integrality assumption ( see Lemma LaTeXMLRef and Theorem LaTeXMLRef ) the contact quotient LaTeXMLMath at LaTeXMLMath is a naturally a co-oriented contact orbifold which is independent of the contact form used to represent the given contact structure .
Removing the regularity and integrality assumption and replacing it with one concerning the existence of a slice , which is satisfied for compact symmetry groups , results in a contact stratified space ; i.e , a stratified space equipped with a line bundle which , when restricted to each stratum , defines a co-oriented contact structure .
This extends the previous work of the author and E. Lerman ( LaTeXMLCite ) .
The earliest notion of a contact quotient that I am aware of is in the work of V. Guillemin and S. Sternberg , where it appears in the guise of co-isotropic reduction of symplectic cones ( LaTeXMLCite ) .
The Guillemin-Sternberg quotient is based on the reduction of foliations and is akin to the Kazhdan-Kostant-Sternberg quotient in symplectic geometry .
The new contact quotient is akin the Marsden-Weinstein-Meyer quotient .
The two models differ , but are related : The Guillemin-Sternberg quotient fibers over a co-adjoint orbit with typical fiber the new quotient .
The Guillemin-Sternberg quotient is described in a later section .
Independently of Guillemin-Sternberg , C. Albert developed a model for contact reduction which is valid at all LaTeXMLMath but depends upon the contact form used to represent the given contact structure ( LaTeXMLCite ) .
The author would like to thank Eugene Lerman for suggesting this line of investigation and innumerable discussions on the material while at the American Institute of Mathematics during the fall of 2000 and for reading several preliminary versions of the paper .
The author would also like to thank Susan Tolman for making several helpful suggestions concerning the presentation of the material .
The notation in this paper will be consistent with that used in LaTeXMLCite .
Before proceeding to the proof of the main theorems of the paper , we first recall the relevant and essential facts about contact manifolds and group actions on contact manifolds .
A contact structure on a manifold LaTeXMLMath of dimension LaTeXMLMath is a co-dimension one distribution LaTeXMLMath on LaTeXMLMath which is given locally by the kernel of a 1-form LaTeXMLMath with LaTeXMLMath .
If there is a global 1-form LaTeXMLMath with LaTeXMLMath , then LaTeXMLMath is called co-orientable , LaTeXMLMath is said to represent LaTeXMLMath , and is called a contact form .
One can define a co-orientable contact structure on a manifold LaTeXMLMath in the following convenient way , which we exploit in a later section .
A co-dimension one distribution LaTeXMLMath is a co-orientable contact structure if its annihilator LaTeXMLMath admits a nowhere vanishing section , LaTeXMLMath , and LaTeXMLMath is a symplectic submanifold of LaTeXMLMath .
Equivalently , a line sub-bundle LaTeXMLMath which is symplectic away from the zero section and equipped with a nowhere vanishing section defines a co-orientable contact structure on LaTeXMLMath .
A choice of a component of LaTeXMLMath is a co-orientation for LaTeXMLMath .
If LaTeXMLMath is a non-vanishing function on LaTeXMLMath , then LaTeXMLMath is another contact form on LaTeXMLMath which represents LaTeXMLMath .
The conformal class of LaTeXMLMath is LaTeXMLEquation .
It should be stressed that the important data on a contact manifold is the contact structure , not the contact form used to represent it .
Observe that for each LaTeXMLMath , LaTeXMLEquation is a symplectic vector space .
On a co-oriented contact manifold LaTeXMLMath there is a distinguished vector field LaTeXMLMath , called the Reeb vector field , which satisifies LaTeXMLEquation .
LaTeXMLMath is unique with respect to these two properties .
This allows us to split the tangent bundle of LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath is the line bundle over LaTeXMLMath spanned by LaTeXMLMath .
Since LaTeXMLMath , the flow of the Reeb vector field preserves LaTeXMLMath .
For the purposes of this article , the most important examples of contact manifolds come via interaction with symplectic manifolds .
A hypersurface LaTeXMLMath of a symplectic manifold , LaTeXMLMath is said to be of contact type if there is a vector field LaTeXMLMath , defined near LaTeXMLMath , which satisfies LaTeXMLMath .
LaTeXMLMath is called a Liouville vector field .
If LaTeXMLMath is a hypersurface of contact type in a symplectic manifold LaTeXMLMath with Liouville vector field LaTeXMLMath , then LaTeXMLMath is a contact manifold .
If LaTeXMLMath is a contact manifold , then away from the zero section the annihilator of LaTeXMLMath in LaTeXMLMath is a symplectic submanifold of LaTeXMLMath .
A connected component of LaTeXMLMath is called the symplectization of LaTeXMLMath .
If LaTeXMLMath is a co-oriented contact manifold , then the symplectization can be identified with LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath coordinate .
Note that every co-oriented contact manifold is a hypersurface of contact type in its symplectization .
Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath and preserves LaTeXMLMath .
The contact moment map associated to the action is denoted by LaTeXMLMath and defined by LaTeXMLEquation for LaTeXMLMath and all LaTeXMLMath .
The contact moment map is equivariant , where LaTeXMLMath acts on LaTeXMLMath through the co-adjoint respresentation ( LaTeXMLCite ) .
The moment map depends upon the choice of an invariant contact form to represent the contact structure : If LaTeXMLMath is a non-vanishing invariant function on LaTeXMLMath , then LaTeXMLMath is an invariant contact form on LaTeXMLMath and LaTeXMLMath .
Suppose LaTeXMLMath is a symplectic manifold equipped with a Hamiltonian LaTeXMLMath -action and let LaTeXMLMath be a corresponding equivariant symplectic moment map .
Let LaTeXMLMath be an invariant hypersurface which has an invariant Liouville vector field and LaTeXMLMath the induced invariant contact form on LaTeXMLMath .
Then the restriction of the LaTeXMLMath -action to LaTeXMLMath preserves LaTeXMLMath and the associated contact moment map LaTeXMLMath is the restriction of LaTeXMLMath to LaTeXMLMath .
Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath preserving LaTeXMLMath and let LaTeXMLMath be the associated contact moment map .
The extension of the action to the symplectization LaTeXMLMath by LaTeXMLMath is Hamiltonian and a corresponding equivariant symplectic moment map LaTeXMLMath is given by LaTeXMLMath .
Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated moment map .
Let LaTeXMLMath be a subset which is invariant under dilations by LaTeXMLMath .
Let LaTeXMLMath be the symplectic moment map for the induced action on LaTeXMLMath .
Then LaTeXMLMath .
If LaTeXMLMath is a positive invariant function on LaTeXMLMath , then LaTeXMLMath .
Because LaTeXMLMath and LaTeXMLMath is LaTeXMLMath invariant , LaTeXMLMath if and only if LaTeXMLMath .
This establishes the first point .
The second one follows in the same manner , using the fact that LaTeXMLMath .
∎ Recall that an action of a Lie group LaTeXMLMath on a manifold LaTeXMLMath is called proper if the map LaTeXMLMath given by LaTeXMLMath is a proper map .
Actions of compact groups are proper .
Proper actions have compact isotropy groups .
If a Lie group LaTeXMLMath acts properly on a ( paracompact ) co-oriented contact manifold LaTeXMLMath and preserves the contact structure and a co-orientation , then we can find an invariant contact form , LaTeXMLMath , on LaTeXMLMath which represents the contact structure ( LaTeXMLCite ) .
In this case , the Reeb vector field is , by uniqueness , LaTeXMLMath -invariant as well .
A slice for an action of a Lie group LaTeXMLMath on a manifold LaTeXMLMath at a point LaTeXMLMath is a LaTeXMLMath invariant submanifold LaTeXMLMath such that LaTeXMLMath is an open subset of LaTeXMLMath and such that the map LaTeXMLMath , LaTeXMLMath descends to a diffeomorphism LaTeXMLMath , LaTeXMLMath .
A theorem of Palais asserts that for smooth proper actions slices exist at every point ( LaTeXMLCite ) .
Suppose a Lie group LaTeXMLMath acts properly on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated contact moment map .
If there exists a slice LaTeXMLMath through LaTeXMLMath for the co-adjoint action which is invariant under dilations by LaTeXMLMath , then LaTeXMLMath is a LaTeXMLMath invariant contact submanifold of LaTeXMLMath and the contact moment map for the LaTeXMLMath action on LaTeXMLMath is given by the restriction of the LaTeXMLMath to LaTeXMLMath followed by the natural projection of LaTeXMLMath onto LaTeXMLMath .
LaTeXMLMath is called a contact cross section .
The proof of Proposition LaTeXMLRef is derived from the corresponding proof of the symplectic cross section theorem of Guillemin and Sternberg ( LaTeXMLCite ) .
Indeed , extend the LaTeXMLMath action to the symplectization of LaTeXMLMath and let LaTeXMLMath be the corresponding symplectic moment map .
The symplectic cross section theorem gives LaTeXMLMath as a symplectic submanifold of the symplectization .
By hypothesis , LaTeXMLMath is invariant under dilations by LaTeXMLMath and hence Lemma LaTeXMLRef implies that LaTeXMLMath .
It follows that the contact cross section is a hypersurface of contact type in the symplectic cross section , which gives the result .
∎ Although Proposition LaTeXMLRef can be proved directly , the direct proof does not diverge significantly from the proof of the symplectic cross section theorem of Guillemin and Sternberg found in LaTeXMLCite .
In this section we prove the first two of our three main theorems .
Suppose a Lie group LaTeXMLMath acts properly on a contact manifold LaTeXMLMath , preserving LaTeXMLMath and let LaTeXMLMath be the associated moment map .
The Reeb flow preserves the level sets of LaTeXMLMath .
For all LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath is an isotropic subspace of the symplectic vector space LaTeXMLMath .
LaTeXMLMath Denote the Reeb vector field by LaTeXMLMath and its flow by LaTeXMLMath .
As noted earlier , by uniqueness LaTeXMLMath is invariant and therefore LaTeXMLMath is equivariant .
Therefore , for any LaTeXMLMath and LaTeXMLMath , it follows that LaTeXMLMath .
Thus LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation This establishes item ( 1 ) .
Since LaTeXMLMath , Cartan ’ s formula gives LaTeXMLMath .
Therefore , LaTeXMLEquation which establishes item ( 2 ) .
The proof of item ( 3 ) is an application of item ( 2 ) .
For any LaTeXMLMath , the second item implies that for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLMath , whence LaTeXMLMath .
The final point follows immediately from the second point .
Note that LaTeXMLMath .
∎ Note that the last item of Proposition LaTeXMLRef implies that LaTeXMLMath is a regular point for LaTeXMLMath if and only if LaTeXMLEquation .
That is , LaTeXMLMath may be critical yet have a discrete stabilizer .
This contrasts sharply with the symplectic category , where a point is regular if and only if its stabilizer is discrete .
Let LaTeXMLMath be a Lie group and choose LaTeXMLMath .
Then LaTeXMLMath is a Lie ideal in LaTeXMLMath .
Hence , there is a unique , connected normal Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
Choose LaTeXMLMath .
Then LaTeXMLMath , where LaTeXMLMath is the differential of the co-adjoint representation .
Thus , for any LaTeXMLMath , we have that LaTeXMLEquation .
Therefore , LaTeXMLMath is a Lie ideal in LaTeXMLMath .
∎ Suppose a Lie group LaTeXMLMath acts properly on a contact manifold , LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated moment map .
Choose LaTeXMLMath .
We define the kernel group of LaTeXMLMath to be the unique connected Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
The kernel group of LaTeXMLMath is denoted by LaTeXMLMath .
We define the contact quotient ( or contact reduction ) of LaTeXMLMath by LaTeXMLMath at LaTeXMLMath to be LaTeXMLEquation .
If LaTeXMLMath is a positive invariant function on LaTeXMLMath , then LaTeXMLMath is another invariant contact form on LaTeXMLMath which is in the same conformal class as LaTeXMLMath .
Since LaTeXMLMath is invariant under dilations by LaTeXMLMath , Lemma LaTeXMLRef implies that LaTeXMLMath and hence the reduced space is topologically independent of the choice of contact forms in the same conformal class .
Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated contact moment map .
Choose LaTeXMLMath and let LaTeXMLMath be the connected Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
Then LaTeXMLMath is transverse to LaTeXMLMath if and only if LaTeXMLMath acts locally freely on LaTeXMLMath .
If LaTeXMLMath is transverse to LaTeXMLMath , then LaTeXMLEquation is a submanifold of LaTeXMLMath .
Choose LaTeXMLMath .
The transversality condition , LaTeXMLEquation is equivalent to the condition LaTeXMLEquation .
Let LaTeXMLMath be the isotropy subgroup of LaTeXMLMath in LaTeXMLMath and LaTeXMLMath its Lie algebra .
Choose LaTeXMLMath and denote the Reeb vector field of LaTeXMLMath by LaTeXMLMath .
Then LaTeXMLMath since LaTeXMLMath .
Because LaTeXMLMath fixes LaTeXMLMath , LaTeXMLMath .
Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , LaTeXMLMath and LaTeXMLMath acts locally freely on LaTeXMLMath .
To establish the second statement , suppose that LaTeXMLMath for some LaTeXMLMath and let LaTeXMLMath .
Then , by Proposition , LaTeXMLRef LaTeXMLMath for some LaTeXMLMath , where LaTeXMLMath is the Reeb vector field .
Therefore , LaTeXMLEquation .
Hence , LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , it follows that LaTeXMLMath .
Since LaTeXMLMath acts locally freely on LaTeXMLMath , LaTeXMLMath and hence LaTeXMLMath is transverse to LaTeXMLMath .
∎ While the contact quotient is defined at any element of LaTeXMLMath , it is not necessarily a contact manifold .
There are two problems .
The first is that there is no guarantee that the kernel group of LaTeXMLMath will act properly on LaTeXMLMath , in which case the resulting quotient may not be Hausdorff .
If LaTeXMLMath is compact and LaTeXMLMath is integral ( i.e , LaTeXMLMath , where LaTeXMLMath is a group map satisfying LaTeXMLMath ) , then LaTeXMLMath is actually compact and no hypothesis is needed .
This integrality condition is required in the Guillemin-Sternberg procedure , described later ( LaTeXMLCite ) .
The second problem is that the kernel and isotropy groups of LaTeXMLMath may coincide .
If LaTeXMLMath is non-zero , then the resulting quotient may fail to be contact , as the below example shows .
Hence , we assume that LaTeXMLMath .
If LaTeXMLMath is compact , then the existence of an invariant metric on LaTeXMLMath implies this condition .
Let LaTeXMLMath and let LaTeXMLMath be the natural contact form on LaTeXMLMath .
For the lift of the natural LaTeXMLMath action the associated contact moment map , LaTeXMLMath , is given by LaTeXMLMath .
Set LaTeXMLEquation .
Then LaTeXMLEquation .
Therefore , LaTeXMLEquation is closed in LaTeXMLMath and acts properly on LaTeXMLMath , but LaTeXMLMath is four dimensional and hence isn ’ t contact .
Let LaTeXMLMath be a vector space and LaTeXMLMath be an antisymmetric bilinear map .
Suppose there is a decomposition LaTeXMLMath which is perpendicular with respect to LaTeXMLMath ; i.e , LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
Choose LaTeXMLMath and LaTeXMLMath .
Write LaTeXMLMath where LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath .
Hence LaTeXMLMath .
∎ Let LaTeXMLMath be a symplectic vector space and suppose that LaTeXMLMath is an isotropic subspace .
Then LaTeXMLEquation where LaTeXMLMath is the symplectic perpendicular of LaTeXMLMath with respect to LaTeXMLMath .
Fix LaTeXMLMath .
Since LaTeXMLMath is isotropic , LaTeXMLMath .
For any LaTeXMLMath , LaTeXMLMath by definition .
Hence LaTeXMLMath .
Conversely , if LaTeXMLMath , then by unravelling the various definitions , one sees that LaTeXMLMath .
∎ Suppose a Lie group LaTeXMLMath acts on a contact manifold , LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated contact moment map .
Choose LaTeXMLMath and let LaTeXMLMath be the connected Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
If LaTeXMLMath acts properly on LaTeXMLMath LaTeXMLMath is transverse to LaTeXMLMath LaTeXMLMath then the quotient LaTeXMLEquation is naturally a contact orbifold ; i.e , LaTeXMLEquation descends to a contact structure on the quotient .
In the special case of LaTeXMLMath C. Albert , H. Geiges , and F. Loose , independently of Guillemin-Sternberg , established the above theorem in various papers ( LaTeXMLCite ) .
Since LaTeXMLMath is transverse to LaTeXMLMath , LaTeXMLMath is a submanifold of LaTeXMLMath and Lemma LaTeXMLRef implies that LaTeXMLMath acts locally freely LaTeXMLMath .
Hence , LaTeXMLMath is an orbifold .
Fix LaTeXMLMath .
For any LaTeXMLMath , we have that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Hence , LaTeXMLMath descends to LaTeXMLMath on LaTeXMLMath , which is contact if and only if LaTeXMLEquation .
Let LaTeXMLMath be the contact moment map associated to the action of LaTeXMLMath on LaTeXMLMath .
Note that LaTeXMLMath where LaTeXMLMath is the natural projection .
Observe that LaTeXMLMath and hence that LaTeXMLMath is an isotropic subspace of the symplectic vector space LaTeXMLMath by Proposition LaTeXMLRef .
Let LaTeXMLMath denote the symplectic perpendicular of LaTeXMLMath in LaTeXMLMath .
A vector LaTeXMLMath is in LaTeXMLMath if and only if for all LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation That is , LaTeXMLMath if and only if LaTeXMLMath .
Hence , LaTeXMLEquation where LaTeXMLMath .
Note that LaTeXMLMath is a submanifold of LaTeXMLMath by the transversality condition .
Indeed , LaTeXMLMath .
Since LaTeXMLMath is transverse to LaTeXMLMath , LaTeXMLMath acts locally freely on LaTeXMLMath by Lemma LaTeXMLRef and thus on a neighborhood of LaTeXMLMath and hence on LaTeXMLMath ( or , at least on a neighborhood of LaTeXMLMath in LaTeXMLMath ) .
Therefore , also by Lemma LaTeXMLRef , LaTeXMLMath is a submanifold of LaTeXMLMath .
Lemma LaTeXMLRef implies that LaTeXMLMath .
We first show that LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , LaTeXMLEquation .
Note that this implies that LaTeXMLMath .
The reverse inclusion is slightly more delicate .
Since LaTeXMLMath , we can choose a splitting LaTeXMLMath where LaTeXMLMath .
Let LaTeXMLMath .
The proof is completed by showing that LaTeXMLMath and LaTeXMLMath are complementary subspaces of LaTeXMLMath which are prependicular with respect to LaTeXMLMath .
Lemma LaTeXMLRef implies the reverse inclusion .
We first show that LaTeXMLMath .
Choose any LaTeXMLMath and let LaTeXMLMath .
Then for some LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation since LaTeXMLMath .
Hence , LaTeXMLMath .
Additionally , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus , LaTeXMLMath .
By equivariance , LaTeXMLMath .
Hence , LaTeXMLMath maps LaTeXMLMath to LaTeXMLMath .
In fact , one can show that this map is an isomorphism .
By definition , LaTeXMLMath .
The assumption LaTeXMLMath is equivalent to LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLMath .
Finally , if LaTeXMLMath , then LaTeXMLMath .
Hence , LaTeXMLMath is a subspace of dimension LaTeXMLMath .
A dimension count implies that LaTeXMLMath and LaTeXMLMath are complementary subspaces of LaTeXMLMath , which completes the proof .
∎ The second main theorem removes the transversality condition above but replaces it with a requirement that a convex , LaTeXMLMath invariant slice exists for LaTeXMLMath .
This condition is satisfied for compact LaTeXMLMath .
It is suspected , but not known to the author , that a stratification theorem for proper group actions ought to hold .
In the particular case of LaTeXMLMath , the author and E. Lerman showed that LaTeXMLMath is topologically a stratified space : Let LaTeXMLMath be a manifold with a co-oriented contact structure LaTeXMLMath .
Suppose a Lie group LaTeXMLMath acts properly on LaTeXMLMath preserving LaTeXMLMath and a co-orientation for LaTeXMLMath .
Choose a LaTeXMLMath -invariant contact form LaTeXMLMath with LaTeXMLMath and let LaTeXMLMath be the corresponding moment map .
Then for every subgroup LaTeXMLMath of LaTeXMLMath , each connected component of the topological space LaTeXMLEquation is a manifold and the partition of the contact quotient LaTeXMLEquation into these manifolds is a stratification .
The symbol LaTeXMLMath stands for the set of points in LaTeXMLMath with the isotropy groups conjugate to LaTeXMLMath .
See LaTeXMLCite for a complete proof .
∎ Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated contact moment map .
Choose LaTeXMLMath .
Suppose The kernel group LaTeXMLMath of LaTeXMLMath acts properly on LaTeXMLMath .
There is a convex slice LaTeXMLMath for LaTeXMLMath which is invariant under dilations by LaTeXMLMath Denote the natural projection , LaTeXMLMath by LaTeXMLMath and let LaTeXMLMath .
Then LaTeXMLMath , where LaTeXMLMath is the reduction of LaTeXMLMath by LaTeXMLMath at LaTeXMLMath .
By Proposition LaTeXMLRef the restriction of LaTeXMLMath to LaTeXMLMath is a contact form and LaTeXMLMath is LaTeXMLMath invariant .
The contact moment map for the LaTeXMLMath action on LaTeXMLMath is given by LaTeXMLMath .
Note that LaTeXMLEquation .
The proof will be completed by showing that LaTeXMLMath .
From this it follows that LaTeXMLMath .
Since LaTeXMLMath , the conclusion follows .
Since LaTeXMLMath is LaTeXMLMath -invariant and contains LaTeXMLMath , LaTeXMLMath .
Recall that LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is tangent to LaTeXMLMath at LaTeXMLMath .
On the other hand , since LaTeXMLMath and LaTeXMLMath is convex , it follows that LaTeXMLMath is also tangent to LaTeXMLMath at LaTeXMLMath .
Hence LaTeXMLMath However , because LaTeXMLMath is a slice , LaTeXMLMath .
Thus , LaTeXMLMath and LaTeXMLMath .
∎ Proposition LaTeXMLRef allows us to always assume that , given that the hypothesis are satisfied , the element at which we are reducing is always fixed by the co-adjoint action of the symmetry group .
The hypothesis of Proposition LaTeXMLRef are satisifed for the action of any compact group .
Indeed , if LaTeXMLMath is compact , we can choose a LaTeXMLMath invariant metric and hence an equivariant splitting LaTeXMLMath , where LaTeXMLMath is embedded as the normal fiber to LaTeXMLMath at LaTeXMLMath .
Take LaTeXMLMath where LaTeXMLMath is a small LaTeXMLMath -ball about LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath is a convex , LaTeXMLMath -invariant slice at LaTeXMLMath .
We can now prove our second main theorem .
Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath , preseving LaTeXMLMath and let LaTeXMLMath be the associated contact moment map .
Choose LaTeXMLMath and let LaTeXMLMath be the connected Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
Suppose LaTeXMLMath acts properly on LaTeXMLMath A convex slice exists for LaTeXMLMath which is invariant under dilations by LaTeXMLMath then the partition of the contact quotient by LaTeXMLMath at LaTeXMLMath by orbit types , LaTeXMLEquation is a stratification .
Here LaTeXMLMath is the set of points whose stabilizer is conjugate to LaTeXMLMath and the indexing set is the set of conjugacy classes of stabilizer subgroups of LaTeXMLMath .
By Proposition LaTeXMLRef , we may assume that LaTeXMLMath is fixed by the co-adjoint action of LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be the moment map for the action of LaTeXMLMath .
Because LaTeXMLMath , where LaTeXMLMath is the natural projection , we have LaTeXMLMath .
Therefore LaTeXMLMath is an open subset of LaTeXMLMath , which is stratified by Theorem LaTeXMLRef .
Open subsets of stratified spaces are naturally stratified .
∎ The previous section established the topological structure of the contact quotient but did not address the geometrical structure of the quotient under stratification .
The point of this section is to show that the contact quotient is a contact stratified space ; i.e , there exists a line bundle over LaTeXMLMath which , when restricted to each stratum , defines a co-oriented contact structure .
Suppose a Lie group LaTeXMLMath acts properly on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated moment map .
Let LaTeXMLMath be a isotropy subgroup of LaTeXMLMath , LaTeXMLEquation the normalizer of LaTeXMLMath in LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation the set of points of LaTeXMLMath whose isotropy group is LaTeXMLMath .
Then LaTeXMLMath is a contact submanifold of LaTeXMLMath , LaTeXMLMath acts freely on LaTeXMLMath , and there is a diffeomorphism , LaTeXMLEquation .
LaTeXMLMath is a submanifold of LaTeXMLMath and for all LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath is the set of LaTeXMLMath -fixed vectors in LaTeXMLMath ( see , for example , Proposition 27.5 of LaTeXMLCite ) .
The Reeb vector field , LaTeXMLMath , of LaTeXMLMath , is LaTeXMLMath -invariant and , since the LaTeXMLMath -action preserves the contact structure , LaTeXMLEquation .
Because LaTeXMLMath is a symplectic subspace of LaTeXMLMath , the restriction of LaTeXMLMath to LaTeXMLMath is a contact form on LaTeXMLMath .
It what follows it is useful to cite Lemma 17 , pg 220 , of LaTeXMLCite , which identifies LaTeXMLMath , the dual of the Lie algebra of LaTeXMLMath , with LaTeXMLMath .
For any LaTeXMLMath and LaTeXMLMath LaTeXMLEquation since LaTeXMLMath .
Hence , the image of LaTeXMLMath in LaTeXMLMath under LaTeXMLMath is contained in LaTeXMLMath .
Because the moment map is equivariant , LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath is defined by LaTeXMLMath , where LaTeXMLMath is the coset containing LaTeXMLMath .
This is free by definition .
The moment map , LaTeXMLMath , for the LaTeXMLMath action on LaTeXMLMath is given by the restriction of LaTeXMLMath to LaTeXMLMath .
Therefore , LaTeXMLEquation .
The natural inclusion , LaTeXMLEquation descends to a map , LaTeXMLEquation defined by LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath denote the orbit through LaTeXMLMath under the LaTeXMLMath and LaTeXMLMath actions respectively .
If LaTeXMLMath , then the stablilizer of LaTeXMLMath in LaTeXMLMath is conjugate to LaTeXMLMath .
This implies that some element of the LaTeXMLMath -orbit through LaTeXMLMath has stabilizer equal to LaTeXMLMath , whence LaTeXMLMath is surjective .
To show that LaTeXMLMath is injective , suppose that LaTeXMLMath and that LaTeXMLMath .
Because LaTeXMLMath and LaTeXMLMath have stabilizer equal to LaTeXMLMath , it follows that LaTeXMLMath and therefore that LaTeXMLMath is injective .
∎ By expressing each stratum of the contact quotient as the reduction at zero of a contact manifold by a freely acting symmetry group , we obtain a contact structure on each stratum .
It is slightly less clear , however , how these structures are related to one another .
Suppose a Lie group LaTeXMLMath acts on a manifold LaTeXMLMath and LaTeXMLMath is an invariant 1-form on LaTeXMLMath .
Then LaTeXMLMath is equivariant as a section , LaTeXMLMath , and hence descends to a section , LaTeXMLMath , where LaTeXMLMath acts on LaTeXMLMath via the lifted action .
Suppose a Lie group LaTeXMLMath acts on a line bundle LaTeXMLMath and LaTeXMLMath is an invariant , non-vanishing section .
Then LaTeXMLMath is equivariantly trivial and hence LaTeXMLMath is a vector bundle .
The trivialization LaTeXMLMath defined by LaTeXMLMath is equivariant ( where LaTeXMLMath acts trivially on LaTeXMLMath ) since LaTeXMLMath is .
∎ Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated moment map .
Assume that LaTeXMLMath acts locally freely and properly on the zero level set of LaTeXMLMath .
Let LaTeXMLMath be the contact distribution on LaTeXMLMath and LaTeXMLMath its annihilator in LaTeXMLMath .
Let LaTeXMLMath .
Then there exists an embedding LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath is the induced section and LaTeXMLMath is the reduced contact form on LaTeXMLMath .
Set LaTeXMLMath .
Then LaTeXMLMath is a submanifold of LaTeXMLMath and the natural inclusion , LaTeXMLMath gives rise to a projection , LaTeXMLMath .
By Proposition LaTeXMLRef , the Reeb vector field is tangent to LaTeXMLMath .
Split the tangent bundle of LaTeXMLMath at LaTeXMLMath as LaTeXMLEquation .
We obtain an induced splitting , LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath .
Therefore , LaTeXMLMath embeds LaTeXMLMath into LaTeXMLMath .
Moreover , this embedding is equivariant .
Denote the symplectic moment map for the lifted LaTeXMLMath action on LaTeXMLMath by LaTeXMLMath .
Then LaTeXMLEquation .
For any LaTeXMLMath , and LaTeXMLMath , we have LaTeXMLEquation since LaTeXMLMath implies LaTeXMLMath for all LaTeXMLMath .
Hence , LaTeXMLMath embeds LaTeXMLMath into LaTeXMLMath .
Let LaTeXMLMath be the induced embedding .
The co-tangent bundle reduction theorem of Abraham-Marsden and Kummer ( LaTeXMLCite ) asserts that there is a symplectomorphsim LaTeXMLMath .
Set LaTeXMLMath .
Recall that the reduced contact form on LaTeXMLMath is defined to be the unique 1-form , LaTeXMLMath , on LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath is the orbit map .
It follows , by definition , that LaTeXMLMath , giving LaTeXMLMath .
∎ Since each stratum in the contact quotient at zero can be expressed as the reduction of a contact manifold under a free group action , our third theorem follows immediately .
Suppose a Lie group LaTeXMLMath acts properly on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated contact moment map .
For each stabilizer subgroup , LaTeXMLMath , of LaTeXMLMath , let LaTeXMLEquation denote the stratum associated to LaTeXMLMath .
Then there exists a line bundle , LaTeXMLMath over LaTeXMLMath equipped with a section LaTeXMLMath such that for each such LaTeXMLMath , the restriction of LaTeXMLMath and LaTeXMLMath to LaTeXMLMath defines a co-oriented contact structure on the stratum .
Set LaTeXMLMath and LaTeXMLEquation .
By Lemma LaTeXMLRef , LaTeXMLMath is a line bundle over LaTeXMLMath .
Let LaTeXMLMath be the normalizer of LaTeXMLMath in LaTeXMLMath and LaTeXMLMath .
Set LaTeXMLEquation .
Then LaTeXMLMath is a contact submanifold of LaTeXMLMath and LaTeXMLMath acts freely on LaTeXMLMath , preserving LaTeXMLMath .
Denote the corresponding moment map by LaTeXMLMath .
Recall that we can identify LaTeXMLMath with LaTeXMLMath and that there is a diffeomorphism LaTeXMLMath .
It follows that , without being too fussy about equivalences versus equalities , LaTeXMLEquation .
Now apply the free case to obtain the result .
∎ Proposition LaTeXMLRef and Theorems LaTeXMLRef and LaTeXMLRef immediately imply Suppose a Lie group LaTeXMLMath acts on a contact manifold LaTeXMLMath , preserving LaTeXMLMath , and let LaTeXMLMath be the associated moment map .
Choose LaTeXMLMath and suppose that LaTeXMLMath acts properly on LaTeXMLMath A slice exists for LaTeXMLMath which is invariant under dilations by LaTeXMLMath LaTeXMLMath Let LaTeXMLMath be the connected Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
For each stabilizer subgroup , LaTeXMLMath , of LaTeXMLMath , let LaTeXMLEquation denote the stratum associated to LaTeXMLMath .
Then there exists a line bundle , LaTeXMLMath over LaTeXMLEquation equipped with a section LaTeXMLMath such that for each such LaTeXMLMath , the restriction of LaTeXMLMath and LaTeXMLMath to LaTeXMLMath defines a co-oriented contact structure on the stratum .
Let LaTeXMLMath be a compact , connected Lie group and LaTeXMLMath a submanifold of LaTeXMLMath which is invariant under the co-adjoint action of LaTeXMLMath on LaTeXMLMath .
Choose LaTeXMLMath and let LaTeXMLMath be the co-normal space of LaTeXMLMath at LaTeXMLMath .
Then LaTeXMLMath is a Lie ideal in LaTeXMLMath .
Let LaTeXMLMath be the unique connected Lie subgroup of LaTeXMLMath with Lie algebra LaTeXMLMath .
Call LaTeXMLMath proper if LaTeXMLMath is closed in LaTeXMLMath for all LaTeXMLMath .
Suppose that LaTeXMLMath acts in a Hamiltonian fashion on a symplectic manifold , LaTeXMLMath and choose a corresponding equivariant moment map , LaTeXMLMath .
Let LaTeXMLMath be a proper , invariant submanifold of LaTeXMLMath .
If LaTeXMLMath is transverse to LaTeXMLMath , then LaTeXMLMath is a co-isotropic submanifold of LaTeXMLMath and the leaf of the null foliation through LaTeXMLMath is identified with the LaTeXMLMath orbit through LaTeXMLMath .
Suppose a compact , connected Lie group LaTeXMLMath acts in a Hamiltonian fashion on a symplectic manifold LaTeXMLMath and let LaTeXMLMath be a corresponding equivariant moment map .
Let LaTeXMLMath is an invariant , proper submanifold of LaTeXMLMath and set LaTeXMLMath .
If LaTeXMLMath is transverse to LaTeXMLMath then the leaf space , LaTeXMLEquation is a symplectic orbifold .
The reader is refered to LaTeXMLCite for a proof of this Proposition .
If LaTeXMLMath , then LaTeXMLMath is the Marsden-Weinstein-Meyer reduced space .
If LaTeXMLMath , a co-adjoint orbit , then LaTeXMLMath is the Kahzdan-Kostant-Sternberg reduced space .
A symplectic manifold LaTeXMLMath is called a symplectic cone if there is a free , proper LaTeXMLMath action LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath be a compact , co-oriented contact manifold and LaTeXMLMath the symplectization of LaTeXMLMath .
Then the action of LaTeXMLMath on LaTeXMLMath given by LaTeXMLMath makes LaTeXMLMath into a symplectic cone .
In fact , one can show that all symplectic cones are of the form given in the above example .
Let LaTeXMLMath be a symplectic cone .
Then LaTeXMLMath where LaTeXMLMath is a co-oriented contact manifold .
The reader is refered to , for example , LaTeXMLCite for a proof of the above proposition and more details on symplectic cones .
Call LaTeXMLMath integral if there is a group homomorphism LaTeXMLMath such that LaTeXMLMath .
Call a co-adjoint orbit LaTeXMLMath an integral orbit if it is an orbit through an integral element of LaTeXMLMath .
Note that if LaTeXMLMath is integral , then LaTeXMLMath , the kernel group of LaTeXMLMath defined earlier .
Integral orbits are automatically proper .
The cone on an integral orbit LaTeXMLMath is a proper submanifold of LaTeXMLMath .
Let LaTeXMLMath be a co-oriented contact manifold equipped with an action of a compact , connected Lie group LaTeXMLMath which preserves LaTeXMLMath and let LaTeXMLMath be the associated contact moment map .
Extend the LaTeXMLMath action to the symplectization LaTeXMLMath by LaTeXMLMath .
A corresponding equivariant symplectic moment map LaTeXMLEquation is given by LaTeXMLMath .
Let LaTeXMLMath be an integral orbit and assume that LaTeXMLMath is transveral to LaTeXMLMath .
By Proposition LaTeXMLRef , LaTeXMLMath is a symplectic orbifold .
The LaTeXMLMath action on LaTeXMLMath described in Example LaTeXMLRef descends to LaTeXMLMath .
By Proposition LaTeXMLRef , the orbit space under the LaTeXMLMath action on LaTeXMLMath is thus a contact manifold and is called the Guillemin-Sternberg reduction of LaTeXMLMath by LaTeXMLMath .
It is denoted by LaTeXMLMath .
The Guillemin-Sternberg reduction should be thought of as a larger form of contact reduction in the following sense .
Since LaTeXMLMath is LaTeXMLMath invariant , LaTeXMLMath .
For any LaTeXMLMath the contact quotient LaTeXMLMath is a contact suborbifold of LaTeXMLMath .
More , in fact , is true .
Since the leaf of the null foliation through LaTeXMLMath is given by the LaTeXMLMath orbit through LaTeXMLMath , the restriction of LaTeXMLMath to a leaf in LaTeXMLMath is constant .
Therefore , LaTeXMLMath descends to LaTeXMLEquation .
For any LaTeXMLMath , let LaTeXMLMath be the map LaTeXMLMath , where LaTeXMLMath .
Denote the graph of LaTeXMLMath by LaTeXMLMath .
It follows that LaTeXMLMath and , since LaTeXMLMath , that LaTeXMLEquation .
In summary , we have shown , Let LaTeXMLMath be a compact , connected , co-oriented contact manifold equipped with an action of a compact , connected Lie group LaTeXMLMath which preserves LaTeXMLMath and let LaTeXMLMath be the associated moment map .
Choose an integral LaTeXMLMath and let LaTeXMLMath be the co-adjoint orbit through LaTeXMLMath .
Then LaTeXMLMath fibers over LaTeXMLMath with typical fiber LaTeXMLMath .
Independent of Guillemin-Sternberg , C. Albert discovered an elegant method for contact reduction .
However , his method depends upon the choice of contact form used to represent the given contact structure .
Let LaTeXMLMath be a co-oriented contact manifold on which a compact group LaTeXMLMath acts by contact transformations .
Let LaTeXMLMath be the associated moment map .
Suppose LaTeXMLMath is a regular value of LaTeXMLMath , so that LaTeXMLMath is a submanifold of LaTeXMLMath .
Consider the map , LaTeXMLMath defined by LaTeXMLMath , where LaTeXMLMath is the Reeb vector field .
It follows that LaTeXMLMath is a map of Lie algebras .
By a theorem of Palais there is a unique map , LaTeXMLMath , where LaTeXMLMath is the universal covering group of LaTeXMLMath , whose differential is LaTeXMLMath .
Set LaTeXMLMath .
Then LaTeXMLMath acts locally freely and effectively on LaTeXMLMath .
The Albert reduction of LaTeXMLMath by LaTeXMLMath at LaTeXMLMath is defined to be LaTeXMLEquation .
If the LaTeXMLMath action is proper , this is an orbifold and , by definition , LaTeXMLMath descends to LaTeXMLMath on LaTeXMLMath .
A standard argument shows that LaTeXMLMath is a contact form .
However LaTeXMLMath , where LaTeXMLMath is a positive invariant function , is an invariant contact form with associated moment map LaTeXMLMath .
This indicates that the Albert quotient will be dependent on the choice of contact form .
The following example illustrates this dependence .
Let LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath act on LaTeXMLMath , LaTeXMLMath , by the restriction of the action on LaTeXMLMath given by the weights LaTeXMLMath .
Since both LaTeXMLMath are star shaped about the origin in LaTeXMLMath , they are isomorphic contact manifolds .
The difference in shape amounts to a choice of different contact forms in the same conformal class .
Let LaTeXMLMath denote the contact moment map for the LaTeXMLMath action on LaTeXMLMath .
A simple calculation yields LaTeXMLEquation .
It is not hard to see that LaTeXMLEquation and that the Albert action is trivial .
Hence the reduction of LaTeXMLMath is the circle , LaTeXMLMath .
A more tedious calculation gives LaTeXMLEquation .
LaTeXMLEquation The symmetry group in the Albert quotient is the circle , acting with weights LaTeXMLMath .
Note that LaTeXMLMath contains a 3-torus , namely LaTeXMLEquation .
Hence , the Albert reduction of LaTeXMLMath is at least 3 dimensional , illustrating the depending of the Albert quotient on the choice of the contact form .
The set theory relations LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath have corollaries in subspace relations .
Geometric Algebra is introduced as the ideal framework to explore these subspace operations .
The relations LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are easily subsumed by Geometric Algebra for Euclidean metrics .
A short computation shows that the meet ( LaTeXMLMath ) and join ( LaTeXMLMath ) are resolved in a projection operator representation with the aid of one additional product beyond the standard Geometric Algebra products .
The result is that the join can be computed even when the subspaces have a common factor , and the meet can be computed without knowing the join .
All of the operations can be defined and computed in any signature ( including degenerate signatures ) by transforming the problem to an analogous problem in a different algebra through a transformation induced by a linear invertible function ( a LIFT to a different algebra ) .
The new results , as well as the techniques by which we reach them , add to the tools available for subspace computations .
Operations on subspaces are useful in applications everywhere .
Geometric Algebra , the intriguing algebra promoted by David Hestenes to unify and simplify many areas of mathematics LaTeXMLCite , is introduced as the ideal framework to explore subspace operations .
A large repertoire of operations to compute subspace operations are made available by Geometric Algebra , but a few holes remain , notably with respect to the meet and join of subspaces .
This paper should resolve the outstanding issues .
A section on preliminaries makes this treatment reasonably self-contained .
Four subspace operations are introduced on equal footing , motivated by the set theory relations ( LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath ) , common usage , and applied needs .
An overview of the four subspace operations for all signatures indicates that they are all fundamental and interconnected .
Experts will note that Geometric Algebra deals with oriented subspaces and that in general there is no oriented solution to the meet and join problem LaTeXMLCite .
This problem is avoided by representing unoriented subspaces by projection operators .
This allows us to extend the meet and join defined in the previous literature LaTeXMLCite to give meaningful ( nonzero ) results for any subspaces .
The price we pay for this extension is that the meet and join presented in this paper are not linear .
This is not a disadvantage , because our meet and join agree with the previous literature except when the previous literature results are zero .
Many operations in this paper have an arbitrary scale and orientation .
This is addressed in the penultimate section , where a geometrical significance can be given to the linear result of zero from the previous literature .
This section includes the definitions and motivation for four subspace operations .
Following the new definitions is a review of Geometric Algebra .
The set theory operations of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath can be applied to subspaces of LaTeXMLMath .
However , in general the outcome will not be a subspace , since the set theory operations do not respect the linear structure of the subspaces .
Four subspaces operations are defined that are motivated by the set theory operations , but that respect the linear structure and thus always produce subspaces .
Let LaTeXMLMath and LaTeXMLMath be two subspaces of LaTeXMLMath .
The Meet Operation The meet of LaTeXMLMath and LaTeXMLMath is the set LaTeXMLMath and LaTeXMLMath .
In words it is the largest common subspace .
It shall be denoted by LaTeXMLMath .
The Join Operation The join of LaTeXMLMath and LaTeXMLMath is the set LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
In words it is the span of the two subspaces ( i.e .
the smallest common superspace ) .
It shall be denoted by LaTeXMLMath .
The Difference Operation The difference of LaTeXMLMath and LaTeXMLMath is the set LaTeXMLMath , provided LaTeXMLMath is a subspace of LaTeXMLMath .
In words it is the orthogonal complement of LaTeXMLMath in LaTeXMLMath .
It shall be denoted by LaTeXMLMath .
The Symmetric Difference Operation The symmetric difference of LaTeXMLMath and LaTeXMLMath is the set LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
In words it is the orthogonal complement of the meet in the join .
It shall be denoted by LaTeXMLMath .
Clearly , LaTeXMLMath .
This paper always uses the symbols LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath to denote subspace operations .
Geometric Algebra is based on Clifford Algebra over a real vector space .
A Clifford Algebra is an algebra generated by the scalars and the elements of a vector space with a metric .
The algebra has a linear , associative , distributive product , and the square of a vector is the squared length determined by the metric ( for more details on Clifford Algebras see LaTeXMLCite ) .
Let LaTeXMLMath be a ( LaTeXMLMath + LaTeXMLMath + LaTeXMLMath ) -dimensional real vector space with a set of linearly independent , mutually orthogonal vectors , LaTeXMLMath , such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
The Clifford Algebra and the Geometric Algebra generated by LaTeXMLMath shall be denoted LaTeXMLMath .
The essential difference between a Geometric Algebra and a Clifford Algebra is that the elements of a Geometric Algebra are given a geometric interpretation .
This leads to a focus on operations that are defined on geometrically meaningful subsets of the algebra and therefore to the introduction of additional structure ( and more products between elements ) , so that a consistent geometric interpretation can be maintained on the results of computations .
To emphasize this difference , we call the elements of the Geometric Algebra multivectors and we call the standard ( Clifford ) product between elements in the Algebra the geometric product .
The geometric product is denoted by juxtaposition of operands , as in LaTeXMLMath .
A summary of the extra features and terminology of Geometric Algebra follows .
Blades If a nonzero multivector can be written as the geometric product of LaTeXMLMath mutually anticommuting vectors , then it is called an LaTeXMLMath -blade .
The word blade refers to an LaTeXMLMath -blade with the value of LaTeXMLMath unspecified .
Real numbers are considered LaTeXMLMath -blades and are often called scalars .
Vectors are considered LaTeXMLMath -blades .
The square of a blade is a scalar .
For a vector the square is the squared length determined by the metric .
Zero is considered an LaTeXMLMath -blade for any value of LaTeXMLMath , but any nonzero blade is an LaTeXMLMath -blade for only one value of LaTeXMLMath LaTeXMLCite .
Steps ( or Grades ) A linear combination of LaTeXMLMath -blades will be called an LaTeXMLMath -vector , and will be said to have step ( or grade ) LaTeXMLMath .
The space of LaTeXMLMath -vectors is a linear subspace of the entire Clifford Algebra .
An arbitrary multivector , LaTeXMLMath , can be uniquely written as LaTeXMLMath where LaTeXMLMath is the LaTeXMLMath -vector part of LaTeXMLMath if LaTeXMLMath is a nonnegative integer and LaTeXMLMath is zero if LaTeXMLMath is not a nonnegative integer .
Outer Product The outer product of an LaTeXMLMath -vector LaTeXMLMath and an LaTeXMLMath -vector LaTeXMLMath is defined to be LaTeXMLMath .
It is denoted by LaTeXMLMath and is extended by linearity to arbitrary multivectors .
The outer product is associative between all multivectors and anti-symmetric between vectors .
The outer product of two blades is a blade ( see the appendix ) .
( Contraction ) Inner Product The ( contraction ) inner product of an LaTeXMLMath -vector LaTeXMLMath and an LaTeXMLMath -vector LaTeXMLMath is defined to be LaTeXMLMath .
It is denoted by LaTeXMLMath and is extended by linearity to arbitrary multivectors .
This inner product differs slightly from the inner product of Hestenes LaTeXMLCite .
It has the useful properties that LaTeXMLMath and LaTeXMLMath for any multivectors LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and any vector LaTeXMLMath .
These details are obvious from considering this definition and looking at the proofs in Hestenes LaTeXMLCite .
The inner product is explicitly expanded for the vectors LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , … , and LaTeXMLMath as LaTeXMLEquation where the inverted circumflex indicates that the LaTeXMLMath th vector was omitted from the product .
The proof in reference LaTeXMLCite carries over to the contraction inner product with no modification .
The inner product of two blades is a blade ( see the appendix ) .
Subspaces One of the geometric interpretations common in geometric algebra is to use blades to represent subspaces .
This works because blades are closely related to subspaces .
Given a nonzero blade , LaTeXMLMath , the set LaTeXMLMath is a subspace of LaTeXMLMath .
Similarly given an oriented basis { LaTeXMLMath } for a LaTeXMLMath -dimensional subspace LaTeXMLMath , there is a nonzero blade that corresponds to that oriented basis .
If LaTeXMLMath then that blade is LaTeXMLMath .
If LaTeXMLMath then that blade is LaTeXMLMath .
The identity LaTeXMLMath is the means by which Geometric Algebra subsumes the operation LaTeXMLMath from set theory .
Since the square of a blade is a scalar , the inverse of a blade ( if it has one ) , is equal to the blade divided by its square , so the inverse is just a scalar multiple of the original blade , and hence a blade and its inverse represent the same ( unoriented ) subspace .
Pseudoscalars Given an algebra , LaTeXMLMath , and a set of LaTeXMLMath vectors LaTeXMLMath , the outer product LaTeXMLMath is called a pseudoscalar .
Often a particular nonzero pseudoscalar is singled out .
This preferred pseudoscalar serves to determine the reference orientation for the vector space and sometimes is used to perform duality operations .
Usually the preferred pseudoscalar is called the pseudoscalar and denoted LaTeXMLMath .
This can be seen as merely a special case of the previous section on subspaces , by noticing that every nonzero LaTeXMLMath -blade is a pseudoscalar for the LaTeXMLMath dimensional subspace it represents .
Projection Operators For a Euclidean metric , blades are closely related to projection operators .
Given a nonzero blade , LaTeXMLMath , representing the subspace LaTeXMLMath , the vector LaTeXMLMath is the orthogonal projection of the vector LaTeXMLMath onto the subspace LaTeXMLMath .
As taken from LaTeXMLCite , the identities LaTeXMLEquation and LaTeXMLEquation hold for any nonzero blades LaTeXMLMath and LaTeXMLMath .
Equation ( LaTeXMLRef ) implies that a projection operator can be decomposed analogously to the way its corresponding blades can be factored .
Outermorphism Given a linear function , LaTeXMLMath , there is an extension of the function to arbitrary multivectors called the outermorphism of LaTeXMLMath .
It is denoted by LaTeXMLMath and is defined to be the identity when restricted to the scalars .
The condition LaTeXMLMath ( LaTeXMLMath ) = LaTeXMLMath ( LaTeXMLMath ) LaTeXMLMath ( LaTeXMLMath ) + LaTeXMLMath ( LaTeXMLMath ) LaTeXMLMath ( LaTeXMLMath ) is then sufficient to define the outermorphism on arbitrary multivectors .
This extension is well-established LaTeXMLCite .
A notation convention is adopted to aid the reader in easily making meaningful distinctions between different multivectors .
Lowercase Greek letters are reserved for scalars .
Lowercase Latin letters are reserved for integers or functions when not in bold face , while lowercase Latin letters are reserved for vectors when in bold face .
Bold face is reserved for blades .
Lastly , uppercase Latin letters are used when it is impossible or unnecessary to be more specific about the nature of a multivector .
This notation convention simplifies the reading of equations and emphasizes that different geometric interpretations are applied to different elements of the algebra .
Since every blade represents an oriented subspace and blades are easy to compute with , they are a natural candidate for subspace computations .
The extra scalar degree of freedom allows the future potential for more precise calculations with subspaces that attach meaning to the magnitude of a blade .
Therefore in this paper four blade operations are introduced to correspond to the four subspace operations .
The blade operations are made first for Euclidean space because the relationship between blades and projection operators is strongest in Euclidean space .
Therefore first a correspondence is made from projection operators to blades , then the four blade operations are defined and shown to faithfully mirror the subspace operations .
Here we give an algorithm to construct a blade from its corresponding projection operator .
The algorithm has an arbitrary scale and orientation inherited from an arbitrarily chosen basis , which is the best that can be expected .
Let LaTeXMLMath be an idempotent linear operator on LaTeXMLMath such that the image of LaTeXMLMath is a LaTeXMLMath -dimensional subspace , LaTeXMLMath .
The algorithm constructs a blade that characterizes LaTeXMLMath as follows .
First we construct a set of candidate blades , and then show that all the nonzero candidate blades represent the subspace LaTeXMLMath .
Finally we show that at least one of the candidate blades is , in fact , nonzero .
Let LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath be LaTeXMLMath LaTeXMLMath -blades such that LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath } is a basis for the space of LaTeXMLMath -vectors .
Let LaTeXMLMath .
The set LaTeXMLMath is our set of candidate blades .
By the properties of the outermorphism , each LaTeXMLMath is a LaTeXMLMath -blade .
Each LaTeXMLMath is clearly the outer product of LaTeXMLMath vectors and each of these LaTeXMLMath vectors is in LaTeXMLMath .
If the LaTeXMLMath vectors are linearly dependent then LaTeXMLMath , if not then they form a basis for LaTeXMLMath so LaTeXMLMath and LaTeXMLMath is exactly the kind of blade to characterize the subspace LaTeXMLMath .
All that remains is to show that at least one of the candidate blades is nonzero .
Since a blade that characterizes LaTeXMLMath exists we know that it is a linear combination of { LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath } , so by the linearity of the outermorphism there must be a LaTeXMLMath such that LaTeXMLMath .
Given this correspondence we can translate operations between projection operators into operations between blades , except for a loss of the scale and orientation .
Consider two nonzero blades , LaTeXMLMath and LaTeXMLMath , that characterize the subspaces LaTeXMLMath and LaTeXMLMath respectively .
When LaTeXMLMath is a subspace of LaTeXMLMath we use the expression LaTeXMLMath to denote the quantity LaTeXMLMath and the call the operation inner division .
The inner division operation is motivated by the difference operation for subspaces .
The justification requires showing two points , first that under such conditions , LaTeXMLMath is a blade , and second that LaTeXMLMath .
Let LaTeXMLMath be an orthogonal basis for LaTeXMLMath .
Let LaTeXMLMath be an orthogonal basis for LaTeXMLMath .
Clearly LaTeXMLMath is an orthogonal basis for LaTeXMLMath .
Since LaTeXMLMath is an orthogonal basis for LaTeXMLMath and since LaTeXMLMath characterizes the subspace LaTeXMLMath it follows that LaTeXMLMath is a nonzero scalar multiple of LaTeXMLMath .
Similarly LaTeXMLMath is a nonzero scalar multiple of LaTeXMLMath .
It then follows that there exists two nonzero scalars LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Thus LaTeXMLMath is a blade and it characterizes the subspace LaTeXMLMath .
A quick look at the step of the output reveals that when LaTeXMLMath is a subspace of LaTeXMLMath then LaTeXMLMath .
Therefore , while it appears that the inner division operation is based on the geometric product it is also just as easily based on the inner product .
It is useful whenever a product between two blades can be written as either of two products , because either definition can be used from line to line of a computation , depending on which product gives simplifications at that particular moment .
An example is the identity , LaTeXMLEquation which is proved by decomposing the inner division first into the geometric product and then into the inner product and looking at the step of the outcome .
Consider two nonzero blades , LaTeXMLMath and LaTeXMLMath , that characterize the subspaces LaTeXMLMath and LaTeXMLMath respectively .
Let LaTeXMLMath and let LaTeXMLMath be any blade characterizing that subspace .
When LaTeXMLMath , LaTeXMLMath .
When LaTeXMLMath , LaTeXMLMath .
In the latter case we can define LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath both represent LaTeXMLMath , which is a subspace of both LaTeXMLMath and LaTeXMLMath , the previous section makes it clear that LaTeXMLMath and LaTeXMLMath are blades .
The intersection of the subspaces characterized by LaTeXMLMath and LaTeXMLMath contains only the element zero , so LaTeXMLMath .
By construction LaTeXMLMath , so the highest step portion of LaTeXMLMath is LaTeXMLMath , and therefore a blade .
This motivates a new product for blades which we call the delta product .
The delta product of two blades , LaTeXMLMath and LaTeXMLMath is denoted LaTeXMLMath and defined to be the highest step portion of LaTeXMLMath .
The delta product is motivated by the symmetric difference operation for subspaces .
The justification requires showing that LaTeXMLMath .
Let LaTeXMLMath be an orthogonal basis for LaTeXMLMath .
Let LaTeXMLMath be an orthogonal basis for LaTeXMLMath .
Similar arguments as in the previous section demonstrate that there exists a nonzero scalar LaTeXMLMath such that LaTeXMLMath , so LaTeXMLMath .
Clearly LaTeXMLMath .
Therefore assume LaTeXMLMath and we will show that LaTeXMLMath .
By definition LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
Clearly LaTeXMLMath .
Clearly LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath , LaTeXMLMath therefore LaTeXMLMath .
This means LaTeXMLMath so LaTeXMLMath .
Similarly for LaTeXMLMath .
Therefore LaTeXMLMath .
The delta product is very different than the inner or outer product in its algebraic properties .
The major difference is that LaTeXMLMath , so the delta product can not be extended by linearity to arbitrary multivectors .
The delta product can only be used on blades .
More care must be taken with implementations of the delta product because even a small change in either LaTeXMLMath or LaTeXMLMath can cause a change in the step of LaTeXMLMath .
Consider two nonzero blades , LaTeXMLMath and LaTeXMLMath , that characterize the subspaces LaTeXMLMath and LaTeXMLMath respectively .
The blade correspondence can be used to define a new product for blades called the meet and denoted LaTeXMLMath .
LaTeXMLMath is defined modulo a scale and an orientation as the blade corresponding to the projection operator LaTeXMLMath , where LaTeXMLMath is defined as : LaTeXMLEquation .
The justification requires that LaTeXMLMath be the orthogonal projection onto LaTeXMLMath .
Let LaTeXMLMath be a blade characterizing the subspace LaTeXMLMath , then define LaTeXMLMath and LaTeXMLMath as above .
Now the result follows from a simple application of equation ( LaTeXMLRef ) .
First note that LaTeXMLMath , so equation ( LaTeXMLRef ) implies that LaTeXMLMath , therefore we have the following identity : LaTeXMLEquation .
Similarly LaTeXMLMath , therefore we have the following identity : LaTeXMLEquation .
Finally LaTeXMLMath , because every vector in LaTeXMLMath is orthogonal to every vector in LaTeXMLMath .
Since LaTeXMLMath represents a subspace of LaTeXMLMath it is just as true that LaTeXMLMath , therefore we have the following identity : LaTeXMLEquation .
The linear combination of the three projection operators has now been reduced to the linear combination of four projection operators .
A quick appeal to equation ( LaTeXMLRef ) implies that for two blades , if their geometric product is a scalar then their projection operators are equal .
Now LaTeXMLMath is a scalar , and so is LaTeXMLMath .
Therefore the terms LaTeXMLMath and LaTeXMLMath cancel and the terms LaTeXMLMath and LaTeXMLMath combine .
The result , ( equation ( LaTeXMLRef ) ) , then follows from equations ( LaTeXMLRef ) , ( LaTeXMLRef ) , and ( LaTeXMLRef ) .
A small commentary is in order .
The first comment is that the calculation of the blade correspondence will be simplified by the fact that if the steps of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath respectively , then the step of the meet is LaTeXMLMath .
The second comment is that the blade correspondence is not precise about the scale and orientation of the blade because the blade correspondence inherits an arbitrary scale and orientation from an arbitrary basis of blades .
Since the meet for blades is defined by the blade correspondence , this lack of precision is then passed on to the meet for blades , except for the disjoint case .
The disjoint case occurs when LaTeXMLMath , and in this case one can choose a basis a priori .
This is possible because in this case the meet for blades is a scalar .
Therefore one can choose the scalar ‘ LaTeXMLMath ’ for the basis , and then since the blade correspondence uses the outermorphism of the projection operator and an outermorphism is the identity when restricted to the scalars , the blade correspondence gives a determinate answer for the meet , namely it gives ‘ LaTeXMLMath ’ back again .
The third comment is that the meet for blades given here is different from previous literature LaTeXMLCite , which only relates nontrivially to our definition when LaTeXMLMath .
When LaTeXMLMath the previous literature gives the zero blade as the result , while we can also treat that case .
The price we pay is linearity .
Like the delta product , the meet is not linear .
Previously we noted that LaTeXMLMath , in fact LaTeXMLMath characterizes LaTeXMLMath , therefore using equation ( LaTeXMLRef ) , we find that LaTeXMLMath .
We have two alternatives .
The first alternative is to define the projection operator for the join directly as LaTeXMLEquation and if the steps of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath respectively , then the step of the join is LaTeXMLMath .
The second alternative is to define the join for blades directly in terms of the meet for blades and the inner division operation through the equation LaTeXMLMath .
Just as the meet had a definite scale and orientation only in the disjoint case , this definition allows the join to inherit the definite scale and orientation from the meet , since in that case LaTeXMLMath .
It bears mentioning that this join for blades only agrees with the previous literature LaTeXMLCite in the disjoint case , but again definitions in the previous literature are merely zero when LaTeXMLMath , so this definition is an extension .
As with the meet the price we pay is linearity .
A new concept is introduced to allow the subspace operations to be performed in any metric .
Each of the four operations is then investigated in turn .
Given two algebras , LaTeXMLMath and LaTeXMLMath , such that LaTeXMLMath + LaTeXMLMath + LaTeXMLMath = LaTeXMLMath + LaTeXMLMath + LaTeXMLMath = LaTeXMLMath , and a linear invertible function , LaTeXMLMath , from the vectors of LaTeXMLMath to LaTeXMLMath then LaTeXMLMath is a linear invertible map between the two algebras .
Call the extended function LaTeXMLMath a LIFT ( ‘ linear invertible function ’ transformation ) of LaTeXMLMath to LaTeXMLMath .
A LIFT can be used to transfer a problem with subspaces to another algebra , preserving incidence relations but allowing the metric to change .
Often a LIFT is taken to a Euclidean space .
After solving the problem in that space , subspace results can be pulled back to the original space .
Examples that extend the previous results on the subspace operations follow .
A minor variation of the LIFT is for LaTeXMLMath to go into an LaTeXMLMath -dimensional subspace of a Clifford Algebra over a larger vector space , then the outermorphism is a linear invertible map onto a subalgebra .
This is especially nice when the outermorphism is an isomorphism between the original algebra and the subalgebra .
Such a LIFT is called an embedding LIFT ( or e-LIFT ) .
This is especially common for degenerate algebras LaTeXMLMath , for which an e-LIFT to LaTeXMLMath always exists .
To see the existence of the e-LIFT , let LaTeXMLMath be an orthogonal basis for the vectors in LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath and let LaTeXMLMath be an orthogonal basis for the vectors in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Then let LaTeXMLMath be a linear function such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then LaTeXMLMath is the promised isomorphism .
If one fixes an arbitrary LIFT , LaTeXMLMath , from LaTeXMLMath to LaTeXMLMath then the meet , LaTeXMLMath , between two blades LaTeXMLMath and LaTeXMLMath can be defined by : LaTeXMLEquation .
The scale and orientation of LaTeXMLMath are indeterminate except when LaTeXMLMath and LaTeXMLMath are disjoint , which only happens when LaTeXMLMath and LaTeXMLMath are disjoint .
The LIFT is an outermorphism , so it is the identity on the scalars , so the meet has a definite scale and orientation in the disjoint case , and they are the same scale and orientation as in the Euclidean space .
More importantly , in the disjoint case , the scale and orientation are independent of which LIFT , LaTeXMLMath , was chosen .
The preservation of the outer product and the scalars makes it clear that this meet corresponds to LaTeXMLMath .
This means that this definition has a well-defined scale and orientation in exactly the cases where the Euclidean definition did , and it has an arbitrary scale and orientation in exactly the cases where the Euclidean definition did .
If one fixes fixes an arbitrary LIFT , LaTeXMLMath , from LaTeXMLMath to LaTeXMLMath then the join , LaTeXMLMath , between two blades LaTeXMLMath and LaTeXMLMath can be defined by : LaTeXMLEquation .
The scale and orientation of LaTeXMLMath are indeterminate except when LaTeXMLMath and LaTeXMLMath are disjoint , which only happens when LaTeXMLMath and LaTeXMLMath are disjoint , in which case the join should reduce to the outer product .
The LIFT is an outermorphism , so it preserves the outer product , so clearly LaTeXMLMath .
Therefore the join has a definite scale and orientation in the disjoint case , and the scale and orientation are independent of which LIFT , LaTeXMLMath , was chosen .
The preservation of the outer product and the scalars makes it clear that this join corresponds to LaTeXMLMath .
This means that this definition has a well-defined scale and orientation in exactly the cases where the Euclidean definition did , and it has an arbitrary scale and orientation in exactly the cases where the Euclidean definition did .
Consider two nonzero blades , LaTeXMLMath and LaTeXMLMath in LaTeXMLMath , that characterize the subspaces LaTeXMLMath and LaTeXMLMath respectively such that LaTeXMLMath .
When LaTeXMLMath exists we can calculate LaTeXMLMath as usual .
Otherwise , we need an e-LIFT , LaTeXMLMath , from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be the pseudoscalar of LaTeXMLMath .
Then define LaTeXMLMath .
This meets the definition for the subspace operation , but now the scale and orientation has an arbitrary dependence on LaTeXMLMath .
Consider two nonzero blades , LaTeXMLMath and LaTeXMLMath in LaTeXMLMath , that characterize the subspaces LaTeXMLMath and LaTeXMLMath respectively .
If LaTeXMLMath has an inverse then the geometric product , LaTeXMLMath , is nonzero and the highest step portion represents the symmetric difference , LaTeXMLMath as usual .
Otherwise , we need an e-LIFT , LaTeXMLMath , from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be the pseudoscalar of LaTeXMLMath .
Then define LaTeXMLMath .
This meets the definition for the subspace operation , but now the scale and orientation has an arbitrary dependence on LaTeXMLMath .
Note that the symmetric difference was used to compute the meet and join for Euclidean signatures , but the meet and join are used to define the symmetric difference for Non-Euclidean signatures .
The meet and join for blades presented in this paper are not linear , ( e.g .
in general LaTeXMLMath , even when LaTeXMLMath is a nonzero blade ) .
The previous literature LaTeXMLCite have linear results for the meet and join .
In our notation the meet and join of the previous literature are LaTeXMLMath and LaTeXMLMath respectively .
The linear versions can operate on any multivector ( by linear extension ) , but the geometric interpretation of the computation becomes confusing .
Also , even when the linear versions operate on nonzero blades , they can disagree with the subspace operations by giving a result of zero .
It is not surprising that the subspace operations disagree with the linear operations , because it was exactly the preponderance of the answer LaTeXMLMath for many meaningful computations that motivated the creation of the new subspace operations of this paper .
However the opposing versions can be reconciled by pursuing a geometric interpretation of the zero blade .
An interpretation of the zero blade that is consistent with the general enterprise of representing oriented subspaces by blades is to have the zero blade represent an indeterminate oriented subspace .
At first glance , it appears that the zero blade represents the whole space ( i.e .
LaTeXMLMath ) , but this interpretation would imply that the zero blade represents a different subspace depending on a particular enveloping pseudoscalar ( a property that destroys natural subalgebra and enveloping algebra relationships ) .
But since this is implicit in stating that the zero blade can represent any subspace this is actually support for the interpretation proposed here .
Furthermore , since the hope of representing subspaces by blades is to eventually be able to deal with uncertainty in geometrical computations , the scale factor would naturally be used to represent how well-determined the blade is .
This implies that the zero blade represents a completely undetermined subspace .
Lastly the linear versions give the zero blade as the result if either of the input blades is the zero blade , this adsorbing property is consistent with the indeterminacy the zero blade represents .
When two oriented subspaces do not span the pseudoscalar , the linear meet gives the result of the zero blade because the orientation of the meet can not be determined .
The linear meet is a fully functional quantitative operation , which can give a quantitative meet , but only if given a quantitative join first ( in the role of the pseudoscalar ) .
Similarly when two oriented subspaces have a nontrivial intersection , the linear join gives the result of the zero blade because the join has an undetermined orientation .
With this interpretation for the zero blade , the linear meet and join can be compared to the versions presented in this paper .
The linearity can be an advantage for implementation for some applications , and if that advantage outweighs the costs of getting the zero blade as a result , then an educated decision to implement the linear versions can be made for that application .
The interpretation for the zero blade can also be used to extend the subspace operations for nonzero blades to the zero blade by declaring the inner division , the delta product , the meet , and the join to be zero if either input of the two input blades is the zero blade .
Blades can represent subspaces , and in applications we need to perform operations on subspaces .
Therefore we would naturally want operations on blades that mirror the results of computations that we would have liked to perform on subspaces .
The four subspace operations ( inner division , delta product , meet , and join ) are the first four operations from nonzero blades to nonzero blades .
The hope is that these operations can contribute to quantitative computations with oriented subspaces .
The four operations are on equal footing because the delta product and the inner division are used to define the meet and join in Euclidean signatures , while the meet and join are used to define the delta product and inner division in Non-Euclidean signatures .
This indicates that they are all fundamental ( though not as fundamental as the geometric product ) and that they are all interconnected .
Standard concepts in geometric algebra needed to be augmented because the meet and join for blades can not be defined LaTeXMLCite with an orientation due to fundamental geometric problems .
This fundamental problem was solved by using projection operators to represent unoriented subspaces .
Within this solution the delta product helps to compute the meet and join for blades .
The price to be paid for this augmentation is that the new blade operations are not linear and can not be extended to arbitrary multivectors .
The authors believe that their non-linearity might be the reason that the operations have not been used previously .
It is only by sacrificing linearity , and thus losing applicability to arbitrary multivectors , that one can solve the meet and join for blades .
The four subspace operations are tools intended for general use in applications , however no applied examples are included in this paper .
Readers looking for examples of the inner division and the delta product need look no farther than the proof of the meet for Euclidean metrics , and readers looking for examples of the meet and join need look no farther than the inner division and delta product for Non-Euclidean metrics .
Beyond the four subspace operations , this paper utilizes another tool of general applicability .
This tool is the LIFT ( ‘ linear invertible function ’ transformation ) .
It is an invertible map between algebras of different signatures .
This tool can be used to advantageously transform problems that are independent of signature to whichever signature is most helpful at any particular moment .
This tool is demonstrated in the paper by extending the results of the meet and join from the Euclidean case to the Non-Euclidean case ( even to degenerate signatures ) .
Lastly some loose ends are resolved .
A geometric interpretation is given to the zero blade that explains the results of the previous literature and facilitates the educated choice between different versions of the subspace operations .
The final loose end is resolved by the appendix , which includes proofs to demonstrate that the inner and outer products go from blades to blades ( even in degenerate signatures ) .
In this Appendix we prove that the outer product of two blades is a blade and that the inner product of two blades is a blade .
To our surprise , this does not appear to have been proved before , but is less trivial than may have been assumed when degenerate algebras are considered .
By associativity of the outer product it suffices to show that the outer product of a vector and an LaTeXMLMath -blade is a blade .
The result is trivial because the ( LaTeXMLMath +1 ) -vector determines an ( LaTeXMLMath +1 ) -dimensional subspace and that subspace has an orthogonal basis .
However a more constructive proof is desirable , to assist in the proof for the inner product and to see how such a factorization can be made .
The proof proceeds by induction on LaTeXMLMath , the base case is LaTeXMLMath , or the outer product of two vectors is a blade .
Let LaTeXMLMath and LaTeXMLMath be two vectors .
Since LaTeXMLMath , this gives a factorization of LaTeXMLMath when LaTeXMLMath .
If LaTeXMLMath then either LaTeXMLMath or LaTeXMLMath is not equal to LaTeXMLMath .
Since LaTeXMLMath , we may assume without loss of generality that LaTeXMLMath .
Then we note that LaTeXMLMath is a vector and that LaTeXMLMath , so this gives a factorization of LaTeXMLMath .
Therefore in both cases the outer product of two vectors can be factored .
Assume that the outer product of LaTeXMLMath vectors is a blade .
Let LaTeXMLMath be LaTeXMLMath + LaTeXMLMath vectors .
The inductive step has three cases .
The first case is when LaTeXMLMath for each LaTeXMLMath .
Then by equation ( LaTeXMLRef ) , LaTeXMLMath .
By the inductive hypothesis , LaTeXMLMath is a blade , so there exist LaTeXMLMath anticommuting vectors , LaTeXMLMath , such that LaTeXMLMath .
Each LaTeXMLMath is a linear combination of the set LaTeXMLMath , so each LaTeXMLMath anticommutes with LaTeXMLMath , so LaTeXMLMath is a factorization of LaTeXMLMath , therefore LaTeXMLMath is an ( LaTeXMLMath + LaTeXMLMath ) -blade .
The next case is when LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLMath .
Now we have guaranteed that LaTeXMLMath for each LaTeXMLMath , so the previous case resolves the factorization .
The last case is when there exists a LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Without loss of generality we may assume LaTeXMLMath since the order of the vectors only affects the sign of the outcome .
By the base case , LaTeXMLMath is a LaTeXMLMath -blade .
Direct computation shows that the square of LaTeXMLMath is LaTeXMLMath , hence nonzero , therefore there exist two invertible vectors LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath for LaTeXMLMath .
Then LaTeXMLMath .
Now since LaTeXMLMath the previous case resolves the factorization .
Let LaTeXMLMath and LaTeXMLMath be two blades in LaTeXMLMath .
By the properties of the inner product , LaTeXMLMath , so it suffices to show that LaTeXMLMath is a blade for any vector LaTeXMLMath and any blade LaTeXMLMath .
If LaTeXMLMath is a scalar then LaTeXMLMath so it is a blade .
Otherwise let LaTeXMLMath .
The nonzero blade LaTeXMLMath characterizes a subspace with signature LaTeXMLMath and hence a subalgebra LaTeXMLMath .
Using equation ( LaTeXMLRef ) , it is clear that LaTeXMLMath resides in LaTeXMLMath .
Let LaTeXMLMath be a LIFT from LaTeXMLMath to LaTeXMLMath .
Since LaTeXMLMath is a blade if and only if LaTeXMLMath is a blade then to show that LaTeXMLMath is a blade it suffices to show that LaTeXMLMath is a blade .
From equation ( LaTeXMLRef ) , it follows that LaTeXMLMath is the sum of ( LaTeXMLMath - LaTeXMLMath ) -blades .
Let LaTeXMLMath be a nonzero pseudoscalar for LaTeXMLMath .
Let LaTeXMLMath be LaTeXMLMath ( LaTeXMLMath - LaTeXMLMath ) -blades in LaTeXMLMath .
Then LaTeXMLMath , where LaTeXMLMath is a vector .
If LaTeXMLMath then the sum , LaTeXMLMath , is zero and trivially a blade .
If not then LaTeXMLMath and therefore LaTeXMLMath has an inverse and by the details of the proof for the outer product , it is clear that LaTeXMLMath can be factored under the geometric product with LaTeXMLMath as a factor .
Then LaTeXMLMath is clearly a blade .
Let LaTeXMLMath denote a compact interval symmetric about LaTeXMLMath and let LaTeXMLMath denote the dilation of LaTeXMLMath by a factor of LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath , be compact oriented smooth manifolds of dimensions LaTeXMLMath and LaTeXMLMath respectively .
We suppose that we have an embedded copy of LaTeXMLMath inside of LaTeXMLMath .
( See Figure LaTeXMLRef ) .
Let LaTeXMLMath denote the complement of the hypersurface LaTeXMLMath .
We consider families LaTeXMLMath of Riemannian metric ( tensors ) on LaTeXMLMath each of whose restriction to LaTeXMLMath is a warped product of the following form LaTeXMLEquation .
Here LaTeXMLMath is a fixed Riemannian metric on LaTeXMLMath , LaTeXMLMath is smooth positive function that is positively homogeneous of degree LaTeXMLMath on LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath are real numbers .
The ‘ limiting metric ’ LaTeXMLMath is singular along the hypersurface LaTeXMLMath provided LaTeXMLMath .
Indeed , since LaTeXMLMath is homogeneous LaTeXMLEquation for some homogeneity constants LaTeXMLMath .
Melrose LaTeXMLCite has observed that the metric LaTeXMLMath is Riemannian complete if and only if LaTeXMLMath , whereas LaTeXMLMath has finite volume if and only if LaTeXMLMath .
( See Figure LaTeXMLRef ) .
Let LaTeXMLMath be a simple closed curve in a compact oriented surface LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be a metric on LaTeXMLMath of constant curvature LaTeXMLMath such that the unique geodesic homotopic to LaTeXMLMath has length LaTeXMLMath .
By the collar lemma , there exists an embedding LaTeXMLMath with LaTeXMLMath such that LaTeXMLEquation where LaTeXMLMath is the usual coordinate on the circle LaTeXMLMath .
Note that the Riemannian surface LaTeXMLMath is a union of hyperbolic cusps .
In this special case , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Henceforth , we will assume that LaTeXMLMath is strictly convex along nonradial lines and that LaTeXMLMath for LaTeXMLMath .
Moreover , we assume that the restriction of LaTeXMLMath to the unit tangent bundle of LaTeXMLMath is bounded .
LaTeXMLMath and LaTeXMLMath as quadratic forms on each tangent space ; thus each may be regarded as function on the unit tangent bundle .
With these assumptions we have Let LaTeXMLMath , and either let LaTeXMLMath or let LaTeXMLMath and LaTeXMLMath .
Suppose that LaTeXMLMath is a real-analytic family of Riemannian metrics on LaTeXMLMath satisfying ( LaTeXMLRef ) .
Then each eigenvalue branch of the associated family of Laplacians , LaTeXMLMath , converges to a finite limit as LaTeXMLMath tends to LaTeXMLMath .
S. Wolpert LaTeXMLCite proved Theorem LaTeXMLRef in the special case of hyperbolic degeneration .
He subsequently used this convergence to produce evidence supporting the belief that Maass cusp forms ‘ disappear ’ under perturbation LaTeXMLCite LaTeXMLCite .
Note that although the present paper does not include a discussion of manifolds with cusps , the methods described here apply equally well to the eigenbranches of a ‘ pseudo-Laplacian ’ associated to a manifold LaTeXMLMath with finitely many cusps .
By combining results of this paper with those of the prequel LaTeXMLCite , we obtain Let LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be an eigenfunction branch whose zeroth Fourier coefficient ( see § LaTeXMLRef ) vanishes identically for small LaTeXMLMath .
Then LaTeXMLMath converges to an LaTeXMLMath -eigenfunction of LaTeXMLMath .
Since real-analytic eigenbranches can ‘ cross ’ , their tracking is far subtler than the continuity of ordered eigenvalues .
For example , consider the family of Laplacians , LaTeXMLMath , associated to the flat tori LaTeXMLMath .
In this case , almost all of the real-analytic eigenvalue branches tend to infinity as LaTeXMLMath tends to zero .
Yet , there are infinitely many branches that tend to zero .
Therefore if for each LaTeXMLMath , one were to label the eigenvalues in increasing order ( with multiplicities ) LaTeXMLEquation then each LaTeXMLMath would tend to zero as LaTeXMLMath tended to zero .
LaTeXMLMath is continuous , it is not real-analytic .
Therefore , although ( LaTeXMLRef ) describes a relatively narrow class of geometric degenerations , the conclusion of Theorem LaTeXMLRef provides a great deal more information concerning spectral behavior than the usual convergence results concerning ordered eigenvalues .
( See , for example , the recent work of Cheeger and Colding LaTeXMLCite . )
Indeed , the geometer ’ s standard tool for estimating the size of eigenvalues—the minimax principle— can not be used to track real-analytic eigenvalue branches due to possible eigenbranch ‘ crossings ’ .
Here , we rely instead on the variational principle LaTeXMLMath .
To illustrate our use of this principle , we prove in § LaTeXMLRef the following general result : t LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Let LaTeXMLMath be a real-analytic family of metrics on a Riemannian manifold LaTeXMLMath .
Then for each real-analytic eigenbranch we have LaTeXMLEquation .
For a family LaTeXMLMath satisfying ( LaTeXMLRef ) , there are vector fields supported in LaTeXMLMath such that the right hand side of ( LaTeXMLRef ) is unbounded as LaTeXMLMath tends to zero .
Thus , in order to use the variational principle to prove Theorem LaTeXMLRef , one must exhibit some control over the size of eigenfunctions in the bicollar LaTeXMLMath .
For large eigenvalues , controlling the size of eigenfunctions is notoriously difficult LaTeXMLCite LaTeXMLCite .
Indeed , for LaTeXMLMath , the central hypersurface LaTeXMLMath is totally geodesic , and hence the correspondence principle of quantum physics leads one to ‘ expect ’ —perhaps erroneously—that the mass of an eigenfunction with large eigenvalue concentrates near LaTeXMLMath .
The possibility of such ‘ scarring ’ on LaTeXMLMath greatly contributes to the delicacy of the proof of Theorem LaTeXMLRef .
Fortunately , the ill-effects of possible ‘ scarring ’ are ameliorated by the inequality LaTeXMLMath , that is , by the completeness of the limiting manifold .
a LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath We devote the remainder of this paper , with the exception of § LaTeXMLRef , to proving Theorem LaTeXMLRef .
We now outline the contents and hence also the proof .
In § LaTeXMLRef we illustrate our use of the variational principle with a proof of Theorem LaTeXMLRef .
In § LaTeXMLRef , we establish some basic facts concerning the warped product ( LaTeXMLRef ) including an integration by parts formula ( Lemma LaTeXMLRef ) on which most of our analysis is based .
Underlying the proof of Theorem LaTeXMLRef is a basic fact : A nonnegative function LaTeXMLMath has a finite limit as LaTeXMLMath tends to LaTeXMLMath provided the negative variation of LaTeXMLMath over LaTeXMLMath tends to zero as LaTeXMLMath tends to zero .
Towards applying this to an eigenbranch LaTeXMLMath , we derive in § LaTeXMLRef lower bounds for the derivative LaTeXMLMath .
As an example of our approach , we use these lower bounds in § LaTeXMLRef to prove For LaTeXMLMath and LaTeXMLMath , each eigenvalue branch converges to a finite limit as LaTeXMLMath tends to LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , each eigenvalue branch remains bounded as LaTeXMLMath tends to LaTeXMLMath .
( Future work will include a more thorough investigation of the cases in Theorem LaTeXMLRef as well as a study of the ‘ adiabatic ’ case LaTeXMLMath . )
Beginning with § LaTeXMLRef , we restrict attention to the case of interest in the present work : LaTeXMLMath and LaTeXMLMath .
We show in § LaTeXMLRef that LaTeXMLMath converges to a finite limit ( Theorem LaTeXMLRef ) .
In § LaTeXMLRef , we find that if LaTeXMLMath remains bounded for some LaTeXMLMath , then LaTeXMLMath converges to a finite limit ( Theorem LaTeXMLRef ) .
In § LaTeXMLRef boundedness for LaTeXMLMath is verified provided LaTeXMLMath is not a positive eigenvalue of the Laplacian LaTeXMLMath for LaTeXMLMath .
Hence in this case the eigenbranch has a finite limit ( Theorem LaTeXMLRef ) .
In § LaTeXMLRef we assume that LaTeXMLMath is a positive eigenvalue of LaTeXMLMath , and obtain a contradiction in the form of two conflicting estimates : Lemmas LaTeXMLRef and LaTeXMLRef .
Theorem LaTeXMLRef follows .
We remark that the condition LaTeXMLMath —and hence the threshold LaTeXMLMath —is intimately tied to ‘ scarring ’ .
Indeed , one finds that the projection of LaTeXMLMath onto the LaTeXMLMath -eigenspace is a scarring mode in the sense of , for example , §7 of LaTeXMLCite .
For the purpose of proving Lemma LaTeXMLRef we need only know that the ‘ width ’ of a scar is LaTeXMLMath as LaTeXMLMath tends to infinity .
This result is given in Appendix LaTeXMLRef .
The reader familiar with §3 in LaTeXMLCite will recognize the thread of the argument outlined above .
Indeed , not only does the case of hyperbolic degeneration serve as motivation for the present work , many of its basic features are representative of the general case .
On the other hand , at this level of generality , we can not avail ourselves of Teichmüller theory nor the Poincaré series estimate of LaTeXMLCite .
Moreover , the peculiar features of the ‘ overcomplete ’ case LaTeXMLMath do not appear in hyperbolic degeneration .
These features add complication to the arguments , especially to those found in § LaTeXMLRef .
I thank my ever-patient wife , Nacy , for her support .
I also thank the referee for generous help wih the exposition .
Let LaTeXMLMath be the space of all ( smooth ) Riemannian inner products on a compact manifold LaTeXMLMath .
To each LaTeXMLMath we associate the Laplacian LaTeXMLMath .
This is a self-adjoint , unbounded operator on LaTeXMLMath defined via the Friedrich ’ s extension with respect to symmetric boundary conditions .
A fixed inner product LaTeXMLMath induces a Banach norm on the space of LaTeXMLMath -tensors LaTeXMLMath .
A family of metric tensors LaTeXMLMath is said to be real-analytic if it defines a real-analytic path in the Banach space LaTeXMLMath .
Using the ratio of Riemannian measures LaTeXMLMath , one constructs a natural family of unitary operators that conjugates LaTeXMLMath into a real-analytic family of compactly resolved operators that are self-adjoint with respect to the fixed sesquilinear form determined by LaTeXMLMath .
It follows from analytic perturbation theory LaTeXMLCite that there exists a countable collection of eigenfunction branches , LaTeXMLMath , such that for each fixed LaTeXMLMath , the set LaTeXMLMath is an orthonomal basis for LaTeXMLMath .
Given a continuous function LaTeXMLMath satisfying LaTeXMLMath for all LaTeXMLMath , let LaTeXMLMath denote the supremum .
An example of such a function is LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is an arbitrary 2-tensor .
Let LaTeXMLMath be a real-analytic family of metrics on LaTeXMLMath .
Then for each real-analytic eigenbranch we have LaTeXMLEquation .
We fix a background metric LaTeXMLMath and write LaTeXMLMath for its volume form .
Define LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be an eigenfunction branch corresponding to LaTeXMLMath .
Supressing subscripts , we have LaTeXMLEquation .
Each object in ( LaTeXMLRef ) is real-analytic in LaTeXMLMath .
By Taylor expanding , collecting first order terms , integrating by parts , and using the eigenequation , we find that LaTeXMLEquation .
Here the symbol LaTeXMLMath denotes the first derivative with respect to LaTeXMLMath evaluated at LaTeXMLMath .
By definition , we have LaTeXMLMath for each fixed vector field LaTeXMLMath and function LaTeXMLMath on LaTeXMLMath .
Differentiating in LaTeXMLMath yields LaTeXMLMath .
Using this identity , ( LaTeXMLRef ) reduces to LaTeXMLEquation .
From LaTeXMLMath , we have LaTeXMLEquation .
By interpreting LaTeXMLMath as the determinant of the matrix representation of LaTeXMLMath with respect to an orthonormal basis of LaTeXMLMath , one finds that the supremum of LaTeXMLMath is bounded by LaTeXMLMath .
The claim follows by applying this bound and ( LaTeXMLRef ) to ( LaTeXMLRef ) .
∎ We record some basic facts concerning the Laplacian , its eigenvalues , and eigenfunctions , on LaTeXMLMath with the metric given in ( LaTeXMLRef ) .
In the following LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , will denote respectively , the Laplacian , gradient , and volume form , associated to the metric LaTeXMLMath on a fibre LaTeXMLMath .
Recall that LaTeXMLMath is the dimension of LaTeXMLMath .
For any LaTeXMLMath LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
The volume form restricted to LaTeXMLMath is LaTeXMLEquation .
If no subscript appears , then the object is associated to LaTeXMLMath .
Given LaTeXMLMath , define LaTeXMLEquation .
Let LaTeXMLMath satisfy LaTeXMLMath .
Then LaTeXMLEquation .
Straightforward computation gives LaTeXMLEquation .
From ( LaTeXMLRef ) and LaTeXMLMath we find that LaTeXMLEquation .
Integrating by parts over LaTeXMLMath gives LaTeXMLEquation .
Also note that from ( LaTeXMLRef ) we have LaTeXMLEquation .
The claim follows .
∎ Let LaTeXMLMath satisfy LaTeXMLMath .
Suppose that for each LaTeXMLMath , LaTeXMLEquation .
Then LaTeXMLEquation .
Apply Proposition LaTeXMLRef and ( LaTeXMLRef ) .
∎ Let LaTeXMLMath be positive and positively homogeneous of degree LaTeXMLMath .
There exist constants LaTeXMLMath , such that for any eigenpair LaTeXMLMath on LaTeXMLMath we have LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLMath denote the dilated interval LaTeXMLMath , and let LaTeXMLMath belong to LaTeXMLMath with LaTeXMLMath on LaTeXMLMath .
By multiplying both sides of ( LaTeXMLRef ) by LaTeXMLMath and integrating over LaTeXMLMath one obtains LaTeXMLEquation .
On the other hand , integration by parts gives LaTeXMLEquation .
We have LaTeXMLMath where LaTeXMLMath in a neighborhood of the origin .
Note that LaTeXMLMath adds LaTeXMLMath to the homogeneity of any function .
Hence LaTeXMLMath , and thus , since LaTeXMLMath , we have LaTeXMLMath .
Therefore , by ( LaTeXMLRef ) LaTeXMLEquation where LaTeXMLMath .
To complete the proof it suffices to show that LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLMath is bounded on the support of LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLEquation .
Let LaTeXMLMath .
Integrating by parts gives LaTeXMLEquation .
Note that the support of LaTeXMLMath belongs to LaTeXMLMath .
Integration by parts in LaTeXMLMath gives LaTeXMLEquation .
Estimate ( LaTeXMLRef ) then follows from the fact that LaTeXMLMath has support in LaTeXMLMath and equals LaTeXMLMath on LaTeXMLMath .
∎ Suppose that LaTeXMLMath .
Then LaTeXMLMath is well-defined , continuous , and homogeneous of degree 0 .
Hence it is bounded .
Thus , there exists a constant LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath .
Since LaTeXMLMath is homogeneous and strictly convex along nonradial lines , there exists LaTeXMLMath such that for each LaTeXMLMath , the function LaTeXMLMath has a unique maximum at LaTeXMLMath .
To prove LaTeXMLRef , without loss of generality , we may assume that LaTeXMLMath .
For otherwise , based on the linear map LaTeXMLMath , one may construct a real-analytic family of diffeomorphisms LaTeXMLMath such that LaTeXMLEquation .
Then one works with the positive , positively homogeneous function LaTeXMLMath .
For each LaTeXMLMath , the maximum of the function LaTeXMLEquation is LaTeXMLMath .
This maximum is uniquely achieved at LaTeXMLMath .
Note that by homogeneity and positivity , LaTeXMLMath for some LaTeXMLMath , and hence LaTeXMLMath .
Therefore , the first claim will follow from the second .
By Remark LaTeXMLRef , we may assume that the function LaTeXMLMath has a uniques minimum at LaTeXMLMath , and thus LaTeXMLMath has a unique maximum there .
To prove the claim , it will suffice to show the same for LaTeXMLMath .
In other words , it is enough to show that LaTeXMLMath is positive for LaTeXMLMath and negative for LaTeXMLMath .
Since LaTeXMLMath is strictly convex , LaTeXMLMath is strictly increasing .
Let LaTeXMLMath .
The derivative LaTeXMLMath is homogeneous of degree 0 , and , therefore , for LaTeXMLMath LaTeXMLEquation .
Hence LaTeXMLMath is decreasing for LaTeXMLMath .
That is , LaTeXMLMath is negative for LaTeXMLMath as desired .
An analogous argument shows that LaTeXMLMath for LaTeXMLMath .
The claim follows .
∎ The purpose of this section is to derive a useful lower bound for LaTeXMLMath .
Towards this end , we define the zeroeth Fourier coefficient , LaTeXMLMath , of a function LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation and the complement , LaTeXMLMath , by LaTeXMLEquation .
Note that LaTeXMLMath is a LaTeXMLMath eigenfunction with eigenvalue LaTeXMLMath , if and only if both LaTeXMLMath and LaTeXMLMath are .
In the sequel , LaTeXMLMath denotes the set complement LaTeXMLMath .
Let LaTeXMLMath .
There exist positive constants LaTeXMLMath such that for each eigenbranch LaTeXMLMath LaTeXMLEquation .
Moreover , if LaTeXMLMath , then the integrand LaTeXMLMath can be replaced with LaTeXMLMath .
Our starting point is formula ( LaTeXMLRef ) : LaTeXMLEquation .
Recall that by ( global ) hypothesis , the supremum of LaTeXMLMath over the unit tangent bundle of LaTeXMLMath is finite .
By applying the argument that immediately follows ( LaTeXMLRef ) to the restriction of LaTeXMLMath to LaTeXMLMath , we obtain LaTeXMLEquation for some positive constant LaTeXMLMath .
By ( LaTeXMLRef ) , the restriction of LaTeXMLMath to LaTeXMLMath equals LaTeXMLMath .
Thus , by combining ( LaTeXMLRef ) and ( LaTeXMLRef ) we obtain LaTeXMLEquation .
We claim that LaTeXMLEquation .
To see this , first note that from ( LaTeXMLRef ) we compute LaTeXMLEquation .
The function LaTeXMLMath is constant on each fibre LaTeXMLMath , and hence LaTeXMLMath .
Therefore , since LaTeXMLMath and LaTeXMLMath , we find that LaTeXMLEquation .
The operator LaTeXMLMath preseves the decomposition LaTeXMLMath .
In particular , LaTeXMLMath and LaTeXMLMath is constant on each fibre .
Therefore , LaTeXMLMath , and it follows that LaTeXMLEquation .
The claimed ( LaTeXMLRef ) follows .
From ( LaTeXMLRef ) we also have that LaTeXMLEquation and hence combined with ( LaTeXMLRef ) we have LaTeXMLEquation .
Substitution into ( LaTeXMLRef ) then yields LaTeXMLEquation .
Since LaTeXMLMath is homogeneous of degree LaTeXMLMath , Lemma LaTeXMLRef applies to give LaTeXMLEquation as well as the analogous estimate with LaTeXMLMath replaced by LaTeXMLMath .
By combining these estimates with ( LaTeXMLRef ) and absorbing constants , we obtain the claim .
∎ Let LaTeXMLMath and LaTeXMLMath .
Then each eigenvalue branch LaTeXMLMath converges to a finite limit as LaTeXMLMath tends to LaTeXMLMath .
We apply Theorem LaTeXMLRef .
Since LaTeXMLMath and LaTeXMLMath are positive and LaTeXMLMath , the first term on the right hand side of ( LaTeXMLRef ) is nonnegative .
Therefore , by Remark LaTeXMLRef we have LaTeXMLEquation for some constants LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , division of both sides by LaTeXMLMath gives LaTeXMLEquation .
Since LaTeXMLMath , the left hand side of ( LaTeXMLRef ) is integrable , and , moreover , the negative variation of LaTeXMLMath over LaTeXMLMath is LaTeXMLMath .
Thus , since LaTeXMLMath , the function LaTeXMLMath has a limit as LaTeXMLMath tends to LaTeXMLMath .
Thus , the claim follows via exponentiation .
∎ Let LaTeXMLMath and LaTeXMLMath .
Then each eigenvalue branch LaTeXMLMath remains bounded as LaTeXMLMath tends to LaTeXMLMath .
Let LaTeXMLMath .
Note that because LaTeXMLMath , we have LaTeXMLMath , and hence LaTeXMLMath for LaTeXMLMath small .
Thus , by using Theorem LaTeXMLRef and Remark LaTeXMLRef , we obtain LaTeXMLEquation .
If LaTeXMLMath , then the right hand side of ( LaTeXMLRef ) is positive for LaTeXMLMath small .
The claim follows .
∎ In the sequel we will assume that LaTeXMLMath and either LaTeXMLMath or LaTeXMLMath and LaTeXMLMath .
The quantity LaTeXMLMath tends to a finite limit as LaTeXMLMath tends to zero .
By Remark LaTeXMLRef and Assumption LaTeXMLRef , we have LaTeXMLMath for LaTeXMLMath small .
Thus , it follows from Theorem LaTeXMLRef and Proposition LaTeXMLRef , that there exist positive constants LaTeXMLMath , LaTeXMLMath such that LaTeXMLEquation .
Since LaTeXMLMath , we have upon letting LaTeXMLMath LaTeXMLEquation .
Dividing by LaTeXMLMath gives LaTeXMLEquation .
Since LaTeXMLMath , we obtain LaTeXMLEquation .
It follows that the negative variation of LaTeXMLMath over the interval LaTeXMLMath is LaTeXMLMath .
It follows that LaTeXMLMath is either finite , in which case LaTeXMLMath is finite , or LaTeXMLMath in which case LaTeXMLMath .
In either case the limit exists .
∎ Let LaTeXMLMath denote the infimum of all LaTeXMLMath such that the function LaTeXMLMath has a limit as LaTeXMLMath tends to zero .
By Theorem LaTeXMLRef , we have LaTeXMLMath .
The purpose of this section is to prove If LaTeXMLMath , then LaTeXMLMath tends to a finite limit as LaTeXMLMath tends to zero .
As a first step towards proving Theorem LaTeXMLRef , we have the following If there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation then LaTeXMLMath tends to a finite limit as LaTeXMLMath tends to zero .
Since LaTeXMLMath is homogeneous of degree zero , it is bounded .
By Assumption LaTeXMLRef , we have LaTeXMLMath , and hence LaTeXMLMath .
Therefore , via Theorem LaTeXMLRef we find that LaTeXMLEquation .
Hence by ( LaTeXMLRef ) and Remark LaTeXMLRef LaTeXMLEquation for some positive LaTeXMLMath and LaTeXMLMath small .
Dividing by LaTeXMLMath gives LaTeXMLEquation .
Since LaTeXMLMath , the right hand side is integrable , and , in particular , the negative variation of LaTeXMLMath is LaTeXMLMath .
Therefore LaTeXMLMath exists , and it follows that LaTeXMLMath has a limit .
∎ To verify ( LaTeXMLRef ) —and thus prove Theorem LaTeXMLRef —we split the domain of integration of the integral on the left hand side according to whether Corollary LaTeXMLRef implies the convexity of LaTeXMLMath or not .
To be precise , let LaTeXMLMath denote the smallest non-zero eigenvalue of LaTeXMLMath .
Define LaTeXMLMath to be set of LaTeXMLMath such that LaTeXMLEquation .
The key idea in what follows is that ( LaTeXMLRef ) and Corollary LaTeXMLRef imply that the function LaTeXMLMath is convex enough to tame the singular behavior of LaTeXMLMath near LaTeXMLMath .
This heuristic will be made precise in Lemma LaTeXMLRef .
In the following LaTeXMLMath will denote the set complement LaTeXMLMath .
We claim that if LaTeXMLMath , then for sufficiently small LaTeXMLMath LaTeXMLEquation .
Indeed , from ( LaTeXMLRef ) , we have LaTeXMLMath if and only if LaTeXMLEquation .
If LaTeXMLMath , then for sufficiently small LaTeXMLMath , we have LaTeXMLMath , and hence for LaTeXMLMath LaTeXMLEquation .
Therefore LaTeXMLMath for LaTeXMLMath and sufficiently small LaTeXMLMath .
The claimed ( LaTeXMLRef ) then follows from homogeneity and integration .
By applying Lemma LaTeXMLRef with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , we obtain the complementary estimate .
Indeed , since LaTeXMLMath is compact , the function LaTeXMLMath —defined by ( LaTeXMLRef ) —is orthogonal to the zero eigenspace of LaTeXMLMath .
Hence for each LaTeXMLMath LaTeXMLEquation .
Thus , Lemma LaTeXMLRef applies to give LaTeXMLEquation .
Combining ( LaTeXMLRef ) and ( LaTeXMLRef ) gives us the desired ( LaTeXMLRef ) for all LaTeXMLMath .
∎ Without loss of generality , we may assume that LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath be a Laplace eigenbranch on LaTeXMLMath .
Let LaTeXMLMath denote the set of LaTeXMLMath that satsify LaTeXMLEquation .
Suppose that LaTeXMLEquation and that for each LaTeXMLMath LaTeXMLEquation .
Then there exist LaTeXMLMath such that for small LaTeXMLMath LaTeXMLEquation .
By Remark LaTeXMLRef we have LaTeXMLMath and hence it follows from ( LaTeXMLRef ) that LaTeXMLMath .
Therefore LaTeXMLEquation .
Here the last inequality follows from ( LaTeXMLRef ) and Corollary LaTeXMLRef .
We wish to apply integration by parts to the last integral in ( LaTeXMLRef ) .
Towards this end , let LaTeXMLMath be a smooth function supported in LaTeXMLMath with LaTeXMLMath on LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is positive , convex , and homogeneous of degree 1 , the set LaTeXMLMath is a closed interval LaTeXMLMath that contains LaTeXMLMath .
For each LaTeXMLMath define LaTeXMLEquation .
Integration by parts shows that the operator LaTeXMLMath is symmetric on LaTeXMLMath with Dirichlet boundary conditions .
Thus , LaTeXMLEquation .
By ( LaTeXMLRef ) and Corollary LaTeXMLRef , we have LaTeXMLMath for LaTeXMLMath .
Thus , since LaTeXMLMath , estimate ( LaTeXMLRef ) and ( LaTeXMLRef ) combine to give LaTeXMLEquation .
Therefore , to verify ( LaTeXMLRef ) , it will suffice to show that LaTeXMLEquation .
By homogeneity , the supremum of LaTeXMLMath over LaTeXMLMath is LaTeXMLMath for any constant LaTeXMLMath .
We compute LaTeXMLEquation where LaTeXMLMath .
The operator LaTeXMLMath adds LaTeXMLMath to the degree of a homogeneous function , and hence LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLMath , and hence LaTeXMLMath .
To estimate the remaining two terms in ( LaTeXMLRef ) , we need to estimate LaTeXMLMath and LaTeXMLMath .
To this end , consider LaTeXMLMath , the inner radius of LaTeXMLMath .
By Lemma LaTeXMLRef , there exists LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath small .
Therefore LaTeXMLMath and , similarly , LaTeXMLMath .
The function LaTeXMLMath appearing in ( LaTeXMLRef ) is homogeneous of degree LaTeXMLMath .
Hence since LaTeXMLMath is homogeneous of degree zero , LaTeXMLEquation .
A similar argument shows that LaTeXMLEquation .
The desired estimate ( LaTeXMLRef ) then follows from ( LaTeXMLRef ) , ( LaTeXMLRef ) , and ( LaTeXMLRef ) .
∎ Let LaTeXMLMath be a Laplace eigenbranch on LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath be defined as in ( LaTeXMLRef ) .
If ( LaTeXMLRef ) holds , then there exists LaTeXMLMath such that LaTeXMLEquation for small LaTeXMLMath .
By definition , LaTeXMLMath satisfies LaTeXMLEquation .
Thus , by homogeneity LaTeXMLEquation .
By Theorem LaTeXMLRef , LaTeXMLMath tends to LaTeXMLMath as LaTeXMLMath tends to zero .
If LaTeXMLMath , then since the right hand side of ( LaTeXMLRef ) is positive , LaTeXMLMath must tend to infinity as LaTeXMLMath tends to zero .
By homogeneity , ( LaTeXMLRef ) implies that LaTeXMLMath .
Thus if LaTeXMLMath , then by ( LaTeXMLRef ) we have LaTeXMLEquation .
By Remark LaTeXMLRef and the condition LaTeXMLMath , the function LaTeXMLMath assumes a unique minimum at LaTeXMLMath .
Therefore , LaTeXMLMath is strictly bounded away from zero .
The claim follows .
∎ By homogeneity LaTeXMLMath , and hence , by Theorem LaTeXMLRef , the limit LaTeXMLEquation exists .
The purpose of this section is to prove If LaTeXMLMath does not equal a positive eigenvalue of LaTeXMLMath , then LaTeXMLMath tends to a finite limit as LaTeXMLMath tends to zero .
To prove Theorem LaTeXMLRef , we will use the LaTeXMLMath spectral decomposition of LaTeXMLMath .
To be precise , the orthogonal projection , LaTeXMLMath , onto the LaTeXMLMath -eigenspace of LaTeXMLMath extends fibre by fibre to an operator LaTeXMLMath .
Set LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are all eigenfunctions of LaTeXMLMath with eigenvalue LaTeXMLMath .
The LaTeXMLMath -eigenspace of LaTeXMLMath consists of the constant functions .
Thus if LaTeXMLMath is not a positive eigenvalue , then LaTeXMLEquation where LaTeXMLMath is defined in ( LaTeXMLRef ) .
Since LaTeXMLMath , by ( LaTeXMLRef ) we have LaTeXMLEquation .
Note that from ( LaTeXMLRef ) we obtain LaTeXMLMath .
Hence , LaTeXMLEquation .
By hypothesis we have LaTeXMLMath and hence LaTeXMLMath .
Applying Parseval ’ s principle for LaTeXMLMath acting on LaTeXMLMath , we obtain LaTeXMLEquation as well as LaTeXMLEquation .
Thus , it follows from Lemmas LaTeXMLRef and LaTeXMLRef that there exists LaTeXMLMath such that LaTeXMLEquation .
Since LaTeXMLMath , combining ( LaTeXMLRef ) with ( LaTeXMLRef ) and ( LaTeXMLRef ) yields LaTeXMLMath such that LaTeXMLEquation for some LaTeXMLMath .
By substituting ( LaTeXMLRef ) into ( LaTeXMLRef ) and applying ( LaTeXMLRef ) to the LaTeXMLMath -term in ( LaTeXMLRef ) , one obtains LaTeXMLEquation for some constants LaTeXMLMath and LaTeXMLMath .
Thus , since LaTeXMLMath , there exists a positive constant LaTeXMLMath such that LaTeXMLEquation .
Division by LaTeXMLMath gives LaTeXMLEquation and hence , by arguing as in the proof of Theorem LaTeXMLRef , one finds that LaTeXMLMath converges as LaTeXMLMath tends to zero .
Therefore the claim follows from Theorem LaTeXMLRef .
∎ There exists LaTeXMLMath such that LaTeXMLEquation .
If LaTeXMLMath is less than the smallest positive eigenvalue of LaTeXMLMath , then LaTeXMLMath and the claim follows .
Otherwise , let LaTeXMLMath be the largest LaTeXMLMath -eigenvalue that is less than LaTeXMLMath .
From the definition of LaTeXMLMath we have LaTeXMLEquation .
Since , by hypothesis , LaTeXMLMath , there exists LaTeXMLMath such that for sufficiently small LaTeXMLMath LaTeXMLEquation .
Hence , using the ( global ) hypothesis that LaTeXMLMath for all LaTeXMLMath , we have LaTeXMLEquation .
Combining this with ( LaTeXMLRef ) gives LaTeXMLEquation .
Therefore , the claim follows from Proposition LaTeXMLRef .
∎ There exists LaTeXMLMath such that LaTeXMLEquation .
By ( LaTeXMLRef ) and ( LaTeXMLRef ) we have LaTeXMLEquation .
Using Lemma LaTeXMLRef , one obtains ( LaTeXMLRef ) with LaTeXMLMath replaced by LaTeXMLMath .
From this and ( LaTeXMLRef ) it follows that the left hand side of ( LaTeXMLRef ) is bounded above by LaTeXMLEquation .
Thus it will suffice to show that LaTeXMLEquation for some LaTeXMLMath .
Towards verifying ( LaTeXMLRef ) , we will apply Lemma LaTeXMLRef .
Namely , let LaTeXMLMath to be the smallest of all eigenvalues that are greater than LaTeXMLMath and let LaTeXMLMath .
Note that from the definition of LaTeXMLMath , for each LaTeXMLMath LaTeXMLEquation .
Note also that ( LaTeXMLRef ) follows in this case from ( LaTeXMLRef ) .
Therefore Lemma LaTeXMLRef provides LaTeXMLMath such that LaTeXMLEquation .
By Lemma LaTeXMLRef , there exists LaTeXMLMath such that LaTeXMLMath .
Define LaTeXMLEquation .
It follows from Proposition LaTeXMLRef that LaTeXMLMath .
By homogeneity , we have LaTeXMLEquation for all LaTeXMLMath .
In particular , estimate ( LaTeXMLRef ) holds for all LaTeXMLMath .
Thus , by integrating this estimate over LaTeXMLMath and combining with ( LaTeXMLRef ) we obtain LaTeXMLEquation .
Therefore , since LaTeXMLMath , the claimed ( LaTeXMLRef ) is proven .
∎ Each eigenvalue branch converges to a finite limit .
We assume that the hypothesis of Theorem LaTeXMLRef is not satisfied and derive a contradiction .
Namely , we assume that LaTeXMLMath is a positive LaTeXMLMath -eigenvalue and obtain a contradiction in the form of two conflicting estimates .
In particular , it is enough to show that there exist constants LaTeXMLMath such that for small LaTeXMLMath LaTeXMLEquation .
This is impossible since LaTeXMLMath and LaTeXMLMath and hence LaTeXMLMath .
The respective sides of ( LaTeXMLRef ) are given below as Lemma LaTeXMLRef and Lemma LaTeXMLRef .
∎ Suppose that LaTeXMLMath is a positive LaTeXMLMath -eigenvalue .
Then there exists a constant LaTeXMLMath such that for all small LaTeXMLMath LaTeXMLEquation .
We first claim that it suffices to show that there exists LaTeXMLMath such that for small LaTeXMLMath LaTeXMLEquation .
Indeed , since LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath for small LaTeXMLMath .
Hence we would have LaTeXMLEquation .
Note that since LaTeXMLMath LaTeXMLEquation .
Thus , since LaTeXMLMath it would follow from ( LaTeXMLRef ) that there exists a constant LaTeXMLMath such that LaTeXMLEquation .
Note that since LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath .
Thus , since LaTeXMLMath , we could then integrate ( LaTeXMLRef ) over LaTeXMLMath and would find that LaTeXMLEquation .
Since LaTeXMLMath , we would then obtain ( LaTeXMLRef ) by dividing both sides of ( LaTeXMLRef ) by LaTeXMLMath .
Recall from ( LaTeXMLRef ) that LaTeXMLMath denotes the fibrewise projection of LaTeXMLMath onto the LaTeXMLMath eigenspace of LaTeXMLMath .
Letting LaTeXMLMath , we have LaTeXMLMath .
We claim that to verify ( LaTeXMLRef ) it suffices to show that LaTeXMLEquation where LaTeXMLEquation and LaTeXMLMath .
To see this , note that by Proposition LaTeXMLRef , there exists LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
It follows that LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLMath , and hence since LaTeXMLMath , for any LaTeXMLMath , we have LaTeXMLMath for all sufficiently small LaTeXMLMath .
Thus , by combining ( LaTeXMLRef ) and ( LaTeXMLRef ) , we would obtain LaTeXMLEquation for LaTeXMLMath small .
Applying the argument in ( LaTeXMLRef ) and ( LaTeXMLRef ) with LaTeXMLMath replaced by LaTeXMLMath , we would have LaTeXMLEquation ( In this and what follows , LaTeXMLMath and LaTeXMLMath represent generic constants . )
Recall that in ( LaTeXMLRef ) we had LaTeXMLMath , and hence we have ( LaTeXMLRef ) with LaTeXMLMath replaced by LaTeXMLMath : LaTeXMLEquation .
As pointed out above , LaTeXMLMath for small LaTeXMLMath .
Hence by applying Parseval ’ s principle as in ( LaTeXMLRef ) and ( LaTeXMLRef ) to LaTeXMLMath , we could combine ( LaTeXMLRef ) and ( LaTeXMLRef ) to find that LaTeXMLEquation .
Combining this with ( LaTeXMLRef ) would yield LaTeXMLMath such that LaTeXMLEquation where LaTeXMLMath denotes a ( generic ) positive constant .
By substituting ( LaTeXMLRef ) into ( LaTeXMLRef ) and applying ( LaTeXMLRef ) to the LaTeXMLMath -term in ( LaTeXMLRef ) and using ( LaTeXMLRef ) , one would obtain LaTeXMLEquation .
Note that LaTeXMLMath for small LaTeXMLMath .
Hence from ( LaTeXMLRef ) one would have LaTeXMLEquation for small LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath LaTeXMLEquation for small LaTeXMLMath .
Thus by choosing LaTeXMLMath and recalling ( LaTeXMLRef ) we would obtain ( LaTeXMLRef ) from ( LaTeXMLRef ) .
And hence ( LaTeXMLRef ) follows from ( LaTeXMLRef ) .
As a first step toward the verification of ( LaTeXMLRef ) , we rescale in LaTeXMLMath .
In particular , let LaTeXMLMath , and LaTeXMLMath , and for each LaTeXMLMath and LaTeXMLMath , define LaTeXMLEquation .
Then using homogeneity , we find that LaTeXMLEquation and LaTeXMLEquation .
Since LaTeXMLMath and since LaTeXMLMath belongs to the LaTeXMLMath eigenspace of LaTeXMLMath , we have LaTeXMLMath from ( LaTeXMLRef ) .
It follows from ( LaTeXMLRef ) and homogeneity that LaTeXMLMath satifies the following ordinary differential equation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is a bounded smooth function .
Hence , by ( LaTeXMLRef ) and ( LaTeXMLRef ) , to prove ( LaTeXMLRef ) it will suffice to prove that there exists LaTeXMLMath such that for any solution LaTeXMLMath to ( LaTeXMLRef ) we have LaTeXMLEquation .
Towards verification of ( LaTeXMLRef ) we apply Lemma LaTeXMLRef to ( LaTeXMLRef ) .
Indeed , by hypothesis LaTeXMLMath is bounded , and hence Lemma LaTeXMLRef applies to give a constant LaTeXMLMath such that LaTeXMLEquation .
It follows that LaTeXMLEquation .
By Proposition LaTeXMLRef and the strict convexity of LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
It follows that LaTeXMLEquation .
By substituting ( LaTeXMLRef ) and choosing LaTeXMLMath we obtain ( LaTeXMLRef ) .
The proof is complete .
∎ Suppose that LaTeXMLMath is a positive LaTeXMLMath -eigenvalue .
Then there exists a constant LaTeXMLMath such that for small LaTeXMLMath LaTeXMLEquation .
It will suffice to show that LaTeXMLEquation .
Indeed , we may suppose that LaTeXMLMath , for otherwise we are done .
Hence ( LaTeXMLRef ) implies LaTeXMLEquation .
By Lemma LaTeXMLRef , we have LaTeXMLMath for small LaTeXMLMath , and thus division would give LaTeXMLEquation .
By integrating over LaTeXMLMath with LaTeXMLMath small we would find that LaTeXMLEquation .
Exponentiation would then give ( LaTeXMLRef ) with LaTeXMLMath .
To verify ( LaTeXMLRef ) we will estimate LaTeXMLMath using the methods of § LaTeXMLRef .
From ( LaTeXMLRef ) and ( LaTeXMLRef ) we have LaTeXMLEquation .
By substituting ( LaTeXMLRef ) into ( LaTeXMLRef ) one obtains that LaTeXMLEquation .
LaTeXMLEquation Thus , by integrating ( LaTeXMLRef ) and using ( LaTeXMLRef ) we find that LaTeXMLEquation .
We may estimate the righthand side of ( LaTeXMLRef ) by splitting the integral over the sum LaTeXMLMath where LaTeXMLMath .
To be precise , apply Parseval ’ s principle—as in ( LaTeXMLRef ) —to find that LaTeXMLEquation as well as LaTeXMLEquation .
It follows that if we let LaTeXMLMath denote the right hand side of ( LaTeXMLRef ) , then LaTeXMLEquation .
By definition LaTeXMLMath and hence LaTeXMLMath .
It follows that LaTeXMLEquation where LaTeXMLEquation .
To estimate LaTeXMLMath , apply ( LaTeXMLRef ) ‘ in reverse ’ to find that LaTeXMLEquation .
Thus by using the assumption LaTeXMLMath , the formula ( LaTeXMLRef ) , and Lemmas LaTeXMLRef and LaTeXMLRef , we obtain LaTeXMLMath such that LaTeXMLEquation .
By combining ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , and ( LaTeXMLRef ) , we find ( generic ) constants LaTeXMLMath such that LaTeXMLEquation .
Hence by substituting this into ( LaTeXMLRef ) and applying ( LaTeXMLRef ) to the middle term in ( LaTeXMLRef ) , we obtain LaTeXMLEquation for possibly different ( generic ) constants .
By using ( LaTeXMLRef ) , we find that LaTeXMLEquation .
We claim that there exists LaTeXMLMath such that for all LaTeXMLMath LaTeXMLEquation for all small LaTeXMLMath .
To see this , first note that we can not have LaTeXMLMath for all small LaTeXMLMath .
For then by ( LaTeXMLRef ) we would have that LaTeXMLMath for all small LaTeXMLMath .
Using the argument of Theorem LaTeXMLRef , we would then find that LaTeXMLMath is bounded and hence LaTeXMLMath would converge by Theorem LaTeXMLRef .
This would contradict the assumption that LaTeXMLMath .
Hence there exists LaTeXMLMath such that ( LaTeXMLRef ) is true for LaTeXMLMath .
A calculation shows that ( LaTeXMLRef ) holds if and only if LaTeXMLEquation .
Thus to prove ( LaTeXMLRef ) for LaTeXMLMath , it suffices to show that LaTeXMLMath is increasing for small LaTeXMLMath .
To see that this is true , note that by homogeneity LaTeXMLMath , and hence LaTeXMLEquation .
By Lemma LaTeXMLRef , LaTeXMLMath for small LaTeXMLMath .
Thus from ( LaTeXMLRef ) we have LaTeXMLMath , and hence LaTeXMLMath as desired .
Therefore , since ( LaTeXMLRef ) holds true , ( LaTeXMLRef ) yields LaTeXMLEquation .
Moreover , for small LaTeXMLMath we have LaTeXMLMath , and hence by ( LaTeXMLRef ) we have LaTeXMLMath .
it follows from ( LaTeXMLRef ) that LaTeXMLEquation .
Into this substitute ( LaTeXMLRef ) , and use both ( LaTeXMLRef ) and the definition of LaTeXMLMath in ( LaTeXMLRef ) to obtain ( LaTeXMLRef ) as desired .
∎ Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , be continuous functions on LaTeXMLMath .
Let LaTeXMLMath be an interval containing LaTeXMLMath , and let LaTeXMLMath and LaTeXMLMath .
Consider the ordinary differntial equation LaTeXMLEquation .
Let LaTeXMLMath be a positive continuous function on LaTeXMLMath .
There exists a constant LaTeXMLMath such that for any solution LaTeXMLMath to ( LaTeXMLRef ) LaTeXMLEquation .
Let LaTeXMLMath .
Then ( LaTeXMLRef ) is equivalent to LaTeXMLEquation .
From ( LaTeXMLRef ) , we see that LaTeXMLMath satisfies LaTeXMLEquation .
If LaTeXMLMath is large and positive , then LaTeXMLMath is uniformly convex on LaTeXMLMath , and ( LaTeXMLRef ) follows .
If LaTeXMLMath is large and negative , then LaTeXMLMath oscillates rapidly .
In particular , by LaTeXMLCite Chapter V §4 , LaTeXMLMath differs from a solution to LaTeXMLMath in LaTeXMLMath -norm by order LaTeXMLMath .
A straightforward calculation shows that ( LaTeXMLRef ) holds uniformly for LaTeXMLMath and hence LaTeXMLMath for LaTeXMLMath positive and sufficiently large .
For LaTeXMLMath bounded , the claim follows from continuous dependence on parameters , the linearity of the equation , and the fact that the LaTeXMLMath -norm can not vanish on any nontrivial interval .
∎
On the hypersurface orbital varieties of LaTeXMLMath Elise Benlolo LaTeXMLMath Département de Mathématiques , Moulin de la Housse , Université de Reims , F-51687 Reims , France , email : Elise.Benlolo @ univ-reims.fr and Yasmine B. Sanderson LaTeXMLMath Department of Mathematics , William Paterson University , Wayne , N.J. 07470 USA , email : sandersony @ wpunj.edu Abstract : We study the structure of hypersurface orbital varieties of LaTeXMLMath ( those that are hypersurfaces in the nilradical of some parabolic subalgebra ) and how information about this structure is encoded in the standard Young tableau associated to it by the Robinson-Schensted algorithm .
We present a conjecture for the exact form of the unique non-linear defining equations of hypersurface orbital varieties and proofs of the conjecture in certain cases .
I .
Introduction Let LaTeXMLMath be a complex semi-simple algebraic group with Lie algebra LaTeXMLMath , on which it acts through the adjoint representation .
A LaTeXMLMath -orbit LaTeXMLMath in LaTeXMLMath is said to be nilpotent if it consists of nilpotent elements .
Fix some Cartan decomposition of LaTeXMLMath .
Then , an irreducible component of LaTeXMLMath is called an orbital variety .
These varieties figure prominently in the primitive ideal theory of LaTeXMLMath and the ongoing attempt to establish an “ orbit correspondence ” for semisimple groups ( see [ B ] , [ BV ] , [ Mc ] , [ J3 ] for example ) .
In the case of orbital varieties , it was shown by Spaltenstein [ Sp ] and Steinberg [ St2 ] that the dimension of an orbital variety is half the dimension of the corresponding nilpotent orbit .
Joseph [ J2 ] showed that this implies that orbital varieties are Lagrangian .
In the orbit method , one would wish to find a correspondence between these Lagrangian subvarieties of ( co- ) adjoint orbits and simple highest weight modules .
Noting that the unions of the closures of orbital varieties arise as associated varieties of simple highest weight modules , Joseph [ J4 ] laid out a program of “ quantization ” : He called an orbital variety LaTeXMLMath weakly quantizable if its closure is the associated variety of a simple highest weight module .
An orbital variety LaTeXMLMath is strongly quantizable if there exists a highest weight module LaTeXMLMath whose formal character ( as LaTeXMLMath -module ) matched that of the coordinate ring of LaTeXMLMath ( see [ Be ] ) .
Benlolo gave two examples of varieties in LaTeXMLMath which were strongly quantizable , but only by non-simple highest weight modules [ Be ] .
Melnikov showed that every variety in LaTeXMLMath is weakly quantizable [ M3 ] .
In [ J4 ] Joseph studies the orbital varieties in the minimal nonzero orbit for a complex semisimple Lie algebra LaTeXMLMath and shows that every such orbit contains at least one strongly quantizable variety .
However , he also finds examples of varieties that are not weakly quantizable and varieties that are weakly but not strongly quantizable .
One stumbling block in the study of orbital varieties and related highest weight modules is simply that the structure of orbital varieties remains quite mysterious .
Except in the ( obvious ) case of the Richardson varieties ( whose defining relations are all linear ) there are no general formulas for the defining equations of orbital varieties .
This , for one , makes studying the character of the coordinate ring of LaTeXMLMath rather difficult if not impossible .
Obtaining an exact description of the ideal of definition of an orbital variety would also greatly benefit the calculation of the characteristic polynomial LaTeXMLMath of LaTeXMLMath .
The importance of characteristic polynomials is revealed through their many characterizations .
As LaTeXMLMath runs over the components of LaTeXMLMath , where LaTeXMLMath is a fixed orbit , the LaTeXMLMath span a LaTeXMLMath -submodule of LaTeXMLMath .
This is the representation of LaTeXMLMath assigned to LaTeXMLMath by the Springer correspondence ( [ J2 ] , [ Ho ] ) .
In addition , the LaTeXMLMath are intimately connected to Goldie rank polynomials [ J2 ] and can also be viewed as equivariant characteristic classes of orbital cone bundles [ BBM ] .
Characteristic polynomials can be calculated from the character of the coordinate ring of LaTeXMLMath [ J2 ] or directly from a recursive algorithm [ J1 ] .
However , knowing the ideal of definition would greatly help in converting the theory into practice .
Our interest in orbital varieties comes mainly from a combinatorial point of view .
We therefore restrict our attention to the orbital varieties of LaTeXMLMath : through the Robinson-Schensted correspondence , the set of orbital varieties so LaTeXMLMath is in bijection with the set of standard Young tableaux with LaTeXMLMath boxes .
This bijection is “ natural ” in the sense that information about an orbital variety LaTeXMLMath can be “ read off ” the associated tableau LaTeXMLMath .
From a standard Young tableau , one can determine the orbit in which an orbital variety lies , its dimension and its LaTeXMLMath -invariant LaTeXMLMath , a certain subset of the set LaTeXMLMath of simple roots .
From LaTeXMLMath one knows the maximal parabolic subgroup LaTeXMLMath of LaTeXMLMath which stabilizes LaTeXMLMath : it is generated by the Borel subgroup LaTeXMLMath and the root vectors LaTeXMLMath where LaTeXMLMath .
Since an orbital variety is , in some sense , determined by a standard Young tableau , one would like to be able to obtain more information about the structure of LaTeXMLMath directly from the combinatorial information in its associated tableau .
With this idea in mind , we concentrated our efforts on the hypersurface orbital varieties of LaTeXMLMath , that is , orbital varieties which are hypersurfaces in the nilradical of some parabolic .
The equations of these varieties are all linear except for one , LaTeXMLMath , where LaTeXMLMath is a homogeneous polynomial in LaTeXMLMath with LaTeXMLMath .
Clearly the linear equations are all of the form LaTeXMLMath where LaTeXMLMath is a sum of simple roots in LaTeXMLMath .
So the real problem was extracting information about LaTeXMLMath from LaTeXMLMath .
Our main idea was to compare the tableau LaTeXMLMath with LaTeXMLMath , the standard Young tableau associated to the Richardson orbital variety LaTeXMLMath with the same LaTeXMLMath -invariant as LaTeXMLMath .
The relationship between LaTeXMLMath and LaTeXMLMath is the following : LaTeXMLMath where LaTeXMLMath is the nilradical of the parabolic subalgebra LaTeXMLMath .
This relationship between varieties translates to the following relationship between tableaux : LaTeXMLMath is obtained from dropping one box of LaTeXMLMath down one row .
This allowed us to determine the minimal connected subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath where LaTeXMLMath is the subalgebra of LaTeXMLMath generated by LaTeXMLMath with LaTeXMLMath .
In other words , we can use LaTeXMLMath to tell us “ where LaTeXMLMath is located ” .
An important tool in our proofs are the so-called power-rank conditions of van Leeuwen [ vanL ] .
These conditions describe relations among the coordinates of a generic matrix in terms of the shapes of the subtableaux of LaTeXMLMath .
We were then able to describe LaTeXMLMath explicitly for many cases of hypersurface orbital varieties .
We also conjecture an explicit formula for LaTeXMLMath when LaTeXMLMath a is an arbitrary hypersurface orbital variety .
Using previous results of [ J1 ] , [ BBM ] we obtain an explicit formula for the characteristic polynomial of such varieties .
We provide extensive examples to illustrate our points .
Acknowledgment : We would like to thank A. Joseph for inspiring us to work on orbital varieties and the referee for making useful comments which improved the final version of this paper .
II .
Some background and notation Let LaTeXMLMath and let LaTeXMLMath denote its Lie algebra .
Let LaTeXMLMath be the Cartan decomposition where LaTeXMLMath denotes the Cartan subalgebra and LaTeXMLMath the nilpotent subalgebra of strictly upper-triangular matrices .
Let LaTeXMLMath denote the set of LaTeXMLMath simple roots LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath denote the positive root LaTeXMLMath .
Let LaTeXMLMath , the set of positive roots .
Let LaTeXMLMath denote the associated root vector .
Note that a generic matrix LaTeXMLMath in the one-dimensional space spanned by LaTeXMLMath satisfies LaTeXMLMath for LaTeXMLMath .
We will denote by LaTeXMLMath the coordinate corresponding to the root LaTeXMLMath .
This allows us to identify LaTeXMLMath .
Let LaTeXMLMath denote the simple reflection with respect to the simple root LaTeXMLMath ( LaTeXMLMath ) .
We also use LaTeXMLMath to denote the transposition LaTeXMLMath .
The LaTeXMLMath generate the Weyl group LaTeXMLMath which , in the case of LaTeXMLMath , is isomorphic to LaTeXMLMath , the symmetric group on LaTeXMLMath letters .
For any subset LaTeXMLMath , let LaTeXMLMath denote the positive roots in LaTeXMLMath which are sums of simple roots of LaTeXMLMath .
Let LaTeXMLMath be the Weyl group element generated by the simple reflections LaTeXMLMath for LaTeXMLMath .
III .
Orbital varieties and Young tableaux LaTeXMLMath acts on LaTeXMLMath by conjugation .
When LaTeXMLMath , the orbit LaTeXMLMath is called a nilpotent orbit .
The set of nilpotent orbits of LaTeXMLMath is in bijection with the set of partitions LaTeXMLMath of LaTeXMLMath .
In fact , to each partition LaTeXMLMath , one can associate the strictly upper-triangular nilpotent matrix LaTeXMLMath with LaTeXMLMath Jordan blocks of size LaTeXMLMath .
Thus , the orbit LaTeXMLMath consists of all nilpotent matrices with Jordan canonical form LaTeXMLMath .
An orbital variety is an irreducible component of the intersection LaTeXMLMath .
A more explicit general description of these varieties , due to [ St1 ] , [ J2 ] , is as follows : Let LaTeXMLMath be a Weyl group element .
Let LaTeXMLMath denote the Borel subgroup of LaTeXMLMath .
Set LaTeXMLMath .
Then LaTeXMLEquation is an orbital variety and the map LaTeXMLMath is a surjection of LaTeXMLMath onto the set of all orbital varieties of LaTeXMLMath .
Since the set of nilpotent orbits is indexed by partitions , it is natural to wonder if this indexing somehow extends to orbital varieties .
Such an extension exists , which we now describe .
We can identify a partition LaTeXMLMath with a Young diagram consisting of LaTeXMLMath boxes with LaTeXMLMath boxes in the LaTeXMLMath row , LaTeXMLMath boxes in the LaTeXMLMath row and so on .
A standard Young tableau is a filling of the LaTeXMLMath boxes with the numbers LaTeXMLMath in such a way that the numbers increase from left to right in every row and from top to bottom in every column .
Example 1 : The Young diagram associated to the partition LaTeXMLMath is below on the left .
On the right are several examples of standard Young tableaux of shape LaTeXMLMath .
LaTeXMLEquation .
Theorem 1 : [ J2 , 9.14 ] The set of orbital varieties in the nilpotent orbit LaTeXMLMath is in bijection with the set LaTeXMLMath of standard Young tableaux of shape LaTeXMLMath .
This bijection is a corollary of the Robinson-Schensted correspondence ( see [ M1 ] , [ vanL ] for nice descriptions ) , which associates to each permutation LaTeXMLMath a certain pair of standard Young tableaux LaTeXMLMath .
The tableau which will be associated to the orbital variety LaTeXMLMath is the tableau LaTeXMLMath associated to LaTeXMLMath by this bijection .
Extensive research has been done studying this connection between orbital varieties .
From a standard Young tableau LaTeXMLMath , one can read information about the associated orbital variety LaTeXMLMath .
In particular , if LaTeXMLMath is the dual partition of LaTeXMLMath , then LaTeXMLMath [ SS ] .
From the tableau LaTeXMLMath , one can also determine the LaTeXMLMath -invariant LaTeXMLMath of an orbital variety LaTeXMLMath .
By definition , LaTeXMLMath is the set of all simple roots LaTeXMLMath such that the subgroup LaTeXMLMath stabilizes LaTeXMLMath .
In other words , if LaTeXMLMath is the maximal parabolic subgroup which stabilizes LaTeXMLMath , then LaTeXMLMath .
It turns out that LaTeXMLMath is above LaTeXMLMath in LaTeXMLMath if and only if LaTeXMLMath [ Ja ] .
To every subset LaTeXMLMath , there exists a ( unique ) orbital variety LaTeXMLMath of maximal dimension whose LaTeXMLMath -invariant is LaTeXMLMath .
LaTeXMLMath is a Richardson variety , that is LaTeXMLMath equals the dimension of the nilradical LaTeXMLEquation of the parabolic subalgebra LaTeXMLMath .
Therefore its standard Young tableau LaTeXMLMath is “ top-heavy ” .
One constructs it by putting the numbered boxes in the topmost row possible such that the restrictions imposed by LaTeXMLMath are respected .
Example 2 : Let LaTeXMLMath .
Then each Young diagram will consist of 8 boxes .
The standard Young tableau associated to a Richardson orbital variety LaTeXMLMath with LaTeXMLMath -invariant LaTeXMLMath is LaTeXMLEquation .
This orbital variety lies in the orbit LaTeXMLMath where LaTeXMLMath .
Then LaTeXMLMath .
We have that LaTeXMLEquation .
Let LaTeXMLMath be the standard Young tableau associated to the Richardson variety LaTeXMLMath .
A chain LaTeXMLMath of LaTeXMLMath is an invariant subset of LaTeXMLMath under the action of LaTeXMLMath .
In other words , it is a set of the form LaTeXMLMath LaTeXMLMath , where ( a ) LaTeXMLMath is in the first row of LaTeXMLMath ( b ) if LaTeXMLMath is also in the first row of LaTeXMLMath , then LaTeXMLMath , i.e .
LaTeXMLMath ( c ) if LaTeXMLMath is not in the first row of LaTeXMLMath , one requires that LaTeXMLMath and , whenever LaTeXMLMath , LaTeXMLMath .
We say that LaTeXMLMath has length LaTeXMLMath and denote this by LaTeXMLMath .
Notice that LaTeXMLMath is completely determined by its chains .
If LaTeXMLMath is of shape LaTeXMLMath , then it has LaTeXMLMath chains and the number of columns of length LaTeXMLMath equals the number of chains of length LaTeXMLMath .
Example 3 : The tableau in our previous example has five chains : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath which are exactly the invariant subsets of LaTeXMLMath under the action of the transpositions LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Consider the orbital varieties LaTeXMLMath and LaTeXMLMath .
We say that LaTeXMLMath is a descendant of LaTeXMLMath if LaTeXMLMath , LaTeXMLMath and if any orbital variety satisfies LaTeXMLMath satisfies LaTeXMLMath , then either LaTeXMLMath or LaTeXMLMath .
If LaTeXMLMath ( resp .
LaTeXMLMath ) is the standard Young tableau associated to LaTeXMLMath ( resp .
LaTeXMLMath ) , then we say that LaTeXMLMath is a descendant of LaTeXMLMath if and only if LaTeXMLMath is a descendant of LaTeXMLMath .
We now consider a hypersurface orbital variety LaTeXMLMath with the same LaTeXMLMath -invariant .
Then LaTeXMLMath is a descendant of LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) be the standard Young tableau associated to LaTeXMLMath ( resp .
LaTeXMLMath ) .
For any tableau LaTeXMLMath , let LaTeXMLMath denote the row of LaTeXMLMath in which the box numbered LaTeXMLMath is located .
Lemma 1 : LaTeXMLMath is obtained from LaTeXMLMath by moving a box containing the maximal element of some chain LaTeXMLMath of length LaTeXMLMath from row LaTeXMLMath to row LaTeXMLMath .
Proof : Let LaTeXMLMath and LaTeXMLMath .
It is a result of Gerstenhaber that LaTeXMLMath implies that LaTeXMLMath ( see [ He ] ) .
Recall that LaTeXMLMath is defined as LaTeXMLMath for all LaTeXMLMath .
Suppose first that LaTeXMLMath differs from LaTeXMLMath by the dropping of one box ( down one or possibly several rows ) .
If LaTeXMLMath and LaTeXMLMath , then either LaTeXMLMath or LaTeXMLMath .
In the latter case , we will set LaTeXMLMath .
Then there exist LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) such that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath when LaTeXMLMath .
So LaTeXMLEquation which equals LaTeXMLMath if and only if LaTeXMLMath .
This can only happen if the box was knocked down one row .
Since LaTeXMLMath , then LaTeXMLMath can not be obtained from LaTeXMLMath by moving more than one box .
Notice that , since LaTeXMLMath , then LaTeXMLMath for all LaTeXMLMath .
Since only one box is moved from LaTeXMLMath in order to obtain LaTeXMLMath , it must contain the maximal element of some chain in LaTeXMLMath .
( Any “ shuffling ” of the boxes would either produce something that is not a standard tableau or would change the LaTeXMLMath -invariant ) .
Example 4 : Consider the Young tableau from Example 2 .
By knocking down box number LaTeXMLMath from the first to the second row , one obtains a Young tableau associated to a hypersurface orbital variety contained in LaTeXMLMath : LaTeXMLEquation .
The corresponding orbital variety LaTeXMLMath is contained in LaTeXMLMath where LaTeXMLMath .
Since LaTeXMLMath , we have that LaTeXMLMath .
Hence LaTeXMLMath is a hypersurface variety .
Notice that there are no other ways that one could move a box down one row without changing the LaTeXMLMath -invariant .
Therefore , in this case , LaTeXMLMath contains only one hypersurface orbital variety with the same LaTeXMLMath -invariant .
Likewise , given a standard Young tableau associated to a hypersurface orbital variety LaTeXMLMath , one can always obtain the tableau associated to the Richardson variety which contains LaTeXMLMath by moving an appropriate box up one row .
IV .
Subtableaux and projections of orbital varieties In the following , for any LaTeXMLMath , we denote by LaTeXMLMath the subalgebra of strictly upper-triangular matrices in LaTeXMLMath .
Let LaTeXMLMath be the projection which , to a generic matrix LaTeXMLMath , assigns the same matrix with the LaTeXMLMath row and column removed .
Let LaTeXMLMath be an orbital variety with standard Young tableau LaTeXMLMath .
It results from work of Schützenberger , Knuth and Melnikov ( See [ M1 ] Lemma 1.1.3 , Theorems 1.3.13 and 4.1.2 ) that LaTeXMLMath is dense in LaTeXMLMath where LaTeXMLMath is a certain orbital variety .
The standard Young tableau LaTeXMLMath associated to LaTeXMLMath is obtained from LaTeXMLMath by removing the box with the largest entry .
( See also [ vanL ] for a discussion of this in terms of flag varieties . )
Example 5 : If LaTeXMLMath then LaTeXMLMath .
Likewise , let LaTeXMLMath be the projection which , to a generic matrix LaTeXMLMath , assigns the same matrix with the LaTeXMLMath row and column removed .
Let LaTeXMLMath be an orbital variety with standard Young tableau LaTeXMLMath .
In the same way , we will associate to this projection a certain standard Young tableau LaTeXMLMath , obtained in the following way : Apply to LaTeXMLMath the Schützenberger “ jeu de taquin ” algorithm ( see [ M1 ] or [ vanL ] , §4 ) : remove the box in the first row and first column to leave an empty square in its place .
Then the following step is repeated until the empty square is a corner of the original tableau : move into the empty square the smaller of the entries located directly to the right of and below it .
Replace each of the entries LaTeXMLMath in this tableau by LaTeXMLMath to obtain a standard Young tableau which we will denote by LaTeXMLMath .
We have that LaTeXMLMath is dense in LaTeXMLMath where LaTeXMLMath is the orbital variety with associated standard Young tableau LaTeXMLMath .
Example 6 : Let LaTeXMLMath .
We show the steps to obtain LaTeXMLMath from LaTeXMLMath using the Schützenberger algorithm .
LaTeXMLEquation .
For LaTeXMLMath we denote by LaTeXMLMath the projection which removes all rows and columns numbered LaTeXMLMath or LaTeXMLMath .
The image under LaTeXMLMath doesn ’ t depend on the order the rows or columns are removed so it is well-defined ( see [ M1 ] 1.3.15 ) .
If LaTeXMLMath is an orbital variety , the image LaTeXMLMath is dense in LaTeXMLMath where LaTeXMLMath is some orbital variety .
We associate to LaTeXMLMath the standard Young tableau LaTeXMLMath , obtained from LaTeXMLMath by removing the entries LaTeXMLMath and by removing the boxes with entries LaTeXMLMath by repeated applications of the above two operations .
Again , the order in which these operations are applied doesn ’ t matter , so LaTeXMLMath is well-defined [ vanL ] .
We will use LaTeXMLMath to denote the shape of LaTeXMLMath .
V. The LaTeXMLMath -set of a hypersurface orbital variety Let LaTeXMLMath be a hypersurface orbital variety and let LaTeXMLMath be a Richardson orbital variety such that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the Richardson tableau ( associated to LaTeXMLMath ) from which one can obtain a hypersurface tableau LaTeXMLMath ( associated to LaTeXMLMath ) .
We now introduce another subset of LaTeXMLMath , which will be crucial in our study of the non-linear generator LaTeXMLMath .
Definition : Let LaTeXMLMath be a hypersurface orbital variety .
The LaTeXMLMath -set LaTeXMLMath of LaTeXMLMath is the smallest connected subset of LaTeXMLMath such that the LaTeXMLMath is contained in LaTeXMLMath where LaTeXMLMath is the subalgebra of LaTeXMLMath generated by LaTeXMLMath with LaTeXMLMath .
We write LaTeXMLMath for LaTeXMLMath when there is no risk of confusion .
We now study the relationship between LaTeXMLMath , LaTeXMLMath and LaTeXMLMath by way of the projections LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Theorem 2 : Assume that box LaTeXMLMath dropped one row to obtain LaTeXMLMath from LaTeXMLMath .
Then LaTeXMLMath if and only if LaTeXMLMath and LaTeXMLMath satisfy the following two properties : 1 .
LaTeXMLMath = LaTeXMLMath .
2 .
Let LaTeXMLMath .
Then LaTeXMLMath .
( In other words , LaTeXMLMath and LaTeXMLMath are the only chains in LaTeXMLMath of length LaTeXMLMath . )
Proof : Since LaTeXMLMath is LaTeXMLMath -stable , then the weight of the non-linear defining polynomial LaTeXMLMath is well-defined ( every monomial of LaTeXMLMath has the same weight with respect to LaTeXMLMath ) .
By the minimality of LaTeXMLMath , this means that for every monomial LaTeXMLMath of LaTeXMLMath , there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are factors of LaTeXMLMath .
Since LaTeXMLMath is the unique nonlinear condition on the LaTeXMLMath ( LaTeXMLMath ) , the only constraints imposed on the coordinate subsets LaTeXMLEquation ( considered as coordinates for either LaTeXMLMath or LaTeXMLMath ) are those linear constraints given by : LaTeXMLMath if LaTeXMLMath .
In terms of projections , this translates to LaTeXMLMath ( resp .
LaTeXMLMath ) and their shapes would be determined uniquely by the restrictions given by LaTeXMLMath , ( resp .
LaTeXMLMath ) .
By definition , LaTeXMLMath is obtained by removing box LaTeXMLMath from LaTeXMLMath .
Since LaTeXMLMath was the block which moved down a row to obtain LaTeXMLMath from LaTeXMLMath , then LaTeXMLMath .
On the other hand , LaTeXMLMath means that removing box LaTeXMLMath from LaTeXMLMath should precipitate in a shift of boxes which results in box LaTeXMLMath moving back up one row .
When box LaTeXMLMath is removed from LaTeXMLMath , the boxes corresponding to the rest of the chain LaTeXMLMath move up one row , leaving a space in row LaTeXMLMath and column LaTeXMLMath .
For each remaining box in the LaTeXMLMath row , the number to its right is smaller than the number directly underneath it ( if it even exists ) .
So , when box LaTeXMLMath moves up one row , then all remaining boxes in the LaTeXMLMath row move over to the left by one space .
The remaining boxes in the LaTeXMLMath column then move up by one row , where LaTeXMLMath is the number of boxes in the LaTeXMLMath row of LaTeXMLMath .
This means that box LaTeXMLMath is part of this series of shifts if and only if it is in the LaTeXMLMath column in LaTeXMLMath .
This can happen if and only if LaTeXMLMath = LaTeXMLMath .
Since box LaTeXMLMath moved down only one row from LaTeXMLMath to LaTeXMLMath , this means that it is in the LaTeXMLMath row and LaTeXMLMath column in LaTeXMLMath and the LaTeXMLMath row and LaTeXMLMath column in LaTeXMLMath which is true if and only if LaTeXMLMath .
From now on , we will call LaTeXMLMath the thickness of LaTeXMLMath and use the notation LaTeXMLMath .
From Lemma 1 and Theorem 2 , we now know how to “ read off ” LaTeXMLMath from any tableau LaTeXMLMath corresponding to a hypersurface orbital variety LaTeXMLMath .
In particular , there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath corresponds to a hypersurface orbital variety in LaTeXMLMath whose LaTeXMLMath -set is all of the simple roots for LaTeXMLMath .
The corresponding tableau ( obtained by an appropriate renumbering of the entries of LaTeXMLMath ) will have the same form as that given in Theorem 2 .
The following corollary also gives the definition for LaTeXMLMath for arbitrary LaTeXMLMath : Corollary 1 ( Obtaining LaTeXMLMath and LaTeXMLMath from LaTeXMLMath ) : Assume that LaTeXMLMath was obtained from LaTeXMLMath by dropping the box with the biggest number LaTeXMLMath of a chain LaTeXMLMath ( LaTeXMLMath ) .
Let LaTeXMLMath .
Let LaTeXMLMath ( LaTeXMLMath ) be the chain in LaTeXMLMath of length LaTeXMLMath such that there is no other length LaTeXMLMath chain in between LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the smallest number in LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath .
Proof : By Lemma 1 , we know that LaTeXMLMath must be the biggest number in some chain LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath be the previous chain of length LaTeXMLMath in LaTeXMLMath and let LaTeXMLMath be the smallest number in LaTeXMLMath .
Then ( the renormalized ) LaTeXMLMath and LaTeXMLMath correspond to , respectively , the first and last chains of LaTeXMLMath and they are the only chains in LaTeXMLMath of length LaTeXMLMath .
In addition LaTeXMLMath is obtained from LaTeXMLMath by dropping the maximal element in the last chain of LaTeXMLMath .
By Theorem 2 , LaTeXMLMath and LaTeXMLMath .
Example 7 : Consider the hypersurface orbital variety LaTeXMLMath with tableau LaTeXMLEquation .
We have LaTeXMLMath , so the tableau LaTeXMLMath associated to LaTeXMLMath is LaTeXMLEquation .
The chains of LaTeXMLMath are LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Box LaTeXMLMath belongs to LaTeXMLMath , which has length 3 .
The preceding chain of length LaTeXMLMath is LaTeXMLMath .
So LaTeXMLMath and LaTeXMLMath .
The hypersurface orbital variety LaTeXMLMath which is determined by LaTeXMLMath ( that is LaTeXMLMath renormalized under the map LaTeXMLMath ) and the same ( renormalized ) LaTeXMLMath as LaTeXMLMath has standard Young tableau LaTeXMLEquation .
VI .
Van Leeuwen ’ s power-rank conditions Now that we have determined the “ location ” of the non-linear generator LaTeXMLMath from the standard Young tableau for a hypersurface orbital variety , we can work on determining LaTeXMLMath itself .
Let LaTeXMLMath be a partition and let LaTeXMLMath .
It is known that LaTeXMLMath is equal to the number of squares beyond the LaTeXMLMath column in the Young diagram LaTeXMLMath .
Equivalently , LaTeXMLMath is equal to the number of squares in the first LaTeXMLMath columns of LaTeXMLMath .
This imposes certain restrictions on the coordinates of LaTeXMLMath .
Similar restrictions can be obtained when considering the diagrams associated to the projections LaTeXMLMath .
These restrictions are the so-called power-rank conditions which were introduced by van Leeuwen [ vanL ] .
Let LaTeXMLMath be an orbital variety with tableau LaTeXMLMath and let LaTeXMLMath be a generic nilpotent matrix in LaTeXMLMath .
For LaTeXMLMath denote by LaTeXMLMath the submatrix of LaTeXMLMath obtained by removing the rows and columns numbered LaTeXMLMath or LaTeXMLMath .
Theorem 3 [ vanL ] : The coordinate vectors of LaTeXMLMath satisfy the power-rank conditions imposed by LaTeXMLMath .
In other words , the coordinate vectors LaTeXMLMath of a generic matrix LaTeXMLMath satisfies all power-rank conditions imposed by all LaTeXMLMath for LaTeXMLMath .
Example 8 : Consider the orbital variety LaTeXMLMath given by LaTeXMLEquation .
Notice that LaTeXMLMath so LaTeXMLMath is obtained from LaTeXMLMath by dropping box LaTeXMLMath which is in the last chain .
It has length LaTeXMLMath .
The only other chain of length LaTeXMLMath in LaTeXMLMath is LaTeXMLMath so , in this case , LaTeXMLMath and LaTeXMLMath .
We show a generic matrix LaTeXMLMath and the associated matrix of LaTeXMLMath : LaTeXMLEquation .
The power-rank condition given by LaTeXMLMath says that the matrix LaTeXMLMath has rank LaTeXMLMath .
This implies that LaTeXMLMath ( which we already know since LaTeXMLMath ) .
Likewise , the power-rank condition given by LaTeXMLMath forces LaTeXMLMath .
The power-rank condition imposed by LaTeXMLMath says that LaTeXMLMath .
We have that every entry of LaTeXMLMath is LaTeXMLMath except for the entry in the LaTeXMLMath row and LaTeXMLMath column which equals LaTeXMLEquation .
The condition LaTeXMLMath implies that LaTeXMLMath .
It is easily checked that the remaining power-rank conditions provide trivial power-rank conditions , that is , they provide no further constraints on the LaTeXMLMath .
We have LaTeXMLMath .
We conclude that LaTeXMLMath is a hypersurface orbital variety and that , since LaTeXMLMath is irreducible , then LaTeXMLMath .
Notice that the LaTeXMLMath that we found in the beginning does correspond to the minimal set such that LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) denote the shape of LaTeXMLMath ( resp .
LaTeXMLMath ) from the previous example .
Then LaTeXMLEquation .
From LaTeXMLMath and LaTeXMLMath we have that LaTeXMLMath and LaTeXMLMath .
The condition LaTeXMLMath results from the fact that a box is dropped down one row in order to obtain LaTeXMLMath from LaTeXMLMath .
We will use power-rank conditions in precisely the same spirit in order to obtain the non-linear defining generator LaTeXMLMath .
Let LaTeXMLMath be a hypersurface orbital variety such that LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath is obtained from LaTeXMLMath by dropping box LaTeXMLMath from row LaTeXMLMath to row LaTeXMLMath .
If box LaTeXMLMath was in column LaTeXMLMath in LaTeXMLMath , then it is in column LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be the number of boxes after the LaTeXMLMath column in LaTeXMLMath .
Then there are LaTeXMLMath boxes after the LaTeXMLMath column in LaTeXMLMath which simply means that LaTeXMLMath .
Consider the LaTeXMLMath submatrix LaTeXMLMath located in the top righthand corner of LaTeXMLMath .
Then LaTeXMLMath when LaTeXMLMath is considered as a submatrix of LaTeXMLMath , but LaTeXMLMath when it is considered as a submatrix of LaTeXMLMath .
This means that the non-linear generator LaTeXMLMath is a factor of LaTeXMLMath .
In the above example , we had that LaTeXMLMath .
However , this is not always the case as the next example will show : Example 9 : Consider the hypersurface orbital variety LaTeXMLMath with tableau LaTeXMLEquation .
We have that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath matrix located in the top righthand corner of LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath is too long to write out .
We can check that LaTeXMLMath is irreducible .
Since every monomial in LaTeXMLMath contains some LaTeXMLMath and some LaTeXMLMath as factors and since LaTeXMLMath in this case , we know that LaTeXMLMath must equal our nonlinear factor LaTeXMLMath .
Remark : Our calculations indicate that LaTeXMLMath if and only if the chains in LaTeXMLMath other than the first and last chains all have length less than LaTeXMLMath or all have length greater than LaTeXMLMath ( but not both ) .
VII .
The exact form of the non-linear generator LaTeXMLMath It is clear that the power-rank conditions do not suffice to give a general formula for LaTeXMLMath .
We now present results and a conjecture concerning its exact form .
Without loss of generality , we can restrict ourselves to the case LaTeXMLMath .
A generic matrix LaTeXMLMath has the following form : LaTeXMLEquation where LaTeXMLMath if and only if LaTeXMLMath .
For a matrix LaTeXMLMath , denote by LaTeXMLMath the LaTeXMLMath submatrix in the top right corner of LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLEquation .
We have LaTeXMLEquation where the LaTeXMLMath are ( up to sign ) sums of LaTeXMLMath minors in LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath satisfy LaTeXMLMath .
Consider the LaTeXMLMath matrix LaTeXMLEquation .
Then LaTeXMLMath where the sum is over all possible LaTeXMLMath -tuples LaTeXMLMath .
Lemma 2 : LaTeXMLMath if and only if LaTeXMLMath .
Proof : Any two LaTeXMLMath -tuples LaTeXMLMath , LaTeXMLMath differ by at least one entry .
Hence any monomial in LaTeXMLMath differs from any monomial in LaTeXMLMath by at least one factor LaTeXMLMath .
So the algebraic independence of the LaTeXMLMath implies that there can be no cancelations of monomials from determinants of different LaTeXMLMath .
Hence LaTeXMLMath is non-zero if and only if there exists an LaTeXMLMath such that LaTeXMLMath .
Notice that LaTeXMLMath if and only if LaTeXMLMath for all LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath ( all coordinates below or to the left of LaTeXMLMath ) .
If at least one of the diagonal elements of LaTeXMLMath is zero , then LaTeXMLMath is an upper-triangular block matrix where at least one of the block matrices has a zero column or zero row .
In this case , LaTeXMLMath .
Clearly , when all of its diagonal elements are non-zero , we have that LaTeXMLMath .
Therefore , LaTeXMLMath if and only if all of its diagonal coordinates are non-zero .
When LaTeXMLMath , all diagonal elements are of the form LaTeXMLMath where LaTeXMLMath or LaTeXMLMath so they are all non-zero and LaTeXMLMath .
Now consider LaTeXMLMath .
Then LaTeXMLMath has LaTeXMLMath diagonal elements of the form LaTeXMLMath .
For any given LaTeXMLMath -tuple LaTeXMLMath , let LaTeXMLMath denote the number of non-zero diagonal elements of the form LaTeXMLMath where LaTeXMLMath .
Let LaTeXMLMath where the maximum is taken over all LaTeXMLMath -tuples LaTeXMLMath .
Let LaTeXMLMath where the maximum is taken over all LaTeXMLMath .
We have that LaTeXMLMath for all LaTeXMLMath -tuples LaTeXMLMath if and only if LaTeXMLMath .
This maximum LaTeXMLMath is attained at the -tuple which contains every non-zero coordinate of the form LaTeXMLMath .
In other words , LaTeXMLMath equals the number of positive roots of length LaTeXMLMath in LaTeXMLMath where LaTeXMLMath .
There are LaTeXMLMath positive roots of length LaTeXMLMath in LaTeXMLMath .
The number of positive roots of length LaTeXMLMath in LaTeXMLMath is the number of boxes after the LaTeXMLMath row of LaTeXMLMath .
In other words , it is LaTeXMLMath .
Therefore , LaTeXMLMath .
We therefore have that LaTeXMLMath if and only if LaTeXMLMath .
For ease of notation , set LaTeXMLMath .
For any LaTeXMLMath -semiinvariant polynomial LaTeXMLMath , we denote the weight of LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be any monomial term of LaTeXMLMath .
Then LaTeXMLMath .
We have that LaTeXMLMath for LaTeXMLMath .
Proposition 1 : When LaTeXMLMath then LaTeXMLMath .
Proof : In this case , the matrices LaTeXMLMath corresponding to the LaTeXMLMath -tuples LaTeXMLMath are diagonal and LaTeXMLEquation .
It follows that LaTeXMLMath is the sum of all possible monomials of this type .
Now , consider LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath row vector LaTeXMLMath , LaTeXMLMath is the LaTeXMLMath matrix LaTeXMLMath , LaTeXMLMath is the LaTeXMLMath column vector LaTeXMLMath , LaTeXMLMath is the LaTeXMLMath square matrix that is left .
The LaTeXMLMath s represent zero matrices of the appropriate size .
For all LaTeXMLMath , we have LaTeXMLEquation .
Notice that LaTeXMLMath .
In other words , LaTeXMLMath is the generic matrix associated to the Richardson orbital variety with standard Young tableau LaTeXMLMath .
The tableau LaTeXMLMath is obtained from LaTeXMLMath by removing the first and last chains , which correspond simply to box LaTeXMLMath , resp .
box LaTeXMLMath .
This means that there are only LaTeXMLMath columns in LaTeXMLMath .
By the power-rank conditions , LaTeXMLMath , but LaTeXMLMath and LaTeXMLMath .
So LaTeXMLMath , but LaTeXMLMath .
There is only one box ( which is box LaTeXMLMath , in fact ) past the LaTeXMLMath column in LaTeXMLMath .
Therefore , when LaTeXMLMath , the matrix LaTeXMLMath has rank LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , we have that LaTeXMLMath is zero everywhere except at the LaTeXMLMath matrix LaTeXMLMath in the top right corner .
Now , LaTeXMLMath so LaTeXMLMath when considered as an entry in LaTeXMLMath .
Hence LaTeXMLMath divides LaTeXMLMath .
This implies that LaTeXMLMath is a positive sum of positive roots .
We have LaTeXMLMath .
We will show that LaTeXMLMath by showing that LaTeXMLMath .
By the definition of LaTeXMLMath , we know that the coefficient of LaTeXMLMath in LaTeXMLMath is non-zero .
If LaTeXMLMath then there is a smallest LaTeXMLMath ( LaTeXMLMath ) such that the coefficient of LaTeXMLMath in LaTeXMLMath is LaTeXMLMath .
We can then write LaTeXMLMath where LaTeXMLMath is not a summand of LaTeXMLMath .
Since every chain ( besides the first and last ) in LaTeXMLMath is of length LaTeXMLMath , then LaTeXMLMath or LaTeXMLMath ( or both ) .
But , LaTeXMLEquation .
Similarly , LaTeXMLMath ( independently of whether or not LaTeXMLMath is a summand of LaTeXMLMath ) .
Therefore , LaTeXMLMath is not invariant under the action of LaTeXMLMath , which brings us to a contradiction .
Finally , since LaTeXMLMath we have LaTeXMLMath .
Example 10 : In the case of Example 8 , we have that LaTeXMLEquation .
We have LaTeXMLMath and indeed LaTeXMLMath is the generator that we had previously found .
Proposition 2 : If LaTeXMLMath then LaTeXMLMath .
Proof : For ease of notation , we will denote the coordinates LaTeXMLMath by LaTeXMLMath for LaTeXMLMath .
Since LaTeXMLMath , then LaTeXMLMath .
Consequently , the only chains of length LaTeXMLMath in LaTeXMLMath are the first and last chains .
This means that the subdiagonal coordinates LaTeXMLMath are not identically LaTeXMLMath .
We show that this implies that LaTeXMLMath is irreducible .
In fact , suppose that LaTeXMLMath where both LaTeXMLMath and LaTeXMLMath .
We claim that LaTeXMLMath and LaTeXMLMath are functions on disjoint sets of row vectors .
In fact , suppose that they aren ’ t .
Then , for some row LaTeXMLMath and some columns LaTeXMLMath and LaTeXMLMath , we have that both LaTeXMLMath and LaTeXMLMath are functions of the coordinates LaTeXMLMath and LaTeXMLMath .
This means that we can write LaTeXMLMath and LaTeXMLMath where the LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) are in LaTeXMLMath and where LaTeXMLMath and LaTeXMLMath do not depend on LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath do not depend on LaTeXMLMath and where LaTeXMLMath and LaTeXMLMath do not depend on either LaTeXMLMath or LaTeXMLMath .
Then LaTeXMLMath where the degree of either LaTeXMLMath or LaTeXMLMath in any term of LaTeXMLMath is at most LaTeXMLMath .
The degrees of LaTeXMLMath and LaTeXMLMath are at most LaTeXMLMath in every term of LaTeXMLMath and no term of LaTeXMLMath contains LaTeXMLMath .
Therefore we have that LaTeXMLMath .
Since LaTeXMLMath is a domain , then either LaTeXMLMath or LaTeXMLMath .
Without loss of generality , we can assume that LaTeXMLMath .
Then either LaTeXMLMath or both LaTeXMLMath .
In either case , one of the factors LaTeXMLMath or LaTeXMLMath does not depend on LaTeXMLMath and LaTeXMLMath .
So LaTeXMLMath and LaTeXMLMath depend on disjoint sets of row vectors .
A similar argument shows that LaTeXMLMath and LaTeXMLMath depend on disjoint sets of column vectors .
Therefore , there exist subsets LaTeXMLMath and LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are the complements of LaTeXMLMath and LaTeXMLMath in LaTeXMLMath .
Without loss of generality , we can assume that LaTeXMLMath is non-empty .
We claim that LaTeXMLMath .
In fact , consider the following specialization LaTeXMLMath of LaTeXMLMath : set LaTeXMLMath for LaTeXMLMath .
Then LaTeXMLMath .
This means that LaTeXMLMath if and only if LaTeXMLMath .
So LaTeXMLMath .
Now we claim that LaTeXMLMath .
In fact , consider the following specialization LaTeXMLMath of LaTeXMLMath : set LaTeXMLMath for LaTeXMLMath not of the form LaTeXMLMath or LaTeXMLMath .
Then LaTeXMLMath .
This means that if some LaTeXMLMath then LaTeXMLMath .
Therefore , LaTeXMLMath and LaTeXMLMath which means that LaTeXMLMath is constant .
Therefore , LaTeXMLMath is irreducible .
The tableau LaTeXMLMath is obtained from LaTeXMLMath by dropping box LaTeXMLMath from the LaTeXMLMath row and LaTeXMLMath column in LaTeXMLMath to the LaTeXMLMath row and LaTeXMLMath column in LaTeXMLMath .
We have that LaTeXMLMath and , by irreducibility of LaTeXMLMath , we have LaTeXMLMath .
On the other hand , there are no boxes in LaTeXMLMath past the LaTeXMLMath th row , so LaTeXMLMath .
Therefore , LaTeXMLMath .
Example 11 : Consider the hypersurface orbital variety LaTeXMLMath with tableau LaTeXMLEquation .
We have LaTeXMLMath , LaTeXMLMath and LaTeXMLEquation .
Then LaTeXMLMath .
We have that LaTeXMLMath is irreducible .
In addition LaTeXMLMath is exactly that constraint given by LaTeXMLMath .
So LaTeXMLMath is the nonlinear generator in LaTeXMLMath .
The results of the previous two propositions and of explicit calculations using MAPLE for most cases up to LaTeXMLMath have led us to believe that LaTeXMLMath is always equal to LaTeXMLMath .
We claim : Conjecture : Let LaTeXMLMath be a hypersurface orbital variety .
Let LaTeXMLMath be the Richardson orbital variety with the same LaTeXMLMath -invariant as LaTeXMLMath .
Assume that LaTeXMLMath and let LaTeXMLMath .
Let LaTeXMLMath represent a generic matrix in LaTeXMLMath .
Let LaTeXMLMath be the coefficient of the smallest power of LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath .
Example 12 : Let LaTeXMLMath be as in Example 7 .
Then LaTeXMLMath and LaTeXMLMath .
In this case LaTeXMLEquation and LaTeXMLMath .
We have LaTeXMLMath .
So LaTeXMLMath .
VIII .
The characteristic polynomial LaTeXMLMath To every orbital variety LaTeXMLMath , one can associate its characteristic polynomial LaTeXMLMath ( [ J1 ] , [ BBM ] ) .
This LaTeXMLMath -harmonic polynomial has degree = LaTeXMLMath [ J2 ] .
Although LaTeXMLMath is explicitly known for a number of examples , there is no known ( explicit ) general formula .
However , when LaTeXMLMath is a hypersurface orbital variety , we can use the following theorem to give an explicit formula for LaTeXMLMath .
This theorem is an immediate consequence of work by Joseph ( see [ J1 , 82 .
] , [ J2 , 2.9 ] ) .
A different proof is given by Borho , Brylinski and MacPherson [ BBM , 4.15 ] .
Theorem 4 : Let LaTeXMLMath be a complete intersection of codimension LaTeXMLMath in LaTeXMLMath , defined by homogeneous equations LaTeXMLMath , LaTeXMLMath of weights LaTeXMLMath .
Then LaTeXMLMath .
In our case , the formula for LaTeXMLMath , where LaTeXMLMath is the unique non-linear defining equation of LaTeXMLMath , is an easy consequence of Propositions 1 and 2 ( or the Conjecture , once proven ) .
Since LaTeXMLMath is homogeneous and LaTeXMLMath we have that LaTeXMLMath .
LaTeXMLMath is simply the determinant of the top right LaTeXMLMath minor in LaTeXMLMath and its weight is equal to the weight of any of its summands .
Therefore , LaTeXMLEquation and we get the desired formula .
Clearly this formula holds for LaTeXMLMath satisfying the assumptions of Propositions 1 and 2 .
Corollary 2 : Let LaTeXMLMath be a hypersurface orbital variety with LaTeXMLMath -invariant LaTeXMLMath , LaTeXMLMath -set LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
Example 13 : Let LaTeXMLMath be as in Examples 7 and 12 .
Then LaTeXMLEquation .
Therefore , LaTeXMLMath .
References [ Be ] E. Benlolo , Sur la quantification de certaines variétés orbitales .
Bull .
Sci .
Math .
118 ( 1994 ) , no .
3 , 225-243 .
[ BV ] D. Barbasch and D. Vogan , Primitive ideals and orbital integrals in complex classical groups , Math .
Ann .
259 ( 1982 ) 153-199 .
[ B ] W. Borho , Nilpotent orbits , primitive ideals , and characteristic classes ( a survey ) , Proceedings of the International Congress of Mathematicians , Vol .
1,2 ( Berkeley , Calif. , 1986 ) , Amer .
Math .
Soc , Providence , ( 1987 ) 350-359 .
[ BBM ] W. Borho , J-L Brylinski and R. MacPherson , Equivariant K-Theory Approach to Nilpotent Orbits , IHES/M/86/13 [ He ] W. Hesselink , Singularities in the nilpotent scheme of a classical group , Trans .
Am .
Math .
Soc .
222 ( 1976 ) 1 - 32 .
[ Ho ] A. Hotta .
On Joseph ’ s construction of Weyl group representations .
Tohuku Math .
J .
36 ( 1984 ) , 49–74 .
[ Ja ] J. C. Jantzen , Einhullende Algebren halbeinfacher Lie-Algebren , Ergebnisse der Mathematik und ihrer Grenzgebiete , 3 , Springer-Verlag , 1983 .
[ J1 ] A. Joseph , On the characteristic polynomials of orbital varieties , Ann .
scient .
Ec .
Norm .
Sup , 4eme série , t.22 ( 1989 ) , 569 - 603 .
[ J2 ] A. Joseph , On the Variety of a Highest Weight Module , J. Algebra , 88 , No .
1 ( 1984 ) , 238-278 .
[ J3 ] A. Joseph , Enveloping Algebras : Problems Old and New , 385-413 , Progress in Mathematics , 123 , Birkhauser , Boston , 1994 .
[ J4 ] A. Joseph , Orbital varietes of the minimal orbit .
Ann .
Sci .
École Norm .
Sup .
( 4 ) 31 ( 1998 ) , no .
1 , 17-45 .
[ Mc ] W. M. McGovern , Dixmier algebras and the orbit method .
Operator algebras , unitary representations , enveloping algebras , and invariant theory ( Paris , 1989 ) .
Progr .
Math , 92 , Birkhauser , Boston , 1990 .
[ M1 ] A. Melnikov , Orbital Varieties and Order Relations on Young Tableaux , ( 1995 ) preprint .
[ M2 ] A. Melnikov , Orbital Varieties in LaTeXMLMath and the Smith Conjecture , J. Algebra 200 ( 1998 ) 1 - 31 .
[ M3 ] A. Melnikov , Irreducibility of the associated varieties of simple highest weight modules in LaTeXMLMath .
C. R. Acad .
Sci .
Paris Sér .
I Math .
316 ( 1993 ) , no .
1 , 53-57 .
[ Sp ] N. Spaltenstein , Classes unipotentes de sous-groupes de Borel .
Lecture Notes in Mathematics , 964 Springer-Verlag , Berlin-New York , 1982 .
[ SS ] T. A. Springer , R. Steinberg , Conjugacy classes , Lecture Notes in Mathematics , 131 , Springer , 1970 , 167-266 .
[ St1 ] R. Steinberg , An Occurrence of the Robinson-Schensted Correspondence , J. Algebra , 113 , ( 1988 ) , 523-528 .
[ St2 ] R. Steinberg , On the desingularization of the unipotent variety , Invent .
Math .
, 36 ( 1976 ) , 209-224 .
[ vanL ] M. A .
A. van Leeuwen , The Robinson-Schensted and Schutzenberger algorithms , Part II : Geometric interpretations , CWI report AM-R9209 ( 1992 ) .
This report is also available electronically via http : //www.cwi.nl/cwi/publications/ # AM .
With a view towards future applications in nuclear physics , the fermion realization of the compact symplectic sp ( 4 ) algebra and its q-deformed versions are investigated .
Three important reduction chains of the sp ( 4 ) algebra are explored in both the classical and deformed cases .
The deformed realizations are based on distinct deformations of the fermion creation and annihilation operators .
For the primary reduction , the su ( 2 ) sub-structure can be interpreted as either the spin , isospin or angular momentum algebra , whereas for the other two reductions su ( 2 ) can be associated with pairing between fermions of the same type or pairing between two distinct fermion types .
Each reduction provides for a complete classification of the basis states .
The deformed induced u ( 2 ) representations are reducible in the action spaces of sp ( 4 ) and are decomposed into irreducible representations .
Symplectic algebras can be used to describe many-particle systems .
The compact , LaTeXMLMath and noncompact versions , LaTeXMLMath of the algebra enter naturally when the number of particles or couplings between the particles change in a pairwise fashion from one configuration to the next .
In this paper we consider the simplest nontrivial case : the compact LaTeXMLMath symplectic algebra which is isomorphic to the Lie algebra of the five-dimensional rotation group LaTeXMLMath LaTeXMLCite .
Applications of LaTeXMLMath are related to different interpretations of the quantum numbers of the fermions used to construct the generators of the LaTeXMLMath group .
Interest in symplectic groups is related to applications to nuclear structure LaTeXMLCite .
In particular , LaTeXMLMath has been used to explore pairing correlations in nuclei LaTeXMLCite .
The reduction chains to different realizations of the LaTeXMLMath subalgebra of LaTeXMLMath yield a complete classification scheme for the basis states .
It is rather easy to generalize this work to higher rank algebras and therefore the algebraic techniques are illustrated by the LaTeXMLMath example LaTeXMLCite .
A further interest in the symplectic algebras is related to their use in mapping methods from the fermion space to the space spanned by collective bosons and ideal fermions LaTeXMLCite .
In these applications the primary purpose is to simplify the Hamiltonian of the initial problem .
In the last decade a lot of effort , from a purely mathematical LaTeXMLCite as well as from the physical point of view LaTeXMLCite , has been concentrated on various deformations of the classical Lie algebras .
The general feature of these deformations is that at the limit of the deformation parameter LaTeXMLMath the LaTeXMLMath -algebra reverts back to the classical Lie algebra .
More than one deformation can be realized for one and the same “ classical ” algebra , which can be chosen in a convenient way in different physical applications .
There are many similarities between the classical Lie algebras and their deformations , especially with respect to their representation .
Deformed algebras introduce a new degree of freedom that can give a better explanation of non-linear effects .
Their study can lead to deeper understanding of the physical significance of the deformation .
In LaTeXMLCite a boson realization of the noncompact LaTeXMLMath and two distinct deformations of it , as well as compact and noncompact subalgebras of each , were investigated and reductions of their action spaces obtained .
As the fermion case has more direct application in nuclear theory than the boson construction , in this work our aim is to investigate in detail the fermion realization of the LaTeXMLMath algebra and its deformations .
Using the methodology from LaTeXMLCite , we begin with the well-known realization of this algebra in terms of “ classical ” fermion creation and annihilation operators and consider all the subalgebras which correspond to different ways of specifying labels of the basis states via the eigenvalues of the operators generating these subalgebras ( Section 2 ) .
Furthermore , we obtain the deformation of this LaTeXMLMath algebra and its subalgebras by introducing a transformation function that deforms the classical fermions into LaTeXMLMath -deformed fermions .
We also introduce another deformation in terms of the standard LaTeXMLMath -fermions and by following the same procedure we investigate the enveloping algebra of LaTeXMLMath and the action of its generators on the deformed basis ( Section 3 ) .
To establish the notation , recall some features of the fermion realization of the LaTeXMLMath algebra LaTeXMLCite , which is isomorphic to LaTeXMLMath LaTeXMLCite , as normally used in the shell-model studies .
The operator LaTeXMLMath creates ( LaTeXMLMath annihilates ) a particle of type LaTeXMLMath in a state of total angular momentum LaTeXMLMath with projection LaTeXMLMath along the LaTeXMLMath axis ( LaTeXMLMath ) .
These operators satisfy Fermi anticommutation relations : LaTeXMLEquation and Hermitian conjugation is given by LaTeXMLMath For a given LaTeXMLMath the dimension of the fermion space is LaTeXMLMath The fermion realization of LaTeXMLMath is given in a standard way by means of the following operators LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation These operators create ( annihilate ) a pair of fermions coupled to total angular momentum LaTeXMLMath LaTeXMLCite and thus constitute boson-like objects according to the Spin-Statistics theorem LaTeXMLCite when the operators LaTeXMLEquation preserve the number of fermions .
Here the normalization constants are LaTeXMLEquation .
The number of the operators LaTeXMLMath and LaTeXMLMath is ten LaTeXMLMath LaTeXMLMath .
Their commutation relations , obtained by means of ( LaTeXMLRef ) , show that these operators generate a fermion realization of the LaTeXMLMath algebra LaTeXMLCite .
An additional index LaTeXMLMath of the creation and annihilation fermion operators is introduced in order to construct non-zero operators LaTeXMLMath and LaTeXMLMath , but the index LaTeXMLMath defines the algebraic properties of the generators LaTeXMLMath and LaTeXMLMath Different interpretations of LaTeXMLMath correspond to different physical meanings for the operators generating the ten-parametric LaTeXMLMath group and therefore different physical models .
These can be used to describe various aspects of the nuclear interaction ( different Hamiltonians ) LaTeXMLCite like charge independent pairing , two level pairing ( Lipkin model ) or two dimensional rotations and vibrations .
The LaTeXMLMath algebra is considered to be the dynamical symmetry algebra in these applications .
Each of the limits is described by a reduction chain of the algebra which serves to label the basis states by eigenvalues of the invariant operators of the subalgebras and gives the corresponding limiting forms of the model Hamiltonian .
The investigation of the subalgebras of LaTeXMLMath contained in its reduction chains is given below .
By using the particle number preserving Weyl generators LaTeXMLMath ( LaTeXMLRef ) , a subalgebra LaTeXMLMath of LaTeXMLMath is realized by the operators : LaTeXMLEquation where LaTeXMLMath are the operators of the total number of fermions of each kind , LaTeXMLEquation .
The action of these operator on the fermion creation and annihilation operators is given by LaTeXMLEquation .
LaTeXMLEquation and the anticommutation relations ( LaTeXMLRef ) yield the equality LaTeXMLEquation .
The operators ( LaTeXMLRef ) satisfy the LaTeXMLMath commutation relations LaTeXMLEquation where LaTeXMLMath close on an algebra LaTeXMLMath that is isomorphic to LaTeXMLMath The operator LaTeXMLMath generates LaTeXMLMath and plays the role of the first order invariant of LaTeXMLMath .
The second order Casimir operator of LaTeXMLMath is given by LaTeXMLEquation and the second order invariant of LaTeXMLMath LaTeXMLCite is simply LaTeXMLEquation .
The algebra LaTeXMLMath plays a very important role in all kinds of different physical applications since it is of the standard spin type , which can be interpreted as spin , isospin or angular momentum in the various models .
Another unitary realization of LaTeXMLMath denoted by LaTeXMLMath is generated by LaTeXMLMath ( LaTeXMLRef ) and the operators LaTeXMLEquation with the following commutation relations : LaTeXMLEquation .
LaTeXMLEquation For this realization the operator LaTeXMLMath acts as a first order invariant of LaTeXMLMath , defining the reduction LaTeXMLMath The second order Casimir invariant of this subgroup is given as LaTeXMLEquation .
The generators of this LaTeXMLMath group are operators pairing particles of two different kinds .
Next , we consider two mutually complementary LaTeXMLMath subalgebras of the algebra LaTeXMLMath denoted by LaTeXMLMath and LaTeXMLMath .
These algebras are generated by the operators LaTeXMLEquation with the following commutation relations : LaTeXMLEquation .
It is simple to see that each of the generators of LaTeXMLMath commutes with all of the LaTeXMLMath generators .
The second order Casimir operators of the LaTeXMLMath are LaTeXMLEquation .
In this case the addition of the operators LaTeXMLMath considered to be generators of the subgroups LaTeXMLMath , extend LaTeXMLMath to LaTeXMLMath LaTeXMLMath LaTeXMLMath .
LaTeXMLMath act as first order Casimir operators of LaTeXMLMath The operators closing the two mutually complementary subalgebras describe pairs of particles of the same kind .
The sum of the generators of the groups LaTeXMLMath and LaTeXMLMath give rise to another unitary realization of LaTeXMLMath subalgebra of LaTeXMLMath denoted by LaTeXMLMath , LaTeXMLEquation with the following commutation relations : LaTeXMLEquation and the second order Casimir invariant LaTeXMLEquation .
In general , the classical fermion operators act in a finite space LaTeXMLMath for a particular LaTeXMLMath -level .
The finite representation is due to the Pauli principle , LaTeXMLMath that allows no more than LaTeXMLMath identical fermions in a single LaTeXMLMath -shell .
In LaTeXMLMath the vacuum LaTeXMLMath is defined by LaTeXMLMath LaTeXMLMath and the scalar product is chosen so that LaTeXMLMath The states that span the LaTeXMLMath spaces consist of different numbers of fermion creation operators acting on the vacuum state .
These satisfy the Pauli principle through their anti-commutation relations ( LaTeXMLRef ) .
They form an orthonormal basis in each space and are the common eigenvectors of the fermion number operators LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath and LaTeXMLMath In this way , they span two subspaces LaTeXMLMath labeled by the eigenvalue of the invariant operator LaTeXMLMath of LaTeXMLMath .
Here we are interested in the even space LaTeXMLMath containing states of coupled fermions , in order to apply the theory to the phenomena like pairing correlations in nuclei .
If we introduce LaTeXMLEquation for operators creating ( LaTeXMLRef ) and annihilating ( LaTeXMLRef ) a pair of particles , it is easy to check that they are components of two conjugated vectors LaTeXMLMath and LaTeXMLMath , LaTeXMLMath with respect to the subgroup LaTeXMLMath ( LaTeXMLRef , LaTeXMLRef ) : LaTeXMLEquation .
In the models where LaTeXMLMath is interpreted as the isospin operator , LaTeXMLMath LaTeXMLMath create ( destroy ) a pair of fermions coupled to a total isospin LaTeXMLMath .
Thus , a linearly independent set of vectors that span the LaTeXMLMath space can be expressed in terms of the ‘ boson creation operators ’ acting on the vacuum state , LaTeXMLEquation .
The basis is obtained by orthonormalization of ( LaTeXMLRef ) .
The operators LaTeXMLMath commute among themselves and therefore form a symmetric representation .
The eigenvalue of the second order Casimir operator LaTeXMLMath labels each representation of LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation The group LaTeXMLMath is of rank two and thus there exist two invariant operators that commute with all the generators of the group LaTeXMLCite .
The other invariant operator is of fourth order and it is linearly dependent on the Casimir operator for this group .
Usually representations of LaTeXMLMath are labeled by the largest eigenvalue of the number operator LaTeXMLMath and the reduced isospin of the uncoupled fermions in the corresponding state LaTeXMLCite .
In each representation of LaTeXMLMath in the vector space spanned over ( LaTeXMLRef ) , the maximum number of particles is LaTeXMLMath and the respective state consists of no uncoupled fermions ( reduced isospin zero ) .
It follows that only one quantum number is needed , LaTeXMLMath Within a representation , LaTeXMLMath is dropped from the labelling of the states .
Another consequence of the symmetric representation is that the vector space consists of states of a system with total angular momentum LaTeXMLMath .
Each representation labelled by LaTeXMLMath is finite , because of the fermion structure of the operators LaTeXMLMath : LaTeXMLMath or LaTeXMLMath Another consequence of the fermion realization is that some of the vectors ( LaTeXMLRef ) of the finite space LaTeXMLMath are linearly dependent , for example LaTeXMLMath The states ( LaTeXMLRef ) are the common eigenvectors of the fermion number operators LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation or of the operators LaTeXMLMath LaTeXMLMath and LaTeXMLMath LaTeXMLMath which are both diagonal in the basis ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation Their eigenvalues can be used to classify the basis within a representation LaTeXMLMath .
The basis states labeled by LaTeXMLMath for LaTeXMLMath are shown in Table 1 , where LaTeXMLMath enumerates the rows and LaTeXMLMath the columns .
LaTeXMLEquation .
The basis vectors are degenerate in the sense that more than one of the common eigenstates of the operators LaTeXMLMath and LaTeXMLMath have one and the same eigenvalues LaTeXMLMath LaTeXMLMath and thus belong to one and the same cell of Table 1 .
They can be distinguished as eigenstates of the Casimir operators of the limiting cases : The basis states can be labeled by the eigenvalues of the invariant operator of each subgroup in the reduction LaTeXMLMath As a first order invariant of LaTeXMLMath , the operator LaTeXMLMath decomposes the spaces LaTeXMLMath into a direct sum of eigensubspaces , defined by the condition that LaTeXMLMath is fixed ( LaTeXMLRef ) , LaTeXMLEquation .
So an irreducible unitary representation ( LaTeXMLMath ) of LaTeXMLMath is realized in each row of Table 1 .
The LaTeXMLMath subgroup provides the other two quantum numbers as a standard label of the basis vectors .
First , it is the eigenvalue of the Casimir operator of second rank in LaTeXMLMath LaTeXMLEquation where LaTeXMLMath LaTeXMLMath ( odd ) or LaTeXMLMath ( even ) LaTeXMLMath LaTeXMLMath , and second it is the eigenvalue of LaTeXMLMath ( LaTeXMLRef ) , where LaTeXMLMath As an example , the orthonormalized basis LaTeXMLMath given in terms of the states LaTeXMLMath is shown in Table 2 for LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLMath In general , the state with the maximum number of particles always has a total isospin zero , LaTeXMLMath , and all the possible states expressed in the basis LaTeXMLMath are equivalent within a normalization factor .
The raising ( lowering ) generators LaTeXMLMath acting LaTeXMLMath times on the lowest LaTeXMLMath ( highest LaTeXMLMath ) weight state give all the basis states of the respective LaTeXMLMath -representation according to the result LaTeXMLEquation .
The reduction chain LaTeXMLMath introduces another labelling scheme for the basis states , namely , LaTeXMLMath The quantum numbers that specify the states are the eigenvalue LaTeXMLMath of LaTeXMLMath LaTeXMLMath LaTeXMLMath the seniority quantum number LaTeXMLMath , and the eigenvalue of the operator LaTeXMLMath ( LaTeXMLRef ) LaTeXMLMath The first invariant of LaTeXMLMath decomposes the spaces LaTeXMLMath into a direct sum of eigensubspaces of the operator LaTeXMLMath at each of its fixed values ( LaTeXMLRef ) .
These subspaces are represented by the columns of Table 1 .
The operator LaTeXMLMath ( LaTeXMLRef ) does not differ essentially from the first invariant operator LaTeXMLMath of LaTeXMLMath and it further reduces the columns of Table 1 to the cells .
The seniority quantum number differs between two states of one and the same LaTeXMLMath and LaTeXMLMath but different coupling scheme , and it is introduced by the eigenvalues of the second Casimir operator for this subgroup : LaTeXMLEquation .
This is a scheme for coupling particles of the two different kinds LaTeXMLMath LaTeXMLMath with LaTeXMLMath or LaTeXMLMath or both LaTeXMLMath These states are the last ones in each of the cells in the Table 1 .
The additional quantum number , LaTeXMLMath , is the maximum number of the remaining pairs coupled as LaTeXMLMath LaTeXMLMath or LaTeXMLMath LaTeXMLMath In that limit , the Casimir operator can be expressed in terms of the eigenvalue of the first order invariant of LaTeXMLMath LaTeXMLEquation .
The raising and lowering generators of the subgroup LaTeXMLMath act along the columns in the following way : LaTeXMLEquation .
LaTeXMLEquation In each column LaTeXMLMath , LaTeXMLMath starts from the lowest LaTeXMLMath weight state LaTeXMLMath or LaTeXMLMath LaTeXMLMath and gives all the basis states within a LaTeXMLMath -representation with LaTeXMLMath .
Similarly , LaTeXMLMath gives the basis states of the representation of the subgroup under consideration , starting with the highest weight state LaTeXMLMath for each LaTeXMLMath The normalized basis states , LaTeXMLEquation can be derived from ( LaTeXMLRef ) .
For the three types of states in this reduction , the normalization coefficients are given by LaTeXMLEquation where the lowest weight state LaTeXMLMath and LaTeXMLMath in each representation can be normalized recursively , LaTeXMLEquation .
The other reduction is described again by the invariants of the subgroups in the reduction chain : LaTeXMLMath Here the labeling is LaTeXMLMath .
First the spaces LaTeXMLMath are decomposed by means of the first order invariants LaTeXMLMath of the respective subalgebras to the subspaces defined by the conditions LaTeXMLMath LaTeXMLMath and represented by the left ( right ) diagonals in Table 1 .
The action of the Casimir operator on the states LaTeXMLEquation provides the LaTeXMLMath quantum number LaTeXMLMath The seniority quantum number LaTeXMLMath is the maximum number of remaining pairs that can be formed by coupling particles of different types , here LaTeXMLMath or LaTeXMLMath The basis states are of the form LaTeXMLMath and LaTeXMLMath and they are placed first in each cell in Table1 .
Furthermore , the operators LaTeXMLMath ( LaTeXMLRef ) , which are equivalent within constants to the operators LaTeXMLMath , give the respective projection of LaTeXMLMath : LaTeXMLMath .
The diagonals are decomposed to the cells belonging to them and defined by the conditions LaTeXMLMath The raising and lowering generators of LaTeXMLMath act along the left/right diagonals : LaTeXMLEquation .
LaTeXMLEquation Starting from the respective lowest or highest weight states , they generate all the states belonging to LaTeXMLMath s of the LaTeXMLMath subgroups of LaTeXMLMath The normalized basis states , LaTeXMLEquation can be derived from ( LaTeXMLRef ) .
For the two types of states LaTeXMLMath or LaTeXMLMath in this reduction , the normalization coefficients are given by LaTeXMLEquation where the results ( LaTeXMLRef ) are consistent with ( LaTeXMLRef ) for LaTeXMLMath and with the lowest weight state in each representation ( LaTeXMLMath ) normalized recursively : LaTeXMLEquation .
Consider LaTeXMLMath -deformed creation and annihilation operators LaTeXMLMath and LaTeXMLMath for a particle of type LaTeXMLMath in a state of a total angular momentum LaTeXMLMath with projection LaTeXMLMath on the LaTeXMLMath axis .
The Hermitian conjugation relation is defined as LaTeXMLMath .
There is a general class of functions , which transform the classical operators into deformed ones LaTeXMLCite .
We use the transformation LaTeXMLEquation where LaTeXMLMath is a complex number with amplitude LaTeXMLMath LaTeXMLMath a real number , and LaTeXMLMath are the classical number operators .
The transformation of ( LaTeXMLRef ) leads to the anticommutation relations for the LaTeXMLMath -deformed fermion operators , LaTeXMLEquation and the identities LaTeXMLEquation .
LaTeXMLEquation The raising and lowering generators of the respective deformed LaTeXMLMath group are given as in the classical case ( LaTeXMLRef - LaTeXMLRef ) but in terms of the LaTeXMLMath -deformed fermion operators : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation where the constants are defined in ( LaTeXMLRef ) .
The operator LaTeXMLMath LaTeXMLMath creates ( destroys ) a LaTeXMLMath -deformed pair of particles of the same kind .
The remaining two Cartan generators LaTeXMLMath used in the deformed commutation relations ( LaTeXMLRef ) , are not deformed .
The transformation ( LaTeXMLRef ) yields the following relations between the deformed ( LaTeXMLRef - LaTeXMLRef ) and the classical operators ( LaTeXMLRef - LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
Since there is a smooth transformation that depends on the Cartan generators of LaTeXMLMath only and maps the LaTeXMLMath -deformed operators LaTeXMLEquation .
LaTeXMLEquation to the classical vectors LaTeXMLMath the LaTeXMLMath -deformed states are equivalent within a phase to the classical ones ( LaTeXMLRef ) LaTeXMLMath All the relations revert back to the classical formulae in the limit LaTeXMLMath .
The important reduction of LaTeXMLMath algebra to compact LaTeXMLMath subalgebra can be used again to obtain classification schemes for the basis states .
The subalgebra LaTeXMLMath of LaTeXMLMath is closed by the number preserving Weyl generators ( LaTeXMLRef ) and LaTeXMLMath defined as : LaTeXMLEquation .
The generators LaTeXMLMath and LaTeXMLMath satisfy the commutation relations : LaTeXMLEquation and the second invariant of LaTeXMLMath is LaTeXMLEquation .
The LaTeXMLMath -deformed operator LaTeXMLMath is defined by LaTeXMLEquation and is related to the classical Casimir operator of LaTeXMLMath ( LaTeXMLRef ) by LaTeXMLEquation .
Thus , the eigenvalues of the Casimir operator are deformed by a phase factor LaTeXMLMath and the eigenvectors are the classical basis states , LaTeXMLMath .
The other subgroup LaTeXMLMath is generated by the operators : LaTeXMLEquation and LaTeXMLMath ( LaTeXMLRef ) , which is the first order invariant LaTeXMLMath The generators of LaTeXMLMath commute in the following way : LaTeXMLEquation where the LaTeXMLMath -commutator is defined as LaTeXMLEquation .
The second order Casimir invariant of LaTeXMLMath is given by LaTeXMLEquation .
The two mutually complementary subalgebras LaTeXMLMath and LaTeXMLMath of the algebra LaTeXMLMath are given by the LaTeXMLMath -deformed operators LaTeXMLEquation and the non-deformed Cartan operators LaTeXMLEquation .
According to the reduction chain LaTeXMLMath , LaTeXMLMath commute with the operators LaTeXMLMath LaTeXMLMath , which close the LaTeXMLMath algebra : LaTeXMLEquation .
LaTeXMLEquation The corresponding Casimir invariant is LaTeXMLEquation .
A similar LaTeXMLMath -deformation is based on the transformation LaTeXMLMath which yields the same relations and identities as above , but with the exchange LaTeXMLMath When LaTeXMLMath is real and positive the deformation parameter is LaTeXMLMath .
Consider another set of LaTeXMLMath -deformed Hermitian conjugate operators LaTeXMLMath and LaTeXMLMath Let the LaTeXMLMath -deformed anticommutation relation holds for every LaTeXMLMath and LaTeXMLMath in the form LaTeXMLCite : LaTeXMLEquation where LaTeXMLMath LaTeXMLMath and LaTeXMLMath are the classical number operators ( LaTeXMLRef ) .
Their action on the deformed fermion operators is defined as in the classical case ( LaTeXMLRef ) : LaTeXMLEquation .
In the previous section we showed that if the transformation function ( LaTeXMLRef ) is used , the anticommutation relations of the deformed fermion operators ( LaTeXMLRef ) depend not only on a single term LaTeXMLMath as in ( LaTeXMLRef ) but rather on the total sum LaTeXMLMath The same dependence , along with the requirement that the deformation is performed only on the LaTeXMLMath index , defines : LaTeXMLEquation .
Using both anticommutation relations , it follows that LaTeXMLMath where LaTeXMLMath which leads to LaTeXMLEquation and LaTeXMLEquation .
In the limit LaTeXMLMath presuming LaTeXMLMath LaTeXMLMath as well , ( LaTeXMLRef , LaTeXMLRef ) revert back to the classical formulas for LaTeXMLMath ( LaTeXMLRef , LaTeXMLRef ) .
This justifies the introduction of the weight coefficient LaTeXMLMath in ( LaTeXMLRef ) .
The remaining anticommutation relations for the LaTeXMLMath -deformed operators can be chosen from among various possibilities LaTeXMLCite : LaTeXMLEquation where the LaTeXMLMath -anticommutator is given by LaTeXMLMath The set of generators for this realization of the deformed LaTeXMLMath algebra is defined as in ( LaTeXMLRef - LaTeXMLRef ) , but in terms of the LaTeXMLMath -deformed creation and annihilation operators LaTeXMLMath LaTeXMLMath fulfilling anticommutation relations ( LaTeXMLRef ) .
The Cartan generators LaTeXMLMath remain the classical number operators .
These ten operators generate the LaTeXMLMath -deformed LaTeXMLMath group and its subgroup structure is investigated in analogy with the classical case .
The subgroup LaTeXMLMath of LaTeXMLMath is generated by the number preserving Weyl operators ( LaTeXMLRef ) and LaTeXMLMath as well as by the equivalent set of the operators LaTeXMLMath and LaTeXMLMath ( LaTeXMLRef ) .
These operators satisfy the commutation relations : LaTeXMLEquation .
The operators LaTeXMLMath close an algebra LaTeXMLMath .
The number operator LaTeXMLMath plays the role of the first order invariant of LaTeXMLMath .
The second order Casimir operator of the subgroup LaTeXMLMath is : LaTeXMLEquation .
Here LaTeXMLMath and the following identity has been used : LaTeXMLEquation .
The Casimir operator coincides with the classical one in the limit LaTeXMLMath ( LaTeXMLRef ) .
The other LaTeXMLMath subalgebra is : LaTeXMLEquation where the generators are defined in ( LaTeXMLRef ) .
The operator LaTeXMLMath ( LaTeXMLRef ) commutes with the generators of LaTeXMLMath ( LaTeXMLRef ) and acts as a first order invariant of LaTeXMLMath The operators LaTeXMLMath couple LaTeXMLMath -deformed particles of two different kinds .
The second order Casimir operator of the subgroup LaTeXMLMath is given by LaTeXMLEquation which coincides with the classical invariant LaTeXMLMath LaTeXMLRef LaTeXMLMath in the limit LaTeXMLMath The two mutually complementary subalgebras LaTeXMLMath and LaTeXMLMath of the algebra LaTeXMLMath are given by the LaTeXMLMath -deformed operators ( LaTeXMLRef ) and the non-deformed Cartan operators ( LaTeXMLRef ) .
They have the following commutation relations : LaTeXMLEquation .
LaTeXMLEquation with LaTeXMLMath It is again true that each of the generators LaTeXMLMath of LaTeXMLMath commutes with all the generators of the other LaTeXMLMath subgroup LaTeXMLMath .
The first order invariants LaTeXMLMath of LaTeXMLMath give the extension of LaTeXMLMath to the subgroup LaTeXMLMath LaTeXMLMath The operator LaTeXMLMath LaTeXMLMath creates ( destroys ) a LaTeXMLMath -deformed pair of particles of the same kind .
The Casimir invariant of the subgroup LaTeXMLMath is : LaTeXMLEquation .
The useful identity ( LaTeXMLRef ) now has the form LaTeXMLEquation .
The Casimir operator coincides with the classical one ( LaTeXMLRef ) in the limit LaTeXMLMath .
The LaTeXMLMath -deformed symplectic algebra reverts back to the classical limit for the rest of the commutation relations between its generators ( LaTeXMLRef ) : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where the constants are defined as follows : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Another set of the same commutation relations can be obtained , which is symmetric with respect to the exchange LaTeXMLMath LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where the functions LaTeXMLMath are defined in the following way : LaTeXMLEquation .
The realization of LaTeXMLMath introduced here is consistent with the algebra of the Chevalley generators of LaTeXMLMath which is given in LaTeXMLCite .
The comparison of the commutation relations yields for the first triplet of the generators corresponding to the long root 1 : LaTeXMLEquation and for the short root 2 : LaTeXMLEquation .
The renormalization of the generators of the second triplet is introduced so that ( LaTeXMLRef ) can be written in the standard LaTeXMLMath form : LaTeXMLEquation .
The rest of the commutation relations of both triplets are consistent within the parameter LaTeXMLMath Comparing ( LaTeXMLRef ) with the other four generators , we obtain LaTeXMLEquation and LaTeXMLEquation which are determined up to an overall multiplicative constant factor .
The results prove the isomorphism of the LaTeXMLMath -fermion realization of LaTeXMLMath and all possible representations of its standard LaTeXMLMath subgroup to the triplets of the Chevalley generators associated with the shorter and longer roots of LaTeXMLMath .
In general , the LaTeXMLMath -deformed fermion operators ( LaTeXMLRef ) act as in the classical case in a finite metric space LaTeXMLMath for each particular LaTeXMLMath -level , with a vacuum LaTeXMLMath defined by LaTeXMLMath LaTeXMLMath The scalar product in LaTeXMLMath is chosen in such a way that LaTeXMLMath is a Hermitian conjugate to LaTeXMLMath and LaTeXMLMath In general the LaTeXMLMath -deformed states are different from the classical ones , but reduce to the classical ones in the limit LaTeXMLMath The LaTeXMLMath -deformed creation ( annihilation ) operators LaTeXMLMath ( LaTeXMLRef ) are components of a tensor of rank LaTeXMLMath with respect to the subgroup LaTeXMLMath ( LaTeXMLRef ) .
These operators create a pair of LaTeXMLMath -fermions coupled to a total angular momentum LaTeXMLMath and a total isospin LaTeXMLMath .
Analogous to the classical limit , a set of vectors that span each space LaTeXMLMath in the LaTeXMLMath -deformed case can be chosen to be of the form LaTeXMLEquation .
The basis is obtained by orthonormalization of ( LaTeXMLRef ) .
The index LaTeXMLMath will be dropped from the notation for the basis states in the following cases which treat only the deformed space .
As in the classical case , LaTeXMLMath labels the representation for each particular LaTeXMLMath -shell .
The basis states are uniquely specified by the classification schemes which use the LaTeXMLMath subalgebras and the relevant Cartan generators .
In the LaTeXMLMath -deformed case the Cartan generators of LaTeXMLMath can be chosen to be the nondeformed operators LaTeXMLMath or their equivalent set of operators LaTeXMLMath and LaTeXMLMath LaTeXMLMath ( LaTeXMLRef ) .
The eigenvalues of these operators that label the basis states coincide with the ones in the classical case and the example of Table 1 can still be used .
The quantum numbers provided by the eigenvalues of the LaTeXMLMath -deformed Casimir invariants have to be taken in the limit LaTeXMLMath We briefly list the reduction chains and compare them to their classical counterparts in order to emphasize the similarity and differences between them .
The basis states together with the second order Casimir operators and their eigenvalues are often used in the physical applications .
It is in this sense that their LaTeXMLMath -deformation may lead to some interesting new results .
In the limit LaTeXMLMath the second order Casimir operator , LaTeXMLMath , of the LaTeXMLMath subgroup has the eigenvalues : LaTeXMLEquation where LaTeXMLMath LaTeXMLMath ( odd ) or LaTeXMLMath ( even ) , where LaTeXMLMath and LaTeXMLMath In the deformed case the eigenvalues of LaTeXMLMath for the lowest and the highest weight states ( LaTeXMLRef ) are LaTeXMLEquation .
The reduction chain LaTeXMLMath describes pairing between fermions of different types and introduces the seniority quantum number LaTeXMLMath in the labelling scheme for the basis states , LaTeXMLMath The eigenvalue of the second order Casimir operator for this LaTeXMLMath -deformed subalgebra is given by LaTeXMLEquation .
Here again , the generators of the subalgebra LaTeXMLMath act along the columns : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The normalized basis states , LaTeXMLEquation can be derived from ( LaTeXMLRef ) .
For the three types of states in this reduction , the normalization coefficients are : LaTeXMLEquation where the LaTeXMLMath -deformed factorial is defined by LaTeXMLMath The normalization coefficients LaTeXMLMath of the lowest weight state LaTeXMLMath and LaTeXMLMath in each representation is derived by means of the generators of the next reduction .
The other reduction LaTeXMLMath introduces deformation in the model of coupled fermions of the same kind .
Here the labeling is LaTeXMLMath , where LaTeXMLMath or LaTeXMLMath is the seniority quantum number .
The action of the Casimir operator on the states is given by LaTeXMLEquation .
In the deformed case the action of the Casimir invariant of LaTeXMLMath differs from that of the Casimir invariant of LaTeXMLMath by the factor LaTeXMLMath The generators of LaTeXMLMath transform the states along the diagonals as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The normalized basis states , LaTeXMLEquation can be derived from ( LaTeXMLRef ) .
For the two types of states LaTeXMLMath or LaTeXMLMath in this reduction , the normalization coefficients are LaTeXMLEquation where for LaTeXMLMath and LaTeXMLMath it follows that LaTeXMLEquation .
It is important to emphasize that this deformation may lead to basis states whose content is very different from the classical case since there is no known simple function that transforms the classical fermion operators LaTeXMLMath and LaTeXMLMath into the LaTeXMLMath -deformed ones LaTeXMLMath and LaTeXMLMath .
Smooth function may not exist when the anticommutation relations ( LaTeXMLRef ) hold simultaneously with both signs for one and the same LaTeXMLMath as they are defined in ( LaTeXMLRef ) .
The deformed basis states are labeled by the classical eigenvalues of the invariant operators of the reduction along each of the cases considered .
The matrix elements , particularly of the raising and lowering generators of LaTeXMLMath and the second order invariants , are also deformed which leads to different results in physical applications .
After obtaining the correspondence between the LaTeXMLMath -fermion realization of LaTeXMLMath and the Chevalley generators of LaTeXMLMath we can compare the two bases for an irreducible representation LaTeXMLMath , which corresponds to the representation LaTeXMLMath at LaTeXMLMath LaTeXMLCite .
In the classical and in the LaTeXMLMath -deformed cases , the first basis considered in LaTeXMLCite is related to the basis states ( LaTeXMLRef ) and ( LaTeXMLRef ) .
In this paper we consider a fermion realization of the LaTeXMLMath algebra and its deformations .
The original algebra , as well as some of its deformed realizations , act in the same finite space LaTeXMLMath The finiteness of the representations is due to the Pauli principle .
The deformed realization of LaTeXMLMath is based on the standard LaTeXMLMath -deformation of the two component Clifford algebra LaTeXMLCite , realized in terms of creation and annihilation fermion operators .
For the LaTeXMLMath case eight of the ten generators are deformed , the fermion number operators LaTeXMLMath and LaTeXMLMath and their linear combinations being the exceptions .
The deformed generators of LaTeXMLMath close different realizations of the compact LaTeXMLMath subalgebra .
The induced representations of each LaTeXMLMath are reducible in the space LaTeXMLMath and decompose into irreducible representations .
In this way we obtain a full description of the irreducible unitary representations of LaTeXMLMath of four different realizations of LaTeXMLMath : LaTeXMLMath and LaTeXMLMath Each reduction into compact subalgebras of LaTeXMLMath and its deformations affords the possibility of a description of a different physical model with different dynamical symmetries .
While within a particular deformation scheme the basis states may either be deformed or not , the generators are always deformed as is their action on basis states .
With a view towards applications , the additional parameter of the deformation gives a richer variety of operators associated with observables , nondeformed as well as deformed .
In a Hamiltonian theory this implies a dependance of the matrix elements on the deformation parameter , leading to the possibility of greater flexibility and richer structures within the framework of algebraic descriptions .
MSC Classification ( 1991 ) : 57 M 50 , 57 M 10 , 57 M 30 .
Keywords and phrases : cobordism , LaTeXMLMath -admissible immersion , extension , cocycle , surgery .
MSC Classification ( 1991 ) : 57 M 50 , 57 M 10 , 57 M 30 .
Keywords and phrases : cobordism , LaTeXMLMath -admissible immersion , extension , cocycle , surgery .
The classification of manifold immersions in codimension at least one up to cobordism was reduced to a homotopy problem by the results of LaTeXMLCite .
These techniques are however awkward to apply if one wants to get effective results .
The classification up to regular homotopy is also a homotopy problem which is closely related to the previous one .
For instance two immersed surfaces in LaTeXMLMath of the same topological type are regularly homotopic if and only if they are cobordant ( see LaTeXMLCite ) .
This subject received recently more attention ( see e.g .
LaTeXMLCite ) .
The group LaTeXMLMath of codimension-one immersions in the LaTeXMLMath -sphere up to cobordism is the the LaTeXMLMath -th stable homotopy group of LaTeXMLMath and they were computed by Liulevicius ( LaTeXMLCite ) for LaTeXMLMath .
Explicit classifications for regular homotopy equivalence of immersed surfaces in 3-manifolds were first obtained by Hass and Hughes in LaTeXMLCite and Pinkall ( see LaTeXMLCite ) .
The cobordism group LaTeXMLMath of immersed surfaces in the 3-manifold LaTeXMLMath was computed geometrically by Benedetti and Silhol ( LaTeXMLCite ) .
Let LaTeXMLMath be a compact oriented 3-manifold and LaTeXMLMath a smooth codimension-one immersion of a ( compact ) surface LaTeXMLMath in LaTeXMLMath .
Fixing a Spin structure on LaTeXMLMath one has a Pin structure induced on LaTeXMLMath which defines a LaTeXMLMath -valued quadratic form on LaTeXMLMath , by counting how the immersion LaTeXMLMath twists the regular neighborhoods of 1-cycles in LaTeXMLMath .
There is then an isomorphism between LaTeXMLMath and LaTeXMLMath , the last being endowed with a twisted product .
The isomorphism sends an immersion LaTeXMLMath into the triple consisting of the homology class of the double points locus , the homology class of the image of LaTeXMLMath , and the Arf invariant of the quadratic form from above .
A similar result holds for nonorientable 3-manifolds LaTeXMLMath , but the factor LaTeXMLMath is replaced now by LaTeXMLMath ( see LaTeXMLCite ) .
Notice that the factor LaTeXMLMath is nothing but LaTeXMLMath so the two results above can be stated in a unitary way by considering LaTeXMLMath .
One would like to have a similar description for the group LaTeXMLMath in all dimensions LaTeXMLMath .
More motivation for that is the result of LaTeXMLCite , which relates the cobordism group LaTeXMLMath to the set LaTeXMLMath of cubulations of the manifold LaTeXMLMath modulo a set of combinatorial moves analogous to Pachner ’ s move on simplicial complexes .
We refer to LaTeXMLCite for an extensive discussion of this problem , due to Habegger ( see problem 5.13 from LaTeXMLCite ) .
One remarks first that there is a natural grading LaTeXMLMath on LaTeXMLMath induced by a cellular decomposition .
The Atiyah-Hirzebruch spectral sequence ( see LaTeXMLCite ) has its second term LaTeXMLMath and converges to the graded LaTeXMLMath .
However one has only very few informations about the differentials in this sequence , hence the direct use of this approach fails .
One develops then a combinatorial way to settle this question .
One obtains thus the extension of the computations of the graded group LaTeXMLMath up to dimension LaTeXMLMath , and up to dimension 7 under a mild homological condition .
However the techniques one uses are different from those of Benedetti and Silhol , though as they are still geometric in nature .
The main theorem is the following : Let LaTeXMLMath be a closed LaTeXMLMath -manifold , LaTeXMLMath .
Then LaTeXMLEquation .
The subgroup LaTeXMLMath is defined in section LaTeXMLRef and is computed for any LaTeXMLMath , and in some cases for LaTeXMLMath , as follows : LaTeXMLEquation where the condition LaTeXMLMath for LaTeXMLMath -manifolds with LaTeXMLMath is that LaTeXMLMath is orientable and LaTeXMLMath .
The whole theory is explicitly developed for closed manifolds .
However the present methods can be applied to non-compact manifolds simply by substituting the ordinary cohomology with the cohomology with compact support .
A sketch of proof .
We briefly summarize the guiding line of the paper .
We introduce in section LaTeXMLRef the natural filtration of LaTeXMLMath that gives rise to the graded group , and prove that LaTeXMLMath can be interpreted geometrically as the subgroup of LaTeXMLMath of those immersions avoiding the LaTeXMLMath -skeleton up to cobordism .
This property is independent of the cellular decomposition .
In section LaTeXMLRef we define in a geometric way an injective homomorphism LaTeXMLEquation where LaTeXMLMath is a subgroup of cocycles related to some particular null-cobordant immersions .
We then introduce an obstruction theory that permits to study the inverse map to LaTeXMLMath , in particular to determine its image .
In section LaTeXMLRef the theory is applied to explicit computations that provide the image of LaTeXMLMath up to LaTeXMLMath , and in section LaTeXMLRef it is proven that LaTeXMLMath for all LaTeXMLMath in all LaTeXMLMath -manifold LaTeXMLMath up to LaTeXMLMath .
These computations prove the main theorem .
Acknowledgements .
We are indebted to Riccardo Benedetti , Octav Cornea , Takuji Kashiwabara , Valentin Poénaru , John Scott-Carter and Pierre Vogel for valuable discussions and suggestions .
Part of this work has been done when the first author visited Tokyo Institute of Technology , which he wishes to thank for the support and hospitality , and especially to Teruaki Kitano and Tomoyoshi Yoshida .
Let LaTeXMLMath be a LaTeXMLMath -dimensional manifold .
Consider the set of immersions LaTeXMLMath with LaTeXMLMath a closed LaTeXMLMath -manifold .
Impose on it the following relation : LaTeXMLMath is cobordant to LaTeXMLMath if there exist a cobordism LaTeXMLMath between LaTeXMLMath and LaTeXMLMath , that is , a compact LaTeXMLMath -manifold LaTeXMLMath with boundary LaTeXMLMath , and an immersion LaTeXMLMath , transverse to the boundary , such that LaTeXMLMath and LaTeXMLMath .
Once the manifold LaTeXMLMath is fixed , the set LaTeXMLMath of cobordism classes of codimension-one immersions in LaTeXMLMath is an abelian group with the composition law given by disjoint union .
In this paper we mainly deal with the cobordism group of immersions in manifolds of dimension less or equal than 7 .
This is due to the fact that the groups LaTeXMLMath , which are always finite , 2-torsion groups ( see LaTeXMLCite ) , are particularly simple for LaTeXMLMath , as is shown in the following table ( see LaTeXMLCite ) : The simplest case is LaTeXMLMath , but the cases LaTeXMLMath are also very easy to handle .
Indeed consider the classical invariant LaTeXMLEquation that associates to an immersion the number of LaTeXMLMath -tuple points modulo 2 , or equivalently , the homology class modulo 2 of the set of LaTeXMLMath -tuple points ( as an element of LaTeXMLMath ) .
It is well-known that LaTeXMLMath is an isomorphism for LaTeXMLMath ( see for example LaTeXMLCite ) .
The group LaTeXMLMath is generated by the immersion 8 which looks like the figure eight in the plane , while the group LaTeXMLMath is generated by an immersion of LaTeXMLMath with a single quadruple point .
By definition of cobordism an immersion LaTeXMLMath of a compact LaTeXMLMath -manifold in LaTeXMLMath represents the trivial element of LaTeXMLMath if and only if there exists an immersion of a compact LaTeXMLMath -manifold with boundary in LaTeXMLMath transverse to the boundary .
It is clear that this is equivalent to ask that LaTeXMLMath bounds in LaTeXMLMath .
We generalize slightly this condition .
Denote by LaTeXMLMath the subgroup of LaTeXMLMath of those immersions in LaTeXMLMath bounding an immersion of a compact LaTeXMLMath -manifold with boundary in a LaTeXMLMath -manifold with boundary LaTeXMLMath .
The reason why LaTeXMLMath being an isomorphism makes computations easier also when dealing with immersions in manifolds that are not spheres amounts to the following easy proposition , lying on the elementary but fundamental fact that compact 1-manifolds with boundary have an even number of points as boundary .
Let LaTeXMLMath be a a codimension-one immersion , and let LaTeXMLMath be such that either LaTeXMLMath is an isomorphism or LaTeXMLMath is trivial .
Then LaTeXMLMath bounds an immersion in a LaTeXMLMath -manifold whose boundary is LaTeXMLMath if and only if it bounds in LaTeXMLMath , that is LaTeXMLMath .
Proof .
If LaTeXMLMath is such that LaTeXMLMath there is nothing to prove .
Assume then that LaTeXMLMath is an isomorphism and let LaTeXMLMath be a immersion in LaTeXMLMath .
Suppose that there exists a LaTeXMLMath -manifold LaTeXMLMath with boundary LaTeXMLMath and a generic codimension-one immersion LaTeXMLMath in LaTeXMLMath transverse to the boundary and such that LaTeXMLMath .
The set of LaTeXMLMath -tuple points of LaTeXMLMath is the immersion in LaTeXMLMath of a compact 1-manifold with boundary , bounding the set of LaTeXMLMath -tuple points of LaTeXMLMath .
But a 1-manifold has an even number of points as boundary , so LaTeXMLMath , that is , LaTeXMLMath represents the trivial element of LaTeXMLMath .
LaTeXMLMath This proposition does not apply , for example , for LaTeXMLMath .
The group LaTeXMLMath is generated by the left Boy immersion , which has a single triple point .
The even elements have no triple points .
Canonical representatives for these classes are the immersions in LaTeXMLMath obtained by rotating an 8 on the LaTeXMLMath plane with the double point in LaTeXMLMath around the LaTeXMLMath axis , while rotating it in its own plane of half a twist , a whole twist and respectively three halves of twists ( see LaTeXMLCite ) .
These immersions , whose Arf invariants are 2,4 and 6 respectively , have a circle of double points .
The immersion similarly constructed that makes no rotations is null-cobordant .
From proposition LaTeXMLRef it immediately follows : For any LaTeXMLMath the group LaTeXMLMath is contained in LaTeXMLMath .
In computing LaTeXMLMath we will make use first of a natural way of producing codimension-one immersions .
Let LaTeXMLMath be a LaTeXMLMath -manifold ( possibly with boundary ) and LaTeXMLMath an embedded codimension- LaTeXMLMath submanifold .
Assume that the structure group of the normal bundle LaTeXMLMath to LaTeXMLMath in LaTeXMLMath can be reduced to the group of symmetries of an element LaTeXMLMath , and choose such a reduction denoted by LaTeXMLMath .
There is then a canonical embedding of LaTeXMLMath in each fiber LaTeXMLMath of LaTeXMLMath giving rise to a sub-fibration of LaTeXMLMath with fiber LaTeXMLMath .
The total space of the last fibration is an immersion in the tubular neighborhood of LaTeXMLMath , hence in LaTeXMLMath , which will be called the immersion obtained by decorating LaTeXMLMath with LaTeXMLMath and will be denoted by LaTeXMLMath or by LaTeXMLMath if the reduction is not relevant to the context .
The cobordism class of this immersion is clearly independent of the representative of the cobordism class of LaTeXMLMath , but might depend on LaTeXMLMath .
For example remark that the symmetry group of the 8 in LaTeXMLMath is the same as the symmetry group of a line , so decorating a codimension-two submanifold with an 8 is the same as choosing a field of lines on the submanifold itself , if any .
In particular the canonical representatives of the even elements of LaTeXMLMath can be considered as LaTeXMLMath , where LaTeXMLMath is the standard circle in LaTeXMLMath and LaTeXMLMath is the line field on the circle making LaTeXMLMath halves of of twist , LaTeXMLMath , hence LaTeXMLMath .
Remark that in an orientable LaTeXMLMath -manifold any simple curve has trivial normal bundle , hence it can be decorated by any element of LaTeXMLMath , the cobordism class of the resulting immersion only depending on the choice of the trivialization .
On the other side in a non-orientable LaTeXMLMath -manifold a simple curve non-trivially intersecting the orientation cycle can not be decorated by an element of LaTeXMLMath not admitting a reflection in its symmetry group .
LaTeXMLMath .
Proof .
Let LaTeXMLMath be the non-orientable LaTeXMLMath bundle on LaTeXMLMath , let LaTeXMLMath be a 4-ball in LaTeXMLMath intersecting LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be LaTeXMLMath .
The normal bundle to LaTeXMLMath is trivial hence one can decorate it with left Boy immersions .
Denote by LaTeXMLMath the resulting immersion and by LaTeXMLMath its intersection with the three sphere LaTeXMLMath .
Remark that when seen in the boundary of LaTeXMLMath ( with any of its possible orientations ) the two connected components of LaTeXMLMath have the same orientation : otherwise the whole of LaTeXMLMath could be decorated by left Boy immersions , which is not possible since the Boy immersions do not admit reflections in their symmetry group .
Hence LaTeXMLMath determines a non-trivial element of LaTeXMLMath that has Arf invariant 2 or 6 , according to the orientation one has chosen on LaTeXMLMath .
But it is clear from its construction that LaTeXMLMath bounds in LaTeXMLMath .
So LaTeXMLMath is in LaTeXMLMath and the same holds for the subgroup of LaTeXMLMath it generates , that is , the whole of LaTeXMLMath .
LaTeXMLMath The proof of this proposition easily extends to the following statement : For any LaTeXMLMath one has LaTeXMLMath .
Moreover any LaTeXMLMath bounds in any non-orientable LaTeXMLMath -manifold with boundary LaTeXMLMath .
LaTeXMLMath Set then LaTeXMLMath for the group of immersions in LaTeXMLMath up to cobordism in manifolds bounding two spheres .
We proved the following : The immersion with invariant 4 bounds in an orientable 4-manifold .
Indeed , let LaTeXMLMath be a sphere corresponding to a complex line in LaTeXMLMath .
There exists a normal vector field LaTeXMLMath on LaTeXMLMath with a single zero .
Let LaTeXMLMath be a 4-ball containing this zero .
The restricted normal field LaTeXMLMath then trivializes the normal bundle to LaTeXMLMath in LaTeXMLMath , so one can define the immersion LaTeXMLMath .
A straightforward computation shows that LaTeXMLMath is an immersion with Arf invariant 4 , and clearly LaTeXMLMath bounds in the orientable manifold LaTeXMLMath .
The point of view from which we are able to tackle the computation of LaTeXMLMath is that of splitting it in pieces .
At first sight the splitting depends on the cellular decomposition of LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -manifold and let LaTeXMLEquation be a skeleton decomposition .
Let LaTeXMLMath , for LaTeXMLMath , be the set of immersions that up to cobordism do not intersect LaTeXMLMath .
An immersion LaTeXMLMath whose class belongs to LaTeXMLMath will be said LaTeXMLMath -admissible if LaTeXMLMath .
Remark that LaTeXMLMath is a subgroup of LaTeXMLMath hence one has a filtration of LaTeXMLMath .
One has LaTeXMLEquation where LaTeXMLMath was added for convenience of notation .
This filtration comes in fact in a natural way from the algebraic-topological definition of LaTeXMLMath .
Under the Pontryagin-Thom construction we have LaTeXMLEquation where LaTeXMLMath means homotopy and LaTeXMLMath is the trivial based loop .
Proof .
If an immersion LaTeXMLMath does not intersect LaTeXMLMath then the map LaTeXMLMath associated by the Pontryagin-Thom construction is constant on LaTeXMLMath .
On the other side , given a map LaTeXMLMath that is null-homotopic when restricted to LaTeXMLMath , consider the homotopy LaTeXMLEquation such that LaTeXMLMath and LaTeXMLMath is the constant map on LaTeXMLMath .
The inclusion of the LaTeXMLMath -skeleton being a cofibration implies ( see LaTeXMLCite ) that LaTeXMLMath extends to LaTeXMLMath .
Thus it gives a homotopy between LaTeXMLMath and a map LaTeXMLMath defined on all of LaTeXMLMath , whose restriction to LaTeXMLMath is constant .
Consider a closed regular neighborhood LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
There exists then a global retraction LaTeXMLMath so that LaTeXMLMath .
The map LaTeXMLMath is thus constant on LaTeXMLMath ( in particular on LaTeXMLMath ) and is homotopic to LaTeXMLMath .
Therefore the Thom-Pontryagin construction for the manifold with boundary LaTeXMLMath associates to the map LaTeXMLMath an immersion in LaTeXMLMath .
When looking at that immersion as contained in LaTeXMLMath it results as a LaTeXMLMath -admissible representative of the class associated to LaTeXMLMath , hence the claim is proved .
We give a second proof introducing a recursive technique that will be often exploited in the sequel .
We assume that the cellular decomposition is in particular a cubulation .
Recall that a cubical complex is a complex LaTeXMLMath consisting of Euclidean cubes , such that the intersection of two cubes is a finite union of cubes from LaTeXMLMath , once a cube is in LaTeXMLMath then all its faces belong to LaTeXMLMath , and no identifications of faces of the same cube are allowed .
A cubulation of a manifold is specified by a cubical complex PL homeomorphic to the manifold .
Consider a class with a representative immersion LaTeXMLMath for which the map LaTeXMLMath defined by the Pontryagin-Thom construction belongs to LaTeXMLMath .
We want to deform LaTeXMLMath up to cobordism in such a way that the new representative does not intersect LaTeXMLMath .
There exists a homotopy LaTeXMLMath of LaTeXMLMath so that the restriction of LaTeXMLMath is null-homotopic .
One uses now a recurrence on the degree LaTeXMLMath .
If LaTeXMLMath then it is obvious since the immersion can miss the LaTeXMLMath -skeleton by general position .
Assume the claim is true for degree at most LaTeXMLMath .
Then there exists a representative immersion so that LaTeXMLMath .
This means that the intersection LaTeXMLMath with any LaTeXMLMath -cell is a closed immersed submanifold lying in the interior of the cell .
By hypothesis we can assume that LaTeXMLMath , where LaTeXMLMath denotes the constant ( trivial ) map .
This means that LaTeXMLMath .
The Thom-Pontryagin theory implies that the immersion LaTeXMLMath is null cobordant .
Consider a small regular neighborhood LaTeXMLMath of LaTeXMLMath in LaTeXMLMath , which is a product LaTeXMLMath .
Since LaTeXMLMath is null cobordant there exists an immersion LaTeXMLMath in LaTeXMLMath providing a null-cobordism for LaTeXMLMath .
One uses LaTeXMLMath to change the immersion in LaTeXMLMath so that the new immersion misses LaTeXMLMath .
LaTeXMLMath Now it is a classical result that such a filtration is independent on the cellular decomposition .
Indeed LaTeXMLMath is the 0-th degree of the generalized cohomology theory associated to the suspension spectrum of LaTeXMLMath LaTeXMLEquation and so the Atiyah-Hirzebruch spectral sequence converges to the graded group associated to the filtration of proposition LaTeXMLRef .
The filtration being independent on the cellular decomposition for realizable generalized cohomology theories is then illustrated in the first and third chapter of the book of Hilton LaTeXMLCite .
From now on we will often choose to use cubical decompositions of the manifold LaTeXMLMath .
This will permit to perform recursive constructions in an easier way because a cubulation of LaTeXMLMath induces in an obvious way one for LaTeXMLMath .
Most of the analysis that follows , and notably the definition of cohomological invariants and the application of obstruction theory , does not make use of any specific property of the generalized cohomology theory of cobordism groups of immersions , except perhaps the fact that it has finite coefficients .
It is possible then that the same definitions apply usefully to other generalized cohomology theories .
Recall that a LaTeXMLMath -admissible immersion LaTeXMLMath is such that LaTeXMLMath .
In particular for any LaTeXMLMath -cell LaTeXMLMath the intersection LaTeXMLMath is contained in LaTeXMLMath .
If LaTeXMLMath is oriented then LaTeXMLMath detects an element of LaTeXMLMath .
One introduces then the following geometric definition .
To any LaTeXMLMath -admissible immersion LaTeXMLMath there is associated a cochain LaTeXMLMath the following way : LaTeXMLEquation .
For LaTeXMLMath -admissible LaTeXMLMath the cochain LaTeXMLMath is a cocycle .
Proof .
Given a LaTeXMLMath -cell LaTeXMLMath recall that LaTeXMLMath holds .
Now denote LaTeXMLMath where LaTeXMLMath is a finite set , LaTeXMLMath are oriented LaTeXMLMath -cells ( not necessarily different from each other ) and LaTeXMLMath .
Consider the cell as a closed LaTeXMLMath -disk attached to LaTeXMLMath by means of an attaching map , that results to be a homeomorphism when restricted to any connected component of the preimage of any LaTeXMLMath .
Since LaTeXMLMath is empty one can then pull back LaTeXMLMath in the disk .
The restriction of the resulting immersion to the boundary of the disk is then LaTeXMLEquation and is trivial since it bounds in the disk .
But this is LaTeXMLMath , hence LaTeXMLMath is a cocycle .
LaTeXMLMath The argument of the previous proof will be repeatedly used .
It is easy to visualize it when the cellular decomposition is a cubulation .
For an immersion LaTeXMLMath and a LaTeXMLMath -cube LaTeXMLMath the intersection LaTeXMLMath is a cobordism to the empty set of LaTeXMLMath .
Thus the last one is trivial as an element of LaTeXMLMath .
When LaTeXMLMath is LaTeXMLMath -admissible LaTeXMLMath splits as the sum ( with signs ) of LaTeXMLMath .
Assume from now on that the cellular decomposition is a cubulation .
It is easy to see that if LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -admissible immersions admitting a cobordism LaTeXMLMath , which does not intersect LaTeXMLMath , then LaTeXMLMath .
In fact for any LaTeXMLMath -cube LaTeXMLMath the intersection LaTeXMLMath is a cobordism between LaTeXMLMath and LaTeXMLMath .
The most natural question is whether LaTeXMLMath and LaTeXMLMath admit a LaTeXMLMath -admissible cobordism , that is a cobordism not intersecting LaTeXMLMath .
This leads to the following result : Let LaTeXMLMath and LaTeXMLMath be LaTeXMLMath -admissible cobordant immersions admitting a cobordism that is LaTeXMLMath -admissible in LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath are cohomologous .
Proof .
Assume LaTeXMLMath and LaTeXMLMath transverse to the cubulation , and take a cobordism LaTeXMLMath between them that is transverse to the standard cubulation of LaTeXMLMath associated with the cubulation of LaTeXMLMath .
Our aim is to define a coboundary between LaTeXMLMath and LaTeXMLMath by means of the same definition we used for LaTeXMLMath , but now applied to LaTeXMLMath .
Remark that for any LaTeXMLMath -cube LaTeXMLMath the intersection LaTeXMLMath is a cobordism to the empty set of LaTeXMLMath , thus the last one is then a trivial element in LaTeXMLMath .
Since LaTeXMLMath is LaTeXMLMath -admissible this element splits as the sum of LaTeXMLMath and of LaTeXMLMath , see figure LaTeXMLRef .
We claim that the last summand is the coboundary of a LaTeXMLMath -cochain of LaTeXMLMath .
Define a cochain LaTeXMLMath in LaTeXMLMath this way .
For any oriented LaTeXMLMath -cell LaTeXMLMath consider the class of the immersion LaTeXMLMath where the orientation of LaTeXMLMath is such that it induces on LaTeXMLMath the opposite of its orientation .
With this convention LaTeXMLMath is a well-defined element of LaTeXMLMath .
It is then easy to see that LaTeXMLMath , hence LaTeXMLMath .
LaTeXMLMath It is immediate that two cobordant immersions which are 1-admissible have a 1-admissible cobordism between them .
For general LaTeXMLMath we are not able to prove the analogous statement .
We will face this problem gradually .
If LaTeXMLMath is a generic cobordism between LaTeXMLMath -admissible immersions , and LaTeXMLMath is an LaTeXMLMath -cube of LaTeXMLMath we denote by LaTeXMLMath the immersion LaTeXMLMath , where the orientation of LaTeXMLMath is such that it induces on LaTeXMLMath the opposite of its orientation .
Now if LaTeXMLMath is LaTeXMLMath -admissible then LaTeXMLMath is actually an immersion in the open LaTeXMLMath -disk LaTeXMLMath , hence LaTeXMLMath represents an element of LaTeXMLMath .
Further if LaTeXMLMath is empty for any LaTeXMLMath -cube then LaTeXMLMath is LaTeXMLMath -admissible .
Let LaTeXMLMath and LaTeXMLMath be 2-admissible cobordant immersions .
Then there exists a 2-admissible cobordism between them .
Proof .
Let LaTeXMLMath be a generic cobordism between LaTeXMLMath and LaTeXMLMath , transverse to the cubulation of LaTeXMLMath associated to a chosen cubulation of LaTeXMLMath .
We want to prove that we can deform LaTeXMLMath until LaTeXMLMath for any LaTeXMLMath .
Remark that LaTeXMLMath represents an element of LaTeXMLMath .
We claim that , up to modify LaTeXMLMath if LaTeXMLMath is compact , this element is trivial for any LaTeXMLMath .
First remark that if LaTeXMLMath and LaTeXMLMath are two vertices of the cubulations that are connected by a edge LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath are the same element of LaTeXMLMath .
This is because LaTeXMLMath provides a cobordism between them and this proves in fact , LaTeXMLMath being connected , that there is a well-defined LaTeXMLMath such that LaTeXMLMath for any vertex LaTeXMLMath of LaTeXMLMath .
Now suppose LaTeXMLMath is not compact .
Since the domain of LaTeXMLMath is compact its image can not intersect all edges of the type LaTeXMLMath , hence LaTeXMLMath is the trivial element of LaTeXMLMath , and the claim is proved in this case .
If LaTeXMLMath is compact , then LaTeXMLMath might be non-trivial .
But then consider a new cobordism obtained by adding a LaTeXMLMath to LaTeXMLMath .
Call LaTeXMLMath the new cobordism and now LaTeXMLMath is trivial as required .
So we are ready to get rid of intersections of type LaTeXMLMath .
For any vertex LaTeXMLMath there is a diffeomorphism of a neighborhood LaTeXMLMath with LaTeXMLMath such that LaTeXMLMath is the inclusion of an even number of disks at levels LaTeXMLMath , since LaTeXMLMath can be assumed to be transverse to LaTeXMLMath .
In this model cut the corresponding disks of radius LaTeXMLMath , and connect the holes in pairs by means of cylinders LaTeXMLMath .
The immersion obtained by repeating this construction in any vertex and then smoothing , that we still call LaTeXMLMath , satisfies LaTeXMLMath for any vertex LaTeXMLMath , hence LaTeXMLMath is 2-admissible .
LaTeXMLMath In general , given a LaTeXMLMath -admissible cobordism LaTeXMLMath the elements LaTeXMLMath for LaTeXMLMath might be nontrivial .
If they are trivial however it is possible to deform LaTeXMLMath to a LaTeXMLMath -admissible cobordism .
Consider the standard cubulation of LaTeXMLMath associated to a cubulation of LaTeXMLMath .
We will say that cubes of the form LaTeXMLMath are vertical , cubes of the form LaTeXMLMath are at the bottom and cubes of the form LaTeXMLMath are at the top .
In general cubes of the form LaTeXMLMath are horizontal .
Let LaTeXMLMath and and LaTeXMLMath be LaTeXMLMath -admissible immersions , let LaTeXMLMath be a cobordism between them that is LaTeXMLMath -admissible and such that for any LaTeXMLMath -cell LaTeXMLMath of LaTeXMLMath the immersion LaTeXMLMath represents a trivial element of LaTeXMLMath .
Then LaTeXMLMath can be deformed to a LaTeXMLMath -admissible cobordism LaTeXMLMath .
If LaTeXMLMath is a LaTeXMLMath -admissible cobordism between LaTeXMLMath and LaTeXMLMath , but LaTeXMLMath alone is LaTeXMLMath -admissible , then LaTeXMLMath can be modified to a LaTeXMLMath -admissible cobordism LaTeXMLMath between LaTeXMLMath and an immersion coinciding with LaTeXMLMath outside a neighborhood of LaTeXMLMath .
Proof .
By a construction analogous to that of proposition LaTeXMLRef it is possible to get rid of intersections .
We put again ourselves in a model , as follows : the normal bundle to LaTeXMLMath in LaTeXMLMath is trivial , take a trivialized neighborhood LaTeXMLMath and take its product with the interval LaTeXMLMath .
From transversality one can suppose that LaTeXMLMath has the structure of a product LaTeXMLMath .
Consider a cobordism to the empty set of LaTeXMLMath , let LaTeXMLMath be the embedding of this cobordism in LaTeXMLMath and take the product LaTeXMLMath .
Remark that the image of this product does not intersect LaTeXMLMath , and that it intersects LaTeXMLMath in LaTeXMLMath .
Thus one can excise LaTeXMLMath and glue back LaTeXMLMath .
After repeating this construction for any LaTeXMLMath -cube and smoothing one gets a cobordism LaTeXMLMath between LaTeXMLMath and LaTeXMLMath satisfying LaTeXMLMath for any LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are both LaTeXMLMath -admissible then LaTeXMLMath is clearly LaTeXMLMath -admissible , since the LaTeXMLMath -skeleton of LaTeXMLMath are the vertical LaTeXMLMath plus the two horizontal copies of LaTeXMLMath .
If LaTeXMLMath is LaTeXMLMath -admissible , but LaTeXMLMath intersects the LaTeXMLMath -skeleton , then we deform LaTeXMLMath further .
Remark that for any LaTeXMLMath -cube LaTeXMLMath of LaTeXMLMath the cobordism LaTeXMLMath is a cobordism to the empty set for LaTeXMLMath .
Hence this last immersion represents the trivial element of LaTeXMLMath .
Then a surgery similar to the one of the first part of the proof leads to the excision of all of the intersections LaTeXMLMath , and this proves the claim .
LaTeXMLMath Since we can not claim than any null-cobordant LaTeXMLMath -admissible immersion admits a LaTeXMLMath -admissible cobordism to the empty set one introduces the following definition .
Set LaTeXMLMath for the subset of LaTeXMLMath of those cohomology classes represented by some LaTeXMLMath -admissible null-cobordant immersions .
LaTeXMLMath is a subgroup of LaTeXMLMath .
Proof .
Given LaTeXMLMath and LaTeXMLMath cohomology classes represented by LaTeXMLMath -admissible null-cobordant immersions it is obvious that LaTeXMLMath is represented the same way .
As for LaTeXMLMath , let LaTeXMLMath be a LaTeXMLMath -admissible immersion such that LaTeXMLMath , and let LaTeXMLMath be a cobordism to the empty set of LaTeXMLMath .
Let LaTeXMLMath be the cobordism between LaTeXMLMath in LaTeXMLMath and the empty set in LaTeXMLMath obtained by composing LaTeXMLMath with the reflection of LaTeXMLMath given by LaTeXMLMath .
In a single LaTeXMLMath consider the following construction .
Put a representative immersion of LaTeXMLMath in LaTeXMLMath , put in LaTeXMLMath the same representative plus two copies of LaTeXMLMath slightly isotoped , and in LaTeXMLMath a single copy of LaTeXMLMath .
Then fill LaTeXMLMath with LaTeXMLMath plus two copies of LaTeXMLMath rescaled of 1/3 , fill LaTeXMLMath with a cobordism between LaTeXMLMath plus a copy of LaTeXMLMath and the empty set and with LaTeXMLMath , and finally fill LaTeXMLMath with LaTeXMLMath ( rescaled of 1/3 ) .
Remark that this immersion is not a cobordism between LaTeXMLMath and LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath possibly intersect LaTeXMLMath .
Now remark that the collection of immersions so defined glue together to a cobordism in LaTeXMLMath , that restricted to LaTeXMLMath represents LaTeXMLMath .
Consider on LaTeXMLMath the empty immersion and in LaTeXMLMath the collection of immersions defined before .
This cobordism can be completed by lemma LaTeXMLRef to a cobordism LaTeXMLMath between an immersion LaTeXMLMath representing LaTeXMLMath and the empty set , hence LaTeXMLMath .
LaTeXMLMath The following lemma , that provides the technical step of the previous proof , will be repeatedly used in this section .
It is an easy algebraic-topological argument , that has however an important geometric interpretation .
An immersion LaTeXMLMath traced in LaTeXMLMath extends to an immersion traced in the whole of LaTeXMLMath .
Moreover if LaTeXMLMath is LaTeXMLMath -admissible , for LaTeXMLMath , the extension is still LaTeXMLMath -admissible .
Proof .
If we see an immersion traced in LaTeXMLMath as a continuous map from LaTeXMLMath to LaTeXMLMath this immediately follows from the fact that LaTeXMLMath is a cofibration .
This construction has however an easy geometrical interpretation , see figure LaTeXMLRef .
Consider a LaTeXMLMath -cube LaTeXMLMath .
We want to extend the immersion traced on LaTeXMLMath to LaTeXMLMath .
First remark that the resulting immersion on the boundary of the cube at the top is null-cobordant , since it bounds in the disk LaTeXMLMath , hence a null-cobordism can be traced on the cube at the top .
The resulting immersion in LaTeXMLMath represents an element of LaTeXMLMath .
Up to adding ( in the interior of the cube at the top ) another immersion we can assume this element is trivial , hence LaTeXMLMath can be extended to the interior of LaTeXMLMath .
Recursively LaTeXMLMath is extended to LaTeXMLMath .
The second statement follows obviously from the construction .
LaTeXMLMath We prove that LaTeXMLMath is a well-defined invariant in the group LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be LaTeXMLMath -admissible cobordant immersions .
Then LaTeXMLMath represents an element in LaTeXMLMath , hence LaTeXMLMath is a well-defined cobordism invariant in the group LaTeXMLMath .
Proof .
The technical step is to modify LaTeXMLMath , far from LaTeXMLMath , in such a way that it doesn ’ t intersect LaTeXMLMath .
This is only possible , in general , up to modifying LaTeXMLMath .
By proposition LaTeXMLRef we might assume that LaTeXMLMath is 2-admissible .
Assume by a recurrence hypothesis that LaTeXMLMath is a LaTeXMLMath -admissible cobordism between LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is a ( null-cobordant ) LaTeXMLMath -admissible immersion .
We want to deform LaTeXMLMath in such a way that LaTeXMLMath becomes trivial for any LaTeXMLMath , in order to apply lemma LaTeXMLRef .
We build up an auxiliary immersion LaTeXMLMath in LaTeXMLMath .
On the vertical cells LaTeXMLMath put a copy LaTeXMLMath , and on the bottom cells LaTeXMLMath fix the empty immersion for any LaTeXMLMath .
Consider then the resulting collection of immersions as an immersion in LaTeXMLMath and extend it to a cobordism LaTeXMLMath by means of lemma LaTeXMLRef .
Then LaTeXMLMath satisfies the following properties : LaTeXMLEquation .
Remark now that there exists an integer LaTeXMLMath ( the order of LaTeXMLMath minus 1 ) such that LaTeXMLMath has the property that LaTeXMLMath is trivial for any LaTeXMLMath .
Call again LaTeXMLMath this cobordism and apply lemma LaTeXMLRef .
The resulting cobordism LaTeXMLMath between LaTeXMLMath and LaTeXMLMath is then LaTeXMLMath -admissible , and LaTeXMLMath is of the form LaTeXMLMath where LaTeXMLMath is ( null-cobordant and ) LaTeXMLMath -admissible .
Repeat this construction until LaTeXMLMath and the resulting LaTeXMLMath -admissible cobordism , that we again call LaTeXMLMath , is such that LaTeXMLEquation where LaTeXMLMath is ( null-cobordant and ) LaTeXMLMath -admissible .
Then by proposition LaTeXMLRef LaTeXMLMath differs from LaTeXMLMath by a coboundary .
But since LaTeXMLEquation one gets the claim .
LaTeXMLMath The invariant LaTeXMLEquation defined by these propositions will be called the LaTeXMLMath -th cohomological invariant .
Remark that if LaTeXMLMath is actually in LaTeXMLMath then LaTeXMLMath is trivial , hence we are left with a well-defined homomorphism from LaTeXMLMath to LaTeXMLMath .
This homomorphism is in fact injective .
Let LaTeXMLMath be an LaTeXMLMath -manifold .
For any LaTeXMLMath the kernel of the LaTeXMLMath -th cohomological invariant is LaTeXMLMath .
Proof .
Fix a cubulation in LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -admissible immersion such that LaTeXMLMath .
This means there is an element LaTeXMLMath such that LaTeXMLMath .
One builds up a cobordism LaTeXMLMath between LaTeXMLMath and a LaTeXMLMath -admissible immersion .
Consider the standard cubulation of LaTeXMLMath associated to the given cubulation of LaTeXMLMath .
Put LaTeXMLMath in the bottom LaTeXMLMath .
For any LaTeXMLMath -cell of LaTeXMLMath , say LaTeXMLMath , put in the vertical cell LaTeXMLMath ( with the orientation that induces on LaTeXMLMath the opposite of its orientation ) the element LaTeXMLMath .
One fixes also the cobordism on the top LaTeXMLMath .
Consider a LaTeXMLMath -cell LaTeXMLMath of LaTeXMLMath .
One defines the cobordism on LaTeXMLMath by remarking that ( from the definition of LaTeXMLMath ) the union of all immersions already defined in LaTeXMLMath is null-cobordant .
One can choose therefore a cobordism to the empty set .
This can be done recursively on the whole of LaTeXMLMath .
Remark that the resulting immersion LaTeXMLMath does not intersect LaTeXMLMath .
Now LaTeXMLMath is defined on LaTeXMLMath , and applying lemma LaTeXMLRef gives rise to a cobordism LaTeXMLMath in LaTeXMLMath , that provides a cobordism between LaTeXMLMath and LaTeXMLMath ; and this last does not intersect the LaTeXMLMath -skeleton , by construction .
A similar construction can be performed if LaTeXMLMath .
Let LaTeXMLMath be a null-cobordant map such that LaTeXMLMath .
Put LaTeXMLMath in the whole bottom LaTeXMLMath , LaTeXMLMath in the intermediate LaTeXMLMath and trace on the vertical LaTeXMLMath the intersection of a cobordism to the empty set of LaTeXMLMath .
The LaTeXMLMath -cochain that cobounds LaTeXMLMath and LaTeXMLMath provides as before a cobordism between LaTeXMLMath and LaTeXMLMath , which we put in the vertical LaTeXMLMath .
Over all this is a cobordism between LaTeXMLMath and the empty set and so it can be extended by lemma LaTeXMLRef to a cobordism LaTeXMLMath between LaTeXMLMath and a map LaTeXMLMath .
By construction the last one does not intersect the LaTeXMLMath -skeleton .
LaTeXMLMath Let LaTeXMLMath be an LaTeXMLMath -manifold .
For any LaTeXMLMath the induced homomorphism LaTeXMLEquation is injective .
This corollary shows that the power of these new invariants is considerable .
Indeed they describe the graded group of LaTeXMLMath associated to the filtration LaTeXMLEquation as a subgroup of LaTeXMLMath , that is : The cohomological invariants induce an injective homomorphism LaTeXMLEquation .
We end this section with an important remark .
The cohomological invariants reduce , under suitable hypothesis , to the restriction of James-Hopf invariants .
These are classical cobordism invariants , see LaTeXMLCite and LaTeXMLCite .
Let LaTeXMLMath be a LaTeXMLMath -manifold and LaTeXMLMath a generic codimension-one immersion .
For LaTeXMLMath consider the locus of LaTeXMLMath -tuple points of LaTeXMLMath , that is , the points of LaTeXMLMath that have a number of preimages equal or bigger than LaTeXMLMath .
This set is in fact a LaTeXMLMath -cycle modulo 2 , whose homology class is invariant up to cobordism .
We denote LaTeXMLMath this class and call it LaTeXMLMath -th James-Hopf invariant .
These invariants are particularly meaningful for those LaTeXMLMath such that LaTeXMLMath is non-trivial .
For example given a codimension- LaTeXMLMath embedded submanifold LaTeXMLMath of an LaTeXMLMath -manifold LaTeXMLMath such that its normal bundle is reducible to the symmetry group of an element LaTeXMLMath satisfying LaTeXMLMath then the following holds LaTeXMLEquation .
Assume that LaTeXMLMath is such that LaTeXMLMath is the reduction modulo 2 of LaTeXMLMath .
Then for any LaTeXMLMath -manifold the LaTeXMLMath -th James-Hopf invariant restricted to LaTeXMLMath is the Poincaré dual to the reduction modulo 2 of LaTeXMLMath .
Proof .
Let LaTeXMLMath be a LaTeXMLMath -admissible immersion generic and transverse to the decomposition of LaTeXMLMath .
We denote by LaTeXMLMath the Poincaré duality isomorphism .
Then for any LaTeXMLMath -cell LaTeXMLMath of LaTeXMLMath the number LaTeXMLMath is the number of LaTeXMLMath -tuple points of LaTeXMLMath , modulo 2 , hence , by the hypothesis on LaTeXMLMath , is the reduction modulo 2 of LaTeXMLMath as an element of LaTeXMLMath , that is LaTeXMLMath .
The following diagram then commutes LaTeXMLEquation and since LaTeXMLMath is reduction modulo 2 in cohomology the claim follows .
LaTeXMLMath This proves at once the following : If LaTeXMLMath is such that LaTeXMLMath is an isomorphism then LaTeXMLMath .
LaTeXMLMath A more detailed study of the groups LaTeXMLMath is in order .
From proposition LaTeXMLRef and lemma LaTeXMLRef one might guess that the vanishing of LaTeXMLMath is correlated with constructions that make the intersections LaTeXMLMath of a cobordism LaTeXMLMath with the vertical walls of LaTeXMLMath null-cobordant .
This was made possible for example in the proof of proposition LaTeXMLRef by means of the construction of an auxiliary immersion LaTeXMLMath with prescribed image on LaTeXMLMath .
As we saw the possibility of obtaining such an auxiliary immersion amounts , at an accurate analysis , to the fact that LaTeXMLMath is a cofibration .
However in order to leave the image of LaTeXMLMath fixed also in LaTeXMLMath we need a more delicate construction , that will be developed in section LaTeXMLRef and applied in section LaTeXMLRef .
A second obvious motivation for developing this theory is the computation of the image of LaTeXMLMath , that is , the subgroup of cohomology classes that are represented by an immersion .
In this section we describe the general obstructions for a cochain in LaTeXMLMath to be realizable as an immersion .
The basic idea is a recursive construction .
Given LaTeXMLMath we first put in the interior of every LaTeXMLMath -cube an immersion representing LaTeXMLMath , then try to extend this codimension-one immersion in LaTeXMLMath to a codimension-one immersion in LaTeXMLMath .
This will be called the first extendibility .
If this construction can be repeated until the LaTeXMLMath -th skeleton i.e .
the immersion can be further extended to a second extension , and so on then the original cochain is said to be realizable or extendible .
If one can reach the LaTeXMLMath -th stage one says that the cochain is LaTeXMLMath -extendible .
We adapt to this context Eilenberg ’ s obstruction theory , see LaTeXMLCite , §V.5 .
A cochain in LaTeXMLMath can be thought of as a map LaTeXMLMath defined from LaTeXMLMath to LaTeXMLMath , that restricted to any LaTeXMLMath -cell is geometrically represented by an element of LaTeXMLMath .
The problem to which we apply Eilenberg ’ s theory is that of extending this map over the next skeleton .
Given a LaTeXMLMath -simple space LaTeXMLMath , a CW-complex LaTeXMLMath and a map LaTeXMLMath the obstruction to extending LaTeXMLMath to the LaTeXMLMath -skeleton is a cochain LaTeXMLMath , assigning to each LaTeXMLMath -cell LaTeXMLMath the map LaTeXMLMath .
Its fundamental properties are stated in the following theorem ( see LaTeXMLCite , §V.5 ) : LaTeXMLMath is extendible to LaTeXMLMath if and only if LaTeXMLMath is the trivial cochain .
LaTeXMLMath is a cocycle , hence represents an element of LaTeXMLMath .
LaTeXMLMath is extendible to LaTeXMLMath if and only if LaTeXMLMath is trivial in LaTeXMLMath .
The problem of further extending LaTeXMLMath is codified in a sequence of obstruction maps .
However for any extension there exists an obstruction cocycle , hence the obstruction to further extend LaTeXMLMath becomes a set of cohomology classes .
Assume that it is extendible to the LaTeXMLMath -th skeleton , and let LaTeXMLMath be the set of cohomology classes given by LaTeXMLEquation .
Assume the map LaTeXMLMath is extendible to the LaTeXMLMath -skeleton .
Then it is extendible to the LaTeXMLMath -skeleton if and only if the 0 class belongs to LaTeXMLMath .
We first recall that LaTeXMLEquation hence the theory applies directly with coefficients in the groups LaTeXMLMath .
Fixed a LaTeXMLMath -cochain LaTeXMLMath we consider it as a map defined on LaTeXMLMath taking values on LaTeXMLMath .
Consider then the obstruction LaTeXMLEquation and remark that , since LaTeXMLMath , the obstruction LaTeXMLMath is nothing but the ordinary coboundary of cochains with coefficients in LaTeXMLMath .
Hence by property 1 of theorem LaTeXMLRef it follows that : LaTeXMLMath is 1-extendible if and only if it is a cocycle .
This condition has the geometrical interpretation that was already illustrated in figure LaTeXMLRef .
Define now LaTeXMLMath , for LaTeXMLMath , to be the map that associates to LaTeXMLMath the set LaTeXMLMath , LaTeXMLMath being any extension of LaTeXMLMath to the LaTeXMLMath -skeleton , and define LaTeXMLMath to be the subset of LaTeXMLMath of cocycles LaTeXMLMath such that LaTeXMLMath contains the trivial element of LaTeXMLMath .
Remark that LaTeXMLMath , and that LaTeXMLMath is in fact only defined on LaTeXMLMath .
From theorem LaTeXMLRef one obtains the following proposition : LaTeXMLMath is LaTeXMLMath -extendible if and only if it belongs to LaTeXMLMath .
In particular it is realizable if and only if it belongs to LaTeXMLMath .
LaTeXMLMath Denote by LaTeXMLMath the subgroup of LaTeXMLMath of extendible cocycles .
Proposition LaTeXMLRef translates into : LaTeXMLEquation .
Let LaTeXMLMath ; then the immersion LaTeXMLMath realizing LaTeXMLMath is a well-defined element in LaTeXMLMath .
Proof .
This follows from proposition LaTeXMLRef .
If LaTeXMLMath and LaTeXMLMath both realize LaTeXMLMath in particular they have the same LaTeXMLMath -th cohomological invariant LaTeXMLMath , so they differ by an element of LaTeXMLMath .
LaTeXMLMath We apply the obstruction theory to the computation of both LaTeXMLMath and LaTeXMLMath ( the subgroup of LaTeXMLMath of null-extendible cocycles cocycles realizable as a null-cobordant LaTeXMLMath -admissible immersion ) .
The computations prove theorem LaTeXMLRef .
The results are summarized in the following table , where LaTeXMLMath denotes the non-extendible cocycles .
In this table LaTeXMLMath denotes the subgroup of LaTeXMLMath defined by LaTeXMLEquation .
Remark that , if LaTeXMLMath is a manifold , this is the quadric of LaTeXMLMath associated with the intersection form .
Moreover for any dimension LaTeXMLMath one has : We first prove the results from the last table .
For any dimension LaTeXMLMath and for any LaTeXMLMath -manifold LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Proof .
Since LaTeXMLMath one can use Poincaré duality with coefficients LaTeXMLMath in both orientable and non-orientable context .
Now represent the Poincaré dual to LaTeXMLMath by an embedding and remark that this embedding realizes LaTeXMLMath .
That LaTeXMLMath and LaTeXMLMath -cohomology classes extend follows immediately from obstruction theory .
LaTeXMLMath This result can be interpreted geometrically .
That every LaTeXMLMath -class is extendible follows easily from the fact that taking a representative cocycle LaTeXMLMath a putting in any LaTeXMLMath -cube LaTeXMLMath an immersion representing LaTeXMLMath already realizes the cocycle .
Let now LaTeXMLMath be an orientable LaTeXMLMath -manifold , let LaTeXMLMath be a cohomology class in LaTeXMLMath , and consider its Poincaré dual LaTeXMLMath .
By the universal coefficient theorem LaTeXMLMath can be thought of as an element of LaTeXMLMath , hence as a combination of the type LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
To each LaTeXMLMath we associate the following immersion .
Take a simple closed loop representing LaTeXMLMath .
By decorating LaTeXMLMath with LaTeXMLMath , which is always possible since the normal bundle of LaTeXMLMath is trivial , one obtains an immersion realizing the Poincaré dual of LaTeXMLMath .
Obviously the sum of such immersions realizes LaTeXMLMath .
Let LaTeXMLMath be such that LaTeXMLMath is either trivial or LaTeXMLMath .
Then if LaTeXMLMath is a non-orientable LaTeXMLMath -manifold the same construction applies .
Indeed , these conditions on LaTeXMLMath both mean that any immersion in LaTeXMLMath ( up to a cobordism ) admits a reflection in its symmetry group , since LaTeXMLMath .
Hence this construction applies , since also curves with non-orientable normal bundle can be always decorated with immersions in LaTeXMLMath .
We are ready now to prove the results from the main table .
Let us first concentrate on codimension-two cocycles .
If LaTeXMLMath there is only one obstruction , namely LaTeXMLMath .
This lives in the cyclic group LaTeXMLMath .
One describes now LaTeXMLMath explicitly as an element of LaTeXMLMath or of LaTeXMLMath , depending on LaTeXMLMath being orientable or not .
Let LaTeXMLMath be any LaTeXMLMath -manifold .
Every LaTeXMLMath is extendible to an immersion defined in LaTeXMLMath , where LaTeXMLMath is an LaTeXMLMath -ball .
Proof .
Perform a first extension LaTeXMLMath of LaTeXMLMath to LaTeXMLMath .
Remark that LaTeXMLMath collapses simplicially on a subset LaTeXMLMath of LaTeXMLMath .
Fix a way of building up LaTeXMLMath from this subset , that is , order the set of LaTeXMLMath -cells in such a way that LaTeXMLMath is attached to LaTeXMLMath and LaTeXMLMath is attached to LaTeXMLMath .
For any LaTeXMLMath call free face of LaTeXMLMath the one to which first a cell of LaTeXMLMath will be attached .
Remark that every LaTeXMLMath -cell but LaTeXMLMath has a free face .
When attaching the first LaTeXMLMath -cell LaTeXMLMath extend the immersion LaTeXMLMath this way .
If LaTeXMLMath is trivial in LaTeXMLMath then the extension is the cobordism to the empty set , if it is non-trivial , see figure LaTeXMLRef , then add a representative of LaTeXMLMath on the free face of LaTeXMLMath .
Call again LaTeXMLMath the new extension of LaTeXMLMath , and perform recursively the same construction .
At the end one is left with LaTeXMLMath defined on LaTeXMLMath .
LaTeXMLMath Let LaTeXMLMath be an LaTeXMLMath -manifold , LaTeXMLMath and LaTeXMLMath be any extension of LaTeXMLMath to LaTeXMLMath .
If LaTeXMLMath is orientable then LaTeXMLMath depends only on LaTeXMLMath .
If LaTeXMLMath is non-orientable then LaTeXMLMath depends only on LaTeXMLMath .
Proof .
Given any extension LaTeXMLMath of LaTeXMLMath to LaTeXMLMath , the set of extensions modulo cobordism relative to LaTeXMLMath is acted on by LaTeXMLMath , by the action LaTeXMLMath .
This action is transitive .
Remark that if two extensions LaTeXMLMath and LaTeXMLMath , both extends to LaTeXMLMath , then their difference LaTeXMLMath must be such that for any LaTeXMLMath -cell LaTeXMLMath it holds LaTeXMLMath .
Assume now that LaTeXMLMath is orientable .
Then at the cochain level LaTeXMLMath , so LaTeXMLMath , and hence LaTeXMLMath .
If LaTeXMLMath is non-orientable the equation LaTeXMLMath only holds modulo 2 , hence one can say that LaTeXMLMath .
Assume now that LaTeXMLMath is represented by a different cocycle , hence by a different immersion in LaTeXMLMath .
Since the two cocycles are cohomologous the two immersions are cobordant , hence there exists a cobordism in LaTeXMLMath between the two representatives .
This cobordism can be extended to a cobordism between extensions to LaTeXMLMath , since any cube LaTeXMLMath has a free face , say , LaTeXMLMath , and in an analogous way to a cobordism between two extensions LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
This proves that LaTeXMLMath and LaTeXMLMath are cobordant .
That LaTeXMLMath does not depend neither on LaTeXMLMath nor on the process of collapsing is then straightforward , hence the claim .
LaTeXMLMath Let LaTeXMLMath be an LaTeXMLMath -manifold and LaTeXMLMath .
Then LaTeXMLMath if and only if , given any extension LaTeXMLMath of LaTeXMLMath to LaTeXMLMath , LaTeXMLEquation .
Proof .
The case LaTeXMLMath orientable follows immediately from the preceding lemmas .
As for the non-orientable case recall the proof of proposition LaTeXMLRef and proposition LaTeXMLRef and remark that if LaTeXMLMath then its opposite LaTeXMLMath bounds in LaTeXMLMath a cobordism not intersecting LaTeXMLMath .
LaTeXMLMath Let LaTeXMLMath be such that LaTeXMLMath .
Then for any LaTeXMLMath -manifold LaTeXMLEquation .
More generally , if LaTeXMLMath is such that LaTeXMLMath , then the same holds true for non-orientable LaTeXMLMath -manifolds .
Proof .
This immediately follows from proposition LaTeXMLRef and the fact that for any extension LaTeXMLMath of LaTeXMLMath to LaTeXMLMath the immersion LaTeXMLMath belongs to LaTeXMLMath .
LaTeXMLMath This yields the claimed values for LaTeXMLMath in all cases but for orientable LaTeXMLMath -manifolds .
In this case a geometric construction is in order .
Let LaTeXMLMath be an orientable 4-manifold and LaTeXMLMath .
Then LaTeXMLMath if and only if LaTeXMLMath .
Proof .
Assume first that LaTeXMLMath , that is , LaTeXMLMath .
Take a smoothly embedded representative LaTeXMLMath of LaTeXMLMath , and take a generic normal field LaTeXMLMath to LaTeXMLMath in LaTeXMLMath .
The hypothesis on LaTeXMLMath implies that LaTeXMLMath has an even number of isolated , simple zeroes LaTeXMLMath .
Around each zero LaTeXMLMath take a small disk LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath has degree 1 or -1 .
Then cut off all the disks and connect the remaining holes in pairs with tubes which are contained in a tubular neighborhood of LaTeXMLMath .
The resulting surface LaTeXMLMath still represents LaTeXMLMath and admits a nowhere vanishing normal field of directions .
Since the group of symmetries of the 8 in LaTeXMLMath is equal to the group of symmetries of a line , the existence of the field of directions means that it is possible to decorate LaTeXMLMath with 8 ’ s .
The resulting codimension-one immersion is in LaTeXMLMath and has second cohomological invariant equal to LaTeXMLMath .
On the opposite direction , assume by absurd that there exists LaTeXMLMath with LaTeXMLMath .
Consider an embedded surface representing LaTeXMLMath , then choose a normal field to LaTeXMLMath .
Up to changing LaTeXMLMath in its homology class we can assume as in the previous step that LaTeXMLMath admits a normal field of directions with a single isolated degenerate point LaTeXMLMath .
Let LaTeXMLMath be a 4-disk around LaTeXMLMath in LaTeXMLMath .
One can extend LaTeXMLMath to an immersion defined in LaTeXMLMath by decorating LaTeXMLMath with 8 ’ s following the normal field , call LaTeXMLMath this immersion .
Now LaTeXMLMath is a non-trivial element of LaTeXMLMath .
If it was trivial , the normal field of directions could be extended to the whole of LaTeXMLMath , which is not possible .
By a local analysis , we can reduce ourselves to the situation of remark LaTeXMLRef , hence one obtains that LaTeXMLMath is in fact the element LaTeXMLMath .
But from proposition LaTeXMLRef , since LaTeXMLMath is orientable , LaTeXMLMath .
LaTeXMLMath We showed in proposition LaTeXMLRef that LaTeXMLMath and in remark LaTeXMLRef that LaTeXMLMath contains immersions that bound in an orientable manifold .
The theory of this section shows that LaTeXMLMath is the subgroup of immersions bounding in an orientable manifold , that is , immersions with invariants 2 and 6 do not bound in any orientable manifold .
This settles the table for LaTeXMLMath .
One proves now that all obstructions involved in the table are trivial except for LaTeXMLMath .
This fact is due to the particular properties of LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
In general , the more the groups LaTeXMLMath are simple the more the extensions LaTeXMLMath are easy to compute .
The easiest case is of course LaTeXMLMath .
For LaTeXMLMath the extension LaTeXMLMath is trivial .
LaTeXMLMath The easiest next step is a property of the first obstruction LaTeXMLMath for some values of LaTeXMLMath .
Let LaTeXMLMath be such that LaTeXMLMath is an isomorphism .
Then the first obstruction LaTeXMLMath is trivial .
Proof .
For LaTeXMLMath consider any first extension LaTeXMLMath to LaTeXMLMath .
Remark that LaTeXMLMath can not be considered as an element of LaTeXMLMath , since LaTeXMLMath is not trivial .
However the number of LaTeXMLMath -tuple points modulo 2 of LaTeXMLMath is well-defined .
Then let LaTeXMLMath be the cochain that associates to LaTeXMLMath this number .
Remark that the composition with LaTeXMLMath induces a natural isomorphism between LaTeXMLMath and LaTeXMLMath .
So we can consider LaTeXMLMath , with the action defined in the proof of lemma LaTeXMLRef .
This immersion extends LaTeXMLMath and is extendible to LaTeXMLMath , since for any LaTeXMLMath -cell LaTeXMLMath LaTeXMLEquation that is , LaTeXMLMath .
LaTeXMLMath The triviality of almost all of the obstructions involved in the table follow then from generalizing the previous results .
The proof of proposition LaTeXMLRef actually extends to the following result , that is in fact the strongest triviality result in this section .
Given an immersion however traced on a LaTeXMLMath -skeleton , if LaTeXMLMath is an isomorphism then in each LaTeXMLMath -cube one can force the parity of LaTeXMLMath -tuple points to be even , and since any LaTeXMLMath -cube has an even number of faces , this permits extension to the LaTeXMLMath -skeleton .
Let LaTeXMLMath be such that LaTeXMLMath is an isomorphism .
Then the obstruction LaTeXMLMath is trivial , for any LaTeXMLMath .
Proof .
The action defined in the proof of proposition LaTeXMLRef can be defined on the set of extensions from any skeleton to the following one , hence the proof applies .
LaTeXMLMath On the other side , the construction of the unique obstruction for codimension-two cocycles can be performed in a more general context .
Specifically , by an easy adaptation of the arguments of lemmas LaTeXMLRef and LaTeXMLRef one proves the following proposition : Let LaTeXMLMath be such that LaTeXMLMath .
Then for any LaTeXMLMath -manifold and any LaTeXMLMath the last obstruction LaTeXMLMath is trivial .
If LaTeXMLMath is such that LaTeXMLMath then the same holds for any non-orientable LaTeXMLMath -manifold .
LaTeXMLMath We are then left to study LaTeXMLMath for LaTeXMLMath -manifolds with LaTeXMLMath , since LaTeXMLMath is the only nontrivial obstruction involved in the table .
Let LaTeXMLMath be a closed orientable LaTeXMLMath -manifold such that LaTeXMLEquation .
Then LaTeXMLEquation .
Proof .
One shows first that LaTeXMLMath implies LaTeXMLMath .
This condition on LaTeXMLMath is in fact a consequence of LaTeXMLMath .
Let LaTeXMLMath .
The hypothesis on LaTeXMLMath implies by the universal coefficient theorem that LaTeXMLMath .
We want to prove that LaTeXMLMath by showing that for all LaTeXMLMath one obtains LaTeXMLMath .
Fix a class LaTeXMLMath , and represent it by an embedded orientable 4-submanifold LaTeXMLMath ( see LaTeXMLCite , theorem II.27 ) .
One obtains easily LaTeXMLMath , where LaTeXMLMath is inclusion of LaTeXMLMath in LaTeXMLMath .
In fact LaTeXMLMath is an extendible element of LaTeXMLMath , and the claim follows from the characterization of proposition LaTeXMLRef .
Hence LaTeXMLMath .
Now we show that LaTeXMLMath implies that LaTeXMLMath .
Let LaTeXMLMath , then LaTeXMLMath .
For any LaTeXMLMath take as before a representative LaTeXMLMath orientable and embedded in LaTeXMLMath .
Remark that LaTeXMLMath , hence LaTeXMLMath .
By functorial properties of obstruction cocycles ( see LaTeXMLCite page 230 ) LaTeXMLMath , and this last is the set LaTeXMLMath evaluated on LaTeXMLMath and composed with Poincaré duality .
We proved therefore that this set contains LaTeXMLMath for any LaTeXMLMath , hence LaTeXMLMath contains the trivial cocycle and so LaTeXMLMath extends over the 4-skeleton .
LaTeXMLMath If LaTeXMLMath is a closed orientable LaTeXMLMath -manifold , LaTeXMLMath , and LaTeXMLMath then LaTeXMLEquation .
Proof .
This is because for LaTeXMLMath the obstruction LaTeXMLMath is the only one that has not been proven to be trivial yet , hence LaTeXMLMath .
So from theorem LaTeXMLRef we obtain the claim .
LaTeXMLMath The results summarized in the table are then proved .
Finally remark that the crucial property of LaTeXMLMath that makes proposition LaTeXMLRef and theorem LaTeXMLRef work can be abstracted to the following definition : For any LaTeXMLMath denote by LaTeXMLMath the group of immersions LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath are cobordant to LaTeXMLMath .
We say LaTeXMLMath is simple if for any LaTeXMLMath there exists an isomorphism LaTeXMLMath such that LaTeXMLEquation .
Hence the following theorem holds : For simple LaTeXMLMath the obstruction LaTeXMLMath is trivial for any LaTeXMLMath .
LaTeXMLMath We apply obstruction theory to computations concerning LaTeXMLMath , the subgroup of cocycles that are extendible to null-cobordant immersions .
We actually obtain the following satisfactory statement : Let LaTeXMLMath be an LaTeXMLMath -manifold , with LaTeXMLMath .
Then for any LaTeXMLMath LaTeXMLEquation .
Proof .
We apply proposition LaTeXMLRef .
We have to prove that , for any LaTeXMLMath , given a cobordism LaTeXMLMath between LaTeXMLMath -admissible immersions , one can obtain from LaTeXMLMath a LaTeXMLMath -admissible cobordism .
By lemma LaTeXMLRef it is enough to show recursively that if LaTeXMLMath is LaTeXMLMath -admissible , for LaTeXMLMath , then one can obtain from it a LaTeXMLMath -admissible cobordism LaTeXMLMath such that for any LaTeXMLMath -cube LaTeXMLMath of LaTeXMLMath the intersection LaTeXMLMath is a trivial element of LaTeXMLMath .
Recall that LaTeXMLMath actually represents an element of LaTeXMLMath since LaTeXMLMath is LaTeXMLMath -admissible .
We already know by proposition LaTeXMLRef that any cobordism LaTeXMLMath between 2-admissible immersions can be considered to be 2-admissible .
Assume then that LaTeXMLMath and LaTeXMLMath are 3-admissible immersions , and let LaTeXMLMath be a 2-admissible cobordism between them .
For any 1-edge LaTeXMLMath of LaTeXMLMath the immersion LaTeXMLMath represents a well-defined element of LaTeXMLMath .
We define a 2-cocycle in LaTeXMLMath this way LaTeXMLEquation .
This means that if LaTeXMLMath is horizontal LaTeXMLMath and if LaTeXMLMath is vertical of the form LaTeXMLMath then LaTeXMLMath .
Remark that LaTeXMLMath is closed as a cocycle in LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath being 3-admissible implies that LaTeXMLMath is empty .
We claim that LaTeXMLMath is an extendible cocycle ( with compact support ) in LaTeXMLMath .
If it is so , then any associated immersion LaTeXMLMath in LaTeXMLMath is such that LaTeXMLMath is a 2-admissible cobordism such that LaTeXMLMath for all LaTeXMLMath , hence can be deformed to a 3-admissible cobordism .
We first build a particular extension of LaTeXMLMath to the vertical 3-skeleton of LaTeXMLMath this way .
Given any vertical 3-cube remark that LaTeXMLMath evaluates non-trivially on an even number of its 4 vertical 2-faces ( possibly none ) .
Trace a vertical 8 on any vertical face with non-trivial LaTeXMLMath , and connect the 8 ’ s in pairs by means of tubes whose section is a vertical 8 .
If all of the 4 faces are traced , the pairs must be of adjacent faces .
Now remark that this first extension is further extendible to the vertical 4-skeleton .
Indeed consider any vertical 4-cube .
The collection of its 6 vertical 3-faces ( that can be visualized as an LaTeXMLMath ) contains a disjoint union of immersions each representing an element of LaTeXMLMath .
But each of these immersions is the trivial element .
Indeed by construction the top of the 8 describes a curve which bounds a disk not intersecting the curve of double points , hence having trivial linking number with it , and the double of this linking number is the Arf invariant of the immersion .
Hence LaTeXMLMath is extendible to the vertical 4-skeleton .
The following obstructions are all trivial by theorem LaTeXMLRef , so LaTeXMLMath is extendible to LaTeXMLMath , as it was claimed .
Now given a LaTeXMLMath -admissible cobordism LaTeXMLMath between LaTeXMLMath -admissible immersions , LaTeXMLMath , define the same way the LaTeXMLMath -cochain LaTeXMLMath closed in LaTeXMLMath and by directly applying theorem LaTeXMLRef extend it to an immersion LaTeXMLMath in LaTeXMLMath .
Up to adding LaTeXMLMath an appropriate number of times one obtains a cobordism LaTeXMLMath that is still LaTeXMLMath -admissible but such that LaTeXMLMath for all LaTeXMLMath -cube LaTeXMLMath of LaTeXMLMath , hence that can be deformed to a LaTeXMLMath -admissible cobordism .
LaTeXMLMath The graded group LaTeXMLMath is isomorphic to LaTeXMLMath as a set , but looses its group structure .
We give a result concerning the group structure when LaTeXMLMath is an orientable 4-manifold .
We first remark that for any LaTeXMLMath -manifold the total James-Hopf invariant LaTeXMLMath , that is , the product of the James-Hopf invariants composed with Poincaré duality , becomes a homomorphism of groups with LaTeXMLMath endowed with the group structure coming from that of algebra LaTeXMLEquation .
In dimension 3 the invariant LaTeXMLMath provides completely the group structure , up to immersions that are contained in a ball .
These last form a subgroup that can be detected by a version of the Arf invariant of LaTeXMLMath ( though Benedetti and Silhol provided a deeper invariant ) .
Up to immersions in a ball , any class can be realized as the decoration of an embedded representative and any immersion can be split in a unique way as the sum of immersions obtained by decorating a submanifold , that is , LaTeXMLMath is surjective and injective .
Neither of those properties hold for orientable 4-manifolds .
Indeed from the main theorem self-intersection of 2-classes is the ( only ) obstruction for decorating an embedded representative .
Moreover decorating a simple curve with an element of LaTeXMLMath provides an immersion in LaTeXMLMath with trivial LaTeXMLMath but non-trivial LaTeXMLMath .
We define the map LaTeXMLMath from LaTeXMLMath in LaTeXMLMath that associates to LaTeXMLMath an embedded curve representing LaTeXMLMath decorated by the canonical immersion with invariant LaTeXMLMath .
This map results to be well-defined , that is , LaTeXMLMath does not depend either on the representative of LaTeXMLMath nor on the trivialization of the normal bundle .
Moreover the image of LaTeXMLMath is the subgroup of LaTeXMLMath whose support is LaTeXMLMath ( see LaTeXMLCite for more details ) .
The following holds .
There is a short exact sequence of groups LaTeXMLEquation .
Proof .
That LaTeXMLMath is injective follows from the fact that images of different cycles have different LaTeXMLMath and this last is injective .
We show exactness in the middle term .
That LaTeXMLMath is trivial for any LaTeXMLMath follows since any representative LaTeXMLMath of LaTeXMLMath has no triple points nor quadruple points , and the locus of its double points is a surface representing the trivial element of LaTeXMLMath .
The whole of LaTeXMLMath retracts on the decorated curve in fact , hence also LaTeXMLMath is trivial .
Now suppose that LaTeXMLMath .
We must prove that LaTeXMLMath is in the image of the map LaTeXMLMath .
Since LaTeXMLMath , from lemma LaTeXMLRef LaTeXMLMath belongs to LaTeXMLMath .
Assume then that LaTeXMLMath is 2-admissible .
Now LaTeXMLMath has trivial second cohomological invariant , since LaTeXMLMath , hence in particular LaTeXMLMath belongs to LaTeXMLMath , see proposition LaTeXMLRef .
Assume that LaTeXMLMath is 3-admissible and consider LaTeXMLMath .
This class has trivial reduction modulo 2 , since LaTeXMLEquation so there is an element LaTeXMLMath such that LaTeXMLMath .
It is easy to see that LaTeXMLMath .
LaTeXMLMath The group of the complex projective plane is LaTeXMLMath generated by a non-trivial immersion in a small ball .
The cobordism group of a 4-manifold is generated by embedded decorated submanifolds .
It is proven in LaTeXMLCite that the cobordism class of a codimension-one embedding only depends on homology modulo 2 , and in LaTeXMLCite that the cobordism class of a decorated curve only depends on the ( oriented ) homology of the curve and the cobordism class of the decorating immersion .
So if LaTeXMLMath is an oriented 4-manifold such that LaTeXMLMath one can choose a the set of generators of LaTeXMLMath by choosing a set LaTeXMLMath of oriented curves generating LaTeXMLMath and a set LaTeXMLMath of codimension-one embeddings generating LaTeXMLMath and considering the set LaTeXMLMath , LaTeXMLMath denoting the left Boy immersion and LaTeXMLMath denoting a generator of LaTeXMLMath .
The image of LaTeXMLMath being LaTeXMLMath implies in particular that for any pair of 1-cocycles modulo 2 , LaTeXMLMath and LaTeXMLMath , the relation LaTeXMLMath holds .
This fact has an elementary geometric proof .
Represent the dual of LaTeXMLMath and LaTeXMLMath by means of embedded hypersurfaces LaTeXMLMath and LaTeXMLMath .
Remark that if LaTeXMLMath is orientable then LaTeXMLMath is 0 , and if LaTeXMLMath is non-orientable its self-intersection is the orientation cycle of LaTeXMLMath , hence can be represented by an orientable surface LaTeXMLMath in LaTeXMLMath .
Call LaTeXMLMath the curve intersection between LaTeXMLMath and LaTeXMLMath .
Then a representative of the dual of LaTeXMLMath is the intersection between LaTeXMLMath and LaTeXMLMath .
But the normal bundle to LaTeXMLMath restricted to LaTeXMLMath is orientable since it is the normal bundle to LaTeXMLMath in LaTeXMLMath , that is trivial since LaTeXMLMath is orientable , and hence LaTeXMLMath .
For a space LaTeXMLMath acted by a finite group LaTeXMLMath , the product space LaTeXMLMath affords a natural action of the wreath product LaTeXMLMath .
The direct sum of equivariant LaTeXMLMath -groups LaTeXMLMath were shown earlier by the author to carry several interesting algebraic structures .
In this paper we study the LaTeXMLMath -groups LaTeXMLMath of LaTeXMLMath -equivariant Clifford supermodules on LaTeXMLMath .
We show that LaTeXMLMath is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra .
Twisted vertex operators make a natural appearance .
The algebraic structures on LaTeXMLMath , when LaTeXMLMath is trivial and LaTeXMLMath is a point , specialize to those on a ring of symmetric functions with the Schur LaTeXMLMath -functions as a linear basis .
As a by-product , we present a novel construction of LaTeXMLMath -theory operations using the spin representations of the hyperoctahedral groups .
Motivated in part by Vafa-Witten LaTeXMLCite and generalizing the work of Segal LaTeXMLCite ( also cf .
Grojnowski LaTeXMLCite ) , we studied in LaTeXMLCite a direct sum , denoted by LaTeXMLMath , of the equivariant LaTeXMLMath -groups LaTeXMLMath associated to a topological LaTeXMLMath -space LaTeXMLMath .
Here LaTeXMLMath is a finite group and the wreath product LaTeXMLMath ( i.e .
the semi-direct product LaTeXMLMath ) acts naturally on the LaTeXMLMath th Cartisian product LaTeXMLMath .
We proved that the space LaTeXMLMath carries several remarkable algebraic structures such as Hopf algebra and Fock representation of a Heisenberg ( super ) algebra etc , and that vertex operators makes a natural appearance as a part of LaTeXMLMath -ring structure .
We in addition pointed out in LaTeXMLCite a new approach to the realization of the Frenkel-Kac-Segal homogeneous vertex representations of affine Lie algebras by using representation rings of the wreath product LaTeXMLMath associated to a finite subgroup LaTeXMLMath of LaTeXMLMath .
This has been subsequently completed in LaTeXMLCite jointly with I. Frenkel and Jing , and further extended in LaTeXMLCite to realize vertex representations of twisted affine and toroidal Lie algebras by using the spin representation rings of a double cover LaTeXMLMath of the wreath product LaTeXMLMath .
In this paper we will introduce a variant of equivariant LaTeXMLMath -theory .
Given a topological space LaTeXMLMath acted upon by a finite supergroup LaTeXMLMath in an appropriate sense , we introduce a category of complex LaTeXMLMath -equivariant spin vector super bundles over LaTeXMLMath , and consider the corresponding Grothendieck group LaTeXMLMath .
The superscript LaTeXMLMath here and below is used in this paper to stand for spin , i.e .
a certain central element LaTeXMLMath in the supergroup LaTeXMLMath acts as LaTeXMLMath .
This LaTeXMLMath -group can be thought as an invariant of orbifolds with ( certain distinguished ) discrete torsion as introduced by Vafa LaTeXMLCite .
Discrete torsion has been a topic of interest from various viewpoints since then , cf .
LaTeXMLCite and the references therein .
We present here a variant of the decomposition theorem of Adem and Ruan LaTeXMLCite for what they call twisted equivariant LaTeXMLMath -theory , which generalizes the decomposition theorem LaTeXMLCite ( also cf .
LaTeXMLCite ) in equivariant LaTeXMLMath -theory .
Our formulation of such LaTeXMLMath -groups is motivated by providing a general framework for the main subject of study in this paper , namely a spin/twisted version of the space LaTeXMLMath studied in LaTeXMLCite .
When the topological space under consideration is a point , our LaTeXMLMath -theory specializes to the theory of supermodules of finite supergroups , cf .
Józefiak LaTeXMLCite .
( In this paper super always means LaTeXMLMath -graded ) .
Such a theory of supermodules has provided , in our opinion , a most natural framework ( cf .
LaTeXMLCite ) for the exposition of the spin representations of a double cover LaTeXMLMath of the symmetric group initiated by Schur LaTeXMLCite .
The spin representation theory of a double cover LaTeXMLMath of the hyperoctahedral group , being almost parallel to and to some extent simpler than the spin representation theory for LaTeXMLMath , can also be treated successfully in terms of supermodules ( cf .
LaTeXMLCite ) .
Given a space LaTeXMLMath acted by a ( non-graded ) finite group LaTeXMLMath , we obtain our main example of such LaTeXMLMath -group LaTeXMLMath by considering the action on the LaTeXMLMath -th Cartesian product LaTeXMLMath by the wreath product LaTeXMLMath which is further extended trivially to the action of a larger finite supergroup LaTeXMLMath .
Here LaTeXMLMath is a double cover of the wreath product LaTeXMLMath .
( We recommend that the reader sets LaTeXMLMath to be the one-element group in their first reading so that the whole picture becomes simpler and more transparent .
In this case LaTeXMLMath reduces to a double cover LaTeXMLMath of the Hyperoctahedral group . )
The category of LaTeXMLMath -equivariant spin vector superbundles over LaTeXMLMath admits an equivalent reformulation which has a perhaps better geometric meaning .
Namely this is the category of LaTeXMLMath -equivariant vector bundles LaTeXMLMath over LaTeXMLMath such that LaTeXMLMath carries a supermodule structure with respect to the complex Clifford algebra of rank LaTeXMLMath which is compatible with the action of LaTeXMLMath .
A fundamental example of LaTeXMLMath -vector superbundles over LaTeXMLMath , which plays an important role in this paper , is given as follows for LaTeXMLMath compact .
Given a LaTeXMLMath -vector bundle LaTeXMLMath over LaTeXMLMath , we consider the vector superbundle LaTeXMLMath over LaTeXMLMath with the natural LaTeXMLMath -grading .
We can endow the LaTeXMLMath th outer tensor LaTeXMLMath a natural LaTeXMLMath -equivariant vector superbundle structure over LaTeXMLMath .
We will show that the direct sum LaTeXMLEquation carries naturally a Hopf algebra structure and it is isomorphic to the Fock space of a twisted Heisenberg superalgebra associated to LaTeXMLMath .
All the algebraic structures are constructed in terms of natural LaTeXMLMath -theory maps .
In particular the dimension of LaTeXMLMath is determined explicitly for all LaTeXMLMath .
We remark that such a twisted Heisenberg algebra has played an important role in the theory of affine Kac-Moody algebras , cf .
LaTeXMLCite .
Roughly speaking , the structure of the space LaTeXMLMath studied in LaTeXMLCite is modeled on the ring LaTeXMLMath of symmetric functions with a basis given by Schur functions ( or equivalently on the direct sum of representation rings of symmetric groups LaTeXMLMath for all LaTeXMLMath ) .
The structure of the space LaTeXMLMath under consideration of this paper is shown to be modeled instead on the ring LaTeXMLMath of symmetric functions with a linear basis given by the so-called Schur LaTeXMLMath -functions ( or equivalently on the direct sum of the spin representation ring of LaTeXMLMath for all LaTeXMLMath ) .
It is well known that the graded dimension of the ring LaTeXMLMath is given by the generating function LaTeXMLEquation where LaTeXMLMath is the Dedekind LaTeXMLMath function .
Just as the LaTeXMLMath ring is modeled on the ring LaTeXMLMath of symmetric functions ( cf .
e.g .
LaTeXMLCite ) , one can introduce a LaTeXMLMath - LaTeXMLMath ring LaTeXMLMath stands for Queer or Schur LaTeXMLMath -functions .
We believe that there exists a rich LaTeXMLMath -mathematical world which is relevant to various twisted , spin , super structures , etc .
structure modeled on the ring LaTeXMLMath with Adams operations of odd degrees only .
We show that as a part of the LaTeXMLMath - LaTeXMLMath ring structure on LaTeXMLMath twisted vertex operators naturally appear in LaTeXMLMath via the LaTeXMLMath -th outer tensor LaTeXMLMath associated to LaTeXMLMath in terms of the Adams operations .
It is of independent interest to see that when we restrict LaTeXMLMath from LaTeXMLMath to its diagonal we are able to obtain various LaTeXMLMath -theory operations on LaTeXMLMath , including supersymmetric power operations and Adams operations of odd degrees , by means of the spin supermodules of LaTeXMLMath .
This is a super analog of Atiyah ’ s construction LaTeXMLCite of LaTeXMLMath -theory operations on LaTeXMLMath by means of the representations of the symmetric groups .
Motivated by Göttsche ’ s formula , Vafa and Witten LaTeXMLCite conjectured that the direct sum LaTeXMLMath of homology groups for Hilbert schemes LaTeXMLMath of LaTeXMLMath points on a ( quasi- ) projective surface LaTeXMLMath should carry the structure of a Fock space of a Heisenberg algebra , which was realized subsequently in a geometric way by Nakajima and Grojnowski ( cf .
LaTeXMLCite ) .
Parallel algebraic structures such as Hopf algebra , vertex operators , and Heisenberg algebra as part of vertex algebra structures LaTeXMLCite , have naturally showed up in LaTeXMLMath as well as in LaTeXMLMath .
When LaTeXMLMath is a suitable resolution of singularities of an orbifold LaTeXMLMath , there appears close connections between LaTeXMLMath and LaTeXMLMath , cf .
LaTeXMLCite and the references therein .
It will be very important to see if one can find a ‘ Hilbert scheme ’ version of the orbifold picture drawn in this paper .
It is also interesting to see if our current work can find some applications in string theory , cf .
LaTeXMLCite .
In fact the special case of our construction for LaTeXMLMath trivial is closely related to an earlier paper of Dijkgraaf LaTeXMLCite .
See the Appendix .
When LaTeXMLMath is a point , the LaTeXMLMath -group LaTeXMLMath becomes the Grothendieck group of spin supermodules of LaTeXMLMath .
In a companion paper LaTeXMLCite joint with Jing , the conjugacy classes and Grothendieck groups of LaTeXMLMath have been studied in detail .
In particular , when LaTeXMLMath is a subgroup of LaTeXMLMath , they are used to realize vertex representations of twisted affine algebras ( cf .
LaTeXMLCite ) and toroidal Lie algebras .
This provides a new group theoretic construction , a somewhat improved version in our opinion than the one in LaTeXMLCite using the groups LaTeXMLMath , of twisted vertex representations .
In the Appendix , we sketch another formulation of our main results in this paper using the groups LaTeXMLMath and LaTeXMLMath instead of LaTeXMLMath and LaTeXMLMath .
The plan of this paper is as follows .
In Sect .
LaTeXMLRef we present the spin representation rings of the finite supergroup LaTeXMLMath .
In Sect .
LaTeXMLRef we introduce the category and LaTeXMLMath -groups of spin vector superbundles which are equivariant with respect to finite supergroups and present a decomposition theorem for such equivariant LaTeXMLMath -groups .
In Sect .
LaTeXMLRef we study LaTeXMLMath -theory operations based on the spin representations of LaTeXMLMath .
The results in this section are not to be further used in this paper .
In Sect .
LaTeXMLRef , which is the heart of the paper , we present the structures of a Hopf algebra and of a LaTeXMLMath - LaTeXMLMath ring on LaTeXMLMath , and relate the latter to the twisted vertex operators .
We further construct a Heisenberg algebra which acts on LaTeXMLMath irreducibly by means of natural LaTeXMLMath -theory maps .
In the Appendix , we outline a somewhat different construction in terms of the group LaTeXMLMath .
In this section we recall briefly some essential points in the theory of supermodules of a finite supergroup ( cf .
LaTeXMLCite ) .
We define the finite supergroup LaTeXMLMath associated to any finite group LaTeXMLMath , and study its conjugacy classes and spin supermodules .
More detail of these can be found in LaTeXMLCite .
Let LaTeXMLMath be a finite group and let LaTeXMLMath be a group epimorphism .
We denote by LaTeXMLMath the kernel of LaTeXMLMath which is a subgroup of LaTeXMLMath of index LaTeXMLMath .
We regard LaTeXMLMath as a parity function on LaTeXMLMath by letting the degree of elements in LaTeXMLMath be LaTeXMLMath and the degree of elements in the complementary LaTeXMLMath be LaTeXMLMath .
Elements in LaTeXMLMath ( resp .
LaTeXMLMath ) will be called even ( resp .
odd ) .
We will often refer to the pair LaTeXMLMath , or simply LaTeXMLMath when there is no ambiguity , as a finite supergroup .
The class of finite supergroups under consideration in this paper has an additional property : it contains a distinguished even central element LaTeXMLMath of order LaTeXMLMath .
We denote by LaTeXMLMath the quotient group homomorphism LaTeXMLMath In this paper the modules over a finite supergroup or a superalgebra ( such as the group superalgebra of a finite supergroup ) will always be LaTeXMLMath -graded ( i.e .
supermodules ) unless otherwise specified .
A general theory of supermodules over finite supergroups was developed by Józefiak LaTeXMLCite .
This was motivated to provide a modern account LaTeXMLCite of Schur ’ s seminal work on spin representations of symmetric groups LaTeXMLCite .
Given two supermodules LaTeXMLMath and LaTeXMLMath over a superalgebra LaTeXMLMath , the linear map LaTeXMLMath between two LaTeXMLMath -supermodules is a homomorphism of degree LaTeXMLMath if LaTeXMLMath and for any homogeneous element LaTeXMLMath and any homogeneous vector LaTeXMLMath we have LaTeXMLEquation .
The degree LaTeXMLMath ( resp .
LaTeXMLMath ) part of a superspace is referred to as the even ( resp .
odd ) part .
We denote LaTeXMLEquation where LaTeXMLMath consists of LaTeXMLMath -homomorphisms of degree LaTeXMLMath from LaTeXMLMath to LaTeXMLMath .
The notions of submodules , tensor product , and irreducibility etc for supermodules are defined similarly .
Given a finite supergroup LaTeXMLMath , a LaTeXMLMath -supermodule LaTeXMLMath is called spin if the central element LaTeXMLMath acts as LaTeXMLMath .
Alternatively , one can associate a LaTeXMLMath -cocycle LaTeXMLMath , such that LaTeXMLMath becomes a projective supermodule of the group LaTeXMLMath associated with LaTeXMLMath , namely , the actions of any two elements LaTeXMLMath on LaTeXMLMath , denoted by LaTeXMLMath , satisfy the relation LaTeXMLEquation .
Among all LaTeXMLMath -supermodules , we will only consider the spin supermodules in this paper .
It is clear that the restriction ( resp .
the induction ) of a spin supermodule to a LaTeXMLMath -graded subgroup ( resp .
a larger supergroup ) with the same distinguished even element LaTeXMLMath remains to be a spin supermodule .
There are two types of complex simple superalgebras according to C.T.C .
Wall : LaTeXMLMath and LaTeXMLMath .
Here LaTeXMLMath is the superalgebra consisting of the linear transformations of the superspace LaTeXMLMath .
The superalgebra LaTeXMLMath is the graded subalgebra of LaTeXMLMath consisting of matrices of the form LaTeXMLEquation .
It is known LaTeXMLCite that the group ( super ) algebra of a finite supergroup is semisimple , i.e .
decomposes into a direct sum of simple superalgebras .
According to the classification of simple superalgebras above , the irreducible supermodules of a finite supergroup are divided into two types , type LaTeXMLMath and type LaTeXMLMath .
We note that the endomorphism algebra of an irreducible supermodule LaTeXMLMath is isomorphic to LaTeXMLMath if LaTeXMLMath is of type LaTeXMLMath and isomorphic to the complex Clifford algebra LaTeXMLMath in one variable if LaTeXMLMath is of type LaTeXMLMath .
Let LaTeXMLMath the group generated by LaTeXMLMath subject to the relations LaTeXMLEquation .
The symmetric group LaTeXMLMath acts on LaTeXMLMath via permutations of the elements LaTeXMLMath , i.e .
LaTeXMLMath for LaTeXMLMath .
We thus form the semi-direct product LaTeXMLMath , which naturally endows a LaTeXMLMath -grading given by the parity LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath .
We may therefore regard LaTeXMLMath as a finite supergroup with a distinguished even central element .
Note that the group superalgebra LaTeXMLMath is exactly the complex Clifford algebra LaTeXMLMath in LaTeXMLMath variables .
Let LaTeXMLMath be a finite group with LaTeXMLMath conjugacy classes .
We denote by LaTeXMLMath the set of complex irreducible characters where LaTeXMLMath denotes the trivial character , and by LaTeXMLMath the set of conjugacy classes .
Let LaTeXMLMath be the space of class functions on LaTeXMLMath , and set LaTeXMLMath .
For LaTeXMLMath let LaTeXMLMath be the order of the centralizer of an element in the conjugacy class LaTeXMLMath , so the order of the class is then LaTeXMLMath .
Given a positive integer LaTeXMLMath , let LaTeXMLMath be the LaTeXMLMath -th direct product of LaTeXMLMath , and let LaTeXMLMath be the trivial group .
The symmetric group LaTeXMLMath naturally acts on LaTeXMLMath by simultaneous permutations of elements in LaTeXMLMath and LaTeXMLMath .
The finite supergroup LaTeXMLMath is then defined to be the semi-direct product of the symmetric group LaTeXMLMath and LaTeXMLMath , with the multiplication given by LaTeXMLEquation .
The order of LaTeXMLMath is clearly LaTeXMLMath .
The LaTeXMLMath -grading on LaTeXMLMath is induced from that on LaTeXMLMath and by letting the elements in LaTeXMLMath be even ( i.e .
of degree LaTeXMLMath ) .
Denoting by LaTeXMLMath , we have the following exact sequence of groups : LaTeXMLEquation .
It is clear that when LaTeXMLMath is trivial LaTeXMLMath reduces to a double cover LaTeXMLMath of the hyperoctahedral group LaTeXMLMath .
The finite supergroup LaTeXMLMath contains LaTeXMLMath and the wreath product LaTeXMLMath as distinguished subgroups .
Letting LaTeXMLMath act trivially on LaTeXMLMath we extend the action of the symmetric group LaTeXMLMath to LaTeXMLMath on LaTeXMLMath .
In this way we may also view LaTeXMLMath as a semi-direct product between LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a finite supergroup and put LaTeXMLMath and LaTeXMLMath as before .
For any conjugacy class LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is either a conjugacy class of LaTeXMLMath or it splits into two conjugacy classes of LaTeXMLMath , cf .
LaTeXMLCite .
If LaTeXMLMath splits , the conjugacy class LaTeXMLMath will be referred to as split , and an element LaTeXMLMath in LaTeXMLMath is also called split , which is equivalent to say that the two preimages of LaTeXMLMath under LaTeXMLMath are not conjugate to each other .
Otherwise LaTeXMLMath is said to be non-split .
In view of ( LaTeXMLRef ) , we have the following easy equivalent formulation .
An element LaTeXMLMath in LaTeXMLMath is split if and only if LaTeXMLEquation defines a trivial character of the centralizer group LaTeXMLMath .
For the study of spin supermodules of LaTeXMLMath , it is crucial to have a detailed description of split conjugacy classes of LaTeXMLMath .
Indeed the characters of spin supermodules vanish on non-split conjugacy classes as well as on odd split classes , cf .
LaTeXMLCite .
Below we will concentrate on the group LaTeXMLMath .
Let LaTeXMLMath be a partition of integer LaTeXMLMath , where LaTeXMLMath .
The integer LaTeXMLMath is called the length of the partition LaTeXMLMath and is denoted by LaTeXMLMath .
We will also make use of another notation for partitions : LaTeXMLEquation where LaTeXMLMath is the number of parts in LaTeXMLMath equal to LaTeXMLMath .
A partition LaTeXMLMath is strict if its parts are distinct integers , namely all the multiplicities LaTeXMLMath are LaTeXMLMath or LaTeXMLMath .
Given a partition LaTeXMLMath of LaTeXMLMath , we define LaTeXMLEquation .
We note that LaTeXMLMath is the order of the centralizer of an element of cycle-type LaTeXMLMath in LaTeXMLMath .
For a finite set LaTeXMLMath and LaTeXMLMath a family of partitions indexed by LaTeXMLMath , we write LaTeXMLEquation .
It is convenient to regard LaTeXMLMath as a partition-valued function on LaTeXMLMath .
We denote by LaTeXMLMath the set of all partitions indexed by LaTeXMLMath and by LaTeXMLMath the set of all partitions in LaTeXMLMath such that LaTeXMLMath .
The total number of parts , denoted by LaTeXMLMath , is called the length of LaTeXMLMath .
Let LaTeXMLMath be the set of partition-valued functions LaTeXMLMath in LaTeXMLMath such that all parts of the partitions LaTeXMLMath are odd integers , and let LaTeXMLMath be the set of partition-valued functions LaTeXMLMath such that each partition LaTeXMLMath is strict .
It is clear that LaTeXMLMath When LaTeXMLMath consists of a single element , we will omit LaTeXMLMath and simply write LaTeXMLMath for LaTeXMLMath , thus LaTeXMLMath or LaTeXMLMath will be used similarly .
We denote by LaTeXMLEquation .
LaTeXMLEquation and define LaTeXMLMath for LaTeXMLMath .
The conjugacy classes of a wreath product is well understood , cf .
LaTeXMLCite .
In particular this gives us the following description of conjugacy classes of the wreath product LaTeXMLMath .
Let LaTeXMLMath be an element in a conjugacy class of LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
We take the convention here that LaTeXMLMath .
For each cycle LaTeXMLMath in the permutation LaTeXMLMath consider the element LaTeXMLMath and LaTeXMLMath ( which is LaTeXMLMath ) corresponding to the cycle LaTeXMLMath .
For each LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath be the number of LaTeXMLMath -cycles in the permutation LaTeXMLMath such that the cycle product LaTeXMLMath lie in the conjugacy class LaTeXMLMath and LaTeXMLMath equals LaTeXMLMath .
Then LaTeXMLMath defines a partition-valued function on LaTeXMLMath for each LaTeXMLMath .
The partition-valued function LaTeXMLMath is called the LaTeXMLMath -type of LaTeXMLMath .
Denote by LaTeXMLMath the set of pairs of partition-valued functions LaTeXMLMath such that LaTeXMLMath .
The pair LaTeXMLMath is called the type of LaTeXMLMath .
One can show that the type only depends on the conjugacy class of LaTeXMLMath in LaTeXMLMath and the conjugacy classes in LaTeXMLMath are parameterized by the types LaTeXMLMath .
We will also say that the conjugacy class containing LaTeXMLMath has conjugacy type LaTeXMLMath and is denoted by LaTeXMLMath if LaTeXMLMath is of type LaTeXMLMath .
Denote by LaTeXMLMath .
The following is established in LaTeXMLCite , Theorem 2.1 .
For LaTeXMLMath , LaTeXMLMath splits into two conjugacy classes in LaTeXMLMath if and only if : ( 1 ) for LaTeXMLMath is even we have LaTeXMLMath and LaTeXMLMath ; ( 2 ) for LaTeXMLMath is odd we have LaTeXMLMath and LaTeXMLMath .
Thus , the set LaTeXMLMath of even split conjugacy classes in LaTeXMLMath is in one-to-one correspondence with the set LaTeXMLMath .
The number of irreducible spin supermodules of LaTeXMLMath equals the number of even split conjugacy classes in LaTeXMLMath , by a general theorem in the supermodule theory of finite supergroups LaTeXMLCite .
The next proposition follows from the equality LaTeXMLMath .
The number of irreducible spin supermodules of LaTeXMLMath is equal to the number LaTeXMLMath of strict partition-valued functions on LaTeXMLMath .
We denote by LaTeXMLMath ( resp .
LaTeXMLMath ) the LaTeXMLMath -span ( resp .
LaTeXMLMath -span ) of the characters of irreducible spin supermodules of LaTeXMLMath .
Let LaTeXMLEquation .
When LaTeXMLMath is trivial , we will simply drop the subscript LaTeXMLMath and write LaTeXMLEquation .
For example , when LaTeXMLMath is trivial and thus LaTeXMLMath reduces to LaTeXMLMath , the irreducible spin supermodules of LaTeXMLMath are parameterized by strict partitions of LaTeXMLMath ( cf .
LaTeXMLCite and LaTeXMLCite ) .
For strict partitions LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , let LaTeXMLMath and LaTeXMLMath denote the corresponding irreducible spin supermodules over LaTeXMLMath .
We have LaTeXMLEquation where the number LaTeXMLMath is LaTeXMLMath for LaTeXMLMath even and LaTeXMLMath otherwise .
That is , the supermodule LaTeXMLMath is of type LaTeXMLMath ( resp .
type LaTeXMLMath ) if and only if LaTeXMLMath is even ( resp .
odd ) .
A most distinguished example of irreducible LaTeXMLMath -supermodule is the so-called basic spin supermodule LaTeXMLMath constructed as follows ( cf .
LaTeXMLCite ) .
As a superspace LaTeXMLMath is isomorphic to the group superalgebra LaTeXMLMath ( i.e .
the Clifford algebra in LaTeXMLMath variables ) .
Denote by LaTeXMLMath the image of LaTeXMLMath .
Then LaTeXMLMath , form a linear basis of LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath is given by LaTeXMLEquation .
Indeed LaTeXMLMath is exactly the LaTeXMLMath -supermodule LaTeXMLMath associated to the one-part partition LaTeXMLMath .
If we denote by LaTeXMLMath the character of LaTeXMLMath , then the character value of LaTeXMLMath is LaTeXMLMath on a conjugacy class of type LaTeXMLMath , where LaTeXMLMath .
For each partition-valued function LaTeXMLMath we define LaTeXMLEquation which is the order of the centralizer of an element in LaTeXMLMath of conjugacy type LaTeXMLMath ( see LaTeXMLCite ) .
For a fixed LaTeXMLMath , we denote by LaTeXMLMath ( LaTeXMLMath ) the even split conjugacy class in LaTeXMLMath of the type LaTeXMLMath , where the partition-valued function LaTeXMLMath takes value the one-part partition LaTeXMLMath at LaTeXMLMath and zero elsewhere .
We denote by LaTeXMLMath the class function of LaTeXMLMath which takes value LaTeXMLMath at the conjugacy class LaTeXMLMath LaTeXMLMath and zero otherwise .
For LaTeXMLMath , we define LaTeXMLEquation and regard it as the class function on LaTeXMLMath which takes value LaTeXMLMath at the conjugacy class LaTeXMLMath and LaTeXMLMath elsewhere .
Then it follows that ( cf .
LaTeXMLCite ) LaTeXMLEquation .
Finally we define the analog for LaTeXMLMath of Young subgroups of the symmetric groups .
Let LaTeXMLMath be the direct product of LaTeXMLMath and LaTeXMLMath with a twisted multiplication LaTeXMLEquation where LaTeXMLMath .
We define the spin product of LaTeXMLMath and LaTeXMLMath by letting LaTeXMLEquation which carries a canonical LaTeXMLMath -grading and can be regarded as a subgroup of the supergroup LaTeXMLMath in a natural way .
For two spin supermodules LaTeXMLMath and LaTeXMLMath of LaTeXMLMath and LaTeXMLMath we define a LaTeXMLMath spin supermodule on the tensor product LaTeXMLMath by letting LaTeXMLEquation .
This induces an isomorphism LaTeXMLMath from LaTeXMLMath to LaTeXMLMath .
The space LaTeXMLMath carries a ( commutative associative ) multiplication which is defined by the composition ( for all LaTeXMLMath ) LaTeXMLEquation .
In this section we introduce a variant of equivariant LaTeXMLMath -theory for a finite supergroup .
We recall a decomposition theorem in the equivariant LaTeXMLMath -theory from LaTeXMLCite , and present a generalization of it in our new setup .
Given a ( non-graded ) finite group LaTeXMLMath and a compact Hausdorff LaTeXMLMath -space LaTeXMLMath , we recall LaTeXMLCite that LaTeXMLMath is the Grothendieck group of LaTeXMLMath -vector bundles over LaTeXMLMath .
One can define LaTeXMLMath in terms of the LaTeXMLMath functor and a certain suspension operation , and one puts LaTeXMLEquation .
In this paper we will be only concerned about LaTeXMLMath , and subsequently we will denote LaTeXMLEquation .
We denote by LaTeXMLMath the dimension of LaTeXMLMath .
If LaTeXMLMath is locally compact , Hausdorff and paracompact LaTeXMLMath -space , take the one-point compactification LaTeXMLMath with the extra point LaTeXMLMath fixed by LaTeXMLMath .
Then we define LaTeXMLMath to be the kernel of the map LaTeXMLEquation induced by the inclusion map LaTeXMLMath .
This definition is equivalent to the earlier one when LaTeXMLMath is compact .
We also define LaTeXMLMath Note that LaTeXMLMath is isomorphic to the Grothendieck ring LaTeXMLMath and LaTeXMLMath is isomorphic to the ring LaTeXMLMath of class functions on LaTeXMLMath .
Let LaTeXMLMath denote the fixed-point set by LaTeXMLMath , which is preserved under the action of the centralizer LaTeXMLMath .
The following decomposition theorem ( cf .
LaTeXMLCite ) gives a description of each direct summand over conjugacy classes of LaTeXMLMath .
We remark that the subspace of invariants LaTeXMLMath is isomorphic to LaTeXMLMath , and it is isomorphic for different choice of LaTeXMLMath in the same conjugacy class LaTeXMLMath .
There is a natural LaTeXMLMath -graded isomorphism LaTeXMLEquation .
Now let LaTeXMLMath be a finite supergroup which contains a distinguished central element LaTeXMLMath of order LaTeXMLMath , and let LaTeXMLMath be the quotient homomorphism LaTeXMLMath Let LaTeXMLMath be a compact ( non-graded ) LaTeXMLMath -space .
We may regard LaTeXMLMath as a LaTeXMLMath -space by letting LaTeXMLMath act by LaTeXMLMath .
We denote by LaTeXMLMath the category whose objects consist of LaTeXMLMath -equivariant complex vector superbundle ( i.e .
LaTeXMLMath -graded bundles ) LaTeXMLMath ( often denoted by LaTeXMLMath ) over LaTeXMLMath on which LaTeXMLMath acts in a LaTeXMLMath -graded manner and LaTeXMLMath acts as LaTeXMLMath .
We will call such an object a spin LaTeXMLMath -vector superbundle .
Given two objects LaTeXMLMath in the category LaTeXMLMath , the space of homomorphisms of LaTeXMLMath -equivariant vector superbundles between LaTeXMLMath and LaTeXMLMath admits a natural LaTeXMLMath -grading : LaTeXMLEquation .
By our definition of finite supergroups , LaTeXMLMath always contains odd elements .
It follows that LaTeXMLMath thanks to the existence of odd automorphisms from the odd elements of LaTeXMLMath .
We denote by LaTeXMLMath the Grothendieck group of the abelian monoid of isomorphism classes LaTeXMLMath -graded and isomorphisms of degree LaTeXMLMath are allowed .
of the vector superbundles in LaTeXMLMath .
As in the ordinary case , we can extend the definition of LaTeXMLMath to locally compact spaces , and define LaTeXMLMath to be LaTeXMLMath where LaTeXMLMath is the real line .
We denote LaTeXMLMath .
In this paper we will be only concerned about the free part LaTeXMLMath , which will be denoted by LaTeXMLMath subsequently .
The following theorem , generalizing Theorem LaTeXMLRef , is a variation of Adem-Ruan ’ s construction LaTeXMLCite for the so-called twisted equivariant LaTeXMLMath -theory .
Recall that the character LaTeXMLMath of the centralizer group LaTeXMLMath was defined in Lemma LaTeXMLRef and LaTeXMLMath acts on LaTeXMLMath .
Let LaTeXMLMath be a finite supergroup which contains a distinguished central element LaTeXMLMath of order LaTeXMLMath , and let LaTeXMLMath be the quotient homomorphism LaTeXMLMath .
Given a locally compact Hausdorff LaTeXMLMath -space LaTeXMLMath and regarding it as a LaTeXMLMath -space , we have a natural LaTeXMLMath -graded isomorphism LaTeXMLEquation where the summation runs over the even conjugacy classes in LaTeXMLMath .
Let us indicate briefly how the map LaTeXMLMath is defined ( also compare LaTeXMLCite ) .
A LaTeXMLMath -equivariant vector superbundle LaTeXMLMath , when restricted to LaTeXMLMath , becomes a spin LaTeXMLMath -vector superbundle , where LaTeXMLMath denotes the subgroup of LaTeXMLMath covering the cyclic subgroup LaTeXMLMath of LaTeXMLMath generated by LaTeXMLMath .
Using Proposition LaTeXMLRef , we can obtain the following isomorphisms of LaTeXMLMath -modules : LaTeXMLEquation .
In this way we obtain a map LaTeXMLMath .
Composing this map with the character evaluation at LaTeXMLMath gives us a map LaTeXMLEquation whose image indeed is LaTeXMLMath -invariant .
Now we claim that this map is zero when LaTeXMLMath is odd ( thanks to the LaTeXMLMath -grading ! )
and thus the summation above does not involve the odd conjugacy classes .
Let LaTeXMLMath be an odd element .
Take an eigenvector LaTeXMLMath of LaTeXMLMath in the fiber of the ungraded subbundle LaTeXMLMath , where LaTeXMLMath .
We see that LaTeXMLMath and LaTeXMLMath It follows that LaTeXMLMath is an eigenvector of LaTeXMLMath with eigenvalue LaTeXMLMath , i.e .
LaTeXMLMath .
Denote by LaTeXMLMath the isomorphism of LaTeXMLMath which is the identity map when restricted to LaTeXMLMath and negative the identity map when restricted to LaTeXMLMath .
Clearly LaTeXMLMath sends LaTeXMLMath to LaTeXMLMath and vice versa .
Thus the map ( LaTeXMLRef ) becomes zero since LaTeXMLMath .
Putting ( LaTeXMLRef ) together for all even conjugacy classes , we obtain the map LaTeXMLMath .
The rest of the proof of the above theorem is the same as in LaTeXMLCite which in turn follows closely the classical case ( cf .
LaTeXMLCite ) .
Below we will single out a certain class of LaTeXMLMath -space LaTeXMLMath with a favorable property .
Assume we are in the setup of Theorem LaTeXMLRef .
We say the LaTeXMLMath -space LaTeXMLMath satisfies a strong vanishing property if for every even non-split conjugacy class LaTeXMLMath in LaTeXMLMath , there exists some element LaTeXMLMath in LaTeXMLMath such that the character LaTeXMLMath of LaTeXMLMath takes non-trivial value ( which has to be -1 ) and the element LaTeXMLMath fixes LaTeXMLMath pointwise .
In view of Lemma LaTeXMLRef , if the LaTeXMLMath -space LaTeXMLMath satisfies the strong vanishing property , the isomorphism ( LaTeXMLRef ) will simplify to the following isomorphism LaTeXMLEquation where the summation runs over all even split conjugacy classes in LaTeXMLMath .
When LaTeXMLMath is a point the isomorphism LaTeXMLMath becomes the map from a spin supermodule of LaTeXMLMath to its character .
As is known LaTeXMLCite , the character of a spin supermodule vanishes on odd conjugacy classes as well on even non-split conjugacy classes .
In our terminology , the one-point space for any LaTeXMLMath automatically satisfies the strong vanishing property .
We shall see that the strong vanishing property holds for other non-trivial examples .
Theorem LaTeXMLRef contains Theorem LaTeXMLRef as a special case .
Indeed , let LaTeXMLMath for some finite group LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -space .
We have an isomorphism LaTeXMLEquation .
On the other hand , we note that LaTeXMLMath is isomorphic to LaTeXMLMath with the LaTeXMLMath -grading given by letting the generator LaTeXMLMath be of degree LaTeXMLMath .
The quotient of LaTeXMLMath by LaTeXMLMath is LaTeXMLMath .
The even conjugacy class in LaTeXMLMath is given by the conjugacy classes in LaTeXMLMath .
Therefore the right-hand side in Theorem LaTeXMLRef reduces to the right-hand side of Theorem LaTeXMLRef .
It is possible to further generalize Theorem LaTeXMLRef along the line of LaTeXMLCite .
In this section , we construct various LaTeXMLMath -theory operations based on the finite supergroup LaTeXMLMath .
This is an analog of Atiyah ’ s construction LaTeXMLCite of LaTeXMLMath -theory operations by using the symmetric groups and implicitly Schur duality .
The role of Schur duality is replaced here by the Sergeev ’ s generalization LaTeXMLCite of the Schur duality involving LaTeXMLMath .
Let LaTeXMLMath be a complex vector space of dimension LaTeXMLMath .
We denote by LaTeXMLMath the superalgebra of linear transformations on the superspace LaTeXMLMath which preserve an odd automorphism LaTeXMLMath such that LaTeXMLMath .
For example , if we take LaTeXMLMath , and LaTeXMLMath to be given by the LaTeXMLMath matrix LaTeXMLEquation then LaTeXMLMath is the Lie superalgebra which is obtained by the associative superalgebra LaTeXMLMath ( see Section LaTeXMLRef ) by taking the supercommutators .
Let us now consider the natural action of LaTeXMLMath on LaTeXMLMath .
We may form the LaTeXMLMath -fold tensor product LaTeXMLMath , on which LaTeXMLMath acts naturally .
In addition we have an action of the finite supergroup LaTeXMLMath : the symmetric group LaTeXMLMath acts on LaTeXMLMath by permutations with appropriate signs ; LaTeXMLMath acts on LaTeXMLMath by means of exchanging the parity of LaTeXMLMath -th copy of LaTeXMLMath via the odd automorphism LaTeXMLMath of LaTeXMLMath .
More explicitly , LaTeXMLMath transforms the vector LaTeXMLMath in LaTeXMLMath into LaTeXMLMath .
According to Sergeev LaTeXMLCite , the actions of LaTeXMLMath and LaTeXMLMath ( super ) commute with each other .
Furthermore , one has LaTeXMLEquation where LaTeXMLMath is the irreducible LaTeXMLMath -supermodule associated to a strict partition LaTeXMLMath , and LaTeXMLMath is an irreducible LaTeXMLMath -module .
The expression LaTeXMLMath above has the following meaning .
Suppose that LaTeXMLMath and LaTeXMLMath are two supergroups or two superalgebras and suppose that LaTeXMLMath and LaTeXMLMath are irreducible supermodules over LaTeXMLMath and LaTeXMLMath of type LaTeXMLMath , namely , LaTeXMLMath and LaTeXMLMath are both isomorphic to the Clifford superalgebra in one odd variable .
It is known ( cf .
e.g .
LaTeXMLCite ) that LaTeXMLMath as a module over LaTeXMLMath is not irreducible , but decomposes into a direct sum of two isomorphic copies ( via an odd isomorphism ) of the same irreducible supermodule .
In our particular setting when LaTeXMLMath is odd both LaTeXMLMath and LaTeXMLMath are such modules ( cf .
LaTeXMLCite ) .
So in this case we mean to take one copy inside their tensor product .
We introduce ( cf .
LaTeXMLCite ) the symmetric functions LaTeXMLMath in the variables LaTeXMLMath by the formula LaTeXMLEquation .
Denote by LaTeXMLMath the subring of symmetric functions generated by LaTeXMLMath and LaTeXMLMath the subspace spanned by symmetric functions in LaTeXMLMath of degree LaTeXMLMath .
Put LaTeXMLMath and LaTeXMLMath .
Recall LaTeXMLCite that a linear basis of LaTeXMLMath is given a distinguished class of symmetric functions LaTeXMLMath , call the Schur LaTeXMLMath -functions , parameterized by strict partitions LaTeXMLMath .
Furthermore LaTeXMLMath where LaTeXMLMath ( LaTeXMLMath odd ) are algebraically independent .
We take the convention that when the set of variables is finite , say LaTeXMLMath , we set LaTeXMLMath .
According to Sergeev LaTeXMLCite , the trace of the diagonal matrix LaTeXMLMath in LaTeXMLMath acting on LaTeXMLMath is LaTeXMLEquation .
Given a spin LaTeXMLMath -supermodule LaTeXMLMath , we define LaTeXMLMath to be the space of LaTeXMLMath -invariants LaTeXMLMath It is easy to check that the correspondence LaTeXMLMath is functorial on LaTeXMLMath .
In particular , if we take a diagonalizable linear transformation LaTeXMLMath with eigenvalues LaTeXMLMath , then the eigenvalues of the induced map LaTeXMLMath are monomials in LaTeXMLMath of degree LaTeXMLMath .
In particular the trace of LaTeXMLMath is a symmetric polynomial in LaTeXMLMath of degree LaTeXMLMath with integer coefficients .
One can argue that this symmetric polynomial ( for LaTeXMLMath ) is the restriction of a unique symmetric function in infinite many variables .
By definition we have LaTeXMLMath It follows the mapping LaTeXMLMath induces a map , denoted by LaTeXMLMath , from LaTeXMLMath to the space of symmetric functions of degree LaTeXMLMath .
Note that LaTeXMLMath since all the character values of LaTeXMLMath are real , i.e .
LaTeXMLMath is self-dual .
It follows from ( LaTeXMLRef ) that if we take LaTeXMLMath to be LaTeXMLMath then LaTeXMLMath sends the class function associated with LaTeXMLMath to LaTeXMLMath .
In this way we have defined a map LaTeXMLMath from LaTeXMLMath to the ring LaTeXMLMath of symmetric functions .
It is possible that one can develop this approach coherently to study the map LaTeXMLMath without referring to the Lie superalgebra LaTeXMLMath and thus essentially independent of the work of Sergeev LaTeXMLCite , as sketched below .
This is a super analog of an approach adopted in Knutson LaTeXMLCite in the setup of symmetric groups .
For example , we can start by arguing that the space LaTeXMLMath associated to the basic spin supermodule LaTeXMLMath ( with character LaTeXMLMath ) is the LaTeXMLMath -th supersymmetric algebra LaTeXMLMath , and thus LaTeXMLMath ( cf .
Ex .
6 ( c ) , pp261 , LaTeXMLCite ) , which is exactly the Schur LaTeXMLMath -function LaTeXMLMath .
Here LaTeXMLMath and LaTeXMLMath , which stand for the LaTeXMLMath -th elementary and complete symmetric function respectively , are the traces of LaTeXMLMath on LaTeXMLMath and LaTeXMLMath .
One can easily show by using the Frobenius reciprocity that LaTeXMLMath is an algebra homomorphism .
The image of LaTeXMLMath contains LaTeXMLMath as we have just seen that it contains LaTeXMLMath for all LaTeXMLMath .
By comparing the dimensions , the characteristic map LaTeXMLMath is indeed an isomorphism .
There is another way to define the characteristic map as follows ( cf .
LaTeXMLCite ) .
Denote by LaTeXMLMath the LaTeXMLMath -th power sum symmetric functions , and define LaTeXMLMath for a partition LaTeXMLMath .
Given a character LaTeXMLMath , we define the characteristic map LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath for LaTeXMLMath and LaTeXMLMath the character value at the even split conjugacy class LaTeXMLMath .
It is known LaTeXMLCite that LaTeXMLMath is an algebra isomorphism and it sends LaTeXMLMath to LaTeXMLMath for all LaTeXMLMath .
Thus LaTeXMLMath coincides with the characteristic map LaTeXMLMath we defined above .
Let LaTeXMLMath is a finite supergroup with LaTeXMLMath , and let LaTeXMLMath be a trivial LaTeXMLMath -space ( and thus LaTeXMLMath -space ) .
Given a LaTeXMLMath -supermodule LaTeXMLMath in LaTeXMLMath and a complex vector bundle LaTeXMLMath on LaTeXMLMath , LaTeXMLMath can be given a natural LaTeXMLMath -equivariant vector superbundle structure over LaTeXMLMath by letting LaTeXMLMath act on only the factor LaTeXMLMath .
Obviously LaTeXMLMath lies in the category LaTeXMLMath .
In this way we obtain a map LaTeXMLMath .
Then we have the following ( compare LaTeXMLCite ) .
Under the above setup , there is a canonical isomorphism LaTeXMLEquation .
One needs some extra care to define the inverse of the isomorphism in the above proposition .
Given a LaTeXMLMath -supermodule LaTeXMLMath in LaTeXMLMath and a complex vector bundle LaTeXMLMath on LaTeXMLMath , consider LaTeXMLMath , where LaTeXMLMath is the trivial LaTeXMLMath -superbundle over LaTeXMLMath associated to LaTeXMLMath .
LaTeXMLMath is isomorphic to LaTeXMLMath if LaTeXMLMath is of type LaTeXMLMath , but isomorphic to LaTeXMLMath if LaTeXMLMath is of type LaTeXMLMath , where LaTeXMLMath is the Clifford algebra in one variable .
It is perhaps more natural to replace LaTeXMLMath in the proposition above by an isomorphic space LaTeXMLMath , cf .
Remark LaTeXMLRef .
Below we will construct various LaTeXMLMath -theory operations based on the construction in the previous subsection .
It is a super analog of an approach due to Atiyah LaTeXMLCite who used the symmetric group representations .
Let LaTeXMLMath be a vector bundle over LaTeXMLMath , consider the LaTeXMLMath -th tensor power LaTeXMLMath of the vector superbundle LaTeXMLMath .
The odd operator LaTeXMLMath ( LaTeXMLMath ) acts on each factor LaTeXMLMath fiberwise and induces an action of the finite supergroup LaTeXMLMath on LaTeXMLMath .
The symmetric group LaTeXMLMath also acts on LaTeXMLMath in a natural way .
The joint action of LaTeXMLMath and LaTeXMLMath then gives rise to an action of the finite supergroup LaTeXMLMath on LaTeXMLMath .
We have the following decomposition LaTeXMLEquation where LaTeXMLMath is a strict partition of LaTeXMLMath , and LaTeXMLMath is a vector bundle on LaTeXMLMath .
Clearly one can extend the definition of the vector bundle LaTeXMLMath associated to any spin supermodule LaTeXMLMath of LaTeXMLMath so that it is additive on LaTeXMLMath .
In this way we obtain a ring homomorphism LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath is the ring of LaTeXMLMath -theory operations on LaTeXMLMath ( cf .
LaTeXMLCite ) .
We note that if LaTeXMLMath is the one-part partition LaTeXMLMath then LaTeXMLMath is the LaTeXMLMath th supersymmetric power LaTeXMLMath : LaTeXMLEquation .
For odd LaTeXMLMath , we denote by LaTeXMLMath the operation corresponding to the class function LaTeXMLMath which takes value LaTeXMLMath in the even split conjugacy class of type LaTeXMLMath and zero elsewhere .
Denote by LaTeXMLMath where LaTeXMLMath is a formal variable .
The operation LaTeXMLMath ( LaTeXMLMath odd ) coincides with the usual LaTeXMLMath th Adams operation .
In particular it is additive .
Furthermore , we have LaTeXMLEquation .
Proof .
We first prove ( LaTeXMLRef ) .
Since the operations LaTeXMLMath ’ s and LaTeXMLMath ’ s are obtained from the ring homomorphism LaTeXMLMath , we only need to show that the corresponding identity holds in LaTeXMLMath , or alternatively in LaTeXMLMath .
But this is a classical identity LaTeXMLCite LaTeXMLEquation .
Now we denote by LaTeXMLMath .
Let us denote by LaTeXMLMath the LaTeXMLMath th Adams operations for the time being .
It is classical that LaTeXMLEquation .
LaTeXMLEquation We have from ( LaTeXMLRef ) that LaTeXMLMath and therefore LaTeXMLEquation .
Comparing with ( LaTeXMLRef ) which we have already established , we have LaTeXMLMath .
LaTeXMLMath It follows that LaTeXMLMath and LaTeXMLMath where LaTeXMLMath , since the Admas operations are additive .
For example , the second equation reads componentwise as follows : LaTeXMLEquation .
The ring LaTeXMLMath of symmetric functions is a basic model for ( free ) LaTeXMLMath rings , and indeed LaTeXMLMath rings can be defined by axiomizing various properties of natural operations on LaTeXMLMath , where the Adams operations play a crucial role ( cf .
LaTeXMLCite ) .
We can define similarly a notion of LaTeXMLMath - LaTeXMLMath ring with Adams operations of odd degrees only , using LaTeXMLMath as a basic model .
Then what we have just shown is that LaTeXMLMath admits a LaTeXMLMath - LaTeXMLMath ring structure .
We will see the structure of a LaTeXMLMath - LaTeXMLMath ring instead of a LaTeXMLMath ring shows up naturally in some fairly non-trivial setup in Section LaTeXMLRef .
In this section , we shall study in detail the LaTeXMLMath -groups LaTeXMLMath of generalized symmetric products for all LaTeXMLMath simultaneously .
Our main examples in this paper are as follows .
Let LaTeXMLMath be a finite group and let LaTeXMLMath be a LaTeXMLMath -space .
The LaTeXMLMath -th Cartesian product LaTeXMLMath is acted by the finite supergroup LaTeXMLMath in a canonical way : LaTeXMLMath acts trivially on LaTeXMLMath ; LaTeXMLMath acts on LaTeXMLMath factorwise while LaTeXMLMath by permutations , and this gives rise to a natural action of the wreath product LaTeXMLMath by letting LaTeXMLEquation where LaTeXMLMath , and LaTeXMLMath .
Note that orbifolds LaTeXMLMath are often called symmetric products .
We may refer to LaTeXMLMath , or LaTeXMLMath with LaTeXMLMath -action , or rather LaTeXMLMath with LaTeXMLMath -action as generalized symmetric products .
Our earlier general construction when applied to the generalized symmetric products gives us the category LaTeXMLMath and its associated LaTeXMLMath -group LaTeXMLMath .
It turns out this category affords an equivalent description below which affords more transparent geometric meaning .
The wreath product LaTeXMLMath acts on the vector space LaTeXMLMath naturally by letting LaTeXMLMath act trivially and LaTeXMLMath act as the permutation representation .
This action preserves the standard quadratic form on LaTeXMLMath .
We denote by LaTeXMLMath such a LaTeXMLMath -vector bundle LaTeXMLMath over LaTeXMLMath .
We denote by LaTeXMLMath the complex Clifford algebra associated to LaTeXMLMath and the standard quadratic form on it .
The action of LaTeXMLMath on LaTeXMLMath induces a natural action on LaTeXMLMath .
We denote by LaTeXMLMath the associated LaTeXMLMath -vector ( super ) bundle LaTeXMLMath on LaTeXMLMath , which is the Clifford module on LaTeXMLMath associated to the vector bundle LaTeXMLMath .
We introduce the following category LaTeXMLMath : the objects consist of complex vector superbundles LaTeXMLMath on LaTeXMLMath equipped with compatible actions of LaTeXMLMath and the Clifford algebra LaTeXMLMath associated to LaTeXMLMath with the standard quadratic form .
That is , LaTeXMLMath is a LaTeXMLMath -graded LaTeXMLMath -module and a LaTeXMLMath -equivariant vector bundle over LaTeXMLMath such that LaTeXMLEquation .
Given two superbundles LaTeXMLMath in LaTeXMLMath , the space of LaTeXMLMath -equivariant homomorphisms of vector superbundles admits a natural LaTeXMLMath -gradation .
Since the twisted group algebra of LaTeXMLMath is isomorphic to the Clifford algebra LaTeXMLMath , the category LaTeXMLMath is then obviously equivalent to the category LaTeXMLMath , and so are the corresponding LaTeXMLMath -groups .
When LaTeXMLMath is a point , LaTeXMLMath reduces to the Grothendieck group LaTeXMLMath of the spin supermodules of LaTeXMLMath .
We may replace the rank LaTeXMLMath vector bundle LaTeXMLMath over LaTeXMLMath above by the LaTeXMLMath th direct sum of a non-trivial line bundle endowed with a quadratic form , and modify the construction of the category LaTeXMLMath accordingly .
We conjecture that the resulting LaTeXMLMath -group is isomorphic to LaTeXMLMath .
We may reverse the above consideration in a more general setup as below .
Take a LaTeXMLMath -space LaTeXMLMath and a LaTeXMLMath -superbundle LaTeXMLMath over LaTeXMLMath whose fiber is the Clifford algebra in LaTeXMLMath variables .
Assume that there exists LaTeXMLMath sections LaTeXMLMath of the superbundle LaTeXMLMath which fiberwise generate the Clifford algebra and LaTeXMLMath permutes LaTeXMLMath .
It is of interest in algebraic topology ( cf .
Karoubi LaTeXMLCite ) to study the category LaTeXMLMath of LaTeXMLMath -equivariant vector superbundles over LaTeXMLMath which are compatible with the the Clifford module structure in the sense of ( LaTeXMLRef ) and study its associated LaTeXMLMath -group .
Then we may form the finite supergroup LaTeXMLMath and reinterpret the category LaTeXMLMath as the category LaTeXMLMath , and then apply Theorem LaTeXMLRef to the study of the associated LaTeXMLMath -group .
In particular if we take LaTeXMLMath and LaTeXMLMath for a LaTeXMLMath -space LaTeXMLMath , we recover our main examples of generalized symmetric products .
Let LaTeXMLMath be a LaTeXMLMath -graded subgroup of a finite supergroup LaTeXMLMath with the same distinguished even element LaTeXMLMath , and let LaTeXMLMath be a LaTeXMLMath -space LaTeXMLMath which is regarded as a LaTeXMLMath -space , where LaTeXMLMath .
we can define the restriction map LaTeXMLMath and the induction map LaTeXMLMath in the same way as in the usual equivariant LaTeXMLMath -theory .
When it is clear from the text , we will often abbreviate LaTeXMLMath as LaTeXMLMath or LaTeXMLMath .
Similar remarks apply to the induction map .
Now we introduce the direct sum of equivariant LaTeXMLMath -groups LaTeXMLEquation where LaTeXMLMath is the one-element group by convention , and LaTeXMLMath is a formal variable counting the graded structure of LaTeXMLMath .
We also set LaTeXMLEquation .
We define a multiplication LaTeXMLMath on the space LaTeXMLMath by a composition of the induction map and the Künneth isomorphism LaTeXMLMath : LaTeXMLEquation .
We denote by LaTeXMLMath the unit in LaTeXMLMath which can be identified with LaTeXMLMath .
On the other hand we can define a comultiplication LaTeXMLMath on LaTeXMLMath to be a composition of the inverse of the Künneth isomorphism and the restriction from LaTeXMLMath to LaTeXMLMath : LaTeXMLEquation .
We define the counit LaTeXMLMath by sending LaTeXMLMath LaTeXMLMath to LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
With various operations defined as above , LaTeXMLMath is a graded Hopf algebra .
The proof is the same as the proof of the Hopf algebra structure on a direct sum of equivariant LaTeXMLMath -groups LaTeXMLMath LaTeXMLCite , where a straightforward generalization to the equivariant LaTeXMLMath -groups of the Mackey ’ s theorem plays a key role .
One easily checks the super version of the Mackey ’ s theorem can be carried over to our LaTeXMLMath -group setup .
In the case when LaTeXMLMath is a point and thus LaTeXMLMath , the Hopf algebra structure above is also treated in LaTeXMLCite .
Take an even split element LaTeXMLMath in LaTeXMLMath of type LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
We define LaTeXMLMath .
Since the subgroup LaTeXMLMath acts on LaTeXMLMath trivially , the orbit space LaTeXMLMath is identified with LaTeXMLMath which has been calculated earlier in Lemmas 4 and 5 of LaTeXMLCite .
We make a convention here to denote the centralizer LaTeXMLMath ( resp .
LaTeXMLMath , LaTeXMLMath ) by LaTeXMLMath ( resp .
LaTeXMLMath , LaTeXMLMath ) by abuse of notations when the choice of a representative LaTeXMLMath in a conjugacy class LaTeXMLMath of a group LaTeXMLMath is irrelevant .
For a fixed LaTeXMLMath , recall that LaTeXMLMath ( LaTeXMLMath ) is the even split conjugacy class in LaTeXMLMath of the type LaTeXMLMath , where the partition-valued function LaTeXMLMath takes value the one-part partition LaTeXMLMath at LaTeXMLMath and zero elsewhere .
Let LaTeXMLMath be an even split element of type LaTeXMLMath , where LaTeXMLMath .
Then the orbit space LaTeXMLMath can be naturally identified with LaTeXMLEquation where LaTeXMLMath denotes the LaTeXMLMath -th symmetric product .
In particular , the orbit space LaTeXMLMath can be naturally identified with LaTeXMLMath In view of Lemma LaTeXMLRef , let us recall how we have classified the split conjugacy classes of LaTeXMLMath in Theorem 1.2 of LaTeXMLCite .
In order to show that a given element LaTeXMLMath in LaTeXMLMath is non-split , an explicit element , say LaTeXMLMath , is constructed in the centralizer LaTeXMLMath such that the character LaTeXMLMath takes value LaTeXMLMath at LaTeXMLMath .
This was achieved in LaTeXMLCite case by case .
On the other hand , we observe that in all cases the element LaTeXMLMath fixes LaTeXMLMath pointwise !
In other words , we have the following .
The LaTeXMLMath -space LaTeXMLMath satisfies the strong vanishing property .
We now give an explicit description of LaTeXMLMath as a graded algebra .
As a LaTeXMLMath -graded algebra LaTeXMLMath is isomorphic to the supersymmetric algebra LaTeXMLMath .
In particular , we have LaTeXMLEquation .
Here the supersymmetric algebra is equal to the tensor product of the symmetric algebra LaTeXMLMath and the exterior algebra LaTeXMLMath .
Proof .
Take an even split element LaTeXMLMath of type LaTeXMLMath , where LaTeXMLMath .
By Lemma LaTeXMLRef and the Künneth formula , we have LaTeXMLEquation .
We now calculate as follows .
The statement concerning LaTeXMLMath follows from this immediately .
LaTeXMLEquation .
LaTeXMLMath Recall that the orbifold Euler number LaTeXMLMath was introduced by Dixon , Harvey , Vafa and Witten LaTeXMLCite in the study of orbifold string theory .
It is subsequently interpreted as the Euler number of the equivariant LaTeXMLMath -group LaTeXMLMath , cf .
e.g .
LaTeXMLCite .
If we define the Euler number of the generalized symmetric product to be the difference LaTeXMLEquation then we obtain the following corollary .
The Euler number LaTeXMLMath is given by the following generating function : LaTeXMLEquation .
In the case when LaTeXMLMath is a point , we obtain the following corollary ( also cf .
LaTeXMLCite ) .
When LaTeXMLMath is a point and thus LaTeXMLMath , we have LaTeXMLEquation .
In the following , we will define various LaTeXMLMath -theory maps appearing in the following diagram ( LaTeXMLMath odd ) : LaTeXMLEquation .
Noting that LaTeXMLMath , we have a canonical isomorphism , denoted by LaTeXMLMath , from LaTeXMLMath to LaTeXMLMath given by LaTeXMLMath .
Given a LaTeXMLMath -equivariant vector bundle LaTeXMLMath , consider the LaTeXMLMath -th outer tensor product LaTeXMLMath which is a vector superbundle over LaTeXMLMath .
The odd operator LaTeXMLMath acting on each factor LaTeXMLMath induces an action of the finite supergroup LaTeXMLMath on LaTeXMLMath while the wreath product LaTeXMLMath acts on LaTeXMLMath by letting LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
It is easy to check the combined action gives rise to an action of the finite supergroup LaTeXMLMath on LaTeXMLMath , and LaTeXMLMath endowed with such an LaTeXMLMath action is an LaTeXMLMath -equivariant vector superbundle over LaTeXMLMath .
On the other hand we can define an LaTeXMLMath action on LaTeXMLMath as follows : LaTeXMLMath acts on the first factor LaTeXMLMath only while LaTeXMLMath acts only on the second factor LaTeXMLMath ; the symmetric group LaTeXMLMath acts diagonally .
One can check that the combined action gives LaTeXMLMath the structure of an LaTeXMLMath -equivariant vector superbundle over LaTeXMLMath .
We easily see that LaTeXMLMath is canonically isomorphic to LaTeXMLMath as a LaTeXMLMath -equivariant superbundle .
Note that the above LaTeXMLMath is precisely the basic LaTeXMLMath -supermodule LaTeXMLMath .
In general for a given LaTeXMLMath -supermodule LaTeXMLMath with character LaTeXMLMath , we can define an LaTeXMLMath -equivariant superbundle structure on LaTeXMLMath when replacing LaTeXMLMath above by LaTeXMLMath .
We will write the corresponding element in LaTeXMLMath as LaTeXMLMath .
This defines an additive map from LaTeXMLMath to LaTeXMLMath by sending LaTeXMLMath to LaTeXMLMath .
Sending LaTeXMLMath to LaTeXMLMath gives rise to the K-theory map LaTeXMLMath .
More explicitly , given LaTeXMLMath two LaTeXMLMath -equivariant vector bundles on LaTeXMLMath , we use LaTeXMLMath itself to denote the corresponding element in LaTeXMLMath by abuse of notation .
Then LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath carry the standard actions of LaTeXMLMath and respectively LaTeXMLMath .
The map LaTeXMLMath is the isomorphism in Theorem LaTeXMLRef given by the summation LaTeXMLMath over the even split conjugacy classes of LaTeXMLMath of type LaTeXMLMath when applying to the case LaTeXMLMath with the action of LaTeXMLMath .
The map LaTeXMLMath is the projection to the direct sum over the even split conjugacy classes LaTeXMLMath of LaTeXMLMath while LaTeXMLMath denotes the inclusion map .
The map LaTeXMLMath denotes the natural identification given by Lemma LaTeXMLRef .
Finally the last map LaTeXMLMath is the isomorphism given in Theorem LaTeXMLRef .
We introduce in addition the following LaTeXMLMath -theory operations .
For LaTeXMLMath , we define the following LaTeXMLMath -theory operations as composition maps : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Recall that the notation LaTeXMLMath ( LaTeXMLMath odd ) was used in Section LaTeXMLRef to denote the LaTeXMLMath th Adams operation .
We shall see the LaTeXMLMath defined here for LaTeXMLMath trivial coincides with the LaTeXMLMath th Adams operation tensored with LaTeXMLMath .
This is why we have chosen to use the same notation .
We list some properties of these K-theory maps which follows directly from defintions .
The following identities hold : LaTeXMLEquation where LaTeXMLMath denotes the identity operator on LaTeXMLMath .
Both LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath odd ) are additive K-theory maps .
In particular , for LaTeXMLMath trivial , the operation LaTeXMLMath given in the definition LaTeXMLRef coincides with the LaTeXMLMath th Adams operations on LaTeXMLMath .
Proof .
We sketch a proof .
By definition LaTeXMLMath is additive and LaTeXMLMath is an isomorphism .
Thanks to the equality LaTeXMLMath by Proposition LaTeXMLRef , to show that LaTeXMLMath is additive , it suffices to check that LaTeXMLMath is additive .
This can be proved in a parallel way by using ( LaTeXMLRef ) as Atiyah LaTeXMLCite proves the additivity of the Adams operations defined in terms of symmetric groups .
Now we set LaTeXMLMath and consider the diagonal embedding LaTeXMLMath .
Since LaTeXMLMath acts on LaTeXMLMath trivially , it follows by Proposition LaTeXMLRef that LaTeXMLMath We have the following commutative diagram ( LaTeXMLMath odd ) : LaTeXMLEquation where LaTeXMLMath is the analog of LaTeXMLMath when LaTeXMLMath is replaced by the diagonal LaTeXMLMath , and the evaluation map LaTeXMLMath is defined to be the character value at the conjugacy class of type LaTeXMLMath .
The map from LaTeXMLMath to itself obtained along the left-bottom route in the above diagram coincides with LaTeXMLMath given in Definition LaTeXMLRef .
The map from LaTeXMLMath to itself obtained along the top-right route in the above diagram gives LaTeXMLMath times the LaTeXMLMath -th Adams operation .
This of course gives another proof that both LaTeXMLMath and LaTeXMLMath are additive when LaTeXMLMath is trivial .
LaTeXMLMath LaTeXMLMath is a LaTeXMLMath - LaTeXMLMath ring with LaTeXMLMath ( LaTeXMLMath odd ) as the LaTeXMLMath th Adams operation .
If LaTeXMLMath is a point and thus LaTeXMLMath then our result reduces to the fact that LaTeXMLMath is a free LaTeXMLMath - LaTeXMLMath ring generated by LaTeXMLMath .
In particular when LaTeXMLMath is trivial this is isomorphic to the model LaTeXMLMath - LaTeXMLMath ring LaTeXMLMath .
Denote by LaTeXMLMath the completion of LaTeXMLMath which allows formal infinite sums .
Given LaTeXMLMath , we introduce LaTeXMLMath as follows : LaTeXMLEquation .
The following lemma is immediate by Definition LaTeXMLRef and Remark LaTeXMLRef .
Given LaTeXMLMath , we have ( for LaTeXMLMath odd ) LaTeXMLEquation .
Given LaTeXMLMath , we can express LaTeXMLMath as follows : LaTeXMLEquation .
Here the right-hand side is understood in terms of the algebra structure on LaTeXMLMath .
Proof .
By ( LaTeXMLRef ) and the above lemma , we have LaTeXMLEquation .
LaTeXMLMath Combining with the additivity of LaTeXMLMath , the proposition implies The following equations hold for LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation The generating function LaTeXMLMath is essentially half the twisted vertex operator , and the other half can be obtained by the adjoint operator to LaTeXMLMath .
Twisted vertex operators have played an important role in the representation theory of infinite dimensional Lie algebras and the moonshine module , cf .
LaTeXMLCite .
When LaTeXMLMath is a point , we can develop the picture more completely ( cf .
LaTeXMLCite ) to provide a group theoretic realization of vertex representations of twisted affine and twisted toroidal Lie algebras ( also compare LaTeXMLCite for a different construction ) .
We see from Theorem LaTeXMLRef that LaTeXMLMath has the same size of the tensor product of the Fock space of an infinite-dimensional twisted Heisenberg algebra of rank LaTeXMLMath and that of an infinite-dimensional twisted Clifford algebra of rank LaTeXMLMath .
In this section we will actually construct such a Heisenberg/Clifford algebra , which we will simply refer to as a twisted Heisenberg ( super ) algebra from now on .
The dual of LaTeXMLMath , denoted by LaTeXMLMath , is naturally LaTeXMLMath -graded as identified with LaTeXMLMath .
Denote by LaTeXMLMath the pairing between LaTeXMLMath and LaTeXMLMath .
For any LaTeXMLMath and LaTeXMLMath , we define an additive map LaTeXMLEquation as the composition LaTeXMLEquation .
On the other hand , we define for any LaTeXMLMath and LaTeXMLMath an additive map LaTeXMLEquation as the composition LaTeXMLEquation .
Let LaTeXMLMath be the linear span of the operators LaTeXMLMath , LaTeXMLMath LaTeXMLMath .
Clearly LaTeXMLMath admits a natural LaTeXMLMath -gradation induced from that on LaTeXMLMath and LaTeXMLMath .
Below we shall use LaTeXMLMath to denote the supercommutator as well .
It is understood that LaTeXMLMath is the anti-commutator LaTeXMLMath when LaTeXMLMath are both odd elements according to the LaTeXMLMath -gradation .
When acting on LaTeXMLMath , LaTeXMLMath satisfies the twisted Heisenberg superalgebra commutation relations , namely for LaTeXMLMath LaTeXMLMath LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Furthermore , LaTeXMLMath is an irreducible representation of the twisted Heisenberg superalgebra .
The proof of the Heisenberg algebra commutation relation can be given in a parallel way as the one for Theorem 4 , LaTeXMLCite .
The irreducibility of LaTeXMLMath as module over the Heisenberg algebra follows from Theorem LaTeXMLRef .
Given a bilinear form on LaTeXMLMath , then we can get rid of LaTeXMLMath in the formulation of the above theorem .
In the special case when LaTeXMLMath is a point , the Heisenberg algebra here specializes to the one given in LaTeXMLCite acting on LaTeXMLMath .
In the case when LaTeXMLMath is a finite subgroup of LaTeXMLMath , we may consider further the space which is the tensor product of LaTeXMLMath with a module of a certain LaTeXMLMath -group which can be constructed out of LaTeXMLMath , and realize in this way a vertex representation of a twisted affine and a twisted toroidal Lie algebra .
This is treated in LaTeXMLCite in detail .
As is well known ( cf .
e.g .
LaTeXMLCite ) , the symmetric group LaTeXMLMath has a double cover LaTeXMLMath : LaTeXMLEquation generated by LaTeXMLMath and LaTeXMLMath and subject to the relations : LaTeXMLEquation .
The map LaTeXMLMath sends LaTeXMLMath ’ s to the simple reflections LaTeXMLMath ’ s in LaTeXMLMath .
The group LaTeXMLMath carries a natural LaTeXMLMath grading by letting LaTeXMLMath ’ s be odd and LaTeXMLMath be even .
Given a finite group LaTeXMLMath , the symmetric group LaTeXMLMath acts on the product group LaTeXMLMath , and the group LaTeXMLMath acts on LaTeXMLMath via LaTeXMLMath .
Thus we can form a semi-direct product LaTeXMLMath , which carries a natural finite supergroup structure by letting elements in LaTeXMLMath be even , cf .
LaTeXMLCite .
We still denote by LaTeXMLMath the quotient map LaTeXMLMath .
Given a LaTeXMLMath -space LaTeXMLMath , we have seen LaTeXMLMath affords a natural LaTeXMLMath action .
Then we can apply the general construction in Sect .
LaTeXMLRef to construct the category LaTeXMLMath and its associated LaTeXMLMath -group LaTeXMLMath .
As before , we denote LaTeXMLMath .
We then form the direct sum LaTeXMLEquation .
LaTeXMLEquation When LaTeXMLMath is a point , LaTeXMLMath reduces to the Grothendieck group LaTeXMLMath of spin supermodules of LaTeXMLMath , and LaTeXMLMath has been studied in detail in LaTeXMLCite .
The purpose of the Appendix is to outline how to modify the various constructions of algebraic structures on LaTeXMLMath for the new space LaTeXMLMath .
As the constructions are very similar to the LaTeXMLMath case , we will be rather sketchy .
Given LaTeXMLMath , we can define a LaTeXMLMath -graded subgroup LaTeXMLMath of LaTeXMLMath , in a way analogous to ( LaTeXMLRef ) .
Then the obvious analog of constructions ( LaTeXMLRef ) and ( LaTeXMLRef ) defines a multiplication and comultiplication on the space LaTeXMLMath .
The following is an analog of Theorem LaTeXMLRef and it generalizes Theorem 3.8 of LaTeXMLCite which is our special case when LaTeXMLMath is a point .
The space LaTeXMLMath carries a natural Hopf algebra structure .
By the analysis of the split conjugacy classes given in the proof of Theorem 2.5 of LaTeXMLCite , we see that the LaTeXMLMath -space LaTeXMLMath satisfies the strong vanishing property , and the analog of Lemma LaTeXMLRef holds .
Therefore we obtain the following theorem which is an analog of Theorem LaTeXMLRef .
As a LaTeXMLMath -graded algebra LaTeXMLMath is isomorphic to the supersymmetric algebra LaTeXMLMath .
Except the first two terms and the first two arrows in the diagram ( LaTeXMLRef ) , the rest of the diagram has a direct analog for LaTeXMLMath .
Note in the definition of the LaTeXMLMath -theory maps LaTeXMLMath and LaTeXMLMath ( see Definition LaTeXMLRef ) only the part of the diagram ( LaTeXMLRef ) which can be directly generalized to the LaTeXMLMath setup has been used .
Therefore analog of LaTeXMLMath and LaTeXMLMath can be defined in our new setup .
This guarantees the analog of annihilation operators ( LaTeXMLRef ) and the creation operators ( LaTeXMLRef ) can be defined in our new setup .
In this way we obtain the following which is an analog of Theorem LaTeXMLRef .
The space LaTeXMLMath affords an action of the twisted Heisenberg algebra LaTeXMLMath in terms of natural additive LaTeXMLMath -theory maps .
Furthermore this representation is irreducible .
We remark that it is much less natural to use LaTeXMLMath to construct various LaTeXMLMath -theory operations on LaTeXMLMath as done in Sect .
LaTeXMLRef using a double cover LaTeXMLMath of the hyperoctahedral group .
The connection between LaTeXMLMath and the LaTeXMLMath - LaTeXMLMath ring in Sect .
LaTeXMLRef does carry over to our new setup .
Keeping Remark LaTeXMLRef in mind and knowing that LaTeXMLMath also has a so-called basic spin supermodule ( cf .
e.g .
LaTeXMLCite ) , we can use it to define the analog of ( LaTeXMLRef ) .
Indeed this also defines an analog of the map LaTeXMLMath ( cf .
the diagram ( LaTeXMLRef ) ) , and thus an analog of LaTeXMLMath .
Therefore , we have an analog of Proposition LaTeXMLRef in our new setup which generalizes Proposition 6.2 in LaTeXMLCite .
However there is an unpleasant square root of LaTeXMLMath in the formula which originates in the spin representation theory of LaTeXMLMath and LaTeXMLMath .
This is another reason why we have preferred the formulation in the main body of the paper using LaTeXMLMath and LaTeXMLMath .
One may wonder that why LaTeXMLMath and LaTeXMLMath are so similar to each other and there are almost parallel constructions on LaTeXMLMath and LaTeXMLMath .
When LaTeXMLMath is a point and thus the LaTeXMLMath -groups reduces to the corresponding Grothendieck groups of spin supermodules , this has been noticed by various different authors ( cf .
e.g .
LaTeXMLCite and the references therein ) .
Yamaguchi LaTeXMLCite explains clearly such a phenomenon by establishing an isomorphism between the group superalgebra LaTeXMLMath and the ( LaTeXMLMath -graded ) tensor product of the group superalgebra LaTeXMLMath with the complex Clifford algebra LaTeXMLMath of LaTeXMLMath variables ( this is not the same copy of LaTeXMLMath associated to the subgroup LaTeXMLMath in LaTeXMLMath ! ) .
It follows that the group superalgebra LaTeXMLMath is isomorphic to the tensor product of the group superalgebra LaTeXMLMath with LaTeXMLMath .
Note that a Clifford algebra admits a unique irreducible supermodule .
As this LaTeXMLMath acts on an LaTeXMLMath -bundle over LaTeXMLMath fiberwise , this isomorphism provides a direct isomorphism between LaTeXMLMath and LaTeXMLMath .
We can also forget about the LaTeXMLMath -gradings ( i.e .
the super structures ) in the group LaTeXMLMath , in the construction of the category LaTeXMLMath and its associated LaTeXMLMath -group LaTeXMLMath .
Let us denote the resulting new LaTeXMLMath -group by LaTeXMLMath .
In particular when LaTeXMLMath is a point and LaTeXMLMath is trivial , this reduces to the Grothendick group LaTeXMLMath of spin ( not super ) modules of LaTeXMLMath where LaTeXMLMath still acts as LaTeXMLMath .
We can then apply the decomposition theorem of Adem-Ruan LaTeXMLCite to calculate LaTeXMLMath in terms of LaTeXMLMath .
The difference here from the calculations in Theorem LaTeXMLRef and Theorem LaTeXMLRef is that the odd split conjugacy classes of LaTeXMLMath will also make contributions .
Recall that the orbifold Euler number LaTeXMLMath defined in LaTeXMLCite is the same as the Euler number of the equivariant LaTeXMLMath -theory LaTeXMLMath .
Using the description of even and odd split conjugacy classes of LaTeXMLMath ( cf .
LaTeXMLCite , Theorem 2.5 ) , we can obtain the Euler number of LaTeXMLMath , denoted by LaTeXMLMath , in terms of the following generating function ( compare Corollary LaTeXMLRef ) : LaTeXMLEquation .
The second summand in the right-hand side of the above equation counts the contributions from odd split conjugacy classes .
When we set LaTeXMLMath to be a point and LaTeXMLMath trivial ( and thus LaTeXMLMath ) , this formula reduces to the classical generating function for the spin Grothendick group LaTeXMLMath ( cf .
Theorem 3.6 , LaTeXMLCite , pp .
213 ; Corollary 3.10 , LaTeXMLCite , pp .
32 ) .
We remark that due to some inaccurate analysis of split conjugacy classes of LaTeXMLMath , the formula ( 6.10 ) given in LaTeXMLCite ( even in the case when LaTeXMLMath is trivial and LaTeXMLMath is a point ) is incompatible with this classical statement .
Department of Math. , North Carolina State University , Raleigh , NC 27695 .
Current address : Department of Mathematics , University of Virginia , Charlottesville , VA 22904 , U.S.A. Email : ww9c @ virginia.edu
Each finite LaTeXMLMath -perfect group LaTeXMLMath ( LaTeXMLMath a prime ) has a universal central LaTeXMLMath -extension coming from the LaTeXMLMath part of its Schur multiplier .
Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups ( LaTeXMLMath ) aimed at geometric covering space problems that included their regular realization for the Inverse Galois Problem .
A special case of a general result is that any finite simple group with a nontrivial LaTeXMLMath part to its Schur multiplier has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime LaTeXMLMath .
Sequences of moduli spaces of curves attached to LaTeXMLMath and LaTeXMLMath , called Modular Towers , capture the geometry of these many appearances of Schur multipliers in degeneration phenomena of Harbater-Mumford cover representatives .
These are modular curve tower generalizations .
So , they inspire conjectures akin to Serre ’ s open image theorem , including that at suitably high levels we expect no rational points .
Guided by two papers of Serre ’ s , these cases reveal common appearance of spin structures producing LaTeXMLMath -nulls on these moduli spaces .
The results immediately apply to all the expected Inverse Galois topics .
This includes systematic exposure of moduli spaces having points where the field of moduli is a field of definition and other points where it is not .
Hurwitz monodromy and Modular Towers ] Hurwitz monodromy , spin separation and higher levels of a Modular Tower P. Bailey ] Paul Bailey M. Fried ] Michael D. Fried [ 2000 ] Primary 11F32 , 11G18 , 11R58 ; Secondary 20B05 , 20C25 , 20D25 , 20E18 , 20F34 Here is one corollary of a result from the early 1990s .
There is an exact sequence ( LaTeXMLCite , discussed in LaTeXMLCite and LaTeXMLCite ) : LaTeXMLEquation .
The group on the left is the profree group on a countable number of generators .
The group on the right is the direct product of the symmetric groups , one copy for each integer : ( LaTeXMLRef ) catches the absolute Galois group LaTeXMLMath of LaTeXMLMath between two known groups .
Suppose a subfield LaTeXMLMath of LaTeXMLMath has LaTeXMLMath a projective ( profinite ) group .
Further , LaTeXMLCite conjectures : Then LaTeXMLMath is pro-free ( on countably many generators ) if and only if LaTeXMLMath is Hilbertian .
The conjecture generalizes ( LaTeXMLRef ) and Shafarevich ’ s conjecture that the cyclotomic closure of LaTeXMLMath has a pro-free absolute Galois group .
So , ( LaTeXMLRef ) is a positive statement about the Inverse Galois Problem .
Still , it works by maneuvering around the nonprojectiveness of LaTeXMLMath as a projective profinite group .
Modular Towers captures , in moduli space properties , implications of this nonprojective nature of LaTeXMLMath .
Its stems from constructing from each finite group LaTeXMLMath , a projective profinite group LaTeXMLMath whose quotients naturally entangle LaTeXMLMath with many classical spaces : modular curves ( § LaTeXMLRef ) and spaces of Prym varieties ( § LaTeXMLRef and § LaTeXMLRef ) , to mention just two special cases .
This LaTeXMLMath started its arithmetic geometry life on a different brand of problem LaTeXMLCite .
LaTeXMLCite gave definitions , motivations and applications surrounding Modular Towers starting from the special case of modular curve towers .
This paper continues that , with a direct attack on properties of higher Modular Tower levels .
Our models for applications include the Open Image Theorem on modular curve towers in LaTeXMLCite ( details in § LaTeXMLRef ) .
In Modular Tower ’ s language this is the ( four ) dihedral group involution realizations case ( LaTeXMLCite and LaTeXMLCite have very elementary explanations ) .
Modular Towers comes from profinite group theory attached to any finite group , a prime dividing its order and a choice of conjugacy classes in the group .
This moduli approach uses a different type of group theory LaTeXMLMath — LaTeXMLMath modular representations LaTeXMLMath — LaTeXMLMath than the homogeneous space approach typical of modular curves or Siegel upper half space .
Still , these spaces appear here with a more elementary look and a new set of applications .
It shows especially when we compare the monodromy groups of the tower levels with those for modular curves .
A key property used in LaTeXMLCite : Levels of a modular curve tower attached to the prime LaTeXMLMath ( LaTeXMLMath ) have a projective sequence of Frattini covers as monodromy groups .
Results similar to this should hold also for Modular Towers ( Ques .
LaTeXMLRef ) .
That is part of our suggestion that there is an Open Image Theorem for Modular Towers .
We first briefly summarize results ; then set up preliminaries to explain them in detail .
The name Harbater-Mumford representatives appears so often , it could well have been part of the title .
Its precise definition is as a type of element in a Nielsen class .
Geometrically its meaning for Hurwitz spaces is an especially detectable degeneration of curves in a family .
The name derives from two papers : LaTeXMLCite and LaTeXMLCite .
§ LaTeXMLRef recognizes there are two uses made of H-M reps .
It suggests generalizations of these based on LaTeXMLMath properties .
The tough question for Modular Towers is not if it is useful or connected well to the rest of mathematics .
Rather , it is if its problems are sufficiently tractable for progress .
By proving our Main Conjecture in convincing cases , with analytic detail often applied to modular curves , we assure rapid progress is possible .
The Main Conjecture shows regular realizations of significant finite groups as Galois groups will not appear serendipitously .
Indeed , the Hurwitz space approach gives meaning to considering where to look for them .
Therefore , we have put a summary statement for those who ask : So , where are those regular realizations ( § LaTeXMLRef ) ?
Four branch point Hurwitz families produce quotients of the upper half plane by finite index subgroups of LaTeXMLMath .
The Hurwitz monodromy group LaTeXMLMath a group LaTeXMLMath , identified with LaTeXMLMath , as a natural quotient .
The structure of a Modular Tower gives sequences of LaTeXMLMath -line covers formed by natural moduli problems .
These are immensely more numerous than the special case of modular curve towers though they have both similarities to them and exotic new properties generally .
There is a Modular Tower for any finite group LaTeXMLMath , prime LaTeXMLMath dividing LaTeXMLMath and collection C of ( LaTeXMLMath ) LaTeXMLMath conjugacy classes .
Then LaTeXMLMath has a universal LaTeXMLMath -Frattini profinite group cover , the profinite limit of a sequence LaTeXMLMath of LaTeXMLMath -extensions of LaTeXMLMath .
Each level LaTeXMLMath has a moduli space LaTeXMLMath in the ( reduced ) Modular Tower .
When LaTeXMLMath all components of the LaTeXMLMath s are LaTeXMLMath -line covers .
Then , reduced inner Hurwitz spaces generalize the modular curve sequence LaTeXMLMath .
Analyzing the action of LaTeXMLMath ( and LaTeXMLMath ) on Nielsen classes attached to each Modular Tower level gives their detailed structure .
If there is LaTeXMLMath with all LaTeXMLMath components of genus at least 2 , then for each number field LaTeXMLMath there is LaTeXMLMath with LaTeXMLMath for LaTeXMLMath ( Thm .
LaTeXMLRef ) .
Main Conjecture ( LaTeXMLCite , Prob .
LaTeXMLRef , § LaTeXMLRef ) : For LaTeXMLMath , LaTeXMLMath centerless and LaTeXMLMath -perfect ( § LaTeXMLRef ) , and an explicit LaTeXMLMath , all LaTeXMLMath components have general type if LaTeXMLMath .
So , if LaTeXMLMath , the genus of LaTeXMLMath components , LaTeXMLMath , exceeds 1 .
§ LaTeXMLRef reminds how finding regular realizations of the collection LaTeXMLMath forces this problem on the levels of a Modular Tower .
The methods of the paper pretty much restrict to the case LaTeXMLMath .
We use LaTeXMLMath and LaTeXMLMath to find considerable about excluding LaTeXMLMath for general LaTeXMLMath .
§ LaTeXMLRef states the unsolved problems for LaTeXMLMath framed by the genus computations of § LaTeXMLRef .
The effect of increasing components in going to higher levels of a Modular Tower justifies concentration on analyzing the component structures at level 1 in our main examples .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , four 3-cycle conjugacy classes , LaTeXMLMath works .
Level LaTeXMLMath has one ( genus 0 ) component .
Level LaTeXMLMath is the moduli of Galois covers with group LaTeXMLMath , a maximal Frattini extension of LaTeXMLMath with exponent 2 kernel , and order 3 branching .
A LaTeXMLMath regular realization of LaTeXMLMath produces a LaTeXMLMath point on level 1 .
Level 1 components ( LaTeXMLMath and and LaTeXMLMath , of genuses 12 and 9 ) correspond to values of a spin cover invariant ( as in LaTeXMLCite and Serre LaTeXMLCite ) .
Pairings on Modular Tower cusps appear in the symmetric sh - incidence matrix .
Each component corresponds to a mapping class orbit on reduced Nielsen classes , whose notation is LaTeXMLMath ( LaTeXMLMath for the level LaTeXMLMath Nielsen class generally ) .
Level 1 has one component of real points containing all Harbater-Mumford ( H-M ) and near H-M reps. ( Thm .
LaTeXMLRef ) .
The finitely many ( LaTeXMLMath equivalence ) LaTeXMLMath regular LaTeXMLMath realizations are on the genus 12 component .
Our computation for real points emphasizes that cusps also interpret as elements of a Nielsen class .
This combines geometry with group computations .
H-M cusps are real points on the reduced moduli space representing total degeneracy of the moduli of curve covers upon approach to the cusp .
The near H-M cusps , also real , are a Modular Tower phenomenon .
The corresponding cusp at one level below in the tower , is an H-M rep .
Above level 0 , H-M and near H-M cusps are the only real cusps .
This holds for any Modular Tower when the prime LaTeXMLMath is 2 .
Given LaTeXMLMath there is a universal sequence of nontrivial elements LaTeXMLMath .
Properties are in Prop .
LaTeXMLRef .
This is a special case of LaTeXMLCite .
An embedding of a group LaTeXMLMath in the alternating group LaTeXMLMath ( LaTeXMLMath ) is spin separating if the lift LaTeXMLMath to the spin cover LaTeXMLMath is nonsplit over LaTeXMLMath ( see Def .
LaTeXMLRef ) .
Such a central extension represents an element LaTeXMLMath .
Given an a priori LaTeXMLMath , we also say the embedding of LaTeXMLMath in LaTeXMLMath is LaTeXMLMath -spin separating .
When LaTeXMLMath , this produces a sequence LaTeXMLMath ( as above for LaTeXMLMath ) .
So , each LaTeXMLMath represents a central extension LaTeXMLMath with LaTeXMLMath as kernel .
Finding spin separating representations for these extensions has applications that include identifying projective systems of components on certain Modular Tower levels .
These conjecturally traces to the evenness and oddness of associated LaTeXMLMath -nulls on these components ( § LaTeXMLRef ) .
Prop .
LaTeXMLRef produces spin separating representations of LaTeXMLMath for which LaTeXMLMath is LaTeXMLMath .
Let LaTeXMLMath be the kernel of LaTeXMLMath and use that LaTeXMLMath factors through the spin cover LaTeXMLMath .
A special case of a general operating principle applies next .
With LaTeXMLMath an H-M rep. at level 0 , let LaTeXMLMath be the subgroup of LaTeXMLMath stabilizing LaTeXMLMath .
This allows us ( a significant special case of a general situation ) to regard LaTeXMLMath acting on LaTeXMLMath with both LaTeXMLMath over LaTeXMLMath and lying in a certain linear subspace of LaTeXMLMath .
There are two orbits : LaTeXMLMath and LaTeXMLMath in the same LaTeXMLMath orbit ( action on LaTeXMLMath ) ; and LaTeXMLMath and LaTeXMLMath in different LaTeXMLMath orbits ( Prop .
LaTeXMLRef ) .
This gives information on all levels of the same tower , and toward the Main Conjecture when LaTeXMLMath for all groups LaTeXMLMath ( § LaTeXMLRef ) .
Finite covers of the LaTeXMLMath -punctured Riemann sphere LaTeXMLMath have an attached group LaTeXMLMath and unordered set of conjugacy classes C of LaTeXMLMath .
The braid group approach to the Inverse Galois Problem combines moduli space geometry with finite group representations to find LaTeXMLMath regular realizations as rational points on Hurwitz spaces .
LaTeXMLCite includes an update on applying this to classical problems without obvious connection to the Inverse Galois Problem .
The history of the Inverse Galois Problem shows it is hard to find LaTeXMLMath realizations of most finite groups LaTeXMLMath .
Such a realization is a quotient of LaTeXMLMath : LaTeXMLMath is LaTeXMLMath with LaTeXMLMath distinct points LaTeXMLMath removed .
To systematically investigate the diophantine difficulties we take large ( maximal Frattini ) quotients LaTeXMLMath of LaTeXMLMath covering LaTeXMLMath instead of one finite group LaTeXMLMath .
Modular Towers are moduli space sequences encoding quotients of LaTeXMLMath isomorphic to LaTeXMLMath as LaTeXMLMath varies .
This generalizes classical modular curve sequences , and their traditional questions , through an Inverse Galois Problem formulation .
The modular curve case starts with a dihedral group LaTeXMLMath , and a natural projective limit of the dihedral groups LaTeXMLMath .
In the generalization , the maximal LaTeXMLMath -Frattini ( universal LaTeXMLMath -Frattini ; § LaTeXMLRef ) cover LaTeXMLMath of any finite group LaTeXMLMath replaces LaTeXMLMath .
Levels of a Modular Tower correspond to characteristic quotients LaTeXMLMath of LaTeXMLMath , replacing LaTeXMLMath .
Finding C regular realizations of the groups LaTeXMLMath is to finding rational points on levels of a Modular Tower as finding regular involution realizations of LaTeXMLMath is to finding rational points on the modular curves LaTeXMLMath ( § LaTeXMLRef ) .
Though the Modular Tower definition supports many general properties of modular curves , when LaTeXMLMath the moduli space levels are algebraic varieties of dimension LaTeXMLMath .
By conjecture ( under the LaTeXMLMath -perfect hypothesis ) we expect a structured disappearance of rational points at high levels in generalization of modular curves .
This holds by proof in many cases beyond those of modular curves from this paper .
Modular Towers joins number theory approaches to LaTeXMLMath actions on one hand to the Inverse Galois Problem braid group approach .
Profinite systems of points on Modular Towers reveal relations in the Lie group of LaTeXMLMath attached to a LaTeXMLMath -adic representation .
Yet , they benefit from the explicit tools of finite group theory ( modular representations ) .
Our concentration is on Modular Towers for groups distinct from dihedral groups .
Many have geometric properties not appearing for towers of modular curves .
Especially they can have several connected components .
The Hurwitz viewpoint applied to towers of modular curves shows how some points of general Modular Towers work .
Examples : § LaTeXMLRef on cusps of the modular curves LaTeXMLMath and LaTeXMLMath ; and § LaTeXMLRef comparing the moduli problems from isogenies of elliptic curves with the Hurwitz space interpretation of these spaces .
Each Modular Tower level maps naturally to a space of abelian varieties , given by the jacobians ( with dimension inductively computed from § LaTeXMLRef ; each far from simple ) of the curve covers parametrized by the points of the moduli space .
For example , level LaTeXMLMath of our LaTeXMLMath Modular Tower maps to a one-dimensional ( nonconstant ) space of Jacobians having dimension LaTeXMLMath .
§ LaTeXMLRef considers how to use this to measure how far is a Modular Tower from being a tower of modular curves .
Let LaTeXMLMath be distinct points ( punctures on LaTeXMLMath ) and denote LaTeXMLMath by LaTeXMLMath .
Riemann ’ s Existence Theorem produces LaTeXMLMath covers .
Classical generators come from paths , LaTeXMLMath , LaTeXMLMath , satisfying these conditions : LaTeXMLMath clockwise bounds a disc LaTeXMLMath around LaTeXMLMath , LaTeXMLMath ; LaTeXMLMath starts at LaTeXMLMath and ends at some point on LaTeXMLMath ; and excluding their beginning and end points , LaTeXMLMath and LaTeXMLMath never meet if LaTeXMLMath .
Let LaTeXMLMath be the respective homotopy classes of LaTeXMLMath .
Use ( LaTeXMLRef LaTeXMLRef ) to attach an order of clockwise emanation to LaTeXMLMath .
Assume it is in the order of their subscript numbering .
Then : LaTeXMLMath : The product-one condition .
A surjective homomorphism LaTeXMLMath produces a cover with monodromy group LaTeXMLMath .
More precise data stipulates that a system of classical generators map to C .
Such a LaTeXMLMath is a geometric LaTeXMLMath representation .
If LaTeXMLMath is an automorphism of LaTeXMLMath , it sends generators to new generators , changing LaTeXMLMath to LaTeXMLMath .
An inner automorphism of LaTeXMLMath produces a cover equivalent to the old cover .
We use moduli of covers , so equivalence two homomorphisms if they differ by an inner automorphism .
Only automorphisms of LaTeXMLMath from the Hurwitz monodromy group LaTeXMLMath send classical generators to classical generators ( ( LaTeXMLRef ) : possibly changing the intrinsic order of the paths ) .
Such automorphisms arise by deforming LaTeXMLMath .
They permute the conjugacy classes of the LaTeXMLMath s in LaTeXMLMath .
Given LaTeXMLMath , C is a well defined LaTeXMLMath invariant .
Suppose LaTeXMLMath and the unordered set LaTeXMLMath have definition field LaTeXMLMath .
Lem .
LaTeXMLRef , the Branch Cycle Lemma , gives a necessary ( rational union ) condition for an arithmetic LaTeXMLMath ( over LaTeXMLMath ) representation ; LaTeXMLMath factors through a representation of the arithmetic fundamental group ( § LaTeXMLRef ) .
Such a cover is a LaTeXMLMath ( regular ) realization ( over LaTeXMLMath ) .
For every finite group LaTeXMLMath and prime LaTeXMLMath dividing LaTeXMLMath , there is a universal LaTeXMLMath -Frattini ( § LaTeXMLRef ) cover LaTeXMLMath of LaTeXMLMath .
Generators of LaTeXMLMath lift automatically to generators of any finite quotient of LaTeXMLMath factoring through LaTeXMLMath .
This and other properties give all finite quotients of LaTeXMLMath a touching resemblance to LaTeXMLMath .
A Modular Tower ( § LaTeXMLRef ) is the system of moduli spaces encoding all LaTeXMLMath -Frattini LaTeXMLMath covers of LaTeXMLMath ( § LaTeXMLRef ) .
Reducing by LaTeXMLMath action gives a reduced Modular Tower .
Let LaTeXMLMath be the modular curve LaTeXMLMath without its cusps .
With LaTeXMLMath a dihedral group LaTeXMLMath ( LaTeXMLMath odd ) and C four repetitions of the involution conjugacy class , the ( reduced ) Modular Tower is the curve sequence LaTeXMLEquation .
Our main results compute properties of an LaTeXMLMath Modular Tower attached to LaTeXMLMath , four repetitions of elements of order 3 as associated conjugacy classes LaTeXMLMath .
§ LaTeXMLRef extends the general discussion of Modular Towers beyond LaTeXMLCite and LaTeXMLCite .
When LaTeXMLMath , reduced Modular Tower levels are curves covering the classical LaTeXMLMath -line .
As they are upper half plane quotients by finite index subgroups of LaTeXMLMath , this case shows how reduced Modular Towers generalize properties of modular curves .
The groups LaTeXMLMath ( Hurwitz monodromy group ) and LaTeXMLMath ( reduced mapping class group ) from Thm .
LaTeXMLRef apply to make reduced Modular Tower computations such as these .
Branch cycle descriptions and cusp data for these LaTeXMLMath -line covers ( § LaTeXMLRef ) .
A Klein 4-group LaTeXMLMath test for reduced Hurwitz spaces being b-fine or fine moduli spaces ( Prop .
LaTeXMLRef ) .
The group LaTeXMLMath is a quotient of a normal quaternion subgroup of LaTeXMLMath .
We use the term b ( irational ) -fine to mean that excluding the locus over LaTeXMLMath or 1 ( or LaTeXMLMath ) , the reduced space is a fine moduli space .
The geometric description of these groups in Prop .
LaTeXMLRef presents arithmetic conclusions when the action of LaTeXMLMath on Nielsen classes is trivial .
These are convenient for applications to the Inverse Galois Problem ( see § LaTeXMLRef and Rem .
LaTeXMLRef ) .
The universal LaTeXMLMath -Frattini cover of a finite group LaTeXMLMath lies behind the construction of Modular Towers .
Any triple LaTeXMLMath with LaTeXMLMath a prime dividing LaTeXMLMath and C some LaTeXMLMath conjugacy classes in LaTeXMLMath produces a tower of moduli spaces .
§ LaTeXMLRef is brief about the relevant group theory , relying on results from LaTeXMLCite .
Still , § LaTeXMLRef reminds of the geometry of inaccessibility that makes Frattini covers intriguing .
The moduli space phenomena in the case LaTeXMLMath and LaTeXMLMath interprets modular representation theory in the geometry of the particular Modular Tower .
§ LaTeXMLRef describes the characteristic quotients of the universal 2-Frattini cover of LaTeXMLMath .
It produces the sequence of groups LaTeXMLMath defining the levels of the Modular Tower of this paper where LaTeXMLMath .
§ LaTeXMLRef uses LaTeXMLMath -line branch cycles ( § LaTeXMLRef ) to draw diophantine conclusions about regular realizations .
The main information comes from analyzing components of real points at level 0 and 1 of the LaTeXMLMath Modular Tower .
If LaTeXMLMath is a ( 4 branch point ) cover over a field LaTeXMLMath in a given Nielsen class Ni ( § LaTeXMLRef ) , then it produces a LaTeXMLMath point on the corresponding reduced Hurwitz space LaTeXMLMath ( Def .
LaTeXMLRef ) .
Call this a LaTeXMLMath -cover point .
The branch locus LaTeXMLMath of a LaTeXMLMath -cover point gives a LaTeXMLMath point on LaTeXMLMath .
Detailed analysis of the case LaTeXMLMath is especially helpful when LaTeXMLMath .
There are three kinds of LaTeXMLMath points on LaTeXMLMath : LaTeXMLMath -cover points ( with LaTeXMLMath understood , call these cover points ) ; LaTeXMLMath -Brauer points ( not LaTeXMLMath -cover points ) given by LaTeXMLMath covers LaTeXMLMath with LaTeXMLMath a conic in LaTeXMLMath over LaTeXMLMath ( § LaTeXMLRef ) ; and LaTeXMLMath points that are neither LaTeXMLMath -cover or LaTeXMLMath - Brauer points .
When LaTeXMLMath is a fine ( resp .
b-fine ) moduli space ( § LaTeXMLRef ) , there are only LaTeXMLMath -cover and LaTeXMLMath -Brauer points ( resp .
excluding points over LaTeXMLMath or 1 ) .
§ LaTeXMLRef interprets how cusps attach to orbits of an element LaTeXMLMath in the Hurwitz monodromy group quotient we call LaTeXMLMath .
It also explains H-M and near H-M representatives ( of a Nielsen class ) .
These correspond to cusps at the end of components of LaTeXMLMath points on levels LaTeXMLMath of our main Modular Tower .
These ( eight ) LaTeXMLMath points are exactly the LaTeXMLMath -cover points at level LaTeXMLMath of the LaTeXMLMath Modular Tower lying above any LaTeXMLMath .
There are also eight LaTeXMLMath -Brauer points , complements of H-M and near H-M reps. ( § LaTeXMLRef ) .
This gives the full set of LaTeXMLMath points on the genus 12 component of LaTeXMLMath we label LaTeXMLMath .
Prop .
LaTeXMLRef shows there is only one component of LaTeXMLMath points on LaTeXMLMath .
So , LaTeXMLMath contains all LaTeXMLMath points at level 1 .
The LaTeXMLMath th level ( LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , of this reduced Modular Tower has a component LaTeXMLMath with both H-M reps. and near H-M reps. ( Prop .
LaTeXMLRef ) .
There is a nontrivial central Frattini extension LaTeXMLMath ( Prop .
LaTeXMLRef ) .
Near H-M reps. also correspond to LaTeXMLMath covers whose field of moduli over LaTeXMLMath is LaTeXMLMath , though LaTeXMLMath is not a field of definition .
There are also LaTeXMLMath -Brauer points on LaTeXMLMath ( Prop .
LaTeXMLRef ) .
The production of an infinite number of significantly different examples of these types of points from one moduli problem is a general application of Modular Towers .
Real points on LaTeXMLMath ( the absolute reduced Hurwitz space ; equivalence classes of degree 5 covers ) and the inner reduced space LaTeXMLMath are all cover points ( Lem .
LaTeXMLRef and Lem .
LaTeXMLRef ) .
This is a special case of the general Prop .
LaTeXMLRef when the group LaTeXMLMath acts trivially on Nielsen classes .
§ LaTeXMLRef lists geometric properties of level 1 of the reduced LaTeXMLMath Modular Tower ( LaTeXMLMath ) .
Level 0 is an irreducible genus 0 curve with infinitely many LaTeXMLMath points LaTeXMLCite .
A dense set of LaTeXMLMath points produces a LaTeXMLMath regular realization of LaTeXMLMath ; each gives a LaTeXMLMath regular realization of the LaTeXMLMath spin cover LaTeXMLMath which identifies with LaTeXMLMath ( Rem .
LaTeXMLRef ) .
The level 1 group LaTeXMLMath covers LaTeXMLMath , factoring through the spin cover .
Thm .
LaTeXMLRef shows level 1 has two absolutely irreducible LaTeXMLMath components of genuses 12 and 9 .
These correspond to two LaTeXMLMath orbits , LaTeXMLMath and LaTeXMLMath , in the Nielsen class LaTeXMLMath ( § LaTeXMLRef ) .
Further , all LaTeXMLMath ( and therefore LaTeXMLMath ) points lie only in the genus 12 component , which has just one connected component of real points ( Prop .
LaTeXMLRef ) .
This starts with showing the 16 H-M and 16 near H-M representatives in the ( inner — not reduced ; see ( LaTeXMLRef ) and § LaTeXMLRef ) Nielsen class fall in one LaTeXMLMath orbit .
Together these account for the eight longest ( length-20 ) cusp widths on the corresponding reduced Hurwitz space component .
Computations in § LaTeXMLRef –§ LaTeXMLRef show how everything organizes around these cusps lying at the end of real components on LaTeXMLMath .
This separates two components of LaTeXMLMath according to the cusps lying on each .
The computer program LaTeXMLCite helped guide our results ( § LaTeXMLRef ) , which include LaTeXMLCite -less arguments accounting for all phenomena of this case .
This reveals the group theoretic obstructions to proving the main conjecture ( Prob .
LaTeXMLRef ; for LaTeXMLMath ) .
We still need Falting ’ s Theorem to conclude there are only finitely many LaTeXMLMath realizations over LaTeXMLMath ( or over any number field ) .
Many Modular Towers should resemble this one , especially in the critical analysis of H-M and near H-M reps .
Detailed forms , however , of the Main Conjecture are unlikely to be easy as we see from sketches of other examples in Ex .
LaTeXMLRef and Ex .
LaTeXMLRef .
H-M and near H-M reps. in these examples work for us , though they differ in detail from our main example .
When LaTeXMLMath , analyzing the LaTeXMLMath action on reduced Nielsen classes has a short summary ( Prop .
LaTeXMLRef ) .
Suppose LaTeXMLMath ( where LaTeXMLMath for some LaTeXMLMath ) represents an element in a reduced Nielsen class .
Then LaTeXMLMath satisfies the product-one condition : LaTeXMLMath .
We have the following actions .
The shift , LaTeXMLMath of § LaTeXMLRef , maps LaTeXMLMath to LaTeXMLMath .
The middle twist LaTeXMLMath maps LaTeXMLMath to LaTeXMLMath .
Orbits of LaTeXMLMath correspond to cusps of a LaTeXMLMath -line cover .
Each LaTeXMLMath ( a mapping class group quotient ) orbit corresponds to a cover of the LaTeXMLMath -line .
§ LaTeXMLRef introduces the ( symmetric ) sh -incidence matrix summarizing a pairing using sh on LaTeXMLMath orbits .
This matrix shows the LaTeXMLMath orbit structure of the genus 12 component of LaTeXMLMath ( § LaTeXMLRef ) .
This orbit contains all H-M ( or near H-M ) orbits and their shifts ( Def .
LaTeXMLRef ) .
This illustrates analyzing the LaTeXMLMath and LaTeXMLMath orbits on reduced Nielsen classes , including arithmetic properties of cusps .
Since components are moduli spaces this means analyzing degeneration of objects in the moduli space on approach ( over LaTeXMLMath or LaTeXMLMath ; LaTeXMLCite ) to the cusps .
It generalizes analysis of elliptic curve degeneration in approaching a cusp of a modular curve .
These are moduli spaces of curves .
Here as in § LaTeXMLRef , Modular Towers involve nilpotent fundamental groups that detect degeneration of curves , not just their Jacobians .
§ LaTeXMLRef shows the idea of sh -incidence works for general LaTeXMLMath .
It is a powerful tool for simplifying the computation of braid orbits on Nielsen classes , though we haven ’ t had time to explore its geometry interpretations for general LaTeXMLMath .
Up to w-equivalence ( LaTeXMLRef LaTeXMLRef ) there are but finitely many four branch point LaTeXMLMath realizations of LaTeXMLMath , a diophantine result ( Thm .
LaTeXMLRef ) .
Should any exist , they correspond to rational points on the genus 12 ( level 1 ) component of the Modular Tower .
From Lem .
LaTeXMLRef , LaTeXMLMath acts faithfully at level 1 .
So , LaTeXMLMath points on LaTeXMLMath correspond either to LaTeXMLMath regular realizations of the representation cover for the Schur multiplier for LaTeXMLMath ( LaTeXMLMath -cover points ) or to LaTeXMLMath -Brauer points ( Lem .
LaTeXMLRef ) .
Which is a diophantine subtlety .
Only the genus 12 orbit LaTeXMLMath is unobstructed for the big LaTeXMLMath invariant LaTeXMLMath of LaTeXMLCite ( § LaTeXMLRef ) .
The component corresponding to LaTeXMLMath is obstructed LaTeXMLCite .
The general meaning of obstructed LaTeXMLMath — LaTeXMLMath applied to a component ( at level , say , LaTeXMLMath ) LaTeXMLMath — LaTeXMLMath is that it has nothing above it at level LaTeXMLMath .
This applies either to the geometric moduli space component or to the LaTeXMLMath orbit on a Nielsen class associated to it .
Prop .
LaTeXMLRef shows LaTeXMLMath , LaTeXMLMath , distinguishes the components .
That is , LaTeXMLMath doesn ’ t contain 1 and the next levels of the Modular Tower have nothing over it .
The LaTeXMLMath invariant separates most Modular Tower components in this paper .
A more general idea of lifting invariant appears in § LaTeXMLRef .
The more general idea is easier to understand , though not so efficient for actually separating components .
LaTeXMLCite interpreted a case of this , for alternating groups .
The ingredients included a Clifford algebra , and half-canonical classes ( LaTeXMLMath -characteristics ) .
Serre ’ s invariant is the special case of § LaTeXMLRef using pullbacks of subgroups of alternating groups to their spin covers .
It appears nontrivially at level 0 of each LaTeXMLMath Modular Tower with LaTeXMLMath and 3-cycle conjugacy classes ( Table LaTeXMLRef , Prop .
LaTeXMLRef , LaTeXMLCite ) .
The phrase spin separating representation refers to presenting a group inside the alternating group so it has a nonsplit central extension through pull back to the spin cover ( Def .
LaTeXMLRef ) .
§ LaTeXMLRef uses a Clifford algebra to produce spin separating representations of LaTeXMLMath .
§ LaTeXMLRef outlines the potential LaTeXMLMath -nulls ( giving automorphic functions on the Modular Tower levels ) coming from spin separating representations ( details in the expansion of LaTeXMLCite ) .
This explains part of this paper ’ s most mysterious phenomenon : The two LaTeXMLMath orbits on LaTeXMLMath have the same image groups , and degrees .
This is despite their being different LaTeXMLMath representations distinguished just by the number and length of their LaTeXMLMath orbits .
Appearance of the Clifford algebra dominates details of our study of this Modular Tower level 1 through component separation .
We haven ’ t yet shown a similar separated components phenomenon happens beyond level 1 of our main Modular Tower ( see § LaTeXMLRef ) .
Still , § LaTeXMLRef relates the subtly different geometry of the ( real ) cusps associated with H-M reps. versus near H-M reps. to the phenomenon of fields of moduli versus field of definitions attached to realizing Frattini central extensions .
This Schur multiplier result ( Prop .
LaTeXMLRef ) happens at all levels of the Modular Tower ( certain to generalize to most Modular Towers ) .
This is a weaker form of the geometry of spin separation and an example of how Frattini central extensions ( Schur multipliers ) of perfect groups affect the geometry of all levels of any Modular Tower .
The two level 1 representations of LaTeXMLMath , with their relation to spin separating representations has ingredients like those in LaTeXMLCite .
So , § LaTeXMLRef uses this analogy to find a precise measure of the difference between the two representations .
Let LaTeXMLMath be a field .
A point on a Modular Tower is a projective system of points LaTeXMLMath : LaTeXMLMath on the LaTeXMLMath st level maps to LaTeXMLMath at level LaTeXMLMath .
Similarly , define points on reduced Modular Towers .
If all LaTeXMLMath s have definition field LaTeXMLMath , then this sequence defines a LaTeXMLMath point .
Suppose LaTeXMLMath is a number field ( more generally , LaTeXMLMath has an infinite number of places with finite residue class field ) .
Then , Thm .
LaTeXMLRef implies a Modular Tower ( of inner Hurwitz spaces ) has no LaTeXMLMath points .
Assume LaTeXMLMath is data for a Modular Tower with LaTeXMLMath centerless and LaTeXMLMath -perfect ( Def .
LaTeXMLRef ) .
For LaTeXMLMath large , show all components of a Modular Tower at level LaTeXMLMath have general type .
Let LaTeXMLMath be a number field .
Find explicit large LaTeXMLMath so the LaTeXMLMath th level of a Modular Tower of inner Hurwitz spaces contains no LaTeXMLMath points ( LaTeXMLCite ; see § LaTeXMLRef ) .
The hypotheses on LaTeXMLMath say that LaTeXMLMath has no LaTeXMLMath quotient .
The centerless condition ensures a Modular Tower of inner Hurwitz spaces consists of fine moduli spaces ( Prop .
LaTeXMLRef ) .
Our Main Example LaTeXMLMath has all inner levels fine moduli spaces .
Its reduced spaces are also , if LaTeXMLMath ( Lem .
LaTeXMLRef ) .
Level 0 , however , of this ( inner and reduced ) Modular Tower is not a fine or b-fine moduli space ( § LaTeXMLRef ) .
Suppose LaTeXMLMath where LaTeXMLMath is a LaTeXMLMath -group ( § LaTeXMLRef ) acting through automorphisms on a LaTeXMLMath -group LaTeXMLMath .
Then , LaTeXMLMath is LaTeXMLMath with LaTeXMLMath the pro-free pro- LaTeXMLMath group on the minimal number of generators for LaTeXMLMath ( Rem .
LaTeXMLRef or LaTeXMLCite ) .
Refer to this as LaTeXMLMath -split data for a Modular Tower .
Assume a finite group LaTeXMLMath acts on LaTeXMLMath ( possibly the integers of a number field ) .
For each prime LaTeXMLMath not dividing LaTeXMLMath , form LaTeXMLMath .
Let LaTeXMLMath be the pro-free pro- LaTeXMLMath group of rank LaTeXMLMath and let C be conjugacy classes in LaTeXMLMath .
Form the Modular Tower for LaTeXMLMath , based on the characteristic LaTeXMLMath -Frattini quotients of LaTeXMLMath ( directly from those of LaTeXMLMath ) .
This has a modular curve-like property : You can vary LaTeXMLMath .
Modular curves are the case LaTeXMLMath , LaTeXMLMath with C four repetitions of the nontrivial element of LaTeXMLMath .
Whenever LaTeXMLMath , the analysis is similar to modular curves .
Compare the case LaTeXMLMath , LaTeXMLMath , C two repetitions each of the conjugacy classes of LaTeXMLMath and LaTeXMLMath with the harder case where LaTeXMLMath , LaTeXMLMath in Ex .
LaTeXMLRef .
Characteristic LaTeXMLMath quotients LaTeXMLMath are perfect and centerless ( Prop .
LaTeXMLRef ) .
For each LaTeXMLMath , the natural map LaTeXMLMath maps through the exponent LaTeXMLMath part of the universal central extension LaTeXMLMath of LaTeXMLMath .
Modular Towers for Ex .
LaTeXMLRef depart from towers of moduli spaces in the literature .
This paper emphasizes new group and geometry phenomena amid comfortable similarities to modular curves even for the many Modular Towers covering the classical LaTeXMLMath -line .
This paper aims to show , when LaTeXMLMath , the genus of reduced Modular Tower components grows as the levels rise .
Modular curve-like aspects in Ex .
LaTeXMLRef guide finding information about cusp behavior .
The phenomenon of obstructed components , appearing in our main example , is the most non-modular curve-like aspect of general Modular Towers .
Suppose in Ex .
LaTeXMLRef , for a given LaTeXMLMath , LaTeXMLMath acts irreducibly on LaTeXMLMath .
Then , the Nielsen class LaTeXMLMath is nonempty ( Lemma LaTeXMLRef ) .
Deciding in Ex .
LaTeXMLRef if there are obstructed components ( LaTeXMLMath orbits ) at high levels above a level 0 LaTeXMLMath orbit seems difficult .
When LaTeXMLMath , this reflects on the genuses of the corresponding LaTeXMLMath -line covers .
Even when LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and C consists of several copies of the two nontrivial conjugacy classes of LaTeXMLMath , obstructed components appear at all level LaTeXMLMath for LaTeXMLMath suitably large ( dependent on LaTeXMLMath ) .
This special case of LaTeXMLCite uses the nontrivial Schur multiplier appearing at each level ( Prop .
LaTeXMLRef generalizing LaTeXMLCite ) .
The case LaTeXMLMath has elements like our main example , because their universal LaTeXMLMath -Frattini covers are close ( § LaTeXMLRef ) .
Yet , for C consisting of LaTeXMLMath ( LaTeXMLMath not large ) 3- cycle conjugacy classes , the early levels of the Modular Towers for LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) are not at all alike ( Ex .
LaTeXMLRef ) .
These calculations collect the key points for considering a full proof of the Main Modular Tower Conjecture when LaTeXMLMath .
Using this paper , LaTeXMLCite will show what to expect of all Modular Tower levels from LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) 3-cycles as conjugacy classes .
A full result on LaTeXMLMath , LaTeXMLMath and LaTeXMLMath would contribute to the following questions on the Inverse Galois Problem .
Does any other reduced LaTeXMLMath Modular Tower have LaTeXMLMath points at level 1 ?
Is there any set of four conjugacy classes C for LaTeXMLMath where the reduced inner Hurwitz space contains infinitely many ( any ? )
LaTeXMLMath points ?
An example best illustrates the serious points about Question ( LaTeXMLRef LaTeXMLRef ) .
Let LaTeXMLMath be the conjugacy class of LaTeXMLMath , LaTeXMLMath the conjugacy class of LaTeXMLMath ( with LaTeXMLMath the conjugacy class of 3-cycles ) .
From the Branch Cycle Lemma , the only positive possibilities for ( LaTeXMLRef LaTeXMLRef ) are Modular Towers for LaTeXMLMath with LaTeXMLMath or LaTeXMLMath .
For both , the level 0 Modular Tower has two components , with exactly one obstructed ( nothing above it at level 1 ) .
The respective unobstructed LaTeXMLMath orbits contain the following representatives : LaTeXMLEquation .
Prop .
LaTeXMLRef explains this and why only the ( LaTeXMLRef b ) Modular Tower has real points at level 1 .
Rem .
LaTeXMLRef explains the one mystery left on answering ( LaTeXMLRef LaTeXMLRef ) , about genus 1 curves at level 1 of the reduced Modular Tower for LaTeXMLMath .
Modular Towers use LaTeXMLMath conjugacy classes .
If we allow conjugacy classes where LaTeXMLMath divides the elements ’ orders , then Hurwitz spaces attached to LaTeXMLMath and these classes must have LaTeXMLMath large if the components are to have definition field LaTeXMLMath ( Thm .
LaTeXMLRef ) .
There appear , however , to be nontrivial cases of LaTeXMLMath components for LaTeXMLMath and LaTeXMLMath in ( LaTeXMLRef LaTeXMLRef ) using conjugacy classes of order 4 .
Completing a yes answer to the following question requires completing these cases .
Are there only finitely many LaTeXMLMath realizations ( over LaTeXMLMath ; up to w-equivalence ) with at most four branch points ?
For , however , the group LaTeXMLMath , if LaTeXMLMath , the finitely many possible realizations can only fall on level 2 of the Modular Tower for LaTeXMLMath or for ( LaTeXMLRef b ) .
LaTeXMLCite produced LaTeXMLMath covers in the Nielsen class LaTeXMLMath when LaTeXMLMath is odd ( see also LaTeXMLCite ) .
LaTeXMLCite shows there is exactly one component here .
So , the results combine to show LaTeXMLMath points are dense in LaTeXMLMath : There are many LaTeXMLMath realizations over LaTeXMLMath that produce LaTeXMLMath realizations .
Does Thm .
LaTeXMLRef generalize to say LaTeXMLMath is finite for LaTeXMLMath odd ?
For LaTeXMLMath even this follows from LaTeXMLCite showing LaTeXMLMath is empty .
Falting ’ s Theorem LaTeXMLCite works when LaTeXMLMath because the moduli spaces are curves .
There is no replacement yet for Falting ’ s Theorem for LaTeXMLMath .
Here is a special case of the problem that would produce positive results for the Inverse Galois Problem .
Is LaTeXMLMath for LaTeXMLMath large ( dependent on LaTeXMLMath ) .
Ex .
LaTeXMLRef describes the complete set of components for LaTeXMLMath ( see Table LaTeXMLRef ) .
The procedure of LaTeXMLCite uses the many embeddings of LaTeXMLMath in LaTeXMLMath for LaTeXMLMath .
Any such embedding extends to an embedding of the level LaTeXMLMath characteristic 2-Frattini cover of LaTeXMLMath into the level LaTeXMLMath characteristic 2-Frattini cover of LaTeXMLMath ( special case of Lem .
LaTeXMLRef ) .
So , Thm .
LaTeXMLRef describing the two components of LaTeXMLMath immediately gives information on components of LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
LaTeXMLCite proves for LaTeXMLMath fixed and LaTeXMLMath large each element of the Schur multiplier of LaTeXMLMath determines a component .
This is a special case of a general result , though here we are precise about the components for all values of LaTeXMLMath .
Combining this with § LaTeXMLRef hints at describing components for higher levels of Modular Towers for all alternating groups .
There are many LaTeXMLMath -line covers .
It is significant to find simple invariants distinguishing Modular Towers from general quotients of the upper half plane by a subgroup of LaTeXMLMath .
As this paper and its consequents shows , they have far more in common with modular curves than does a general such quotient .
The structure of a symmetric integral sh -incidence matrix makes such a distinction , effectively capturing complicated data about Modular Tower levels .
Example : It displays each connected component ( Lem .
LaTeXMLRef ) .
The cusp pairing for LaTeXMLMath extends to the case LaTeXMLMath ( § LaTeXMLRef ) .
Thm .
LaTeXMLRef results from information about the level 1 cusps of this Modular Tower .
Most interesting are cusps attached to H-M representatives .
LaTeXMLCite explains the moduli manifestation of total degeneration around such a cusp .
The compactification is through equivalence classes of specialization sequences .
Especially short specialization sequences detect H-M reps. from total degeneration .
This is a very different compactification from that of the stable compactification theorem cover versions , like the log structures LaTeXMLCite uses .
Specialization sequences require no extension of the base field .
So they are compatible with absolute Galois group actions .
H-M reps. give the type of moduli degeneration useful for a tangential basepoint ( language of Ihara-Matsumoto LaTeXMLCite and Nakamura LaTeXMLCite ) .
These also use compactifications not quite like stable compactification ( see LaTeXMLCite ) .
Likely for LaTeXMLMath there is not a huge difference between these compactifications , though that is unlikely for LaTeXMLMath .
For Modular Towers over LaTeXMLMath with LaTeXMLMath and LaTeXMLMath ( or higher ) consider pull back to the configuration space LaTeXMLMath of branch points of these covers .
Then , a real point on LaTeXMLMath with LaTeXMLMath corresponds to a cover with complex conjugate pairs of branch points ( Thm .
LaTeXMLRef ) .
So , one must recast relations on the image of LaTeXMLMath coming from the Grothendieck-Teichmüller group .
More appropriate for this situation than four real points is complex conjugate pairs of branch points ( compare with LaTeXMLCite ) .
For other problems one would want configurations with a complex conjugate pair and two real points of branch points ( as in § LaTeXMLRef ) .
This adapts for LaTeXMLMath acting on quotients of the profinite completion of LaTeXMLMath corresponding to projective limits of the monodromy groups of monodromy group cover from Modular Towers with LaTeXMLMath .
Here we connect to LaTeXMLCite at level 0 .
An example result gives exactly three degree five LaTeXMLMath covers of LaTeXMLMath ( up to w- equivalence ( LaTeXMLRef LaTeXMLRef ) ) , over LaTeXMLMath with branch points in LaTeXMLMath .
There are three non-cusp points in the order 12 group generated by the cusps on the absolute reduced space attached to LaTeXMLMath .
These three covers correspond to those three non-cusp points .
Mazur ’ s explicit bound on torsion points shows , among all LaTeXMLMath realizations ( over LaTeXMLMath ) , exactly three w-equivalence classes can have rational branch points ( LaTeXMLCite ) .
A result of Serre ( Prop .
LaTeXMLRef ; LaTeXMLCite ) allows computing cusp widths in the LaTeXMLMath case .
This includes the length-20 cusps attached to H-M and near H-M representatives .
Computing the genus of curves at all levels of a Modular Tower depends on such formulas .
We show many phenomena for Modular Tower levels of simple groups not appearing in modular curves .
There is a precise dividing line between two types of Modular Towers , with modular curves a model for one type .
LaTeXMLCite told the story of how modular curves are essentially the moduli for dihedral group realizations with four involution conjugacy classes .
The absolute ( resp .
inner ) Hurwitz spaces correspond to the curves LaTeXMLMath ( resp .
LaTeXMLMath ) .
The case of general LaTeXMLMath is about Hurwitz spaces associated to modular curve like covers of the moduli of hyperelliptic curves of genus LaTeXMLMath .
A generalization of this situation would include LaTeXMLMath with LaTeXMLMath a ( n abelian ) subgroup of LaTeXMLMath ( acting naturally on LaTeXMLMath ) and a collection of conjugacy classes C whose elements generate LaTeXMLMath .
For , however , inner Hurwitz spaces to be over LaTeXMLMath , C must be a rational union of conjugacy classes ( trivial case of the branch cycle argument ) .
This puts a lower bound on LaTeXMLMath .
For example , if LaTeXMLMath is cyclic and order LaTeXMLMath , then C must contain a minimum of LaTeXMLMath conjugacy classes .
That is why one rarely sees these spaces in the classical context , though they share with modular curves the attribute of having 1-dimensional LaTeXMLMath -Frattini modules .
For a LaTeXMLMath -perfect centerless group , the characteristic LaTeXMLMath -Frattini LaTeXMLMath module LaTeXMLMath has dimension 1 if and only if LaTeXMLMath never appears in the Loewy layers of LaTeXMLMath for any LaTeXMLMath ( § LaTeXMLRef and § LaTeXMLRef ) .
Appearance of those LaTeXMLMath s could mean obstructed components , uncertain location of cusps in components and related moduli interpretations of spaces for covers whose field of moduli is not a field of definition of representing points of the levels of a Modular Tower .
These useful geometric phenomena , present Modular Towers as a new tool for investigating still untouched mysteries .
That is our major theme .
Further , given LaTeXMLMath , if the conjugacy classes in C repeat often enough ( as a function of LaTeXMLMath ) , LaTeXMLCite implies these complications will occur at level LaTeXMLMath of the Modular Tower .
A big mystery is whether they occur at infinitely many levels of any Modular Tower for a LaTeXMLMath -perfect centerless group .
§ LaTeXMLRef formulates how to extend part of Serre ’ s Open Image Theorem LaTeXMLCite ( for just the prime LaTeXMLMath ) to Modular Towers .
This extension would be a tool for finding a precise lower bound on LaTeXMLMath for which higher levels of a Modular Tower have no rational points .
Ex .
LaTeXMLRef is appropriate for generalizations of Mazur-Merel .
Consider LaTeXMLMath with LaTeXMLMath running over primes not dividing LaTeXMLMath and integers LaTeXMLMath .
Find LaTeXMLMath so LaTeXMLMath has general type if LaTeXMLMath exceeds LaTeXMLMath .
If LaTeXMLMath , and LaTeXMLMath , find explicit LaTeXMLMath with LaTeXMLMath empty for LaTeXMLMath and LaTeXMLMath .
Mazur-Merel is the case LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath acting by multiplication ) , LaTeXMLMath and classes all the nontrivial element in LaTeXMLMath .
We continue with the case LaTeXMLMath .
What is the analog of Serre ’ s result for nonintegral LaTeXMLMath .
We interpret that to say , for LaTeXMLMath not an algebraic integer , the action of LaTeXMLMath on projective systems of points on LaTeXMLMath lying over LaTeXMLMath has an open orbit .
The ingredients in our generalizing statement ( still a conjecture ) is a Modular Tower meaning to LaTeXMLMath being suitably ( LaTeXMLMath -adically ) close to LaTeXMLMath and the Ihara-Matsumoto-Wewers approach to using tangential base points .
One topic we couldn ’ t resist was what we call LaTeXMLMath -awareness .
It would be trivial to generalize the Open Image Theorem if each appropriate Modular Tower was close to being a tower of modular curves .
That would mean there is some cover LaTeXMLMath so the pullback of a tower of modular curves dominates the levels of the given Modular Tower .
Checking this out on the main examples of this paper ( with LaTeXMLMath ) repeatedly called for analyzing Prym varieties ( Ex .
LaTeXMLRef and § LaTeXMLRef ) .
We compare our approach to LaTeXMLMath points with that of LaTeXMLCite .
Recall the complex conjugate of a complex manifold .
Suppose LaTeXMLMath is an atlas for a complex manifold LaTeXMLMath .
For simplicity , assume LaTeXMLMath is a 1-dimensional complex manifold .
Create a new manifold by composing each LaTeXMLMath with complex conjugation .
Call the resulting map LaTeXMLMath .
The atlas LaTeXMLMath is a new complex manifold structure LaTeXMLMath on the set LaTeXMLMath .
Denote by LaTeXMLMath the value of LaTeXMLMath .
Compare the transition functions LaTeXMLMath with the function LaTeXMLMath as LaTeXMLMath varies over the complex conjugate of LaTeXMLMath running over LaTeXMLMath .
The effect of the former is this : LaTeXMLEquation .
Suppose LaTeXMLMath is a local expression of the transition function LaTeXMLMath as a power series ( about the origin ) in LaTeXMLMath .
Then , the power series expressions for LaTeXMLMath comes by applying complex conjugation to the coefficients of LaTeXMLMath , and so the resulting function is analytic in LaTeXMLMath .
∎ A complex manifold LaTeXMLMath has an LaTeXMLMath structure if LaTeXMLMath is analytically isomorphic to LaTeXMLMath .
Apply this to a cover LaTeXMLMath using the complex structure from the LaTeXMLMath -sphere , as in Prop .
LaTeXMLRef .
You can also use the complex structure from uniformization by the upper half plane LaTeXMLMath as in LaTeXMLCite .
Keep in mind : LaTeXMLMath is the same set as LaTeXMLMath .
For LaTeXMLMath , there is a natural map LaTeXMLMath where we regard LaTeXMLMath as the complex conjugate of LaTeXMLMath , though in this formulation LaTeXMLMath is the same point on the set LaTeXMLMath .
Also , the lower half plane LaTeXMLMath naturally uniformizes LaTeXMLMath .
An LaTeXMLMath structure induces LaTeXMLMath , which in turn induces LaTeXMLMath .
Of course , LaTeXMLMath induces LaTeXMLMath .
Mochizuki makes simplifying assumptions : There is LaTeXMLMath with LaTeXMLMath and LaTeXMLMath the identity map .
Take LaTeXMLMath to be all meromorphic algebraic functions in a neighborhood of LaTeXMLMath that extend analytically along each path in LaTeXMLMath ( § LaTeXMLRef ) .
Then , LaTeXMLMath is Galois with profinite group LaTeXMLMath forming a split sequence LaTeXMLMath ( as in Prop .
LaTeXMLRef ) .
The virtues of this LaTeXMLCite inspired approach are these .
A splitting of LaTeXMLMath given by LaTeXMLMath , an anti-holomorphic involution LaTeXMLMath ( determinant -1 ) acting on LaTeXMLMath by LaTeXMLMath .
So , LaTeXMLMath acts through the matrix group LaTeXMLMath extending LaTeXMLMath .
Extend LaTeXMLMath by LaTeXMLMath to consider the LaTeXMLMath bundle over LaTeXMLMath with a natural flat connection ( algebraic from Serre ’ s GAGA ) .
Mochizuki calls this uniformization construction the canonical indigenous bundle .
Compare to our Prop .
LaTeXMLRef with several complex conjugation operators LaTeXMLMath , corresponding to a type of degenerate behavior at a cusp ( and the location of a set of branch points over LaTeXMLMath on LaTeXMLMath ) .
His simplifying assumption ( LaTeXMLMath ) applied to Modular Towers when LaTeXMLMath would reduce to considering LaTeXMLMath that appear as H-M reps. ( say , in Cor .
LaTeXMLRef ) , just one case we must treat in describing the full real locus for a result like Prop .
LaTeXMLRef on a Modular Tower level .
Mochizuki calls his prime LaTeXMLMath , though in our context it would be LaTeXMLMath prime to LaTeXMLMath , since we use LaTeXMLMath as a prime for the construction of a Modular Tower .
His construction is for any family of curves over any base .
It extends to work for LaTeXMLMath -adic uniformization giving a notion of LaTeXMLMath -ordinary points on a family of curves over LaTeXMLMath .
The Modular Tower goal would be to detect ordinary LaTeXMLMath ( with LaTeXMLMath ) points on a Modular Tower , generalizing ( from LaTeXMLMath a dihedral group ) ordinary elliptic curves over LaTeXMLMath LaTeXMLCite .
Only the genus 12 component at level 1 of the LaTeXMLMath Modular Tower supports totally degenerate cusps , associated with the name Harbater-Mumford representatives ( § LaTeXMLRef ) .
An analysis that combines Harbater patching and Mumford ’ s theory of total LaTeXMLMath -adic degeneration shows most points on such components will exhibit ordinary LaTeXMLMath -adic behavior .
Nilpotent fundamental groups enter directly in all Modular Tower definitions .
Wewers LaTeXMLCite uses the degeneration behavior type ( a tangential basepoint ) to consider LaTeXMLMath -adic ( LaTeXMLMath ) information for level 0 of our main Modular Tower .
His goal is to detect fields of moduli versus fields of definition ( following LaTeXMLCite ) .
§ LaTeXMLRef explains why the geometry of H-M reps. and near H-M reps. should have an LaTeXMLMath -adic analog at all levels of our LaTeXMLMath Modular Tower .
His analysis is also a tool for our approach to generalizing Serre ’ s Open Image Theorem ( § LaTeXMLRef ) .
Denote the Artin Braid group on LaTeXMLMath strings by LaTeXMLMath .
It has generators LaTeXMLMath satisfying these relations : LaTeXMLEquation .
LaTeXMLEquation We start from Bohnenblust LaTeXMLCite ( or LaTeXMLCite ) .
Let LaTeXMLMath be the free group of rank LaTeXMLMath on generators LaTeXMLMath , and denote LaTeXMLMath by LaTeXMLMath .
Then LaTeXMLMath embeds in LaTeXMLMath via this right action of LaTeXMLMath : LaTeXMLEquation .
Also , LaTeXMLMath consists of automorphisms of LaTeXMLMath fixing LaTeXMLMath and mapping LaTeXMLMath to permutations of conjugates of these generators .
Let LaTeXMLMath be the quotient of LaTeXMLMath by the relation LaTeXMLMath .
The Hurwitz monodromy group LaTeXMLMath is the quotient of LaTeXMLMath by adding the relation LaTeXMLEquation .
It is also the fundamental group of an open subset of LaTeXMLMath ( § LaTeXMLRef ) .
Denote inner automorphisms of a group LaTeXMLMath by LaTeXMLMath .
Then , LaTeXMLMath induces automorphisms of LaTeXMLMath mapping the images of LaTeXMLMath to permutations of conjugates of these generators , modulo LaTeXMLMath .
If LaTeXMLMath , denote the normalizer of LaTeXMLMath in LaTeXMLMath by LaTeXMLMath .
The next lemma is obvious .
Suppose LaTeXMLMath .
Define an induced action of LaTeXMLMath on the collection of such LaTeXMLMath s by LaTeXMLMath .
The action of ( LaTeXMLRef ) commutes with conjugation on an LaTeXMLMath -tuple by an element of LaTeXMLMath .
For LaTeXMLMath suppose LaTeXMLMath .
The induced action of LaTeXMLMath on LaTeXMLMath commutes with conjugation by LaTeXMLMath .
The mapping class group LaTeXMLMath is the quotient by LaTeXMLMath of automorphisms of LaTeXMLMath mapping LaTeXMLMath to permutations of conjugates .
This maps LaTeXMLMath to LaTeXMLMath factoring through LaTeXMLMath .
Further , LaTeXMLMath is the quotient of LaTeXMLMath by the following relations ( LaTeXMLCite or LaTeXMLCite ) : LaTeXMLEquation .
This complicated presentation is oblivious to the map LaTeXMLMath dominating this paper .
Using it conceptualizes the LaTeXMLMath kernel , LaTeXMLMath .
The switch of emphasis shows in Prop .
LaTeXMLRef .
With LaTeXMLMath write generators of LaTeXMLMath as follows : LaTeXMLEquation .
Thm .
LaTeXMLRef presents both LaTeXMLMath and LaTeXMLMath memorably .
An appearance of LaTeXMLMath in § LaTeXMLRef offers a place for spin separation to explain similarities of the two LaTeXMLMath orbits in level 1 of our main example .
The Riemann sphere uniformized by the variable LaTeXMLMath is LaTeXMLMath .
Relation ( LaTeXMLRef ) comes from the geometry of LaTeXMLMath distinct points on LaTeXMLMath .
We sometimes drop the notation for uniformizing by LaTeXMLMath .
Similarly , copies of LaTeXMLMath ( affine LaTeXMLMath -space ) and LaTeXMLMath ( projective LaTeXMLMath -space ) come equipped with coordinates suitable for a diagram : LaTeXMLEquation .
The upper left copy of LaTeXMLMath has coordinates LaTeXMLMath .
Then , LaTeXMLMath is the fat diagonal of LaTeXMLMath -tuples with two or more of the LaTeXMLMath s equal .
Embed LaTeXMLMath in LaTeXMLMath by LaTeXMLMath .
In LaTeXMLMath the LaTeXMLMath s may take on the value LaTeXMLMath .
Regard LaTeXMLMath as the quotient action of LaTeXMLMath permuting the coordinates of LaTeXMLMath .
As with LaTeXMLMath , put this action on the right .
Thus , the lower copy of LaTeXMLMath has coordinates LaTeXMLMath , with LaTeXMLMath equal to LaTeXMLMath times the LaTeXMLMath th symmetric function in LaTeXMLMath .
Regard this as giving a monic polynomial of degree LaTeXMLMath in LaTeXMLMath ( with zeros LaTeXMLMath ) .
Then , complete the commutative diagram by taking LaTeXMLMath as all monic ( nonzero ) polynomials of degree at most LaTeXMLMath .
The map LaTeXMLMath on the right sends each LaTeXMLMath -tuple LaTeXMLMath to LaTeXMLMath .
When LaTeXMLMath replace the factor LaTeXMLMath by 1 .
The image of LaTeXMLMath is the discriminant locus of polynomials with two or more equal zeros .
For coordinates around LaTeXMLMath , regard a monic polynomial of degree LaTeXMLMath as having LaTeXMLMath zeros at LaTeXMLMath .
This interprets LaTeXMLMath as the space of LaTeXMLMath distinct unordered points in LaTeXMLMath , the image of LaTeXMLMath .
Thus , LaTeXMLMath is an unramified Galois cover with group LaTeXMLMath .
The alternating group of degree LaTeXMLMath is LaTeXMLMath .
The dihedral group of degree LaTeXMLMath ( and order LaTeXMLMath ) is LaTeXMLMath .
Denote the LaTeXMLMath -adic integers by LaTeXMLMath , with LaTeXMLMath for its quotient field .
Suppose LaTeXMLMath is a prime and C is a conjugacy class in a group LaTeXMLMath .
Call LaTeXMLMath a LaTeXMLMath group if LaTeXMLMath .
Call C a LaTeXMLMath -conjugacy class ( or a LaTeXMLMath class ) if elements in C have orders prime to LaTeXMLMath .
This applies to conjugacy classes in profinite groups .
Each Hurwitz space ( our main moduli spaces ) comes from a finite group LaTeXMLMath , a set of conjugacy classes C in LaTeXMLMath and an equivalence relation on the corresponding Nielsen class .
§ LaTeXMLRef reviews this definition while enhancing traditional equivalences in LaTeXMLCite , LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
The fundamental group of LaTeXMLMath ( resp .
LaTeXMLMath ) is the Artin braid group LaTeXMLMath ( resp .
the Hurwitz monodromy group LaTeXMLMath ) ( LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
Embedding LaTeXMLMath into LaTeXMLMath in ( LaTeXMLRef ) gives the lower row surjective homomorphism from LaTeXMLMath to LaTeXMLMath .
The result is a commutative diagram of fundamental groups induced from a geometric commutative diagram .
Fundamental groups in the ( LaTeXMLRef ) upper row are the straight ( or pure ) Artin braid and Hurwitz monodromy groups .
That is , LaTeXMLMath is the kernel of LaTeXMLMath mapping LaTeXMLMath to LaTeXMLMath , LaTeXMLMath .
It increases precision to use capitals for generators LaTeXMLMath of LaTeXMLMath and small letters for their images LaTeXMLMath in LaTeXMLMath .
Consider LaTeXMLMath , an LaTeXMLMath -tuple of distinct points in LaTeXMLMath .
Let LaTeXMLMath be classical generators of LaTeXMLMath as in § LaTeXMLRef .
Suppose LaTeXMLMath is the universal covering space of LaTeXMLMath .
The fiber LaTeXMLMath over LaTeXMLMath consists of isotopy classes of classical generators on LaTeXMLMath ; it is a homogeneous space for LaTeXMLMath .
Use the isotopy class of LaTeXMLMath as a designated point LaTeXMLMath in the fiber , so each LaTeXMLMath is LaTeXMLMath for some LaTeXMLMath .
The corresponding isotopy takes a base point for classical generators for LaTeXMLMath to a basepoint for classical generators for LaTeXMLMath .
Further , a path in LaTeXMLMath from LaTeXMLMath to LaTeXMLMath induces a canonical isomorphism LaTeXMLMath : LaTeXMLMath ( resp .
LaTeXMLMath ) is the basepoint for classical generators for LaTeXMLMath ( resp .
LaTeXMLMath ) .
Equivalence two points LaTeXMLMath if some representing isomorphism LaTeXMLMath ( with LaTeXMLMath ) induces an inner automorphism on LaTeXMLMath .
This equivalence gives the fiber over LaTeXMLMath for an unramified cover LaTeXMLMath corresponding to LaTeXMLMath .
Prop .
LaTeXMLRef interprets LaTeXMLMath as isotopy classes of ( orientation preserving ) diffeomorphisms of LaTeXMLMath .
Classes here means to mod out by diffeomorphisms deforming ( through diffeomorphisms on LaTeXMLMath ) to the identity map .
Then , LaTeXMLMath acts through equivalence classes of orientation preserving diffeomorphisms on LaTeXMLMath .
This gives a natural map from it to LaTeXMLMath : LaTeXMLMath has group LaTeXMLMath .
Equivalence LaTeXMLMath and its image under LaTeXMLMath .
Elements of LaTeXMLMath that fix LaTeXMLMath may equivalence some isotopy classes of diffeomorphisms not previously equivalenced in LaTeXMLMath .
Map these new equivalence classes to LaTeXMLMath ( § LaTeXMLRef ) giving a ramified cover with group LaTeXMLMath .
When LaTeXMLMath : LaTeXMLMath has trivial fundamental group while LaTeXMLMath is far from trivial .
There is a precise Teichmüller space LaTeXMLMath for this situation .
It is isotopy classes of orientation preserving diffeomorphisms LaTeXMLMath with LaTeXMLMath .
Here LaTeXMLMath is equivalent to LaTeXMLMath if LaTeXMLMath is isotopic to an analytic map LaTeXMLMath ( from LaTeXMLMath ) .
Then LaTeXMLMath , the quotient of this Teichmüller space ( a ball by a famous theorem ) by LaTeXMLMath , is a moduli space .
For all LaTeXMLMath , we interpret it as the moduli of LaTeXMLMath branch point covers of LaTeXMLMath modulo LaTeXMLMath ( see the proof of Prop .
LaTeXMLRef ) .
For LaTeXMLMath it would typically be considered the moduli of hyperelliptic curves of genus LaTeXMLMath .
The following proposition geometrically interprets the group LaTeXMLMath appearing in Thm .
LaTeXMLRef .
As elsewhere ( see § LaTeXMLRef and Lem .
LaTeXMLRef ) , change the LaTeXMLMath variable linearly so 0 and 1 are the locus of ramification of the degree six cover LaTeXMLMath .
LaTeXMLCite gave an approximation to Prop .
LaTeXMLRef ( as in Prop .
LaTeXMLRef ) sufficient for that paper .
Its proof comprises § LaTeXMLRef .
When LaTeXMLMath , Thm .
LaTeXMLRef shows LaTeXMLMath is the center of LaTeXMLMath .
Then , LaTeXMLMath .
Two points LaTeXMLMath , respectively over LaTeXMLMath and LaTeXMLMath , induce an isomorphism LaTeXMLMath .
Equivalence LaTeXMLMath and LaTeXMLMath if some LaTeXMLMath taking LaTeXMLMath to LaTeXMLMath induces LaTeXMLMath inverse to LaTeXMLMath .
These equivalence classes form a space LaTeXMLMath , inducing LaTeXMLMath an analytic ( ramified ) cover with automorphism group LaTeXMLMath ( Thm .
LaTeXMLRef ) .
The group LaTeXMLMath is a Klein 4-group .
All points in the cover LaTeXMLMath over LaTeXMLMath ramify of order 3 .
All points over LaTeXMLMath ramify of order 2 .
Any Hurwitz space cover LaTeXMLMath induces a ( ramified ) cover LaTeXMLMath through a permutation representation of LaTeXMLMath .
Assume LaTeXMLMath and LaTeXMLMath have definition field LaTeXMLMath and LaTeXMLMath .
Suppose LaTeXMLMath acts trivially on a Nielsen class .
Then , each geometric point of the fiber LaTeXMLMath on LaTeXMLMath over LaTeXMLMath goes one-one to the fiber LaTeXMLMath with both points having exactly the same fields of definition over LaTeXMLMath .
Relate the homotopy classes of classical generators for LaTeXMLMath to those for LaTeXMLMath through applying an element LaTeXMLMath ( LaTeXMLCite or LaTeXMLCite ) .
The action is that given by ( LaTeXMLRef ) .
The map LaTeXMLMath explicitly makes this identification with an element of LaTeXMLMath .
From Lem .
LaTeXMLRef , if LaTeXMLMath acts as an inner automorphism , then it must be in the center of LaTeXMLMath .
Apply Thm .
LaTeXMLRef : LaTeXMLMath is in the center of LaTeXMLMath and acts as an inner automorphism of LaTeXMLMath .
Further , since LaTeXMLMath generates the whole center of LaTeXMLMath , LaTeXMLMath .
Now consider the equivalence from composing equivalence classes defining LaTeXMLMath with equivalence of maps on fundamental groups of LaTeXMLMath from elements of LaTeXMLMath .
This lies over the equivalence classes for the action of LaTeXMLMath on LaTeXMLMath .
The LaTeXMLMath equivalence class of LaTeXMLMath has a representative of form LaTeXMLMath .
For LaTeXMLMath not lying over LaTeXMLMath or 1 , we show the elements LaTeXMLMath fixing the set LaTeXMLMath form a Klein 4-group .
These computations undoubtedly occur in the literature , so we give only an outline .
First : LaTeXMLMath contains a Klein 4-group .
Example : The LaTeXMLMath switching 0 and 1 , and switching LaTeXMLMath and LaTeXMLMath is LaTeXMLMath .
Second : To see that no LaTeXMLMath acts like a 4-cycle on the support of LaTeXMLMath unless LaTeXMLMath is special , assume with no loss , LaTeXMLEquation .
Then , LaTeXMLMath .
From the last condition LaTeXMLMath or LaTeXMLMath .
Third : To see that no LaTeXMLMath acts like a 3-cycle on the support of LaTeXMLMath unless LaTeXMLMath is special , assume LaTeXMLMath is one of the permutations of LaTeXMLMath and LaTeXMLMath is an outside fixed point of LaTeXMLMath .
Example : LaTeXMLMath also fixes the two roots of LaTeXMLMath , which are sixth roots of 1 .
These computations show for each LaTeXMLMath , the fiber of equivalence classes is the same as the fiber of LaTeXMLMath over LaTeXMLMath modulo the faithful action of a Klein 4-group .
Over LaTeXMLMath the fiber is the same as the fiber of LaTeXMLMath over LaTeXMLMath modulo the faithful action of a group isomorphic to LaTeXMLMath , and over LaTeXMLMath it is the fiber over LaTeXMLMath modulo the action of a group isomorphic to LaTeXMLMath , the dihedral group of order 8 .
Complete the identification of the group of LaTeXMLMath with LaTeXMLMath from Thm .
LaTeXMLRef .
This only requires LaTeXMLMath to induce on LaTeXMLMath the Klein 4- group in LaTeXMLMath that sits in LaTeXMLMath above : Those switching the support of LaTeXMLMath in pairs .
This could be a case-by-case identification .
Since , however , Thm .
LaTeXMLRef shows LaTeXMLMath is the minimal normal subgroup of LaTeXMLMath containing LaTeXMLMath , it suffices to achieve LaTeXMLMath through an element of LaTeXMLMath .
It is convenient to take LaTeXMLMath , though this LaTeXMLMath is larger than a Klein 4-group .
Crucial to the proof of LaTeXMLCite is LaTeXMLMath and its effect on covers LaTeXMLMath in a Nielsen class branched over LaTeXMLMath .
Under any reduced equivalence of covers in this paper , LaTeXMLMath and LaTeXMLMath are equivalent .
The argument uses a set of paths LaTeXMLMath based at LaTeXMLMath .
These give classical generators ( § LaTeXMLRef ) of LaTeXMLMath with product LaTeXMLMath homotopic to 1 .
The effect of LaTeXMLMath ( up to homotopy ) is to switch LaTeXMLMath and LaTeXMLMath , and to map LaTeXMLMath to a path LaTeXMLMath for which LaTeXMLMath is homotopic to LaTeXMLMath .
Conjugating the resulting collection of paths by LaTeXMLMath gives the effect of LaTeXMLMath as follows .
It takes a representative LaTeXMLMath of the Nielsen class of LaTeXMLMath to LaTeXMLMath conjugated by LaTeXMLMath .
This fills in LaTeXMLCite and corrects a typo giving LaTeXMLMath as LaTeXMLMath ( which doesn ’ t fix LaTeXMLMath ) .
The points of a Hurwitz space LaTeXMLMath over LaTeXMLMath correspond to equivalence classes of homomorphisms from LaTeXMLMath where LaTeXMLMath is a finite group attached to the Hurwitz space .
While LaTeXMLMath may have several connected components , easily reduce to constructing a map from LaTeXMLMath to any one of them .
Start with a base point LaTeXMLMath over LaTeXMLMath and a base point LaTeXMLMath over LaTeXMLMath .
Then , relative to the classical generators given by LaTeXMLMath , LaTeXMLMath corresponds to a specific homomorphism LaTeXMLMath up to inner automorphism or some stronger equivalence ( § LaTeXMLRef ) .
Any other point LaTeXMLMath lying over LaTeXMLMath comes with an isomorphism from LaTeXMLMath .
This isomorphism takes classical generators of LaTeXMLMath to classical generators of LaTeXMLMath .
Relative to this new set of generators , form the same homomorphism into LaTeXMLMath .
Interpret as a canonically given homomorphism LaTeXMLMath .
This homomorphism determines the image of LaTeXMLMath in LaTeXMLMath and an unramified cover LaTeXMLMath .
This map respects equivalence classes modulo LaTeXMLMath action .
So it produces LaTeXMLMath .
For LaTeXMLMath , the analog of LaTeXMLMath is trivial ( § LaTeXMLRef ( .
Then , if a component LaTeXMLMath of LaTeXMLMath has a dense set of rational points , it is automatically true for LaTeXMLMath .
Assume LaTeXMLMath , LaTeXMLMath acts faithfully and LaTeXMLMath .
If LaTeXMLMath or LaTeXMLMath is a fine moduli space ( over LaTeXMLMath ) , then its LaTeXMLMath points correspond to equivalence classes of covers represented by an actual cover over LaTeXMLMath .
Prop .
LaTeXMLRef , efficiently tests for LaTeXMLMath points on LaTeXMLMath being cover or Brauer points ( Cor .
LaTeXMLRef ) .
Suppose LaTeXMLMath and a component LaTeXMLMath of LaTeXMLMath ( image of a component LaTeXMLMath of the Hurwitz space LaTeXMLMath ) has genus 0 and dense LaTeXMLMath points .
If any conjugacy class in C is LaTeXMLMath rational and distinct from the others , then Lem .
LaTeXMLRef shows LaTeXMLMath is also dense in LaTeXMLMath .
This , however , leaves many cases : all four conjugacy classes are distinct , none LaTeXMLMath rational ; or each conjugacy class appears at least twice in C .
Further , LaTeXMLMath can act neither faithfully , nor trivially , on Nielsen classes through LaTeXMLMath ( see Ex .
LaTeXMLRef ) .
As in Lem .
LaTeXMLRef , it is useful in all cases to distinguish between LaTeXMLMath -cover and LaTeXMLMath -Brauer points on LaTeXMLMath .
For the Inverse Galois Problem , LaTeXMLMath is special for the possibility of direct computations , especially when the simplest case of Braid rigidity ( for LaTeXMLMath ) does not apply .
So , Prop .
LaTeXMLRef is valuable for applications to it .
Here is an example long in the literature .
The Hurwitz space LaTeXMLMath ( over LaTeXMLMath ) has genus 0 and a rational point LaTeXMLCite .
From Ex .
LaTeXMLRef , LaTeXMLMath acts trivially .
Prop .
LaTeXMLRef shows LaTeXMLMath has a dense set of LaTeXMLMath points .
Lem .
LaTeXMLRef and Lem .
LaTeXMLRef illustrate this phenomena .
So , even though the reduced Hurwitz space is not a fine moduli space , its LaTeXMLMath points are still realized by LaTeXMLMath -covers .
This also happens when the reduced spaces are modular curves ( as in the moduli dilemma of § LaTeXMLRef ) .
It is more subtle to test when LaTeXMLMath gives a LaTeXMLMath realization .
Any LaTeXMLMath produces a cover sequence LaTeXMLMath with LaTeXMLMath geometrically Galois with group LaTeXMLMath having the following two properties .
LaTeXMLMath is an LaTeXMLMath cover over LaTeXMLMath .
LaTeXMLMath is unramified : from the unique LaTeXMLMath orbit ( § LaTeXMLRef ) on LaTeXMLMath having lifting invariant 1 ( it contains an H-M rep. or from Prop .
LaTeXMLRef ) .
If LaTeXMLMath is over LaTeXMLMath , this gives a regular LaTeXMLMath realization .
LaTeXMLCite summarizes a sufficient condition for this when the branch cycles have odd order ( as here ) .
It is that LaTeXMLMath has a LaTeXMLMath point .
See this directly by applying Lem .
LaTeXMLRef to LaTeXMLMath .
The resulting cover LaTeXMLMath has degree 2 , is therefore Galois , and LaTeXMLMath produces the LaTeXMLMath realization .
Yet , LaTeXMLMath won ’ t have a LaTeXMLMath point for all LaTeXMLMath points LaTeXMLMath .
This fails even if we replace LaTeXMLMath by LaTeXMLMath .
Those LaTeXMLMath covers with a nontrivial complex conjugation operator don ’ t achieve a LaTeXMLMath realization ; those with a trivial conjugation operator do .
See Lem .
LaTeXMLRef and LaTeXMLRef to see both happen .
This is a special case of Prop .
LaTeXMLRef : The same phenomenon occurs at all levels of the LaTeXMLMath Modular Tower for LaTeXMLMath .
That is , there are real points achieving regular realization of a nontrivial central extension of LaTeXMLMath ( H-M reps. ) , and real points that don ’ t ( near H-M reps. ; see Prop .
LaTeXMLRef ) .
An elliptic curve over LaTeXMLMath ( genus 1 with a LaTeXMLMath point ) is a degree two cover of the sphere over LaTeXMLMath , ramified at four distinct unordered points LaTeXMLMath .
The equivalence class of LaTeXMLMath modulo the action of LaTeXMLMath determines the isomorphism class of the elliptic curve .
Isotopy classes of ( orientation preserving ) diffeomorphisms of a complex torus LaTeXMLMath identifies with LaTeXMLMath , appearing from its action on LaTeXMLMath .
Isotopy classes of diffeomorphisms of LaTeXMLMath ( as elements in LaTeXMLMath : Prop .
LaTeXMLRef ) relate to diffeomorphisms of the torus attached to LaTeXMLMath .
The difference ?
The complex torus has a chosen origin ( a canonical rational point ) preserved by diffeomorphisms ; there is no chosen point in LaTeXMLMath .
This appears in mapping a space of involution realizations of dihedral groups to a modular curve ( LaTeXMLCite ; the most elementary example from § LaTeXMLRef ) .
Basics about LaTeXMLMath expressed in LaTeXMLMath and LaTeXMLMath : LaTeXMLMath and LaTeXMLMath generate LaTeXMLMath .
From ( LaTeXMLRef ) , LaTeXMLMath .
Further : LaTeXMLEquation .
This identifies LaTeXMLMath .
Characterize the dicyclic ( or quaternion ) group LaTeXMLMath of order LaTeXMLMath as having generators LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
LaTeXMLCite has a memorable matrix representation of LaTeXMLMath in LaTeXMLMath : LaTeXMLMath and LaTeXMLMath with LaTeXMLMath the rotation in 2-space through the angle LaTeXMLMath , LaTeXMLMath the LaTeXMLMath zero matrix and LaTeXMLMath .
For LaTeXMLMath : LaTeXMLMath and LaTeXMLMath ; and LaTeXMLMath is the dicyclic group of order 12 .
For each integer LaTeXMLMath , LaTeXMLMath in LaTeXMLMath has order 4 : LaTeXMLMath .
For LaTeXMLMath , every proper subgroup of LaTeXMLMath contains LaTeXMLMath , and LaTeXMLMath is the dihedral group LaTeXMLMath of order LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
From this point LaTeXMLMath .
We give a convenient presentation of LaTeXMLMath and various subgroups and quotients .
Computations in both LaTeXMLMath and in LaTeXMLMath use the letters LaTeXMLMath for the images of LaTeXMLMath in either .
Distinguishing between their images in these two groups requires care about the context .
Here is how we intend to do that .
When LaTeXMLMath , the extra relations for LaTeXMLMath ( beyond those for LaTeXMLMath ) are LaTeXMLEquation and LaTeXMLMath .
Lemma LaTeXMLRef shows adding these relations to LaTeXMLMath is equivalent to adding LaTeXMLMath to LaTeXMLMath .
This produces new equations : LaTeXMLEquation .
Then , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath satisfy LaTeXMLEquation .
The relation LaTeXMLMath is not automatic from ( LaTeXMLRef ) .
Crucial , however , is how LaTeXMLMath acts on reduced Nielsen classes ( § LaTeXMLRef ) .
This action does factor through the relation LaTeXMLMath ( § LaTeXMLRef and Prop .
LaTeXMLRef ) .
Therefore it factors through the induced quotient LaTeXMLMath of Thm .
LaTeXMLRef , from LaTeXMLMath acting on Nielsen classes ( Prop .
LaTeXMLRef ) .
The upper half plane appears as a classical ramified Galois cover of the LaTeXMLMath -line minus LaTeXMLMath .
The elements LaTeXMLMath and LaTeXMLMath in LaTeXMLMath generate the local monodromy of this cover around 0 and 1 ( § LaTeXMLRef ) .
Sometimes LaTeXMLMath acts significantly different when viewed in LaTeXMLMath .
On these occasions denote it LaTeXMLMath : representing the local monodromy action corresponding to the cusp LaTeXMLMath of the LaTeXMLMath -line ( § LaTeXMLRef ) .
Similarly , denote LaTeXMLMath as sh : the shift .
From the above , sh and LaTeXMLMath are the same in LaTeXMLMath ( § LaTeXMLRef ) .
Let LaTeXMLMath .
The following hold .
LaTeXMLMath has one nontrivial involution LaTeXMLMath generating its center .
LaTeXMLMath and LaTeXMLMath is the quaternion group LaTeXMLMath ( Lem .
LaTeXMLRef ) .
LaTeXMLMath .
Exactly two conjugacy classes of LaTeXMLMath subgroups LaTeXMLMath and LaTeXMLMath ( both containing LaTeXMLMath ) are isomorphic to LaTeXMLMath .
From ( LaTeXMLRef LaTeXMLRef ) , LaTeXMLMath is the smallest normal subgroup of LaTeXMLMath containing either LaTeXMLMath or LaTeXMLMath .
So , from ( LaTeXMLRef ) , LaTeXMLMath and LaTeXMLMath is LaTeXMLMath modulo the relation LaTeXMLMath .
Use § LaTeXMLRef generators LaTeXMLMath of LaTeXMLMath , etc .
Since LaTeXMLMath , use LaTeXMLMath for LaTeXMLMath .
As above , reserve the symbols LaTeXMLMath for the generators of the LaTeXMLMath , and LaTeXMLMath for their images in LaTeXMLMath .
For convenience , arrange the LaTeXMLMath s action in a table .
The LaTeXMLMath th column has LaTeXMLMath acting on the 4-tuple of LaTeXMLMath s : LaTeXMLEquation .
Let LaTeXMLMath map LaTeXMLMath to the inner automorphism it induces .
Initially we work with the image of LaTeXMLMath in LaTeXMLMath where LaTeXMLMath and LaTeXMLMath commute .
Denote LaTeXMLMath by LaTeXMLMath : it maps LaTeXMLMath to LaTeXMLEquation .
Further Notation : Let LaTeXMLMath be the quotient of LaTeXMLMath by the relation LaTeXMLEquation .
While LaTeXMLMath is a free group on 3 generators , LaTeXMLMath computations require LaTeXMLMath to appear symmetrically .
So , we use this quotient presentation .
Let LaTeXMLMath be the automorphisms of LaTeXMLMath .
Then LaTeXMLMath is the normal subgroup of inner automorphisms of LaTeXMLMath .
Recall : The image of LaTeXMLMath , and all its conjugates , in LaTeXMLMath is trivial .
Both LaTeXMLMath and LaTeXMLMath map LaTeXMLMath to LaTeXMLEquation .
Action of LaTeXMLMath on LaTeXMLMath induces LaTeXMLMath with the center LaTeXMLMath of LaTeXMLMath generating the kernel .
This induces LaTeXMLMath .
The kernel of LaTeXMLMath is the direct product of the free group LaTeXMLMath and LaTeXMLMath .
Above , LaTeXMLMath is identical to an element of LaTeXMLMath .
Consider the image LaTeXMLMath of LaTeXMLMath in LaTeXMLMath : LaTeXMLEquation .
In particular , combining this with ( LaTeXMLRef ) identifies LaTeXMLMath with the image of LaTeXMLMath in LaTeXMLMath .
So , the image of LaTeXMLMath in LaTeXMLMath is 1 .
The map LaTeXMLMath induces LaTeXMLMath .
Action of LaTeXMLMath preserves LaTeXMLMath .
Thus , it induces a homomorphism of LaTeXMLMath into LaTeXMLMath where LaTeXMLMath goes to the automorphism LaTeXMLMath , LaTeXMLMath ( as in ( LaTeXMLRef ) ) .
Conclude : Modulo inner automorphisms of LaTeXMLMath , LaTeXMLMath acts trivially , producing the desired homomorphism LaTeXMLEquation .
Now consider the explicit formulas .
Most of the calculation is in ( LaTeXMLRef ) : LaTeXMLMath cycles entries of LaTeXMLMath back 1 , and conjugates all entries by the first entry ’ s inverse .
So LaTeXMLMath leaves entries of LaTeXMLMath untouched except for conjugating them by LaTeXMLEquation .
Also , LaTeXMLMath cycles the entries of LaTeXMLMath forward 1 .
The new first entry is the old 4th entry conjugated by the inverse of the product of the old first three entries .
Thus , LaTeXMLMath and LaTeXMLMath act the same .
Add that LaTeXMLMath maps to 1 in LaTeXMLMath to see LaTeXMLEquation .
Let LaTeXMLMath .
Extending the calculation above gives the next list : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Add the relation LaTeXMLMath to deduce , in order , these relations in LaTeXMLMath : LaTeXMLEquation .
So , the image of LaTeXMLMath in LaTeXMLMath is 1 .
The kernel of LaTeXMLMath contains elements of LaTeXMLMath inducing inner automorphisms commuting with conjugation by LaTeXMLMath .
Since LaTeXMLMath generates conjugations commuting with LaTeXMLMath , LaTeXMLMath generates the kernel of LaTeXMLMath .
Generators of LaTeXMLMath act on LaTeXMLMath : Respectively , LaTeXMLMath induce conjugation by LaTeXMLMath .
These conjugations on LaTeXMLMath form a free group .
So LaTeXMLMath is a free group on these generators .
∎ The remainder of the proof of Thm .
LaTeXMLRef is in § LaTeXMLRef .
First we show LaTeXMLMath is a normal subgroup isomorphic to LaTeXMLMath .
Then we list combinatorics contributing to ( LaTeXMLRef ) .
From ( LaTeXMLRef d ) , LaTeXMLMath is a word in LaTeXMLMath and LaTeXMLMath .
So , ( LaTeXMLRef e ) shows LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath .
From LaTeXMLEquation .
LaTeXMLMath commutes with LaTeXMLMath .
Apply ( LaTeXMLRef a ) : LaTeXMLMath commutes with LaTeXMLMath , LaTeXMLMath .
Thus , Lemma LaTeXMLRef shows LaTeXMLMath is a central involution .
§ LaTeXMLRef shows LaTeXMLMath and illustrates its significance .
§ LaTeXMLRef reformulates property ( LaTeXMLRef LaTeXMLRef ) : LaTeXMLMath is the only involution of LaTeXMLMath .
Consider LaTeXMLMath .
Set LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
These generators simplify presenting LaTeXMLMath .
Rewrite ( LaTeXMLRef a–d ) using LaTeXMLMath , LaTeXMLMath and LaTeXMLMath to give LaTeXMLEquation .
Further , LaTeXMLMath .
Since LaTeXMLMath is a central involution , ( LaTeXMLRef b–d ) show LaTeXMLMath is LaTeXMLMath and LaTeXMLMath is the dihedral group of order 8 ( Lem .
LaTeXMLRef ) .
From relations for LaTeXMLMath , LaTeXMLEquation and LaTeXMLMath .
Thus LaTeXMLMath normalizes LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath also normalizes LaTeXMLMath .
We now show LaTeXMLMath does also .
Apply ( LaTeXMLRef a ) in the form LaTeXMLMath to get LaTeXMLEquation .
Also , since LaTeXMLMath is a central involution , conjugate by LaTeXMLMath to get LaTeXMLMath .
With LaTeXMLMath , denote the subgroup LaTeXMLMath , the cusp group in LaTeXMLMath , by LaTeXMLMath .
It is LaTeXMLMath with LaTeXMLMath switching the two factors on the copy of the Klein LaTeXMLMath .
Nontrivial calculations with LaTeXMLMath figure in computing the genus of reduced Hurwitz space components ( see § LaTeXMLRef ) .
From ( LaTeXMLRef ) , the image of LaTeXMLMath and LaTeXMLMath give elements of respective orders 3 and 2 generating LaTeXMLMath .
A well-known abstract characterization of LaTeXMLMath is as the free product of elements of order 3 and 2 .
Further , LaTeXMLMath is a free amalgamation of cyclic groups of order 6 and 4 along their common subgroups of order 2 LaTeXMLCite .
Thus , LaTeXMLMath is isomorphic to a quotient of LaTeXMLMath .
The proceeding arguments , however , have derived a presentation for LaTeXMLMath , and so for LaTeXMLMath .
Thus ( LaTeXMLRef LaTeXMLRef ) holds .
§ LaTeXMLRef and § LaTeXMLRef give ( LaTeXMLRef LaTeXMLRef ) and ( LaTeXMLRef LaTeXMLRef ) .
In any quotient of the group LaTeXMLMath where LaTeXMLMath , all involutions are conjugate to LaTeXMLMath .
Thus , the only possible involutions in LaTeXMLMath are in LaTeXMLMath , and this contains just one .
From ( LaTeXMLRef ) ( expression c ) ) , LaTeXMLMath has order 4 .
This gives ( LaTeXMLRef LaTeXMLRef ) .
It remains to discuss LaTeXMLMath being nontrivial ( as above ) and ( LaTeXMLRef LaTeXMLRef ) .
From ( LaTeXMLRef LaTeXMLRef ) , each subgroup of LaTeXMLMath containing ( a copy of ) LaTeXMLMath contains LaTeXMLMath .
Thus , there is a one-one correspondence between subgroups LaTeXMLMath of LaTeXMLMath isomorphic to LaTeXMLMath and subgroups of LaTeXMLMath isomorphic to LaTeXMLMath via LaTeXMLMath .
So it suffices to show LaTeXMLMath contains precisely two conjugacy classes of subgroups isomorphic to LaTeXMLMath .
( These join together by the relation LaTeXMLMath . )
Expression ( LaTeXMLRef ) identifies two such groups , the images of LaTeXMLMath and LaTeXMLMath .
Consider LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Note that LaTeXMLMath is a Klein 4-group .
Denote the group LaTeXMLMath by LaTeXMLMath .
It is isomorphic to LaTeXMLMath and a complement to LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be the centralizer , LaTeXMLMath , of LaTeXMLMath ( regard all as subgroups of LaTeXMLMath ) .
From ( LaTeXMLRef ) , the quotient LaTeXMLMath is isomorphic to LaTeXMLMath .
This identifies LaTeXMLMath with a well-known rank 2 free subgroup of LaTeXMLMath .
Also , LaTeXMLMath is absolutely irreducible as an LaTeXMLMath module .
Complements of LaTeXMLMath in LaTeXMLMath correspond to elements of LaTeXMLMath LaTeXMLCite .
Conclude ( LaTeXMLRef LaTeXMLRef ) if this cohomology group has order 2 .
Let LaTeXMLMath be the largest elementary abelian 2- group quotient of the rank 2 free group LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath are isomorphic as LaTeXMLMath modules .
For any group LaTeXMLMath with normal subgroup LaTeXMLMath acting on a module LaTeXMLMath , there is an exact sequence LaTeXMLCite : LaTeXMLEquation .
Apply this with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath .
Restrict to a 2-Sylow LaTeXMLMath of LaTeXMLMath .
As in § LaTeXMLRef , if LaTeXMLMath is trivial , so is LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath is the regular representation on two copies of LaTeXMLMath , so LaTeXMLMath is a free LaTeXMLMath module and the cohomology groups LaTeXMLMath and LaTeXMLMath are trivial .
Conclude : LaTeXMLMath .
The nontrivial action of LaTeXMLMath appears in many Nielsen classes .
Especially in the action of LaTeXMLMath on LaTeXMLMath , LaTeXMLMath : all orbits have length four on LaTeXMLMath ( Lem .
LaTeXMLRef ) .
We make much , however , of the trivial action of LaTeXMLMath when LaTeXMLMath .
Branch cycles for the LaTeXMLMath -line cover from the Nielsen class LaTeXMLMath and LaTeXMLMath are in LaTeXMLCite .
Conjugate by LaTeXMLMath to assume LaTeXMLMath representatives for LaTeXMLMath have LaTeXMLMath .
Further conjugation by elements of LaTeXMLMath gives the LaTeXMLMath entries of absolute Nielsen class representatives ( Table LaTeXMLRef ) .
Then , LaTeXMLMath acts trivially on everything in Table LaTeXMLRef .
Compute : LaTeXMLEquation .
Even on LaTeXMLMath , LaTeXMLMath is trivial : LaTeXMLMath maps LaTeXMLMath to LaTeXMLMath , conjugate by LaTeXMLMath to LaTeXMLMath .
From Lem .
LaTeXMLRef , LaTeXMLMath in the LaTeXMLMath quotient .
Acting on an inner Nielsen class ( § LaTeXMLRef ; conjugating by LaTeXMLMath ) , LaTeXMLMath will always be trivial .
The following self-contained computation shows it is nontrivial in LaTeXMLMath ( compare with LaTeXMLCite quoted in LaTeXMLCite ) .
Suppose LaTeXMLMath in LaTeXMLMath .
Apply this to LaTeXMLEquation .
Project LaTeXMLMath onto the first three factors ( as in ( LaTeXMLRef ) ) .
Then , LaTeXMLMath goes to LaTeXMLMath with LaTeXMLMath the generators of LaTeXMLMath .
This implies LaTeXMLMath contrary to LaTeXMLMath in ( LaTeXMLRef ) .
Consider LaTeXMLMath with LaTeXMLMath a group from § LaTeXMLRef and its corresponding curve cover LaTeXMLMath .
Denote the subgroup of LaTeXMLMath fixing LaTeXMLMath by LaTeXMLMath .
LaTeXMLCite produces an explicit monodromy action of LaTeXMLMath on LaTeXMLMath .
We ask if LaTeXMLMath appears nontrivially in a structure from a Nielsen class , as in the situation of § LaTeXMLRef .
Since LaTeXMLMath fixes each element of the inner Nielsen classes , when is it nontrivial on each representative cover of the corresponding Hurwitz space ?
What are its LaTeXMLMath eigenspaces on the first homology of curves in the family ?
Add the relations LaTeXMLMath , LaTeXMLMath , to LaTeXMLMath .
Let LaTeXMLMath be the quotient of LaTeXMLMath by the relations LaTeXMLMath , LaTeXMLMath .
Denote the quotient of LaTeXMLMath by the ( image of ) the relations LaTeXMLMath , LaTeXMLMath , by LaTeXMLMath .
Then , LaTeXMLMath is a ( finite ) nonsplit extension of LaTeXMLMath by LaTeXMLMath .
The image of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath in LaTeXMLMath identify with classical generators of LaTeXMLMath ( § LaTeXMLRef ) .
Each LaTeXMLMath has image an involution LaTeXMLMath in LaTeXMLMath .
The product-one condition , LaTeXMLMath implies these elements generate a Klein 4-group LaTeXMLMath ( Lem .
LaTeXMLRef ) .
The group LaTeXMLMath is LaTeXMLMath ( in its regular representation ) .
To check how LaTeXMLMath acts on the LaTeXMLMath , note the element of order LaTeXMLMath ( image of LaTeXMLMath ) acts nontrivially on LaTeXMLMath ( by conjugation ) .
Here is the effect ( in LaTeXMLMath ) of conjugating LaTeXMLMath by LaTeXMLMath : LaTeXMLEquation .
Since LaTeXMLMath , the action is nontrivial and LaTeXMLMath is LaTeXMLMath .
To see these relations don ’ t kill LaTeXMLMath , verify the normalizing action of LaTeXMLMath on LaTeXMLMath gives LaTeXMLMath acting trivially , LaTeXMLMath .
Use conjugation by LaTeXMLMath as in § LaTeXMLRef .
Finally , to see it is nonsplit , consider the element LaTeXMLMath whose image in LaTeXMLMath is LaTeXMLMath ( image of LaTeXMLMath ) of order 2 .
A splitting of LaTeXMLMath would give a lift of this to an element of order 2 , contrary to Lem .
LaTeXMLRef giving no such element in LaTeXMLMath .
∎ The action of LaTeXMLMath ( from Prop .
LaTeXMLRef ) on LaTeXMLMath is the same as LaTeXMLMath .
It sends LaTeXMLMath to LaTeXMLMath modulo conjugation by LaTeXMLMath .
Sometimes we want LaTeXMLMath , the shift , instead of LaTeXMLMath , on LaTeXMLMath .
Here also , use sh .
The ( middle ) twist LaTeXMLMath and sh ( on LaTeXMLMath ) entwine by LaTeXMLEquation .
Knowing sh is an involution and ( LaTeXMLRef ) holds gives a presentation of LaTeXMLMath .
Let LaTeXMLMath be a finite group with conjugacy classes LaTeXMLMath .
Consider : LaTeXMLEquation .
Consider the twist LaTeXMLMath mapping LaTeXMLMath to LaTeXMLMath with LaTeXMLMath .
For LaTeXMLMath let LaTeXMLMath be those LaTeXMLMath with LaTeXMLMath .
Then , LaTeXMLMath maps one-one from LaTeXMLMath to LaTeXMLMath .
Also , LaTeXMLMath .
The structure constant formula ( § LaTeXMLRef ) calculates LaTeXMLMath using complex representations of LaTeXMLMath .
Calculations in § LaTeXMLRef compute the length of LaTeXMLMath orbits on LaTeXMLMath .
For LaTeXMLMath in a group , denote the centralizer of LaTeXMLMath by LaTeXMLMath .
Assume LaTeXMLMath .
Let LaTeXMLMath .
The orbit of LaTeXMLMath containing LaTeXMLMath is LaTeXMLMath , LaTeXMLMath .
So , the orbit of LaTeXMLMath has length LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath and LaTeXMLMath .
Assume : LaTeXMLMath is odd ; and LaTeXMLMath has order 2 .
Then , the orbit of LaTeXMLMath has length LaTeXMLMath .
Otherwise , the orbit of LaTeXMLMath containing LaTeXMLMath has length LaTeXMLMath .
For LaTeXMLMath an integer , LaTeXMLEquation .
The minimal LaTeXMLMath with LaTeXMLMath is LaTeXMLMath .
Further , the minimal LaTeXMLMath with LaTeXMLMath divides any other integer with this property .
So LaTeXMLMath and if LaTeXMLMath is odd , LaTeXMLMath .
From the above , if the orbit of LaTeXMLMath does not have length LaTeXMLMath , it has length LaTeXMLMath .
Use the notation around ( LaTeXMLRef ) .
The expressions LaTeXMLMath and LaTeXMLMath are tautologies .
If LaTeXMLMath is odd , then LaTeXMLMath .
Assume this equals LaTeXMLMath , which is true if and only if LaTeXMLMath .
The expression LaTeXMLMath and LaTeXMLMath are equivalent .
Conclude LaTeXMLMath .
So long as the order of LaTeXMLMath is not 1 , this shows ( LaTeXMLRef ) holds .
If , however , LaTeXMLMath , then LaTeXMLMath , contrary to hypothesis .
This reversible argument shows the converse : LaTeXMLMath follows from ( LaTeXMLRef ) .
This concludes the proof .
∎ Prop .
LaTeXMLRef applies to compute the LaTeXMLMath orbit length of LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is centerless ( see Lem .
LaTeXMLRef ) .
With LaTeXMLMath , replace LaTeXMLMath in the proposition by the entries LaTeXMLMath .
To give examples where LaTeXMLMath , suppose LaTeXMLMath is a vector space and LaTeXMLMath is an endomorphism fixing no nontrivial element of LaTeXMLMath .
Entries of LaTeXMLMath generate a centerless group .
The structure constant formula for LaTeXMLMath ( for example , LaTeXMLCite ) shows the result of multiplying elements from conjugacy classes .
Let LaTeXMLMath be a function constant on conjugacy classes LaTeXMLMath and LaTeXMLMath .
Let the common value of LaTeXMLMath on LaTeXMLMath be LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath .
If LaTeXMLMath is an irreducible character of LaTeXMLMath , LaTeXMLEquation .
List the irreducible complex characters , LaTeXMLMath , of LaTeXMLMath .
Write LaTeXMLMath .
Then , LaTeXMLMath .
Consider LaTeXMLMath : 1 at LaTeXMLMath and 0 otherwise .
So , LaTeXMLMath counts solutions of LaTeXMLMath with LaTeXMLMath : LaTeXMLEquation .
We do many computations of LaTeXMLMath orbits on absolute , inner and reduced Nielsen classes .
For general LaTeXMLMath an obvious , but valuable , listing of LaTeXMLMath orbits appears in the next lemma .
Let LaTeXMLMath be a Nielsen class representative .
With LaTeXMLMath , the orbit of LaTeXMLMath on LaTeXMLMath is the collection LaTeXMLEquation .
Producing modular curves from Hurwitz spaces illustrates modding out by various groups of automorphisms of LaTeXMLMath on LaTeXMLMath ( LaTeXMLCite or LaTeXMLCite ) .
The Nielsen class in this case is for s-equivalence ( § LaTeXMLRef ) of dihedral involution realizations .
Regard LaTeXMLMath , the dihedral group of order LaTeXMLMath with LaTeXMLMath an odd prime , as LaTeXMLMath matrices .
Start with absolute Nielsen classes ( § LaTeXMLRef ) from the degree LaTeXMLMath representation of LaTeXMLMath .
Take LaTeXMLMath the conjugacy class of LaTeXMLMath ; and LaTeXMLMath the standard representation .
The upper-right element LaTeXMLMath determines elements of LaTeXMLMath .
Let LaTeXMLMath be the normalizer in LaTeXMLMath of LaTeXMLMath .
List LaTeXMLMath as LaTeXMLMath modulo LaTeXMLMath .
Count these using representatives with these properties : LaTeXMLMath , LaTeXMLMath ; and LaTeXMLMath or LaTeXMLMath and LaTeXMLMath .
Then : LaTeXMLMath .
Modding out only by LaTeXMLMath LaTeXMLMath — LaTeXMLMath instead of LaTeXMLMath LaTeXMLMath — LaTeXMLMath gives LaTeXMLMath with LaTeXMLMath the Euler LaTeXMLMath -function .
Renormalize : Use LaTeXMLMath with LaTeXMLMath in place of LaTeXMLMath ; conjugate by LaTeXMLMath to take LaTeXMLMath .
This allows further conjugation with LaTeXMLMath .
We pose some computations as an exercise .
The superscript LaTeXMLMath means to add the quotient action of LaTeXMLMath .
Show LaTeXMLMath and LaTeXMLMath : LaTeXMLMath action is trivial .
Hint : Use H-M rep. LaTeXMLMath .
Find a length 1 ( resp .
LaTeXMLMath ) LaTeXMLMath orbit on LaTeXMLMath : LaTeXMLMath ( resp .
LaTeXMLMath ) .
For LaTeXMLMath count LaTeXMLMath orbits on LaTeXMLMath with LaTeXMLMath .
Hints on ( LaTeXMLRef LaTeXMLRef ) : Let LaTeXMLMath have LaTeXMLMath and LaTeXMLMath .
Use : LaTeXMLMath corresponds to the pair LaTeXMLMath .
Compute the minimal LaTeXMLMath so the 2nd entry of LaTeXMLMath equals LaTeXMLMath , LaTeXMLMath .
Complete the set of LaTeXMLMath s : prime to LaTeXMLMath , modulo LaTeXMLMath up to multiplication by LaTeXMLMath .
List LaTeXMLMath orbits on LaTeXMLMath .
Let LaTeXMLMath be the subgroup stabilizing ( LaTeXMLRef ) .
Then , LaTeXMLMath and this gives a LaTeXMLMath orbit of length LaTeXMLMath .
Each LaTeXMLMath , LaTeXMLMath contributes LaTeXMLMath orbits of LaTeXMLMath of this length .
A traditional way to compute LaTeXMLMath cusp data is in LaTeXMLCite .
Note : In ( LaTeXMLRef LaTeXMLRef ) , an H-M rep. ( from LaTeXMLMath , see § LaTeXMLRef ) gives the longest LaTeXMLMath orbit .
Its shift gives the shortest LaTeXMLMath orbit .
This is the exact analog of Prop .
LaTeXMLRef .
Any LaTeXMLMath represents a cover in the Nielsen class LaTeXMLMath : LaTeXMLMath .
Its four branch cycles all have order two , each fixing just one integer in the representation .
This gives LaTeXMLMath from Riemann-Hurwitz ( as in LaTeXMLCite or LaTeXMLCite ) .
The geometric ( over LaTeXMLMath ) Galois closure of LaTeXMLMath is a nondiagonal component LaTeXMLMath of the LaTeXMLMath times fiber product LaTeXMLMath ( § LaTeXMLRef has the fiber product construction ) : LaTeXMLMath .
The copy of LaTeXMLMath is the ( unique ) LaTeXMLMath -Sylow of the automorphism group of LaTeXMLMath .
Let LaTeXMLMath be the elliptic curve of degree 0 divisor classes on LaTeXMLMath .
Let LaTeXMLMath be LaTeXMLMath without its cusps ( points over LaTeXMLMath ) .
Points of LaTeXMLMath are elliptic curve isogeny classes LaTeXMLMath with kernel isomorphic to LaTeXMLMath .
So , LaTeXMLMath gives a point in LaTeXMLMath : LaTeXMLMath .
Produce LaTeXMLMath : If LaTeXMLMath lies over LaTeXMLMath , then LaTeXMLMath identifies LaTeXMLMath with LaTeXMLMath .
Pick a generator LaTeXMLMath .
Let LaTeXMLMath be the origin on LaTeXMLMath .
This gives a LaTeXMLMath -division point on LaTeXMLMath : LaTeXMLMath .
For the Hurwitz interpretation of LaTeXMLMath add markings to the Nielsen classes LaTeXMLCite .
§ LaTeXMLRef produces a LaTeXMLMath rational isomorphism of LaTeXMLMath and LaTeXMLMath .
Their moduli problems , however , are different .
Points of LaTeXMLMath correspond to pairs LaTeXMLMath with LaTeXMLMath an elliptic curve and LaTeXMLMath a LaTeXMLMath division point .
Since LaTeXMLMath is a fine moduli space , a LaTeXMLMath point LaTeXMLMath automatically produces the pair LaTeXMLMath over LaTeXMLMath .
This gives LaTeXMLMath : LaTeXMLMath is the unique involution of LaTeXMLMath fixing the origin ; and LaTeXMLMath is shorthand for translating points on LaTeXMLMath by LaTeXMLMath .
By contrast , LaTeXMLMath is the moduli space of genus 1 Galois covers of LaTeXMLMath with these further properties .
The covering group has an isomorphism with LaTeXMLMath , up to conjugation .
Branch cycles for the cover are in LaTeXMLMath .
Covers LaTeXMLMath and LaTeXMLMath for LaTeXMLMath are equivalent if pullback by LaTeXMLMath respects the isomorphism LaTeXMLMath to LaTeXMLMath ( in § LaTeXMLRef ) .
In ( LaTeXMLRef LaTeXMLRef ) , LaTeXMLMath acts trivially on inner Nielsen classes .
So LaTeXMLMath is not a fine ( or even b-fine ) moduli space ( Prop .
LaTeXMLRef ) .
We explain further .
Let LaTeXMLMath .
Let LaTeXMLMath be a smooth family of curves with an attached algebraic map LaTeXMLMath .
Denote the fiber of LaTeXMLMath over LaTeXMLMath by LaTeXMLMath .
Suppose LaTeXMLMath acts as a group scheme on LaTeXMLMath preserving each fiber LaTeXMLMath .
Write this as LaTeXMLMath .
Then , form the quotient map LaTeXMLMath .
Assume for LaTeXMLMath : An isomorphism of LaTeXMLMath with LaTeXMLMath presents LaTeXMLMath in the inner Nielsen class of LaTeXMLMath with branch points in the equivalence class LaTeXMLMath .
By assumption , LaTeXMLMath induces a natural map LaTeXMLMath .
If LaTeXMLMath were a fine moduli space , the family LaTeXMLMath would pull back from a unique total family on LaTeXMLMath .
Though it isn ’ t , there is at least one total family on LaTeXMLMath ( identified with LaTeXMLMath ) .
For some total family LaTeXMLMath : LaTeXMLMath acts on the total space commuting with LaTeXMLMath ; and the fiber over LaTeXMLMath is LaTeXMLMath ( notation above ) with restriction of LaTeXMLMath producing the cover LaTeXMLMath .
Any LaTeXMLMath point on LaTeXMLMath produces a LaTeXMLMath regular realization in the inner Nielsen class of LaTeXMLMath .
It only remains to show the last statement .
Prop .
LaTeXMLRef gives this because ( LaTeXMLRef LaTeXMLRef ) shows LaTeXMLMath acts trivially on the Nielsen class .
A classical argument , however , suffices in this case by contrast to § LaTeXMLRef .
Given LaTeXMLMath , consider the LaTeXMLMath cover LaTeXMLMath .
If LaTeXMLMath is isomorphic to LaTeXMLMath over LaTeXMLMath , this gives a regular LaTeXMLMath realization in the inner Nielsen class of LaTeXMLMath .
Such an isomorphism occurs if and only if LaTeXMLMath has a LaTeXMLMath point .
The image from origin in LaTeXMLMath is such a LaTeXMLMath point .
∎ Suppose LaTeXMLMath were a fine ( resp .
b-fine ) moduli space .
Then the family LaTeXMLMath ( resp .
off of LaTeXMLMath or 1 ) would be the pullback of this family .
There would then be a section LaTeXMLMath for LaTeXMLMath from pulling back the section for the origin on the fibers of LaTeXMLMath .
That section would have the same field of definition as does LaTeXMLMath .
The structural maps , including LaTeXMLMath , have field of definition LaTeXMLMath .
We don ’ t assume the genus zero curve LaTeXMLMath is isomorphic to LaTeXMLMath over LaTeXMLMath ; this would give a LaTeXMLMath regular realization LaTeXMLMath of LaTeXMLMath .
It is exactly in this way the next proposition says the family LaTeXMLMath is special .
Prop .
LaTeXMLRef would be most satisfying if there was a diophantine consequence for LaTeXMLMath not being a fine ( or b-fine ) moduli space .
Best of all would be values of LaTeXMLMath with LaTeXMLMath not isomorphic over LaTeXMLMath to LaTeXMLMath .
For example , suppose LaTeXMLMath is a LaTeXMLMath -adic field and LaTeXMLMath is an elliptic curve over LaTeXMLMath with a LaTeXMLMath division point LaTeXMLMath .
From this form LaTeXMLMath as in the Prop .
LaTeXMLRef proof .
Let LaTeXMLMath be the LaTeXMLMath module of all roots of 1 and let LaTeXMLMath be the dual abelian variety to LaTeXMLMath .
Tate has a theory ( LaTeXMLCite , LaTeXMLCite ) identifying homogeneous spaces for LaTeXMLMath with LaTeXMLMath .
We suggest a project .
Let LaTeXMLMath be a nontrivial homogeneous space for LaTeXMLMath , and LaTeXMLMath its 2-fold symmetric product .
Map positive divisors on LaTeXMLMath of degree 2 to divisor classes as LaTeXMLMath .
Fibers of LaTeXMLMath are forms of LaTeXMLMath over LaTeXMLMath with a degree 2 map from LaTeXMLMath .
Assume further that LaTeXMLMath has LaTeXMLMath points ; it is isomorphic to LaTeXMLMath .
LaTeXMLCite shows there is a LaTeXMLMath -fiber LaTeXMLMath of LaTeXMLMath not isomorphic to LaTeXMLMath over LaTeXMLMath .
Assume LaTeXMLMath or 1 ( the special values for our normalized LaTeXMLMath ) .
Does LaTeXMLMath have a LaTeXMLMath invariant LaTeXMLMath fiber LaTeXMLMath not isomorphic to LaTeXMLMath ?
A positive solution gives a sequence LaTeXMLMath ( with LaTeXMLMath a degree LaTeXMLMath map ) that is a LaTeXMLMath involution realization over LaTeXMLMath ( LaTeXMLMath agrees with § LaTeXMLRef ) .
Despite Châtelet ’ s interest in Brauer-Severi varieties and homogeneous spaces for elliptic curves ( Rem .
LaTeXMLRef ) he seems not to have come across this problem .
§ LaTeXMLRef looks at LaTeXMLMath orbits .
§ LaTeXMLRef applies to LaTeXMLMath a preferred rubric for computing LaTeXMLMath orbits .
Let LaTeXMLMath be the twist operator of § LaTeXMLRef .
Take LaTeXMLMath .
Then , the LaTeXMLMath orbit of LaTeXMLMath has length 5 : LaTeXMLEquation .
Similarly , take LaTeXMLMath .
On this element LaTeXMLMath produces an orbit of length 3 : LaTeXMLMath .
The following lemma from LaTeXMLCite appears often in our computations .
For any finite group LaTeXMLMath , let LaTeXMLMath be the first characteristic quotient of the universal LaTeXMLMath -Frattini cover LaTeXMLMath of LaTeXMLMath ( § LaTeXMLRef ) .
Suppose LaTeXMLMath has order divisible by LaTeXMLMath .
Then , any lift of LaTeXMLMath to LaTeXMLMath has order LaTeXMLMath times the order of LaTeXMLMath .
Given LaTeXMLMath , an extension LaTeXMLMath has the LaTeXMLMath -lifting property if the conclusion of Lem .
LaTeXMLRef holds ( for LaTeXMLMath ) .
Often a much smaller cover than LaTeXMLMath will suffice for the LaTeXMLMath -lifting property .
For example , LaTeXMLMath has one conjugacy class of elements of order LaTeXMLMath .
That lifts to an element of order 4 in LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , be distinct order 2 elements in LaTeXMLMath .
Denote lifts of them to LaTeXMLMath by LaTeXMLMath , LaTeXMLMath .
Then , LaTeXMLMath is a quaternion group .
Consider LaTeXMLMath , the Spin cover of LaTeXMLMath .
Write any LaTeXMLMath of order 2 ( LaTeXMLMath ) as a product LaTeXMLMath of two commuting elements LaTeXMLMath of order 2 with the LaTeXMLMath s having lifts LaTeXMLMath of order 4 .
Then , LaTeXMLMath is isomorphic to LaTeXMLMath if and only if LaTeXMLMath has order 2 ( otherwise it is isomorphic to LaTeXMLMath ) .
Further , LaTeXMLMath has order 4 if and only if LaTeXMLMath is a product of LaTeXMLMath transpositions , with LaTeXMLMath odd .
The opening statement follows because the quaternion group of order 8 is only group of order 8 with a Klein 4-group as quotient having all its elements lifting to have order 4 .
The last statement is a variant on that using Prop .
LaTeXMLRef .
∎ Let LaTeXMLMath be LaTeXMLMath for LaTeXMLMath , the first characteristic quotient of the universal 2-Frattini cover of LaTeXMLMath ( § LaTeXMLRef ) .
Suppose LaTeXMLMath is a lift of LaTeXMLMath to elements of order 3 in LaTeXMLMath in either of these cases .
Then , LaTeXMLMath on LaTeXMLMath gives orbits of length LaTeXMLMath .
For example , let LaTeXMLMath .
Suppose the orbit has length LaTeXMLMath .
Then , some lift of LaTeXMLMath to LaTeXMLMath conjugates LaTeXMLMath to LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
Such an element , however , would have order 2 .
Since 2 divides the order LaTeXMLMath , this contradicts Lemma LaTeXMLRef : LaTeXMLMath has order 2 times the order of LaTeXMLMath .
Similarly for LaTeXMLMath .
We don ’ t know how to compute LaTeXMLMath in general as in Lemma LaTeXMLRef .
When LaTeXMLMath is a subgroup of an alternating group and LaTeXMLMath , Prop .
LaTeXMLRef contributes considerably .
Here is an immediate example .
§ LaTeXMLRef uses it on indirect examples .
Take LaTeXMLMath and LaTeXMLMath .
Suppose LaTeXMLMath is a pair of elements of order 3 lifting to LaTeXMLMath either LaTeXMLMath or LaTeXMLMath .
Then , LaTeXMLMath .
Further , assume LaTeXMLMath is in Table LaTeXMLRef .
Then , if LaTeXMLMath or 5 , the conclusion holds for LaTeXMLMath .
Improve Table LaTeXMLRef to get a complete orbit for LaTeXMLMath on LaTeXMLMath in 2-steps .
First : List LaTeXMLMath orbits for the H-M reps. LaTeXMLEquation .
Here are LaTeXMLMath , LaTeXMLMath : LaTeXMLEquation .
A LaTeXMLMath means repeat that position from the previous element .
Conjugate these by LaTeXMLMath for list LaTeXMLMath , LaTeXMLMath .
The middle product ( order ) of a 4-tuple LaTeXMLMath is LaTeXMLMath .
Second : List elements LaTeXMLMath where LaTeXMLMath is in ( LaTeXMLRef ) and LaTeXMLMath ( LaTeXMLMath ) or 3 ( LaTeXMLMath , LaTeXMLMath ) and close under LaTeXMLMath action .
Those with LaTeXMLMath are LaTeXMLMath and LaTeXMLMath .
Those with LaTeXMLMath are in LaTeXMLMath , LaTeXMLMath : LaTeXMLEquation .
Note : Each row of ( LaTeXMLRef ) contains two elements which sh maps to each of the type LaTeXMLMath orbits ( 5 is the order of both the middle twist and LaTeXMLMath orbits ; as in § LaTeXMLRef ) .
Example : Let LaTeXMLMath be LaTeXMLMath as above .
Then , LaTeXMLEquation so LaTeXMLMath is in the LaTeXMLMath orbit of LaTeXMLMath .
The 3rd elements of each ( LaTeXMLRef ) row map to each other by sh .
Conclude : Two steps produce a list of 4-tuples of 3-cycles ( modulo conjugation by LaTeXMLMath ) closed under sh and LaTeXMLMath .
For a general reduced Nielsen class , list LaTeXMLMath orbits as LaTeXMLMath .
The sh -incidence matrix LaTeXMLMath has LaTeXMLMath term LaTeXMLMath .
Since sh has order two on reduced Nielsen classes , this is a symmetric matrix .
Equivalence LaTeXMLMath matrices LaTeXMLMath and LaTeXMLMath running over permutation matrices LaTeXMLMath ( LaTeXMLMath is its transpose ) associated to elements of LaTeXMLMath .
List LaTeXMLMath orbits as LaTeXMLEquation corresponding to LaTeXMLMath orbits .
Choose LaTeXMLMath to assume LaTeXMLMath is arranged in blocks along the diagonal .
If LaTeXMLMath is the LaTeXMLMath th block of LaTeXMLMath , then LaTeXMLMath doesn ’ t break into smaller blocks .
So , LaTeXMLMath orbits form irreducible blocks in the sh -incidence matrix .
With no loss assume one LaTeXMLMath orbit and two blocks , with orbit listings as LaTeXMLMath .
As , however , there is one orbit , for some LaTeXMLMath , LaTeXMLMath for some LaTeXMLMath .
This contradicts there being two blocks .
∎ In practice it is difficult to list the LaTeXMLMath orbits .
So , we start with the H-M reps. , apply sh , then complete the LaTeXMLMath orbits and check LaTeXMLMath .
Sometimes we ’ ll then be done .
The case LaTeXMLMath illustrates this .
Denote ( as above ) the LaTeXMLMath orbits of LaTeXMLMath and LaTeXMLMath by LaTeXMLMath and LaTeXMLMath ; LaTeXMLMath orbits of LaTeXMLEquation by LaTeXMLMath and LaTeXMLMath ; and of LaTeXMLMath by LaTeXMLMath .
For general LaTeXMLMath , denote LaTeXMLMath at the Hurwitz monodromy level to be the shift LaTeXMLMath , so LaTeXMLMath is what we call the shift above .
Ideas for LaTeXMLMath generalize to indicate cusp geometry for general LaTeXMLMath .
The element LaTeXMLMath plays the role of a shift in two ways .
Consider an LaTeXMLMath -tuple LaTeXMLMath of free generators of LaTeXMLMath .
The effect of LaTeXMLMath on LaTeXMLMath is to give LaTeXMLEquation .
In specializing to a Nielsen class the effect of LaTeXMLMath is to the shift the Nielsen class representative entries by 1 .
Iterate this LaTeXMLMath times to see the effect of LaTeXMLMath is conjugation on LaTeXMLMath by the product LaTeXMLMath of these generators .
Such a conjugation commutes with the action of the braid group .
So we have an interesting interpretation for the action of conjugating by LaTeXMLMath on the generators LaTeXMLMath .
Define LaTeXMLMath to be LaTeXMLMath .
Then , conjugation by LaTeXMLMath on the left of the array LaTeXMLMath maps its entries to LaTeXMLEquation .
To see the effect of conjugation of LaTeXMLMath on LaTeXMLMath use that LaTeXMLMath is in the center of LaTeXMLMath ( or of LaTeXMLMath ) .
Then , LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath .
Notice LaTeXMLMath has exactly the same effect on LaTeXMLMath as does LaTeXMLMath .
In LaTeXMLMath use that LaTeXMLMath to see LaTeXMLMath , so LaTeXMLMath has its square equal to 1 .
When LaTeXMLMath the group LaTeXMLMath is exactly LaTeXMLMath .
An especially handy description of LaTeXMLMath in this case is LaTeXMLMath .
In general there is a LaTeXMLMath -incidence matrix .
As in the case LaTeXMLMath , it suffices to choose the image of LaTeXMLMath in LaTeXMLMath for some value of LaTeXMLMath .
It doesn ’ t make any difference which LaTeXMLMath , though for LaTeXMLMath it was convenient to take LaTeXMLMath .
Call the resulting element LaTeXMLMath .
List the LaTeXMLMath reduced orbits as LaTeXMLMath and define LaTeXMLMath to be the matrix with LaTeXMLMath term LaTeXMLMath .
For general LaTeXMLMath it won ’ t be symmetric .
Let LaTeXMLMath be a centerless transitive subgroup of LaTeXMLMath .
Assume C is LaTeXMLMath conjugacy classes of LaTeXMLMath .
This data produces several canonical moduli spaces .
Four occur often : inner and absolute ( adding a permutation representation on LaTeXMLMath ) Hurwitz spaces and their reduced versions ( quotients by a natural LaTeXMLMath action ) .
We review their definitions and relation to LaTeXMLMath .
When LaTeXMLMath , the reduced versions are curves : upper half plane quotients by finite index subgroups of LaTeXMLMath covering the LaTeXMLMath -line and ramified at traditional points .
Consider covers LaTeXMLMath , LaTeXMLMath .
There are two natural equivalences between isomorphisms LaTeXMLMath .
Call LaTeXMLMath an s-equivalence if LaTeXMLMath .
Call LaTeXMLMath a w-equivalence if there exists LaTeXMLMath with LaTeXMLMath .
In the notation above , call LaTeXMLMath and LaTeXMLMath s-equivalent ( resp .
w-equivalent ) if ( LaTeXMLRef LaTeXMLRef ) ( resp .
( LaTeXMLRef LaTeXMLRef ) ) holds for some LaTeXMLMath .
Moduli spaces formed from equivalence ( LaTeXMLRef LaTeXMLRef ) support a natural LaTeXMLMath action .
The quotient is a moduli space for equivalence ( LaTeXMLRef LaTeXMLRef ) .
Let LaTeXMLMath be one of these reduced moduli spaces .
If LaTeXMLMath , its projective completion gives a ramified cover LaTeXMLMath .
We refine LaTeXMLCite for computing a branch cycle description of LaTeXMLMath .
This clarifies that a geometric equivalence from a Klein 4-group comes exactly from the group labeled LaTeXMLMath in § LaTeXMLRef .
Consider any smooth connected family of LaTeXMLMath ( fixed ) branch point covers of LaTeXMLMath .
Denote the degree of the covers in the family by LaTeXMLMath .
It has this attached data : an associated group LaTeXMLMath ; a permutation representation LaTeXMLMath ; and a set of conjugacy classes LaTeXMLMath of LaTeXMLMath .
Denote the subgroup of LaTeXMLMath normalizing LaTeXMLMath and permuting the conjugacy classes C by LaTeXMLMath .
If a cover LaTeXMLMath in this family has definition field LaTeXMLMath , then the Galois closure of the cover has Galois ( arithmetic monodromy ) group a subgroup of LaTeXMLMath .
Below use the notation LaTeXMLMath for the elements LaTeXMLMath with LaTeXMLMath : LaTeXMLMath is the stabilizer of 1 in the representation .
Relate covers with data LaTeXMLMath to homomorphisms of fundamental groups using the associated Nielsen class : LaTeXMLEquation .
Writing LaTeXMLMath means the conjugacy classes of the LaTeXMLMath s in LaTeXMLMath are , in some possibly rearranged order , the same ( with multiplicity ) as those in C .
Suppose given LaTeXMLMath , LaTeXMLMath branch points , and a choice LaTeXMLMath of classical generators for LaTeXMLMath ( § LaTeXMLRef ) .
Then , LaTeXMLMath lists all homomorphisms from LaTeXMLMath to LaTeXMLMath giving a cover with branch points LaTeXMLMath associated to LaTeXMLMath .
A Galois cover ( over LaTeXMLMath ) with attached data LaTeXMLMath is a LaTeXMLMath realization , suppressing LaTeXMLMath .
Analogous results for any characteristic 0 field ( including non-Galois covers ) have easy formulations as in LaTeXMLCite .
Denote the subgroup of automorphisms LaTeXMLMath of LaTeXMLMath permuting elements in C by LaTeXMLMath .
An LaTeXMLMath respects the multiplicity of a conjugacy class in C .
It acts on LaTeXMLMath by LaTeXMLEquation .
Given LaTeXMLMath form LaTeXMLMath in the obvious way .
Suppose LaTeXMLMath is a permutation representation .
Let LaTeXMLMath be a subgroup of LaTeXMLMath ( § LaTeXMLRef ) .
Conjugation on LaTeXMLMath gives a homomorphism LaTeXMLMath .
This is an isomorphism into if and only if LaTeXMLMath contains no nontrivial centralizer of LaTeXMLMath .
Abusing notation , denote the quotient of LaTeXMLMath action by LaTeXMLMath .
The braid group LaTeXMLMath acts from LaTeXMLMath acting on LaTeXMLMath as in ( LaTeXMLRef ) .
If LaTeXMLMath contains LaTeXMLMath , this LaTeXMLMath action induces an LaTeXMLMath action ( § LaTeXMLRef ) .
Assume LaTeXMLMath .
Permutation representations of fundamental groups produce covers .
For this LaTeXMLMath action , denote the ( Hurwitz space ) cover of LaTeXMLMath by LaTeXMLMath .
Use notation from § LaTeXMLRef for a cover LaTeXMLMath of degree LaTeXMLMath with LaTeXMLMath and a permutation representation LaTeXMLMath attached .
The Galois closure of LaTeXMLMath over any defining field for LaTeXMLMath has a geometric formulation .
Take the fiber product LaTeXMLMath of LaTeXMLMath , LaTeXMLMath times .
Points on LaTeXMLMath consist of LaTeXMLMath -tuples LaTeXMLMath of points on LaTeXMLMath satisfying LaTeXMLMath .
This variety will be singular around LaTeXMLMath -tuples where LaTeXMLMath and LaTeXMLMath are both ramified through LaTeXMLMath .
Replace LaTeXMLMath by its normalization to make it now a non-singular cover .
Retain the notation LaTeXMLMath .
Then , LaTeXMLMath has components where at least two of the coordinates are identical .
These form the fat diagonal .
Remove components of this fat diagonal to give LaTeXMLMath .
Over the algebraic closure LaTeXMLMath has as many components as LaTeXMLMath .
List one of these components over LaTeXMLMath as LaTeXMLMath .
The stabilizer in LaTeXMLMath of LaTeXMLMath is a conjugate of LaTeXMLMath .
Normalize by choosing LaTeXMLMath so the stabilizer is actually LaTeXMLMath .
Now , choose any LaTeXMLMath component LaTeXMLMath of LaTeXMLMath containing LaTeXMLMath .
Then , LaTeXMLMath is Galois ( over LaTeXMLMath ) with group LaTeXMLMath having LaTeXMLMath as a normal subgroup .
Also , LaTeXMLMath has the same conjugacy classes C attached to the branch points LaTeXMLMath and it factors through LaTeXMLMath ( project on any coordinate of LaTeXMLMath ) .
The Galois cover LaTeXMLMath has group LaTeXMLMath where LaTeXMLMath is the coset representation of LaTeXMLMath on LaTeXMLMath .
The following is elementary LaTeXMLCite .
The centralizer of LaTeXMLMath in LaTeXMLMath induces the automorphisms of LaTeXMLMath that commute with LaTeXMLMath .
Consider any permutation representation LaTeXMLMath .
This provides LaTeXMLMath ; LaTeXMLMath is the quotient LaTeXMLMath ( with LaTeXMLMath as in § LaTeXMLRef ) .
The regular version of the Inverse Galois Problem asks if every finite LaTeXMLMath is the group of a cover of LaTeXMLMath , for some integer LaTeXMLMath , with equations for its automorphisms and the cover over LaTeXMLMath .
Call such an LaTeXMLMath -branch point realization over LaTeXMLMath .
To find LaTeXMLMath , LaTeXMLMath and such a cover needs structure .
Form the profinite completion LaTeXMLMath of LaTeXMLMath with respect to all its finite index subgroups .
Suppose LaTeXMLMath and LaTeXMLMath is a LaTeXMLMath stable set .
There is a natural short exact sequence LaTeXMLEquation .
For an analog substituting LaTeXMLMath for LaTeXMLMath , replace LaTeXMLMath by LaTeXMLMath in ( LaTeXMLRef ) .
Let LaTeXMLMath be all meromorphic algebraic functions in a neighborhood of LaTeXMLMath that extend analytically along each path in LaTeXMLMath .
Regard them as in the Laurent series LaTeXMLMath about LaTeXMLMath .
Each is meromorphic in a disk about LaTeXMLMath having the nearest point of LaTeXMLMath on its boundary .
These are the extensible functions in LaTeXMLMath .
To each such LaTeXMLMath associate a definition field LaTeXMLMath .
This has generators ( over LaTeXMLMath ) the ratios of coefficients of any ( nonzero ) polynomial LaTeXMLMath in LaTeXMLMath and LaTeXMLMath with LaTeXMLMath near LaTeXMLMath .
The field LaTeXMLMath that LaTeXMLMath generates over LaTeXMLMath is the field of meromorphic functions of a cover LaTeXMLMath over LaTeXMLMath ( restricting to an unramified cover LaTeXMLMath ) .
Suppose LaTeXMLMath is a finite ( unramified ) cover .
Then , LaTeXMLMath is s- equivalent to LaTeXMLMath for some LaTeXMLMath with the following properties .
LaTeXMLMath is in the algebraic closure of LaTeXMLMath .
LaTeXMLMath and each analytic continuation of it ( around a closed path based at LaTeXMLMath ) have nonzero differential at LaTeXMLMath .
So , these analytic continuations have LaTeXMLMath distinct values at LaTeXMLMath .
Their power series around LaTeXMLMath have coefficients in the algebraic closure of LaTeXMLMath .
With no loss , when LaTeXMLMath and LaTeXMLMath are over LaTeXMLMath , assume LaTeXMLMath consists of power series around LaTeXMLMath with coefficients in LaTeXMLMath .
There is an ordering on elements of LaTeXMLMath : LaTeXMLMath if LaTeXMLMath .
Analytic continuation over a path , as an operation on LaTeXMLMath , determines the path ’ s analytic continuation on LaTeXMLMath .
So , paths acting on these equivalence classes respect this ordering .
Each equivalence class represents a specific function field , with all functions in it expanded around LaTeXMLMath .
It is the exact data one expects from a cover and a point on the cover over LaTeXMLMath .
This action by paths in LaTeXMLMath based at LaTeXMLMath on the points above LaTeXMLMath respects the action on points of a higher covering .
For LaTeXMLMath , let LaTeXMLMath be the analytic continuation of LaTeXMLMath to the endpoint of LaTeXMLMath .
Define how LaTeXMLMath acts on LaTeXMLMath by how it acts on extensible algebraic functions : LaTeXMLMath .
In words : Apply LaTeXMLMath to the coefficients of LaTeXMLMath , continue LaTeXMLMath around LaTeXMLMath and then apply LaTeXMLMath to the coefficients of the result .
The effect of LaTeXMLMath on algebraic functions determines it .
This defines LaTeXMLMath .
Unless LaTeXMLMath is complex conjugation , no actual path gives LaTeXMLMath .
With our right action of LaTeXMLMath replacing a left action , the following is compatible with LaTeXMLCite .
Fiber compatible action on a projective system of points LaTeXMLMath determines paths in LaTeXMLMath .
Then , LaTeXMLMath also acts on LaTeXMLMath .
Define LaTeXMLMath as the projective completion of this action from finite Galois covers .
Then , LaTeXMLMath is the projective limit of LaTeXMLMath quotients from all normal subgroups of finite index .
The previous action of LaTeXMLMath on LaTeXMLMath extends to LaTeXMLMath , giving a splitting of the sequence ( LaTeXMLRef ) .
The Branch Cycle Lemma ( LaTeXMLCite or the case in Lem .
LaTeXMLRef ) gives a necessary condition for a LaTeXMLMath realization ( over LaTeXMLMath ) through a quotient of LaTeXMLMath .
Let LaTeXMLMath be classical generators of LaTeXMLMath ( § LaTeXMLRef ) .
Any homomorphism LaTeXMLMath produces a geometrically Galois cover LaTeXMLMath .
Though construction of LaTeXMLMath is canonical , it depends on the base point .
Remove this by equivalencing homomorphisms LaTeXMLMath differing by conjugation of LaTeXMLMath .
§ LaTeXMLRef explains this and related equivalences .
For LaTeXMLMath and LaTeXMLMath ( a profinite integer ) , LaTeXMLMath acts on LaTeXMLMath by putting it to the LaTeXMLMath th power .
Then , invertible elements LaTeXMLMath map LaTeXMLMath to another generator of LaTeXMLMath .
Call a set of conjugacy classes C in LaTeXMLMath a rational union if LaTeXMLMath ( both sides counted with multiplicity ) for all LaTeXMLMath .
Given conjugacy classes C , there is a natural rationalization LaTeXMLMath : The minimal rational collection of conjugacy classes containing C .
Suppose LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are respectively the conjugacy classes of the 5-cycles in LaTeXMLMath given by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath again .
Then , LaTeXMLMath is not a rational union because the conjugacy class of LaTeXMLMath appears with multiplicity 2 , while its square appears only with multiplicity 1 .
The collection LaTeXMLMath is its rationalization .
The image LaTeXMLMath of LaTeXMLMath gives conjugacy classes C in LaTeXMLMath , with LaTeXMLMath attached to LaTeXMLMath , LaTeXMLMath .
Suppose the following : LaTeXMLMath ( with its automorphisms ) has equations over LaTeXMLMath .
If ( LaTeXMLRef ) holds then LaTeXMLMath is a LaTeXMLMath set .
A necessary and sufficient condition for ( LaTeXMLRef ) is that LaTeXMLMath extends to a homomorphism LaTeXMLMath ( as in ( LaTeXMLRef ) ) .
For an analog with any LaTeXMLMath of characteristic 0 ( including LaTeXMLMath or LaTeXMLMath ) replace LaTeXMLMath with LaTeXMLMath in the definition of LaTeXMLMath .
Assume LaTeXMLMath maps to LaTeXMLMath and LaTeXMLMath satisfies LaTeXMLMath .
Then , ( LaTeXMLRef ) implies LaTeXMLEquation .
So , when LaTeXMLMath and ( LaTeXMLRef ) holds , C is a rational union ( Def .
LaTeXMLRef ) .
Prop .
LaTeXMLRef illustrates using Lem .
LaTeXMLRef , to answer question ( LaTeXMLRef LaTeXMLRef ) .
Suppose LaTeXMLMath over LaTeXMLMath has degree LaTeXMLMath with attached LaTeXMLMath .
Then , LaTeXMLMath produces a permutation representation LaTeXMLMath .
Sometimes the goal is to find LaTeXMLMath over LaTeXMLMath , with less concern for the definition field of the Galois closure of the cover ( as in LaTeXMLCite and LaTeXMLCite ) .
The necessary condition for this requires a choice of Galois closure group LaTeXMLMath between LaTeXMLMath and LaTeXMLMath as in § LaTeXMLRef .
For each such LaTeXMLMath , replace ( LaTeXMLRef ) by LaTeXMLEquation .
A general Branch Cycle Lemma , including for non-Galois covers is in LaTeXMLCite .
LaTeXMLMath , it pays to stipulate orbits of LaTeXMLMath on the branch points precisely .
For a general LaTeXMLMath , assume LaTeXMLMath acts on the support of LaTeXMLMath as LaTeXMLMath .
Regard LaTeXMLMath as a subgroup of LaTeXMLMath .
Consider : LaTeXMLEquation .
Let LaTeXMLMath be in the Nielsen class LaTeXMLMath have definition field LaTeXMLMath .
Call LaTeXMLMath attached to LaTeXMLMath if its branch points LaTeXMLMath are a LaTeXMLMath set with the given action LaTeXMLMath and its arithmetic monodromy group over LaTeXMLMath is a subgroup of LaTeXMLMath .
Then , the fixed field LaTeXMLMath of LaTeXMLMath is a subfield of LaTeXMLMath ( similar to LaTeXMLCite ) .
When LaTeXMLMath drop the appearance of LaTeXMLMath .
Example : LaTeXMLMath becomes LaTeXMLMath .
The history of the Inverse Galois Problem shows it is hard to find LaTeXMLMath and LaTeXMLMath producing LaTeXMLMath realizations .
To systematically investigate the diophantine nature of these difficulties we take large ( maximal Frattini ) quotients of LaTeXMLMath , rather than finite groups .
Start with a finite group LaTeXMLMath .
Call LaTeXMLMath a Frattini cover if for any subgroup LaTeXMLMath , LaTeXMLMath implies LaTeXMLMath .
This exactly translates the cover property from § LaTeXMLRef .
There is a ( uni ) versal Frattini group LaTeXMLMath covering LaTeXMLMath LaTeXMLCite .
We review properties of LaTeXMLMath by analogy with the universal Frattini cover of the dihedral group LaTeXMLMath ( LaTeXMLMath odd ) .
Consider LaTeXMLEquation as pieces of the universal Frattini profinite cover of LaTeXMLMath .
Patch these as a fiber product over LaTeXMLMath : LaTeXMLEquation .
This generalizes : For each prime LaTeXMLMath , LaTeXMLMath , there is a universal LaTeXMLMath -Frattini cover LaTeXMLMath with these properties LaTeXMLCite .
LaTeXMLMath is the fiber product of LaTeXMLMath ( over LaTeXMLMath ) over LaTeXMLMath primes dividing LaTeXMLMath .
Both LaTeXMLMath and a LaTeXMLMath -Sylow of LaTeXMLMath are pro-free pro- LaTeXMLMath groups , and LaTeXMLMath is the minimal profinite cover of LaTeXMLMath with this property .
LaTeXMLMath has a characteristic sequence of finite quotients LaTeXMLMath .
Each LaTeXMLMath -conjugacy class of LaTeXMLMath lifts uniquely to a LaTeXMLMath class of LaTeXMLMath .
If LaTeXMLMath has image in LaTeXMLMath all of LaTeXMLMath , then LaTeXMLMath .
To simplify notation we suppress the appearance of LaTeXMLMath in forming the characteristic sequence LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath .
For any pro- LaTeXMLMath group , the Frattini subgroup is the closed subgroup that commutators and LaTeXMLMath th powers generate .
Let LaTeXMLMath be the Frattini subgroup of LaTeXMLMath .
Continue inductively to form LaTeXMLMath as the Frattini subgroup of LaTeXMLMath .
Then , LaTeXMLMath .
Use of modular representation theory throughout this paper is from the action of LaTeXMLMath on LaTeXMLMath , a natural LaTeXMLMath module .
Rem .
LaTeXMLRef notes some points about this .
We occasionally use LaTeXMLMath with LaTeXMLMath a field for the absolute Galois group of LaTeXMLMath .
The context and use of capital letters for fields in that rare event should cause no confusion with this LaTeXMLMath notation .
Let LaTeXMLMath be a cover of profinite groups with the kernel of LaTeXMLMath a pro- LaTeXMLMath group .
Such a cover is LaTeXMLMath -projective if for any cover LaTeXMLMath with kernel a pro- LaTeXMLMath group , there is a homomorphism LaTeXMLMath with LaTeXMLMath .
With the same setup as above and LaTeXMLMath a positive integer , we say LaTeXMLMath is a LaTeXMLMath -projective cover if for any LaTeXMLMath with kernel a LaTeXMLMath -group of exponent LaTeXMLMath , there exists LaTeXMLMath with LaTeXMLMath .
The next proposition captures the most significant property in the list ( LaTeXMLRef ) .
Equivalent to ( LaTeXMLRef LaTeXMLRef ) is that LaTeXMLMath is LaTeXMLMath -projective .
The group LaTeXMLMath is the minimal LaTeXMLMath -projective cover of LaTeXMLMath .
Also , LaTeXMLMath ( with its morphism to LaTeXMLMath is the minimal LaTeXMLMath -projective cover of LaTeXMLMath .
An analogous definition without reference to covers is of LaTeXMLMath -projective .
That would be a profinite group LaTeXMLMath such that any cover LaTeXMLMath is LaTeXMLMath -projective .
Many fields relevant to arithmetic and algebraic geometry have absolute Galois groups that are LaTeXMLMath -projective ( call the field itself LaTeXMLMath -projective ) .
In particular that includes function fields over algebraically closed fields , and the PAC fields from LaTeXMLCite that play a role in the exact sequence for LaTeXMLMath in ( LaTeXMLRef ) .
Further , many function fields have geometric fundamental groups that are LaTeXMLMath -projective .
If LaTeXMLMath is an algebraic curve ( affine or projective ) over an algebraically closed field of characteristic LaTeXMLMath , then its ( profinite ) fundamental group is LaTeXMLMath -projective .
This LaTeXMLMath -projective result is due to Grothendieck , though there are many recent revisitings of this topic ( for example LaTeXMLCite ) .
We know fundamental groups of projective curves in 0 characteristic , though not in positive characteristic ; LaTeXMLMath -projectivity is a very good hint why not .
The LaTeXMLMath quotient of such fundamental groups ( affine or projective ) are like their characteristic 0 counterparts from Grothendieck ’ s enhancement of Riemann ’ s existence theorem ( see § LaTeXMLRef ) .
Non LaTeXMLMath quotients are not , though we know the group is LaTeXMLMath projective .
Clearly the group is not pro-free for you can ’ t put those two properties together in a pro-free group .
We don ’ t know how to characterize profinite groups that do put those properties together .
General automorphisms of LaTeXMLMath may not preserve LaTeXMLMath .
That is , the phrase characteristic quotients of LaTeXMLMath doesn ’ t imply LaTeXMLMath is a characteristic subgroup of LaTeXMLMath .
Still , the following easy lemma often applies .
If LaTeXMLMath is a characteristic subgroup of the LaTeXMLMath -Sylow of LaTeXMLMath , then by definition all LaTeXMLMath s are characteristic subgroups .
This holds for the universal LaTeXMLMath -Frattini cover of LaTeXMLMath .
If all LaTeXMLMath -Sylows of LaTeXMLMath intersect in LaTeXMLMath ( as when LaTeXMLMath is a simple group ) , then LaTeXMLMath is the intersection of all LaTeXMLMath -Sylows of LaTeXMLMath .
So , LaTeXMLMath is also characteristic : All automorphisms of LaTeXMLMath preserve the intersection of the LaTeXMLMath -Sylows .
Suppose LaTeXMLMath is a LaTeXMLMath -Sylow of LaTeXMLMath .
Since it is a pro-free pro- LaTeXMLMath group , its characteristic subgroups appear from the filtrations by the LaTeXMLMath th power subgroups and the lower central series .
In principle this allows determining if LaTeXMLMath is a characteristic subgroup of LaTeXMLMath .
The fiber product characterization of ( LaTeXMLRef ) allows dealing with one LaTeXMLMath -Frattini cover LaTeXMLMath of LaTeXMLMath at a time .
Fix LaTeXMLMath and LaTeXMLMath .
Here is a naive diophantine goal referencing the groups LaTeXMLMath of ( LaTeXMLRef LaTeXMLRef ) .
For each LaTeXMLMath find the following : LaTeXMLMath distinct points LaTeXMLMath ( possibly varying with LaTeXMLMath ) ; and LaTeXMLMath factoring through LaTeXMLMath .
The Branch Cycle Lemma limits conjugacy classes LaTeXMLMath to satisfy ( LaTeXMLRef ) .
Assume C consists of LaTeXMLMath -conjugacy classes .
Suppose LaTeXMLMath is simultaneously a quotient of LaTeXMLMath factoring through LaTeXMLMath and of LaTeXMLMath .
Call the corresponding cover of LaTeXMLMath a LaTeXMLMath -Frattini LaTeXMLMath cover if classical generators ( § LaTeXMLRef ) of LaTeXMLMath also map to C .
According to ( LaTeXMLRef LaTeXMLRef ) , if LaTeXMLMath passes the Branch Cycle Lemma test , then so does LaTeXMLMath for all LaTeXMLMath .
This illustrates the groups LaTeXMLMath are similar .
The guiding question asks this .
Will all the LaTeXMLMath fall to the Inverse Galois Problem with a bound , independent of LaTeXMLMath , on LaTeXMLMath in Prob .
LaTeXMLRef ?
A Yes !
to Question LaTeXMLRef is equivalent to the following ( Thm .
LaTeXMLRef ; also with any number field LaTeXMLMath replacing LaTeXMLMath ) .
There is a rational set of LaTeXMLMath classes C ( with cardinality LaTeXMLMath for some LaTeXMLMath ) so ( LaTeXMLRef ) holds with LaTeXMLMath ( and LaTeXMLMath ) for all LaTeXMLMath .
So , with no loss , LaTeXMLMath has exactly LaTeXMLMath points in its support for all LaTeXMLMath .
The result is exactly the same if we replace LaTeXMLMath by any number field LaTeXMLMath , which we now do .
As in LaTeXMLCite ( or Prob .
LaTeXMLRef ) , if LaTeXMLMath is LaTeXMLMath -perfect and centerless we conjecture the answer is No !
.
The Hurwitz spaces LaTeXMLMath for the inner LaTeXMLMath Modular Tower of covers up to inner equivalence ( § LaTeXMLRef ) generalize the modular curves LaTeXMLMath .
Assume there exists LaTeXMLMath with LaTeXMLMath for all positive integers LaTeXMLMath , and let C be the LaTeXMLMath conjugacy class in ( LaTeXMLRef ) .
Then , Prop .
LaTeXMLRef shows the LaTeXMLMath -perfect and centerless hypothesis on LaTeXMLMath guarantees ( LaTeXMLRef ) is equivalent to producing LaTeXMLMath points on each level LaTeXMLMath of the inner LaTeXMLMath Modular Tower .
Given LaTeXMLMath with LaTeXMLMath , denote the set of integers LaTeXMLMath where there is an LaTeXMLMath -tuple of LaTeXMLMath conjugacy classes in LaTeXMLMath with a LaTeXMLMath point on every level of the LaTeXMLMath Modular Tower by LaTeXMLMath .
The goal of the Main Conjecture is is to show LaTeXMLMath is empty .
The methods of the paper pretty much restrict to showing LaTeXMLMath does not contain LaTeXMLMath or 4 .
When LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we succeed in showing this , and in finding considerable about excluding LaTeXMLMath for general LaTeXMLMath .
§ LaTeXMLRef states the unsolved problems left from § LaTeXMLRef in excluding LaTeXMLMath from LaTeXMLMath .
One sees there the effect of increasing components in going to higher levels of a Modular Tower .
This justifies concentration on analyzing the component structures at level 1 in our main examples .
A rubric for including all Modular Tower levels would regard C as LaTeXMLMath conjugacy classes .
This produces the Nielsen class LaTeXMLMath .
For Modular Towers defined by absolute equivalence , rational point conclusions at high levels are likely weaker .
For absolute equivalence there is an analog set LaTeXMLMath .
As with the modular curves LaTeXMLMath , the case LaTeXMLMath , many times the answer may also be No rational points at high levels !
To get closer to a problem in LaTeXMLCite , let LaTeXMLMath be a normal subgroup of LaTeXMLMath .
An example would be LaTeXMLMath as in § LaTeXMLRef .
Let LaTeXMLMath be LaTeXMLMath .
Then , instead of the full Modular Tower ( for absolute or inner equivalence ) , consider LaTeXMLMath ( or the inner case ) .
We regard this as a quotient Modular Tower .
Similarly , consider values LaTeXMLMath in LaTeXMLMath ( or LaTeXMLMath .
The phrase Thompson tuples is from LaTeXMLCite .
Consider the groups LaTeXMLMath with LaTeXMLMath a prime .
To find regular realizations of this group , take LaTeXMLMath , and what Völklein calls Thompson triples .
These produce regular realizations with LaTeXMLMath branch points , where the attendant Hurwitz space is a nearly trivial cover of LaTeXMLMath .
This is surprising , generalizing Belyi tuples .
It works only for Chevalley groups using extremely special conjugacy classes .
The concept that gives the realizations they call weak rigidity .
LaTeXMLCite describes the data that displays Thompson tuples in LaTeXMLMath .
Let C be the corresponding conjugacy class for these , and let LaTeXMLMath be the least common multiple of orders of element in C .
They consider such Thompson tuples where LaTeXMLMath and LaTeXMLMath are fixed , yet there is an infinite set LaTeXMLMath of corresponding primes LaTeXMLMath ( LaTeXMLCite ; the C changes with LaTeXMLMath , though we suppress the extra notation ) .
The following points are no contradiction to our Main Conjecture .
Still , they reflect on it with LaTeXMLMath .
There is a value of LaTeXMLMath and a finite extension LaTeXMLMath of LaTeXMLMath with LaTeXMLMath regular realizations over LaTeXMLMath for all LaTeXMLMath .
There is also a cyclic cover LaTeXMLMath , over LaTeXMLMath , ramified at LaTeXMLMath , so pullback of the regular realizations of ( LaTeXMLRef ) over LaTeXMLMath are unramified Galois ( over LaTeXMLMath ) covers of LaTeXMLMath .
Consequence : Using that special point LaTeXMLMath , LaTeXMLCite creates a curve LaTeXMLMath over a number field whose geometric fundamental group has an infinite quotient that is a projective limit of Galois unramified covers over the number field .
One of the authors suggested to replace LaTeXMLMath , LaTeXMLMath varying , with LaTeXMLMath ( for suitable LaTeXMLMath ) with LaTeXMLMath varying .
Analogous to the above , there would be LaTeXMLMath consisting of values of LaTeXMLMath with corresponding realizations of LaTeXMLMath .
It appears an argument similar to LaTeXMLCite , would then produce C , a collection of LaTeXMLMath conjugacy classes , a number field LaTeXMLMath and a point LaTeXMLMath with the following properties .
There is a projective system of covers LaTeXMLEquation .
LaTeXMLMath has branch points LaTeXMLMath and LaTeXMLMath is unramified , LaTeXMLMath .
A point of LaTeXMLMath , LaTeXMLMath , gives LaTeXMLMath .
LaTeXMLCite offers a systematic approach through the BC Functor to discussing projective systems of Chevalley groups , and so one may consider the whole apparatus as a challenge to the Main Conjecture .
We conclude , however , by noting why ( LaTeXMLRef ) does not contradict it .
It is true that LaTeXMLMath is a LaTeXMLMath -Frattini cover , if LaTeXMLMath , LaTeXMLMath is odd , LaTeXMLMath and if LaTeXMLMath , LaTeXMLMath LaTeXMLCite .
The group LaTeXMLMath , however , is not the universal LaTeXMLMath -Frattini cover of LaTeXMLMath : The kernel to LaTeXMLMath is not a pro- free pro- LaTeXMLMath group as is the kernel of LaTeXMLMath .
Even the rank of the latter is larger than that of LaTeXMLMath .
Example : LaTeXMLMath has rank 3 , while the kernel of LaTeXMLMath has rank 6 LaTeXMLCite .
Make a choice LaTeXMLMath of classical generators for LaTeXMLMath ( § LaTeXMLRef ) .
Let LaTeXMLMath be a permutation representation .
Any surjective homomorphism LaTeXMLMath produces a degree LaTeXMLMath cover LaTeXMLMath canonically .
Points of LaTeXMLMath are homotopy classes of paths LaTeXMLMath with LaTeXMLMath modulo this relation : LaTeXMLMath and LaTeXMLMath are equivalent if LaTeXMLMath and LaTeXMLMath .
To be canonical requires the covering space have a base point .
For this take the constant path from LaTeXMLMath .
To avoid running into a base point LaTeXMLMath , deforming LaTeXMLMath requires moving LaTeXMLMath .
So the cover is independent of LaTeXMLMath only up to inner automorphism .
As in § LaTeXMLRef , consider any group LaTeXMLMath between LaTeXMLMath and LaTeXMLMath .
We now explain different Hurwitz spaces as equivalence classes of covers corresponding to points LaTeXMLMath .
The results we quote in § LaTeXMLRef and § LaTeXMLRef are refinements of LaTeXMLCite suitable for this paper .
Throughout we equivalence only LaTeXMLMath covers , with LaTeXMLMath and C fixed .
The structure on LaTeXMLMath is the moduli space of inner equivalence classes , denoted LaTeXMLMath .
Interpret LaTeXMLMath as an equivalence class of pairs : LaTeXMLEquation with LaTeXMLMath the automorphisms of a Galois cover and LaTeXMLMath a group isomorphism .
Then , LaTeXMLMath is equivalent to ( LaTeXMLRef ) if a continuous one-one map LaTeXMLMath induces the latter from the former .
Example of an equivalent pair : Let LaTeXMLMath and compose LaTeXMLMath with conjugation by LaTeXMLMath to get LaTeXMLMath .
Composing LaTeXMLMath , however , with an outer automorphism of LaTeXMLMath gives a new equivalence class of pairs .
For LaTeXMLMath , automorphisms of LaTeXMLMath identify with the centralizer of LaTeXMLMath in LaTeXMLMath ( Lem .
LaTeXMLRef ) with LaTeXMLMath the regular representation .
A map between two pairs as in ( LaTeXMLRef ) is unique if and only if LaTeXMLMath has no center .
This is equivalent to there being a unique total family LaTeXMLMath .
For LaTeXMLMath , points LaTeXMLMath over LaTeXMLMath form a Galois cover of LaTeXMLMath with group LaTeXMLMath representing LaTeXMLMath .
This makes LaTeXMLMath a fine moduli space .
Then , the ( minimal ) definition field of LaTeXMLMath and the automorphisms given by LaTeXMLMath is LaTeXMLMath , generated over LaTeXMLMath by coordinates of LaTeXMLMath LaTeXMLCite .
The total family may exist even if LaTeXMLMath has a center , though it won ’ t be unique .
Even if LaTeXMLMath has a center , for any point LaTeXMLMath , there is a cover LaTeXMLMath with definition field LaTeXMLMath .
It may not , however , be Galois over LaTeXMLMath .
More generally , suppose LaTeXMLMath is a Galois cover of nonsingular projective curves with LaTeXMLMath over LaTeXMLMath and LaTeXMLMath is unramified in LaTeXMLMath .
Assume LaTeXMLMath is s-equivalent to LaTeXMLMath for each LaTeXMLMath .
Then there exists LaTeXMLMath over LaTeXMLMath , s-equivalent to LaTeXMLMath ( over LaTeXMLMath ) with a rational point over LaTeXMLMath .
A version of this is in LaTeXMLCite .
Consider any Galois cover LaTeXMLMath ( over some algebraic closure LaTeXMLMath of LaTeXMLMath ) in the equivalence class of LaTeXMLMath .
Then , choose any LaTeXMLMath over LaTeXMLMath unramified in LaTeXMLMath .
From the moduli property , for LaTeXMLMath , there is an isomorphism LaTeXMLMath commuting with the maps to LaTeXMLMath .
Compose such a map with the unique automorphism of the Galois cover assuring LaTeXMLMath takes LaTeXMLMath to LaTeXMLMath .
Apply Weil ’ s cocycle condition to LaTeXMLMath for an equivalent pair LaTeXMLMath over LaTeXMLMath LaTeXMLCite .
We may , however , lose the automorphisms : LaTeXMLMath defines a cover in LaTeXMLMath where LaTeXMLMath is the regular representation of LaTeXMLMath .
The cover , however , is special , for it has a rational point LaTeXMLMath over LaTeXMLMath .
The proof works with a general curve replacing LaTeXMLMath .
With LaTeXMLMath replacing LaTeXMLMath and LaTeXMLMath , this construction works uniformly to give a fine moduli space of geometrically Galois covers with a point over LaTeXMLMath .
∎ For LaTeXMLMath , LaTeXMLMath action on LaTeXMLMath gives LaTeXMLMath ( as in § LaTeXMLRef ) .
Refer back to § LaTeXMLRef for the notation for Galois closure of a cover .
Given LaTeXMLMath , we chose LaTeXMLMath to be a geometric Galois closure of LaTeXMLMath .
This depended on choosing a coset of LaTeXMLMath in LaTeXMLMath .
A particular choice determined an isomorphism of LaTeXMLMath with LaTeXMLMath .
A point LaTeXMLMath corresponds to a cover LaTeXMLMath up to a choice of LaTeXMLMath determined by a coset of LaTeXMLMath in LaTeXMLMath .
Formally : LaTeXMLMath is LaTeXMLMath - equivalent to the corresponding expression for LaTeXMLMath if some continuous one-one map LaTeXMLMath induces the latter up to conjugation of LaTeXMLMath by LaTeXMLMath from the former .
This generalizes inner equivalence , the case LaTeXMLMath .
When LaTeXMLMath denote LaTeXMLMath by LaTeXMLMath , absolute equivalence classes of covers LaTeXMLMath with associated permutation representation LaTeXMLMath .
From § LaTeXMLRef this requires no choice in LaTeXMLMath .
So , two covers are LaTeXMLMath -equivalent if there is a map between them commuting with their maps to LaTeXMLMath .
Assume LaTeXMLMath has no center , so LaTeXMLMath has a unique total representing family .
Then , LaTeXMLMath acting on LaTeXMLMath and LaTeXMLMath produces LaTeXMLMath .
Rem .
LaTeXMLRef gives the cyclotomic field LaTeXMLMath ( resp .
LaTeXMLMath ) as the definition field of LaTeXMLMath ( resp .
LaTeXMLMath and LaTeXMLMath ) .
If C is a rational union of conjugacy classes , this field is LaTeXMLMath .
The following interprets the main technical result of LaTeXMLCite .
Recall previous notation for the fibers of a family LaTeXMLMath ( or LaTeXMLMath ) .
If LaTeXMLMath , then LaTeXMLMath is the set of points of LaTeXMLMath over LaTeXMLMath and LaTeXMLMath is restriction of LaTeXMLMath .
Suppose LaTeXMLMath has no nontrivial centralizer in LaTeXMLMath .
Then there are total representing families LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath over LaTeXMLMath , the covers LaTeXMLMath and LaTeXMLMath corresponding to these points have the following properties .
LaTeXMLMath ( resp .
LaTeXMLMath ) has field of definition LaTeXMLMath ( resp .
LaTeXMLMath ) .
LaTeXMLMath is an absolutely irreducible component of a LaTeXMLMath ( arithmetic ) component of the Galois closure of LaTeXMLMath ( via the § LaTeXMLRef construction ) .
LaTeXMLMath is a Galois extension with group naturally isomorphic to a subgroup of LaTeXMLMath .
Let LaTeXMLMath be the ( standard ) representation of LaTeXMLMath , LaTeXMLMath .
LaTeXMLCite lists the complete set of inner and absolute Hurwitz space components at level 0 of the LaTeXMLMath Modular Tower .
Table LaTeXMLRef displays these for inner spaces ( the result is nontrivially the same for absolute spaces ) .
All these components have definition field LaTeXMLMath .
Locations in this diagram have an attached integer pair LaTeXMLMath .
In each case the LaTeXMLMath inner component LaTeXMLMath maps to the absolute LaTeXMLMath component LaTeXMLMath by a degree 2 ( Galois ) map with group identified with LaTeXMLMath .
Similarly for the corresponding LaTeXMLMath components .
In Table LaTeXMLRef , the notation for components corresponds to lifting invariant values as in Prop .
LaTeXMLRef ( or § LaTeXMLRef , specifically in ( LaTeXMLRef LaTeXMLRef ) ) .
The genus at LaTeXMLMath of a degree LaTeXMLMath cover is LaTeXMLMath .
Start with any equivalence between covers of LaTeXMLMath .
The Hurwitz space representing these equivalences is a fine moduli space if it has a unique total family representing the equivalence classes of its points .
Prop .
LaTeXMLRef considers only Hurwitz spaces of type LaTeXMLMath or LaTeXMLMath .
For any LaTeXMLMath module LaTeXMLMath , consider a group extension LaTeXMLMath with LaTeXMLMath and the lifted conjugation action of LaTeXMLMath on LaTeXMLMath is that given .
Then , LaTeXMLMath corresponds to automorphisms of LaTeXMLMath trivial on LaTeXMLMath and LaTeXMLMath modulo automorphisms induced by conjugation by LaTeXMLMath LaTeXMLCite .
Suppose LaTeXMLMath , with LaTeXMLMath acting trivially .
Then , LaTeXMLMath is just the homomorphisms of LaTeXMLMath into LaTeXMLMath .
For LaTeXMLMath a prime , a group LaTeXMLMath is LaTeXMLMath -perfect if it has no LaTeXMLMath quotient .
That is , LaTeXMLMath is trivial .
Let LaTeXMLMath be the LaTeXMLMath part of the Schur multiplier of LaTeXMLMath ( this may be trivial ) .
That LaTeXMLMath is LaTeXMLMath -perfect interprets as LaTeXMLMath having a central extension LaTeXMLMath with this property .
LaTeXMLMath and LaTeXMLMath is universal for central extensions of LaTeXMLMath with LaTeXMLMath -group kernel .
Prop .
LaTeXMLRef illustrates the necessity of this condition .
Any finite group has a centerless cover LaTeXMLCite .
No cover , however , of LaTeXMLMath can be LaTeXMLMath -perfect , unless LaTeXMLMath is .
Here is another characterization of LaTeXMLMath -perfect .
The LaTeXMLMath elements in LaTeXMLMath generate if and only LaTeXMLMath is LaTeXMLMath -perfect .
Let LaTeXMLMath be the ( normal ) subgroup of LaTeXMLMath generated by its LaTeXMLMath elements .
If LaTeXMLMath is a proper subgroup of LaTeXMLMath , then LaTeXMLMath is a nontrivial LaTeXMLMath -group , and any LaTeXMLMath -group has a LaTeXMLMath quotient .
Conversely , given LaTeXMLMath , the kernel of LaTeXMLMath contains all the LaTeXMLMath elements of LaTeXMLMath .
∎ Note : Perfect groups are exactly those LaTeXMLMath that are LaTeXMLMath -perfect for every prime dividing LaTeXMLMath .
Let LaTeXMLMath be the characteristic quotients of LaTeXMLMath , the universal LaTeXMLMath -Frattini cover of LaTeXMLMath and LaTeXMLMath as in ( LaTeXMLRef ) .
Prop .
LaTeXMLRef uses notation from the Loewy display of LaTeXMLMath as a LaTeXMLMath module ( § LaTeXMLRef ) .
Suppose LaTeXMLMath is LaTeXMLMath -perfect , centerless , and has a nontrivial LaTeXMLMath part in its Schur multiplier .
Let LaTeXMLMath be the first characteristic Frattini cover of LaTeXMLMath as in ( LaTeXMLRef LaTeXMLRef ) .
Then the canonical map LaTeXMLMath factors through a nontrivial central extension of LaTeXMLMath .
Further , this automatically replicates at all levels .
For all LaTeXMLMath , LaTeXMLMath is LaTeXMLMath -perfect and centerless ( Prop .
LaTeXMLRef ) and LaTeXMLMath factors through a nontrivial central extension of LaTeXMLMath ( Prop .
LaTeXMLRef ) .
Subexample : The universal exponent 2-Frattini extension LaTeXMLMath of LaTeXMLMath factors through the spin cover of LaTeXMLMath .
The next proposition is a characterization for all levels of a Modular Tower having fine moduli .
Suppose a Hurwitz space is of type LaTeXMLMath with associated permutation representation LaTeXMLMath .
It is a fine moduli space if LaTeXMLMath has image with no centralizer in LaTeXMLMath ( see § LaTeXMLRef ) .
A Hurwitz space of type LaTeXMLMath is a fine moduli space if LaTeXMLMath has no center .
Assume LaTeXMLMath is a centerless LaTeXMLMath -perfect group .
Then , for each LaTeXMLMath , so is LaTeXMLMath : LaTeXMLMath does not appear at the far left of the Loewy display of LaTeXMLMath .
Let C be a set of LaTeXMLMath classes of LaTeXMLMath .
Then , the Hurwitz spaces LaTeXMLMath are all fine moduli spaces .
The first part is a subset of Thm .
LaTeXMLRef .
Assume LaTeXMLMath is centerless and LaTeXMLMath -perfect .
We inductively show LaTeXMLMath also has these properties for all LaTeXMLMath .
LaTeXMLCite shows LaTeXMLMath is centerless if the following hold .
LaTeXMLMath has no center .
LaTeXMLMath has no LaTeXMLMath subquotient of Loewy type LaTeXMLMath .
The module in ( LaTeXMLRef LaTeXMLRef ) is distinct from LaTeXMLMath .
It comes from a nontrivial representation of LaTeXMLMath of form : LaTeXMLMath .
The map LaTeXMLMath is a homomorphism of LaTeXMLMath into the LaTeXMLMath .
By hypothesis this doesn ’ t exist .
That leaves showing LaTeXMLMath has no quotient isomorphic to LaTeXMLMath , assuming LaTeXMLMath is centerless and has no such quotient .
Suppose LaTeXMLMath is surjective with kernel LaTeXMLMath .
Consider the map from LaTeXMLMath to LaTeXMLMath induced by the canonical map LaTeXMLMath .
This is a Frattini cover .
So , LaTeXMLMath is not onto LaTeXMLMath .
Then , LaTeXMLMath has image an index LaTeXMLMath normal subgroup of LaTeXMLMath .
This is contrary to our assumptions .
Finally , since LaTeXMLMath is centerless , LaTeXMLMath has a center if and only if LaTeXMLMath stabilizes some nontrivial element of LaTeXMLMath ; if and only if LaTeXMLMath appears to the far left in the Loewy display of LaTeXMLMath ) .
∎ We comment on a Hurwitz space topic that arises in LaTeXMLCite .
Let LaTeXMLMath be a subgroup of LaTeXMLMath .
Use the diagram of ( LaTeXMLRef ) for LaTeXMLMath .
So , LaTeXMLMath defines a quotient LaTeXMLMath : LaTeXMLMath is then the canonical map .
A component LaTeXMLMath of a Hurwitz space LaTeXMLMath has an LaTeXMLMath -ordering on its branch points if LaTeXMLMath factors through LaTeXMLMath .
Up to whatever equivalence defines the moduli problem for LaTeXMLMath , this means for any cover LaTeXMLMath representing the equivalence class of LaTeXMLMath , the effect of LaTeXMLMath on orderings LaTeXMLMath of the branch points LaTeXMLMath of LaTeXMLMath is conjugate to a subgroup of LaTeXMLMath .
The notion depends only on the conjugacy class of LaTeXMLMath in LaTeXMLMath .
This topic arises naturally in considering the Branch Cycle Lemma ( see § LaTeXMLRef ) .
For LaTeXMLMath , and LaTeXMLMath a component of a Hurwitz space LaTeXMLMath , let LaTeXMLMath be a component of the fiber product LaTeXMLMath .
Such an LaTeXMLMath is an LaTeXMLMath -ordering ( of the branch points ) of LaTeXMLMath .
When LaTeXMLMath this is the traditional meaning of an ordering the branch points .
There is a simple Nielsen class interpretation for LaTeXMLMath having an LaTeXMLMath - ordering .
Let LaTeXMLMath be the LaTeXMLMath orbit on LaTeXMLMath ( as in § LaTeXMLRef ) corresponding to LaTeXMLMath .
We also say , an LaTeXMLMath -ordering of LaTeXMLMath .
Recall LaTeXMLMath from § LaTeXMLRef .
For LaTeXMLMath , let LaTeXMLMath be the subgroup of LaTeXMLMath stabilizing the equivalence class of LaTeXMLMath .
Then , there is an LaTeXMLMath -ordering of LaTeXMLMath if and only if some conjugate of LaTeXMLMath is in LaTeXMLMath .
Suppose LaTeXMLMath factors through LaTeXMLMath .
Then a point LaTeXMLMath has image LaTeXMLMath .
The ( geometric ) decomposition group for LaTeXMLMath in the cover LaTeXMLMath is a subgroup of LaTeXMLMath .
It must contain the image LaTeXMLMath since LaTeXMLMath is a subgroup of the fundamental group of LaTeXMLMath .
The argument is reversible .
∎ Suppose LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then , the minimal ( up to conjugacy in LaTeXMLMath ) group for which there is an LaTeXMLMath -ordering of an LaTeXMLMath orbit on LaTeXMLMath is LaTeXMLMath .
If , however , LaTeXMLMath and LaTeXMLMath are conjugate in a group LaTeXMLMath between LaTeXMLMath and LaTeXMLMath , then the minimal LaTeXMLMath -ordering of an LaTeXMLMath orbit on LaTeXMLMath is LaTeXMLMath .
Use the notation from § LaTeXMLRef : Denote LaTeXMLMath by LaTeXMLMath and LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be one of the groups from § LaTeXMLRef .
Action of LaTeXMLMath on LaTeXMLMath produces an unramified cover LaTeXMLMath ( LaTeXMLRef ) .
Pull this cover back to LaTeXMLMath as the fiber product LaTeXMLMath .
Consider LaTeXMLMath acting diagonally on LaTeXMLMath copies of LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
This action commutes with LaTeXMLMath permuting coordinates ; put LaTeXMLMath on the left .
So , LaTeXMLMath generalizes the LaTeXMLMath -line minus LaTeXMLMath from modular curves .
Also , LaTeXMLMath generalizes the LaTeXMLMath -line minus LaTeXMLMath .
Both spaces have complex dimension LaTeXMLMath .
For any strong equivalence of covers ( from § LaTeXMLRef , including absolute or inner equivalence ) , composing LaTeXMLMath with LaTeXMLMath preserves the Nielsen class .
Further , the equivalence classes of LaTeXMLMath and LaTeXMLMath lie in the same component of the corresponding Hurwitz space .
A Nielsen class is given by LaTeXMLMath with LaTeXMLMath a faithful permutation representation .
The equivalence depends on some subgroup of LaTeXMLMath , containing LaTeXMLMath and normalizing it .
A cover in the Nielsen class is of degree LaTeXMLMath and has LaTeXMLMath the natural permutation representation associated to it .
When , it is an inner class , we also attach an isomorphism between the group of the cover and the group LaTeXMLMath .
The monodromy groups of the covers LaTeXMLMath and LaTeXMLMath are exactly the same .
If LaTeXMLMath are the branch points for LaTeXMLMath , then LaTeXMLMath are the branch points of LaTeXMLMath , with LaTeXMLMath having the conjugacy class LaTeXMLMath attached to it .
This shows LaTeXMLMath preserves Nielsen classes .
The action of LaTeXMLMath mapping on s-equivalence classes of covers in a Nielsen class is continuous .
Given LaTeXMLMath , map LaTeXMLMath to the s- equivalence class of LaTeXMLMath .
Since LaTeXMLMath is connected , the LaTeXMLMath orbit of LaTeXMLMath lies in one connected component of the Hurwitz space .
The orbit contains LaTeXMLMath ; the component is that of LaTeXMLMath .
∎ LaTeXMLCite shows LaTeXMLMath action extends to LaTeXMLMath and to LaTeXMLMath .
This produces affine schemes LaTeXMLMath and LaTeXMLMath covering ( usually ramified ) LaTeXMLMath .
These are reduced Hurwitz spaces .
Take LaTeXMLMath .
Four unordered distinct points , LaTeXMLMath , are the branch points of a unique degree two cover LaTeXMLMath .
With the right choice of inhomogeneous coordinate on LaTeXMLMath , the image of LaTeXMLMath in LaTeXMLMath is the classical elliptic curve LaTeXMLMath -invariant .
Take the elliptic curve to be degree 0 divisor classes on LaTeXMLMath : LaTeXMLMath .
Identify LaTeXMLMath with LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath is a reduced Hurwitz space cover .
Assume also that a general point LaTeXMLMath corresponds to the equivalence class of a cover LaTeXMLMath whose Galois closure maps surjectively to the elliptic curve with invariant LaTeXMLMath .
We say LaTeXMLMath is LaTeXMLMath -aware .
Many Hurwitz spaces are LaTeXMLMath -aware .
Assume each class in C is in LaTeXMLMath .
Let LaTeXMLMath lie over LaTeXMLMath .
The regular representation of LaTeXMLMath gives a map LaTeXMLMath .
The cover LaTeXMLMath naturally factors through LaTeXMLMath : Quotient LaTeXMLMath by LaTeXMLMath .
( This works for any even LaTeXMLMath ; LaTeXMLMath is then hyperelliptic . )
Suppose LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
Choose LaTeXMLMath .
Then , LaTeXMLMath satisfies the product-one condition .
It produces a Nielsen class ( for some new group ) with moduli problem directly recognizing the LaTeXMLMath -line as parameterizing elliptic curves .
In summary we have the following LaTeXMLCite .
Further remarks on LaTeXMLMath -awareness appear in § LaTeXMLRef .
The ( unramified ) cover LaTeXMLMath modulo LaTeXMLMath produces the classical ( ramified ) map LaTeXMLMath .
This extends to a ( ramified ) cover LaTeXMLMath .
Prop .
LaTeXMLRef gives the precise action of LaTeXMLMath on reduced Nielsen classes .
This produces a branch cycle description of the cover LaTeXMLMath .
Many computations of this paper depend on this .
Similar to § LaTeXMLRef , Let LaTeXMLMath be a subgroup of LaTeXMLMath ( the group of the Galois cover LaTeXMLMath ) .
A component LaTeXMLMath of a reduced Hurwitz space LaTeXMLMath has an LaTeXMLMath -ordering on its branch points if LaTeXMLMath factors through LaTeXMLMath .
Up to the reduced equivalence defining the moduli problem for LaTeXMLMath , for any cover LaTeXMLMath representing the reduced equivalence class of LaTeXMLMath , the effect of LaTeXMLMath on the orderings LaTeXMLMath , of LaTeXMLMath of LaTeXMLMath is conjugate to a subgroup of LaTeXMLMath .
§ LaTeXMLRef is an application of the next lemma .
Suppose LaTeXMLMath , and for LaTeXMLMath a component LaTeXMLMath of a Hurwitz space LaTeXMLMath has an LaTeXMLMath -ordering ( as in § LaTeXMLRef ) of its branch points .
Then , the corresponding reduced Hurwitz space component LaTeXMLMath has an LaTeXMLMath ordering of its branch points .
Particularly , if LaTeXMLMath , then LaTeXMLMath factors through the natural map LaTeXMLMath .
The argument of § LaTeXMLRef shows how to identify the group of the cover LaTeXMLMath with the action of LaTeXMLMath on the points of LaTeXMLMath modulo a Klein 4-group .
∎ As in Ex .
LaTeXMLRef , assume LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath are conjugate in a group LaTeXMLMath between LaTeXMLMath and LaTeXMLMath .
Then , the minimal groups with the reduced Nielsen classes LaTeXMLMath and LaTeXMLMath having an ordering of the branch points are the same : LaTeXMLEquation .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) be a component of a Hurwitz ( resp .
reduced Hurwitz ) space covering LaTeXMLMath ( resp .
LaTeXMLMath ) .
The geometric monodromy ( Galois closure ) groups LaTeXMLMath and LaTeXMLMath of these covers are invariants of the component ( and of the Nielsen class ) describing this cover .
We call attention to delicate points useful outside the area of this paper for investigating rational points on these spaces .
Example : Formula ( LaTeXMLRef ) for how complex conjugation acts on the branch cycle description of a reduced Hurwitz space may not determine LaTeXMLMath .
Recognition , however , of how to locate one H-M rep. gives the correct determination .
Apply complex conjugation directly to paths representing LaTeXMLMath , the braid generators of LaTeXMLMath .
For example , suppose LaTeXMLMath consists of real points LaTeXMLMath arranged around the real circle .
As in LaTeXMLCite let LaTeXMLMath be a clockwise circle on LaTeXMLMath with a marked diameter on the real axis having LaTeXMLMath and LaTeXMLMath as endpoints .
( One of these has LaTeXMLMath and LaTeXMLMath at the endpoints of the directed diameter . )
Parametrize the top of LaTeXMLMath with LaTeXMLMath on LaTeXMLMath , so LaTeXMLMath and LaTeXMLMath .
Similarly , parametrize the bottom of LaTeXMLMath with LaTeXMLMath on LaTeXMLMath so LaTeXMLMath and LaTeXMLMath .
Consider the path LaTeXMLEquation .
The range of ( LaTeXMLRef ) in LaTeXMLMath represents the braid group generator LaTeXMLMath .
The inverse of path ( LaTeXMLRef ) is LaTeXMLEquation where the notation LaTeXMLMath indicates complex conjugation applied to the coordinate .
Apply this when LaTeXMLMath to compute complex conjugation on the branch cycles for reduced Hurwitz covers of the LaTeXMLMath -line .
Use LaTeXMLEquation for paths in LaTeXMLMath , images by LaTeXMLMath reduction from the paths above .
Let LaTeXMLMath and LaTeXMLMath on the interval LaTeXMLMath .
Assume LaTeXMLMath is the cover from an absolutely irreducible component of a reduced Hurwitz with LaTeXMLMath over LaTeXMLMath .
Let LaTeXMLMath be its geometric monodromy group and take LaTeXMLMath .
Then , an involution LaTeXMLMath gives the effect of complex conjugation on the points of LaTeXMLMath lying over LaTeXMLMath .
Suppose LaTeXMLMath are the branch cycles from Prop .
LaTeXMLRef from the action on reduced Nielsen classes .
Then , LaTeXMLEquation .
The hypotheses are for the situation of a 3-branch point real cover of the sphere .
The element LaTeXMLMath will be independent of the value of LaTeXMLMath .
Since LaTeXMLMath is the image of LaTeXMLMath with LaTeXMLMath , Lem .
LaTeXMLRef says LaTeXMLMath .
The formula for computing complex conjugation is the special case of Prop .
LaTeXMLRef where LaTeXMLMath and all branch points are real .
The effect of complex conjugation on the LaTeXMLMath s take them to their inverse .
This induces the effect of taking LaTeXMLMath to its inverse LaTeXMLMath .
The effect , however , of this image element on reduced Nielsen classes is an element of order 2 .
Since the permutation effect of complex conjugation is to conjugate by an involution LaTeXMLMath , this gives the formula LaTeXMLMath .
Similarly , LaTeXMLMath .
∎ Suppose LaTeXMLMath are generators of a group LaTeXMLMath , and LaTeXMLMath is an involution .
Let LaTeXMLMath be the centralizer of LaTeXMLMath in LaTeXMLMath .
If LaTeXMLMath contains no involutions , then conjugation by LaTeXMLMath on LaTeXMLMath determines it .
Assume LaTeXMLMath is transitive .
Then , with no assumptions on LaTeXMLMath , if LaTeXMLMath fixes 1 , then its conjugation effect on LaTeXMLMath determines c. Suppose a ( n irreducible ) cover LaTeXMLMath over a field LaTeXMLMath with LaTeXMLMath has monodromy group LaTeXMLMath over LaTeXMLMath .
Then , the group LaTeXMLMath of LaTeXMLMath automorphisms of LaTeXMLMath commuting with LaTeXMLMath identifies with LaTeXMLMath .
Suppose LaTeXMLMath contains no involutions .
Let LaTeXMLMath be two involutions with the same conjugation effect on LaTeXMLMath .
Then LaTeXMLMath is an involution that centralizes LaTeXMLMath , and so it is trivial .
Now assume nothing about LaTeXMLMath , that LaTeXMLMath and LaTeXMLMath have the same effect on LaTeXMLMath , both fix 1 and G is transitive .
Then , LaTeXMLMath fixes 1 .
From transitivity , for any LaTeXMLMath , there is LaTeXMLMath with LaTeXMLMath .
Conclude : LaTeXMLMath , and LaTeXMLMath is the identify .
The statement on LaTeXMLMath comes from identifying it with LaTeXMLMath .
List the right cosets LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
Elements of LaTeXMLMath that permute these by left multiplication on these cosets are in LaTeXMLMath .
Those acting trivially are in LaTeXMLMath .
Left action commutes with the right action of LaTeXMLMath , thus producing elements in LaTeXMLMath .
LaTeXMLCite has complete details .
∎ § LaTeXMLRef uses ( LaTeXMLRef ) to detect all real points on LaTeXMLMath and LaTeXMLMath .
Then , § LaTeXMLRef does the same for the space LaTeXMLMath which is level 1 of the reduced Modular Tower for LaTeXMLMath .
Lem .
LaTeXMLRef shows the arithmetic ( resp .
geometric ) monodromy group LaTeXMLMath ( resp .
LaTeXMLMath ) of LaTeXMLMath over LaTeXMLMath ( resp .
LaTeXMLMath ) is LaTeXMLMath ( resp .
LaTeXMLMath ) in LaTeXMLMath .
Further , LaTeXMLMath is isomorphic to LaTeXMLMath .
Here it identifies with the element LaTeXMLMath generating the center of LaTeXMLMath .
Therefore , ( LaTeXMLRef ) does not determine the complex conjugation LaTeXMLMath .
We now show this complication is common , occurring at all levels of many Modular Towers .
§ LaTeXMLRef , however , gives a satisfying remedy for it .
Recall the normalizer LaTeXMLMath from § LaTeXMLRef .
Let LaTeXMLMath be a transitive subgroup and C a collection of conjugacy classes from LaTeXMLMath .
Use this permutation representation for absolute Nielsen classes .
Let LaTeXMLMath .
Suppose LaTeXMLMath ( resp .
LaTeXMLMath ) is an absolutely irreducible component of LaTeXMLMath ( resp .
LaTeXMLMath ) with LaTeXMLMath ( over a field LaTeXMLMath ) from the natural map LaTeXMLMath .
Then , LaTeXMLMath is Galois with group LaTeXMLMath a subgroup of LaTeXMLMath .
Denote the monodromy group of LaTeXMLMath ( over LaTeXMLMath ) by LaTeXMLMath .
Then the centralizer of LaTeXMLMath in its natural permutation representation of degree LaTeXMLMath contains a subgroup isomorphic to LaTeXMLMath .
Once we know that LaTeXMLMath is Galois and LaTeXMLMath is a subgroup of LaTeXMLMath , the centralizer statement follows from Princ .
LaTeXMLRef .
That identification is in LaTeXMLCite .
∎ Consider the map of Hurwitz spaces LaTeXMLMath ( LaTeXMLMath ) from Ex .
LaTeXMLRef .
As noted there , each component of LaTeXMLMath maps by a degree 2 map to a component of LaTeXMLMath .
This gives cases when complex conjugation LaTeXMLMath on the monodromy group of the cover does not determine its action on the fibers of a Hurwitz ( resp .
reduced Hurwitz ) space over LaTeXMLMath ( resp .
LaTeXMLMath ) .
Often at level 0 , it is easy to compute elements that look like complex conjugation operators ( as in Prop .
LaTeXMLRef ) by inspection .
A few tricks , however , are needed if ( LaTeXMLRef ) does not determine the effect LaTeXMLMath of complex conjugation on a LaTeXMLMath -line cover .
The simplest remedy is to identify an LaTeXMLMath -cover point .
Suppose all branch points LaTeXMLMath of a cover LaTeXMLMath are real .
We may choose whatever paths we desire on LaTeXMLMath to compute Nielsen representatives of covers .
For the next lemma , choose them to detect covers defined over LaTeXMLMath with LaTeXMLMath as real branch points using the LaTeXMLMath in Prop .
LaTeXMLRef .
Call these the LaTeXMLMath -paths .
For any set LaTeXMLMath there are such LaTeXMLMath -paths .
Let LaTeXMLMath be branch cycles in LaTeXMLMath for covers passing the test of Prop .
LaTeXMLRef .
Suppose LaTeXMLMath is the image of LaTeXMLMath corresponding to a cover LaTeXMLMath branched over LaTeXMLMath consisting of four real points on LaTeXMLMath .
Then , the complex conjugation operator LaTeXMLMath for the cover fixes the reduced Nielsen classes coming from any elements of LaTeXMLMath .
Consider LaTeXMLMath .
It has definition field LaTeXMLMath .
§ LaTeXMLRef computes LaTeXMLMath for it .
This cover has geometric ( resp .
arithmetic monodromy ) LaTeXMLMath ( resp .
LaTeXMLMath ) in LaTeXMLMath and LaTeXMLEquation generates LaTeXMLMath ( § LaTeXMLRef ) .
The criterion of Prop .
LaTeXMLRef gives two possible complex conjugation operators corresponding to LaTeXMLMath , that labeled LaTeXMLMath in ( LaTeXMLRef ) and LaTeXMLMath .
Only , however , the former fixes an H-M rep. ( represented by the integers 1 and 10 ) as is necessary from Lem .
LaTeXMLRef .
Using that the real components form a 1-dimensional manifold often is effective to handle the intervals outside LaTeXMLMath .
§ LaTeXMLRef illustrates this .
At levels beyond the first it is usually prohibitive to produce the complex conjugation operators directly from the monodromy of reduced Hurwitz space covers .
For example , LaTeXMLCite couldn ’ t do it for level 1 of our main example .
Yet , with reasonable computation ability with the group LaTeXMLMath of the level , it suffices to check what LaTeXMLMath ( for four real branch points ) does to the elements of LaTeXMLMath .
§ LaTeXMLRef illustrates by showing the genus 12 component of LaTeXMLMath has one component of real points , while the genus 9 component has no real points .
Consider Hurwitz spaces LaTeXMLMath attached to LaTeXMLMath as in Prop .
LaTeXMLRef .
This is the Modular Tower for LaTeXMLMath ( or for LaTeXMLMath LaTeXMLCite .
Reduce elements of LaTeXMLMath modulo the kernel of LaTeXMLMath .
This induces LaTeXMLMath .
The LaTeXMLMath action is compatible with these maps .
This produces the sequence for the reduced Modular Tower for LaTeXMLMath : LaTeXMLEquation .
Call a sequence of representations LaTeXMLMath compatible if LaTeXMLMath goes to a conjugate of LaTeXMLMath by the canonical map LaTeXMLMath .
A sequence of absolute Hurwitz spaces requires a compatible system of representations LaTeXMLCite .
Regular representations of each group LaTeXMLMath give one example .
Another example appears when LaTeXMLMath .
Apply Schur-Zassenhaus to the inverse image of LaTeXMLMath in LaTeXMLMath to conclude LaTeXMLMath embeds compatibly in all the LaTeXMLMath s. Take LaTeXMLMath the action of LaTeXMLMath on LaTeXMLMath cosets .
Example : With LaTeXMLMath in its standard representation with LaTeXMLMath odd , LaTeXMLMath is cyclic of order 2 ( § LaTeXMLRef ) .
Quotient by LaTeXMLMath to produce LaTeXMLEquation .
We suppress the appearance of LaTeXMLMath when the representation is obvious .
We don ’ t know example sequences of representations LaTeXMLMath compatible for the characteristic quotients of LaTeXMLMath .
For example , suppose LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath includes the whole 2-Sylow of LaTeXMLMath .
Then , Prop .
LaTeXMLRef shows LaTeXMLMath must be the pullback of LaTeXMLMath in LaTeXMLMath .
Such an example ( giving not faithful representations ) is useless for most purposes .
The faithful representations of LaTeXMLMath in Prop .
LaTeXMLRef have LaTeXMLMath lying over LaTeXMLMath , a group with 2-Sylow of order 2 .
These representations give spin separation ( Def .
LaTeXMLRef ) .
We suspect there are extending compatible LaTeXMLMath , one for each LaTeXMLMath , giving spin separation at all levels , though we haven ’ t found them yet .
Compatible sequences of permutation representations , suggest considering compatible sequences of subgroups LaTeXMLMath .
Compatibility would require that LaTeXMLMath map to LaTeXMLMath .
The next lemma notes this is not automatic .
To induce an action on the cosets of LaTeXMLMath in LaTeXMLMath requires knowing LaTeXMLMath normalizes the kernel of LaTeXMLMath .
This holds automatically if LaTeXMLMath is a characteristic subgroup of LaTeXMLMath : a common event ( see Lem .
LaTeXMLRef ) .
Let LaTeXMLMath correspond to an LaTeXMLMath orbit LaTeXMLMath in its action on LaTeXMLMath ( resp .
on LaTeXMLMath ) .
Thus , LaTeXMLMath is an absolutely irreducible component of a Hurwitz space LaTeXMLMath ( over some number field LaTeXMLMath ) , equivalence classes of LaTeXMLMath branch point covers .
Prop .
LaTeXMLRef produces a finite cover LaTeXMLEquation .
Complete this to a cover LaTeXMLMath .
Let LaTeXMLMath be the images of LaTeXMLMath ( or LaTeXMLMath ) in LaTeXMLMath as in ( LaTeXMLRef ) .
Form one further equivalence on LaTeXMLMath ( or LaTeXMLMath ) .
Recall : For LaTeXMLMath , LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath has this effect : LaTeXMLEquation .
Similarly , with LaTeXMLMath , LaTeXMLMath has this effect : LaTeXMLEquation .
LaTeXMLCite used the normal subgroup of LaTeXMLMath that LaTeXMLMath generates acting on LaTeXMLMath .
It simplifies computations to make these observations from Thm .
LaTeXMLRef .
The action of LaTeXMLMath on LaTeXMLMath factors through the Klein 4-group LaTeXMLMath .
LaTeXMLMath is the minimal normal subgroup of LaTeXMLMath containing either LaTeXMLMath or LaTeXMLMath .
Denote the LaTeXMLMath orbits on LaTeXMLMath by LaTeXMLMath : reduced classes .
Apply ( LaTeXMLRef ) : Action of LaTeXMLMath on LaTeXMLMath ( as in § LaTeXMLRef ) induces LaTeXMLMath acting on LaTeXMLMath .
For LaTeXMLMath ( resp .
LaTeXMLMath ) there is the quotient set LaTeXMLMath ( resp .
LaTeXMLMath ) .
Continue using LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath acting on LaTeXMLMath .
Then , LaTeXMLCite computes branch cycles for LaTeXMLMath .
Consider LaTeXMLMath , generators of LaTeXMLMath , with relations LaTeXMLMath ( LaTeXMLRef ) .
Further , with LaTeXMLMath , the product-one condition LaTeXMLMath holds ( LaTeXMLRef ) .
Then , LaTeXMLMath orbits on LaTeXMLMath ( resp .
LaTeXMLMath ) correspond one-one to LaTeXMLMath orbits on LaTeXMLMath ( resp .
LaTeXMLMath ) .
Let LaTeXMLMath be the orbit in the discussion above .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be respective actions of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath on the image of LaTeXMLMath in LaTeXMLMath ( resp .
LaTeXMLMath ) .
Then LaTeXMLMath is a branch cycle description of the cover LaTeXMLMath .
Suppose LaTeXMLMath gets killed in the LaTeXMLMath quotient of all moduli spaces of LaTeXMLMath branch point covers .
This happens if for every LaTeXMLMath , there exists LaTeXMLMath fixing LaTeXMLMath and inducing on LaTeXMLMath the same effect ( modulo inner automorphisms ) as does LaTeXMLMath .
Prop .
LaTeXMLRef identifies the group LaTeXMLMath as the group of such LaTeXMLMath , and LaTeXMLMath as the quotient of LaTeXMLMath by a Klein 4-group , LaTeXMLMath .
∎ Let LaTeXMLMath as in Thm .
LaTeXMLRef .
We start with results assuring some kind of fine moduli condition for reduced Hurwitz spaces .
§ LaTeXMLRef then illustrates why we can not escape considering situations where it does not hold .
We interpret the phrase fine ( resp .
b-fine ) reduced Hurwitz space for inner equivalence ( absolute equivalence is similar , Rem .
LaTeXMLRef ) .
Let LaTeXMLMath be the reduced space for inner equivalence on Nielsen classes of covers attached to LaTeXMLMath .
Consider any smooth family LaTeXMLMath of curves with an analytic map LaTeXMLMath .
Denote the fiber of LaTeXMLMath over LaTeXMLMath by LaTeXMLMath .
Assume LaTeXMLMath acts as a group scheme on LaTeXMLMath preserving each fiber LaTeXMLMath : LaTeXMLMath .
For the quotient LaTeXMLMath , assume for LaTeXMLMath : An isomorphism of LaTeXMLMath with LaTeXMLMath presents LaTeXMLMath in LaTeXMLMath with branch points in the equivalence class LaTeXMLMath .
By assumption this induces LaTeXMLMath : LaTeXMLMath is a target for such maps .
The quotient LaTeXMLMath is a geometric LaTeXMLMath bundle over LaTeXMLMath .
§ LaTeXMLRef briefly discusses the obstruction to fibers being LaTeXMLMath over their definition field .
Call LaTeXMLMath a fine moduli space ( has fine moduli ) if for every such family , there is a unique family LaTeXMLMath satisfying ( LaTeXMLRef ) inducing LaTeXMLMath by pullback from LaTeXMLMath .
Consider LaTeXMLMath and LaTeXMLMath , the restriction of LaTeXMLMath over LaTeXMLMath .
The weaker notion b-fine is that LaTeXMLMath restricted to LaTeXMLMath is the pullback by LaTeXMLMath of LaTeXMLMath restricted to LaTeXMLMath .
Each notion applies separately to any component of LaTeXMLMath .
Further , there is an obvious generalization to LaTeXMLMath , though we will not be able to be so precise about testing for it .
The action of LaTeXMLMath on Nielsen classes gives an if and only if test for a reduced Hurwitz space being a b-fine moduli space ( Prop .
LaTeXMLRef for inner equivalence , Rem .
LaTeXMLRef for absolute equivalence ) .
The LaTeXMLMath th level of the LaTeXMLMath Modular Tower passes this test for b-fine moduli , LaTeXMLMath ( Prop .
LaTeXMLRef ) ; even for fine moduli ( Ex .
LaTeXMLRef and Lem .
LaTeXMLRef ) .
It is not even b-fine for level LaTeXMLMath .
Let LaTeXMLMath be a Hurwitz space with inner Nielsen class LaTeXMLMath .
A component LaTeXMLMath of the reduced space LaTeXMLMath has fine moduli if and only if there is a unique total space LaTeXMLMath with a LaTeXMLMath action on LaTeXMLMath satisfying ( LaTeXMLRef ) when LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath has definition field LaTeXMLMath .
Assume LaTeXMLMath and LaTeXMLMath has fine moduli ( as in Prop .
LaTeXMLRef ) .
Let LaTeXMLMath be an LaTeXMLMath orbit on the Nielsen class corresponding to a component LaTeXMLMath of LaTeXMLMath with LaTeXMLMath its image in LaTeXMLMath .
Then , LaTeXMLMath orbits on LaTeXMLMath have length 4 if and only if LaTeXMLMath has b-fine moduli .
Further , assuming b-fine moduli , LaTeXMLMath has fine moduli if and only if all its points over LaTeXMLMath and LaTeXMLMath ramify ( Prop .
LaTeXMLRef : LaTeXMLMath and LaTeXMLMath have no fixed points ) .
If LaTeXMLMath , and LaTeXMLMath has fine moduli , then all components of LaTeXMLMath have b-fine moduli .
Suppose LaTeXMLMath has a LaTeXMLMath divisor with odd degree image in LaTeXMLMath .
( Examples : a branch point conjugacy class is LaTeXMLMath rational and distinct from other branch cycle conjugacy classes ; or LaTeXMLMath is a finite field . )
Then , LaTeXMLMath is LaTeXMLMath isomorphic to LaTeXMLMath .
The field of definition statement follows from LaTeXMLMath defined over LaTeXMLMath .
Then , the divisor hypothesis produces an odd degree LaTeXMLMath divisor on LaTeXMLMath .
Since LaTeXMLMath has genus 0 and an odd degree LaTeXMLMath divisor , it is LaTeXMLMath isomorphic to LaTeXMLMath .
If LaTeXMLMath is a finite field , a homogeneous space for any Brauer-Severi variety always has a rational point ( the Brauer group of a finite field being trivial LaTeXMLCite ) .
Let LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be in the LaTeXMLMath orbit of LaTeXMLMath .
Consider a given set of branch points LaTeXMLMath and classical generators LaTeXMLMath of LaTeXMLMath .
Assume the image of LaTeXMLMath in LaTeXMLMath is different from 0 or 1 .
Denote by LaTeXMLMath and LaTeXMLMath the covers from the homomorphisms sending LaTeXMLMath respectively to LaTeXMLMath and to LaTeXMLMath .
Suppose for some LaTeXMLMath there exists an isomorphism LaTeXMLMath for which LaTeXMLMath .
As in § LaTeXMLRef and § LaTeXMLRef , LaTeXMLMath lies in a Klein 4-group identified with LaTeXMLMath .
The b-fine hypotheses implies LaTeXMLMath is unique .
So , the subgroup of LaTeXMLMath fixing LaTeXMLMath extends to a faithful action of LaTeXMLMath on LaTeXMLMath over LaTeXMLMath .
Quotient action gives the total space representing w-equivalence classes of covers for points of LaTeXMLMath .
The fine moduli hypothesis say LaTeXMLMath fixed on LaTeXMLMath over LaTeXMLMath or 1 extends to LaTeXMLMath without fixed points over that value of LaTeXMLMath .
For LaTeXMLMath , the b-fine property follows from § LaTeXMLRef .
∎ Let LaTeXMLMath be the reduced space for absolute equivalence on Nielsen classes of covers attached to LaTeXMLMath .
Consider any smooth family LaTeXMLMath with an analytic map LaTeXMLMath with LaTeXMLMath a ( geometric ) LaTeXMLMath bundle .
Assume also the fiber LaTeXMLMath over LaTeXMLMath satisfies this : An isomorphism of LaTeXMLMath with LaTeXMLMath presents LaTeXMLMath in LaTeXMLMath with branch points in the equivalence class LaTeXMLMath .
The analogs of b-fine and fine moduli are clear , and the proof and conclusion of Prop .
LaTeXMLRef hold with little adjustment .
Dilemma : LaTeXMLMath is LaTeXMLMath without its cusps .
The former space , however , isn ’ t a fine ( even b-fine ) moduli space according to Prop .
LaTeXMLRef .
Yet , it is classical the latter is a fine moduli space .
Resolution : It is for elliptic curves with a LaTeXMLMath division point , though not for ( inner ) w-equivalence of dihedral group Galois covers ( § LaTeXMLRef ) .
An observation of Serre says an automorphism of an elliptic curve fixing a LaTeXMLMath division point must be the identity .
This assures LaTeXMLMath is a fine moduli space .
An appropriate generalization is to decide for LaTeXMLMath , when LaTeXMLMath and LaTeXMLMath have no fixed points on the level LaTeXMLMath Nielsen classes LaTeXMLMath for a Modular Tower , when LaTeXMLMath is large ( see § LaTeXMLRef ) .
Let LaTeXMLMath represent a cover in a Nielsen class Ni for inner or absolute equivalence .
Assume it has definition field in the algebraic closure of a field LaTeXMLMath ( assume char .
0 for simplicity ) .
Apply each LaTeXMLMath to LaTeXMLMath , denoting the result LaTeXMLMath .
Let LaTeXMLMath be the collection of LaTeXMLMath for which LaTeXMLMath is equivalent to LaTeXMLMath .
The field of moduli , LaTeXMLMath is the fixed field of LaTeXMLMath .
For reduced equivalence , we have a corresponding field LaTeXMLMath .
Some circumstances might use a cover both as an inner cover , and as an absolute cover .
Reflecting this the notation would be LaTeXMLMath .
Having a fine moduli space for the Inverse Galois Problem assures for LaTeXMLMath ( or LaTeXMLMath or LaTeXMLMath , etc . )
there is a cover representing that point over LaTeXMLMath .
In both the inner case ( when LaTeXMLMath has a center ) , or the absolute case ( when LaTeXMLMath is nontrivial ) , there has been work to make Hurwitz spaces useful .
Modular Towers include all information about Frattini central extensions , as in Ex .
LaTeXMLRef .
Lem .
LaTeXMLRef reminds of the Harbater-Coombes argument : In the inner case , even with a center , there is an absolute cover over LaTeXMLMath .
This representing cover , however , may have no automorphisms over LaTeXMLMath .
The extension of this result to any base ( in place of a point ) does not hold LaTeXMLCite .
This recognizes existence of a family over LaTeXMLMath and existence of a representing cover over LaTeXMLMath as part of the same problem , though changing the base makes a difference .
Both problems suit the language of stacks or gerbes ( as in first version of this question LaTeXMLCite ) .
LaTeXMLCite gives an exposition on the gerbe approach of LaTeXMLCite and attempts to compute the obstructions to these problems .
LaTeXMLCite uses an approach like § LaTeXMLRef relating cusps and complex conjugation operators .
Consider again our main example LaTeXMLMath , LaTeXMLMath .
There is a nontrivial central Frattini extension LaTeXMLMath .
Then , LaTeXMLMath contains , among its components , the components of LaTeXMLMath ( the moduli for inner covers with group LaTeXMLMath ) .
For each LaTeXMLMath , Prop .
LaTeXMLRef notes there are two kinds of real points on LaTeXMLMath : those for H-M reps. and those for near H-M reps. H-M rep. points in LaTeXMLMath have representing covers over LaTeXMLMath .
Near H-M rep. points in LaTeXMLMath have no representing cover over LaTeXMLMath : for these points the field of moduli is not a field of definition .
This interprets as different degeneration at cusps on LaTeXMLMath attached to H-M reps. from those for near H-M reps .
The phenomena above also happens at LaTeXMLMath , though there are no near H-M reps. there .
LaTeXMLCite starts there to explain a LaTeXMLMath -adic theory .
LaTeXMLCite provides a Hurwitz space context for the Drinfeld-Ihara-Grothendieck relations ( that apply to elements of the absolute Galois group ; call these DIG relations ) .
The approach was through tangential base points .
This is the LaTeXMLCite approach , though he does not take the exact same tangential base points .
For example , he often uses complex conjugate pairs of branch points , while they always used sets of real branch points .
Ihara ’ s use of the DIG relations has been primarily to describe the Lie algebra of the absolute Galois group acting through various pronilpotent braid groups , especially on the 3 punctured LaTeXMLMath -line .
LaTeXMLCite proposed Modular Towers , though a profinite construction , as suitably like finite representations of the fundamental group to see the DIG relations at a finite level .
There is an analogy with Ihara in that modular curves are close to considerations about the LaTeXMLMath -line , we know no direct phenomenon for modular curves suggesting the DIG relations .
Still , modular curves are just one case of Modular Towers .
In Prop .
LaTeXMLRef , the Modular Tower attached to LaTeXMLMath and four 3- cycles produces a system of Serre obstruction situations from covers of LaTeXMLMath , typical for a Modular Tower .
For LaTeXMLMath , LaTeXMLMath -adic points on these Modular Towers levels should have a similar tangential base point ( cusp geometry ) analysis to the near H-M and H-M reps. over LaTeXMLMath .
We expect the H-M rep. analysis appplied to LaTeXMLMath -adic fields to give a tower of LaTeXMLMath realizations .
The near H-M rep. analysis , done LaTeXMLMath -adically adds even more interest .
We expect these to provide examples at each level LaTeXMLMath , over LaTeXMLMath -adic fields of a function field extension with group a central Frattini extension of LaTeXMLMath whose field of moduli is not a field of definition .
Further , this geometry should reveal the DIG relations on actual covers , instead of as a Lie algebra relation .
Though there are some points that must be handled to do this , the computations of Cor .
LaTeXMLRef for level 1 of this LaTeXMLMath Modular Tower should give a precise analog of LaTeXMLCite ( for level 0 ) .
As in § LaTeXMLRef , let LaTeXMLMath be a Hurwitz space and LaTeXMLMath its reduced version .
We assume LaTeXMLMath has ( at least ) b-fine moduli ( for inner or absolute equivalence ) .
For LaTeXMLMath a field , a LaTeXMLMath point on LaTeXMLMath produces a LaTeXMLMath point on LaTeXMLMath .
§ LaTeXMLRef interprets the subtlety of the converse .
Then , § LaTeXMLRef applies this to points on a Modular Tower .
§ LaTeXMLRef and § LaTeXMLRef formulate Serre ’ s Open Image Theorem for LaTeXMLMath points on a Modular Tower .
Use the setup for inner Hurwitz spaces in ( Prop .
LaTeXMLRef ) .
Suppose LaTeXMLMath is a fine ( resp .
b-fine ) moduli space , and LaTeXMLMath ( resp .
also , doesn ’ t lie over LaTeXMLMath or 1 ) .
To simplify notation , refer to the total family over LaTeXMLMath ( resp .
LaTeXMLMath ) as LaTeXMLMath .
To simplify further , also assume fine moduli , for the adjustments to b-fine are obvious .
The ( geometric ) LaTeXMLMath bundle LaTeXMLMath is algebraic from Serre ’ s GAGA .
We may even cover LaTeXMLMath with LaTeXMLMath affine sets LaTeXMLMath so the restriction of LaTeXMLMath over LaTeXMLMath is a conic bundle in LaTeXMLMath .
That is , for each LaTeXMLMath , the fiber over LaTeXMLMath is a conic in LaTeXMLMath .
We drop the subscript referring to LaTeXMLMath .
This gives the reduced class of a cover LaTeXMLMath over LaTeXMLMath , with group LaTeXMLMath , in the Nielsen class .
To simplify , Then , if LaTeXMLMath has a LaTeXMLMath point , LaTeXMLMath is a LaTeXMLMath -cover ( of LaTeXMLMath ) in the Nielsen class .
Compatible with definitions of § LaTeXMLRef , call each LaTeXMLMath a LaTeXMLMath -cover point ( the structure of the Nielsen class is over LaTeXMLMath ) .
Otherwise , LaTeXMLMath is isomorphic to a conic over LaTeXMLMath .
When LaTeXMLMath has no LaTeXMLMath point , call LaTeXMLMath a LaTeXMLMath -Brauer point ( of LaTeXMLMath ) .
The conic attached to a point LaTeXMLMath defines an element of the group of 2-torsion elements in the Brauer group LaTeXMLMath of the field LaTeXMLMath .
Any LaTeXMLMath component LaTeXMLMath of the Hurwitz space defines an element of LaTeXMLMath by the same argument for a generic point of LaTeXMLMath .
When the closure of LaTeXMLMath is isomorphic to LaTeXMLMath , regard the conic bundle as an element LaTeXMLMath .
This case arises often in the Inverse Galois Problem .
Given a rational function LaTeXMLMath in a new variable LaTeXMLMath , consider LaTeXMLMath as a subfield of LaTeXMLMath by setting LaTeXMLMath .
This induces LaTeXMLMath .
LaTeXMLCite discusses ( what is in our notation ) if for some choice of LaTeXMLMath the image of LaTeXMLMath in LaTeXMLMath vanishes .
There is a natural notion of poles of LaTeXMLMath , and LaTeXMLCite shows that if there are at most four poles , then such a LaTeXMLMath exists .
As in Prop .
LaTeXMLRef , when LaTeXMLMath acts trivially on Nielsen classes , we can take LaTeXMLMath of degree 1 , regarding it as a section for the LaTeXMLMath -invariant .
So , in the next lemma we emphasize the crucial case when LaTeXMLMath acts faithfully on the Nielsen classes for the component LaTeXMLMath .
Let LaTeXMLMath be a fine moduli space for the Nielsen class .
Assume LaTeXMLMath , a component of LaTeXMLMath , is a fine moduli space and LaTeXMLMath as above for LaTeXMLMath .
This produces a cover LaTeXMLMath over LaTeXMLMath and a unique cocycle class in LaTeXMLMath .
These fit in a cocycle of Nielsen class covers .
The cocycle class is trivial if and only if LaTeXMLMath is isomorphic to LaTeXMLMath over LaTeXMLMath .
In turn this holds if and only if LaTeXMLMath has an odd degree LaTeXMLMath divisor .
Conversely , given a cocycle of LaTeXMLMath Nielsen class covers attached to LaTeXMLMath , there is LaTeXMLMath cover LaTeXMLMath , which over LaTeXMLMath is in the Nielsen class .
Suppose LaTeXMLMath has a LaTeXMLMath point .
Let LaTeXMLMath be the image in LaTeXMLMath of LaTeXMLMath .
Then , there is a one-one association between LaTeXMLMath points in the fiber LaTeXMLMath of LaTeXMLMath and LaTeXMLMath points LaTeXMLMath over LaTeXMLMath .
The linear system LaTeXMLMath attached to LaTeXMLMath gives an isomorphism of the genus 0 curve LaTeXMLMath with LaTeXMLMath .
Take LaTeXMLMath to be the composition of LaTeXMLMath and this isomorphism .
Apply each LaTeXMLMath to LaTeXMLMath to get LaTeXMLMath .
The isomorphism here is that given by replacing LaTeXMLMath by LaTeXMLMath .
Since LaTeXMLMath is a fine moduli space , there is a unique LaTeXMLMath and LaTeXMLMath satisfying LaTeXMLMath .
The cocycle condition follows from the uniqueness conditions .
Call this data a cocycle of Nielsen class covers attached to LaTeXMLMath .
It is standard the cocycle is trivial if and only if LaTeXMLMath has a LaTeXMLMath point .
Since LaTeXMLMath has genus 0 , this is equivalent to LaTeXMLMath having a degree one LaTeXMLMath divisor .
Since the canonical class on LaTeXMLMath is a degree -2 class over LaTeXMLMath , this is equivalent to LaTeXMLMath having an odd degree LaTeXMLMath divisor .
Now suppose we have such a cocycle of Nielsen class covers .
This produces LaTeXMLMath as LaTeXMLMath .
We have only to check LaTeXMLMath is well-defined , independent of LaTeXMLMath .
The cocycle condition guarantees this .
Finally , if there is a LaTeXMLMath point on LaTeXMLMath , this gives LaTeXMLMath lying over LaTeXMLMath .
Composing the cover LaTeXMLMath with elements of LaTeXMLMath gives the correspondence between the LaTeXMLMath points of LaTeXMLMath and the LaTeXMLMath points of LaTeXMLMath over LaTeXMLMath .
∎ § LaTeXMLRef considers the Nielsen class is LaTeXMLMath .
Then , a point LaTeXMLMath has a LaTeXMLMath point of LaTeXMLMath over it .
This is because an elliptic curve isogeny LaTeXMLMath over LaTeXMLMath represents LaTeXMLMath .
The quotient of LaTeXMLMath by LaTeXMLMath has a rational point from the image of the rational point on LaTeXMLMath .
This gives LaTeXMLMath representing LaTeXMLMath lying over LaTeXMLMath .
Suppose LaTeXMLMath is not a fine moduli space .
We can still ask which LaTeXMLMath are LaTeXMLMath -cover points or LaTeXMLMath -Brauer points as in Rem .
LaTeXMLRef and Rem .
LaTeXMLRef .
Let LaTeXMLMath be a projective system of points on a reduced Modular Tower .
Call this a point on a reduced Modular Tower .
Suppose LaTeXMLMath is the absolutely irreducible component containing LaTeXMLMath .
Then LaTeXMLMath is a projective sequence of algebraic varieties .
This works with LaTeXMLMath a projective system of points on Hurwitz spaces LaTeXMLMath rather than reduced Hurwitz spaces .
Components of LaTeXMLMath are manifolds and moduli spaces .
Conclude : LaTeXMLMath being a LaTeXMLMath point implies LaTeXMLMath contains a definition field for LaTeXMLMath .
Assume LaTeXMLMath lies over the level 0 point LaTeXMLMath of LaTeXMLMath .
Then , this produces a point LaTeXMLMath on the Modular Tower with its level 0 point equal LaTeXMLMath .
For LaTeXMLMath any algebraic variety , and LaTeXMLMath , denote the pro- LaTeXMLMath completion of LaTeXMLMath by LaTeXMLMath .
It is the closure of LaTeXMLMath in the diagonal of the product of all finite LaTeXMLMath -group quotients of LaTeXMLMath .
When considering homomorphisms involving it , defined up to conjugation by an element of this group , with no loss drop the LaTeXMLMath decoration .
We concentrate now on inner Hurwitz spaces .
For LaTeXMLMath , not in LaTeXMLMath , consider classical generators LaTeXMLMath for LaTeXMLMath ( § LaTeXMLRef ) .
Let LaTeXMLMath be a projective system of points over LaTeXMLMath with LaTeXMLMath a representing cover .
Any projective system LaTeXMLMath of points over LaTeXMLMath gives a compatible system of homomorphisms LaTeXMLMath factoring through LaTeXMLMath ( LaTeXMLRef ) .
This produces LaTeXMLMath with LaTeXMLMath mapping into C ( § LaTeXMLRef ) , not depending on LaTeXMLMath .
Restriction of LaTeXMLMath to the kernel of LaTeXMLMath factors through LaTeXMLMath ( Thm .
LaTeXMLRef , as in § LaTeXMLRef ) .
Factor LaTeXMLMath by the kernel of this map , and denote the result by LaTeXMLMath .
The group LaTeXMLMath fits in a natural exact sequence LaTeXMLEquation .
Keep the notation LaTeXMLMath for the image of LaTeXMLMath in LaTeXMLMath .
Points on the Modular Tower for LaTeXMLMath over LaTeXMLMath correspond one-one with elements of LaTeXMLMath mapping LaTeXMLMath into C .
In turn these correspond with elements of LaTeXMLMath .
Then , the action of LaTeXMLMath on LaTeXMLMath induces an action of LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath , a point on the projective sequence of components LaTeXMLMath containing points over LaTeXMLMath , as above .
Suppose this corresponds to LaTeXMLMath .
Then , the collection of points LaTeXMLMath above LaTeXMLMath correspond to an LaTeXMLMath orbit of LaTeXMLMath .
Let LaTeXMLMath be the genus of LaTeXMLMath .
For LaTeXMLMath , too , there is a notion of classical generators .
These are topological generators , LaTeXMLMath and LaTeXMLMath , satisfying these properties .
The only relation in LaTeXMLMath is the commutator product LaTeXMLMath .
In LaTeXMLMath , the cup product pairing maps LaTeXMLMath to 0 , LaTeXMLMath to 0 , and LaTeXMLMath to LaTeXMLMath ( Kronecker LaTeXMLMath function ) for all LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath .
As in Prop .
LaTeXMLRef , consider the collection LaTeXMLMath of homomorphisms LaTeXMLMath ( up to inner action ) having this property : LaTeXMLMath induces the identity map LaTeXMLMath .
Since LaTeXMLMath is a Frattini cover , such a LaTeXMLMath is necessarily surjective .
Denote the surjective homomorphisms in LaTeXMLMath by LaTeXMLMath .
Thus , LaTeXMLMath maps into this space .
Also , let LaTeXMLMath denote the ( dual of the ) Tate module of LaTeXMLMath .
Let LaTeXMLMath be a subfield of LaTeXMLMath .
Assume LaTeXMLMath and LaTeXMLMath has fine moduli .
Then , LaTeXMLMath acts naturally on LaTeXMLMath .
This permutes elements of LaTeXMLMath , compatible ( according to § LaTeXMLRef ) with acting on coordinates of points LaTeXMLMath .
It induces a LaTeXMLMath action on LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath ( § LaTeXMLRef ) induces an action on any closed subgroup of LaTeXMLMath .
For LaTeXMLMath denote its fixed field by LaTeXMLMath .
Let LaTeXMLMath , so it defines a homomorphism LaTeXMLMath .
Then , LaTeXMLMath is a projective system of points over LaTeXMLMath of LaTeXMLMath if and only if LaTeXMLMath normalizes the kernel of LaTeXMLMath , and the induced action of LaTeXMLMath on the quotient is trivial .
Since levels of the Modular Tower are fine moduli spaces , this produces the desired sequence of Galois covers over LaTeXMLMath .
The argument reverses .
Further , LaTeXMLMath acts on LaTeXMLMath , a characteristic subgroup of LaTeXMLMath ( on which LaTeXMLMath acts by hypothesis ) .
The image of LaTeXMLMath in LaTeXMLMath is into LaTeXMLMath .
This induces a LaTeXMLMath action on LaTeXMLMath .
∎ Thm .
LaTeXMLRef says LaTeXMLMath has no fixed points on LaTeXMLMath .
More generally , there are no LaTeXMLMath points LaTeXMLMath if LaTeXMLMath induces the Frobenius on LaTeXMLMath for some prime LaTeXMLMath not dividing LaTeXMLMath .
This topic continues in § LaTeXMLRef .
A Modular Tower has levels corresponding to a sequence of groups LaTeXMLEquation .
If LaTeXMLMath is centerless and LaTeXMLMath -perfect , each LaTeXMLMath is a centerless ( Prop .
LaTeXMLRef ) Frattini extension of LaTeXMLMath with LaTeXMLMath -group as kernel .
Frattini extensions of perfect groups are perfect : Commutators of the covering group generate the image , so they generate the covering group .
§ LaTeXMLRef recounts the geometry behind a Frattini cover .
§ LaTeXMLRef extends the discussion of LaTeXMLCite on the universal LaTeXMLMath -Frattini cover of LaTeXMLMath , starting with the case when its LaTeXMLMath -Sylow is normal .
§ LaTeXMLRef describes the groups LaTeXMLMath when LaTeXMLMath .
LaTeXMLCite describes some aspects of the universal Frattini cover LaTeXMLMath of LaTeXMLMath , especially the ranks of the Universal LaTeXMLMath -Frattini kernels .
Then , LaTeXMLMath has three pieces LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , one for each prime LaTeXMLMath dividing LaTeXMLMath ( § LaTeXMLRef ) .
We use LaTeXMLCite to enhance and simplify properties of the characteristic modules LaTeXMLMath ( LaTeXMLMath in § LaTeXMLRef ) of LaTeXMLMath : LaTeXMLMath when LaTeXMLMath and LaTeXMLMath .
§ LaTeXMLRef shows only two Modular Towers for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath have LaTeXMLMath points at level 1 .
Then , § LaTeXMLRef explains H-M and near H-M representatives .
When LaTeXMLMath , these describe connected components of real points on levels 1 and above of a Modular Tower .
As a prelude for § LaTeXMLRef , § LaTeXMLRef uses LaTeXMLMath -line branch cycles ( § LaTeXMLRef ) for diophantine conclusions about regular realizations .
This describes real point components on level 0 of the LaTeXMLMath Modular Tower .
This paper uses properties of LaTeXMLMath as a LaTeXMLMath module .
Let LaTeXMLMath be a Galois cover with group LaTeXMLMath , and let LaTeXMLMath be a cover for which LaTeXMLMath is Galois with group LaTeXMLMath .
Call LaTeXMLMath a Frattini extension of LaTeXMLMath if the following holds .
For any sequence LaTeXMLMath , of ( not necessarily Galois ) covers with LaTeXMLMath , there is always a proper cover of LaTeXMLMath through which both LaTeXMLMath and LaTeXMLMath factor .
A Frattini extension of LaTeXMLMath has no differentials and functions that are pullbacks from covers disjoint from LaTeXMLMath .
For a field theoretic restatement of the property let LaTeXMLMath be a chain of fields with LaTeXMLMath ( resp .
LaTeXMLMath ) Galois with group LaTeXMLMath ( resp .
LaTeXMLMath ) .
This is a Frattini chain if the only subfield LaTeXMLMath for which LaTeXMLMath , is LaTeXMLMath .
Denote by LaTeXMLMath the natural map .
Let LaTeXMLMath be the fixed field of a subgroup LaTeXMLMath of LaTeXMLMath .
Then , LaTeXMLMath is equivalent to LaTeXMLMath .
Hint : LaTeXMLMath allows extending any automorphism of LaTeXMLMath to LaTeXMLMath to be the identity on LaTeXMLMath .
The group theoretic restatement is that LaTeXMLMath and LaTeXMLMath implies LaTeXMLMath : LaTeXMLMath is a Frattini cover .
A Frattini cover LaTeXMLMath always has a nilpotent kernel .
Let LaTeXMLMath be the normalizer of a LaTeXMLMath -Sylow LaTeXMLMath of any finite group LaTeXMLMath .
Apply Schur-Zassenhaus to write LaTeXMLMath as LaTeXMLMath with LaTeXMLMath having order prime to LaTeXMLMath .
Let LaTeXMLMath be the pro-free pro- LaTeXMLMath group on LaTeXMLMath generators , with LaTeXMLMath the rank ( minimal number of generators ) of LaTeXMLMath .
Then , the universal LaTeXMLMath -Frattini cover of LaTeXMLMath is LaTeXMLMath ; extend LaTeXMLMath acting on LaTeXMLMath to LaTeXMLMath through the map LaTeXMLMath as in Remark LaTeXMLRef .
For any group LaTeXMLMath , LaTeXMLMath is the pro-free pro- LaTeXMLMath kernel of LaTeXMLMath .
Commutators and LaTeXMLMath th powers in LaTeXMLMath generate LaTeXMLMath , the Frattini subgroup of LaTeXMLMath .
Iterating this produces LaTeXMLMath .
Let LaTeXMLMath be the first characteristic quotient of the universal LaTeXMLMath -Frattini cover of LaTeXMLMath .
Then LaTeXMLMath is a LaTeXMLMath module and LaTeXMLMath is a Frattini extension of LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be any subgroup of LaTeXMLMath .
Then , LaTeXMLMath embeds in LaTeXMLMath .
Further , for each LaTeXMLMath , LaTeXMLMath naturally embeds in LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath appears from the universal LaTeXMLMath -Frattini cover LaTeXMLMath as LaTeXMLMath .
For LaTeXMLMath this applies if LaTeXMLMath .
A LaTeXMLMath -Sylow of LaTeXMLMath contains a LaTeXMLMath -Sylow of LaTeXMLMath .
So , the latter is profree .
LaTeXMLCite characterizes LaTeXMLMath as the minimal cover of LaTeXMLMath with pro-free LaTeXMLMath -Sylow .
So , there is a natural map LaTeXMLMath commuting with the map to LaTeXMLMath .
As LaTeXMLMath is a LaTeXMLMath -Frattini cover of LaTeXMLMath the map is surjective .
Since the natural map LaTeXMLMath has a pro- LaTeXMLMath group as kernel , the natural map LaTeXMLMath produces LaTeXMLMath commuting with the projections to LaTeXMLMath .
The composition LaTeXMLMath ( commuting with the projections to LaTeXMLMath ) is an endomorphism of LaTeXMLMath .
The image of LaTeXMLMath is a closed subgroup of LaTeXMLMath mapping surjectively to LaTeXMLMath .
So , from the Frattini property , LaTeXMLMath is onto .
An onto endomorphism of finitely generated profinite groups is an isomorphism LaTeXMLCite .
In particular , LaTeXMLMath is an injection .
The characteristic quotients have maps between them induced by LaTeXMLMath , and so LaTeXMLMath injects into LaTeXMLMath , inducing an injection of LaTeXMLMath .
If ( for LaTeXMLMath ) , LaTeXMLMath and LaTeXMLMath have the same dimension , they are isomorphic .
As these groups characterize LaTeXMLMath and LaTeXMLMath , that implies they are equal .
This gives an isomorphism of LaTeXMLMath and LaTeXMLMath in the special case .
∎ Since LaTeXMLMath is a pro-free pro- LaTeXMLMath group , it is easy to create a profinite group LaTeXMLMath whose action on LaTeXMLMath extends LaTeXMLMath on LaTeXMLMath .
Any LaTeXMLMath lifts to an automorphism LaTeXMLMath of LaTeXMLMath commuting with the map to LaTeXMLMath .
The ( profinite ) automorphism group of LaTeXMLMath is the projective limit of automorphism groups of finite group quotients ( by finite index characteristic subgroups ) of LaTeXMLMath .
Let LaTeXMLMath be any generators of LaTeXMLMath .
Choose LaTeXMLMath to be the closure of LaTeXMLMath in the automorphism group of LaTeXMLMath .
By construction it maps surjectively to LaTeXMLMath .
The problem is to split off a copy of LaTeXMLMath .
Automorphisms of LaTeXMLMath trivial on its Frattini quotient have LaTeXMLMath -power order LaTeXMLCite .
The kernel from LaTeXMLMath will be trivial on the Frattini quotient of LaTeXMLMath ( which equals the Frattini quotient of LaTeXMLMath ) .
So Schur-Zassenhaus ( for profinite groups , LaTeXMLCite ) always allows splitting off a copy of LaTeXMLMath in LaTeXMLMath .
Since , however , it depends on LaTeXMLMath , it is an art to do this explicitly .
Again by Schur-Zassenhaus , this copy of LaTeXMLMath is unique up to conjugacy .
That LaTeXMLMath in § LaTeXMLRef is a LaTeXMLMath module appears in using modular representation theory at each level of a Modular Tower .
Significantly , LaTeXMLMath is not a LaTeXMLMath module ( though it may have nontrivial quotients that are ) .
For each LaTeXMLMath , however , LaTeXMLMath is an LaTeXMLMath module through this lifted action .
Let LaTeXMLMath be a LaTeXMLMath -Sylow of LaTeXMLMath and LaTeXMLMath ( resp .
LaTeXMLMath ) the restriction of LaTeXMLMath to LaTeXMLMath ( resp .
the normalizer of LaTeXMLMath in LaTeXMLMath ) .
Recall the Loewy display of composition factors for a LaTeXMLMath module LaTeXMLMath .
It derives from the radical submodules of LaTeXMLMath , LaTeXMLEquation .
LaTeXMLMath is the minimal LaTeXMLMath submodule of LaTeXMLMath with LaTeXMLMath semi-simple .
The display consists of writing the simple module summands of LaTeXMLMath at the LaTeXMLMath position with arrows indicating relations between modules at different levels .
Warning !
A Loewy display is not a sequence of module homomorphisms ; it indicates the relation between modules at different levels in the Krull decomposition .
We write this display right to left , compatible with the quotients from an exact sequence , instead of top to bottom ( as group theorists often do ) .
A Loewy display of all information on subquotients of a module often requires several arrows between layers ; there may be several arrows from a simple level LaTeXMLMath module toward level LaTeXMLMath modules ( see the examples of Prop .
LaTeXMLRef ) .
Note : Such a display uniquely determines a module .
The Loewy display of LaTeXMLMath consists of copies of LaTeXMLMath , the only simple LaTeXMLMath -group module .
Jenning ’ s Theorem LaTeXMLCite is an efficient tool to figure the dimension of the Loewy layers of LaTeXMLMath .
( The proof of Prop .
LaTeXMLRef has an example of its use . )
Further , since the action of LaTeXMLMath respects this construction , it efficiently reveals how LaTeXMLMath acts on the Poincaré-Witt basis of the LaTeXMLMath group ring ’ s universal enveloping algebra .
The inductive detection of LaTeXMLMath on Loewy layers of LaTeXMLMath often comes through LaTeXMLMath appearing from a previous Loewy layer , or from tensors products of representations from previous Loewy layers ( like a representation and its complex conjugate appearing juxtaposed ) .
The following argument ( a collaboration with D. Semmen ) for the LaTeXMLMath -split case shows LaTeXMLMath usually appears quickly unless the LaTeXMLMath -Sylow is cyclic .
Princ .
LaTeXMLRef already shows LaTeXMLMath appears at infinitely many levels if LaTeXMLMath has a center .
Let LaTeXMLMath be centerless , with LaTeXMLMath a LaTeXMLMath -group and LaTeXMLMath .
If LaTeXMLMath is not cyclic , LaTeXMLMath appears infinitely often .
Otherwise , LaTeXMLMath is the same cyclic LaTeXMLMath module for each LaTeXMLMath .
So , LaTeXMLMath does not appear .
By replacing LaTeXMLMath by LaTeXMLMath , with LaTeXMLMath its Frattini subgroup , form LaTeXMLMath .
The universal LaTeXMLMath -Frattini cover of LaTeXMLMath is the same as that of LaTeXMLMath .
So , Princ .
LaTeXMLRef shows LaTeXMLMath occurring at some level for LaTeXMLMath implies it occurs at infinitely many levels for LaTeXMLMath .
So , for our question , with no loss take LaTeXMLMath to assume LaTeXMLMath is an elementary LaTeXMLMath -group and a LaTeXMLMath module .
Then , with LaTeXMLMath the projective indecomposable for LaTeXMLMath , consider it as an LaTeXMLMath ( therefore a LaTeXMLMath ) module .
Since LaTeXMLMath , Higman ’ s criterion says it is the projective indecomposable for LaTeXMLMath LaTeXMLCite .
Its most natural display might by LaTeXMLMath .
In its Loewy display LaTeXMLMath appears to the far right .
Write the next Loewy layer as LaTeXMLMath , with the LaTeXMLMath s irreducible LaTeXMLMath ( LaTeXMLMath ) modules .
Forming LaTeXMLMath ( * is the LaTeXMLMath dual ) returns LaTeXMLMath .
So the dual of the LaTeXMLMath th socle layer of LaTeXMLMath is the LaTeXMLMath th Loewy layer of LaTeXMLMath .
Conclude that to the far left there is LaTeXMLMath ( 1st socle layer ) with LaTeXMLMath immediately to the right of that ( 2nd socle layer ) .
Now we apply that LaTeXMLMath is an elementary LaTeXMLMath -group .
The LaTeXMLMath module LaTeXMLMath is the group ring LaTeXMLMath .
Let LaTeXMLMath be a basis for LaTeXMLMath and identify LaTeXMLMath in the vector space LaTeXMLMath as the space spanned by LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath preserves this space modulo the second power of the augmentation ideal of LaTeXMLMath .
Conclude : LaTeXMLMath as LaTeXMLMath ( or LaTeXMLMath ) modules .
As in Prop .
LaTeXMLRef use the notation of LaTeXMLCite .
So , the kernel LaTeXMLMath of LaTeXMLMath starts with LaTeXMLMath .
Now consider a minimal projective LaTeXMLMath mapping surjectively to LaTeXMLMath .
Higman ’ s criterion again implies the LaTeXMLMath module LaTeXMLMath , being projective for LaTeXMLMath , is also projective for LaTeXMLMath .
The Loewy layers of LaTeXMLMath are just the Loewy layers of LaTeXMLMath tensored over LaTeXMLMath with LaTeXMLMath modules .
Reason : a Loewy layer of LaTeXMLMath tensored with LaTeXMLMath is semisimple .
Therefore LaTeXMLMath .
Then , LaTeXMLEquation modulo projective summands ( as in LaTeXMLCite , called LaTeXMLMath there ) .
The 2nd socle layer of LaTeXMLMath is LaTeXMLMath with one copy of LaTeXMLMath removed ( from the end of LaTeXMLMath .
The module LaTeXMLMath has exactly one appearance of LaTeXMLMath for each absolutely irreducible factor in LaTeXMLMath ( Maschke ’ s Theorem ; since LaTeXMLMath acts trivially on LaTeXMLMath , the same is true of LaTeXMLMath ) .
Thus , LaTeXMLMath appears in the Loewy display of LaTeXMLMath as a LaTeXMLMath module unless LaTeXMLMath , and LaTeXMLMath is absolutely irreducible .
Now suppose LaTeXMLMath .
Instead of looking at LaTeXMLMath , look at LaTeXMLMath as an LaTeXMLMath module .
By our hypotheses , LaTeXMLMath is not a cyclic module .
Therefore , a computation of the rank of LaTeXMLMath comes directly from Schreier ’ s formula for ranks of subgroups of pro-free groups ( see § LaTeXMLRef ) .
The rank of LaTeXMLMath , as LaTeXMLMath increases , exceeds the degree of any irreducible LaTeXMLMath module .
Replace LaTeXMLMath with a suitable LaTeXMLMath to revert to the case LaTeXMLMath is not absolutely irreducible .
This completes showing the appearance of LaTeXMLMath .
∎ The structure constant formula ( § LaTeXMLRef ) can detect LaTeXMLMath appearing here , as LaTeXMLMath means it applies in characteristic LaTeXMLMath .
This topic continues in § LaTeXMLRef .
Lemma LaTeXMLRef hypotheses imply LaTeXMLMath is a LaTeXMLMath module .
In the proof of Lemma LaTeXMLRef , LaTeXMLMath induces a surjective LaTeXMLMath module map LaTeXMLMath .
Suppose LaTeXMLMath ( LaTeXMLMath a LaTeXMLMath -Sylow of LaTeXMLMath ) and LaTeXMLMath is a LaTeXMLMath module extending the LaTeXMLMath action so the following holds .
There is an extension LaTeXMLMath with kernel LaTeXMLMath so the pullback of LaTeXMLMath in LaTeXMLMath is the natural quotient LaTeXMLMath .
Then , the Lemma LaTeXMLRef conclusion holds .
This applies with LaTeXMLMath and LaTeXMLMath .
The hypothesis says the morphism LaTeXMLMath , a priori an LaTeXMLMath module homomorphism , is actually a LaTeXMLMath module homomorphism .
Here is why .
Suppose LaTeXMLMath is a proper subgroup of LaTeXMLMath mapping surjectively to LaTeXMLMath .
Then , the pullback LaTeXMLMath of LaTeXMLMath in LaTeXMLMath is a proper subgroup of the pullback of LaTeXMLMath in LaTeXMLMath .
Further , LaTeXMLMath maps surjectively to LaTeXMLMath , contrary to LaTeXMLMath being a Frattini cover .
Conclude that LaTeXMLMath is also a Frattini cover .
So , there is a natural map from LaTeXMLMath inducing a surjective LaTeXMLMath module homomorphism LaTeXMLMath .
Since LaTeXMLMath is the 1st characteristic quotient of LaTeXMLMath , universal for covers with elementary LaTeXMLMath -group kernel , there is an LaTeXMLMath module splitting of LaTeXMLMath .
Higman ’ s Theorem LaTeXMLCite says , since LaTeXMLMath , an LaTeXMLMath splitting of this LaTeXMLMath map gives a LaTeXMLMath splitting .
This is contrary to LaTeXMLMath being an indecomposable LaTeXMLMath module unless this is an isomorphism ( LaTeXMLCite or LaTeXMLCite ) .
Suppose a LaTeXMLMath -Sylow LaTeXMLMath of LaTeXMLMath has this property : Either LaTeXMLEquation for each LaTeXMLMath .
LaTeXMLCite shows LaTeXMLEquation is an isomorphism ( so both have dimension 1 ) guaranteeing ( LaTeXMLRef ) .
This holds for LaTeXMLMath and its 2-Sylows since they are distinguished by which integer from LaTeXMLMath each element in the 2-Sylow fixes .
∎ Suppose LaTeXMLMath , LaTeXMLMath is a higher characteristic quotient of LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -Sylow of LaTeXMLMath .
Then , the hypotheses of Prop .
LaTeXMLRef automatically hold .
Even when LaTeXMLMath , given LaTeXMLMath , suppose LaTeXMLMath contains LaTeXMLMath and is maximal for this property .
For some extension LaTeXMLMath with kernel LaTeXMLMath , the pullback of LaTeXMLMath in LaTeXMLMath is the natural quotient LaTeXMLMath .
As in LaTeXMLCite , let LaTeXMLMath be the LaTeXMLMath module induced from LaTeXMLMath acting on LaTeXMLMath .
Apply Shapiro ’ s Lemma LaTeXMLCite : LaTeXMLMath .
So there is an extension of LaTeXMLMath with kernel LaTeXMLMath whose pullback over LaTeXMLMath has LaTeXMLMath as a quotient .
LaTeXMLCite uses LaTeXMLMath to produce the characteristic LaTeXMLMath -Frattini module LaTeXMLMath from the LaTeXMLMath -split case using indecomposability of LaTeXMLMath LaTeXMLCite .
As in LaTeXMLCite , producing LaTeXMLMath for LaTeXMLMath either 3 or 5 requires using the LaTeXMLMath module induced from the LaTeXMLMath module LaTeXMLMath ( generalizing Prop .
LaTeXMLRef ) .
Prop .
LaTeXMLRef applies immediately to LaTeXMLMath when LaTeXMLMath .
Let LaTeXMLMath .
Then , LaTeXMLMath identifies with the LaTeXMLMath module generated by the six cosets of a LaTeXMLMath in LaTeXMLMath , modulo the module generated by the sum of the cosets .
Any LaTeXMLMath in LaTeXMLMath has a unique LaTeXMLMath lying in LaTeXMLMath .
So , the action of LaTeXMLMath on LaTeXMLMath cosets extends to an LaTeXMLMath action on cosets of a dihedral group .
Thus , Prop .
LaTeXMLRef gives LaTeXMLMath as an extension of LaTeXMLMath by LaTeXMLMath .
Suppose LaTeXMLMath is pro-free pro- LaTeXMLMath group on LaTeXMLMath generators .
Let LaTeXMLMath be a surjective homomorphism , with LaTeXMLMath any ( finite ) LaTeXMLMath group .
Schreier ’ s construction gives explicit generators of the kernel of LaTeXMLMath LaTeXMLCite .
Apply this with LaTeXMLMath and LaTeXMLMath , the Klein 4-group and LaTeXMLMath .
Let LaTeXMLMath be a generator of LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath generators of LaTeXMLMath let LaTeXMLMath act on LaTeXMLMath by mapping LaTeXMLMath to LaTeXMLMath .
Use LaTeXMLMath as coset representatives for LaTeXMLMath in LaTeXMLMath .
Form the set LaTeXMLMath of elements in LaTeXMLMath having the form LaTeXMLMath or LaTeXMLMath with LaTeXMLMath .
Toss from LaTeXMLMath those that equal 1 .
Now consider the images of LaTeXMLMath and LaTeXMLMath on LaTeXMLMath .
This produces LaTeXMLMath and LaTeXMLMath .
Consider LaTeXMLMath on LaTeXMLMath .
Recall : Modulo LaTeXMLMath any two elements in LaTeXMLMath commute .
Apply this to get LaTeXMLEquation .
Apply LaTeXMLMath again to get LaTeXMLMath .
The action of LaTeXMLMath on the LaTeXMLMath s is the same as the action on the six cosets of an element of order 2 .
Denote a commutator of two elements LaTeXMLMath by LaTeXMLMath .
Modulo LaTeXMLMath there are relations among the LaTeXMLMath s : LaTeXMLMath ; and LaTeXMLMath .
So , the product of the LaTeXMLMath s is 1 .
The proof follows from associating a LaTeXMLMath in LaTeXMLMath with a dihedral in LaTeXMLMath as in the statement of the proposition .
∎ We finish a self-contained treatment of much of LaTeXMLCite .
The first characteristic quotient of the universal 2-Frattini cover of LaTeXMLMath is a ( nonsplit ) extension of LaTeXMLMath by an irreducible module V using LaTeXMLMath .
Prop .
LaTeXMLRef shows this as follows .
As previously , let LaTeXMLMath be the kernel of LaTeXMLMath .
Sums of LaTeXMLMath cosets , LaTeXMLMath , represent its elements .
The augmentation map sends such an element to the sum LaTeXMLMath .
Let LaTeXMLMath be the 4-dimensional kernel of the augmentation map .
Nonzero elements of LaTeXMLMath have representatives LaTeXMLMath with two of the LaTeXMLMath s nonzero .
Let LaTeXMLMath be the subgroup LaTeXMLMath in the proof of Prop .
LaTeXMLRef .
The action of LaTeXMLMath on LaTeXMLMath is two copies of the 2-dimensional irreducible of LaTeXMLMath .
So , no element of LaTeXMLMath centralizes LaTeXMLMath and the centralizer of LaTeXMLMath in LaTeXMLMath is a LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -Sylow of LaTeXMLMath with LaTeXMLMath , a Klein 4-group , its image in LaTeXMLMath .
Denote the module for restriction of LaTeXMLMath to a subgroup LaTeXMLMath by LaTeXMLMath .
Besides the origin there are three conjugacy classes ( orbits for LaTeXMLMath action ) in LaTeXMLMath .
These are ( 15 ) elements of LaTeXMLMath ; 10 representing sums LaTeXMLMath with exactly three LaTeXMLMath s nonzero ( in LaTeXMLMath ) ; and six representing sums LaTeXMLMath with exactly one LaTeXMLMath nonzero ( in LaTeXMLMath ) .
Call the second set LaTeXMLMath and the third LaTeXMLMath .
Elements of LaTeXMLMath ( resp .
LaTeXMLMath ) are exactly those in LaTeXMLMath some 3-cycle ( resp .
5-cycle ) stabilizes .
Correspond to each the respective 3 or 5-Sylow of LaTeXMLMath that stabilizes it .
The action of LaTeXMLMath on LaTeXMLMath has a module presentation LaTeXMLEquation .
Extending this action to LaTeXMLMath gives a Loewy display LaTeXMLEquation with LaTeXMLMath the two dimensional LaTeXMLMath module on which LaTeXMLMath acts irreducibly .
Restrict LaTeXMLMath to LaTeXMLMath for an exact sequence LaTeXMLMath .
The Loewy display of LaTeXMLMath ( resp .
LaTeXMLMath ) is LaTeXMLMath ( resp .
LaTeXMLMath ) .
If LaTeXMLMath has order 3 or 5 and LaTeXMLMath then , giving LaTeXMLMath and stipulating LaTeXMLMath determines LaTeXMLMath .
Suppose LaTeXMLMath has order 4 .
Then , the centralizer LaTeXMLMath of LaTeXMLMath on LaTeXMLMath has dimension 3 and on V has dimension 2 , LaTeXMLMath and LaTeXMLMath contains two elements each from LaTeXMLMath and from LaTeXMLMath .
The action of LaTeXMLMath preserves cosets .
So , orbits for conjugation by LaTeXMLMath are clear if LaTeXMLMath , LaTeXMLMath and LaTeXMLMath each consist of one LaTeXMLMath orbit .
This follows from triple transitivity of LaTeXMLMath in the standard representation .
The exact sequence ( LaTeXMLRef ) comes from writing the right cosets of LaTeXMLMath in LaTeXMLMath .
Use LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath has generators LaTeXMLMath and LaTeXMLMath .
Images of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath generate LaTeXMLMath .
Consider LaTeXMLMath of order 4 .
To be explicit assume LaTeXMLMath lifts LaTeXMLMath .
Then , LaTeXMLMath fixes LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , so it acts like LaTeXMLMath on the cosets .
The centralizer of LaTeXMLMath in LaTeXMLMath is therefore LaTeXMLEquation .
An element in LaTeXMLMath is in LaTeXMLMath if and only if LaTeXMLMath .
Note : LaTeXMLMath fixes LaTeXMLMath .
Sum both with LaTeXMLMath to get two elements in LaTeXMLMath .
The observation that LaTeXMLMath is a special case of Prop .
LaTeXMLRef .
Now consider the sequence ( LaTeXMLRef ) .
Write LaTeXMLMath as LaTeXMLMath with LaTeXMLMath .
First : LaTeXMLMath is an LaTeXMLMath module .
As LaTeXMLMath sends LaTeXMLMath to LaTeXMLMath , it preserves the leftmost layer of the Loewy display .
In the quotient LaTeXMLMath , LaTeXMLMath and LaTeXMLMath generate a copy of LaTeXMLMath .
Then , LaTeXMLMath fixes LaTeXMLMath and LaTeXMLMath generates a copy of LaTeXMLMath .
The module LaTeXMLMath has Loewy display LaTeXMLMath .
If there is a subquotient of LaTeXMLMath having a Loewy display LaTeXMLMath , it must be LaTeXMLMath .
Clearly this is invariant under LaTeXMLMath .
The action of LaTeXMLMath is to take LaTeXMLMath to LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath differ by an element of LaTeXMLMath , this shows LaTeXMLMath is invariant under LaTeXMLMath .
∎ We take advantage of the special coset aspect of module LaTeXMLMath throughout the study of our main example .
The seed , however , is an explicit presentation of LaTeXMLMath as a LaTeXMLMath module , with LaTeXMLMath a LaTeXMLMath -Sylow of LaTeXMLMath .
The proof of Prop .
LaTeXMLRef shows , with illustration from LaTeXMLMath ( versus LaTeXMLMath above ) an effective way to compute the analog of ( LaTeXMLRef ) .
As with modular curves , the Prop .
LaTeXMLRef branch cycle description tells much about the reduced cover of the LaTeXMLMath -line , especially about its points over LaTeXMLMath , the cusps .
Cusp widths are the lengths of these orbits .
An analysis of possible cusp widths starts with a general group computation .
We refer to computations in § LaTeXMLRef .
Suppose LaTeXMLMath has odd order ( so necessarily LaTeXMLMath ) .
Write LaTeXMLMath as a product LaTeXMLMath of disjoint cycles with LaTeXMLMath of length LaTeXMLMath , LaTeXMLMath .
Define LaTeXMLMath to be LaTeXMLMath .
Serre notes : LaTeXMLMath if and only if LaTeXMLMath .
Let LaTeXMLMath be the universal central exponent 2 extension of LaTeXMLMath .
Regard the kernel of this map as LaTeXMLMath .
Assume entries of LaTeXMLMath ( in LaTeXMLMath ) have odd order .
Lift to LaTeXMLMath the entries of LaTeXMLMath , preserving their respective orders .
Denote the lifted LaTeXMLMath -tuple by LaTeXMLMath .
Suppose the following hold .
Product-one condition : LaTeXMLMath .
Transitivity : LaTeXMLMath is a transitive subgroup of LaTeXMLMath .
Genus 0 condition : LaTeXMLMath .
Then , LaTeXMLMath .
The transitivity hypothesis in Prop .
LaTeXMLRef is not serious .
Restrict LaTeXMLMath to each orbit of LaTeXMLMath .
If the genus 0 condition applies in each such restriction , then LaTeXMLMath is the product of the LaTeXMLMath values appearing in each restriction .
The genus 0 condition , however , is very serious .
Examples of § LaTeXMLRef or LaTeXMLCite show how to use Prop .
LaTeXMLRef to go beyond the genus 0 condition .
A preliminary example !
Take LaTeXMLMath .
With LaTeXMLMath , the lifting invariant LaTeXMLMath is clearly LaTeXMLMath : LaTeXMLMath with LaTeXMLMath of order 3 .
The conclusion , however , of Prop .
LaTeXMLRef seems to be LaTeXMLMath .
It , however , doesn ’ t apply because the genus 0 condition doesn ’ t hold .
§ LaTeXMLRef and Princ .
LaTeXMLRef have cases of this computation without the genus 0 hypothesis ; still by using Prop .
LaTeXMLRef .
Suppose LaTeXMLMath has even order , and LaTeXMLMath is a central extension with kernel an elementary abelian 2- group .
Let LaTeXMLMath be a lift of LaTeXMLMath to LaTeXMLMath .
Then , the order of LaTeXMLMath is independent of the choice of the lift .
The following applies the technique of proof from Prop .
LaTeXMLRef to analyze the orders of lifts of elements of order 2 to LaTeXMLMath .
We compare this to an analog question with LaTeXMLMath , the first characteristic quotient of LaTeXMLMath replacing LaTeXMLMath ( Lem .
LaTeXMLRef ) .
Assume LaTeXMLMath , and LaTeXMLMath of order 2 is a product of LaTeXMLMath disjoint 2-cycles .
Any lift LaTeXMLMath of LaTeXMLMath has order 4 if LaTeXMLMath is odd and 2 if LaTeXMLMath is even .
We review the Clifford algebra setup used in LaTeXMLCite .
Let LaTeXMLMath be the Clifford algebra on LaTeXMLMath with generators LaTeXMLMath subject to relations LaTeXMLEquation .
In the Clifford algebra , write LaTeXMLMath .
Then , LaTeXMLMath and LaTeXMLMath .
The collection of LaTeXMLMath under multiplication generate a subgroup LaTeXMLMath .
Characterization : It is the central nonsplit extension of LaTeXMLMath whose restriction to transpositions splits , and whose restriction to products of two disjoint transpositions is nontrivial LaTeXMLCite .
The map LaTeXMLMath appears from LaTeXMLMath .
So , LaTeXMLMath if LaTeXMLMath .
That LaTeXMLMath is nontrivial if LaTeXMLMath shows from lifts of certain elements of order 2 .
Example : LaTeXMLMath lifts to have order 4 : LaTeXMLEquation .
Of course the order of a lift is conjugacy class invariant .
Similarly , with LaTeXMLMath , LaTeXMLEquation .
By induction , the result is LaTeXMLMath : LaTeXMLMath has order LaTeXMLMath .
∎ Prop .
LaTeXMLRef gives a partial answer to ( LaTeXMLRef LaTeXMLRef ) about existence of rational points at level 1 of other LaTeXMLMath Modular Towers .
There are three LaTeXMLMath classes : the two classes of 5-cycles , LaTeXMLMath ( the class of LaTeXMLMath ) and LaTeXMLMath ( the class of LaTeXMLMath ) and 3-cycles .
From the Branch Cycle Lemma , excluding LaTeXMLMath , to give a Galois cover over LaTeXMLMath with LaTeXMLMath requires C be LaTeXMLMath or LaTeXMLMath .
Both give Modular Towers with two components at level 0 , one obstructed ( nothing above it at level 1 ) , one not .
The only Modular Towers for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath with possible LaTeXMLMath components at level 1 ( for realizations of LaTeXMLMath ) have LaTeXMLMath , LaTeXMLMath or LaTeXMLMath .
Each of the latter two has exactly two level 0 components , one obstructed , one not .
There are no LaTeXMLMath ( so , no LaTeXMLMath ) points at level 1 of the LaTeXMLMath Modular Tower .
The next four subsections prove this by applying a lift to LaTeXMLMath in the two new cases .
For computational reasons , start with LaTeXMLMath .
Ex .
LaTeXMLRef notes the two genus 1 components at level 1 on the Modular Tower for LaTeXMLMath over the level 0 unobstructed component .
If have definition field LaTeXMLMath , they might have infinitely many LaTeXMLMath realizations over LaTeXMLMath .
This is the only hope for infinitely many ( up to LaTeXMLMath equivalence ) LaTeXMLMath regular realizations of LaTeXMLMath with LaTeXMLMath .
Applying § LaTeXMLRef to this to see there will be at most finitely many realizations LaTeXMLMath for LaTeXMLMath .
Suppose LaTeXMLMath is in the LaTeXMLMath orbit LaTeXMLMath .
Then , there is LaTeXMLMath over LaTeXMLMath if and only if there is something from LaTeXMLMath over each element of LaTeXMLMath .
If this holds , call LaTeXMLMath unobstructed ( § LaTeXMLRef ) .
We list inner Nielsen classes LaTeXMLMath for LaTeXMLMath restricting to pieces where it is easy to demonstrate they lie in one LaTeXMLMath orbit .
With no loss use representatives for LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Conjugate the collection of 5-cycles by LaTeXMLMath .
There are four length one and four length five orbits .
Two length five orbits of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , are 5- cycles in LaTeXMLMath .
There are two cases : LaTeXMLMath or LaTeXMLMath , LaTeXMLMath and LaTeXMLMath have 1 in their common support ; or LaTeXMLMath is a representative from LaTeXMLMath or LaTeXMLMath .
Here are LaTeXMLMath representatives for ( LaTeXMLRef LaTeXMLRef ) : LaTeXMLEquation .
We show the orbit of LaTeXMLMath satisfying ( LaTeXMLRef LaTeXMLRef ) is obstructed .
Example : LaTeXMLMath .
Suppose LaTeXMLMath lies over LaTeXMLMath .
Consider the image LaTeXMLMath in LaTeXMLMath of LaTeXMLMath , as in proof of Lem .
LaTeXMLRef .
Denote the product of the entries of LaTeXMLMath by LaTeXMLMath .
Then , LaTeXMLMath .
The product-one condition , LaTeXMLMath , holds for LaTeXMLMath if and only if it holds for LaTeXMLMath .
This is a lift of LaTeXMLMath , by Riemann-Hurwitz a branch cycle description of a genus 0 cover .
Apply Prop .
LaTeXMLRef .
Entries of LaTeXMLMath have product LaTeXMLEquation .
The product-one condition doesn ’ t hold for LaTeXMLMath .
So , it can ’ t hold in LaTeXMLMath .
This is necessary for a Nielsen class element , concluding the proof of obstruction .
Now consider which elements of ( LaTeXMLRef LaTeXMLRef ) are obstructed .
Check : LaTeXMLMath limits LaTeXMLMath to be conjugates of LaTeXMLMath by the Klein four group centralizer of LaTeXMLMath .
LaTeXMLMath limits LaTeXMLMath to be one of two types : conjugates of LaTeXMLMath by the centralizer of LaTeXMLMath ; or both are in LaTeXMLMath .
As above , consider the unique lift LaTeXMLMath to LaTeXMLMath of LaTeXMLMath that could be an image from LaTeXMLMath for the second of ( LaTeXMLRef LaTeXMLRef ) .
Since the lifts are unique , LaTeXMLMath .
Typically : LaTeXMLMath would lift LaTeXMLEquation .
Apply Prop .
LaTeXMLRef to LaTeXMLMath : LaTeXMLMath .
Therefore , LaTeXMLMath .
One last application of the same computation shows LaTeXMLMath .
To conclude LaTeXMLMath is unobstructed , use the following from LaTeXMLCite .
The Loewy display of LaTeXMLMath refers to the LaTeXMLMath action in the LaTeXMLMath -Frattini cover LaTeXMLMath .
Obstruction can occur from level LaTeXMLMath to level LaTeXMLMath in a Modular Tower only where LaTeXMLMath appears in the Loewy display of LaTeXMLMath .
Translate this to say there is a ( Frattini cover ) sequence LaTeXMLMath with LaTeXMLMath and LaTeXMLMath is a trivial LaTeXMLMath module .
The appearance of LaTeXMLMath for any value of LaTeXMLMath implies that LaTeXMLMath appears in LaTeXMLMath for finitely many LaTeXMLMath .
If LaTeXMLMath is LaTeXMLMath -perfect and centerless , then so is LaTeXMLMath ( argument of Prop .
LaTeXMLRef ) .
Everything has an explanation already except the automatic appearance of LaTeXMLMath in LaTeXMLMath for finitely many LaTeXMLMath given its appearance at level LaTeXMLMath .
Apply Prop .
LaTeXMLRef with LaTeXMLMath replacing LaTeXMLMath .
Then , use that the universal LaTeXMLMath -Frattini cover of both LaTeXMLMath and LaTeXMLMath is the same as that of LaTeXMLMath .
So , the characteristic quotients for the universal LaTeXMLMath -Frattini of LaTeXMLMath ( in place of LaTeXMLMath ) are cofinal in the projective system LaTeXMLMath .
Therefore the simple modules ( including copies of the identity representation ) appearing in the analog for LaTeXMLMath of the modules LaTeXMLMath also appear as simple modules for a cofinal collection from LaTeXMLMath .∎ Since LaTeXMLMath appears only at the head ( in the LaTeXMLMath Schur multiplier ) of LaTeXMLMath ( § LaTeXMLRef ) , LaTeXMLMath passes the lifting test to LaTeXMLMath exactly if there is LaTeXMLMath over it .
The principle from § LaTeXMLRef is the following , where LaTeXMLMath is as in Prop .
LaTeXMLRef .
Suppose for LaTeXMLMath , the product-one condition holds and there exists LaTeXMLMath with LaTeXMLMath a 3-cycle , LaTeXMLMath a 5-cycle and their product ( in either order ) is a 3-cycle .
To compute LaTeXMLMath we may assume LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath if and only if LaTeXMLMath is unobstructed .
With no loss assume LaTeXMLMath .
The exists LaTeXMLMath braiding LaTeXMLMath to LaTeXMLMath in the same Nielsen class with LaTeXMLMath and LaTeXMLMath .
From LaTeXMLCite , LaTeXMLMath .
( More generally , LaTeXMLMath where LaTeXMLMath is the big braid invariant of a Nielsen class . )
Apply the argument of § LaTeXMLRef to LaTeXMLMath to reduce computing LaTeXMLMath to computing LaTeXMLMath .
∎ Consider a case of ( LaTeXMLRef LaTeXMLRef ) : LaTeXMLMath .
Princ .
LaTeXMLRef applies to LaTeXMLMath and LaTeXMLMath .
The product LaTeXMLMath is a 3-cycle .
Thus , LaTeXMLMath where LaTeXMLMath is a 3-tuple consisting of two 3-cycles and a 5-cycle .
As previously , LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is unobstructed .
Finally , consider the first case of ( LaTeXMLRef LaTeXMLRef ) .
Note : The genus 0 hypothesis of Prop .
LaTeXMLRef doesn ’ t hold .
For each ordering of the conjugacy classes LaTeXMLMath , the nielsen class LaTeXMLMath has exactly one element , for a total of six elements .
All representatives LaTeXMLMath have LaTeXMLMath .
That there is only one LaTeXMLMath with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath follows from the data in ( LaTeXMLRef ) .
The computation LaTeXMLMath comes from LaTeXMLMath in ( LaTeXMLRef ) .
∎ Let LaTeXMLMath .
As above , if LaTeXMLMath , then this is obstructed .
Princ .
LaTeXMLRef shows LaTeXMLMath .
Since the entries satisfy the genus 0 hypothesis , Prop .
LaTeXMLRef gives this value as -1 .
Thm .
LaTeXMLRef shows level 1 of the Modular Tower for LaTeXMLMath has no real points ; that would require H-M reps. at level 0 ( see ( LaTeXMLRef ) ) .
It has none , for the inverse of an element of LaTeXMLMath is in LaTeXMLMath .
This brings a new issue .
There are H-M representatives : LaTeXMLMath .
So some elements are unobstructed .
For an obstructed element in the Nielsen class use LaTeXMLMath , three repetitions of the LaTeXMLMath conjugacy class .
( Note : ( LaTeXMLRef ) shows LaTeXMLMath is empty . )
Write the first entry LaTeXMLMath as LaTeXMLMath .
Take LaTeXMLMath , and LaTeXMLMath to produce LaTeXMLMath .
Clearly LaTeXMLMath .
Example : LaTeXMLEquation .
The next principle gives an obstructed element in LaTeXMLMath .
The Nielsen class LaTeXMLMath has one element .
For LaTeXMLMath in this class , LaTeXMLMath .
Write LaTeXMLMath with LaTeXMLMath and LaTeXMLMath a conjugate of LaTeXMLMath not in LaTeXMLMath .
Choose LaTeXMLMath in one of the orbits LaTeXMLMath or LaTeXMLMath ( for conjugation by LaTeXMLMath ; § LaTeXMLRef ) in LaTeXMLMath .
If we choose LaTeXMLMath to be the orbit of LaTeXMLMath ( in LaTeXMLMath ) , then conjugate by LaTeXMLMath to see LaTeXMLMath is also in LaTeXMLMath .
The other LaTeXMLMath orbit of 5-cycles , represented by LaTeXMLMath , doesn ’ t give an element in the Nielsen class : The product LaTeXMLMath is a 3-cycle .
Calculate LaTeXMLMath by considering LaTeXMLMath .
Then , LaTeXMLMath .
Use Princ .
LaTeXMLRef , then Princ .
LaTeXMLRef to see LaTeXMLMath equals LaTeXMLEquation .
One last application of Princ .
LaTeXMLRef now shows LaTeXMLMath .
∎ Noting the position of a 3-cycle gives LaTeXMLMath .
A representative for the LaTeXMLMath orbit is LaTeXMLMath .
Put LaTeXMLMath on the left and LaTeXMLMath on the right as in Princ .
LaTeXMLRef .
Conclude its lifting invariant is - 1 .
Let LaTeXMLMath be LaTeXMLMath where LaTeXMLMath is a LaTeXMLMath -group acting irreducibly ( and nontrivially ) through LaTeXMLMath .
This presents LaTeXMLMath as a primitive affine group where LaTeXMLMath are the letters of the permutation representation .
As in Ex .
LaTeXMLRef , LaTeXMLMath is LaTeXMLMath with LaTeXMLMath the pro-free pro- LaTeXMLMath group on LaTeXMLMath generators .
( Action of LaTeXMLMath extends to LaTeXMLMath LaTeXMLMath — LaTeXMLMath Remark LaTeXMLRef LaTeXMLMath — LaTeXMLMath uniquely up to conjugacy .
Still , it will rarely be easy to find . )
Let LaTeXMLMath be the LaTeXMLMath th characteristic quotient of LaTeXMLMath , LaTeXMLMath .
Assume given LaTeXMLMath conjugacy classes C in LaTeXMLMath .
The next lemma shows how to find lifts of an element LaTeXMLMath to LaTeXMLMath , ensuring LaTeXMLMath is nonempty .
As above , let LaTeXMLMath have LaTeXMLMath with LaTeXMLMath , LaTeXMLMath .
Then , LaTeXMLMath if and only if there is no LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
So , there is a unique solution for LaTeXMLMath given any elements LaTeXMLMath .
Condition ( LaTeXMLRef LaTeXMLRef ) is the product one condition for LaTeXMLMath .
Since LaTeXMLMath is an irreducible LaTeXMLMath module , given LaTeXMLMath , there is a unique solution for LaTeXMLMath .
From the same argument , LaTeXMLMath if and only if the projection of LaTeXMLMath onto LaTeXMLMath has no kernel ; otherwise the kernel is all of LaTeXMLMath since it is a nontrivial LaTeXMLMath invariant submodule of LaTeXMLMath .
Conclude , LaTeXMLMath is a splitting of LaTeXMLMath into LaTeXMLMath .
From Schur-Zassenhaus , it is conjugate to the canonical copy of LaTeXMLMath in LaTeXMLMath .
Condition ( LaTeXMLRef LaTeXMLRef ) is exactly the computation for that .
∎ Levels 0 and 1 of the LaTeXMLMath Modular Tower ( Ex .
LaTeXMLRef ) show the LaTeXMLMath -split situation , for LaTeXMLMath , is a serious challenge for our Main Conjecture .
These general comments on the universal LaTeXMLMath -Frattini cover appear here for reference in LaTeXMLCite .
Let LaTeXMLMath be the maximal LaTeXMLMath normal subgroup of any finite group LaTeXMLMath .
Let LaTeXMLMath be any LaTeXMLMath subgroup of LaTeXMLMath and let LaTeXMLMath be restriction of LaTeXMLMath on LaTeXMLMath to LaTeXMLMath .
Then , this action extends to an action of LaTeXMLMath on LaTeXMLMath , unique up to conjugation inside LaTeXMLMath .
Such an LaTeXMLMath acts trivially on LaTeXMLMath if and only if it acts trivially on LaTeXMLMath .
If LaTeXMLMath , then this action is trivial .
The extending action of LaTeXMLMath to LaTeXMLMath is a special case of Rem .
LaTeXMLRef applied to LaTeXMLMath .
The extending group LaTeXMLMath is the universal LaTeXMLMath -Frattini of LaTeXMLMath .
If the action on LaTeXMLMath is trivial , then the trivial extension of LaTeXMLMath to LaTeXMLMath is an extending action .
Consider the case LaTeXMLMath .
We have the natural short exact sequence LaTeXMLMath .
Form the ( group ) fiber product LaTeXMLEquation .
By the basic property of the LaTeXMLMath -Frattini cover , LaTeXMLMath is the minimal LaTeXMLMath -projective cover of LaTeXMLMath .
Since LaTeXMLMath has LaTeXMLMath kernel , LaTeXMLMath is the minimal LaTeXMLMath -projective cover of LaTeXMLMath .
So , by Prop .
LaTeXMLRef , it is LaTeXMLMath .
∎ Warning !
Each LaTeXMLMath subgroup LaTeXMLMath of LaTeXMLMath does have its action extend to LaTeXMLMath , and that extending action is unique up conjugacy .
Assuming a group is LaTeXMLMath -perfect then means we can extend the action of generators on LaTeXMLMath .
Those actions , however , won ’ t fit together to have LaTeXMLMath act on LaTeXMLMath ( as in § LaTeXMLRef ) .
In Prop .
LaTeXMLRef we found many explicit appearances of LaTeXMLMath in LaTeXMLMath in the LaTeXMLMath -split case , excluding the case that LaTeXMLMath is cyclic ( with nontrivial LaTeXMLMath ) action .
The following result , basically from LaTeXMLCite , characterizes when LaTeXMLMath is faithful on LaTeXMLMath .
It is an ingredient in Thm .
LaTeXMLRef , giving an asymptotic formula for the multiplicity of appearance of any simple LaTeXMLMath module in LaTeXMLMath for large LaTeXMLMath .
Suppose the dimension of LaTeXMLMath exceeds 1 .
Then , LaTeXMLMath acts faithfully on LaTeXMLMath , for all LaTeXMLMath .
Once we know the rank of LaTeXMLMath exceeds 1 , then so does the rank of LaTeXMLMath on LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath is a LaTeXMLMath normal subgroup , there is a splitting of the pullback of LaTeXMLMath in LaTeXMLMath .
That identifies LaTeXMLMath as the maximal normal LaTeXMLMath subgroup of LaTeXMLMath for each LaTeXMLMath .
Given the Griess-Schmid result for LaTeXMLMath therefore gives it for all LaTeXMLMath .
∎ With the previous notation , assume LaTeXMLMath appears in the Loewy display of LaTeXMLMath .
Princ .
LaTeXMLRef shows LaTeXMLMath appears ( even where ) in the Loewy display for infinitely many integers LaTeXMLMath .
Thm .
LaTeXMLRef is in LaTeXMLCite .
As a special case it shows that LaTeXMLMath appears with an explicit positive density in LaTeXMLMath for LaTeXMLMath large .
The result is effective , though for small values of LaTeXMLMath it is subtle to predict the appearance of LaTeXMLMath .
Further , it is imprecise on the place in the Loewy display of the appearance of the LaTeXMLMath s. So , Princ .
LaTeXMLRef and the very effective LaTeXMLMath -split case remain valuable .
Recall : Over an algebraically closed field the set of simple LaTeXMLMath modules has the same cardinality as the set of LaTeXMLMath conjugacy classes .
Let LaTeXMLMath be any simple LaTeXMLMath module .
Let LaTeXMLMath be an algebraically closed field and retain the notation LaTeXMLMath after tensoring with LaTeXMLMath .
We use LaTeXMLMath , and related compatible notation , for the total multiplicity of LaTeXMLMath in all Loewy layers of the LaTeXMLMath module LaTeXMLMath .
The condition LaTeXMLMath is equivalent to LaTeXMLEquation .
Let LaTeXMLMath be the maximal normal LaTeXMLMath -subgroup of LaTeXMLMath .
Recall that a group LaTeXMLMath is LaTeXMLMath -supersolvable if LaTeXMLMath is abelian of exponent dividing LaTeXMLMath .
LaTeXMLCite has the following characterization of LaTeXMLMath : G is a LaTeXMLMath -supersolvable group with a cyclic LaTeXMLMath -Sylow .
Specifically , if LaTeXMLMath is simple , then LaTeXMLMath .
Investigations of modular curves often analyze behavior of functions near cusps .
Degeneration behavior of the curve covers associated to points on Hurwitz spaces , as the points approach the cusps , hints at diophantine properties .
This section illustrates with detailed analysis of the easiest case : Degeneration behavior of real points on Hurwitz and reduced Hurwitz spaces .
So doing , it points to special cusps named for this degeneration .
The word cusp implies some compactification of the Hurwitz space .
Therefore , after generalities on Hurwitz spaces for any LaTeXMLMath , we concentrate on cusp behavior of reduced Hurwitz spaces for the case LaTeXMLMath .
Then the spaces are curves , and the compactification behavior stays within the confines of this paper .
In Modular Tower higher levels for LaTeXMLMath , real points on Hurwitz spaces at those levels associate to branch cycles we call H-M and near H-M reps. Cusps at the end of the corresponding components of real points then inherit the H-M and near H-M moniker .
In Thm .
LaTeXMLRef , LaTeXMLMath is level LaTeXMLMath of a Modular Tower of inner Hurwitz spaces .
In assuming LaTeXMLMath is centerless and LaTeXMLMath -perfect ( Def .
LaTeXMLRef ) , these are fine moduli spaces ( Prop .
LaTeXMLRef ) .
As a diophantine corollary , over any field LaTeXMLMath with LaTeXMLMath ( see LaTeXMLCite or LaTeXMLCite ) , LaTeXMLMath if and only if some cover ( in the inner equivalence class ) associated to LaTeXMLMath has definition field LaTeXMLMath .
A Harbater-Mumford ( H-M ) representative of Nielsen class LaTeXMLMath is an LaTeXMLMath -tuple LaTeXMLMath with the following property : LaTeXMLMath , LaTeXMLMath , with LaTeXMLMath .
Call a component of LaTeXMLMath an H-M component if it corresponds to an LaTeXMLMath orbit of an H-M rep ( resentative ) .
The definition of a near H-M representative applies to any Modular Tower at level LaTeXMLMath if LaTeXMLMath , and LaTeXMLMath is even .
It uses an operator LaTeXMLMath on Nielsen classes , the special case LaTeXMLMath for the operator LaTeXMLMath in ( LaTeXMLRef ) .
Define the effect of LaTeXMLMath on LaTeXMLMath by consecutively listing entries of LaTeXMLMath .
List entries of LaTeXMLMath as LaTeXMLMath .
As in ( LaTeXMLRef ) , denote LaTeXMLMath by LaTeXMLMath .
LaTeXMLEquation .
Then , LaTeXMLMath , LaTeXMLMath , is a near H-M rep. if the following hold .
LaTeXMLMath isn ’ t an H-M representative .
There exists an involution LaTeXMLMath satisfying LaTeXMLMath .
LaTeXMLMath is an H-M representative .
We use § LaTeXMLRef for direct calculation of real points on reduced Hurwitz spaces .
Reference to LaTeXMLMath means conjugacy classes varying with LaTeXMLMath , not necessarily consisting of LaTeXMLMath classes .
For any prime LaTeXMLMath , let LaTeXMLMath denote the field with all LaTeXMLMath th roots of 1 , LaTeXMLMath , adjoined .
If LaTeXMLMath , denote the complement of LaTeXMLMath pulled back to LaTeXMLMath by LaTeXMLMath .
Assume LaTeXMLMath is centerless and LaTeXMLMath -perfect .
Fix LaTeXMLMath and a subfield LaTeXMLMath .
Suppose there are LaTeXMLMath realizations ( over LaTeXMLMath ) with LaTeXMLMath conjugacy classes in LaTeXMLMath as the entries of LaTeXMLMath , for each LaTeXMLMath .
Assume LaTeXMLMath .
Then , the only possibility for such LaTeXMLMath realizations is there exists LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath classes C with LaTeXMLMath realizations ( over LaTeXMLMath ) for all LaTeXMLMath .
Assume there is a prime LaTeXMLMath , LaTeXMLMath , satisfying the following .
At some place LaTeXMLMath of LaTeXMLMath over LaTeXMLMath the residue class field LaTeXMLMath is finite .
Then , reduction of LaTeXMLMath at LaTeXMLMath has no LaTeXMLMath points for LaTeXMLMath large .
Suppose there is a proper algebraic subset LaTeXMLMath with this property .
LaTeXMLMath is finite for each LaTeXMLMath .
Then , for some LaTeXMLMath , LaTeXMLMath is empty for LaTeXMLMath .
Assume LaTeXMLMath and all components of LaTeXMLMath have genus at least 2 for some LaTeXMLMath .
Then , there is LaTeXMLMath with no LaTeXMLMath realizations over LaTeXMLMath for LaTeXMLMath .
Assume LaTeXMLMath and LaTeXMLMath .
Then all LaTeXMLMath realizations over LaTeXMLMath for LaTeXMLMath appear in components of LaTeXMLMath for LaTeXMLMath orbits containing H-M or near H-M reps. ( This holds even replacing C by LaTeXMLMath varying with LaTeXMLMath . )
A LaTeXMLMath corresponds to a LaTeXMLMath realization over LaTeXMLMath .
A connected component of LaTeXMLMath associated to an H-M rep. has a connected component of LaTeXMLMath above it .
A connected component of LaTeXMLMath for a near H-M rep. has nothing in LaTeXMLMath above it .
Explicit formulas ( as in ( LaTeXMLRef ) , Lem .
LaTeXMLRef and § LaTeXMLRef ) give us confidence in deciding LaTeXMLCite when LaTeXMLMath .
If the LaTeXMLCite conjecture is correct , that LaTeXMLMath has no rational points for LaTeXMLMath large , then LaTeXMLMath realizations of LaTeXMLMath require increasingly large sets of conjugacy classes as LaTeXMLMath grows .
This is subtler than information from the Branch Cycle Lemma .
§ LaTeXMLRef succinctly interprets points on a Modular Tower as a statement on quotients of a fundamental group .
Thm .
LaTeXMLRef says if LaTeXMLMath , real points on a Modular Tower ( versus points at various levels ) appear only on projective sequences of H-M components .
In § LaTeXMLRef we see this gives a tool for progress on the Main Conjecture .
The proof of Thm .
LaTeXMLRef appears in two subsections .
§ LaTeXMLRef gives the argument bounding points at high levels on a Modular Tower of inner Hurwitz spaces .
§ LaTeXMLRef gives details on LaTeXMLMath points .
Note : Hurwitz spaces LaTeXMLMath are affine varieties .
The theorem says nothing about rational points on ( any ) closure of LaTeXMLMath including the boundary .
LaTeXMLCite gives the first statement of Thm .
LaTeXMLRef .
Bounding LaTeXMLMath with LaTeXMLMath having LaTeXMLMath realizations and LaTeXMLMath not consisting of LaTeXMLMath classes requires only the Branch Cycle Lemma .
This is effective , though dependent on data about LaTeXMLMath for a result referencing only LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Consider any suitable Modular Tower for inner equivalence where LaTeXMLMath is LaTeXMLMath -perfect and centerless .
Then , the inner Hurwitz spaces are fine moduli spaces ( Prop .
LaTeXMLRef ) .
Also , they and their reduced versions have good reduction modulo any prime LaTeXMLMath , if LaTeXMLMath ( LaTeXMLCite or with more details LaTeXMLCite ) .
Use LaTeXMLEquation for the sequence of reduced spaces .
Assume each LaTeXMLMath has a LaTeXMLMath point .
Let LaTeXMLMath be a prime of LaTeXMLMath over LaTeXMLMath with residue class field LaTeXMLMath .
Suppose , contrary to the conclusion of the theorem , LaTeXMLMath has an LaTeXMLMath rational point for each integer LaTeXMLMath .
The set LaTeXMLMath is finite for each LaTeXMLMath .
So , these finite sets form a nonempty projective system .
Conclude : There is a projective sequence of points LaTeXMLMath on them .
Let LaTeXMLMath be a finite extension of LaTeXMLMath over which the point LaTeXMLMath produces an inner cover LaTeXMLMath in the Nielsen class corresponding to LaTeXMLMath on the level 0 Hurwitz space lying over LaTeXMLMath .
Apply Princ .
LaTeXMLRef when LaTeXMLMath is a finite field .
That translates to a projective system of LaTeXMLMath regular realizations with definition field LaTeXMLMath .
With no loss , take this finite extension to be LaTeXMLMath .
Let LaTeXMLMath be the Tate module for LaTeXMLMath formed by taking an abelian quotient of the Modular Tower as in Prop .
LaTeXMLRef .
Suppose the kernel of the universal LaTeXMLMath -Frattini cover has rank LaTeXMLMath .
For a general Modular Tower this implies the Frobenius for LaTeXMLMath acts trivially on a rank LaTeXMLMath quotient of LaTeXMLMath .
Contradiction : The Frobenius has eigenvalues of absolute value LaTeXMLMath .
This contradiction shows there can not be LaTeXMLMath points on the reduced space LaTeXMLMath for all LaTeXMLMath .
Now assume ( LaTeXMLRef ) holds .
There are finitely many points on LaTeXMLMath .
So , the argument above shows that either LaTeXMLMath is empty for some value of LaTeXMLMath or there is a projective system of LaTeXMLMath points .
Taking a suitably large prime LaTeXMLMath allows reducing the covers for the projective system modulo LaTeXMLMath , and the proceeding argument completes the proof of this part of the theorem .
The argument of Thm .
LaTeXMLRef does not hold for a Modular Tower of absolute Hurwitz spaces .
Modular curves give an example of this .
Consider a fixed prime LaTeXMLMath , and all of the curves LaTeXMLMath ( § LaTeXMLRef ) as in a Modular Tower .
Let LaTeXMLMath or 2 be a prime , and reduce modulo LaTeXMLMath .
The following facts are in LaTeXMLCite .
There are many supersingular elliptic curves in characteristic LaTeXMLMath , roughly LaTeXMLMath of them .
Further , all have field of definition LaTeXMLMath .
Characterize a supersingular curve ( over LaTeXMLMath ) by its having no LaTeXMLMath division points .
So , any curve isogenous to it , say by a cyclic degree LaTeXMLMath isogeny , is also supersingular .
Since both curves have definition field LaTeXMLMath , the isogeny has definition field LaTeXMLMath ( though it is not a Galois cover over LaTeXMLMath ) .
This gives a large number of points on LaTeXMLMath , as many as can be expected for LaTeXMLMath large .
This is Ihara ’ s first example for producing many points on high genus curves over a finite field of square order .
Ihara uses Shimura curves to do the same over finite fields of order LaTeXMLMath for LaTeXMLMath .
( LaTeXMLCite has an exposition on this result ’ s scattered literature . )
The Shimura curves he uses are compact families of abelian varieties , unlike levels of a Modular Tower which have cusps .
We apply LaTeXMLCite using the pattern of LaTeXMLCite .
This explicitly described real points on any of the fine moduli Hurwitz spaces LaTeXMLMath appearing in this paper .
Let LaTeXMLMath be a cover over LaTeXMLMath , branched over LaTeXMLMath , with LaTeXMLMath .
Specific classical generators of LaTeXMLMath produce an explicit uniformization of any real points on LaTeXMLMath over LaTeXMLMath .
This uses a complex conjugation operator LaTeXMLMath from ( LaTeXMLRef ) on Nielsen classes .
It includes giving a combinatorial restatement for LaTeXMLMath having definition field LaTeXMLMath .
So , fixed points of this complex conjugation operator produce points in LaTeXMLMath .
With a fine moduli assumption these are exactly the points of LaTeXMLMath .
The process works efficiently for unramified Frattini extensions of the covers LaTeXMLMath appearing in an inner family ( using the language of § LaTeXMLRef ) .
So , this gives a useful description of real points on a Modular Tower .
The approach falls into cases from how the support of LaTeXMLMath behaves under complex conjugation .
As LaTeXMLMath has definition field LaTeXMLMath , it has LaTeXMLMath complex conjugate pairs and LaTeXMLMath real points .
To simplify , call the latter LaTeXMLMath , arranged left to right on the real line and the former , LaTeXMLMath .
With no loss ( see § LaTeXMLRef ) assume LaTeXMLMath appear in that order on the real line ( circle on the Riemann sphere ) .
Then , LaTeXMLCite produces paths based at LaTeXMLMath , LaTeXMLEquation with explicit complex conjugation action LaTeXMLMath on these paths and on points over LaTeXMLMath .
Represent LaTeXMLMath as LaTeXMLMath , by its permutation effect on points over LaTeXMLMath .
With these conditions this description is uniform in LaTeXMLMath .
Under LaTeXMLMath the paths LaTeXMLMath go to a new set of paths , LaTeXMLMath , LaTeXMLMath .
From LaTeXMLCite : If LaTeXMLMath and LaTeXMLMath are one of the LaTeXMLMath pairs of paths , then LaTeXMLMath sends LaTeXMLMath ( resp .
LaTeXMLMath ) to a conjugate of LaTeXMLMath ( resp .
LaTeXMLMath ) .
Branch cycles for the complex conjugate cover LaTeXMLMath relative to the LaTeXMLMath s are also LaTeXMLMath .
A formula ( dependent on the particular paths ) computes branch cycles for LaTeXMLMath relative to the LaTeXMLMath s : Call this LaTeXMLMath ( as in ( LaTeXMLRef ) ) .
If LaTeXMLMath is s-equivalent to LaTeXMLMath ( Def .
LaTeXMLRef ) , then LaTeXMLMath conjugates the last branch cycle description to LaTeXMLMath .
Assume LaTeXMLMath as in § LaTeXMLRef .
Prop .
LaTeXMLRef produces for each LaTeXMLMath a test for a connected component of LaTeXMLMath .
Let LaTeXMLMath be a cover with branch cycle description LaTeXMLMath ( relative to the chosen classical generators for LaTeXMLMath ) .
The test succeeds if some component has a point corresponding to LaTeXMLMath .
The basic formula ( LaTeXMLRef ) is from LaTeXMLCite .
Suppose LaTeXMLMath .
Then , LaTeXMLMath represents LaTeXMLMath over LaTeXMLMath ( using the paths ( LaTeXMLRef ) ) .
Denote LaTeXMLMath by LaTeXMLMath and LaTeXMLMath by LaTeXMLMath .
A connected component of LaTeXMLMath goes through LaTeXMLMath if for some LaTeXMLMath , LaTeXMLMath with LaTeXMLMath as follows .
For such a LaTeXMLMath , LaTeXMLMath centralizes LaTeXMLMath .
LaTeXMLEquation .
If a cover over LaTeXMLMath represents LaTeXMLMath ( using the paths ( LaTeXMLRef ) ) , then some involution LaTeXMLMath satisfies LaTeXMLMath .
If LaTeXMLMath is a fine moduli space , such an involution LaTeXMLMath exists if and only if a connected component of LaTeXMLMath goes through LaTeXMLMath .
The operator LaTeXMLMath acts as an involution : LaTeXMLMath and LaTeXMLMath commutes with conjugation by LaTeXMLMath .
So LaTeXMLMath acts trivially on LaTeXMLMath .
The condition for LaTeXMLMath to have fine moduli is that the centralizer of LaTeXMLMath in LaTeXMLMath is trivial ( Thm .
LaTeXMLRef ) .
∎ Prop .
LaTeXMLRef includes an archetype example for an inner Hurwitz family with group LaTeXMLMath having a nontrivial center : It is a Frattini central extension of a centerless group LaTeXMLMath .
If LaTeXMLMath is branch cycles for a LaTeXMLMath cover , we can usually decide if LaTeXMLMath is an involution .
So , we can decide if there is a cover over LaTeXMLMath realizing LaTeXMLMath .
Suppose LaTeXMLMath lies over LaTeXMLMath and LaTeXMLMath is a real base point relative to which ( LaTeXMLRef ) is a branch cycle description of LaTeXMLMath .
Assume : Complex conjugation LaTeXMLMath is trivial , and LaTeXMLMath in ( LaTeXMLRef ) .
Then , the branch cycle description for LaTeXMLMath is an H-M representative .
This is in the LaTeXMLMath orbit on LaTeXMLMath corresponding to the component of LaTeXMLMath containing LaTeXMLMath .
To complete the proof of Thm .
LaTeXMLRef requires showing two things : ( LaTeXMLRef ) is equivalent to having an H-M representative .
If LaTeXMLMath and LaTeXMLMath with LaTeXMLMath , then LaTeXMLMath is a near H-M representative .
First assume LaTeXMLMath .
Suppose LaTeXMLMath is the identity .
Then ( LaTeXMLRef ) says LaTeXMLMath , or LaTeXMLMath has order 2 .
This can ’ t hold if this conjugacy class is LaTeXMLMath .
It also can ’ t hold if LaTeXMLMath .
The reason is this : The hypotheses remain the same for the reduction of LaTeXMLMath modulo LaTeXMLMath .
Apply Lem .
LaTeXMLRef : Any lift to LaTeXMLMath of an element of order divisible by 2 in LaTeXMLMath increases its order .
So , LaTeXMLMath can ’ t have order 2 .
Let LaTeXMLMath be complex conjugation at level 0 ( reduction modulo LaTeXMLMath ) .
The same applies to it .
So , LaTeXMLMath has order two ( not 1 ) if LaTeXMLMath .
For inner classes , LaTeXMLMath .
Apply Lem .
LaTeXMLRef again so LaTeXMLMath has order exceeding that of LaTeXMLMath .
This contradicts that LaTeXMLMath , a complex conjugation operator , has order 2 and is a lift of LaTeXMLMath .
Now we know LaTeXMLMath .
Further , this shows LaTeXMLMath is the identity .
This argument applies in going from level LaTeXMLMath to level LaTeXMLMath , LaTeXMLMath .
Two possibilities happen .
Assume LaTeXMLMath lies below LaTeXMLMath , where LaTeXMLMath defines a connected component of real points on LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) be the complex conjugation operator associated with LaTeXMLMath ( resp .
LaTeXMLMath ) .
Then : Either LaTeXMLMath is the identity and LaTeXMLMath is an H-M representative ; or LaTeXMLMath is the identity , LaTeXMLMath is an H-M rep. , LaTeXMLMath is not the identity , and nothing in LaTeXMLMath is over the LaTeXMLMath component corresponding to LaTeXMLMath in LaTeXMLMath .
Near H-M representatives ( satisfying ( LaTeXMLRef LaTeXMLRef ) ) occur at level LaTeXMLMath of the Modular Tower for LaTeXMLMath ( § LaTeXMLRef , especially Prop .
LaTeXMLRef ) .
§ LaTeXMLRef gives an elementary lemma about LaTeXMLMath values of cover points over LaTeXMLMath on a Hurwitz space .
Then , § LaTeXMLRef shows how real components over critical intervals fit together .
We note the LaTeXMLMath invariant separates four branch point covers over LaTeXMLMath according to a configuration of their branch points .
Suppose LaTeXMLMath is a four branch point cover over LaTeXMLMath with either 0 or 4 real branch points .
Then , the corresponding LaTeXMLMath value under the representative of LaTeXMLMath on a reduced Hurwitz space is in the interval LaTeXMLMath along the real line .
If LaTeXMLMath has , instead , two complex conjugate and two real branch points , then the corresponding LaTeXMLMath value is in the interval LaTeXMLMath .
Recall the cross ratio of distinct points LaTeXMLMath : LaTeXMLMath .
The basics are in LaTeXMLCite .
Four points in complex conjugate pairs ( or on the real line ) lie on a circle and the cross ratio is real .
The cross-ratio is invariant under a transform of the points LaTeXMLMath by LaTeXMLMath .
Since there is an LaTeXMLMath that takes two complex conjugate pairs of points to four points in the reals , with no loss assume LaTeXMLMath has either two or four real points in its support .
For these cases apply LaTeXMLMath to assume LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath .
In the former case LaTeXMLMath runs over the unit circle ( excluding LaTeXMLMath ) and in the latter case over all real numbers ( excluding LaTeXMLMath , 1 and LaTeXMLMath ) .
The LaTeXMLMath value corresponding to LaTeXMLMath is LaTeXMLMath with LaTeXMLCite .
( Classically this is without the LaTeXMLMath .
We chose it so the ramified LaTeXMLMath -values are 0 , 1 , LaTeXMLMath . )
For LaTeXMLMath the connected range of LaTeXMLMath includes large positive values and is bounded away from 0 .
So the range of LaTeXMLMath for real LaTeXMLMath is LaTeXMLMath .
For LaTeXMLMath in the unit circle ( minus 1 ) , the range of LaTeXMLMath includes both sides of 0 .
Also , for LaTeXMLMath close to 1 , the numerator of LaTeXMLMath is positive and bounded , while the denominator is approximately LaTeXMLMath .
Therefore the range is the interval LaTeXMLMath .
∎ Suppose LaTeXMLMath is a reduced Hurwitz space cover defined over LaTeXMLMath .
Lem .
LaTeXMLRef shows the intervals LaTeXMLMath and LaTeXMLMath on LaTeXMLMath lie under real points on LaTeXMLMath ( the original Hurwitz space ) coming from covers over LaTeXMLMath with two different styles of branch points .
The interval LaTeXMLMath goes through LaTeXMLMath , though this is a branch point for the cover LaTeXMLMath .
This is because ramification over 0 has order 3 ( or 1 ) , and a unique 3rd root of 1 ( or -1 ) is real .
The same simple observation gives the next dessins d ’ enfant type lemma .
Denote by LaTeXMLMath ( resp .
LaTeXMLMath ) the real points of LaTeXMLMath over the interval LaTeXMLMath ( resp .
LaTeXMLMath ) of LaTeXMLMath .
The closure of each component LaTeXMLMath of LaTeXMLMath has endpoints LaTeXMLMath over 1 and LaTeXMLMath over LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) be the ramification order of LaTeXMLMath ( resp .
LaTeXMLMath ) over 1 ( resp .
LaTeXMLMath ) .
Note : LaTeXMLMath or 2 .
The same attachment of endpoints applies to components of LaTeXMLMath .
Suppose for a given LaTeXMLMath in LaTeXMLMath , LaTeXMLMath ( resp .
LaTeXMLMath ) is odd .
Then there is a unique LaTeXMLMath in LaTeXMLMath with LaTeXMLMath ( resp .
LaTeXMLMath ) .
If LaTeXMLMath ( resp .
LaTeXMLMath ) is even , then there is a unique LaTeXMLMath in LaTeXMLMath with LaTeXMLMath ( resp .
LaTeXMLMath ) .
So , no LaTeXMLMath in LaTeXMLMath has LaTeXMLMath ( resp .
LaTeXMLMath ) .
Suppose LaTeXMLMath is a local uniformizing parameter for LaTeXMLMath in a neighborhood of LaTeXMLMath .
The argument for LaTeXMLMath is the same , so we do only the former case .
In local analytic coordinates over LaTeXMLMath , choose LaTeXMLMath so LaTeXMLMath and there is a parametrization of the neighborhood of LaTeXMLMath using power series in LaTeXMLMath with real coefficients .
If LaTeXMLMath is odd , then real points around LaTeXMLMath map one-one to real points around LaTeXMLMath .
If LaTeXMLMath is even , then real points around LaTeXMLMath map two-one to the positive number side of LaTeXMLMath ( no points falling on the negative side of LaTeXMLMath ) .
That interprets the lemma ’ s statement in local coordinates .
∎ Assume LaTeXMLMath .
Prop .
LaTeXMLRef shows , if LaTeXMLMath acts trivially on Nielsen classes , LaTeXMLMath -cover points produce all the points on a reduced Hurwitz space LaTeXMLMath ( with any equivalence on Nielsen classes ) except possibly in the fibers of LaTeXMLMath lying over LaTeXMLMath or LaTeXMLMath .
Our next result extends Prop .
LaTeXMLRef , by combining it with Lem .
LaTeXMLRef , to consider any LaTeXMLMath action .
Assume , as above , LaTeXMLMath ( resp .
LaTeXMLMath or LaTeXMLMath ) and LaTeXMLMath is the operator of Prop .
LaTeXMLRef for complex conjugate pairs of points LaTeXMLMath ( resp .
a complex conjugate point and two real points ) .
Let LaTeXMLMath lie over LaTeXMLMath .
Let LaTeXMLMath be a listing of representatives from the equivalence classes of Nielsen classes .
Then , LaTeXMLMath points of LaTeXMLMath over LaTeXMLMath correspond one-one with reduced equivalence classes LaTeXMLMath that upon containing LaTeXMLMath also contain LaTeXMLMath .
Suppose LaTeXMLMath has fine moduli , and LaTeXMLMath has b-fine ( resp .
fine ) moduli ( Prop .
LaTeXMLRef ) .
Then LaTeXMLMath over LaTeXMLMath ( resp .
LaTeXMLMath ) is either an LaTeXMLMath -cover point , or an LaTeXMLMath -Brauer point .
When LaTeXMLMath is an inner Hurwitz space , and LaTeXMLMath corresponds to LaTeXMLMath , then LaTeXMLMath if and only if LaTeXMLMath corresponds to an H-M rep .
This result applies to the main Modular Tower of this paper .
Each level LaTeXMLMath of the LaTeXMLMath Modular Tower has an absolutely irreducible component LaTeXMLMath with both H-M and near H-M reps. Each point of LaTeXMLMath of either type produces LaTeXMLMath -cover points in the Nielsen class .
Suppose LaTeXMLMath is a branch cycle description of such a cover with respect to LaTeXMLMath having complex conjugate pairs of branch points .
Then , the complement LaTeXMLMath of LaTeXMLMath ( Def .
LaTeXMLRef ) corresponds to an LaTeXMLMath -Brauer point of LaTeXMLMath .
Consider the nontrivial central Frattini extension LaTeXMLMath from Cor .
LaTeXMLRef .
The natural one-one ( not necessarily onto ) map LaTeXMLMath gives a component of LaTeXMLMath isomorphic ( equivalent as covers of LaTeXMLMath ) to LaTeXMLMath .
An LaTeXMLMath -cover point LaTeXMLMath for an H-M ( resp .
near H-M ) rep. in the Nielsen class LaTeXMLMath corresponds to a cover in LaTeXMLMath with minimal field of definition LaTeXMLMath ( resp .
LaTeXMLMath ) .
So , an LaTeXMLMath -cover point for a near H-M rep. ( for LaTeXMLMath ) corresponds to a LaTeXMLMath cover with field of moduli LaTeXMLMath , but minimal definition field LaTeXMLMath .
Prop .
LaTeXMLRef gives the component containing both H-M and near H-M reps .
The statement on LaTeXMLMath is from Prop .
LaTeXMLRef .
Nontriviality of LaTeXMLMath is from the discussion prior to Cor .
LaTeXMLRef , an inductive consequence of Prop .
LaTeXMLRef and that LaTeXMLMath .
The complex conjugation operator for a near H-M rep. in this case gives conjugation by an element whose lift to LaTeXMLMath has order 4 .
From this , Prop .
LaTeXMLRef shows a near H-M LaTeXMLMath ( regarding it in LaTeXMLMath ) has no cover realizing it over LaTeXMLMath .
If , however , LaTeXMLMath corresponds to an H-M rep. , then regarding it as the inner class of an LaTeXMLMath cover , it has a trivial complex conjugation operator .
So , we have LaTeXMLMath with group LaTeXMLMath over LaTeXMLMath and geometrically LaTeXMLMath ( LaTeXMLMath unramified ) with field of moduli LaTeXMLMath , and the group of LaTeXMLMath is LaTeXMLMath .
Apply Lem .
LaTeXMLRef to LaTeXMLMath ; the resulting cover LaTeXMLMath has degree 2 .
It is therefore Galois , and LaTeXMLMath produces the LaTeXMLMath realization over LaTeXMLMath .
∎ Assume LaTeXMLMath has two conjugate pairs of points .
Choose LaTeXMLMath to map LaTeXMLMath to LaTeXMLMath , four points in LaTeXMLMath .
If LaTeXMLMath is a cover over LaTeXMLMath with branch points LaTeXMLMath , then LaTeXMLMath is in the reduced equivalence class of LaTeXMLMath .
It is not , however , a cover over LaTeXMLMath as it stands , because LaTeXMLMath is not a field of definition of LaTeXMLMath .
Sometimes , however , LaTeXMLMath is strong equivalent ( so it has the same branch points LaTeXMLMath ) to a cover over LaTeXMLMath .
An example arising in this paper is the absolute and inner reduced equivalence classes for LaTeXMLMath .
Consider the cover with H-M description LaTeXMLMath and LaTeXMLMath as branch points .
With a base point LaTeXMLMath , the effect of complex conjugation from Prop .
LaTeXMLRef is the identity .
The points on LaTeXMLMath over LaTeXMLMath are all real .
Transform the paths giving LaTeXMLMath above by LaTeXMLMath .
This gives the same branch cycle description of LaTeXMLMath with respect to these new paths .
Since , however , LaTeXMLMath consists of four real points , use the corresponding complex conjugation operator .
As LaTeXMLMath may not be real , this may require adjustment in getting a base point LaTeXMLMath .
Still , with this choice , LaTeXMLMath relative to LaTeXMLMath transformed paths gives complex conjugation on the points above LaTeXMLMath .
Thus , the cover ( either degree 5 or the Galois closure of the degree 5 cover ) is equivalent to one over LaTeXMLMath .
This might happen whenever LaTeXMLMath is not faithful on a Nielsen class , especially one with an H-M rep. , LaTeXMLMath .
Then , LaTeXMLMath and LaTeXMLMath is equivalent to the phenomenon above .
For , however , H-M reps. in LaTeXMLMath , there is no such LaTeXMLMath .
So the corresponding cover with real branch points is not equivalent to a cover over LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath is suitably general ( lying off an explicit Zariski closed subset of LaTeXMLMath , dependent on LaTeXMLMath ) .
Then , there is no nonidentity LaTeXMLMath with LaTeXMLMath .
The argument of § LaTeXMLRef for ( fixed LaTeXMLMath ) , shows the set of LaTeXMLMath , with LaTeXMLMath acting on it as an element of order exceeding 3 , is a Zariski closed subset .
If LaTeXMLMath has order LaTeXMLMath , assume with no loss LaTeXMLMath is LaTeXMLMath .
This reduces the problem to noting there are only two other specific fixed points and no other cycles of order 3 ( § LaTeXMLRef ) .
We show only a proper Zariski closed subset of LaTeXMLMath is fixed by some involution .
With no loss , consider four values LaTeXMLMath transposed in pairs .
This determines LaTeXMLMath , so other elements of LaTeXMLMath , either fixed or transposed in pairs , are special .
Denote this exceptional subset by LaTeXMLMath .
Suppose LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is a Hurwitz space over LaTeXMLMath .
Assume LaTeXMLMath .
Then , points of the fiber LaTeXMLMath go one-one to the points of the fiber LaTeXMLMath , so they have exactly the same fields of definition over LaTeXMLMath .
There is a proper algebraic subset LaTeXMLMath of LaTeXMLMath so that if LaTeXMLMath is a LaTeXMLMath point , then the fiber of LaTeXMLMath over LaTeXMLMath has LaTeXMLMath points .
For LaTeXMLMath , a LaTeXMLMath point on LaTeXMLMath over LaTeXMLMath is the image of a LaTeXMLMath point on LaTeXMLMath for some LaTeXMLMath mapping to LaTeXMLMath .
So , for LaTeXMLMath , some problems for LaTeXMLMath don ’ t appear in comparing LaTeXMLMath and LaTeXMLMath .
Particularly , from Prop .
LaTeXMLRef , if LaTeXMLMath is a fine moduli space , each component of LaTeXMLMath is a b-fine moduli space ; and avoiding LaTeXMLMath over LaTeXMLMath , we can lift LaTeXMLMath points from LaTeXMLMath to LaTeXMLMath .
Still , the points of LaTeXMLMath , including the orbifold points of LaTeXMLMath require a refined analysis to extend the full version of Prop .
LaTeXMLRef when LaTeXMLMath .
§ LaTeXMLRef produces a ramified cover LaTeXMLMath from LaTeXMLMath acting on reduced absolute Nielsen classes .
Closure of this cover over the LaTeXMLMath -line gives LaTeXMLMath with branch cycles LaTeXMLMath LaTeXMLCite : LaTeXMLEquation .
This is a cover with branch points LaTeXMLMath .
We explicitly uniformize the real points on this cover , and on the inner version of this space , including the points over branch points .
From Lem .
LaTeXMLRef , the geometric ( resp .
arithmetic ) monodromy group of the cover is LaTeXMLMath ( resp .
LaTeXMLMath ) .
So , ( LaTeXMLRef ) uniquely determines the effect of complex conjugation .
For a reduced Hurwitz space ( over LaTeXMLMath ) covering LaTeXMLMath , there are three complex conjugation operators LaTeXMLMath , LaTeXMLMath and LaTeXMLMath corresponding to the three intervals over which the cover is unramified .
As in § LaTeXMLRef , LaTeXMLMath , LaTeXMLMath will produce the same number of real points lying over a corresponding LaTeXMLMath value .
Apply ( LaTeXMLRef ) .
With LaTeXMLMath , write the operator as LaTeXMLMath ( suppressing the LaTeXMLMath ) .
Its characterization is LaTeXMLMath , LaTeXMLMath .
As LaTeXMLMath acts trivially on LaTeXMLMath , Prop .
LaTeXMLRef shows all elements of LaTeXMLMath are LaTeXMLMath -cover points .
We see this directly in the next lemma .
All three points of LaTeXMLMath lying over LaTeXMLMath are LaTeXMLMath -cover points .
The same is true for each point over the intervals LaTeXMLMath and LaTeXMLMath .
From the above , LaTeXMLMath conjugates both LaTeXMLMath and LaTeXMLMath to their inverses .
The unique element doing this is LaTeXMLMath .
So there are three real points above LaTeXMLMath .
Check : LaTeXMLMath and LaTeXMLMath .
Now we show cover points account for all LaTeXMLMath points on LaTeXMLMath .
The fixed points of LaTeXMLMath point to three covers in Table LaTeXMLRef labeled LaTeXMLMath .
Each attaches ( by the LaTeXMLMath operator ) to a distinct cusp over LaTeXMLMath , of respective widths 5 , 1 and 3 .
That , however , comes from choosing classical generators of LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath .
It doesn ’ t show what the description of branch cycles will be for a set of paths on LaTeXMLMath supporting a complex conjugation operator of Prop .
LaTeXMLRef .
Actual covers LaTeXMLMath having these real structures come from two types of LaTeXMLMath s using the paths ( LaTeXMLRef ) .
For LaTeXMLMath with real entries : LaTeXMLMath has complex operator LaTeXMLMath ; LaTeXMLMath has LaTeXMLMath and LaTeXMLMath has LaTeXMLMath .
∎ Let LaTeXMLMath be a perfect field with LaTeXMLMath isomorphic to LaTeXMLMath over LaTeXMLMath .
For LaTeXMLMath a projective nonsingular curve , assume LaTeXMLMath is over LaTeXMLMath .
If LaTeXMLMath has genus 0 , then LaTeXMLMath defines an element LaTeXMLMath of order 2 LaTeXMLCite .
A simple formula relates these : LaTeXMLMath with LaTeXMLMath LaTeXMLCite .
Further , for any odd integer LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , there is such a LaTeXMLMath over LaTeXMLMath with LaTeXMLMath of genus 0 LaTeXMLCite .
Given LaTeXMLMath and LaTeXMLMath odd , which Nielsen classes of genus 0 curve covers have LaTeXMLMath over LaTeXMLMath of degree LaTeXMLMath in the Nielsen class ?
See LaTeXMLCite or LaTeXMLCite for background on these comments .
Châtalet knew that LaTeXMLMath a Brauer-Severi variety of dimension LaTeXMLMath defines a central simple algebra over LaTeXMLMath of dimension LaTeXMLMath .
We recognize this as LaTeXMLMath .
If LaTeXMLMath with LaTeXMLMath , then it possesses a LaTeXMLMath point ( from the corestriction-restriction sequence of Galois cohomology LaTeXMLCite ) .
Also , if LaTeXMLMath for each completion LaTeXMLMath of LaTeXMLMath , then Chǎtelet knew LaTeXMLMath .
This gives the easy direction of LaTeXMLCite when their images in the completions of a field determine the Brauer-Severi variety .
According to LaTeXMLCite , Weil LaTeXMLCite gave the modern Galois cohomology interpretation .
Now consider branch cycles for LaTeXMLMath from inner classes LaTeXMLMath ( LaTeXMLCite , LaTeXMLCite ) : LaTeXMLEquation .
For LaTeXMLMath , inspection gives LaTeXMLMath conjugating LaTeXMLMath and LaTeXMLMath to their inverses : LaTeXMLEquation .
The centralizer of the monodromy group ( LaTeXMLMath in Ex .
LaTeXMLRef ) moves 1 .
To show LaTeXMLMath is the correct complex conjugation , it suffices to check the complex conjugation operator for four real points fixes 1 and 10 ( the H-M reps. ) .
Similarly , for LaTeXMLMath , by inspection find LaTeXMLEquation conjugates LaTeXMLMath and LaTeXMLMath to their inverses .
Note : LaTeXMLMath fixes 2 and 11 .
If the complex conjugation operator for LaTeXMLMath fixes either , the set of real points would not be a 1-dimensional manifold in the neighborhood of the index 2 branch point over LaTeXMLMath .
As LaTeXMLMath fixes 2 and 11 , LaTeXMLMath is the correct complex conjugation operator .
Applying LaTeXMLCite gives the following information on points over LaTeXMLMath LaTeXMLMath — LaTeXMLMath cusps LaTeXMLMath — LaTeXMLMath in each cover .
Again , since LaTeXMLMath acts trivially on LaTeXMLMath , Prop .
LaTeXMLRef shows all elements of LaTeXMLMath are LaTeXMLMath - cover points .
Our computations above confirm this directly .
( Ex .
LaTeXMLRef has more on using this example with LaTeXMLMath covers . )
For each LaTeXMLMath ( resp .
LaTeXMLMath ) there are four ( resp .
two ) points of LaTeXMLMath .
There are three LaTeXMLMath points over LaTeXMLMath in the cover LaTeXMLMath of ramification indices 1 , 3 and 5 .
Denote these LaTeXMLMath .
Real points over LaTeXMLMath in the cover LaTeXMLMath correspond to disjoint cycles in LaTeXMLMath that conjugation by LaTeXMLMath maps to their inverses .
These are the disjoint cycles LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
There are three real points over LaTeXMLMath , one over LaTeXMLMath and two over LaTeXMLMath .
This section and § LaTeXMLRef describe level LaTeXMLMath components of the LaTeXMLMath Modular Tower , LaTeXMLMath and its reduced version LaTeXMLMath .
As usual , LaTeXMLMath is four repetitions of the class of 3-cycles in LaTeXMLMath .
For this noncongruence tower our detail is comparable to literature describing some specific modular curves .
The idea is to apply Thm .
LaTeXMLRef to inspect those cusps at the end of real components on LaTeXMLMath .
This information about the cusps eventually separates the two components of LaTeXMLMath according to the cusps lying in each .
When LaTeXMLMath , LaTeXMLMath action produces the phenomenon LaTeXMLMath orbit shortening affecting precise genus calculations .
§ LaTeXMLRef and § LaTeXMLRef develop lemmas anticipating the general case .
Let LaTeXMLMath be a standard fundamental domain for LaTeXMLMath on the upper half plane .
Branch cycle data from Prop .
LaTeXMLRef allows computing a fundamental domain for the subgroup of LaTeXMLMath defining LaTeXMLMath ( with compatible notation for inner and absolute reduced spaces ) .
The element LaTeXMLMath ( resp .
LaTeXMLMath ) identifies with the standard element of order 3 ( resp .
order 2 ) in LaTeXMLMath .
The action of LaTeXMLMath and LaTeXMLMath on LaTeXMLMath ( or LaTeXMLMath depending on the type of Hurwitz space ) gives LaTeXMLMath and LaTeXMLMath , generating a permutation group LaTeXMLMath whose orbits correspond to LaTeXMLMath -line covers .
For further reference , let LaTeXMLMath be one of these orbits .
Take LaTeXMLMath to be the subgroup of LaTeXMLMath stabilizing an element in LaTeXMLMath .
A fundamental domain for a cover for the orbit LaTeXMLMath comes from translating LaTeXMLMath around by coset representatives for LaTeXMLMath in LaTeXMLMath .
We give the geometric and arithmetic monodromy of both LaTeXMLMath -line covers .
Denote LaTeXMLMath by LaTeXMLMath , and LaTeXMLMath by LaTeXMLMath .
Consider any sequence of ( separable ) absolutely irreducible covers LaTeXMLMath over a field LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath ) be the group of the galois closure of LaTeXMLMath ( resp .
LaTeXMLMath ) over LaTeXMLMath .
Then , the group LaTeXMLMath of the Galois closure ( over LaTeXMLMath ) of the cover LaTeXMLMath is a subgroup of the wreath product LaTeXMLMath .
Further , it has the following properties LaTeXMLCite .
The projection of LaTeXMLMath to LaTeXMLMath is surjective .
The kernel of LaTeXMLMath contains a group isomorphic to LaTeXMLMath .
The geometric monodromy LaTeXMLMath of LaTeXMLMath is LaTeXMLMath .
Let LaTeXMLMath .
The arithmetic monodromy group LaTeXMLMath is LaTeXMLMath .
The two groups fit in a natural short exact sequence LaTeXMLEquation .
The geometric monodromy group LaTeXMLMath of LaTeXMLMath is LaTeXMLMath with LaTeXMLMath acting as permutation of the coordinates of LaTeXMLMath .
The arithmetic monodromy group LaTeXMLMath maps to LaTeXMLMath producing the extension LaTeXMLEquation .
Use LaTeXMLMath in ( LaTeXMLRef ) to see that LaTeXMLMath contains a 5-cycle , LaTeXMLMath , so it must primitive .
It also contains a 3- cycle LaTeXMLMath .
A well-known argument says a primitive subgroup of LaTeXMLMath containing a 3-cycle is LaTeXMLMath : LaTeXMLMath .
Apply LaTeXMLCite .
The gist : As the branch points of the cover are rational , the arithmetic monodromy group contains the character field of the conjugacy class of any element of form ( 1 ) ( 3 ) ( 5 ) .
Elements that are products of distinct odd disjoint cycle lengths , form two conjugacy classes in LaTeXMLMath .
The outer automorphism of LaTeXMLMath ( conjugation by LaTeXMLMath ) permutes these two conjugacy classes .
For LaTeXMLMath , the degree 18 cover breaks into a chain of degree 9 and degree 2 covers .
The largest possible group for the geometric closure is LaTeXMLMath and for the arithmetic closure it is LaTeXMLMath .
To complete the proof only requires showing the kernel of the geometric closure to LaTeXMLMath contains one of the factors of LaTeXMLMath .
Since LaTeXMLMath generates such a factor , we are done .
∎ To illustrate , consider ( LaTeXMLRef ) for LaTeXMLMath ( or ( LaTeXMLRef ) for LaTeXMLMath ) .
The representation has degree 9 ( or 18 ) .
An explicit orbit comes as follows .
Label 9 words LaTeXMLMath , LaTeXMLMath , in the elements LaTeXMLEquation so these words applied to 1 give the complete set LaTeXMLMath .
That is , they are coset representatives .
Example : LaTeXMLMath the trivial word , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Then , LaTeXMLEquation .
Finally , apply these , with LaTeXMLMath and LaTeXMLMath replacing LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
Take the union of the images as a fundamental domain for the LaTeXMLMath -line cover .
Every group with two generators , LaTeXMLMath , with respective orders 3 and 2 , is a quotient of LaTeXMLMath .
LaTeXMLCite calls these ( 2,3 ) generated groups .
Finding which such finite simple groups do so occur has been a problem with much literature for over 100 years .
The groups LaTeXMLMath have this property for LaTeXMLMath , a result of Miller from 1901 , and all simple ( 2,3 ) generated groups are known LaTeXMLCite .
The group LaTeXMLMath does not so occur , though LaTeXMLMath ( a simple group ) does give a primitive subgroup of LaTeXMLMath that is LaTeXMLMath generated .
Though to us it is icing on the cake that the main example of this paper starts at the border of this old area , we consider it more evidence that there is a cake .
Let LaTeXMLMath .
Recall : Cusps are equivalence classes for LaTeXMLMath on LaTeXMLMath for the LaTeXMLMath action .
These equivalence classes correspond to orbits of LaTeXMLMath .
For example , in ( LaTeXMLRef ) , LaTeXMLMath .
So , there are three cusps , with widths ( lengths ) 1 , 3 and 5 .
By contrast , LaTeXMLMath , with LaTeXMLMath a prime , has two cusps of widths 1 and LaTeXMLMath .
With no loss , LaTeXMLMath corresponds to the cycle of length 1 in LaTeXMLMath .
( Let the image of LaTeXMLMath corresponding to the integer 2 to have LaTeXMLMath as its cusp . )
Then , there exist elements LaTeXMLMath as follows : LaTeXMLEquation .
To get explicit LaTeXMLMath and LaTeXMLMath from this calculation choose an expression in the LaTeXMLMath s taking an integer in the respective 3 and 5 cycles to the integer 2 .
The following summarizes this explicitness result .
Start with any Nielsen class having LaTeXMLMath .
Prop .
LaTeXMLRef produces a fundamental domain ( in the upper half plane ) and basic data on the cusp points of the cover LaTeXMLMath coming from that Nielsen class .
The main result of § LaTeXMLRef shows LaTeXMLMath has two absolutely irreducible ( LaTeXMLMath -curve ) components , of respective genuses 12 and 9 .
The former component corresponds to a braid orbit ( LaTeXMLMath below ) on LaTeXMLMath containing all H-M and near H-M representatives ( § LaTeXMLRef ) .
So , Thm .
LaTeXMLRef implies only the genus 12 component contains LaTeXMLMath points .
Branch cycles for these components come from applying the program LaTeXMLCite to compute LaTeXMLMath on LaTeXMLMath .
As in Prop .
LaTeXMLRef , this gives LaTeXMLMath for a cover from each LaTeXMLMath orbit .
Especially important are the lengths of LaTeXMLMath orbits .
§ LaTeXMLRef isolates an expected general structure of fundamental domains for higher levels of a Modular Tower .
It is that the monodromy groups of the higher reduced levels over level 0 should be near LaTeXMLMath -groups ( 2-groups in our example ) .
Denote the image of LaTeXMLMath in LaTeXMLMath by LaTeXMLMath .
Refined calculations differentiate between LaTeXMLMath orbits on inner Nielsen classes LaTeXMLMath and LaTeXMLMath orbits on LaTeXMLMath .
The latter attach to cusps on LaTeXMLMath -line covers .
We often denote LaTeXMLMath orbits on LaTeXMLMath by notation like LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is the middle product ( § LaTeXMLRef ) of LaTeXMLMath .
Also , LaTeXMLMath is the width of LaTeXMLMath .
The decoration LaTeXMLMath distinguishes orbits with specific LaTeXMLMath .
If it is cumbersome , or we don ’ t know LaTeXMLMath , a briefer notation suffices .
Given LaTeXMLMath , refer to its orbit type as LaTeXMLMath .
When the context is clear , use LaTeXMLMath as the type of a LaTeXMLMath orbit on LaTeXMLMath .
The LaTeXMLMath orbit type of LaTeXMLMath may be different from its LaTeXMLMath orbit type if LaTeXMLMath orbit shortening reduces the value of LaTeXMLMath ( § LaTeXMLRef ) .
Consider LaTeXMLMath lifting LaTeXMLMath with LaTeXMLMath or 5 .
According to Lem .
LaTeXMLRef , LaTeXMLMath has a complementary element LaTeXMLMath .
This is the unique element in the LaTeXMLMath orbit of LaTeXMLMath , distinct from LaTeXMLMath , that maps to LaTeXMLMath .
Denote this by LaTeXMLMath .
Prop .
LaTeXMLRef generalizes the case LaTeXMLMath to define LaTeXMLMath as LaTeXMLMath for LaTeXMLMath , LaTeXMLMath .
§ LaTeXMLRef generalizes further .
Thm .
LaTeXMLRef says LaTeXMLMath acts on an inner Nielsen class through a Klein 4-group , LaTeXMLMath .
As in LaTeXMLCite or LaTeXMLCite , the Nielsen class LaTeXMLMath contains exactly 18 elements .
Just two are H-M representatives .
Branch cycles for these are LaTeXMLEquation .
These two are equivalent in the absolute Nielsen classes by LaTeXMLMath conjugation , an outer automorphism of LaTeXMLMath .
Further , LaTeXMLMath is transitive on LaTeXMLMath .
So , the number of lifts of any LaTeXMLMath to LaTeXMLMath is independent of the choice of LaTeXMLMath .
We first show faithful action of LaTeXMLMath on LaTeXMLMath .
As usual , LaTeXMLMath are the characteristic quotients of LaTeXMLMath , with C a collection of LaTeXMLMath conjugacy classes of LaTeXMLMath .
Write LaTeXMLMath when we regard it as a subgroup of LaTeXMLMath .
So , LaTeXMLMath acts on LaTeXMLMath ( no equivalence by conjugation by LaTeXMLMath ) .
The notation LaTeXMLMath means this Klein 4-group is acting on inner classes .
Use LaTeXMLMath for the stabilizer in LaTeXMLMath of LaTeXMLMath ( for absolute equivalence , mod out further by LaTeXMLMath ) .
For any group LaTeXMLMath , express the orbit of LaTeXMLMath under LaTeXMLMath as LaTeXMLMath .
Let LaTeXMLMath be a Nielsen class for LaTeXMLMath with LaTeXMLMath centerless .
Then , LaTeXMLMath is invariant under LaTeXMLMath if and only if LaTeXMLMath is an H-M rep .
If LaTeXMLMath is an H-M rep. , then LaTeXMLMath acts on LaTeXMLMath as a quotient of the Klein 4-group LaTeXMLMath .
The LaTeXMLMath action is faithful unless LaTeXMLMath and LaTeXMLMath are involutions .
Let LaTeXMLMath be the LaTeXMLMath th level inner Nielsen class for any Modular Tower with LaTeXMLMath .
Length of the LaTeXMLMath orbit on LaTeXMLMath depends only on the LaTeXMLMath ( or LaTeXMLMath ) orbit of LaTeXMLMath .
Faithful action of LaTeXMLMath on an LaTeXMLMath level LaTeXMLMath orbit extends to faithful action on an LaTeXMLMath level LaTeXMLMath orbit above it .
Similarly , if LaTeXMLMath , then LaTeXMLMath for LaTeXMLMath above LaTeXMLMath .
Assume LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , is the Nielsen class for centerless 2-perfect LaTeXMLMath .
Let LaTeXMLMath be an LaTeXMLMath orbit on LaTeXMLMath containing an H-M rep. Then all LaTeXMLMath orbits on LaTeXMLMath have length four .
So , LaTeXMLMath , the reduced component corresponding to LaTeXMLMath has b-fine moduli ( Prop .
LaTeXMLRef ) .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is faithful on all LaTeXMLMath orbits in LaTeXMLMath , LaTeXMLMath .
Since LaTeXMLMath is conjugation of LaTeXMLMath by LaTeXMLMath , if LaTeXMLMath is centerless , and LaTeXMLMath fixes LaTeXMLMath , then LaTeXMLMath .
So , LaTeXMLMath is an H-M rep .
Squares of elements in LaTeXMLMath act trivially on an H-M rep .
This shows LaTeXMLMath acts on the set LaTeXMLMath as a quotient of LaTeXMLMath .
Assume LaTeXMLMath is an H-M rep. and LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath : Both are involutions .
Similarly , if LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath is cyclic ( in particular abelian ) .
Also , LaTeXMLMath implies LaTeXMLMath , contradicting LaTeXMLMath is centerless .
Assume LaTeXMLMath .
As LaTeXMLMath ( Prop .
LaTeXMLRef ) , for LaTeXMLMath , LaTeXMLEquation .
So , LaTeXMLMath depends only on the LaTeXMLMath orbit of LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
This gives the statement on faithful action at level LaTeXMLMath .
Now assume LaTeXMLMath and LaTeXMLMath is LaTeXMLMath -perfect and centerless , so these hypotheses apply at all levels ( Prop .
LaTeXMLRef ) .
With no loss , on the general statement on an LaTeXMLMath orbit containing LaTeXMLMath , assume LaTeXMLMath .
Suppose for LaTeXMLMath , LaTeXMLMath for some LaTeXMLMath .
From the above , LaTeXMLMath is not the identity .
As LaTeXMLMath acts trivially on LaTeXMLMath , LaTeXMLMath with LaTeXMLMath ( Lem .
LaTeXMLRef ) .
This contradicts LaTeXMLMath being centerless .
Now assume LaTeXMLMath .
We show LaTeXMLMath is faithful on any LaTeXMLMath orbit LaTeXMLMath in LaTeXMLMath .
Any orbit at level 1 has elements lying over any element of the unique orbit at level 0 .
Anything in LaTeXMLMath above an H-M rep. LaTeXMLMath looks like LaTeXMLMath with LaTeXMLMath , and some LaTeXMLMath generating LaTeXMLMath .
We handled when LaTeXMLMath is in the centralizer of LaTeXMLMath , so assume it is not .
Apply the previous argument when LaTeXMLMath with LaTeXMLMath .
Conclude : If LaTeXMLMath , then LaTeXMLMath lifts an element of LaTeXMLMath having order 2 .
From Prop .
LaTeXMLRef , LaTeXMLMath .
Compute the first two entries of LaTeXMLMath to be LaTeXMLMath .
The remainder of the argument uses Cor .
LaTeXMLRef .
Let LaTeXMLMath be the generator of the centralizer of LaTeXMLMath .
Multiply LaTeXMLMath if necessary by LaTeXMLMath to assume LaTeXMLMath .
Then , LaTeXMLMath .
Conclude LaTeXMLMath contrary to our assumption .
For other elements in LaTeXMLMath , the argument is similar , though no easier .
∎ Two involutions generate a dihedral group .
So , LaTeXMLMath stabilizing an element of Ni in Lem .
LaTeXMLRef comes precisely from the Hurwitz version of modular curves in § LaTeXMLRef .
Suppose LaTeXMLMath , LaTeXMLMath is a level LaTeXMLMath Nielsen class for LaTeXMLMath , centerless and LaTeXMLMath -perfect .
Assume LaTeXMLMath , an LaTeXMLMath orbit on LaTeXMLMath , contains an H-M rep .
These are the hypotheses of Lem .
LaTeXMLRef , except LaTeXMLMath .
Then , there may be H-M reps. LaTeXMLMath with LaTeXMLMath ( an involution , or the identity if these are involutions ) with LaTeXMLMath , LaTeXMLMath and all LaTeXMLMath .
We don ’ t know if LaTeXMLMath s and LaTeXMLMath s as in ( LaTeXMLRef ) exist in this case : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath lie over the pair LaTeXMLMath .
Again LaTeXMLMath , with Nielsen classes of any type ( though we continue to simplify notation by using inner classes ) .
Slightly abusing the notation of § LaTeXMLRef use LaTeXMLMath and LaTeXMLMath as generators of LaTeXMLMath .
Recall the LaTeXMLMath subgroup LaTeXMLMath with LaTeXMLMath switching the two factors on the copy of LaTeXMLMath .
In this subsection LaTeXMLMath is a LaTeXMLMath orbit with LaTeXMLMath for LaTeXMLMath .
Then , LaTeXMLMath is the LaTeXMLMath orbit length : If LaTeXMLMath is odd and ( LaTeXMLRef ) holds , then LaTeXMLMath , otherwise LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath is the subgroup of LaTeXMLMath stabilizing LaTeXMLMath : LaTeXMLEquation .
Denote the set of LaTeXMLMath orbits on LaTeXMLMath by LaTeXMLMath .
We speak of the LaTeXMLMath orbit type of LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath ; or LaTeXMLMath ( or with LaTeXMLMath replacing LaTeXMLMath ) and LaTeXMLMath is even .
Then LaTeXMLMath .
Suppose LaTeXMLMath with LaTeXMLMath odd .
Then , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath and LaTeXMLMath .
In all cases LaTeXMLMath is a LaTeXMLMath orbit of type LaTeXMLMath .
We do all computations on inner Nielsen classes LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath or LaTeXMLMath ( or with LaTeXMLMath replacing LaTeXMLMath ) and LaTeXMLMath is even .
As LaTeXMLMath commutes with even powers of LaTeXMLMath , this gives LaTeXMLMath : LaTeXMLMath .
Suppose LaTeXMLMath with LaTeXMLMath odd .
Apply LaTeXMLMath to both sides .
From § LaTeXMLRef conclude LaTeXMLEquation .
So , LaTeXMLMath .
Apply LaTeXMLMath to both sides of LaTeXMLMath .
This gives LaTeXMLEquation .
Inductively , this shows LaTeXMLMath .
So , LaTeXMLEquation and LaTeXMLMath , or LaTeXMLMath .
∎ When LaTeXMLMath , the LaTeXMLMath orbit type of LaTeXMLMath is LaTeXMLMath if LaTeXMLMath ( two-shortening ) ; of type LaTeXMLMath otherwise .
When LaTeXMLMath , the LaTeXMLMath orbit type of LaTeXMLMath is LaTeXMLMath ( no shortening ) .
Assume LaTeXMLMath .
When LaTeXMLMath , the LaTeXMLMath orbit type of LaTeXMLMath is LaTeXMLMath ( total- shortening ) .
This is equivalent to LaTeXMLMath with LaTeXMLMath odd .
When LaTeXMLMath , the LaTeXMLMath orbit type of LaTeXMLMath is LaTeXMLMath ( two-shortening ; then LaTeXMLMath with LaTeXMLMath even ) .
The subsections of this section together comprise the complete the proof of Prop .
LaTeXMLRef , the main goal of this section .
The first two subsections describe the H-M reps. Then § LaTeXMLRef describes how the H-M reps. produce the near H-M reps. Denote LaTeXMLMath by LaTeXMLMath and LaTeXMLMath by LaTeXMLMath .
There are 2304 elements in LaTeXMLMath .
Exactly 16 are H-M reps. ; 16 others are near H-M reps. ( as in ( LaTeXMLRef ) ) .
The LaTeXMLMath orbit of an H-M ( resp .
near H-M ) rep. contains exactly one H-M ( resp .
near H-M ) rep. All orbits of LaTeXMLMath on LaTeXMLMath have length four , and LaTeXMLMath maps H-M ( resp .
near H-M ) reps. among themselves .
In particular , LaTeXMLMath orbits in LaTeXMLMath containing either H-M or near H-M reps. have type LaTeXMLMath .
From Lem .
LaTeXMLRef , there are four orbits of LaTeXMLMath on LaTeXMLMath containing H-M ( resp .
near H-M ) reps. giving eight LaTeXMLMath orbits on LaTeXMLMath containing H-M or near H-M reps .
Choose LaTeXMLMath by lifting LaTeXMLMath and LaTeXMLMath to LaTeXMLMath and LaTeXMLMath of order 3 .
Take LaTeXMLMath .
As LaTeXMLMath is transitive on LaTeXMLMath , it suffices to count lifts of LaTeXMLMath .
Multiply by LaTeXMLMath to count elements in LaTeXMLMath .
Use notation from § LaTeXMLRef .
Let LaTeXMLMath and LaTeXMLMath be the unique order 3 lifts to LaTeXMLMath of LaTeXMLMath and LaTeXMLMath .
Lifts of LaTeXMLMath to LaTeXMLMath correspond exactly to lifts of LaTeXMLMath to LaTeXMLMath .
Count these by counting conjugates of an element of order 3 by the kernel LaTeXMLMath from LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath has order 3 , then LaTeXMLMath ( as in § LaTeXMLRef ) .
As LaTeXMLMath is an irreducible LaTeXMLMath module ( Cor .
LaTeXMLRef ) , the set LaTeXMLMath gives the complete set of conjugates of LaTeXMLMath by LaTeXMLMath .
Three entries of a Nielsen class 4-tuple determine the 4th from the product-one condition by rewriting entries LaTeXMLMath .
Divide by inner automorphisms from the kernel from LaTeXMLMath .
As LaTeXMLMath is centerless ( Prop .
LaTeXMLRef ) , there are LaTeXMLMath such lifts of LaTeXMLMath to LaTeXMLMath .
This gives LaTeXMLMath total inner Nielsen classes .
Continue using LaTeXMLMath .
Fix any H-M representative mapping to LaTeXMLMath .
Modulo inner action of LaTeXMLMath , select representatives with LaTeXMLMath in the first two positions .
Other H-M representatives come from conjugating LaTeXMLMath by LaTeXMLMath .
As in § LaTeXMLRef or Cor .
LaTeXMLRef , take lifts by conjugating LaTeXMLMath by elements of LaTeXMLMath .
Or should we choose , by conjugating LaTeXMLMath by elements of LaTeXMLMath .
The centralizer of LaTeXMLMath in LaTeXMLMath is a LaTeXMLMath acting nontrivially on LaTeXMLMath .
This cuts from 16 to 8 the inner classes that are H-M representatives and lifts of LaTeXMLMath .
This concludes the part of Prop .
LaTeXMLRef counting H-M representatives .
Since the LaTeXMLMath and LaTeXMLMath positions determine an H-M rep. LaTeXMLMath , the LaTeXMLMath orbit of LaTeXMLMath can contain only one H-M rep. By inspection , LaTeXMLMath maps H-M reps. to H-M reps .
Unless otherwise said , this subsection is about inner and inner reduced Nielsen classes .
The definition of near H-M reps. is in ( LaTeXMLRef ) .
A modular representation observation produces near H-M reps. LaTeXMLMath by tweaking an H-M rep. LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , the notation LaTeXMLMath is shorthand for LaTeXMLMath .
This is the same as the right action of LaTeXMLMath in LaTeXMLMath acting on LaTeXMLMath .
Let LaTeXMLMath be the complex conjugation operator in ( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath as complex conjugate pairs .
Assume LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath has order 10 ( 5 LaTeXMLMath ) .
It fixes a unique nontrivial LaTeXMLMath ( Cor .
LaTeXMLRef ) .
Let LaTeXMLMath .
Then LaTeXMLMath if and only if LaTeXMLMath .
In this case , the LaTeXMLMath orbit of LaTeXMLMath in LaTeXMLMath has length 20 .
That LaTeXMLMath has order 10 is a special case of Lemma LaTeXMLRef .
An element of order 5 in LaTeXMLMath acts on LaTeXMLMath by right multiplying cosets of a LaTeXMLMath in LaTeXMLMath ( Prop .
LaTeXMLRef ) .
So LaTeXMLMath fixes one nontrivial element .
It must be LaTeXMLMath .
That LaTeXMLMath satisfies the product-one condition is exactly that LaTeXMLMath .
Since LaTeXMLMath has no center , what a conjugation on LaTeXMLMath does to the 1st and 4th elements determines it .
Therefore , the LaTeXMLMath orbit of LaTeXMLMath has length 2 times the order of LaTeXMLMath : The orbit has length 20 .
∎ Let LaTeXMLMath be the LaTeXMLMath th characteristic quotient of LaTeXMLMath .
We use the following proposition in generality .
Its special case with LaTeXMLMath appears in precise calculations for level 1 of this Modular Tower .
The symbol LaTeXMLMath in Prop .
LaTeXMLRef is the complex conjugation operator from Prop .
LaTeXMLRef from complex conjugate pairs of branch points .
Suppose LaTeXMLMath lie over LaTeXMLMath with LaTeXMLMath having order LaTeXMLMath or 5 .
Then , LaTeXMLMath has order LaTeXMLMath .
If LaTeXMLMath , let LaTeXMLMath .
Then , LaTeXMLEquation satisfies the product-one condition and LaTeXMLMath ( so is a near H-M rep. from ( LaTeXMLRef ) at level LaTeXMLMath ) .
For LaTeXMLMath , H-M reps. give Galois covers LaTeXMLMath ( in LaTeXMLMath ) over LaTeXMLMath with branch points LaTeXMLMath in complex conjugate pairs and LaTeXMLMath .
For near H-M reps. there are such covers LaTeXMLMath over LaTeXMLMath , but LaTeXMLMath .
When LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are in the LaTeXMLMath orbit labeled LaTeXMLMath in Cor .
LaTeXMLRef .
Conversely , given LaTeXMLMath satisfying LaTeXMLMath , an H-M rep. LaTeXMLMath exists giving LaTeXMLMath by ( LaTeXMLRef ) .
That ( LaTeXMLRef ) holds is a simple check .
The order of the product LaTeXMLMath is from Lem .
LaTeXMLRef .
Apply Prop .
LaTeXMLRef to an H-M rep. cover LaTeXMLMath with respect to the LaTeXMLMath operator for two pairs of complex conjugate branch points to compute the effect of complex conjugation over LaTeXMLMath .
It is given by LaTeXMLMath equal the identity .
So all points on LaTeXMLMath over LaTeXMLMath are real .
For a near H-M rep. the effect of complex conjugation is given by LaTeXMLMath as in the statement of the proposition .
So , LaTeXMLMath moves all points over LaTeXMLMath .
There are no real points on LaTeXMLMath .
Now assume LaTeXMLMath and LaTeXMLMath satisfies LaTeXMLMath ( automatically associated with LaTeXMLMath for LaTeXMLMath and LaTeXMLMath as complex conjugate pairs ) .
Write LaTeXMLMath : LaTeXMLEquation .
From Cor .
LaTeXMLRef , assume LaTeXMLMath .
Apply LaTeXMLMath to see LaTeXMLEquation .
The product-one condition for LaTeXMLMath is equivalent to LaTeXMLMath .
Then , LaTeXMLMath gives four conditions according to the entries of LaTeXMLMath , with the last two automatic .
The second gives LaTeXMLEquation showing LaTeXMLMath ( Cor .
LaTeXMLRef ) : the first condition is automatic .
The product-one condition says LaTeXMLMath : LaTeXMLMath generates the centralizer of LaTeXMLMath .
∎ To conclude proving Prop .
LaTeXMLRef requires two points .
A unique near H-M rep. is in the LaTeXMLMath orbit of a near H-M rep. LaTeXMLMath is stable on the set of near H-M reps. As with H-M reps. the 1st and 4th positions determine them .
So the former is clear .
Apply LaTeXMLMath to LaTeXMLMath to get LaTeXMLEquation .
Write LaTeXMLMath and LaTeXMLMath , and compute that LaTeXMLMath centralizes LaTeXMLMath .
Thus , LaTeXMLMath maps LaTeXMLMath to a near H-M representative .
Similarly for LaTeXMLMath .
Assume LaTeXMLMath , and LaTeXMLMath is a Nielsen class .
Distinguishing between absolute and inner Nielsen classes is cumbersome , though computations using them both are invaluable ( as in Thm .
LaTeXMLRef ) and similar ( add the action of some group LaTeXMLMath as in § LaTeXMLRef ) .
For simplicity , assume inner Nielsen classes so the decoration LaTeXMLMath means reduced inner classes ( unless said otherwise ) .
Following two preliminary subsections , for the LaTeXMLMath Modular Tower this section lists cusps of a given width from LaTeXMLMath action on LaTeXMLMath .
Possible widths are 2 , 4 , 6 , 8 , 10 , 12 and 20 .
The aim is to relate all LaTeXMLMath orbits to orbits of width 20 ( especially to H-M reps. ) .
Lem .
LaTeXMLRef and Prop .
LaTeXMLRef report precisely on H-M and near H-M cusps .
Prop .
LaTeXMLRef gives the spin separation ingredient that establishes the distribution of cusps between two LaTeXMLMath orbits LaTeXMLMath and LaTeXMLMath on LaTeXMLMath .
When no further ramification occurs from level LaTeXMLMath to level 1 over the elliptic points LaTeXMLMath and LaTeXMLMath , the genus of level 1 components comes just from the story of LaTeXMLMath and LaTeXMLMath orbits .
§ LaTeXMLRef observations on this continue in § LaTeXMLRef .
§ LaTeXMLRef has subtle conjugation computations for LaTeXMLMath shortening of LaTeXMLMath orbits to LaTeXMLMath orbits .
For much of this subsection , the prime LaTeXMLMath is arbitrary .
Suppose LaTeXMLMath is an orbit of LaTeXMLMath acting on LaTeXMLMath .
Use § LaTeXMLRef notation for LaTeXMLMath orbits : For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the number of fixed points of LaTeXMLMath on LaTeXMLMath , LaTeXMLMath .
From Prop .
LaTeXMLRef , the Riemann-Hurwitz formula gives the genus LaTeXMLMath of the reduced Hurwitz space component LaTeXMLMath : LaTeXMLEquation .
Use this to rephrase for LaTeXMLMath our Main Problem LaTeXMLRef on Modular Towers .
For LaTeXMLMath centerless and LaTeXMLMath -perfect , show for each LaTeXMLMath orbit LaTeXMLMath on LaTeXMLMath , LaTeXMLMath if LaTeXMLMath is large .
If this holds , we say LaTeXMLMath is in the hyperbolic range .
Thm .
LaTeXMLRef says this implies there are no LaTeXMLMath points ( LaTeXMLMath a number field ) on LaTeXMLMath if LaTeXMLMath is large ( possibly larger than the beginning of the hyperbolic range ) .
Assume LaTeXMLMath ( resp .
LaTeXMLMath ) is a LaTeXMLMath orbit in LaTeXMLMath ( resp .
LaTeXMLMath over LaTeXMLMath ) .
So , cusps of LaTeXMLMath lie over cusps of LaTeXMLMath : LaTeXMLMath orbits on LaTeXMLMath lie over LaTeXMLMath orbits on LaTeXMLMath .
LaTeXMLMath ; and for LaTeXMLMath over LaTeXMLMath , if LaTeXMLMath not an H-M rep. , then LaTeXMLMath .
The condition LaTeXMLMath not an H-M rep. in ( LaTeXMLRef LaTeXMLRef ) is equivalent to LaTeXMLMath , so is absolutely necessary .
For a LaTeXMLMath orbit LaTeXMLMath use the notation LaTeXMLMath for cusps of LaTeXMLMath under LaTeXMLMath .
If ( LaTeXMLRef LaTeXMLRef ) holds ( for LaTeXMLMath ) , then it holds with LaTeXMLMath replacing LaTeXMLMath .
For LaTeXMLMath ( LaTeXMLMath ) , both ( LaTeXMLRef LaTeXMLRef ) and ( LaTeXMLRef LaTeXMLRef ) hold for any LaTeXMLMath orbit LaTeXMLMath , LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath .
Reduce all expressions modulo LaTeXMLMath to conclude LaTeXMLMath fixes LaTeXMLMath .
This is a contradiction .
The same argument works for LaTeXMLMath .
By inspection ( LaTeXMLRef ) shows ( LaTeXMLRef LaTeXMLRef ) holds for LaTeXMLMath , and so at all levels in this LaTeXMLMath Modular Tower .
If LaTeXMLMath or 5 , ( LaTeXMLRef LaTeXMLRef ) follows from the opening statement of Prop .
LaTeXMLRef .
The only other possibility is that LaTeXMLMath is an H-M rep. , but LaTeXMLMath above it is not .
Lem .
LaTeXMLRef says if LaTeXMLMath , then LaTeXMLMath for LaTeXMLMath over LaTeXMLMath .
That completes the proof .
∎ Assume at level LaTeXMLMath of a Modular Tower there are still components of genus 0 or 1 .
Lem .
LaTeXMLRef inspects the contribution of cusp ramification toward the genus of components at level LaTeXMLMath .
Three phenomena play a role in this contribution for each cusp in going from level LaTeXMLMath to level LaTeXMLMath .
Detecting if condition ( LaTeXMLRef ) in Prop .
LaTeXMLRef changes .
Deciding if there is a multiplication by LaTeXMLMath factor as in ( LaTeXMLRef LaTeXMLRef ) .
Computing LaTeXMLMath orbit shortening changes from one level to another .
As above , LaTeXMLMath is a LaTeXMLMath orbit in LaTeXMLMath ; LaTeXMLMath is a LaTeXMLMath orbit in LaTeXMLMath above it .
For LaTeXMLMath , assume LaTeXMLMath so the LaTeXMLMath orbit type is LaTeXMLMath unless LaTeXMLMath is odd and ( LaTeXMLRef ) holds ( Prop .
LaTeXMLRef ) .
For each LaTeXMLMath orbit LaTeXMLMath let LaTeXMLMath in the former case , 1 in the latter .
If LaTeXMLMath , then LaTeXMLMath for LaTeXMLMath over LaTeXMLMath .
Let LaTeXMLMath be the index of ramification of LaTeXMLMath over LaTeXMLMath .
Also , let LaTeXMLMath ( 1 or LaTeXMLMath ; automatically LaTeXMLMath if LaTeXMLMath from Lem .
LaTeXMLRef ) .
Depending on the amount of LaTeXMLMath orbit shortening ( § LaTeXMLRef ) , let LaTeXMLMath ( no shortening ) , 2 ( two-shortening ) or 4 ( total shortening ) .
Then , LaTeXMLEquation .
Suppose LaTeXMLMath .
Then LaTeXMLMath if for some LaTeXMLMath orbit LaTeXMLMath over LaTeXMLMath , LaTeXMLMath .
Suppose LaTeXMLMath .
Then , LaTeXMLEquation .
If LaTeXMLMath , then equality holds in ( LaTeXMLRef ) .
The references explain most of this lemma .
Given LaTeXMLMath over LaTeXMLMath the ramification index of the respective cusps is exactly LaTeXMLMath which by previous comments is LaTeXMLMath as in ( LaTeXMLRef ) .
It is well-known that if a LaTeXMLMath is a covering of projective nonsingular curves with the lower curve of genus 1 , then the genus of LaTeXMLMath is 1 if and only if the cover is unramified .
Formula ( LaTeXMLRef ) expresses the Riemann-Hurwitz formula applies to the relative curve covering LaTeXMLMath , when the latter has genus 0 .
∎ We use Lem .
LaTeXMLRef to show what goes into computing the genus of the two LaTeXMLMath components .
There are two orbits LaTeXMLMath and LaTeXMLMath of LaTeXMLMath on LaTeXMLMath , each of degree 16 over the unique LaTeXMLMath orbit LaTeXMLMath .
The genus of the orbit LaTeXMLMath containing H-M reps. is 12 .
The other orbit LaTeXMLMath has genus 9 .
Apply ( LaTeXMLRef ) to LaTeXMLMath .
From Prop .
LaTeXMLRef , one LaTeXMLMath orbit on LaTeXMLMath contains all H-M and near H-M reps .
Call this orbit LaTeXMLMath .
Prop .
LaTeXMLRef says there are exactly eight LaTeXMLMath orbits with the following properties : 5 divides mp ; they are in the LaTeXMLMath orbit of a near H-M rep .
Prop .
LaTeXMLRef says the LaTeXMLMath cusp type of an H-M or near H-M rep. is LaTeXMLMath .
So , a cusp of LaTeXMLMath lying over an H-M rep. of LaTeXMLMath has ramification index 4 .
Let LaTeXMLMath be the component of LaTeXMLMath corresponding to LaTeXMLMath .
Conclude : Each cusp ( eight total ) of LaTeXMLMath with mp divisible by 5 has ramification index 4 over the cusp below them at level 0 .
Together they contribute LaTeXMLMath to the right side of ( LaTeXMLRef ) .
Similarly , consider cusps of LaTeXMLMath over cusps of LaTeXMLMath with mp equal 3 .
Prop .
LaTeXMLRef gives a similar conclusion about cusps at level 0 with middle product 3 .
Cusps above them on LaTeXMLMath have ramification index 4 ( LaTeXMLMath does not shorten them ) .
They also contribute LaTeXMLMath to the right side of ( LaTeXMLRef ) .
Now apply Prop .
LaTeXMLRef for the contribution of cusps on LaTeXMLMath over sh applied to level 0 H-M reps .
This contributes LaTeXMLMath for the shift of H-M reps. and LaTeXMLMath for the others to the right side of ( LaTeXMLRef ) .
So , the right side of ( LaTeXMLRef ) is 54 .
The expression LaTeXMLMath gives LaTeXMLMath .
Let LaTeXMLMath be the collection of cusps at level 1 not in LaTeXMLMath .
First assume they all lie in one LaTeXMLMath component for the computation of the genus of this orbit .
From Prop .
LaTeXMLRef there are eight cusps in LaTeXMLMath with width 6 , and four with width 12 .
Similarly , there eight cusps in LaTeXMLMath with width 10 , and four with width 20 .
Finally , Lem .
LaTeXMLRef gives 8 type ( 2,4 ) LaTeXMLMath orbits in LaTeXMLMath .
To complete the calculation above for LaTeXMLMath , list respective contributions to the right of ( LaTeXMLRef ) : Type ( 2,4 ) contribute 8 ; type ( 6,6 ) contribute 8 ; type ( 6,12 ) contribute LaTeXMLMath ; type ( 10,10 ) contribute 8 and type ( 10,20 ) contribute LaTeXMLMath .
So , LaTeXMLMath gives LaTeXMLMath .
Since it is true at level 0 , it is also true at level 1 that every LaTeXMLMath not in LaTeXMLMath is in the LaTeXMLMath orbit of an element LaTeXMLMath with LaTeXMLMath .
From Prop .
LaTeXMLRef ( and its notation ) any cusp of LaTeXMLMath ( resp .
LaTeXMLMath ) connects to each cusp of LaTeXMLMath ( resp .
LaTeXMLMath ) .
So , to prove all elements of LaTeXMLMath lie in one LaTeXMLMath , it suffices to connect some cusp of LaTeXMLMath ( resp .
LaTeXMLMath ) to some cusp of LaTeXMLMath ( resp .
LaTeXMLMath ) .
This shows a component LaTeXMLMath containing a cusp of LaTeXMLMath has degree sixteen over a cusp at level 0 with LaTeXMLMath .
So LaTeXMLMath has degree sixteen everywhere .
For the degree over every cusp to be 16 forces including all cusps in LaTeXMLMath .
To join something in LaTeXMLMath to something in LaTeXMLMath consider a LaTeXMLMath type ( 2,4 ) orbit in LaTeXMLMath .
A representative for such an orbit has LaTeXMLMath .
Lem .
LaTeXMLRef shows each type ( 2,4 ) LaTeXMLMath orbit in LaTeXMLMath is such an element .
This concludes the proof of the corollary .
∎ Consider orbits of length one for LaTeXMLMath and LaTeXMLMath ( of respective orders 3 and 2 ) acting on LaTeXMLMath .
Suppse LaTeXMLMath is a situation for absolute equivalence , as in § LaTeXMLRef .
Knowing the length one orbits tells what is the contribution of LaTeXMLMath and LaTeXMLMath to the genus of components of LaTeXMLMath in ( LaTeXMLRef ) .
Consider LaTeXMLMath ; LaTeXMLMath is similar .
Let LaTeXMLMath be a set of branch points representing the elliptic point for LaTeXMLMath .
Then LaTeXMLMath cycles the set LaTeXMLMath .
A fixed set of classical generators of LaTeXMLMath produces a list of covers ( up to LaTeXMLMath -equivalence ) LaTeXMLMath , LaTeXMLMath .
Composing the LaTeXMLMath s with LaTeXMLMath permutes them .
For some choice of classical generators , this action is LaTeXMLMath modulo the action of LaTeXMLMath .
Generalize Rem .
LaTeXMLRef to the general case of inner reduced Hurwitz spaces when LaTeXMLMath .
Further , if LaTeXMLMath is a Modular Tower , with LaTeXMLMath -perfect and centerless LaTeXMLMath , when can there be a projective system of Nielsen classes fixed by LaTeXMLMath for LaTeXMLMath or 1 ?
The Modular Tower version of this question makes sense for any LaTeXMLMath applied to the orbifold points in LaTeXMLMath : When can a Modular Tower have a projective system of Nielsen class representatives fixed by a nontrivial element in LaTeXMLMath associated with an orbifold stabilizer ?
According to § LaTeXMLRef , LaTeXMLMath acts trivially on the list of Table LaTeXMLRef ( § LaTeXMLRef ) .
Note : LaTeXMLMath fixes LaTeXMLMath from LaTeXMLMath .
Yet , LaTeXMLMath fixes no item of LaTeXMLMath .
Since LaTeXMLMath acts trivially , this means , from the list LaTeXMLMath , LaTeXMLMath , of degree 5 covers , exactly one of LaTeXMLMath is equivalent to LaTeXMLMath .
Here is a fact about the list : Suppose an item from it has two 3-cycles ( not necessarily consecutive ) with exactly two integers of common support .
Up to conjugation by LaTeXMLMath these 3-cycles are LaTeXMLMath where the common support integers ( 1 and 2 here ) appear in opposite order in the second 3-cycle .
Further , for LaTeXMLMath , this is true for all consecutive pairs of 3-cycles , especially including the 4th and 1st , taken in that order .
Now , apply LaTeXMLMath as the shift of § LaTeXMLRef .
Conjugation by an element of LaTeXMLMath is determined by its action on two generators LaTeXMLMath of order 3 .
Several computations require the precise effect of that conjugation given the generators and what the conjugation does modulo LaTeXMLMath .
Use the notation of Lem .
LaTeXMLRef for elements of LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath be its LaTeXMLMath orbit .
Let LaTeXMLMath .
If LaTeXMLMath , we say LaTeXMLMath shortens LaTeXMLMath ( as in § LaTeXMLRef ) .
Variants : LaTeXMLMath shortens LaTeXMLMath or LaTeXMLMath shortens LaTeXMLMath .
Use LaTeXMLMath and LaTeXMLMath as representatives of the two H-M reps. in LaTeXMLMath .
§ LaTeXMLRef has notation for the type of orbits .
Let LaTeXMLMath be order 3 lifts of LaTeXMLMath .
Denote LaTeXMLMath , LaTeXMLMath and LaTeXMLMath respectively by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Each LaTeXMLMath with LaTeXMLMath has the form LaTeXMLMath with LaTeXMLMath .
( So , LaTeXMLMath . )
If LaTeXMLMath ( resp .
LaTeXMLMath , LaTeXMLMath ) is conjugate to LaTeXMLMath , then the conjugation is by a lift of LaTeXMLMath ( resp .
LaTeXMLMath , LaTeXMLMath ) .
Suppose LaTeXMLMath is a near H-M rep. Then , LaTeXMLMath shortens LaTeXMLMath .
Thus , shifted complements of near H-M reps. fall into two pairs of LaTeXMLMath type ( 2,2 ) orbits .
All other LaTeXMLMath s with LaTeXMLMath fall into type ( 2,4 ) LaTeXMLMath orbits .
Three subsections cover the proof of Prop .
LaTeXMLRef .
The first establishes the relevance of LaTeXMLMath , LaTeXMLMath , to the existence of appropriate conjugations .
§ LaTeXMLRef shows neither LaTeXMLMath nor LaTeXMLMath shorten any LaTeXMLMath with LaTeXMLMath .
§ LaTeXMLRef shows LaTeXMLMath shortens the complement of a near H-M rep. As this is an even power of LaTeXMLMath , Rem .
LaTeXMLRef notes this suffices to determine exactly the shortening type : It is two-shortening .
With LaTeXMLMath , LaTeXMLMath as above : LaTeXMLEquation .
Then , LaTeXMLMath .
If this is conjugate to ( LaTeXMLRef ) , it is by a lift of an LaTeXMLMath conjugation switching LaTeXMLMath and LaTeXMLMath .
The element LaTeXMLMath gives this conjugation .
Consider if LaTeXMLMath is conjugate to ( LaTeXMLRef ) .
Such a conjugation is by a lift of an LaTeXMLMath conjugation mapping LaTeXMLMath to LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
Thus , the conjugation is a lift of LaTeXMLMath .
Now consider if LaTeXMLMath is conjugate to LaTeXMLMath .
If so , it is by a lift of an LaTeXMLMath conjugation mapping each of LaTeXMLMath and LaTeXMLMath to their inverses .
The element LaTeXMLMath gives this conjugation .
Suppose LaTeXMLMath normalizes a subgroup LaTeXMLMath .
Denote the centralizer of LaTeXMLMath in LaTeXMLMath by LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath , LaTeXMLMath .
From the last statement in Cor .
LaTeXMLRef , with no loss LaTeXMLMath uniquely determines LaTeXMLMath and LaTeXMLMath knowing also LaTeXMLMath .
Suppose LaTeXMLMath is conjugate to ( LaTeXMLRef ) .
Then some lift LaTeXMLMath of LaTeXMLMath conjugates as follows : LaTeXMLEquation .
The 1st , 2nd and 3rd expressions give the effect of conjugating LaTeXMLMath ( Lemma LaTeXMLRef ) on LaTeXMLMath two ways : LaTeXMLEquation .
Conclude LaTeXMLMath .
Similarly , figure the effect of LaTeXMLMath conjugating LaTeXMLMath : LaTeXMLEquation .
Now we show LaTeXMLMath satisfying ( LaTeXMLRef ) does not exist .
From ( LaTeXMLRef ) and the product one condition ( LaTeXMLMath ) LaTeXMLMath and LaTeXMLMath , an expression invariant under LaTeXMLMath .
From ( LaTeXMLRef ) and ( LaTeXMLRef ) , LaTeXMLMath .
This is false : The nontrivial element LaTeXMLMath would centralize LaTeXMLMath .
As LaTeXMLMath has no center ( Prop .
LaTeXMLRef ) , LaTeXMLMath doesn ’ t exist .
The following shows there is a natural braid taking the reduced class of a complement of a near H-M rep. over LaTeXMLMath to one over LaTeXMLMath .
So , for reduced classes , complements of near H-M reps. are similar to a pair of H-M reps. LaTeXMLMath and LaTeXMLMath naturally paired as being over LaTeXMLMath and LaTeXMLMath .
Suppose LaTeXMLMath is conjugate to ( LaTeXMLRef ) .
A lift LaTeXMLMath of LaTeXMLMath conjugates as follows : LaTeXMLEquation .
Here are the analogs of ( LaTeXMLRef ) and ( LaTeXMLRef ) : LaTeXMLEquation .
Conjugate the latter by LaTeXMLMath to conclude LaTeXMLMath and LaTeXMLMath together are equivalent to LaTeXMLMath shortening LaTeXMLMath .
The expression LaTeXMLMath is automatic from LaTeXMLMath and conjugating LaTeXMLMath by LaTeXMLMath .
As in the proof of Cor .
LaTeXMLRef , compute the effect of LaTeXMLMath on LaTeXMLMath .
By explicit computation LaTeXMLMath acts as LaTeXMLMath .
Similarly , LaTeXMLMath acts as LaTeXMLMath .
Finally , LaTeXMLMath acts as LaTeXMLMath .
The remaining list of possible LaTeXMLMath values is in the following lemma .
With LaTeXMLMath , LaTeXMLMath shortens the LaTeXMLMath orbit of LaTeXMLMath if and only if LaTeXMLMath is invariant under LaTeXMLMath .
Given a value of LaTeXMLMath ( or LaTeXMLMath ) with this property , all others arise by running over all possible lifts of LaTeXMLMath .
There is an orbit shortening for LaTeXMLMath only if there is an LaTeXMLMath ( lifting LaTeXMLMath ) with LaTeXMLMath .
This implies LaTeXMLMath is invariant under LaTeXMLMath .
Given one such lift LaTeXMLMath , Cor .
LaTeXMLRef shows you get all others by multiplying this LaTeXMLMath by LaTeXMLMath as LaTeXMLMath runs over LaTeXMLMath .
Given a lift LaTeXMLMath giving LaTeXMLMath , multiplying LaTeXMLMath by LaTeXMLMath produces LaTeXMLMath .
Producing one such LaTeXMLMath is the final step .
The Cor .
LaTeXMLRef proof gives explicit action of LaTeXMLMath on the cosets of the LaTeXMLMath .
Respectively : LaTeXMLEquation .
Consider a near H-M rep. as in ( LaTeXMLRef ) : LaTeXMLEquation with LaTeXMLMath centralizing LaTeXMLMath .
For reasons coming up , we take LaTeXMLMath , a lift of LaTeXMLMath and LaTeXMLMath , a lift of LaTeXMLMath .
Let LaTeXMLMath be the centralizer of LaTeXMLMath .
Then , compute the shift of the complement of LaTeXMLMath to get LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath and LaTeXMLMath centralizing LaTeXMLMath .
Since LaTeXMLMath , the desired shortening amounts to showing LaTeXMLMath and ( using that by definition LaTeXMLMath ) LaTeXMLMath centralizes LaTeXMLMath .
Explicit computations are reassuring : LaTeXMLMath and so LaTeXMLMath .
Similarly , LaTeXMLMath , so LaTeXMLMath .
Thus , LaTeXMLMath and LaTeXMLMath .
The other check works as easily .
∎ From § LaTeXMLRef and § LaTeXMLRef there is a concise relation between LaTeXMLMath orbits of length two and H-M and near H-M reps. As previously , use LaTeXMLMath for LaTeXMLMath .
Continue the notation from § LaTeXMLRef for LaTeXMLMath , LaTeXMLMath and the types of cusps .
There are LaTeXMLMath total LaTeXMLMath with LaTeXMLMath ; all have LaTeXMLMath an H-M rep .
This gives two LaTeXMLMath orbits LaTeXMLMath and LaTeXMLMath of type LaTeXMLMath .
LaTeXMLMath total LaTeXMLMath have these properties : LaTeXMLMath has order LaTeXMLMath ; and LaTeXMLMath is in a LaTeXMLMath orbit of type LaTeXMLMath .
Such LaTeXMLMath have LaTeXMLMath a complement of a near H-M rep .
These account for the LaTeXMLMath orbits LaTeXMLMath and LaTeXMLMath of type LaTeXMLMath , giving all width 2 cusps .
There are LaTeXMLMath total LaTeXMLMath with LaTeXMLMath having LaTeXMLMath .
From these there are 14 LaTeXMLMath orbits in LaTeXMLMath of type LaTeXMLMath .
All LaTeXMLMath orbits LaTeXMLMath in this proposition have LaTeXMLMath modulo LaTeXMLMath , mapping surjectively to LaTeXMLMath .
All H-M and near H-M reps. lie in one LaTeXMLMath orbit containing the width two cusps .
Each LaTeXMLMath with LaTeXMLMath is the shift of an H-M rep. LaTeXMLMath .
Further , applying LaTeXMLMath to such a LaTeXMLMath gives another such element .
By inspection LaTeXMLMath is stable on this set .
So , according to Prop .
LaTeXMLRef , the shift applied to H-M reps. contributes a total of two length 2 orbits for LaTeXMLMath on reduced classes .
Now , consider the case LaTeXMLMath has LaTeXMLMath : LaTeXMLMath .
With no loss , LaTeXMLMath and LaTeXMLMath .
Further , assume LaTeXMLMath or LaTeXMLMath .
Fix LaTeXMLMath .
There are LaTeXMLMath choices of LaTeXMLMath , all lifts of LaTeXMLMath modulo conjugation by the centralizer of LaTeXMLMath .
Then , there are LaTeXMLMath lifts of LaTeXMLMath , now determining LaTeXMLMath .
So , there are LaTeXMLMath total LaTeXMLMath with LaTeXMLMath having LaTeXMLMath .
Since LaTeXMLMath acts faithfully ( Prop .
LaTeXMLRef ) , there are LaTeXMLMath elements of LaTeXMLMath in LaTeXMLMath orbits of length 2 or 4 .
§ LaTeXMLRef completes showing all H-M and near H-M reps. fall in one LaTeXMLMath orbit .
∎ Lem .
LaTeXMLRef and Lem .
LaTeXMLRef explain the eight length 20 orbits of LaTeXMLMath on LaTeXMLMath containing H-M and near H-M reps. Part of the next lemma applies to any Modular Tower with LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath th characteristic quotient of LaTeXMLMath .
For an H-M rep. LaTeXMLMath , LaTeXMLMath is LaTeXMLMath .
The LaTeXMLMath orbit of of LaTeXMLMath has length LaTeXMLMath .
Further , only one H-M rep. is in a given LaTeXMLMath orbit .
Generally , let LaTeXMLMath be a projective system of Nielsen class representatives in the LaTeXMLMath Modular Tower .
If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath .
For LaTeXMLMath with LaTeXMLMath , its LaTeXMLMath orbit has length 4 , 12 or 20 .
Let LaTeXMLMath be the LaTeXMLMath th level Nielsen class in any Modular Tower ( any LaTeXMLMath ) .
For LaTeXMLMath , a projective system of Nielsen class representatives , either : LaTeXMLMath for all LaTeXMLMath ; or some smallest LaTeXMLMath satisfies LaTeXMLMath and LaTeXMLEquation .
We show the conclusion of ( LaTeXMLRef ) first .
Consider the projective system LaTeXMLMath in the Modular Tower for LaTeXMLMath .
Let LaTeXMLMath be the second and 3rd entries of LaTeXMLMath .
Suppose LaTeXMLMath exists so LaTeXMLMath .
By assumption , LaTeXMLMath is a lift of LaTeXMLMath to LaTeXMLMath .
Thus , Lem .
LaTeXMLRef says LaTeXMLMath .
Inductively apply this for the conclusion of the lemma .
For LaTeXMLMath the value of LaTeXMLMath is either 1 , 3 or 5 .
If LaTeXMLMath , then LaTeXMLMath is an H-M rep .
If LaTeXMLMath , then LaTeXMLMath , and the 3-tuple LaTeXMLMath satisfies the product-one and genus 0 ( LaTeXMLMath implies LaTeXMLMath ) conditions of Prop .
LaTeXMLRef .
Conclude that LaTeXMLMath .
Equivalently : If LaTeXMLMath are the lifts of LaTeXMLMath to LaTeXMLMath , then LaTeXMLMath .
Similarly , if LaTeXMLMath , then LaTeXMLMath , the genus 0 condition holds , and the conclusion from Prop .
LaTeXMLRef is that LaTeXMLMath has order 10 .
From Lemma LaTeXMLRef , if LaTeXMLMath , then the length of the orbit of LaTeXMLMath on LaTeXMLMath is twice LaTeXMLMath with LaTeXMLMath written as a perturbation of an H-M rep. LaTeXMLMath , LaTeXMLMath .
∎ When LaTeXMLMath , LaTeXMLMath is equivalent to LaTeXMLMath is an H-M rep .
Suppose LaTeXMLMath is a projective system of Nielsen class representatives as in Lem .
LaTeXMLRef with LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath is the smallest integer LaTeXMLMath with LaTeXMLMath not an H-M rep .
If all the LaTeXMLMath s are H-M reps. , then LaTeXMLMath .
If LaTeXMLMath is an H-M ( or near H-M ) rep. , refer to its LaTeXMLMath orbit as an H-M ( or near H-M ) rep. orbit .
Call a LaTeXMLMath orbit of an element in LaTeXMLMath with LaTeXMLMath an H-M ( or near H-M ) orbit the shift of an H-M ( or near H-M ) orbit .
Prop .
LaTeXMLRef braids a near H-M rep. , LaTeXMLEquation to the H-M rep. LaTeXMLMath .
That LaTeXMLMath satisfies the product one condition is equivalent to LaTeXMLMath centralizes LaTeXMLMath .
As in § LaTeXMLRef , denote LaTeXMLMath by LaTeXMLMath and LaTeXMLMath .
The central extension LaTeXMLMath appears above Cor .
LaTeXMLRef .
Its characterization is that LaTeXMLMath and any lift to LaTeXMLMath of the nontrivial element in LaTeXMLMath has order LaTeXMLMath .
Suppose LaTeXMLMath with LaTeXMLMath ( and LaTeXMLMath of order 5 ) .
Then LaTeXMLMath satisfies the product one condition if and only if LaTeXMLMath centralizes LaTeXMLMath .
If LaTeXMLMath , it is LaTeXMLMath .
Then , LaTeXMLMath is the H-M rep. LaTeXMLMath .
Also , LaTeXMLMath is a different H-M rep. lying over the same H-M rep. at level 0 as does LaTeXMLMath Then , if LaTeXMLMath is an H-M rep. , LaTeXMLMath represents the reduced class of a near H-M rep. Each type ( 2,2 ) cusp has Nielsen class representatives consisting of sh applied to elements lying over either LaTeXMLMath or LaTeXMLMath .
Conclude : All near H-M reps. and elements in width 2 cusps fall in one LaTeXMLMath orbit .
More generally , consider LaTeXMLMath , and suppose the image of LaTeXMLMath in LaTeXMLMath has order 5 .
So , LaTeXMLMath has order LaTeXMLMath ( Lem .
LaTeXMLRef ) and LaTeXMLMath has order 2 .
Further , any lift of LaTeXMLMath in the cover LaTeXMLMath has order LaTeXMLMath .
There is a braid from LaTeXMLMath to a near H-M rep. LaTeXMLMath .
Complex conjugation LaTeXMLMath ( for two pairs of complex conjugate branch points ) applied to LaTeXMLMath gives LaTeXMLMath with LaTeXMLMath lifting to LaTeXMLMath to have order LaTeXMLMath .
The statement on LaTeXMLMath comes to noting the product-one condition is equivalent to LaTeXMLMath .
Apply LaTeXMLMath to both sides to see LaTeXMLMath fixes LaTeXMLMath .
Prop .
LaTeXMLRef shows LaTeXMLMath is the unique nontrivial element of LaTeXMLMath in LaTeXMLMath .
Lem .
LaTeXMLRef shows LaTeXMLMath braids LaTeXMLMath to LaTeXMLMath .
Apply LaTeXMLMath to LaTeXMLMath to get LaTeXMLEquation .
Use LaTeXMLMath to rewrite this as LaTeXMLMath with LaTeXMLMath .
Apply the first part of this lemma with LaTeXMLMath : An element of LaTeXMLMath takes LaTeXMLMath to the H-M representative LaTeXMLMath .
As in Lem .
LaTeXMLRef , LaTeXMLMath centralizes LaTeXMLMath .
Use that LaTeXMLMath and its conjugates don ’ t centralize LaTeXMLMath ( Cor .
LaTeXMLRef ) to see LaTeXMLEquation and LaTeXMLMath give two H-M reps. at level 1 in one LaTeXMLMath orbit lying over the same level 0 H-M rep. We already know how to braid from LaTeXMLMath to LaTeXMLMath .
Since , there are only two H-M reps. over LaTeXMLMath , it suffices that we have this braid from one to the other .
So , § LaTeXMLRef gives the braiding between all the width two cusps .
The last expression relating H-M reps. to complements of near H-M reps. comes from writing LaTeXMLMath as LaTeXMLEquation to express this in standard generators of LaTeXMLMath .
The relation between H-M reps. LaTeXMLMath and near H-M reps. given by LaTeXMLMath works at any level ( with LaTeXMLMath ) .
Prop .
LaTeXMLRef has already established that near H-M reps. have the complex conjugation properties stated here .
The element LaTeXMLMath has the form LaTeXMLMath .
Lift it to LaTeXMLMath by lifting LaTeXMLMath and LaTeXMLMath to ( respectively ) LaTeXMLMath and LaTeXMLMath of order 3 in LaTeXMLMath .
Then form LaTeXMLMath .
Let LaTeXMLMath .
The characterizing property of LaTeXMLMath implies LaTeXMLMath has order LaTeXMLMath or LaTeXMLMath has order 4 .
So , any lift of LaTeXMLMath to LaTeXMLMath has order 4 .
Since LaTeXMLMath is just a conjugate of LaTeXMLMath , it applies to LaTeXMLMath as well .
∎ By now it is clear H-M reps. figure in many computations .
Denote the collection of these by LaTeXMLMath .
The effect of LaTeXMLMath on LaTeXMLMath is conjugation of LaTeXMLMath by LaTeXMLMath ( Lem .
LaTeXMLRef ) leaving LaTeXMLMath and LaTeXMLMath fixed .
A LaTeXMLMath orbit of length six implies a length three orbit of LaTeXMLMath on LaTeXMLMath .
The group LaTeXMLMath is the pullback in LaTeXMLMath of a copy of LaTeXMLMath in LaTeXMLMath ( Table LaTeXMLRef , items 3,6,7 or one of the conjugates of these by LaTeXMLMath ) .
So , LaTeXMLMath has no nontrivial centralizer in LaTeXMLMath .
Conclude : Three divides the order of LaTeXMLMath , and with LaTeXMLMath , LaTeXMLMath does not equal LaTeXMLMath in LaTeXMLMath ( Prop .
LaTeXMLRef ) .
Note : LaTeXMLMath generates the centralizer in LaTeXMLMath of LaTeXMLMath .
So , the LaTeXMLMath orbit of LaTeXMLMath has length six if and only if LaTeXMLMath is conjugate to LaTeXMLMath for some LaTeXMLMath .
Prop .
LaTeXMLRef shows no element of LaTeXMLMath fixes any inner classes .
There are eight LaTeXMLMath orbits on LaTeXMLMath of length six .
Let LaTeXMLMath be in a length three LaTeXMLMath orbit , LaTeXMLMath .
Then there are exactly six length twelve and four length six LaTeXMLMath orbits on LaTeXMLMath lying over LaTeXMLMath .
They have these further properties .
For each H-M and near H-M orbit LaTeXMLMath : four length twelve orbits LaTeXMLMath over LaTeXMLMath satisfy LaTeXMLMath ; and the remaining length twelve or six over LaTeXMLMath satisfy LaTeXMLMath .
Denote the collection of length 6 orbits and those of length 12 not meeting H-M or near H-M reps. , respectively by LaTeXMLMath and LaTeXMLMath .
Similarly , there are eight LaTeXMLMath orbits on LaTeXMLMath of length 10 and four of length 20 that are not H-M or near H-M orbits .
Denote the former by LaTeXMLMath and the latter by LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath or for LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
This accounts for all entries in the sh -incidence matrix for orbits with 3 dividing mp paired with orbits with 5 dividing mp .
Subsections § LaTeXMLRef and § LaTeXMLRef report on respective length six and twelve orbits to conclude the proof of Prop LaTeXMLRef .
The calculation shows that various sh -incidence intersections are 0 or 1 ( at most 1 ) .
These incidence intersections divide according to how the orbits relate to H-M and near H-M reps .
This is a consequence of the lifting invariant from Prop .
LaTeXMLRef .
Calculations for ( LaTeXMLRef ) are similar to those in the last half of the statement .
So , we only give details for the former .
Use the previous notation for LaTeXMLMath with LaTeXMLMath lying over LaTeXMLMath .
Then LaTeXMLMath has length six exactly when LaTeXMLMath is inner equivalent to LaTeXMLMath for some LaTeXMLMath .
Modulo LaTeXMLMath , LaTeXMLMath .
Since nontrivial elements of LaTeXMLMath have order 2 , an element of order 4 must give any conjugation of LaTeXMLMath to LaTeXMLMath .
The only possibility is that LaTeXMLMath is conjugate in LaTeXMLMath to one of the following : LaTeXMLEquation .
Try these by cases .
Suppose LaTeXMLMath conjugates LaTeXMLMath to LaTeXMLMath .
Then , LaTeXMLMath .
There is a contradiction : When LaTeXMLMath divides three , the images of LaTeXMLMath and LaTeXMLMath in LaTeXMLMath don ’ t commute ( check Table LaTeXMLRef ) .
If LaTeXMLMath conjugates LaTeXMLMath to LaTeXMLMath , then LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath .
These generate the pullback of LaTeXMLMath in LaTeXMLMath which has no center ( Cor .
LaTeXMLRef ) .
This implies LaTeXMLMath , contrary to LaTeXMLMath having order 4 .
This leaves the serious case .
Consider if LaTeXMLMath is conjugate to LaTeXMLEquation .
That is , for some LaTeXMLMath : LaTeXMLEquation .
Apply conjugation by LaTeXMLMath to these expressions with LaTeXMLMath .
Conclude : Conjugation by LaTeXMLMath on LaTeXMLMath gives LaTeXMLMath .
Opening arguments in § LaTeXMLRef imply LaTeXMLMath .
Yet , LaTeXMLMath generates the centralizer of LaTeXMLMath .
So , LaTeXMLMath stabilizes LaTeXMLMath .
There are two cases .
If the image of LaTeXMLMath in LaTeXMLMath is transitive on four letters , then the image is isomorphic to LaTeXMLMath : its pullback in LaTeXMLMath ( as above ) has no center .
Otherwise , the image of LaTeXMLMath in LaTeXMLMath is isomorphic to LaTeXMLMath .
§ LaTeXMLRef gives the only allowable shape for elements LaTeXMLMath in a length six orbit of LaTeXMLMath .
We prove there are in total eight such orbits .
To be explicit , assume LaTeXMLMath ( with LaTeXMLMath in Table LaTeXMLRef ) .
If LaTeXMLMath lies over LaTeXMLMath , conjugating by LaTeXMLMath has the effect of switching LaTeXMLMath with LaTeXMLMath ( resp .
LaTeXMLMath with LaTeXMLMath ) .
Apply Lem .
LaTeXMLRef to find LaTeXMLMath having an additional property : LaTeXMLMath generates the centralizer LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
Fix some LaTeXMLMath in its conjugacy class , lifting LaTeXMLMath .
Let LaTeXMLMath be arbitrary in its conjugacy class lifting LaTeXMLMath .
Define LaTeXMLMath to be LaTeXMLMath and LaTeXMLMath to be LaTeXMLMath .
The resulting LaTeXMLMath gives a LaTeXMLMath orbit of length six if LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath ( as in Prop .
LaTeXMLRef ) implies LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath if and only if LaTeXMLMath : LaTeXMLMath fixes LaTeXMLMath .
Conjugate LaTeXMLMath by LaTeXMLMath and replace LaTeXMLMath by LaTeXMLMath with LaTeXMLMath to construct LaTeXMLMath from LaTeXMLMath .
Compute LaTeXMLMath as LaTeXMLEquation .
For each LaTeXMLMath , there is a unique LaTeXMLMath giving product 1 in ( LaTeXMLRef ) .
Representatives LaTeXMLMath for distinct length six orbits of LaTeXMLMath over LaTeXMLMath correspond to the four elements LaTeXMLMath that centralize LaTeXMLMath ( Cor .
LaTeXMLRef ) .
The first sentence follows if LaTeXMLMath is one-one .
Consider LaTeXMLMath as an element in the group ring LaTeXMLMath .
It suffices to show LaTeXMLMath is invertible on LaTeXMLMath .
Use notation from the proof of Lem .
LaTeXMLRef : LaTeXMLMath and LaTeXMLMath on LaTeXMLMath cosets act by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , compute LaTeXMLMath to be LaTeXMLMath .
Then LaTeXMLMath , in the group ring , acts on the six cosets generating LaTeXMLMath as follows : LaTeXMLEquation or as LaTeXMLMath .
Recall LaTeXMLMath ( in LaTeXMLMath .
Check LaTeXMLMath on a basis for LaTeXMLMath : LaTeXMLEquation .
The range vectors form a basis .
So , LaTeXMLMath is invertible on LaTeXMLMath .
The remainder of the proof follows from the setup to this lemma .
∎ We apply the higher lifting invariant of Prop .
LaTeXMLRef to compute the sh -incidence entries for H-M and near H-M orbits ( Def .
LaTeXMLRef ) and LaTeXMLMath orbits of elements with mp six .
Suppose LaTeXMLMath is the LaTeXMLMath orbit of LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is in the LaTeXMLMath orbit of an H-M or near H-M rep. Then , for each LaTeXMLMath orbit LaTeXMLMath of an H-M or near H-M rep. , LaTeXMLMath .
Suppose LaTeXMLMath is the LaTeXMLMath orbit of LaTeXMLMath with LaTeXMLMath with LaTeXMLMath not the orbit of either an H-M or near H-M rep. Then , LaTeXMLMath .
Suppose LaTeXMLMath is an H-M or near H-M rep. , or a complement of one of these .
Then , LaTeXMLMath has mp six if only if LaTeXMLMath .
So , for values of LaTeXMLMath , there are eight distinct elements LaTeXMLMath with mp six .
We show no two are on the same LaTeXMLMath orbit .
First consider level 0 from Table LaTeXMLRef and LaTeXMLMath and LaTeXMLMath .
Conjugation by LaTeXMLMath takes one of the two LaTeXMLMath orbits to the other .
So , it suffices to show LaTeXMLMath , LaTeXMLMath are in distinct LaTeXMLMath orbits .
That requires showing LaTeXMLMath and LaTeXMLMath are in distinct LaTeXMLMath orbits .
This is saying the complements shift to distinct LaTeXMLMath orbits .
The computations are similar for H-M and near H-M reps. , so we do the former only .
For simplicity write LaTeXMLMath as LaTeXMLMath .
With LaTeXMLMath and LaTeXMLMath , we show LaTeXMLMath and LaTeXMLMath are in distinct LaTeXMLMath orbits .
By reducing modulo LaTeXMLMath , the only possibility is LaTeXMLMath is in the same reduced equivalence class as LaTeXMLMath .
Check easily this isn ’ t so .
The level one lifting invariant of Prop .
LaTeXMLRef has value LaTeXMLMath on sh applied to the LaTeXMLMath orbits of elements LaTeXMLMath with mp six described above .
From Prop .
LaTeXMLRef , such an element LaTeXMLMath is in the LaTeXMLMath orbit of an H-M or near H-M rep .
Prop .
LaTeXMLRef also says that orbits LaTeXMLMath in the statement of the lemma have lifting invariant LaTeXMLMath .
Since the lifting invariant is an LaTeXMLMath invariant , they can not intersect the set from sh applied to the LaTeXMLMath orbit of LaTeXMLMath .
∎ Table LaTeXMLRef has the sh -incidence matrix for the reduced Nielsen classes LaTeXMLMath .
We have all the data in place to compute the sh -incidence matrix for LaTeXMLMath ( replacing LaTeXMLMath by LaTeXMLMath ) .
As at level 0 , a 2-stage process gives the block for orbit LaTeXMLMath : List H-M and near H-M reps. and compute their LaTeXMLMath orbits ; then , apply sh to these , and compute their LaTeXMLMath orbits .
What results is closed under sh , so the process completes after two stages .
Use the notation for LaTeXMLMath and LaTeXMLMath in the above discussion .
Label the type ( 10,20 ) LaTeXMLMath orbits as LaTeXMLMath , LaTeXMLMath for those containing H-M reps. and LaTeXMLMath , LaTeXMLMath for those containing near H-M reps. Refinement : LaTeXMLMath with LaTeXMLMath odd ( resp .
even ) lies over the LaTeXMLMath orbit at level 0 containing LaTeXMLMath ( resp .
LaTeXMLMath ) .
Further refinement : LaTeXMLMath , LaTeXMLMath ; LaTeXMLMath , LaTeXMLMath .
According to Prop .
LaTeXMLRef , complements of near H-M reps. produce LaTeXMLMath , LaTeXMLMath ; and LaTeXMLMath , LaTeXMLMath .
Two type ( 2,4 ) LaTeXMLMath orbits , LaTeXMLMath , LaTeXMLMath , contain sh applied to two near H-M reps. and two complements of H-M reps .
So , sh applied to LaTeXMLMath covers LaTeXMLMath evenly , LaTeXMLMath .
Let LaTeXMLMath be an H-M rep. Let LaTeXMLMath be the near H-M rep. attached to LaTeXMLMath by Prop .
LaTeXMLRef .
Let LaTeXMLMath be the near H-M rep. attached to the H-M rep. LaTeXMLMath .
Then , LaTeXMLMath is different from LaTeXMLMath .
Let LaTeXMLMath be an H-M rep .
The relationship LaTeXMLMath already says LaTeXMLMath takes the complement of an H-M rep. to a near H-M rep. Now use LaTeXMLMath in LaTeXMLMath .
Let LaTeXMLMath be LaTeXMLMath , and apply sh to it , to get LaTeXMLEquation is the shift of a near H-M rep .
This shows why a type ( 2,4 ) orbit containing sh applied to a complement of an H-M rep. also contains sh applied to a near H-M rep .
The remaining conclusions are similar .
∎ As in Prop .
LaTeXMLRef use LaTeXMLMath and LaTeXMLMath for the two H-M reps. in LaTeXMLMath .
Beyond Lem .
LaTeXMLRef , there are four type ( 2,4 ) LaTeXMLMath orbits LaTeXMLMath , LaTeXMLMath , with lifting invariant +1 ( Prop .
LaTeXMLRef ) .
For LaTeXMLMath , LaTeXMLEquation .
For LaTeXMLMath an H-M rep. , LaTeXMLMath and LaTeXMLMath lie over the same element of LaTeXMLMath .
Each of these LaTeXMLMath sets contains exactly one reduced equivalence class of form LaTeXMLMath .
That leaves eight type ( 2,4 ) LaTeXMLMath orbits with lifting invariant -1 ( from Prop .
LaTeXMLRef ) .
A representative for such an orbit has LaTeXMLMath .
The shift applied to the LaTeXMLMath orbit of LaTeXMLMath intersects exactly two type ( 10,20 ) LaTeXMLMath orbits , with these covering LaTeXMLMath .
This accounts for all 16 of the elements with shifts of type ( 2,4 ) in the four ( 10,20 ) orbits with lifting invariant -1 .
Shifts of the remaining two elements from each ( 2,4 ) orbit meet type ( 10,10 ) LaTeXMLMath orbits , covering LaTeXMLMath .
Let LaTeXMLMath be an H-M rep. of the form LaTeXMLMath .
Then , LaTeXMLMath and LaTeXMLMath lie over the same element of LaTeXMLMath .
In each width 20 LaTeXMLMath orbit , there are four elements to which sh gives an element with LaTeXMLMath or 2 .
Apply the shift to the remaining ( from 16 ) elements in the orbits containing H-M and near H-M reps. that lie over LaTeXMLMath with LaTeXMLMath .
There are LaTeXMLMath elements in the width 20 cusps with shifts of this type .
The remaining elements with mp 2 must have lifting invariant -1 .
A representative has the form LaTeXMLMath with LaTeXMLMath as in Prop .
LaTeXMLRef .
With LaTeXMLMath the centralizer of LaTeXMLMath in LaTeXMLMath , the LaTeXMLMath orbit of LaTeXMLMath includes LaTeXMLEquation .
An analog the Lem .
LaTeXMLRef proof shows that if LaTeXMLMath , then LaTeXMLMath .
As there , this argument uses the special presentation of LaTeXMLMath from Cor .
LaTeXMLRef .
∎ Let LaTeXMLMath be one of the ( 6,12 ) type LaTeXMLMath orbits with lifting invariant +1 .
From § LaTeXMLRef , LaTeXMLMath , LaTeXMLMath .
Further , LaTeXMLMath hits two other LaTeXMLMath LaTeXMLMath orbits with multiplicity two .
The argument is similar to that of Lem .
LaTeXMLRef : Use the LaTeXMLMath operator applied to LaTeXMLMath with mp six with LaTeXMLMath also having mp six .
This gives the LaTeXMLMath elements in ( 6,12 ) type orbits with lifting invariant LaTeXMLMath .
The only unaccounted nonzero entries in the sh -incidence matrix for orbit LaTeXMLMath have the form LaTeXMLMath with LaTeXMLMath and LaTeXMLMath running over H-M and near H-M orbits .
From symmetry , it suffices to fix LaTeXMLMath an H-M rep. orbit , and account for the intersections as LaTeXMLMath varies .
As LaTeXMLMath runs over H-M and near H-M orbits , LaTeXMLMath .
If LaTeXMLMath is an H-M rep. , then the H-M orbit LaTeXMLMath of LaTeXMLMath meets it with multiplicity four .
Two near H-M orbits LaTeXMLMath and LaTeXMLMath meet LaTeXMLMath with multiplicity two .
Lemmas LaTeXMLRef and LaTeXMLRef account for all of the intersections of sh applied to orbits with LaTeXMLMath with LaTeXMLMath .
There are four such intersections .
§ LaTeXMLRef shows that sh applied to orbits with LaTeXMLMath contributes eight intersections with LaTeXMLMath .
Since there are 20 elements in LaTeXMLMath , LaTeXMLMath orbits LaTeXMLMath account for all the remaining intersections with LaTeXMLMath .
Now we show why LaTeXMLMath give the indicated intersections .
Such intersections lie above corresponding level 0 intersections .
§ LaTeXMLRef gives the formula LaTeXMLMath acting on reduced inner Nielsen classes .
Suppose LaTeXMLMath lies over LaTeXMLMath and LaTeXMLMath has middle product 2 .
For example , this would hold if LaTeXMLMath is an H-M rep. Then ( Lem .
LaTeXMLRef ) , LaTeXMLMath lies over LaTeXMLMath , on an H-M or near H-M rep. orbit , and its shift also has middle product 2 .
So LaTeXMLMath , sh applied to LaTeXMLMath is an intersection of LaTeXMLMath and LaTeXMLMath .
If we use instead LaTeXMLMath , this gives another intersection of LaTeXMLMath and LaTeXMLMath : LaTeXMLMath .
If LaTeXMLMath is the H-M rep. on LaTeXMLMath , then LaTeXMLMath is an H-M rep .
Suppose LaTeXMLEquation ( LaTeXMLMath ; the complement of an H-M rep. ) , then LaTeXMLMath is a near H-M rep. as given by Prop .
LaTeXMLRef .
Similarly , replacing LaTeXMLMath by LaTeXMLMath and LaTeXMLMath replaces LaTeXMLMath by the complement of LaTeXMLMath and the other near H-M rep. over LaTeXMLMath .
∎ Lem .
LaTeXMLRef fully accounts for the sh -incidence matrix restricted to the block corresponding to the pairings of LaTeXMLMath orbits in LaTeXMLMath orbit LaTeXMLMath having lifting invariant +1 .
The LaTeXMLMath block of pairings of LaTeXMLMath orbits in LaTeXMLMath orbit LaTeXMLMath having lifting invariant -1 is more complicated .
§ LaTeXMLRef discusses why there may be a representation theory connection between the two orbits , though it would not be numerically simple .
To conclude this section we illustrate the lemmas above with a contribution to the complete sh -incidence matrix at level 1 for having LaTeXMLMath lifting invariant .
From here to the end of this subsection , LaTeXMLMath orbits refer to suborbits of LaTeXMLMath .
The part of the sh -incidence matrix pairing LaTeXMLMath and LaTeXMLMath with respective middle products 6 and 10 is significant .
Still , this LaTeXMLMath block consists of just one ’ s ( Prop .
LaTeXMLRef ) .
More interesting is the LaTeXMLMath block of intersections from ( 10,20 ) type LaTeXMLMath orbits .
For economy , replace LaTeXMLMath as the LaTeXMLMath orbit type in Table LaTeXMLRef by the symbol LaTeXMLMath .
We show the genus 12 component of LaTeXMLMath has one component of real points , while the genus 9 component has no real points .
We check the effect of LaTeXMLMath at level 1 extending the use of the complex conjugation operator at level 0 as Lem .
LaTeXMLRef reports .
Note : LaTeXMLMath acts trivially on LaTeXMLMath satisfying the product-one condition .
For LaTeXMLMath , while LaTeXMLMath acts trivially on a Nielsen class , it usually is a nontrivial conjugation dependent on the LaTeXMLMath representing a Nielsen class .
First , check the effect over LaTeXMLMath .
Corresponding to LaTeXMLMath , we know there are eight cover points corresponding to the four H-M reps. and the four near H-M reps. lying on LaTeXMLMath ( Prop .
LaTeXMLRef ) and the complex conjugation operator LaTeXMLMath .
The Nielsen classes for the corresponding components all lie in distinct LaTeXMLMath orbits .
For each , moreover , their complementary Nielsen class gives another point over LaTeXMLMath from Lem .
LaTeXMLRef .
These 16 real components over the interval LaTeXMLMath all lie on the component LaTeXMLMath ( as in Prop .
LaTeXMLRef ) .
Denote the complete set of real points ( corresponding to reduced Nielsen classes ) lying on LaTeXMLMath over LaTeXMLMath by LaTeXMLMath .
For LaTeXMLMath , elements of LaTeXMLMath lie on exactly 16 components of LaTeXMLMath lying over LaTeXMLMath .
The closure in LaTeXMLMath of these components consists of one connected component of real points .
There are no LaTeXMLMath points on LaTeXMLMath over LaTeXMLMath , and no LaTeXMLMath points at all on LaTeXMLMath .
The proof of Prop .
LaTeXMLRef takes up the next five subsections .
Its arrangement starts with an equivalent equation defining a point on LaTeXMLMath for a reduced equivalence class of covers represented by a particular cover with branch points all in LaTeXMLMath or in complex conjugate pairs .
Assume a representing cover has branch cycles LaTeXMLMath .
Then , LaTeXMLMath with LaTeXMLMath the complex conjugation operator corresponding to the branch point configuration , LaTeXMLMath and LaTeXMLMath .
Also , LaTeXMLMath , LaTeXMLMath acts trivially modulo conjugation .
The level 0 real point data for the absolute Hurwitz space appears at the end of the proof of Lem .
LaTeXMLRef .
For four points LaTeXMLMath in LaTeXMLMath : LaTeXMLMath has complex operator LaTeXMLMath ; LaTeXMLMath has LaTeXMLMath .
The involution LaTeXMLMath acts on Nielsen classes in Table LaTeXMLRef , revealing LaTeXMLMath and LaTeXMLMath as the first two items in that table .
Since the complex conjugation operator for LaTeXMLMath is LaTeXMLMath , the Galois closure of that cover is not over LaTeXMLMath .
So , points corresponding to these branch cycle descriptions on the absolute space have no real points on the inner space above them .
§ LaTeXMLRef has the level 0 inner Hurwitz space data summarized in the complex conjugation operator from ( LaTeXMLRef ) : LaTeXMLMath .
The fixed points correspond to real points , with branch cycle descriptions LaTeXMLMath and LaTeXMLMath and their conjugates by LaTeXMLMath .
Let LaTeXMLMath be the complex conjugation operator for four real branch points .
It suffices to handle the defining equation , when LaTeXMLMath .
With no loss , take LaTeXMLMath equal either LaTeXMLMath or LaTeXMLMath .
First consider the case LaTeXMLMath , which is an H-M rep. at level 0 .
Here and below we use the normalization that writes LaTeXMLMath , a perturbation of an H-M rep. With no loss LaTeXMLMath lies over LaTeXMLMath or LaTeXMLMath with LaTeXMLMath .
It is convenient to use past data by replacing LaTeXMLMath by LaTeXMLMath , the complex conjugation operator for complex conjugate pairs of branch points .
As in Rem .
LaTeXMLRef , this doesn ’ t change the real points representing reduced classes .
Assume LaTeXMLMath covers LaTeXMLMath .
Double the number of components corresponding to using LaTeXMLMath in place of LaTeXMLMath .
Use the notation around Prop .
LaTeXMLRef to distinguish the LaTeXMLMath orbit of LaTeXMLMath by whether LaTeXMLMath and LaTeXMLMath are in the same LaTeXMLMath orbit ( acting on LaTeXMLMath ) .
Suppose LaTeXMLMath corresponds to LaTeXMLMath in different LaTeXMLMath orbits .
We run through nontrivial elements LaTeXMLMath , showing that LaTeXMLMath is not possible for any LaTeXMLMath .
The product of 3rd and 4th entries of LaTeXMLMath is LaTeXMLMath .
So , LaTeXMLMath maps LaTeXMLMath to LaTeXMLMath .
The components LaTeXMLMath and LaTeXMLMath have field of definition LaTeXMLMath and so LaTeXMLMath gives branch cycles for a cover in LaTeXMLMath .
Prop .
LaTeXMLRef implies it is attached to LaTeXMLMath as a perturbation of an H-M rep. One case is easy : LaTeXMLMath .
On the right side these produce LaTeXMLMath conjugate to LaTeXMLMath .
Though LaTeXMLMath is still a perturbation of an H-M rep. , it corresponds to LaTeXMLMath with LaTeXMLMath ( resp .
LaTeXMLMath ) in the opposite LaTeXMLMath orbit to LaTeXMLMath ( resp .
LaTeXMLMath ) .
Similarly for LaTeXMLMath .
Yet , when LaTeXMLMath , LaTeXMLMath .
A conjugation of LaTeXMLMath that gives LaTeXMLMath is by a lift LaTeXMLMath of LaTeXMLMath with square the centralizer LaTeXMLMath of LaTeXMLMath in LaTeXMLMath , and LaTeXMLMath the centralizer LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
The last is equivalent to LaTeXMLMath , or LaTeXMLMath .
Cor .
LaTeXMLRef notes that LaTeXMLMath fixes exactly four elements in LaTeXMLMath and two elements of LaTeXMLMath in the conjugacy class LaTeXMLMath .
They are LaTeXMLMath and LaTeXMLMath .
So , LaTeXMLMath is possible for exactly four elements in LaTeXMLMath .
( The proof of Cor .
LaTeXMLRef has explicit computations , for a different LaTeXMLMath , corroborating this . )
The expression LaTeXMLMath then has a unique solution LaTeXMLMath from which we derive values of LaTeXMLMath .
We already know four values of LaTeXMLMath that satisfy this .
These correspond to the complements of H-M and near H-M reps. lying above the real component of LaTeXMLMath ( over LaTeXMLMath ) corresponding to the cover points with branch cycles LaTeXMLMath and complex conjugation operator LaTeXMLMath .
Thus , Prop .
LaTeXMLRef shows for LaTeXMLMath in different LaTeXMLMath orbits on LaTeXMLMath , LaTeXMLMath is not possible .
Adopt the idea of using a perturbation of an H-M rep. here by writing LaTeXMLMath .
The product-one condition gives LaTeXMLMath .
We show LaTeXMLMath is impossible for LaTeXMLMath .
All computations are similar .
We do just LaTeXMLMath .
This will complete that there are exactly 16 components of LaTeXMLMath over the LaTeXMLMath -interval LaTeXMLMath .
Compute LaTeXMLMath : LaTeXMLEquation .
The effect of LaTeXMLMath on the 3rd and 4th entries shows LaTeXMLMath is the identity .
Since , however , LaTeXMLMath is a lift of LaTeXMLMath , it has order 4 ( Lem .
LaTeXMLRef ) .
Now we show these 16 components of real points close up to one connected component on LaTeXMLMath .
That is , if we add the endpoints over LaTeXMLMath ( cusps § LaTeXMLRef ) and 1 to these components , they form one connected set .
Branch cycles for an H-M rep. and its complement give real components on LaTeXMLMath meeting ( each pair ) at a cusp of type LaTeXMLMath ( Prop .
LaTeXMLRef ) .
The same is true for near H-M reps .
Apply sh to these to get another real component ( over LaTeXMLMath ) .
This is the result of reflection off the LaTeXMLMath boundary of the entering real locus .
Apply Prop .
LaTeXMLRef to decipher the labeling of branch cycles to components on applying sh to the H-M , near H-M reps. and their complements .
The result is a cusp of type ( 2,2 ) or ( 2,4 ) .
In the former case it reflects off the LaTeXMLMath boundary to give another real component .
In the latter case an application of LaTeXMLMath gives a real component reflection .
Applying Prop .
LaTeXMLRef to these reflections shows there is one connected component of real points .
To finish Prop .
LaTeXMLRef we show there are no points on LaTeXMLMath over LaTeXMLMath .
According to Lem .
LaTeXMLRef , this is a check that LaTeXMLMath is impossible under the following conditions .
LaTeXMLMath is the complex conjugation operator for two real and a complex conjugate pair of branch points .
LaTeXMLMath and LaTeXMLMath .
As in § LaTeXMLRef , consider branch cycles for real points on LaTeXMLMath .
Write LaTeXMLMath .
Running through entries of Table LaTeXMLRef gives the existence of LaTeXMLMath in LaTeXMLMath with LaTeXMLMath only for LaTeXMLMath with LaTeXMLMath , LaTeXMLMath with LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
As in § LaTeXMLRef , only the real component corresponding to LaTeXMLMath lies below real components of LaTeXMLMath .
This is compatible with LaTeXMLMath from § LaTeXMLRef fixing two integers .
It suffices to consider LaTeXMLMath lying over LaTeXMLMath .
Write LaTeXMLMath as LaTeXMLMath with LaTeXMLMath or a perturbation of this of the form LaTeXMLMath with LaTeXMLMath .
The calculation to show that for no LaTeXMLMath and LaTeXMLMath can this give an LaTeXMLMath point on LaTeXMLMath is similar to that of § LaTeXMLRef .
We leave it to the reader .
Cor .
LaTeXMLRef computes genuses of the two components LaTeXMLMath and LaTeXMLMath of level of the LaTeXMLMath Modular Tower .
Prop .
LaTeXMLRef shows only LaTeXMLMath has real points ( one component of them on LaTeXMLMath ) .
§ LaTeXMLRef describes data we got from LaTeXMLCite .
Following that , all but the last subsection is group theory for the spin structure explanation that nails the two components in Cor .
LaTeXMLRef .
The LaTeXMLCite data was reassuring at several stages .
Still , it was crude compared to our final arguments .
Despite the detail developed for our special case , there remains the mysterious similarity , amidst the differences , of the two level 1 orbits .
§ LaTeXMLRef gives ideas for that appropriate for testing the speculative Lie algebra guess in § LaTeXMLRef .
The paper ( excluding appendices ) concludes with lessons on computing ( genuses of ) components for all Modular Towers ( § LaTeXMLRef ) .
Orbits of LaTeXMLMath correspond to cusps of a LaTeXMLMath -line cover .
Orbits of LaTeXMLMath correspond to components of the LaTeXMLMath -line cover .
Finding discrete invariants of a Modular Tower means analyzing the LaTeXMLMath and LaTeXMLMath orbits on reduced Nielsen classes .
That includes analyzing arithmetic properties of the cusps .
Since components are moduli spaces this means analyzing degeneration of objects in the moduli space on approach ( over LaTeXMLMath or LaTeXMLMath ) to the cusps .
This generalizes analysis of elliptic curve degeneration in approaching a cusp of a modular curve ( LaTeXMLCite introduces using LaTeXMLMath functions to make this analysis ; § LaTeXMLRef ) .
Even for the one Modular Tower for LaTeXMLMath , it would be a major event to prove the analog of Prop .
LaTeXMLRef for LaTeXMLMath -adic points ( see § LaTeXMLRef ) .
Apply Falting ’ s Theorem LaTeXMLCite to these genus 12 and 9 components to conclude the following from Thm .
LaTeXMLRef .
There are only finitely many LaTeXMLMath realizations over any number field LaTeXMLMath .
For LaTeXMLMath large there are no LaTeXMLMath realizations over LaTeXMLMath .
If LaTeXMLMath is the level k characteristic quotient for LaTeXMLMath , then the pullback of LaTeXMLMath in LaTeXMLMath is LaTeXMLMath , the level LaTeXMLMath characteristic quotient of LaTeXMLMath .
The notation LaTeXMLMath is ambiguous .
There are two conjugacy classes of 3-cycles in LaTeXMLMath .
Compatible with Ex .
LaTeXMLRef , we mean each pair of conjugacy classes appears twice , or what we might label as LaTeXMLMath .
From the Branch Cycle Lemma this is equivalent to the levels of the Modular Tower have definition field LaTeXMLMath .
Level 0 is a special case of LaTeXMLCite ( Ex .
LaTeXMLRef ) .
There are two components , LaTeXMLMath and LaTeXMLMath , corresponding to two LaTeXMLMath orbits LaTeXMLMath and LaTeXMLMath of the lifting invariant from LaTeXMLMath to LaTeXMLMath ( analogous to Prop .
LaTeXMLRef ) .
Note : LaTeXMLMath acts through LaTeXMLMath on these inner classes ( § LaTeXMLRef has details ) .
For absolute equivalence these spaces are families of genus 1 curves .
It is a special case of LaTeXMLCite that the map of each component to the moduli space of curves of genus 1 is generically surjective ( see § LaTeXMLRef ) .
As in Prop .
LaTeXMLRef , there is nothing above the component LaTeXMLMath at level 1 .
Applying this paper as with the LaTeXMLMath Modular Tower , the first author ’ s thesis will show level 1 for LaTeXMLMath , LaTeXMLMath has six components .
From § LaTeXMLRef , some differences between this case and LaTeXMLMath are clear .
For LaTeXMLMath , LaTeXMLMath is not the complete set of conjugates of LaTeXMLMath , though one can still choose conjugation by LaTeXMLMath or LaTeXMLMath .
Also , there are level 0 cases where LaTeXMLMath .
The same lifting invariant from Prop .
LaTeXMLRef applies .
At level 1 there are components that together comprise a moduli space LaTeXMLMath , and another LaTeXMLMath .
Unlike , however , for LaTeXMLMath , each of these has several ( three ) connected ( reduced , inner ) components .
For LaTeXMLMath the two H-M components have genus 1 , and the 3rd , which contains near H-M reps. , has genus 3 .
So , all these components have real points .
Two distinct components of the inner classes at level 1 contain H-M reps. Further , the two orbits differ by an outer automorphism of LaTeXMLMath .
We don ’ t know if the H-M rep. components are conjugate over LaTeXMLMath .
They come from an outer automorphism ( as in Thm .
LaTeXMLRef ) .
So , if they are conjugate , it must be from an extension of LaTeXMLMath in the Galois closures of the absolute covers in either one of these H-M families .
For LaTeXMLMath , there are no real points .
It has two genus 0 components , with corresponding LaTeXMLMath orbits differing by an outer automorphism of LaTeXMLMath ( different from that binding the two H-M components ) .
The genus 0 components are conjugate over LaTeXMLMath by the complex conjugation operator .
The last component of LaTeXMLMath has genus 3 .
Since the genus 0 components are on LaTeXMLMath , they are obstructed : there is nothing above them at level 2 .
The Main Conjecture for this case ( Prob .
LaTeXMLRef ) therefore follows by assuring any component at level 2 above the genus 1 H-M components has genus at least 2 .
This , however , follows easily from Lem .
LaTeXMLRef , the situation where one of the orbits ( at level LaTeXMLMath ) has genus 1 .
Use that at level 1 , 2 divides all mp values , except for H-M reps. Then , use that an H-M rep. orbit contains elements with 3 dividing their mp , so it contains non-H-M reps. above which are elements with larger mp .
Prop .
LaTeXMLRef discusses level 0 for this Modular Tower , while the level 1 components and genuses are exactly as for Ex .
LaTeXMLRef .
Here automorphisms join the two H-M and two genus 0 orbits at level 1 , generating the full group of LaTeXMLMath outer automorphisms .
There are two puzzles .
Why are Exs .
LaTeXMLRef and LaTeXMLRef so alike ?
If the two H-M components in either case have definition field LaTeXMLMath ?
We had LaTeXMLCite list each inner Nielsen class , computing the action of the LaTeXMLMath s as elements of LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the braid orbits in LaTeXMLMath .
They each have cardinality 1152 .
For LaTeXMLMath , mod out on LaTeXMLMath by the action of LaTeXMLMath .
In each LaTeXMLMath there are 288 orbits of LaTeXMLMath , each of size 4 .
LaTeXMLCite automatically computes the action of LaTeXMLMath on LaTeXMLMath .
This gives the branch cycles of both degree 288 covers of LaTeXMLMath .
LaTeXMLCite doesn ’ t seem to have facility with split extensions , which meant we interpreted LaTeXMLMath as embedded in LaTeXMLMath for some integer LaTeXMLMath .
We started with a representation of degree 192 , and eventually used a degree 80 representation LaTeXMLMath allowing LaTeXMLCite faster computation .
In LaTeXMLMath let LaTeXMLMath have order 3 , and let LaTeXMLMath be a Klein 4-group in LaTeXMLMath on which LaTeXMLMath acts irreducibly .
The group LaTeXMLMath , LaTeXMLMath and the centralizer of LaTeXMLMath generate has order 24 .
Its cosets give LaTeXMLMath .
By inspection below : LaTeXMLMath has ninety-six 3-cycles ; LaTeXMLMath , one hundred and forty-four 2-cycles ; and LaTeXMLMath , four 2-cycles , six 4-cycles , eight 12-cycles and eight 20-cycles .
From Riemann-Hurwitz , the genus LaTeXMLMath of a cover with these branch cycles satisfies LaTeXMLMath : LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By inspection as above : LaTeXMLMath has ninety-six 3-cycles ; LaTeXMLMath , one hundred and forty-four 2-cycles ; and LaTeXMLMath , eight 4-cycles , eight 6-cycles , eight 10-cycles , four 12-cycles and four 20-cycles .
Riemann-Hurwitz gives LaTeXMLMath : LaTeXMLMath or LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
From Lem .
LaTeXMLRef , there are 16 H-M representatives in LaTeXMLMath .
§ LaTeXMLRef shows all LaTeXMLMath orbits on LaTeXMLMath have length four .
LaTeXMLCite identifies the H-M representatives in LaTeXMLMath as corresponding to the integers LaTeXMLMath in the permutations LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Further , LaTeXMLMath and LaTeXMLMath lie over LaTeXMLMath while LaTeXMLMath and LaTeXMLMath lie over LaTeXMLMath .
Also , it also shows that LaTeXMLMath , the near H-M representatives satisfying ( LaTeXMLRef ) , lie in this orbit .
Let LaTeXMLMath be an LaTeXMLMath orbit on a Nielsen class Ni .
LaTeXMLCite uses an invariant LaTeXMLMath , the big invariant of LaTeXMLMath , a union of conjugacy classes in LaTeXMLMath .
We adapt this for an LaTeXMLMath orbit on LaTeXMLMath .
For LaTeXMLMath define LaTeXMLEquation .
We use a quotient invariant of LaTeXMLMath .
Prop .
LaTeXMLRef , with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath produces LaTeXMLMath with LaTeXMLMath , and any lift of LaTeXMLMath to LaTeXMLMath has order 2 ( Cor .
LaTeXMLRef ) .
There are two LaTeXMLMath orbits on LaTeXMLMath .
Denote these by LaTeXMLMath ; use LaTeXMLMath for their corresponding Hurwitz space components , LaTeXMLMath .
Then , LaTeXMLMath , but LaTeXMLMath .
Therefore , LaTeXMLMath is an obstructed component , having nothing above it in the Modular Tower .
Suppose LaTeXMLMath is a LaTeXMLMath point for any number field LaTeXMLMath .
Then , this corresponds to a LaTeXMLMath cover LaTeXMLMath with the following properties .
LaTeXMLMath is in the Nielsen class LaTeXMLMath .
LaTeXMLMath extends to a LaTeXMLMath sequence of covers LaTeXMLMath with group LaTeXMLMath .
LaTeXMLMath is unramified of degree 2 .
Each LaTeXMLMath satisfies ( LaTeXMLRef LaTeXMLRef ) , but not the combination of ( LaTeXMLRef LaTeXMLRef ) and ( LaTeXMLRef LaTeXMLRef ) .
Prop .
LaTeXMLRef gives the essential ingredient of the proof : The two orbits separate by the values of invariants called LaTeXMLMath and LaTeXMLMath , quotients of LaTeXMLMath and LaTeXMLMath .
Prop .
LaTeXMLRef generalizes LaTeXMLCite .
Assume LaTeXMLMath is LaTeXMLMath -perfect and centerless with LaTeXMLMath the LaTeXMLMath th characteristic quotient of LaTeXMLMath .
Let LaTeXMLMath be the universal exponent LaTeXMLMath central extension of LaTeXMLMath ; this exists from Def .
LaTeXMLRef .
Write LaTeXMLMath as LaTeXMLMath , and the closure of LaTeXMLMath in LaTeXMLMath as LaTeXMLMath .
The canonical morphism LaTeXMLMath factors through LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath .
Then , LaTeXMLMath acts trivially on LaTeXMLMath .
For LaTeXMLMath , the LaTeXMLMath module map of LaTeXMLMath to LaTeXMLMath by the LaTeXMLMath th power is injective .
So , a lift of LaTeXMLMath to LaTeXMLMath has order LaTeXMLMath if and only if the image of LaTeXMLMath in LaTeXMLMath is nontrivial .
This characterizes LaTeXMLMath as a quotient of LaTeXMLMath .
Since LaTeXMLMath is a Frattini cover with kernel of exponent LaTeXMLMath , LaTeXMLMath factors through LaTeXMLMath .
There are two types of generators of LaTeXMLMath : LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath .
By assumption , there are LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
From LaTeXMLMath , LaTeXMLEquation .
Similarly , LaTeXMLMath .
Conclude : LaTeXMLMath acts trivially on LaTeXMLMath .
Finally , suppose LaTeXMLMath .
Since LaTeXMLMath is a pro-free , pro- LaTeXMLMath group , LaTeXMLMath is nontrivial if and only if LaTeXMLMath is not in LaTeXMLMath .
This translates the last sentence .
∎ § LaTeXMLRef combines the next corollary with the computation of Prop .
LaTeXMLRef .
Prop .
LaTeXMLRef , with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath produces LaTeXMLMath .
In this case LaTeXMLMath is the spin cover of LaTeXMLMath .
Use Prop .
LaTeXMLRef to inductively define a quotient LaTeXMLMath of LaTeXMLMath , LaTeXMLMath so that LaTeXMLMath ; all lifts of the nontrivial element of LaTeXMLMath to LaTeXMLMath have order LaTeXMLMath .
Let LaTeXMLMath , so LaTeXMLMath .
As previously , LaTeXMLMath and LaTeXMLMath is the pullback of LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be the representation cover of LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath is a natural LaTeXMLMath ( acting trivially ) quotient module of LaTeXMLMath .
With LaTeXMLMath the image of LaTeXMLMath in LaTeXMLMath , LaTeXMLMath is a single element .
Let LaTeXMLMath be a lift of LaTeXMLMath to LaTeXMLMath where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then , each of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is isomorphic to LaTeXMLMath .
Since LaTeXMLMath is a Frattini cover of LaTeXMLMath it is a quotient of LaTeXMLMath .
This defines LaTeXMLMath as the maximal quotient of LaTeXMLMath on which LaTeXMLMath acts trivially .
So , there is Frattini cover LaTeXMLMath , a central extension of LaTeXMLMath with kernel LaTeXMLMath .
Each LaTeXMLMath element of LaTeXMLMath has a unique LaTeXMLMath lift to LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath .
Then , the unique lifts of the entries of LaTeXMLMath to LaTeXMLMath determine the image of LaTeXMLMath in LaTeXMLMath .
Further , this element only depends on the LaTeXMLMath orbit of LaTeXMLMath .
The two given generators in the groups LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath have product of order 2 .
The resulting groups have order 8 , the Klein 4-group as a quotient and a pair of generators with one of order 4 and one of order 2 .
Only the group LaTeXMLMath has these properties : elements of order 4 don ’ t generate the dihedral group , and the quaternion group ’ s only element of order 2 is in its center .
∎ The next proposition gives the key invariant separating the two orbits LaTeXMLMath and LaTeXMLMath .
Together with the sh -incidence matrix of § LaTeXMLRef it explains precisely the two LaTeXMLMath orbits on the reduced Nielsen class LaTeXMLMath .
Let LaTeXMLMath be an H-M representative .
Consider LaTeXMLEquation .
With no loss , assume LaTeXMLMath ( Cor .
LaTeXMLRef ) .
Let LaTeXMLMath have as entries the unique lifts of LaTeXMLMath entries to LaTeXMLMath .
Then , LaTeXMLMath equals LaTeXMLMath .
With no loss , take LaTeXMLMath as a representative of a given LaTeXMLMath orbit LaTeXMLMath .
Then , if LaTeXMLMath : LaTeXMLMath ; and LaTeXMLMath or LaTeXMLMath .
The corresponding statement for LaTeXMLMath is LaTeXMLMath , with LaTeXMLMath or LaTeXMLMath .
A LaTeXMLMath orbit satisfying either ( LaTeXMLRef ) or ( LaTeXMLRef ) contains 8 reduced classes of such LaTeXMLMath s. If LaTeXMLMath has order 3 , and LaTeXMLMath , both LaTeXMLMath and LaTeXMLMath are order 3 lifts to LaTeXMLMath of LaTeXMLMath .
So , they are equal .
The first computation is almost trivial : LaTeXMLEquation .
The lifting invariant is a braid invariant .
The formulas of ( LaTeXMLRef ) and ( LaTeXMLRef ) come to computing LaTeXMLMath when without the hats it is 1 .
Each element in the pullback LaTeXMLMath of LaTeXMLMath in LaTeXMLMath has order 2 .
Therefore , LaTeXMLMath is an elementary 2-group and a LaTeXMLMath module .
Suppose LaTeXMLMath has order 3 .
For LaTeXMLMath , LaTeXMLMath is the trivial element if and only if LaTeXMLMath .
So , as LaTeXMLMath varies , LaTeXMLMath ( or LaTeXMLMath ) runs over an LaTeXMLMath module LaTeXMLMath of LaTeXMLMath mapping to LaTeXMLMath by the canonical map LaTeXMLMath .
Further , LaTeXMLMath has the same image as LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath differ by the element generating the centralizer of LaTeXMLMath .
In particular , LaTeXMLMath also maps to LaTeXMLMath under this map .
Now suppose LaTeXMLMath .
Then , there are LaTeXMLMath with LaTeXMLMath having the same image in LaTeXMLMath .
Conclude : There exists LaTeXMLMath with LaTeXMLMath if and only if LaTeXMLMath .
If , however , LaTeXMLMath , then LaTeXMLMath is a proper LaTeXMLMath module of LaTeXMLMath mapping surjectively to LaTeXMLMath .
So , as a LaTeXMLMath module , LaTeXMLMath is the direct sum of LaTeXMLMath and LaTeXMLMath .
Consider LaTeXMLMath .
Cor .
LaTeXMLRef shows this is a LaTeXMLMath module .
Then , LaTeXMLMath is a LaTeXMLMath module of order 4 .
Thus , it is either LaTeXMLMath or LaTeXMLMath .
Suppose LaTeXMLMath , and LaTeXMLMath acts nontrivially here .
This would force LaTeXMLMath to have a nontrivial LaTeXMLMath quotient , a contradiction to LaTeXMLMath being 2-perfect .
Now assume LaTeXMLMath and LaTeXMLMath acts trivially .
Then , LaTeXMLMath is a central Frattini cover with kernel LaTeXMLMath .
Exclude this because the Schur multiplier of LaTeXMLMath is LaTeXMLMath .
So LaTeXMLMath holds .
We may choose LaTeXMLMath to map trivially and LaTeXMLMath to map nontrivially by LaTeXMLMath to LaTeXMLMath .
Thus , as an abelian group we can can write LaTeXMLMath as LaTeXMLMath .
Since LaTeXMLMath maps surjectively to LaTeXMLMath and has the same order , It is isomorphic to LaTeXMLMath .
Contrary to how we formed LaTeXMLMath , each element in LaTeXMLMath would have order 2 .
This concludes showing LaTeXMLMath : Both LaTeXMLMath orbits are nonempty .
We show there are eight values of LaTeXMLMath with LaTeXMLMath ( resp .
-1 ) .
Consider pairs LaTeXMLMath with LaTeXMLMath over the trivial element in LaTeXMLMath .
These form an LaTeXMLMath module LaTeXMLMath consisting of the two cosets of the submodule LaTeXMLMath where LaTeXMLMath .
So , each of the two invariants contribute eight elements .
Now we show the distinguishing conditions of ( LaTeXMLRef ) and ( LaTeXMLRef ) hold .
If LaTeXMLMath is in an H-M rep. orbit , then LaTeXMLEquation .
Otherwise LaTeXMLMath and LaTeXMLMath .
Toward the former , consider LaTeXMLMath when LaTeXMLMath is an H-M , a near H-M rep. or a complement of such .
Let LaTeXMLMath centralize LaTeXMLMath , LaTeXMLMath .
Then , LaTeXMLMath .
Let LaTeXMLMath centralize LaTeXMLMath .
For the complement LaTeXMLMath of LaTeXMLMath , LaTeXMLMath .
Next , suppose LaTeXMLMath where LaTeXMLMath centralizes LaTeXMLMath .
As in Prop .
LaTeXMLRef , this is a near H-M rep .
Here LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath for the complement LaTeXMLMath of LaTeXMLMath .
To show LaTeXMLMath , use the Lem .
LaTeXMLRef proof .
Assume LaTeXMLMath is a lift of LaTeXMLMath and LaTeXMLMath is a lift of LaTeXMLMath .
As usual write LaTeXMLMath as LaTeXMLMath : LaTeXMLMath ( resp .
LaTeXMLMath ) permutes the coordinates as LaTeXMLMath ( resp .
LaTeXMLMath ) .
Since LaTeXMLMath ( resp .
LaTeXMLMath ) , LaTeXMLMath and LaTeXMLMath .
Also , LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath and LaTeXMLMath is LaTeXMLEquation .
Let LaTeXMLMath be the subgroup mapping isomorphically to LaTeXMLMath by the action of LaTeXMLMath .
To simplify exponents replace LaTeXMLMath by LaTeXMLMath .
Similarly , form LaTeXMLMath .
The above shows that for all LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are in the same conjugacy class of LaTeXMLMath .
To finish it suffices that the remaining eight pairs , with LaTeXMLMath , LaTeXMLMath are in different conjugacy classes in LaTeXMLMath .
Do an explicit calculation using LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Elements in LaTeXMLMath : LaTeXMLEquation .
Apply LaTeXMLMath ( resp .
LaTeXMLMath ) to this list , to get two new lists LaTeXMLMath , LaTeXMLMath .
For example , the 6th item in LaTeXMLMath ( resp .
the 10th item in List LaTeXMLMath ) is LaTeXMLMath ( resp .
LaTeXMLMath ) .
Treat these two lists as giving a permutation LaTeXMLMath of LaTeXMLMath , where LaTeXMLMath fixes LaTeXMLMath , LaTeXMLMath .
The rules for LaTeXMLMath : LaTeXMLMath if the ith element of LaTeXMLMath equals the LaTeXMLMath th element of LaTeXMLMath .
The result : LaTeXMLEquation .
The final step is to check the collection of pairs LaTeXMLMath where LaTeXMLMath .
Eight of these pairs are in LaTeXMLMath or in LaTeXMLMath .
Examples : LaTeXMLMath while LaTeXMLMath .
∎ There are two LaTeXMLMath orbits on LaTeXMLMath .
These correspond to two components LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , and in turn to the two values achieved by LaTeXMLMath as LaTeXMLMath runs over LaTeXMLMath .
Points on the genus 12 component LaTeXMLMath have branch cycle descriptions LaTeXMLMath for which LaTeXMLMath ( Prop .
LaTeXMLRef ) .
Alternating groups come with a standard faithful permutation representation .
Some characteristic Frattini covers have faithful ( less obvious ) permutation representations .
This subsection has our main illustration of spin separation ( Def .
LaTeXMLRef ) .
It is that the lifting invariant LaTeXMLMath for an LaTeXMLMath orbit LaTeXMLMath on LaTeXMLMath ( from LaTeXMLMath ) comes from the spin cover of LaTeXMLMath ( or of either LaTeXMLMath or LaTeXMLMath ) .
Suppose LaTeXMLMath and LaTeXMLMath is a permutation representation of a finite group LaTeXMLMath of degree LaTeXMLMath .
Assume the image of LaTeXMLMath is in LaTeXMLMath , and let LaTeXMLMath be the central nonsplit extension of LaTeXMLMath with kernel LaTeXMLMath .
Finally , let LaTeXMLMath be the fiber product of LaTeXMLMath and LaTeXMLMath .
If elements of odd order generate LaTeXMLMath , then any permutation representation LaTeXMLMath has range in LaTeXMLMath .
Also , LaTeXMLMath is nonsplit if some LaTeXMLMath of order LaTeXMLMath has a lift of order 4 in LaTeXMLMath ( Prop .
LaTeXMLRef applies ; involution lifting property ) .
Elements of odd order in LaTeXMLMath have a presentation as a product of an even number of disjoint 2-cycles , so they are always in LaTeXMLMath .
If LaTeXMLMath is split , then any element LaTeXMLMath has a lift LaTeXMLMath having the same order .
Since LaTeXMLMath is a central extension with order 2 kernel , the order of a lift of an element of even order is independent of the lift .
This concludes the proof .
∎ Let LaTeXMLMath be a finite group , with LaTeXMLMath denoting the trivial LaTeXMLMath module of dimension 1 over LaTeXMLMath .
Consider LaTeXMLMath .
This always defines a central extension of LaTeXMLMath , and it does so canonically if LaTeXMLMath is LaTeXMLMath -perfect ( Def .
LaTeXMLRef and the following comments ) .
Suppose LaTeXMLMath .
Call a faithful LaTeXMLMath an LaTeXMLMath -spin separating representation if LaTeXMLMath realizes LaTeXMLMath .
When reference to LaTeXMLMath is clear refer to the representation as spin separating .
In our application to LaTeXMLMath the extension LaTeXMLMath is from Prop .
LaTeXMLRef , spin separating representations of LaTeXMLMath interpret that LaTeXMLMath is LaTeXMLMath .
Even if each involution of LaTeXMLMath lifts to an involution of LaTeXMLMath ( as in Lem .
LaTeXMLRef ) , LaTeXMLMath may still not split ( § LaTeXMLRef ) .
Use the notation of Cor .
LaTeXMLRef .
For LaTeXMLMath , a faithful permutation representation , LaTeXMLMath .
If a lift of LaTeXMLMath to LaTeXMLMath has order 4 , then the following hold .
Each LaTeXMLMath lifts to have order 4 .
Each LaTeXMLMath lifts to have order 2 .
LaTeXMLMath is the extension LaTeXMLMath .
Therefore , with LaTeXMLMath , compute the lifting invariant LaTeXMLMath of the LaTeXMLMath orbit of LaTeXMLEquation ( from Prop .
LaTeXMLRef ) as LaTeXMLMath .
Apply Lem .
LaTeXMLRef .
Elements of order 3 generate LaTeXMLMath .
Now suppose LaTeXMLMath lifts to an element of order 4 .
Then , the nonsplit central extension LaTeXMLMath must be the quotient of the universal central extension of the perfect group LaTeXMLMath that characterizes orders of lifts from LaTeXMLMath in Prop .
LaTeXMLRef .
The criterion for this is that LaTeXMLMath is a product of LaTeXMLMath disjoint 2-cycles with LaTeXMLMath odd ( Prop .
LaTeXMLRef ) .
∎ Consider LaTeXMLMath as an LaTeXMLMath module from Prop .
LaTeXMLRef .
Let LaTeXMLMath , and denote the corresponding ( right ) coset representation by LaTeXMLMath with LaTeXMLMath .
We list faithful spin separating representations LaTeXMLMath .
This is equivalent to LaTeXMLMath lifts to order 4 elements of LaTeXMLMath .
Prop .
LaTeXMLRef characterizes this with two properties .
LaTeXMLMath contains no nontrivial subgroup normal in LaTeXMLMath .
LaTeXMLMath acting on LaTeXMLMath cosets has LaTeXMLMath fixed points with LaTeXMLMath and LaTeXMLMath odd .
A faithful representation of LaTeXMLMath is not compatible ( as in Def .
LaTeXMLRef ) with the standard degree 5 representation on LaTeXMLMath .
Reason : Any lift of LaTeXMLMath ( which contains a 2-Sylow of LaTeXMLMath ) contains a 2-Sylow of LaTeXMLMath .
So , it does not give a faithful representation .
Denote a LaTeXMLMath -Sylow of LaTeXMLMath ( resp .
LaTeXMLMath ) by LaTeXMLMath ( resp .
LaTeXMLMath ) .
Notation for the centralizer in LaTeXMLMath of the set LaTeXMLMath is LaTeXMLMath .
Let LaTeXMLMath be any permutation representation with LaTeXMLMath defining the cosets .
For LaTeXMLMath , denote by LaTeXMLMath the subset of LaTeXMLMath ( not necessarily subgroup ) whose inverses conjugate LaTeXMLMath into LaTeXMLMath .
Since multiplication by LaTeXMLMath on the left maps LaTeXMLMath into LaTeXMLMath , LaTeXMLMath consists of right cosets of LaTeXMLMath .
Use LaTeXMLMath as the number of these .
Then LaTeXMLMath fixes LaTeXMLMath integers .
Denote the orbit of LaTeXMLMath under LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be the orbits of LaTeXMLMath on LaTeXMLMath .
Choose LaTeXMLMath , LaTeXMLMath .
Then , LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath is spin separating .
Then , some LaTeXMLMath is in LaTeXMLMath .
Further , the 2-Sylow of LaTeXMLMath has order at least LaTeXMLMath and at most LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath , and LaTeXMLMath .
From our previous notation LaTeXMLMath conjugates LaTeXMLMath into LaTeXMLMath .
Right cosets of LaTeXMLMath in LaTeXMLMath are the LaTeXMLMath cosets that LaTeXMLMath fixes .
Suppose LaTeXMLMath conjugates LaTeXMLMath to LaTeXMLMath .
Running over LaTeXMLMath gives elements whose inverses conjugate LaTeXMLMath into LaTeXMLMath .
If LaTeXMLMath conjugates LaTeXMLMath to LaTeXMLMath , then LaTeXMLMath with LaTeXMLMath .
So , the cosets with representatives conjugating LaTeXMLMath to LaTeXMLMath are in one-one correspondence with LaTeXMLMath .
The test for spin separation does not depend the choice of LaTeXMLMath .
If LaTeXMLMath contains no LaTeXMLMath , then LaTeXMLMath contains the 2-Sylow of LaTeXMLMath and LaTeXMLMath fixes no cosets : LaTeXMLMath .
This implies LaTeXMLMath with LaTeXMLMath odd .
Also , the 2-Sylow of LaTeXMLMath can ’ t contain LaTeXMLMath , or else the representation won ’ t be faithful .
So the 2-Sylow LaTeXMLMath of LaTeXMLMath has order at most 8 .
This implies LaTeXMLMath is divisible by LaTeXMLMath , a contradiction to ( LaTeXMLRef LaTeXMLRef ) .
Now assume the 2-Sylow of LaTeXMLMath has order at most 8 .
As above , let LaTeXMLMath be the number of cosets LaTeXMLMath fixes .
Then , LaTeXMLMath and LaTeXMLMath with LaTeXMLMath odd , implies LaTeXMLMath with LaTeXMLMath odd .
The centralizer of LaTeXMLMath has a 2-Sylow of order LaTeXMLMath , and this centralizer is in LaTeXMLMath .
This implies LaTeXMLMath , a contradiction .
Now suppose the 2-Sylow LaTeXMLMath of LaTeXMLMath has order at least LaTeXMLMath .
Assume first that LaTeXMLMath surjects onto a 2-Sylow LaTeXMLMath of LaTeXMLMath .
Then LaTeXMLMath is the whole 2-Sylow LaTeXMLMath of LaTeXMLMath , for LaTeXMLMath is a Frattini cover .
If , however , LaTeXMLMath does not surject onto a 2-Sylow of LaTeXMLMath , then the kernel of the map has order LaTeXMLMath .
So , LaTeXMLMath contains LaTeXMLMath , and LaTeXMLMath is not faithful .
Conclude : LaTeXMLMath .
∎ Prop .
LaTeXMLRef lists properties of the spin separating representations of LaTeXMLMath of degree 120 .
Prop .
LaTeXMLRef does the same for the degree 60 and 40 representations .
Let LaTeXMLMath have order 4 .
Its image LaTeXMLMath and LaTeXMLMath determine LaTeXMLMath .
Given LaTeXMLMath , possible LaTeXMLMath are elements of LaTeXMLMath fixed by LaTeXMLMath .
Let LaTeXMLMath be a Klein 4-group on which LaTeXMLMath acts .
With LaTeXMLMath , LaTeXMLMath spin separates if and only if LaTeXMLMath is odd : LaTeXMLMath is nontrivial on LaTeXMLMath .
If LaTeXMLMath ( resp .
LaTeXMLMath ) , then LaTeXMLMath ( resp .
LaTeXMLMath ) , LaTeXMLMath .
Further , corresponding to the two choices : LaTeXMLMath ( resp .
LaTeXMLMath ) if LaTeXMLMath ; LaTeXMLMath ( resp .
LaTeXMLMath ) if LaTeXMLMath ; and LaTeXMLMath if LaTeXMLMath .
Four conjugacy classes of subgroups LaTeXMLMath correspond to these choices .
They give all spin separating representations of degree 120 .
Suppose LaTeXMLMath .
Then , LaTeXMLMath acts imprimitively on ten sets of integers of cardinality 12 .
Each set consists of the integers fixed by an element of LaTeXMLMath .
This induces the degree 10 representation of LaTeXMLMath on ordered pairs of distinct integers from LaTeXMLMath .
If LaTeXMLMath has order 10 , then LaTeXMLMath is a product of twelve 5-cycles and six 10-cycles .
If LaTeXMLMath has order 6 , then LaTeXMLMath is a product of four 3-cycles and eighteen 6-cycles .
Use the notation for LaTeXMLMath of order 4 in the statement .
To be explicit assume LaTeXMLMath — LaTeXMLMath up to conjugacy LaTeXMLMath — LaTeXMLMath LaTeXMLMath lifts LaTeXMLMath .
Given one lift LaTeXMLMath , all others have the shape LaTeXMLMath .
Map LaTeXMLMath by LaTeXMLMath .
Cor .
LaTeXMLRef says the kernel of LaTeXMLMath has rank 3 , so the image LaTeXMLMath of LaTeXMLMath is a homogeneous space for the squares of lifts of LaTeXMLMath .
Conclude : LaTeXMLMath and LaTeXMLMath determine LaTeXMLMath ; and LaTeXMLMath runs over the subset LaTeXMLMath of LaTeXMLMath that LaTeXMLMath fixes .
Suppose LaTeXMLMath has order 16 and LaTeXMLMath is spin separating .
Then , some LaTeXMLMath has order 4 , LaTeXMLMath and LaTeXMLMath stabilizes LaTeXMLMath .
Write LaTeXMLMath .
Eight divides the degree of the representation .
As in Lem .
LaTeXMLRef , spin separation is equivalent to LaTeXMLMath being four times an odd number .
First case : Suppose LaTeXMLMath is trivial on the Klein 4-group LaTeXMLMath .
Cor .
LaTeXMLRef says LaTeXMLMath contains exactly two conjugates of LaTeXMLMath , and all of LaTeXMLMath centralizes LaTeXMLMath .
Let LaTeXMLMath be 3 ( resp .
5 ) if LaTeXMLMath ( resp .
LaTeXMLMath ) .
So , LaTeXMLMath fixes LaTeXMLEquation integers .
Therefore , 8 divides LaTeXMLMath , and LaTeXMLMath is not spin separating .
Second Case : Suppose LaTeXMLMath is nontrivial on LaTeXMLMath .
If LaTeXMLMath is the only element in LaTeXMLMath in its conjugacy class , then the computation just concluded gives LaTeXMLEquation .
So , 4 exactly divides LaTeXMLMath and LaTeXMLMath is spin separating .
If three elements in LaTeXMLMath are conjugate to LaTeXMLMath , then LaTeXMLMath .
Now suppose LaTeXMLMath .
As LaTeXMLMath is nontrivial on LaTeXMLMath , conjugates ( under LaTeXMLMath ) of LaTeXMLMath in LaTeXMLMath fall in two LaTeXMLMath orbits .
Let LaTeXMLMath be a representative of one of these orbits .
Then , LaTeXMLMath , so LaTeXMLMath .
The argument is similar for 3 conjugates of LaTeXMLMath in LaTeXMLMath .
Now we count subgroups LaTeXMLMath , up to conjugacy in LaTeXMLMath , giving spin separating representations of degree 120 .
Giving LaTeXMLMath is equivalent to giving pairs LaTeXMLMath with LaTeXMLMath subject to these conditions .
LaTeXMLEquation .
Note : LaTeXMLMath and LaTeXMLMath are in the same conjugacy class of LaTeXMLMath ; this asks exactly that LaTeXMLMath and LaTeXMLMath are in different conjugacy classes of LaTeXMLMath .
Count elements of LaTeXMLMath from the proof of Cor .
LaTeXMLRef .
With the cosets of LaTeXMLMath labeled as there , LaTeXMLMath represents LaTeXMLMath .
Thus , its centralizer in LaTeXMLMath is LaTeXMLEquation .
Calculate : LaTeXMLMath .
Uniquely determine a representative for LaTeXMLMath by taking LaTeXMLMath and LaTeXMLMath .
The elements LaTeXMLMath run over all choices LaTeXMLMath .
So , LaTeXMLMath equals the number of pairs LaTeXMLMath and LaTeXMLMath with one having three nonzero entries , and the other not .
Given any pair LaTeXMLMath , there are two choices for LaTeXMLMath satisfying this condition .
Conjugate the set of groups LaTeXMLMath attached to these choices by any lift of LaTeXMLMath ( centralizing LaTeXMLMath ) .
The result is a distinct subgroup .
Conclude there are four such subgroups , falling in pairs according to LaTeXMLMath or LaTeXMLMath .
Now , suppose LaTeXMLMath .
As above , for LaTeXMLMath , LaTeXMLMath fixes twelve integers .
Running over the ten elements of LaTeXMLMath , transitivity of the representation guarantees some element of LaTeXMLMath fixes any given integer in LaTeXMLMath .
As LaTeXMLMath , these sets of fixed integers are disjoint , and they form a set of imprimitivity .
With no loss assume LaTeXMLMath , so LaTeXMLMath .
Elements of LaTeXMLMath are exactly those permuting the fixed integers for LaTeXMLMath .
Acting on LaTeXMLMath cosets gives the degree 10 representation of LaTeXMLMath on pairs of distinct integers from LaTeXMLMath .
Continue the hypothesis LaTeXMLMath .
Suppose LaTeXMLMath has order 10 .
Then , LaTeXMLMath has order 5 , and LaTeXMLMath fixes no integers .
On the other hand , LaTeXMLMath .
From above , LaTeXMLMath fixes exactly sixty integers .
Conclude : LaTeXMLMath is a product of twelve 5-cycles and six 10-cycles .
For LaTeXMLMath of order 6 , similarly deduce LaTeXMLMath is a product of four 3-cycles and eighteen 6-cycles .
∎ The following is similar to Prop .
LaTeXMLRef .
We leave details to the reader .
Continue Prop .
LaTeXMLRef notation ( for LaTeXMLMath ) .
Assume LaTeXMLMath has order three , LaTeXMLMath has order LaTeXMLMath and the image of LaTeXMLMath is an LaTeXMLMath .
Then , with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath spin separates .
There are two conjugacy classes of such groups LaTeXMLMath corresponding to these choices giving all spin separating representations of degree 40 .
If LaTeXMLMath are distinct , the four fixed integers of LaTeXMLMath are distinct from those fixed by LaTeXMLMath .
Let LaTeXMLMath be one of the groups with LaTeXMLMath of degree 120 , and let LaTeXMLMath .
Then , LaTeXMLMath has 4 or 6 elements .
Exactly in the former case LaTeXMLMath spin separates .
There are two conjugacy classes of such groups LaTeXMLMath corresponding to these choices giving all spin separating representations of degree 60 .
If LaTeXMLMath , as in Prop .
LaTeXMLRef , the conditions on LaTeXMLMath are that it centralizes this element of LaTeXMLMath and it acts on LaTeXMLMath in the standard Klein 4-group representation .
For this case , LaTeXMLMath ( resp .
20 and 8 ) if LaTeXMLMath ( resp .
LaTeXMLMath and LaTeXMLMath ) .
∎ Prop .
LaTeXMLRef and Prop .
LaTeXMLRef include the complete list of spin separating permutation representations of the LaTeXMLMath orbits on LaTeXMLMath .
Assume for LaTeXMLMath , LaTeXMLMath gives a faithful spin separation .
According to Lem .
LaTeXMLRef , the 2-Sylow LaTeXMLMath of LaTeXMLMath has order LaTeXMLMath or LaTeXMLMath and it contains an element LaTeXMLMath .
Prop .
LaTeXMLRef lists all cases where LaTeXMLMath is 120 , 40 or 60 .
The only degrees left as dangling possibilities are where LaTeXMLMath has order divisible by 5 or where it has order LaTeXMLMath .
In the former case , since LaTeXMLMath contains nontrivial elements of LaTeXMLMath , it must contain all of LaTeXMLMath .
So it is not faithful .
Similarly for the latter case : The action of an element of order 3 on LaTeXMLMath forces this to be all of LaTeXMLMath .
∎ Restriction of an element from LaTeXMLMath to LaTeXMLMath is injective if LaTeXMLMath is the LaTeXMLMath -Sylow of LaTeXMLMath LaTeXMLCite .
Sometimes this allows taking LaTeXMLMath to be a LaTeXMLMath -group .
We use this and produce a 2-group extension LaTeXMLMath with the following properties .
The kernel of LaTeXMLMath is LaTeXMLMath .
LaTeXMLMath is nonsplit and spin separating : LaTeXMLMath , pullback from an embedding LaTeXMLMath for some LaTeXMLMath .
Each involution of LaTeXMLMath lifts to an involution of LaTeXMLMath .
The quaternion group presentation is in Def .
LaTeXMLRef : LaTeXMLMath of order LaTeXMLMath has generators LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the group from dropping the condition LaTeXMLMath .
Then , the only involution LaTeXMLMath in LaTeXMLMath lifts to an involution in LaTeXMLMath .
This extension , however , does not split .
The calculations below apply for LaTeXMLMath even to all the groups LaTeXMLMath .
For simplicity we do the case LaTeXMLMath and LaTeXMLMath , using LaTeXMLMath and LaTeXMLMath ; LaTeXMLMath has generators LaTeXMLMath and LaTeXMLMath , with relations LaTeXMLMath .
For LaTeXMLMath even , LaTeXMLMath doesn ’ t have an extension satisfying ( LaTeXMLRef ) , though LaTeXMLMath does .
( This observation should start with the Klein 4-group whose LaTeXMLMath extension in Lem .
LaTeXMLRef is an example of spin separation . )
The extension LaTeXMLMath has no spin separating extension .
Still , LaTeXMLMath has an extension LaTeXMLMath satisfying all properties of ( LaTeXMLRef ) .
The group LaTeXMLMath has only one faithful transitive permutation representation as a subgroup of LaTeXMLMath .
In this representation elements of order 4 have the shape LaTeXMLMath .
So these generators are in LaTeXMLMath : LaTeXMLMath .
This is the only embedding we must test to check if LaTeXMLMath has a spin separating representation of any kind .
Form a central extension of LaTeXMLMath by mapping LaTeXMLMath to LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
In the extension , the square of LaTeXMLMath is cleaved away from the squares LaTeXMLMath ( they remain equal ) .
Let LaTeXMLMath be the trivial LaTeXMLMath module .
Recall : LaTeXMLMath has rank LaTeXMLMath and order LaTeXMLMath .
So , the 1st characteristic 2-Frattini module LaTeXMLMath ( notation of § LaTeXMLRef ) of LaTeXMLMath has dimension LaTeXMLMath LaTeXMLCite : Schreier ’ s formula for the number of generators of a subgroup of a free group of rank LaTeXMLMath and index LaTeXMLMath .
As a LaTeXMLMath module it is indecomposable LaTeXMLCite .
The maximal quotient of it on which LaTeXMLMath acts trivially is LaTeXMLMath .
We ouline our computation that LaTeXMLMath has dimension two .
Use the notation of LaTeXMLCite .
The augmentation map gives LaTeXMLMath .
Denote its kernel by LaTeXMLMath .
The version of Jenning ’ s Theorem ( LaTeXMLCite or LaTeXMLCite ) in LaTeXMLCite gives an effective tool for computing the Loewy layers of any LaTeXMLMath -group ring using the Poincaré-Witt basis of its universal enveloping algebra .
The Loewy layer at the top of LaTeXMLMath ( the second radical layer in LaTeXMLMath ) is LaTeXMLMath .
The projective module LaTeXMLMath maps naturally and surjectively to LaTeXMLMath extending the direct sum of two augmentation maps LaTeXMLMath .
The kernel of this map is LaTeXMLMath LaTeXMLCite .
The explicit basis of LaTeXMLMath from the group elements produces a natural interpretation of LaTeXMLMath as a matrix .
Row reduce and compute explicitly a basis of LaTeXMLMath from that of LaTeXMLMath .
This gives LaTeXMLMath a LaTeXMLMath module structure .
Rational canonical form of the matrix action LaTeXMLMath of LaTeXMLMath ( resp .
LaTeXMLMath and LaTeXMLMath ) on this module determines the maximal quotient on which LaTeXMLMath ( resp .
LaTeXMLMath ) acts like the identity matrix , giving the maximal quotient on which LaTeXMLMath acts like the identity .
We let GAP do this computation .
All non-split extensions of LaTeXMLMath are isomorphic as groups though not as extensions .
As in § LaTeXMLRef , the isomorphisms between them come from LaTeXMLMath which is isomorphic to a Klein 4-group .
Extensions correspond to which order 4 subgroup of LaTeXMLMath cleaves from the other order 4 subgroups .
The proof of Prop .
LaTeXMLRef reviews the generators and relations for LaTeXMLMath .
In that notation , symbols LaTeXMLMath , LaTeXMLMath ( in the multiplicative group of units in the Clifford algebra ) , generate subject to the relations LaTeXMLMath and LaTeXMLMath .
The map LaTeXMLMath appears from LaTeXMLMath .
Finally , LaTeXMLMath .
Like LaTeXMLMath , LaTeXMLMath has no non-trivial coreless subgroup .
So , only its regular representation is faithful and its generators LaTeXMLMath and LaTeXMLMath , of order 4 , have the shape LaTeXMLMath ; they are in LaTeXMLMath .
Involutions in LaTeXMLMath are the products of 8 disjoint two cycles .
So Prop .
LaTeXMLRef implies they lift to the same order on pullback in LaTeXMLMath .
Now we show LaTeXMLMath ( the pullback of LaTeXMLMath to LaTeXMLMath ) does not split off LaTeXMLMath .
If the extension splits , the lifts would retain the relation LaTeXMLMath .
So , it suffices that the square of any lift of LaTeXMLMath differs from the square of any lift of LaTeXMLMath .
With no loss : LaTeXMLEquation .
Write each 4-cycle ’ s lift to LaTeXMLMath as the product of three obvious generators .
Example : LaTeXMLMath lifts to LaTeXMLMath .
Square the lifts of LaTeXMLMath and LaTeXMLMath using the relations from the previous paragraph .
Then multiply the two squares together to get -1 , the generator of the kernel of LaTeXMLMath .
This also shows -1 is in the Frattini subgroup , so the extension is Frattini , and therefore nonsplit .
The same procedure for LaTeXMLMath shows the squares of the lifts of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are the same .
Every non-split extension of LaTeXMLMath has one of these cleaved off from the rest .
This shows the spin extension of LaTeXMLMath splits .
∎ Representations in Prop .
LaTeXMLRef give spin separation of elements in LaTeXMLMath .
§ LaTeXMLRef gives the lifting invariant that separates the two LaTeXMLMath orbits on LaTeXMLMath .
§ LaTeXMLRef and § LaTeXMLRef together suggest why the two orbits have the same LaTeXMLMath monodromy groups .
§ LaTeXMLRef concludes the paper with lessons for similar computations at higher levels .
First we characterize commuting pairs of elements of order 2 in LaTeXMLMath .
Denote the elements of LaTeXMLMath that are of products of LaTeXMLMath disjoint 2-cycles by LaTeXMLMath .
In the situation where LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath commute , we will identify the product .
The most complicated standard situation is where LaTeXMLMath is a LaTeXMLMath -pair : LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Given LaTeXMLMath as above , let LaTeXMLMath be the number of disjoint 2-cycles appearing in LaTeXMLMath having no support in LaTeXMLMath .
Note : LaTeXMLMath may not equal LaTeXMLMath .
Let LaTeXMLMath by the disjoint 2-cycles appearing in both LaTeXMLMath and LaTeXMLMath .
Suppose LaTeXMLMath is a LaTeXMLMath -pair , LaTeXMLMath and LaTeXMLMath ( resp .
LaTeXMLMath and LaTeXMLMath ) have no common support , and LaTeXMLMath and LaTeXMLMath commute .
We say the LaTeXMLMath -pair LaTeXMLMath appears in LaTeXMLMath .
Finally , let LaTeXMLMath be the number of LaTeXMLMath -pairs appearing in LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath with LaTeXMLMath .
Write LaTeXMLMath as LaTeXMLMath with LaTeXMLMath ( resp .
LaTeXMLMath ) the disjoint cycles in LaTeXMLMath that don ’ t ( resp .
do ) appear in LaTeXMLMath .
Similarly , write LaTeXMLMath as LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath commute , so do LaTeXMLMath and LaTeXMLMath .
Proving the lemma amounts to extracting a LaTeXMLMath -pair LaTeXMLMath that appears in LaTeXMLMath .
Then , an induction characterizes LaTeXMLMath as given by products of disjoint LaTeXMLMath -pairs .
Suppose LaTeXMLMath appears in a disjoint cycle LaTeXMLMath in LaTeXMLMath .
By assumption , LaTeXMLMath appears in a disjoint cycle LaTeXMLMath in LaTeXMLMath .
Since we have extracted LaTeXMLMath , LaTeXMLMath .
Suppose LaTeXMLMath is not in a disjoint cycle of LaTeXMLMath .
Then , LaTeXMLMath applied to LaTeXMLMath has the effect LaTeXMLMath .
Thus , LaTeXMLMath has order larger than 2 .
So , contrary to a previous deduction , LaTeXMLMath and LaTeXMLMath don ’ t commute .
Therefore , LaTeXMLMath appears in a disjoint cycle LaTeXMLMath of LaTeXMLMath .
Calculate : LaTeXMLMath applied to LaTeXMLMath has the effect LaTeXMLMath .
Now compute the effect of LaTeXMLMath on LaTeXMLMath to conclude LaTeXMLMath contains the disjoint cycle LaTeXMLMath .
That is , LaTeXMLMath appears in LaTeXMLMath .
∎ Suppose LaTeXMLMath , as in Prop .
LaTeXMLRef .
Suppose LaTeXMLMath are distinct .
For this case , LaTeXMLMath and LaTeXMLMath : LaTeXMLMath and LaTeXMLMath .
Integers not in the support of LaTeXMLMath appear in 2-cycles in LaTeXMLMath , and give LaTeXMLMath .
Conclude LaTeXMLMath : LaTeXMLMath .
Use that LaTeXMLMath to conclude LaTeXMLMath .
Let LaTeXMLMath have order 5 , and denote by LaTeXMLMath a lift of it to an element of order 10 in LaTeXMLMath .
With LaTeXMLMath giving the degree 40 representation used above , let LaTeXMLMath be its image ( isomorphic to LaTeXMLMath ) in LaTeXMLMath .
Then , we can speak of the 20 cosets of LaTeXMLMath lying above the cosets LaTeXMLMath , LaTeXMLMath , of LaTeXMLMath .
Analogous to previous calculations , LaTeXMLMath consists of four 5-cycles and two 10-cycles .
The integers moved ( resp .
fixed ) by LaTeXMLMath are in the support of the 10-cycles ( resp .
5-cycles ) .
Suppose LaTeXMLMath .
Then , multiplying LaTeXMLMath on the right of these cosets gives four 5-cycles of LaTeXMLMath .
Two 10-cycles comprise the remaining action of LaTeXMLMath .
If LaTeXMLMath , then the description of the 5-cycles and 10-cycles of LaTeXMLMath switches .
If LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
The number of integers in the support of 2-cycles from LaTeXMLMath is the same as in the 10-cycles of LaTeXMLMath .
From Prop .
9.12 this means there are twenty such integers , and therefore two 10-cycles in LaTeXMLMath .
Since this element fixes no integers there are also four 5-cycles in LaTeXMLMath .
Without loss , take the 20 cosets of LaTeXMLMath above the cosets LaTeXMLMath , LaTeXMLMath , of LaTeXMLMath to be LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
The LaTeXMLMath s are four representatives of LaTeXMLMath cosets in LaTeXMLMath .
First assume LaTeXMLMath .
For each LaTeXMLMath , LaTeXMLMath cycles LaTeXMLMath , LaTeXMLMath ( use that LaTeXMLMath commutes with LaTeXMLMath ) .
The argument is the same if LaTeXMLMath : Write the distinct cosets above as LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Now inspect the relation between two elements of LaTeXMLMath .
There are six elements in LaTeXMLMath .
An element LaTeXMLMath of order 5 that stabilizes LaTeXMLMath is transitive on the remaining five elements of LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , these values are independent of their arguments .
Use that LaTeXMLMath and LaTeXMLMath .
Thus , LaTeXMLMath , and LaTeXMLMath , 1 or 0 .
If LaTeXMLMath and LaTeXMLMath have eight common fixed integers , then they also move eight common integers .
So we show the former and conclude LaTeXMLMath and LaTeXMLMath .
With no loss , assume LaTeXMLMath and find the cosets from among LaTeXMLMath fixed by LaTeXMLMath .
Since LaTeXMLMath , as LaTeXMLMath varies , LaTeXMLMath runs over all five other elements of LaTeXMLMath .
Use LaTeXMLMath to see LaTeXMLMath fixes exactly two of these cosets as LaTeXMLMath varies , and eight cosets in all .
∎ The two LaTeXMLMath orbits on LaTeXMLMath have the same image groups , and degrees .
This is despite their being considerably different as representations .
They are clearly permutation inequivalent representations , or else LaTeXMLMath would have the same shape on each orbit .
More strikingly , however , the permutation representations are inequivalent as representations ( have different traces ) .
The number of LaTeXMLMath orbits on each of the LaTeXMLMath orbits LaTeXMLMath and LaTeXMLMath ( resp .
26 and 32 LaTeXMLMath orbits ) is different .
This shows the two representations are different ( see for example LaTeXMLCite ) despite their similarities .
§ LaTeXMLRef speculates on representation theory ( based on LaTeXMLCite ) that might explain this and also measure the difference between these two representations .
From Lem .
LaTeXMLRef , the direct product of the free group LaTeXMLMath and LaTeXMLMath equals LaTeXMLMath .
Denote LaTeXMLMath ( LaTeXMLMath in ( LaTeXMLRef ) ) by LaTeXMLMath .
The following presentation of LaTeXMLMath is superior for our purposes .
It easily follows from the notation and proof of Lemma LaTeXMLRef .
Recall : The effect of LaTeXMLMath on LaTeXMLMath is conjugation by LaTeXMLMath , LaTeXMLMath .
We state this only for LaTeXMLMath , though it works for any value of LaTeXMLMath .
These are generators of LaTeXMLMath : LaTeXMLEquation .
Any representative element LaTeXMLMath in a Nielsen class LaTeXMLMath produces an evaluation homomorphism LaTeXMLMath mapping LaTeXMLMath to 1 .
Conversely , any homomorphism LaTeXMLMath mapping LaTeXMLMath to 1 produces an associated Nielsen class representative .
For LaTeXMLMath , act on LaTeXMLMath by applying evalution of LaTeXMLMath to LaTeXMLMath .
An orbit on Ni is equivalent to a LaTeXMLMath orbit on the homomorphisms LaTeXMLMath ( up to conjugation by LaTeXMLMath ) .
For the computation of orbits of LaTeXMLMath , consider the induced action of LaTeXMLMath on LaTeXMLMath .
Denote images of LaTeXMLMath by LaTeXMLMath , so these generate a free group with one relation LaTeXMLMath .
An obvious reformulation of Prop .
LaTeXMLRef has a rephrasing in LaTeXMLMath orbits using LaTeXMLMath .
Use the same notation LaTeXMLMath for the corresponding evaluation homomorphisms .
Referring to the generators LaTeXMLMath allows us give the Nielsen class of a homomorphism by what conjugacy classes the generators hit .
Let LaTeXMLMath be an automorphism of LaTeXMLMath .
We say LaTeXMLMath preserves a Nielsen class LaTeXMLMath if LaTeXMLMath is in the same Nielsen class as LaTeXMLMath .
If LaTeXMLMath preserves the Nielsen class Ni , then it permutes LaTeXMLMath orbits .
Any two LaTeXMLMath orbits under LaTeXMLMath have the same length .
Further , the monodromy groups ( permutation groups for the action of LaTeXMLMath ) on these orbits will be the same .
Note : While the group on two LaTeXMLMath orbits in Lem .
LaTeXMLRef are the same , the permutation representations may be inequivalent .
Denote the automorphism of LaTeXMLMath by LaTeXMLMath by LaTeXMLMath .
Among the six LaTeXMLMath orbits on LaTeXMLMath ) the automorphism LaTeXMLMath fixes four of them , but permutes the other two .
Is there a Nielsen class preserving automorphism interchanging the two LaTeXMLMath orbits on LaTeXMLMath ?
Explicit formulas ( ( LaTeXMLRef ) and Lem .
LaTeXMLRef ) list properties of LaTeXMLMath characteristic quotients helpful to decide Prob .
LaTeXMLRef and other properties of Modular Towers when LaTeXMLMath .
By example , the paper establishes it is difficult to predict component genuses at level 0 of a Modular Tower .
As in all our LaTeXMLMath ( and LaTeXMLMath ) examples , genus 0 components are common .
For many applications this is excellent ; rational points from such a moduli space produce Galois realizations .
So , establishing that at some level in the Modular Tower all components will have large genus requires exploiting the module representation theory of the universal LaTeXMLMath -Frattini cover .
Whatever the genus of the components at level 0 , it is properties of the universal LaTeXMLMath -Frattini cover that force sufficient ramification at higher levels to have the genus go up ( as in ( LaTeXMLRef ) ) .
Lem .
LaTeXMLRef exploited the hypotheses ( LaTeXMLRef ) , valid in the main Modular Tower LaTeXMLMath .
A more general hypothesis , with LaTeXMLMath a projective system of LaTeXMLMath orbits , is the following : LaTeXMLMath ; and LaTeXMLMath .
These two assumptions give a positive conclusion to Prob .
LaTeXMLRef ( when LaTeXMLMath ) from Lem .
LaTeXMLRef in that there is no bound on the genus of components corresponding to the projective system LaTeXMLMath .
Rem .
LaTeXMLRef and Prob .
LaTeXMLRef view fixed points of LaTeXMLMath and LaTeXMLMath on projective systems of Nielsen classes in a Modular Tower as generalizing a classical observation about fine moduli for the modular curve spaces LaTeXMLMath .
Ex .
LaTeXMLRef for LaTeXMLMath is a valuable test case .
The genus situation from the view of Prob .
LaTeXMLRef is even worse at level 0 and level 1 than for our main example .
There are components , some with low genuses .
Yet , the universal 2- Frattini situations produce the desired ramification .
In all examples , perturbation of level 0 H-M reps. starts the story .
Spin separating representations are the most intriguing element in structured description of Modular Towers levels , like listing real points in Prop .
LaTeXMLRef .
Serre ’ s Prop .
LaTeXMLRef is indispensable , with § LaTeXMLRef using it indirectly even when the initial hypotheses don ’ t apply .
Still , this is only about the prime LaTeXMLMath , and only about universal 2-Frattini covers that have an initial relation with alternating groups .
An elementary approach to producing the desired ramification comes from the conjugacy class counting formula of § LaTeXMLRef .
Suppose LaTeXMLMath is the LaTeXMLMath th characteristic quotient of the universal LaTeXMLMath -Frattini cover of a finite LaTeXMLMath -perfect group LaTeXMLMath .
Assume LaTeXMLMath have LaTeXMLMath order .
Princ .
LaTeXMLRef reminds that obstruction ( from LaTeXMLCite ; as in § LaTeXMLRef ) occurs only when LaTeXMLMath appears in the Loewy display of LaTeXMLMath : LaTeXMLMath with LaTeXMLMath centerless and LaTeXMLMath the trivial LaTeXMLMath module .
With no loss for the question of concern , assume LaTeXMLMath .
Suppose also that LaTeXMLMath does not divide the order of LaTeXMLMath .
Then , consider the triple of conjugacy classes LaTeXMLMath defined by LaTeXMLMath .
Consider ( any ) lifts LaTeXMLMath .
Let LaTeXMLMath be the pullback of LaTeXMLMath in LaTeXMLMath .
Asking if LaTeXMLMath divides the order of LaTeXMLMath is equivalent to asking if element LaTeXMLMath is obstructed in going to LaTeXMLMath .
We continue this topic in § LaTeXMLRef .
§ LaTeXMLRef found the data for computing the sh -incidence matrix for orbit LaTeXMLMath in two stages .
Going to higher levels gives further reason to carefully organize information from LaTeXMLMath .
It makes sense to combine sh -incidence data and the use of Riemann-Hurwitz as in Lem .
LaTeXMLRef .
Prop .
LaTeXMLRef finds all degree 120 spin separating representations of LaTeXMLMath .
That gave subgroups LaTeXMLMath and LaTeXMLMath with the natural map LaTeXMLMath inducing LaTeXMLMath .
Suppose we have a sequence LaTeXMLMath , with each LaTeXMLMath inducing a spin separating representation of LaTeXMLMath , LaTeXMLMath .
Regard the projective limit LaTeXMLMath as giving the compatible system of spin separating representations of LaTeXMLMath .
The proof of Prop .
LaTeXMLRef gives hints of the following sort .
We can expect to find pro-2 LaTeXMLMath containing a pro-cyclic subgroup LaTeXMLMath with generator LaTeXMLMath where LaTeXMLMath is conjugate ( in LaTeXMLMath ) to no other element of LaTeXMLMath .
Lem .
LaTeXMLRef and Lem .
LaTeXMLRef has difficult arguments , respectively about shortening the respective type ( 6,12 ) and type ( 10,20 ) LaTeXMLMath orbits .
These used the precise form of the level 1 characteristic module LaTeXMLMath from Cor .
LaTeXMLRef .
We have yet to capture the essence of LaTeXMLMath shortening from LaTeXMLMath -group representations .
Lem .
LaTeXMLRef says an LaTeXMLMath orbit containing an H-M rep. , at level LaTeXMLMath when LaTeXMLMath , has faithful LaTeXMLMath action .
Note , too , the relation between near H-M and H-M reps. exploited in such places as Prop .
LaTeXMLRef .
Further , the notion of complements also goes with H-M reps .
Consider any Nielsen classes LaTeXMLMath attached to a Modular Tower ( for the prime LaTeXMLMath ) .
Let LaTeXMLMath be a projective system of Nielsen class representatives .
Suppose the conclusion of Prop .
LaTeXMLRef holds for this system : LaTeXMLMath for LaTeXMLMath .
If LaTeXMLMath , Def .
LaTeXMLRef would give LaTeXMLMath as the complement of LaTeXMLMath in LaTeXMLMath .
Both elements lie over the same element of LaTeXMLMath .
A similar notion with a general prime LaTeXMLMath replacing 2 would give LaTeXMLMath such elements as LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath .
These are examples of H-M reps. indicating a moduli ( boundary ) behavior that informs about the components in which they appear .
The following problem appears in § LaTeXMLRef in a special case .
To start with assume LaTeXMLMath is the univeral LaTeXMLMath -Frattini cover of LaTeXMLMath , with notation as usual for the characteristic quotients .
Let LaTeXMLMath be LaTeXMLMath conjugacy classes in LaTeXMLMath .
Refer to LaTeXMLMath as LaTeXMLMath -divisible at level LaTeXMLMath if all lifts LaTeXMLMath to LaTeXMLMath have LaTeXMLMath dividing the order of LaTeXMLMath .
Refer to LaTeXMLMath as LaTeXMLMath -divisible if it is LaTeXMLMath -divisible at some level LaTeXMLMath .
If LaTeXMLMath is a LaTeXMLMath group , then there is a lift of the whole group to LaTeXMLMath for each LaTeXMLMath ( special case of Schur-Zassenhaus ) .
That gives LaTeXMLMath as necessary condition for LaTeXMLMath -divisibility .
Is it sufficient ?
Is LaTeXMLMath dividing LaTeXMLMath sufficient for LaTeXMLMath being LaTeXMLMath -divisible ?
The following says the LaTeXMLMath -divisible property holds for lifting to the universal LaTeXMLMath -Frattini cover of LaTeXMLMath if and only if it holds for lifting to the universal LaTeXMLMath -Frattini cover of LaTeXMLMath .
Recall the universal LaTeXMLMath -Frattini cover of a LaTeXMLMath group is just the group itself , so the lemma applies even if LaTeXMLMath is a LaTeXMLMath group .
That LaTeXMLMath is not LaTeXMLMath -divisible is equivalent to their being no lift LaTeXMLMath of LaTeXMLMath to LaTeXMLMath with LaTeXMLMath having the same order as LaTeXMLMath .
Suppose LaTeXMLMath are LaTeXMLMath elements .
Let LaTeXMLMath be the LaTeXMLMath th characteristic quotient of the universal LaTeXMLMath -Frattini cover of LaTeXMLMath .
Then , LaTeXMLMath is LaTeXMLMath -divisible at level LaTeXMLMath for lifts to LaTeXMLMath if and only if it is LaTeXMLMath -divisible at level LaTeXMLMath for lifts to LaTeXMLMath .
Suppose at each level LaTeXMLMath there are lifts LaTeXMLMath to LaTeXMLMath as above , where LaTeXMLMath has the same order as LaTeXMLMath .
The sets LaTeXMLMath of such lifts in LaTeXMLMath form a projective system of nonempty closed subsets .
So , there is an element in the projective limit of the LaTeXMLMath s giving the desired lift LaTeXMLMath .
Now consider the relation between lifts to LaTeXMLMath to that of lifts to LaTeXMLMath .
Apply Prop .
LaTeXMLRef to the pullback LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
Then , LaTeXMLMath is a LaTeXMLMath -projective cover of LaTeXMLMath .
So , it maps surjectively to the LaTeXMLMath -Frattini cover of LaTeXMLMath .
A lifting LaTeXMLMath to LaTeXMLMath , as in the statement where LaTeXMLMath has the same order as LaTeXMLMath , will map to such a lifting in the image .
Conversely , the universal LaTeXMLMath -Frattini cover of LaTeXMLMath has a map back through LaTeXMLMath over LaTeXMLMath .
Again , a lifting of the type in the lemma to LaTeXMLMath produces one in LaTeXMLMath .
∎ Consider LaTeXMLMath at level 0 .
The two components separate by lifting invariant values .
In the argument for Princ .
LaTeXMLRef , LaTeXMLMath has lifting invariant +1 .
This was a nontrivial case of nonobstruction for LaTeXMLMath .
We still , however , don ’ t know if in the LaTeXMLMath Modular Tower there is obstruction at some level .
In the language here , is the following pair 2- divisible : LaTeXMLMath ?
Reminder of the positive effect of obstructed components : They have nothing above them at higher levels , so that eliminates the component from consideration in the Main Conjecture for any value of LaTeXMLMath .
In the problem here the negative effect of the appearance of LaTeXMLMath s is toward providing sufficient ramification over cusps in the Main Conjecture .
Here is a projective version of it .
Let LaTeXMLMath be a projective system of Nielsen class representatives as in Lem .
LaTeXMLRef .
Characterize when LaTeXMLMath .
What generalization of Serre ’ s Prop .
could deduce for a projective system above LaTeXMLMath what might be the values of LaTeXMLMath .
Even if the genus 0 condition does not hold in Prop .
LaTeXMLRef , is there a general procedure for checking the possible values of the lifting invariant by coalescing branch cycles , as does § LaTeXMLRef , to the case of genus 0 ?
The main effect of Harbater-Mumford representatives on our Main Conjecture has been to give lifts of pairs from level 0 to level LaTeXMLMath where the order of the product does not change .
This has the effect of giving projective systems of cusps on a Modular Tower with relative ramification of degree 1 .
For this property ( depending on the outcome of Prob .
LaTeXMLRef ) , we suggest the natural generalization of H-M reps. is to consider elements LaTeXMLMath with having the following form : LaTeXMLEquation where , LaTeXMLMath is a LaTeXMLMath group , LaTeXMLMath , and for all LaTeXMLMath , LaTeXMLMath .
Consider LaTeXMLMath , LaTeXMLMath and LaTeXMLMath from Table LaTeXMLRef .
None are H-M reps .
Yet , for LaTeXMLMath , each satisfies the LaTeXMLMath property ( LaTeXMLRef ) ; LaTeXMLMath has LaTeXMLMath and LaTeXMLMath isomorphic to LaTeXMLMath ( 5 not dividing the order ) .
H-M reps. were simple enough to allude to them by just the shape of their Nielsen class representives .
Some geometry applications of H-M reps. require not only that ( LaTeXMLRef ) holds .
Suppose also for each LaTeXMLMath , LaTeXMLMath has entries with product 1 , satisfying the genus 0 condition ( on each orbit ) .
Such situations aid in analyzing geometry behind components of reduced Hurwitz spaces .
As an example need for treating Prob .
LaTeXMLRef , note the difficulties in Lem .
LaTeXMLRef in establishing that all LaTeXMLMath orbits at level 1 of LaTeXMLMath have faithful LaTeXMLMath action .
This used details from Cor .
LaTeXMLRef .
§ LaTeXMLRef suggests finding good cosets to go from fundamental a domain for level 0 of the LaTeXMLMath to a fundamental domain for level 1 .
This includes inspecting the monodromy groups LaTeXMLMath of the Galois closure of LaTeXMLMath for LaTeXMLMath large .
This will mostly be a 2-group ( for a general Modular Tower , mostly a LaTeXMLMath -group ) .
When LaTeXMLMath the map has degree 16 .
The monodromy group , LaTeXMLMath has these properties .
LaTeXMLMath and its center has order 2 .
It has a subnormal series with abelian quotients : LaTeXMLMath with LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Note : LaTeXMLMath is not entirely a 2-group , having a copy of LaTeXMLMath at the top .
This is from a LaTeXMLMath inducing an automorphism on a 2-Sylow of LaTeXMLMath , though LaTeXMLMath does not act as an automorphism of LaTeXMLMath through its Nielsen class action .
LaTeXMLCite documents this very Galois-like ( as in LaTeXMLCite ) and subtle phenomenon .
For any a projective curve LaTeXMLMath , denote the divisor class of a divisor LaTeXMLMath on LaTeXMLMath by LaTeXMLMath .
Our concern here is for divisors LaTeXMLMath that are half-canonical : LaTeXMLMath is the canonical class ( of divisors of meromorphic differentials ) on LaTeXMLMath .
Consider the lifting invariant LaTeXMLMath for a cover LaTeXMLMath with a branch cycle description LaTeXMLMath .
Prop .
LaTeXMLRef discusses the special case of the lifting invariant that § LaTeXMLRef calls LaTeXMLMath .
For that case , LaTeXMLMath comes from a spin separating representation and LaTeXMLMath is a lift of LaTeXMLMath conjugacy classes .
Particular cases are in Prop .
LaTeXMLRef and Prop .
LaTeXMLRef .
Much of our technical work is to establish the former proposition is a case of spin separation .
For such a spin separating representation , under special hypotheses , LaTeXMLCite gives a formula for computing LaTeXMLMath , and § LaTeXMLRef gives examples of loosening the genus 0 hypothesis to make this computation .
The formula of LaTeXMLCite ties the lifting invariant to an invariant of Riemann for describing many delicacies about Riemann surfaces .
We explain its potential to create automorphic functions , produced by LaTeXMLMath -nulls , on certain Hurwitz space components .
Such functions produced by natural moduli properties will reveal properties of the moduli space , especially about the nature of the cusps and divisors supported in the cusps .
Assume the cover LaTeXMLMath , LaTeXMLMath has odd order ramification .
At each point LaTeXMLMath , lying over LaTeXMLMath , express LaTeXMLMath locally as LaTeXMLMath with LaTeXMLMath ( if LaTeXMLMath use LaTeXMLMath ) with LaTeXMLMath odd .
As the multiplicity of zeros and poles of LaTeXMLMath is independent of the local uniformizing paramater , the differential LaTeXMLMath has all multiplicities of its zeros and poles even .
Serre treats the case with a general curve LaTeXMLMath replacing LaTeXMLMath .
Then , the monodromy group of the cover may not be in LaTeXMLMath .
That requires extra care describing the precise group cover LaTeXMLMath that one uses for computing the lifting invariant .
Back to LaTeXMLMath .
Let LaTeXMLMath be the divisor LaTeXMLMath , the divisor of a half-canonical class .
Replacing LaTeXMLMath by LaTeXMLMath with LaTeXMLMath replaces LaTeXMLMath be a divisor linearly equivalent to LaTeXMLMath .
The formula is LaTeXMLMath .
Notations are as follows : LaTeXMLMath is the dimension of the linear system of the divisor ; and LaTeXMLMath is the sum over ramification indices LaTeXMLMath of LaTeXMLMath of ramified LaTeXMLMath .
Formulas using ramification indices depend on the permutation representation giving a cover .
For either of the Nielsen classes LaTeXMLMath or LaTeXMLMath ( absolute classes here mean for the standard representations of LaTeXMLMath , LaTeXMLMath or 5 ) , LaTeXMLMath is even .
So , LaTeXMLMath ( see § LaTeXMLRef ) .
This also holds for inner classes in this case .
More significantly , it holds in replacing LaTeXMLMath ( resp .
LaTeXMLMath ) by LaTeXMLMath , the LaTeXMLMath th characteristic quotient of the universal 2-Frattini cover of LaTeXMLMath , and the standard representation of LaTeXMLMath by some spin separating representation LaTeXMLMath of LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath a group with a faithful representation , LaTeXMLMath the stabilizer of 1 ( § LaTeXMLRef ) , we have always to keep in mind the relation of the Galois cover LaTeXMLMath and the cover LaTeXMLMath on the absolute space LaTeXMLMath .
It is the latter we are attaching a half-canonical divisor class ( and LaTeXMLMath function below ) .
Still , we suppress LaTeXMLMath in the continuing discussion .
Regard LaTeXMLMath as a point of LaTeXMLMath .
To simplify notation , refer to a cover associated with LaTeXMLMath as LaTeXMLMath .
Each representation LaTeXMLMath from Prop .
LaTeXMLRef and Prop .
LaTeXMLRef provides an even LaTeXMLMath function LaTeXMLMath on the cover of degree LaTeXMLMath corresponding to a point LaTeXMLMath .
Similarly , there is an even LaTeXMLMath function LaTeXMLMath corresponding to the Galois cover LaTeXMLMath with group LaTeXMLMath attached to LaTeXMLMath ( see § LaTeXMLRef ) .
The final version of LaTeXMLCite gives normalizations of these theta functions ( in their dependency on LaTeXMLMath ) .
That will refer to this exposition for its setup .
Typically the normalization is to integrate the LaTeXMLMath function with respect to a volume form , and set it equal to 1 .
Prop .
LaTeXMLRef implies the geometric realization of the unramified spin cover of LaTeXMLMath comes from a LaTeXMLMath invariant nontrivial 2-torsion point LaTeXMLMath on LaTeXMLMath ; with this description LaTeXMLMath is unique .
In some cases , the ratio LaTeXMLMath ( or its log ) appears in a nontrivial measure of the distance from an H-M cusp , so long as that ratio makes sense , the technical point of the duration of this section .
Let LaTeXMLMath be the moduli space of curves of genus LaTeXMLMath .
Suppose LaTeXMLMath is a Hurwitz space of covers LaTeXMLMath having genus LaTeXMLMath .
Then , let LaTeXMLMath denote the isomorphism class of the curve .
A corollary of one of the main theorems on moduli of curves is that the map LaTeXMLMath is a morphism of quasi-projective varieties .
For our applications , consider LaTeXMLMath : An even function on the universal covering space LaTeXMLMath of LaTeXMLMath , the Jacobian ( divisor classes of degree 0 ) of LaTeXMLMath .
Denote by LaTeXMLMath the positive divisors on LaTeXMLMath of degree LaTeXMLMath ( different than in § LaTeXMLRef where it was a fiber product ) .
LaTeXMLCite guided this , though detecting which bits of this construction depend only on LaTeXMLMath is less obvious .
We need a basis LaTeXMLMath for global holomorphic differentials on LaTeXMLMath to see LaTeXMLMath in its classical coordinates LaTeXMLMath .
Choose a positive degree LaTeXMLMath divisor LaTeXMLMath .
Then , consider any LaTeXMLMath -tuple of paths LaTeXMLMath on LaTeXMLMath where LaTeXMLMath starts at LaTeXMLMath ( denote its endpoint LaTeXMLMath ) , LaTeXMLMath , so LaTeXMLMath is a positive divisor of degree LaTeXMLMath .
The Jacobi inversion theorem says the following about the map LaTeXMLMath .
LaTeXMLMath is surjective .
It factors through LaTeXMLMath .
The induced map LaTeXMLMath identifies LaTeXMLMath with the universal covering space LaTeXMLMath .
This is the classical version of Jacobi Inversion LaTeXMLCite .
If we know LaTeXMLMath is a projective variety , then this complex analytic map LaTeXMLMath is algebraic ( Chow ’ s Lemma LaTeXMLCite or LaTeXMLCite ) .
It is a birational morphism , identifying the two function fields .
The differentials LaTeXMLMath identify with the differentials LaTeXMLMath , which live on LaTeXMLMath .
Further , for any integer LaTeXMLMath and any divisor LaTeXMLMath of degree LaTeXMLMath , a similar map LaTeXMLEquation factors through LaTeXMLMath .
Taking LaTeXMLMath and LaTeXMLMath gives an embedding LaTeXMLMath ( assuming LaTeXMLMath ) .
Again by Chow ’ s Lemma , this gives coordinates on LaTeXMLMath compatible with those of LaTeXMLMath .
Recover the basis LaTeXMLMath as restriction of LaTeXMLMath to LaTeXMLMath .
Finally , let LaTeXMLMath be a positive divisor of degree LaTeXMLMath .
Then , LaTeXMLMath is a divisor on LaTeXMLMath .
Forming this data , however , seemed to require choices in LaTeXMLMath and the positive divisors .
Under what circumstances can one use this to construct objects analytically varying in the coordinates of LaTeXMLMath ( alone LaTeXMLMath — LaTeXMLMath not dependent on our choices ) ?
We are looking for a global construction tightly tying the curves LaTeXMLMath in the family to coordinates for LaTeXMLMath .
That is the topic for the rest of this section .
Even the most relaxed conditions won ’ t give explicit LaTeXMLMath functions in classical coordinates from a uniform construction .
Here are reasons .
A nontrivial family of curves will have nontrivial monodromy on the 1st homology ( or cohomology ) of its fibers .
Analytic coordinates in LaTeXMLMath for an embedding LaTeXMLMath come from an analytic choice LaTeXMLMath of divisor class of degree LaTeXMLMath .
From ( LaTeXMLRef LaTeXMLRef ) , in nontrivial algebraic families it is impossible to get a uniform basis LaTeXMLMath of holomorphic differentials .
Since LaTeXMLMath naturally embeds in LaTeXMLMath , use LaTeXMLMath to give analytic coordinates in ( LaTeXMLRef LaTeXMLRef ) .
It is especially significant for LaTeXMLMath and LaTeXMLMath .
A LaTeXMLMath analytic in LaTeXMLMath need not have a representative divisor ( positive or not ) with coordinates in LaTeXMLMath .
The divisor class is sufficient .
Finding , however , such an analytically varying divisor usually requires it appear from the map LaTeXMLMath through ramification .
That isn ’ t a formal definition , though several illustrative situations occur in LaTeXMLCite .
An example of it for ( LaTeXMLRef LaTeXMLRef ) when LaTeXMLMath is having an analytically varying half-canonical divisor class .
In turn that produces a precise copy in LaTeXMLMath of the divisor LaTeXMLMath .
Let LaTeXMLMath be any meromorphic function on LaTeXMLMath of degree LaTeXMLMath .
Then , LaTeXMLMath has zeros LaTeXMLMath and poles LaTeXMLMath .
Suppose we have an embedding of LaTeXMLMath in LaTeXMLMath .
So , each zero LaTeXMLMath and pole of LaTeXMLMath produces a point in LaTeXMLMath .
List these as LaTeXMLMath , LaTeXMLMath .
Yet , finding these LaTeXMLMath s doesn ’ t require giving LaTeXMLMath .
It only needs points LaTeXMLMath and LaTeXMLMath on LaTeXMLMath viewed as inside LaTeXMLMath .
Define LaTeXMLEquation to be the sum of all the LaTeXMLMath s minus all the LaTeXMLMath s on LaTeXMLMath .
To say LaTeXMLMath is zero means it is the origin of LaTeXMLMath .
Existence of LaTeXMLMath with these zeros and poles characterizes exactly when LaTeXMLMath is zero .
If LaTeXMLMath exists , consider the logarithmic derivative LaTeXMLMath .
This is a meromorphic differential of 3rd kind with pure imaginary periods .
Even if LaTeXMLMath doesn ’ t exist , given the divisor LaTeXMLMath above , the following always holds .
There is a unique differential LaTeXMLMath with residue divisor LaTeXMLMath having pure imaginary periods .
Suppose LaTeXMLMath is not zero , but LaTeXMLMath is zero on LaTeXMLMath for some integer LaTeXMLMath .
Then , repeating all the zeros and poles LaTeXMLMath times produces a function LaTeXMLMath on LaTeXMLMath .
The LaTeXMLMath th root of LaTeXMLMath defines a cyclic unramified degree LaTeXMLMath cover LaTeXMLMath .
Riemann produced LaTeXMLMath functions to provide uniformizing coordinates for this construction .
Torelli space LaTeXMLMath is a cover of LaTeXMLMath .
The points of LaTeXMLMath lying over LaTeXMLMath consists of a canonical basis for LaTeXMLMath for which we borrow the notation of LaTeXMLMath and LaTeXMLMath generators from ( LaTeXMLRef ) .
See LaTeXMLCite for the full abelian variety context of the next statement .
The fiber of LaTeXMLMath over LaTeXMLMath is a homogeneous space for LaTeXMLMath the group LaTeXMLEquation .
The standard normalization is to choose LaTeXMLMath so the matrix LaTeXMLMath with LaTeXMLMath entry LaTeXMLMath is the identity matrix .
The matrix LaTeXMLMath with LaTeXMLMath entry LaTeXMLMath is the period matrix .
Then LaTeXMLMath is symmetric and has positive definite imaginary part ( Riemann ’ s bilinear relations : LaTeXMLCite ) .
Given this , there is a unique LaTeXMLMath function LaTeXMLMath attached to LaTeXMLMath .
Many mathematical items on LaTeXMLMath appear constructively from this .
This includes functions and meromorphic differentials ( with particular zeros and poles ) .
This was a central goal in generalizing Abel ’ s Theorem : To provide Abel ( -Jacobi ) constructions for a general Riemann surface .
For the function LaTeXMLMath it has this look : LaTeXMLEquation .
In LaTeXMLMath you see LaTeXMLMath coordinates ; the LaTeXMLMath th entry is LaTeXMLMath , an integral over a path on LaTeXMLMath .
The integrals make sense up to integration around closed paths .
So , they define a point in LaTeXMLMath .
Even if LaTeXMLMath doesn ’ t exist , the logarithmic differential of ( LaTeXMLRef ) does .
It gives the third kind differential from ( LaTeXMLRef ) .
This gives the differential equation defining LaTeXMLMath functions .
In expressing LaTeXMLMath , replace LaTeXMLMath by a vector LaTeXMLMath in the universal covering space of the Jacobian .
Form the logarithmic differential of it : LaTeXMLMath .
To construct his LaTeXMLMath functions , Riemann following Abel ’ s case for cubic equations , imposed the condition that translations by periods would change the logarithmic differential only by addition of a constant .
With LaTeXMLMath the gradient in LaTeXMLMath , LaTeXMLMath is invariant under the lattice of periods .
The main point about LaTeXMLMath functions ( of 1st order for dimension LaTeXMLMath ) is that there is essentially just one : LaTeXMLMath where LaTeXMLMath runs over symmetric LaTeXMLMath matrices with imaginary part positive definite .
The LaTeXMLMath divisor determines LaTeXMLMath for fixed LaTeXMLMath up to a constant multiple , though we can always translate the LaTeXMLMath divisor on LaTeXMLMath .
The space of translated divisors is a homogeneous space for LaTeXMLMath .
In varying the LaTeXMLMath function with LaTeXMLMath , our concerns include that choice of constant , and also the analytic choice in LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
It depends on our goals ( one element of depth in Riemann ’ s results ) what type of choice we want for LaTeXMLMath .
We illustrate with Riemann ’ s original goal .
To represent all functions LaTeXMLMath appropriately in ( LaTeXMLRef ) , the following is necessary : LaTeXMLMath must be an odd function so as to give 0 at LaTeXMLMath with multiplicity 1 .
LaTeXMLCite makes a choice to replace the matrices LaTeXMLMath and LaTeXMLMath respectively by LaTeXMLMath and LaTeXMLMath is symmetric and its real part is negative definite .
This scaling doesn ’ t change the isomorphism class of LaTeXMLMath and it has numerous advantages .
With this notation , and all these choices , rewrite Riemann ’ s classical function LaTeXMLMath as LaTeXMLEquation .
One can see the difference in the formulas by contrasting with the notation of LaTeXMLCite .
The LaTeXMLMath factor appears in their expression for a LaTeXMLMath function , and their version of the heat equation .
LaTeXMLCite only considers the case LaTeXMLMath .
From our viewpoint , that is because they only consider moduli spaces of genus 1 curves where the LaTeXMLMath -invariant for a point LaTeXMLMath is the LaTeXMLMath -invariant of the curve it represents : These are modular curves .
Their LaTeXMLMath -nulls are close to automorphic functions on these curves , and the relations among the LaTeXMLMath - nulls with various characteristics are algebraic equations describing these curves .
LaTeXMLCite : In fact , partial motivation for the results discussed so far is to better understand the multivariable case .
Even for LaTeXMLMath , Modular Towers forces relating moduli spaces of curves of infinitely many genuses parametrized by reduced 1-dimensional Hurwitz spaces .
When LaTeXMLMath , the moduli spaces are upper half plane quotients .
So comparing with LaTeXMLMath -nulls from modular curves is inevitable .
Our period matrices LaTeXMLMath for the curve LaTeXMLMath are therefore ( matrix ) functions of LaTeXMLMath , rather than just being LaTeXMLMath .
Notice : Because a lattice is invariant under multiplication by -1 , LaTeXMLMath is an even function of LaTeXMLMath .
This isn ’ t the function that works in ( LaTeXMLRef ) .
Since the divisors for the LaTeXMLMath functions differ only by a translation , giving an appropriate function for ( LaTeXMLRef ) comes by translating LaTeXMLMath .
Further , the even LaTeXMLMath from Prop .
LaTeXMLRef , for LaTeXMLMath , is not likely to be this one .
As in § LaTeXMLRef use LaTeXMLMath for the transpose of a matrix .
Let LaTeXMLMath with LaTeXMLMath , represent an arbitrary point of LaTeXMLMath .
A special notation LaTeXMLMath — LaTeXMLMath given by characteristics LaTeXMLMath — LaTeXMLMath expresses the LaTeXMLMath translated by LaTeXMLMath ( then multiplied by an exponential factor ) : LaTeXMLMath LaTeXMLEquation .
There are LaTeXMLMath choices of LaTeXMLMath giving even LaTeXMLMath s ( LaTeXMLMath LaTeXMLMath — LaTeXMLMath even characteristics ) and LaTeXMLMath choices of LaTeXMLMath giving odd LaTeXMLMath s. Riemann ’ s Theorem identifies the LaTeXMLMath from Prop .
LaTeXMLRef with an even characteristic LaTeXMLCite .
Again consider LaTeXMLMath , and let LaTeXMLMath be a closed path in LaTeXMLMath based at LaTeXMLMath .
Suppose LaTeXMLMath is a choice of canonical homology basis for LaTeXMLMath .
It is classical that analytic continuation around LaTeXMLMath induces a linear transformation LaTeXMLMath on LaTeXMLMath : LaTeXMLMath .
Using Lem .
LaTeXMLRef , this represents LaTeXMLMath as an element LaTeXMLMath .
For the modular curve case , the identification of LaTeXMLMath with LaTeXMLMath ( Fay ’ s notation ) makes this transparent because each path identifies with an element of LaTeXMLMath .
LaTeXMLCite , LaTeXMLCite and LaTeXMLCite use a transformation formula that explains how LaTeXMLMath functions change with an application of LaTeXMLMath a homology basis .
Here are is a qualitative description .
Change LaTeXMLMath to LaTeXMLMath so analytic continuation around LaTeXMLMath gives the new normalized period matrix LaTeXMLMath .
Express a translate of LaTeXMLMath that gives the new LaTeXMLMath , a function of LaTeXMLMath , as an explicit multiple of LaTeXMLMath .
Express LaTeXMLMath as LaTeXMLMath .
Denote LaTeXMLMath by LaTeXMLMath .
To simplify , refer to LaTeXMLMath as LaTeXMLMath .
Then , LaTeXMLMath and LaTeXMLMath .
For a LaTeXMLMath matrix LaTeXMLMath , use bracket notation LaTeXMLMath for the vector of diagonal elements ( as in LaTeXMLCite ) .
The result : LaTeXMLEquation .
Here LaTeXMLMath , having absolute value 1 , depends on which branch of LaTeXMLMath we have chosen .
When LaTeXMLMath , LaTeXMLCite has LaTeXMLMath explicitly , recognizing LaTeXMLMath as an 8th root of 1 .
View LaTeXMLMath as producing the matrix whose LaTeXMLMath entry is the partial with respect to the variable in the LaTeXMLMath position of LaTeXMLMath .
Suppose LaTeXMLMath is a Hurwitz family ( any LaTeXMLMath ) for some absolute Nielsen class from LaTeXMLMath ( or inner cases ) .
Call LaTeXMLMath g ( alois ) c ( losure ) -ramified if for any cover LaTeXMLMath in the Nielsen class , there is no unramified cover LaTeXMLMath , of degree exceeding 1 , fitting in a diagram LaTeXMLMath with the Galois closure LaTeXMLMath of LaTeXMLMath .
The notion gc-ramified ( without the name ) is from LaTeXMLCite which characterizes it using a branch cycle description LaTeXMLMath for LaTeXMLMath .
It is usual that LaTeXMLMath is gc- ramified ( true in our cases ) , though nonpathological examples abound where it is not LaTeXMLCite .
Let LaTeXMLMath be any subgroup of LaTeXMLMath .
Consider the permutation representation LaTeXMLMath on LaTeXMLMath cosets .
There is a map from disjoint cycles of LaTeXMLMath , LaTeXMLMath , to disjoint cycles of LaTeXMLMath : Any LaTeXMLMath orbit of LaTeXMLMath cosets goes to the corresponding orbit of LaTeXMLMath cosets .
Call LaTeXMLMath unramified in LaTeXMLMath if the lengths of the disjoint cycle of LaTeXMLMath and the corresponding disjoint cycle of LaTeXMLMath are equal .
Let LaTeXMLMath be minimal among LaTeXMLMath with LaTeXMLMath unramified in LaTeXMLMath , LaTeXMLMath .
Then , LaTeXMLMath is gc-ramified if and only if LaTeXMLMath .
In the notation above , let LaTeXMLMath be the branch points of LaTeXMLMath , LaTeXMLMath a set of classical generators of LaTeXMLMath ( § LaTeXMLRef ) , and LaTeXMLMath the genus of LaTeXMLMath .
By a LaTeXMLMath -null we mean a function of the form LaTeXMLMath , a LaTeXMLMath with characteristic evaluated at LaTeXMLMath .
Let LaTeXMLMath be the multiplicative group of complex numbers having absolute value 1 .
The expression LaTeXMLMath refers to a function at LaTeXMLMath up to multiplication by elements in LaTeXMLMath .
Assume LaTeXMLMath is gc-ramified .
Then there is an effective procedure for computing LaTeXMLMath on LaTeXMLMath .
Assume LaTeXMLMath .
This action extends to LaTeXMLMath acting on the reduced Nielsen classes .
For LaTeXMLMath , let LaTeXMLMath be the stabilizer in LaTeXMLMath of the reduced class of LaTeXMLMath .
Then , there is an effective procedure for computing LaTeXMLMath acting on LaTeXMLMath ( extending the action of § LaTeXMLRef and Prop .
LaTeXMLRef ) .
This induces a map LaTeXMLMath through which LaTeXMLMath acts as in ( LaTeXMLRef ) .
Suppose elements in the classes C all have odd order and there is an even half-canonical class on LaTeXMLMath given by LaTeXMLMath according to the conclusion of Prop .
LaTeXMLRef .
Let LaTeXMLMath be the characteristic corresponding to it ( relative to LaTeXMLMath ) at the point LaTeXMLMath .
If LaTeXMLMath is not 0 at all points LaTeXMLMath , then the action of LaTeXMLMath gives a LaTeXMLMath -null mod LaTeXMLMath , an automorphic function on LaTeXMLMath .
The effective procedure for computing the monodromy is LaTeXMLCite .
For LaTeXMLMath , consider LaTeXMLMath the corresponding elements in LaTeXMLMath , so and LaTeXMLMath and LaTeXMLMath are the corresponding LaTeXMLMath factors .
Then , LaTeXMLEquation .
So , mod LaTeXMLMath , the LaTeXMLMath s are factors of automorphy .
From LaTeXMLCite , however , finding factors of automorphy without modding out by LaTeXMLMath improves if we know LaTeXMLMath is in the level 2 congruence subgroup of LaTeXMLMath .
Then , the 4th power of the LaTeXMLMath -null is an automorphic function .
∎ Again , suppose elements in the classes C all have odd order and LaTeXMLMath gives an even half-canonical class on LaTeXMLMath as in the conclusion of Prop .
LaTeXMLRef .
Let LaTeXMLMath be the spin cover .
Let LaTeXMLMath represent the unique LaTeXMLMath invariant nontrivial 2-division point on LaTeXMLMath corresponding to the unramified spin cover LaTeXMLMath corresponding to spin separation .
Denote the genus of LaTeXMLMath by LaTeXMLMath , global holomorphic differentials on LaTeXMLMath by LaTeXMLMath .
As in LaTeXMLCite , LaTeXMLMath is isogenous to LaTeXMLMath with LaTeXMLMath ( the Prym ) an abelian variety of dimension LaTeXMLMath .
Properties of this construction : The tangent space at LaTeXMLMath identifies with the LaTeXMLMath eigenspace , of the involution of LaTeXMLMath commuting with LaTeXMLMath , on LaTeXMLMath .
The isogenies between LaTeXMLMath and LaTeXMLMath have degree LaTeXMLMath in each direction , composing to multiplication by 2 , with LaTeXMLMath generating LaTeXMLMath .
More than just the involution of LaTeXMLMath , all of LaTeXMLMath acts on this construction .
Analysis of this prym varying canonically with LaTeXMLMath will be in the final version of LaTeXMLCite ( see § LaTeXMLRef for the easiest appearance of a Prym ) .
Consider again LaTeXMLMath .
Is there a nondegenerate LaTeXMLMath -null ( nonzero and varying with LaTeXMLMath ) through LaTeXMLMath represented as a point on the moduli space of curves of genus 21 ?
This is what Prop .
LaTeXMLRef has as an hypothesis .
This requires at least that the image LaTeXMLMath of the curve for LaTeXMLMath vary nontrivially in the moduli space LaTeXMLMath as LaTeXMLMath varies .
LaTeXMLCite shows the moduli of all families of inner LaTeXMLMath covers , LaTeXMLMath as in Ex .
LaTeXMLRef , varies nontrivially .
Our case , however , is inner covers with LaTeXMLMath , so this does not apply directly .
It is more elementary .
Let LaTeXMLMath be an inner reduced Hurwitz space with LaTeXMLMath , where the genus LaTeXMLMath of the covers exceeds 1 .
The moduli map LaTeXMLMath is nonconstant .
Restrict the argument to any geometrically connected component of LaTeXMLMath .
Assume the result is false and LaTeXMLMath is the constant image of the moduli map .
So , with LaTeXMLMath running over LaTeXMLMath , there are an infinite number of inequivalent Galois covers LaTeXMLMath with group LaTeXMLMath ( isomorphic to LaTeXMLMath , though not necessarily equal to it ) .
Consider the group LaTeXMLMath these LaTeXMLMath s generate in the automorphism group of LaTeXMLMath .
Since LaTeXMLMath , this is a finite group .
So there are only finitely many covers up to equivalence in this class .
The family is connected : There is only one cover up to equivalence , contradicting the above .
∎ LaTeXMLCite provides coordinates for period matrix degeneration as points on LaTeXMLMath approach its boundary .
The simplest corollary considers being near that part of the boundary with period matrices close to products of period matrices of elliptic curves .
All the even LaTeXMLMath -nulls are nonzero and all the odd LaTeXMLMath -nulls are nondegenerate on an elliptic curve .
( § LaTeXMLRef therefore shows LaTeXMLMath has a nonzero even LaTeXMLMath -null . )
So , these coordinates show the same holds for jacobians close to such points .
For H-M components of Modular Tower levels , although the degeneration isn ’ t quite so simple , this type of analysis also works .
This is the Mumford half of the reason for calling them Harbater-Mumford representatives ( continuing in LaTeXMLCite ) .
Computations toward Main Conjecture Prob .
LaTeXMLRef LaTeXMLMath — LaTeXMLMath showing the genus of components of levels of a reduced Modular Tower go up LaTeXMLMath — LaTeXMLMath require recognizing the organization of LaTeXMLMath orbits within the LaTeXMLMath orbits on the Nielsen class .
Every curve cover in a Nielsen class deforms to a cover associated with collections of cusps .
Equivalencing covers according to which cusps are in their deformation range ( call this d ( eformation ) - equivalence ) is equivalent to finding LaTeXMLMath orbits .
Acting with a braid generator on Nielsen classes is quite elementary .
Yet , as the main examples of this paper show , finding LaTeXMLMath orbits is usually difficult .
Further , even when we find from LaTeXMLMath action that there are several components , we need to know if these components are conjugate under LaTeXMLMath .
In significant cases we expect intrinsic geometry ( of covers in our case ) to inform LaTeXMLMath about d-equivalence .
Our examples show that augmenting bare braid group computations with lifting invariants often reveals such intrinsic geometry .
Two covers certainly have differences in their d-equivalence classes if one has unramified covers with a given group the other does not have .
While there are other ways to say they have different geometries , this and its generalizations are d-equivalence invariants .
The remainder of this section formulates this purely group theoretically .
Up to d-equivalence , assume two covers ( in the same Nielsen class ) whose LaTeXMLMath orbits we wish to compare have the same branch points LaTeXMLMath so can compare them as covers by their branch cycles using the same classical generators of LaTeXMLMath .
If the monodromy group of the cover is LaTeXMLMath , let these branch cycles be LaTeXMLMath and LaTeXMLMath ( both in LaTeXMLMath ; § LaTeXMLRef ) .
Consider LaTeXMLMath , and a choice of LaTeXMLMath : LaTeXMLMath conjugacy classes lifting to LaTeXMLMath those of C .
For LaTeXMLMath , consider the LaTeXMLMath -lifting invariant : LaTeXMLEquation .
Use LaTeXMLMath for the collection of LaTeXMLMath running over all LaTeXMLMath to separate d-inequivalent elements in LaTeXMLMath .
Two cases appear often in this paper .
LaTeXMLMath for some integer LaTeXMLMath and LaTeXMLMath is the pullback of LaTeXMLMath to LaTeXMLMath in Prop .
LaTeXMLRef .
LaTeXMLMath is the LaTeXMLMath characteristic quotient of the universal LaTeXMLMath -Frattini cover of the group LaTeXMLMath , and LaTeXMLMath in Prop .
LaTeXMLRef .
When LaTeXMLMath , and LaTeXMLMath , we may compare ( LaTeXMLRef LaTeXMLRef ) with ( LaTeXMLRef LaTeXMLRef ) , and ask if the latter is a case of the former through some embedding of LaTeXMLMath in an alternating group .
That is exactly what happens in our main example at level 1 ( Prop .
LaTeXMLRef ) .
The following easy lemma slightly generalizes LaTeXMLCite LaTeXMLMath is an LaTeXMLMath invariant : LaTeXMLEquation any LaTeXMLMath .
It is a d-equivalence invariant .
Call LaTeXMLMath and LaTeXMLMath in LaTeXMLMath Nielsen separated if the collection LaTeXMLMath differs from LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are in separate d-equivalence classes , are LaTeXMLMath and LaTeXMLMath provably ( computationally , significantly ) different so as to detect this ?
The case of Nielsen separation from this paper is where LaTeXMLMath is a Frattini central extension with LaTeXMLMath of order prime to the orders of elements in C .
Construct LaTeXMLMath as the conjugacy classes C of LaTeXMLMath , each lifted ( uniquely ) to LaTeXMLMath to have the same order .
The next lemma follows easily from the technique of LaTeXMLCite .
For LaTeXMLMath , LaTeXMLMath is a single element .
Assume LaTeXMLMath and conjugacy classes in C appear many times .
Then LaTeXMLMath .
So , there are at least LaTeXMLMath orbits for LaTeXMLMath on LaTeXMLMath .
General Nielsen separation uses complicated witnessing pairs LaTeXMLMath .
In the last example , the LaTeXMLMath witnessing separation is a small cover of LaTeXMLMath .
Lem .
LaTeXMLRef shows that embeddings in alternating groups are only a tiny part of the story of general Nielsen separation of orbits of LaTeXMLMath .
For instance , consider the universal LaTeXMLMath -Frattini cover LaTeXMLMath of any simple group LaTeXMLMath ( LaTeXMLMath ; Thm .
LaTeXMLRef ) or LaTeXMLMath -split , LaTeXMLMath -perfect group with non-cyclic LaTeXMLMath -Sylow ( Prop .
LaTeXMLRef ) .
Then , LaTeXMLMath has infinitely many quotients LaTeXMLMath with LaTeXMLMath a central ( Frattini ) extension with LaTeXMLMath .
In any of these circumstances , consider such a group LaTeXMLMath and corresponding LaTeXMLMath conjugacy classes LaTeXMLMath satisfying the hypotheses of Lem .
LaTeXMLRef .
Then , the Hurwitz space LaTeXMLMath has at least LaTeXMLMath components .
For , however , a fixed LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath ( say , LaTeXMLMath ) consider for which LaTeXMLMath -tuples of LaTeXMLMath conjugacy classes C , does the collection of covers LaTeXMLMath Nielsen separate LaTeXMLMath ( equivalently LaTeXMLMath ) orbits on LaTeXMLMath ( with LaTeXMLMath running over the mentioned quotients of LaTeXMLMath ) .
Use the notation of Prop .
LaTeXMLRef .
Figuring the image of LaTeXMLMath ( up to subgroups of finite index ) in LaTeXMLMath or in LaTeXMLMath generalizes the goal of Serre ’ s Theorem in LaTeXMLCite .
One goal would be to decide ( find situations ) when this action has an open orbit .
From the Hurwitz viewpoint , LaTeXMLCite is the case LaTeXMLMath , where LaTeXMLMath is an elliptic curve .
It says , the image of LaTeXMLMath is an open subgroup of the general linear group acting on the Tate module of LaTeXMLMath , if LaTeXMLMath has no complex multiplications .
Otherwise , its image is an open subgroup of the Tate module automorphisms commuting with the ring of endomorphisms .
Much of LaTeXMLCite describes the complex multiplication case .
Still , the guiding theorem is LaTeXMLCite : If LaTeXMLMath is not an algebraic integer ( at LaTeXMLMath ) , then the image of LaTeXMLMath in the Tate module of LaTeXMLMath is the full general linear group .
Further , in its closing pages , LaTeXMLCite outlines the case when LaTeXMLMath is integral and LaTeXMLMath has no complex multiplication .
Prop .
LaTeXMLRef has a framework for considering these results applied to a Modular Tower where direct analogs would start with one prime LaTeXMLMath ( with LaTeXMLMath not dividing LaTeXMLMath ) .
We ’ ll use LaTeXMLMath for explicit comments .
Since LaTeXMLMath covers the LaTeXMLMath -line , interpret LaTeXMLMath as being LaTeXMLMath -adically close to LaTeXMLMath if its LaTeXMLMath -invariant is sufficiently negative for the LaTeXMLMath -adic valuation .
A first goal is to find the largest ( up to equivalence according to the Lie algebra of the image ) possible action of LaTeXMLMath on projective systems of points lying over LaTeXMLMath .
Finding how to estimate how LaTeXMLMath -adically close to LaTeXMLMath would be part of this .
We make simplifying assumptions to propose a program .
We haven ’ t yet found a good analog on a general ( LaTeXMLMath -perfect ) Modular Tower .
There are two parts .
§ LaTeXMLRef reminds of the place where a LaTeXMLMath -Frattini covering property enters in Serre ’ s Open Image Theorem .
§ LaTeXMLRef gives a precise test case for starting from Serre ’ s situation using LaTeXMLMath values LaTeXMLMath -adically close to LaTeXMLMath for some prime LaTeXMLMath .
Let LaTeXMLMath be the subset of projective systems LaTeXMLMath ( points ) on the LaTeXMLMath Modular Tower lying over LaTeXMLMath ( so they correspond to elements of LaTeXMLMath ( Prop .
LaTeXMLRef ) with this additional property : Each LaTeXMLMath lies in a component at level LaTeXMLMath containing H-M reps. An absolute Galois group preserves the collection of H-M components LaTeXMLCite .
So , it is a special case of the problem to describe the H-M components at every level ( see Rem .
LaTeXMLRef ) .
Let LaTeXMLMath be a number field .
Consider an arbitrary reduced Modular Tower ( possibly inner , or absolute ) LaTeXMLMath .
Apply the notation of § LaTeXMLRef to a projective system of LaTeXMLMath components LaTeXMLMath .
Specifically , let LaTeXMLMath be the ( geometric ) monodromy group of the map LaTeXMLMath with LaTeXMLMath the ( geometric ) monodromy group from LaTeXMLMath .
The LaTeXMLCite case is LaTeXMLMath , the arithmetic monodromy groups are LaTeXMLMath while the geometric monodromy groups are LaTeXMLMath LaTeXMLCite .
Lem .
LaTeXMLRef computes the geometric and arithmetic monodromy groups for level 0 of the LaTeXMLMath Modular Tower .
The modular curve tower has LaTeXMLMath -groups for LaTeXMLMath .
§ LaTeXMLRef notes that for the LaTeXMLMath Modular Tower , LaTeXMLMath ( for either component ) is not a 2-group as its order is divisible by 1 power of 3 .
A basic lemma is that LaTeXMLMath is a Frattini cover if LaTeXMLMath LaTeXMLCite .
For LaTeXMLMath : LaTeXMLMath is LaTeXMLMath LaTeXMLMath — LaTeXMLMath acting on the 4 points of the projective plane .
So , it is not 3- perfect and its universal 3-Frattini cover is LaTeXMLMath .
For LaTeXMLMath , the result is a very strong Hilbert ’ s Irreducibility Theorem for this situation hidden in LaTeXMLCite .
It starts by assuming LaTeXMLMath , and the Galois closure of LaTeXMLMath contains LaTeXMLMath .
The conclusion : For any LaTeXMLMath and LaTeXMLMath over LaTeXMLMath , the Galois closure of LaTeXMLMath equals the full arithmetic monodromy group of LaTeXMLMath .
This is the strongest conclusion supporting the open image theorem .
Considering Modular Towers requires addressing the next problem , especially in the LaTeXMLMath -split case ( the topic of LaTeXMLCite ) .
Assume LaTeXMLMath is LaTeXMLMath -perfect .
Characterize when for some LaTeXMLMath , LaTeXMLMath is a LaTeXMLMath -Frattini cover of LaTeXMLMath ?
To understand a general Modular Tower requires considering geometric components other than H-M types that could provide projective ( unobstructed ) systems of components .
Here is an example .
Suppose a collection of conjugacy classes C satisfies the following properties .
LaTeXMLMath contains no H-M reps ( say , if LaTeXMLMath is odd ) .
LaTeXMLMath is nonempty for all LaTeXMLMath .
Example : Suppose LaTeXMLMath with LaTeXMLMath odd and with all conjugacy classes equal to C , a rational LaTeXMLMath conjugacy class of an element of order LaTeXMLMath .
Assume also : LaTeXMLMath with LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath is a lift of LaTeXMLMath to C in LaTeXMLMath , then LaTeXMLEquation .
It would help if there were just one H-M component at each level of a Modular Tower .
For a general Modular Tower a simple assumption guarantees this LaTeXMLCite .
This assumption , however , holds in none of the cases when LaTeXMLMath in this paper .
The conclusion can even be false : Ex .
LaTeXMLRef has two H-M components at level 1 .
LaTeXMLCite gives an explicit LaTeXMLMath for the Modular Towers associated to LaTeXMLMath so that each level of the Modular Tower has exactly one H-M component ( though possibly many components if the level LaTeXMLMath is large ) if LaTeXMLMath .
If LaTeXMLMath or LaTeXMLMath , however , the number of H-M components is still a mystery .
Consider LaTeXMLMath over the field LaTeXMLMath of Laurent series .
Regard LaTeXMLMath as acting as in § LaTeXMLRef ( except extending to the LaTeXMLMath component ) .
Let LaTeXMLMath be a squarefree integer .
Start with a tangential base point convenient for such an LaTeXMLMath -adic investigation .
A convenient example is that from LaTeXMLCite : LaTeXMLEquation by using the unordered four points LaTeXMLMath in the first position and LaTeXMLMath ( a fixed base point ) in the last position .
To extend the map to LaTeXMLMath requires forming a special fiber in place of LaTeXMLMath over LaTeXMLMath .
Extending the Stable Compactification Theorem appropriately to covers replaces LaTeXMLMath by three copies LaTeXMLMath , LaTeXMLMath and LaTeXMLMath tied together as on the left side of Table 5 .
Each of the two pairs of points coming together ( complex conjugate pairs ) is wandering in LaTeXMLMath space .
Add a line of directions for the approach of each pair .
Each pair degenerates to the common limit of the two points with the addition of the direction of their coming together in the copy of LaTeXMLMath .
The downward pointing tree on the right in Table 5 has root 0 labeled for the component LaTeXMLMath , one edge each for LaTeXMLMath and LaTeXMLMath ( with corresponding vertices 5 and 6 ) and four leaf edges terminating in vertices corresponding to the points LaTeXMLMath .
Computing with this amounts to finding explicit action of LaTeXMLMath on the elements of the Nielsen classes LaTeXMLMath representing germs of covers approaching the special fiber .
For example , in the complex conjugate pairs business above , identify each component with a choice of coordinate LaTeXMLMath .
That identifies the function field of the copy of LaTeXMLMath ( modulo the action of LaTeXMLMath ) .
For the root component use LaTeXMLMath .
For LaTeXMLMath , the component from LaTeXMLMath and LaTeXMLMath coming together , choose LaTeXMLMath .
For LaTeXMLMath choose LaTeXMLMath .
Then , LaTeXMLEquation .
As LaTeXMLMath , LaTeXMLMath and LaTeXMLMath have limit LaTeXMLMath on LaTeXMLMath .
On , however , LaTeXMLMath , the LaTeXMLMath values representing the limit of LaTeXMLMath and LaTeXMLMath are LaTeXMLMath and LaTeXMLMath .
Similarly for LaTeXMLMath and LaTeXMLMath coming together , given by LaTeXMLMath on LaTeXMLMath and LaTeXMLMath values LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the generator of LaTeXMLMath .
The coordinates on each component give the LaTeXMLMath action on the tree .
Extend it to the fundamental group of the tree .
The assumptions for this explicit action are in LaTeXMLCite .
First : Identify LaTeXMLMath as a braid by recognizing its action on the tree .
As in our example , this is from writing LaTeXMLMath explicitly as LaTeXMLMath .
You get the image of the braid in LaTeXMLMath just from its action on the power series for LaTeXMLMath .
Vertices LaTeXMLMath of the tree consist of LaTeXMLMath together with the components of the special fiber .
Edges are from a point of LaTeXMLMath lying on a component , or from the meeting of two components .
Suppose an edge LaTeXMLMath represents the component meeting at a singular point LaTeXMLMath ( like LaTeXMLMath and LaTeXMLMath meeting ) .
Then , the local ring around that point is LaTeXMLMath for some well-determined LaTeXMLMath where LaTeXMLMath defines the two components , LaTeXMLMath .
Fix LaTeXMLMath , and form a neighborhood LaTeXMLMath using the coordinates LaTeXMLMath to include the arc LaTeXMLEquation .
Let LaTeXMLMath .
For a leaf edge LaTeXMLMath take the clockwise path around LaTeXMLMath .
For LaTeXMLMath , set LaTeXMLMath and let LaTeXMLMath be the connected component of LaTeXMLMath containing the annulus LaTeXMLEquation .
This is the topological realization of the ordered tree .
Reminder of those details : LaTeXMLCite produces the presentation compatible with a given graph .
For LaTeXMLMath , let the unique path LaTeXMLMath connecting LaTeXMLMath to LaTeXMLMath be denoted LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Choose a simple path LaTeXMLMath leading from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath , and LaTeXMLMath ( an element of LaTeXMLMath ) .
As usual , use classical generators LaTeXMLMath , clockwise around the LaTeXMLMath from LaTeXMLMath .
Add these paths : LaTeXMLMath .
We act on the opposite side of Wewers , following the notation of § LaTeXMLRef .
So , LaTeXMLMath acts as LaTeXMLEquation .
Then , with LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath , LaTeXMLEquation .
The action of LaTeXMLMath on the subscripts of the LaTeXMLMath s is the obvious one from our the action of LaTeXMLMath on the Puiseux expression of the paths in LaTeXMLMath .
Further , with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
LaTeXMLCite shows that if LaTeXMLMath is prime to LaTeXMLMath and the branch locus doesn ’ t degenerate at LaTeXMLMath , we may take LaTeXMLMath in the maximal ideal of the valuation ring for LaTeXMLMath and specialize all the actions .
There would be two steps in the program corresponding to two parts of Serre ’ s result .
Still , we restrict to the case of H-M components for the LaTeXMLMath Modular Tower .
For LaTeXMLMath , giving LaTeXMLMath suitably LaTeXMLMath -adically close to LaTeXMLMath draw deductions about the action of LaTeXMLMath on LaTeXMLMath for LaTeXMLMath lying over LaTeXMLMath in an H-M component .
Estimate from this the LaTeXMLMath -adic domain in the LaTeXMLMath -line to which the one may analytically continue from LaTeXMLMath the Lie algebra action of LaTeXMLMath on the LaTeXMLMath -adic Grassmanian of § LaTeXMLRef .
Toward ( LaTeXMLRef LaTeXMLRef ) , an accomplishment would be to differentiate between the action of LaTeXMLMath on H-M reps. and on near H-M reps. , exactly as we did with LaTeXMLMath at all levels of the Modular Tower in Prop .
LaTeXMLRef , and Wewers did LaTeXMLMath -adically at level 0 of the same Modular Tower .
Two situations about a Modular Tower require extending considerations of this paper .
The most classical situation doesn ’ t satisfy LaTeXMLMath -perfectness .
§ LaTeXMLRef discusses a reasonable remedy .
Further , given a reduced Modular Tower LaTeXMLMath , some significant part of the tower may come from a tower of modular curves .
While monodromy groups of the levels over the LaTeXMLMath -line often preclude this , it is difficult to compute the monodromy groups even should that be sufficient to set the matter straight .
§ LaTeXMLRef illustrates by example the problems .
The pairing of dihedral groups and modular curves generalizes neatly into Modular Towers to include , for example , simple groups and any prime that divides the order of such a group .
Still , the most classical case of moduli of covers does not : simple branching LaTeXMLMath with LaTeXMLMath being LaTeXMLMath involutions in LaTeXMLMath with LaTeXMLMath .
Since LaTeXMLMath has kernel LaTeXMLMath , LaTeXMLMath is not 2-perfect , and 2 divides the orders of the conjugacy classes .
LaTeXMLCite explains this situation in more generality ( as in Ex .
LaTeXMLRef ) .
The remedy , however , is clear .
Regard each level 0 curve in the reduced Hurwitz space as coming with a map to a hyperelliptic curve .
Then view the ( ramified ) LaTeXMLMath cover of hyperelliptic curves ( of genus LaTeXMLMath ) as the starting point for unramified projective systems of covers coming from the universal 2-Frattini cover of LaTeXMLMath .
Prop .
LaTeXMLRef applies to this situation extending its application to Modular Towers starting with covers of LaTeXMLMath .
LaTeXMLCite illustrates this with LaTeXMLMath using just the group theory from this paper .
Continue with LaTeXMLMath as at the beginning of this section .
Ex .
LaTeXMLRef gives situations automatically implying the jacobian of LaTeXMLMath , LaTeXMLMath maps through the elliptic curve with LaTeXMLMath -invariant LaTeXMLMath .
( This is equivalent to LaTeXMLMath itself mapping onto this elliptic curve . )
This is the situation we call LaTeXMLMath -aware .
Even if LaTeXMLMath maps through an elliptic curve , it may not be the elliptic curve with LaTeXMLMath -invariant LaTeXMLMath .
For a general Modular Tower the jacobians of the corresponding curves at level LaTeXMLMath go up in rank very quickly .
Given the rank LaTeXMLMath of LaTeXMLMath , inductively compute the rank of LaTeXMLMath as LaTeXMLMath from the Shreier formula .
Given the genus LaTeXMLMath of LaTeXMLMath ( a curve at level 0 ) , computed from the branch cycles as a cover of LaTeXMLMath , then LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath , LaTeXMLMath , … .
§ LaTeXMLRef continues the Nielsen class LaTeXMLMath from Ex .
LaTeXMLRef .
We show level 0 is not a modular curve .
The natural map from modding out by the Klein 4-group in LaTeXMLMath gives LaTeXMLMath .
Regard LaTeXMLMath as a multiplicative group of order 8 .
Any element of LaTeXMLMath ( the level 0 Nielsen class ) gives a symbol LaTeXMLMath where two entries are +1 and two entries are -1 .
Let LaTeXMLMath be the H-M component of this reduced Hurwitz space at level 0 ( as in Ex .
LaTeXMLRef ) .
The absolute and inner ( nonreduced ) spaces , LaTeXMLMath and LaTeXMLMath , at level 0 , have a natural degree 2 map LaTeXMLMath .
The element LaTeXMLMath takes the 4-tuple LaTeXMLMath to the 4-tuple LaTeXMLMath .
The latter is conjugation of the former by LaTeXMLMath .
This is exactly why LaTeXMLMath has degree 2 ( illustrating Thm .
LaTeXMLRef ) .
Then , LaTeXMLMath is a family of genus 1 curves , while the curves of LaTeXMLMath have genus LaTeXMLMath with LaTeXMLMath : LaTeXMLMath .
Since , however , we mod out by LaTeXMLMath to get reduced Nielsen classes , this induces an isomorphism of LaTeXMLMath with LaTeXMLMath .
As in § LaTeXMLRef and Prop .
LaTeXMLRef , the spaces still represent different moduli problems .
Neither reduced space is a fine ( or even b-fine ) moduli space according to Prop .
LaTeXMLRef .
Still , to simplify , refer to both as LaTeXMLMath .
Choose a basepoint LaTeXMLMath from LaTeXMLMath ( as in Lem .
LaTeXMLRef ) .
Then , LaTeXMLMath acts on allowable symbols from LaTeXMLMath , giving a cover LaTeXMLMath with this branch cycle description : LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
One H-M rep. LaTeXMLMath lies over LaTeXMLMath and and one over LaTeXMLMath , LaTeXMLMath : LaTeXMLMath while LaTeXMLMath ( as in § LaTeXMLRef ) .
With LaTeXMLMath over LaTeXMLMath as a basepoint for the cover LaTeXMLMath , the fiber over this point has representatives LaTeXMLEquation .
It is easy from this to form a branch cycle description for LaTeXMLMath : LaTeXMLEquation .
The group generated by these has size 648 and is centerless .
Denote LaTeXMLMath by LaTeXMLMath .
As the geometric monodromy group of a chain of two degree 3 extensions , it would naturally be a subgroup of the wreath product LaTeXMLMath , as in the argument of Lem .
LaTeXMLRef .
The full wreath product has order 1296 , while the kernel for the projection LaTeXMLMath is the subgroup of LaTeXMLMath for which LaTeXMLMath .
Our problem was to decide , for a general cover LaTeXMLMath in this family , lying over LaTeXMLMath , if the Galois closure LaTeXMLMath of the cover could factor through the elliptic curve with LaTeXMLMath invariant LaTeXMLMath .
This is equivalent to LaTeXMLMath having that elliptic curve as an isogeny factor .
The generality of the next argument in LaTeXMLMath will appear in later applications .
Let LaTeXMLMath be the copy of the Klein 4-group in LaTeXMLMath .
For LaTeXMLMath generic , LaTeXMLMath is isogenous to LaTeXMLEquation with LaTeXMLMath a simple abelian variety of dimension 2 , and LaTeXMLMath and LaTeXMLMath Prym ( elliptic ) varieties for the hyperelliptic curve LaTeXMLMath .
Riemann-Hurwitz shows LaTeXMLMath has genus 2 .
LaTeXMLCite shows for any LaTeXMLMath , the general cyclic degree 3 triple cover with LaTeXMLMath branch points has a simple Jacobian .
This result applies to available for LaTeXMLMath covers no matter the number of branch points .
The proof considers LaTeXMLMath , the function field LaTeXMLMath of the branch point locus , with the field of definition for the generic cover LaTeXMLMath and its group of automorphisms adjoined .
Then , for LaTeXMLMath a prime , LaTeXMLMath acts on the LaTeXMLMath -division points LaTeXMLMath according to this recipe : If LaTeXMLMath then LaTeXMLMath with LaTeXMLMath and LaTeXMLMath a standard isotropic decomposition with respect to the Weil pairing , LaTeXMLMath fixing LaTeXMLMath and LaTeXMLMath , inducing a group between LaTeXMLMath and LaTeXMLMath ; and if LaTeXMLMath , then LaTeXMLMath makes LaTeXMLMath an LaTeXMLMath vector space , with LaTeXMLMath inducing a group between LaTeXMLMath and LaTeXMLMath .
Suppose LaTeXMLMath is not simple .
Let LaTeXMLMath be a finite extension over which LaTeXMLMath is isogenous to a product of abelian varieties , so LaTeXMLMath acts through a reducible normal subgroup of one of the simple groups occurring in ( LaTeXMLRef ) .
Then , the finite group LaTeXMLMath would contain infinitely many nonisomorphic simple groups , according to the composition factors appearing in the cases of ( LaTeXMLRef ) .
Finally , the cover LaTeXMLMath is unramified , and for each copy LaTeXMLMath of a LaTeXMLMath in LaTeXMLMath , LaTeXMLMath is a Prym cover .
So , LaTeXMLMath has a degree 2 isogeny to LaTeXMLMath with LaTeXMLMath an elliptic curve ( § LaTeXMLRef and LaTeXMLCite ) .
If LaTeXMLMath and LaTeXMLMath generate LaTeXMLMath , then their associated elliptic curves LaTeXMLMath , LaTeXMLMath , give the curves appearing in the statement of the lemma .
These isogeny factors account for LaTeXMLMath up to isogeny .
∎ Many different maps from LaTeXMLMath to LaTeXMLMath induce projective systems of covers of the LaTeXMLMath -line having the same geometric monodromy as the projective system of curve covers called modular curves .
Even , however , if some of these covers did have significant moduli problems describing them , they would not be the same as those describing modular curves .
For example , having monodromy group LaTeXMLMath does not imply an upper half plane quotient is the LaTeXMLMath -line cover of the LaTeXMLMath -line .
That is the case with our space LaTeXMLMath .
The cover LaTeXMLMath does not factor through LaTeXMLMath .
So , LaTeXMLMath is not a modular curve .
Since the moduli space LaTeXMLMath is a family of LaTeXMLMath branch point covers , Ex .
LaTeXMLRef shows it does not satisfy the moduli interpretation for factoring through LaTeXMLMath .
As the monodromy group has LaTeXMLMath as a quotient , this means LaTeXMLMath is not a quotient of a modular curve .
∎ Suppose LaTeXMLMath defines the degree 2 cover LaTeXMLMath .
The other degree two covers ( for example , LaTeXMLMath ) in Lem .
LaTeXMLRef have the form LaTeXMLMath with LaTeXMLMath having order 3 .
So , if for a generic LaTeXMLMath value LaTeXMLMath is isogenous to LaTeXMLMath , then so will be LaTeXMLMath .
According to Lem .
LaTeXMLRef , to decide if LaTeXMLMath is LaTeXMLMath -aware requires answering the following question .
For general LaTeXMLMath , is either LaTeXMLMath or LaTeXMLMath isogenous to the elliptic curve with LaTeXMLMath -invariant LaTeXMLMath .
We leave this question after the following observations .
Since the question is for general LaTeXMLMath , and we know from LaTeXMLCite we may assume LaTeXMLMath has no complex multiplication .
The Riemann-Roch Theorem for genus 1 curves gives the dimension of linear systems of 2-division points as the dimension of LaTeXMLMath where LaTeXMLMath is the canonical class , which on a genus 1 curve is trivial .
This dimension is 1 ( odd ) if LaTeXMLMath is linearly equivalent to 0 , and 0 ( even ) if not .
So , according to Prop .
LaTeXMLRef , the linear equivalence class of LaTeXMLMath defines a non-trivial LaTeXMLMath 2-division point on LaTeXMLMath .
The second author saw an S. Sternberg talk in Jerusalem based on LaTeXMLCite .
In that talk , three semi-simple lie group representations had appeared as associated with the existence of elementary particles , and the point was to see them as a natural multiplet .
The shape of the final result was this .
They have LaTeXMLMath two complex Lie algebras with g semi-simple , r reductive and both having the same Carton subalgebra h .
So the following properties hold .
r and g have the same rank .
The Weyl group of r is a subgroup of the Weyl group of g .
With a choice of positive roots for each assume the positive Weyl chamber of g is in the positive Weyl chamber of r .
The killing form on g gives an orthogonal splitting as LaTeXMLMath .
The last item allows embedding r in the orthogonal Lie algebra acting on po .
Further , this embedding has two representations LaTeXMLMath and LaTeXMLMath associated with the plus and minus representations of the orthogal Lie algebra on the even and odd part of the Clifford algebra for the orthogonal product ( as in Prop .
LaTeXMLRef ) .
Consider the elements LaTeXMLMath of the Weyl group of g that map the positive Weyl chamber of g into the positive Weyl chamber of r .
The highest weights LaTeXMLMath in the positive Weil chamber of g correspond to irreducible representations LaTeXMLMath of g LaTeXMLCite .
In the Grothendieck group of r representations , form LaTeXMLMath as a representation of r .
Linear algebra magic occurs here : For each LaTeXMLMath , LaTeXMLCite constructs a highest weight for r from the pair LaTeXMLMath .
Denote the corresponding representation by LaTeXMLMath .
Their formula is LaTeXMLEquation with LaTeXMLMath indicating some choice of sign .
The analogy for us starts with a group LaTeXMLMath , and LaTeXMLMath representing an H-M rep. at level 0 .
Let LaTeXMLMath be the subgroup of LaTeXMLMath stabilizing the reduced class of LaTeXMLMath .
Then , Prop .
LaTeXMLRef gives two representations LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , respectively corresponding to the actions on the sets of ( LaTeXMLRef ) and ( LaTeXMLRef ) .
The analogy would go further if there was a representation LaTeXMLMath of LaTeXMLMath that fit in the formula LaTeXMLMath to give the difference between the two representations on ( LaTeXMLRef ) and ( LaTeXMLRef ) .
The analogy misses a lot , though we can easily construct LaTeXMLMath -adic Lie algebras from the projective situation .
Since there are infinitely many levels of a Modular Tower , our goal would be to understand if there are such paired representations at higher levels of the Modular Tower .
This would give a direct translation of the Clifford ( Spin ) invariant of Prop .
LaTeXMLRef , and its relation to LaTeXMLMath -nulls for each component of LaTeXMLMath ( as in § LaTeXMLRef ) .
Let LaTeXMLMath the characteristic quotients of the universal LaTeXMLMath -Frattini cover LaTeXMLMath of a LaTeXMLMath -perfect group LaTeXMLMath .
Our contributions to Main Conjecture ( LaTeXMLCite , Prob .
LaTeXMLRef , § LaTeXMLRef ) with LaTeXMLMath branch points suggest that no bounded set of branch points will allow regular realizations over a fixed number field of the complete set LaTeXMLMath .
Further , suppose for some LaTeXMLMath , there are realizations of LaTeXMLMath , each requiring no more than LaTeXMLMath branch points .
Then , Thm .
LaTeXMLRef says there must exist a Modular Tower for LaTeXMLMath with C a set of LaTeXMLMath conjugacy classes having cardinality LaTeXMLMath , and rational points at every level .
The tower levels are manifolds .
So , having LaTeXMLMath points at every level implies the Modular Tower levels have a projective system of LaTeXMLMath components .
LaTeXMLCite shows that if there is just one absolutely irreducible H-M component at every level of a LaTeXMLMath Modular Tower , then this will be defined over LaTeXMLMath .
There are variants on this , though this is the clearest case establishing this necessary condition .
Further , assume LaTeXMLMath contains H-M reps. For any integers LaTeXMLMath , call LaTeXMLMath an H-M pair if LaTeXMLMath is inverse to the class LaTeXMLMath .
In this situation let LaTeXMLMath be C with LaTeXMLMath and LaTeXMLMath removed .
The only successful hypotheses guaranteeing one H-M component at every level are mildly variant on the other half of LaTeXMLCite .
There will be just one H-M component if these ( H-Mgcomplete ) hypotheses hold : For each H-M pair LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
For example , this would hold if for each H-M pair LaTeXMLMath , LaTeXMLMath contains every nontrivial conjugacy class of LaTeXMLMath .
There are results that encourage using H-Mgcomplete reps. : For any prime LaTeXMLMath , LaTeXMLCite finds a LaTeXMLMath regular realization of LaTeXMLMath using the H-Mgcomplete condition .
The suggested generalization of the Open Image Theorem in § LaTeXMLRef would apply especially well to a sequence of H-Mgcomplete components .
Reminder : Shafarevich ’ s method with solvable groups does not give regular realizations .
Only some version of the braid rigidity method has systematically produced regular realizations , and going beyond solvable groups the story has been regular realizations all the way .
That even includes Shih ’ s results using modular curves : These came from the first version of the method the second author told Shimura when he first worked it out at the Institute for Advanced Study at Princeton in 1967-69 .
We consider Modular Towers a general invention , extending through the analogy with modular curves , the potential for many applications .
What , however , of someone who wants such regular realizations ?
Does this say anything of the next reasonable step in the search for them ?
For a large integer LaTeXMLMath , where would one find a regular realization of LaTeXMLMath .
From Thm .
LaTeXMLRef combined with the conclusion of the Main Conjecture , you will need many branch points , and so you need a model for where to get an indication of how many , and what kind of conjugacy classes ( with elements of order divisible by LaTeXMLMath ) you will use .
When LaTeXMLMath is LaTeXMLMath -split , with cyclic LaTeXMLMath -Sylow , it is tempting to imitate realization of dihedral groups .
No one knows how to get regular involution realizations of dihedral groups ( when LaTeXMLMath , the moduli of these relates special Hurwitz spaces and modular curves § LaTeXMLRef ) .
Yet , any elementary algebra book shows how to get regular realizations of dihedral groups .
You must use LaTeXMLMath -cycles as branch cycles , and you require many branch points ( see the implications of this elementary analysis in LaTeXMLCite or LaTeXMLCite ) .
Yet , for a general finite group LaTeXMLMath and prime LaTeXMLMath , no one has produced similar branch cycles .
Results from § LaTeXMLRef show divide the case where LaTeXMLMath is cyclic from those where it is not .
We refer to the former case as dihedral-like .
It is the latter case that is serious .
We consider finding such branch cycles when the situation is not dihedral-like , our basic assumption from here .
So , we are seeking dihedral-like realizations in non-dihedral-like situations .
Recorded in LaTeXMLCite is the following possibility : Given LaTeXMLMath , there might be infinitely many LaTeXMLMath formed by suitably increasing the multiplicity of appearance of conjugacy classes in C , so that LaTeXMLMath will be a uni- rational variety .
( That result would make a mockery of the inverse Galois problem . )
Our replacement would consider using special realizations at level 0 , formed , say , from understanding simple groups .
We acknowledge the Main Conjecture might be false ; it has phenomenal implications for LaTeXMLMath ( see the comments about LaTeXMLCite in § LaTeXMLRef ) .
An approach of Thompson-Völklein ( LaTeXMLCite has an example , or the Thompson tuples of Ex .
LaTeXMLRef ) uses intricate knowledge of Chevalley simple groups to locate conjugacy classes that show some Chevalley series over each finite field LaTeXMLMath has regular realizations excluding finitely many LaTeXMLMath .
This has worked for many series , extending Völklein ’ s production of high rank Chevalley group realizations .
( Prior to his approach , a less abstract use of the main idea of LaTeXMLCite , there had been essentially no higher rank realizations . )
We consider as a starting point some successful realization cases where the Hurwitz space LaTeXMLMath is uni-rational ( it might even be an abelian cover of LaTeXMLMath as in the Thompson-Völklein examples ) .
Take a prime LaTeXMLMath not dividing the orders of elements in C .
Form , for each LaTeXMLMath , a sequence LaTeXMLMath of conjugacy classes , using the following principles .
Reminder : Def .
LaTeXMLRef explains rationalization of any conjugacy classes LaTeXMLMath .
LaTeXMLMath is a rational union of conjugacy classes in LaTeXMLMath with LaTeXMLMath ( possibly empty ) conjugacy classes whose elements have LaTeXMLMath power order , C are LaTeXMLMath classes , LaTeXMLMath are conjugacy classes of LaTeXMLMath -power order , and LaTeXMLMath is a rational ( or uni-rational ) variety over LaTeXMLMath .
LaTeXMLMath consists of LaTeXMLMath where LaTeXMLMath consists of the rationalization of LaTeXMLMath whose terms consist of a lift to LaTeXMLMath of the conjugacy class for LaTeXMLMath for each member of LaTeXMLMath , and LaTeXMLMath is a rational union of conjugacy classes ( for LaTeXMLMath ) in LaTeXMLMath .
Since LaTeXMLMath is a Frattini cover , any lifts of generators of LaTeXMLMath to LaTeXMLMath generate LaTeXMLMath .
So , as with the universal LaTeXMLMath -Frattini cover , it is only the product 1 condition that governs the possibility of braid orbits .
There will be no obvious minimal choice .
Given ( nontrivial ) LaTeXMLMath , any lift LaTeXMLMath to LaTeXMLMath will have LaTeXMLMath times the order of LaTeXMLMath ( Lem .
LaTeXMLRef ) .
Further , since the situation is not dihedral-like , there will be many LaTeXMLMath orbits on the collection of lifts of LaTeXMLMath .
We use the notation established in the rest of the paper for Nielsen classes at levels of a Modular Tower .
Consider LaTeXMLMath with LaTeXMLMath the conjugacy class of an element LaTeXMLMath in LaTeXMLMath .
For LaTeXMLMath to be nonempty , the image of LaTeXMLMath in LaTeXMLMath must be LaTeXMLMath ( see Ex .
LaTeXMLRef ) .
When LaTeXMLMath we know this is sufficient to guarantee LaTeXMLMath is nonempty ; the argument right after Princ .
LaTeXMLRef .
( For LaTeXMLMath , there are two conjugacy class choices for such LaTeXMLMath , and several reasons ( like Prop .
LaTeXMLRef ) to see them as significantly different , though not for this particular obstruction . )
For , however , LaTeXMLMath , appearance of LaTeXMLMath beyond the first Loewy layer of LaTeXMLMath might obstruct LaTeXMLMath from being nonempty for some choices of conjugacy class LaTeXMLMath .
That is , we don ’ t know for certain if LaTeXMLMath fills out all conjugacy classes lying over the conjugacy class of LaTeXMLMath in LaTeXMLMath .
Assume the non-dihedral-like case , with the usual LaTeXMLMath -perfect assumption , and consider the list ( LaTeXMLRef ) .
The next problem asks if it is possible to remove choices after level 1 and still be certain higher level Nielsen classes are nonempty .
Is it always possible to find some LaTeXMLMath so the following holds .
For LaTeXMLMath , with LaTeXMLMath empty and any choice of a single lift of each class in LaTeXMLMath to LaTeXMLMath to give LaTeXMLMath and its rationalization LaTeXMLMath , then LaTeXMLMath is nonempty ?
The aim of this thesis is to define a geometric , explicitly computable compactification of the moduli space LaTeXMLMath of smooth plane curves of degree LaTeXMLMath .
The basic idea is to regard a plane curve LaTeXMLMath as a log surface LaTeXMLMath .
Then there is a compactification given by a moduli space LaTeXMLMath of log surfaces LaTeXMLMath where LaTeXMLMath is semi-log-canonical and ample , the log analogue of the moduli space of surfaces of general type constructed by Kollár and Shepherd-Barron LaTeXMLCite .
For the definitions of semi-log-canonical ( slc ) and semi-log-terminal ( slt ) singularities see LaTeXMLRef .
I initially tried to compute these compactifications , with little success — the LaTeXMLMath case was calculated by Hassett LaTeXMLCite , but as LaTeXMLMath increases the problem quickly becomes unmanageable .
We consider instead the family of compactifications given by moduli spaces LaTeXMLMath of log surfaces LaTeXMLMath where LaTeXMLMath is slc and ample , where LaTeXMLMath .
Note that we require LaTeXMLMath so that LaTeXMLMath is ample .
The compactification is simpler for lower LaTeXMLMath : roughly , the boundary has fewer components , fewer types of degenerate surfaces LaTeXMLMath occur .
The compactification LaTeXMLMath we compute below can be described as the limit of LaTeXMLMath as LaTeXMLMath approaches LaTeXMLMath from above .
However , we don ’ t define it in this way to avoid technical difficulties and give a clearer presentation .
LaTeXMLMath is a moduli space of stable pairs LaTeXMLMath .
Passing from LaTeXMLMath to LaTeXMLMath affords three major simplifications : For LaTeXMLMath a stable pair , LaTeXMLMath is a slc del Pezzo surface , that is , LaTeXMLMath is ample and LaTeXMLMath has slc singularities .
For LaTeXMLMath a stable pair , LaTeXMLMath is Cartier and linearly equivalent to zero .
This makes many computations with stable pairs easy ( e.g. , calculating the possible slt singularities on LaTeXMLMath for LaTeXMLMath a stable pair of given degree LaTeXMLMath ) .
For LaTeXMLMath a relative stable pair , both LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier ( whereas for LaTeXMLMath we only know that LaTeXMLMath is LaTeXMLMath -Cartier ) .
In particular , we need only consider LaTeXMLMath -Gorenstein deformations of LaTeXMLMath for LaTeXMLMath a stable pair .
We now give a map of this thesis .
In Section LaTeXMLRef we write down the definition of a stable pair of degree LaTeXMLMath , and define a moduli stack LaTeXMLMath of these objects .
We prove that LaTeXMLMath is separated and proper using the valuative criterion and the relative MMP as in LaTeXMLCite .
Thus LaTeXMLMath gives a compactification of LaTeXMLMath .
We delay the discussion of some technical points until Sections LaTeXMLRef , LaTeXMLRef and LaTeXMLRef .
In Sections LaTeXMLRef and LaTeXMLRef we explain the definition of an allowable family LaTeXMLMath of stable pairs .
In particular , in Section LaTeXMLRef we develop a theory of LaTeXMLMath -Gorenstein deformations for slc surfaces which is of independent interest .
We apply these results in Section LaTeXMLRef to prove that LaTeXMLMath is a Deligne–Mumford stack , using the methods of Artin LaTeXMLCite .
We remark that we don ’ t need to appeal to Alexeev ’ s results in order to bound the index of LaTeXMLMath here — we give an explicit bound using elementary methods .
The aim of the remainder of this thesis is to classify stable pairs .
Essentially it is enough to classify the degenerate surfaces LaTeXMLMath occurring in stable pairs LaTeXMLMath .
For , given LaTeXMLMath , the divisor LaTeXMLMath satisfies LaTeXMLMath and LaTeXMLMath slc for some LaTeXMLMath , so LaTeXMLMath is a member of a given linear system with specified singularities .
In Section LaTeXMLRef we give a rough classification of the surfaces LaTeXMLMath that occur .
They are of 4 types A , B , C , D. Type A are the normal surfaces — the log terminal cases have already been classified by Manetti LaTeXMLCite .
In Section LaTeXMLRef we show that the only strictly log canonical cases are the elliptic cones of degree LaTeXMLMath .
Type B are the nonnormal slt cases — they have two components meeting in a LaTeXMLMath .
We give a classification of these in Section LaTeXMLRef .
Types C and D are the nonnormal strictly log canonical cases .
Type C have a degenerate cusp , type D have a LaTeXMLMath quotient of a degenerate cusp , and in both cases the surface is slt outside these points .
We remark that there is one more type of slc del Pezzo , denoted B* , which is irreducible , slt and has a folded double curve .
However we show in Section LaTeXMLRef that a surface of type B* never admits a smoothing to LaTeXMLMath , so does not occur in a stable pair .
In Section LaTeXMLRef , we show that if LaTeXMLMath is a stable pair of degree LaTeXMLMath where LaTeXMLMath then LaTeXMLMath is slt .
The main point is that in the case LaTeXMLMath the condition LaTeXMLMath shows that LaTeXMLMath is 3-divisible in LaTeXMLMath — this is a strong restriction on LaTeXMLMath .
So LaTeXMLMath is either a Manetti surface or a surface of type B and we have a classification in each case .
In particular , LaTeXMLMath has either LaTeXMLMath or LaTeXMLMath components .
We also show that LaTeXMLMath is smooth if LaTeXMLMath using deformation theory developed in Sections LaTeXMLRef and LaTeXMLRef .
The case LaTeXMLMath is much more involved — for example if LaTeXMLMath there is an LaTeXMLMath with 18 components ( see Example LaTeXMLRef ) .
Moreover LaTeXMLMath is not smooth if LaTeXMLMath ( see Example LaTeXMLRef ) .
In Section LaTeXMLRef we give the complete classification of the LaTeXMLMath that occur in degrees 4 and 5 , and identify the possible singularities of LaTeXMLMath on LaTeXMLMath .
We also present partial results in the degree 6 case — we give the complete list of surfaces of types A and B , and give a list of candidates for the surfaces of types C and D. We have yet to determine which of these candidates are smoothable .
I ’ d like to thank my supervisor Alessio Corti , for invaluable guidance throughout the course of my PhD .
Many of the new ideas contained in this thesis grew out of discussions with Alessio , I believe that he always had an excellent intuitive feel for the problem which was a great help to me .
I ’ m indebted to Brendan Hassett , whose preprint of October 1998 ( c.f .
LaTeXMLCite ) suggested that one should consider the moduli spaces LaTeXMLMath for LaTeXMLMath .
He also provides an outline of the LaTeXMLMath -Gorenstein deformation theory of slc surfaces set out in Section LaTeXMLRef .
Finally , I ’ d like to thank Tom Fisher , Jan Wierzba and Nick Shepherd-Barron for various helpful conversations .
We compactify the space of smooth plane curves LaTeXMLMath of degree LaTeXMLMath .
Here LaTeXMLMath , where LaTeXMLMath is the open subscheme of the Hilbert scheme of curves of degree LaTeXMLMath in LaTeXMLMath corresponding to smooth curves .
We do this by constructing a moduli stack LaTeXMLMath of stable pairs LaTeXMLMath as defined below .
We use the relative MMP , see LaTeXMLCite for details .
We always work over LaTeXMLMath .
Let LaTeXMLMath denote the spectrum of a DVR , LaTeXMLMath the generic point , and LaTeXMLMath the geometric generic point .
If LaTeXMLMath is a local scheme , we use script letters to denote flat families over LaTeXMLMath and normal letters to denote the special fibres .
We say LaTeXMLMath is a family of pairs over LaTeXMLMath if LaTeXMLMath is a flat family of CM reduced surfaces and LaTeXMLMath is a relative Weil divisor on LaTeXMLMath .
For the definition of a relative Weil divisor , see Section LaTeXMLRef .
Let LaTeXMLMath be a proper connected surface .
Let LaTeXMLMath be an effective LaTeXMLMath -divisor on LaTeXMLMath .
The pair LaTeXMLMath ( or LaTeXMLMath ) is semi-log-canonical ( respectively semi-log-terminal ) if LaTeXMLMath is Cohen–Macaulay and normal crossing in codimension LaTeXMLMath .
LaTeXMLMath is LaTeXMLMath -Cartier .
Let LaTeXMLMath be the normalisation of LaTeXMLMath .
Let LaTeXMLMath be the double curve of LaTeXMLMath and write LaTeXMLMath , LaTeXMLMath for the inverse images on LaTeXMLMath .
Then LaTeXMLMath is log canonical ( respectively log terminal ) .
We use the abbreviations slc and slt for semi-log-canonical and semi-log-terminal .
Note LaTeXMLMath .
Let LaTeXMLMath be a proper connected surface over an algebraically closed field LaTeXMLMath .
Let LaTeXMLMath be an effective Weil divisor on LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath .
We say that LaTeXMLMath is a semistable pair of degree LaTeXMLMath if LaTeXMLMath is normal and log terminal .
LaTeXMLMath is slc .
LaTeXMLMath , and moreover LaTeXMLMath if LaTeXMLMath .
There exists a smoothing LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
We write ‘ LaTeXMLMath ’ to denote linear equivalence of Weil divisors ( not LaTeXMLMath -divisors ) .
Thus if LaTeXMLMath then in particular LaTeXMLMath is Cartier .
This will be important later in the proof of Theorem LaTeXMLRef .
We use ‘ LaTeXMLMath ’ to denote the weaker notion of numerical equivalence ( of LaTeXMLMath -Cartier LaTeXMLMath -divisors ) .
If LaTeXMLMath is a semistable pair , LaTeXMLMath is a normal degeneration of LaTeXMLMath with log terminal singularities .
These have been classified by Manetti LaTeXMLCite .
In particular , LaTeXMLMath is projective and LaTeXMLMath is ample .
LaTeXMLMath , LaTeXMLMath as in Notation LaTeXMLRef .
Let LaTeXMLMath be a smooth curve of degree LaTeXMLMath defined over LaTeXMLMath .
Then there exists a base change LaTeXMLMath and a family LaTeXMLMath of semistable pairs completing LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
We give the proof of the Theorem at the end of this section .
Let LaTeXMLMath be a proper connected surface over an algebraically closed field LaTeXMLMath .
Let LaTeXMLMath be an effective Weil divisor on LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath is a stable pair of degree LaTeXMLMath if There exists LaTeXMLMath such that LaTeXMLMath is slc and ample .
LaTeXMLMath , and moreover LaTeXMLMath if LaTeXMLMath .
There exists a smoothing LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
Note that ( 1 ) and ( 2 ) imply that LaTeXMLMath is ample .
This is the main reason that stable pairs are easy to classify .
Let LaTeXMLMath be a smooth curve of degree LaTeXMLMath .
Then there exists a base change LaTeXMLMath and a family LaTeXMLMath of stable pairs completing LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
We give the proof of the Theorem at the end of this section .
For each LaTeXMLMath there exists LaTeXMLMath such that for every stable pair LaTeXMLMath of degree LaTeXMLMath , LaTeXMLMath is Cartier .
It follows by general theory that stable pairs of degree LaTeXMLMath are bounded for each LaTeXMLMath , see the proof of Theorem LaTeXMLRef .
Let LaTeXMLMath denote the least such LaTeXMLMath for each LaTeXMLMath .
Let LaTeXMLMath be a family of stable pairs of degree LaTeXMLMath .
That is , LaTeXMLMath is flat over LaTeXMLMath , LaTeXMLMath is an effective relative Weil divisor on LaTeXMLMath , and for every geometric point LaTeXMLMath of LaTeXMLMath , the fibre LaTeXMLMath is a stable pair of degree LaTeXMLMath .
We say that LaTeXMLMath is an allowable family if LaTeXMLMath and LaTeXMLMath commute with base change for all LaTeXMLMath .
The theory of the conditions ‘ LaTeXMLMath and LaTeXMLMath commute with base change ’ is developed in detail in Sections LaTeXMLRef and LaTeXMLRef .
In particular LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier since LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier for each closed point LaTeXMLMath , using Lemma LaTeXMLRef .
Moreover , if LaTeXMLMath is the spectrum of a DVR , with generic point LaTeXMLMath , and LaTeXMLMath is canonical , then LaTeXMLMath is allowable iff LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier by Proposition LaTeXMLRef .
We shall only require this case below .
Let LaTeXMLMath be a stable pair of degree LaTeXMLMath .
Let LaTeXMLMath be a versal allowable deformation of the pair LaTeXMLMath , where LaTeXMLMath is of finite type over LaTeXMLMath ( we prove the existence of such a deformation in Section LaTeXMLRef ) .
Let LaTeXMLMath be the open subscheme where the geometric fibres of LaTeXMLMath are isomorphic to LaTeXMLMath .
Let LaTeXMLMath be the scheme theoretic closure of LaTeXMLMath in LaTeXMLMath .
An allowable deformation of LaTeXMLMath is smoothable if it is obtained by pullback from the deformation LaTeXMLMath .
Example LaTeXMLRef and Example LaTeXMLRef describe two examples of this construction .
Given the smoothability assumption for stable pairs LaTeXMLMath , we require the relative smoothability assumption for families LaTeXMLMath of stable pairs in order to obtain an algebraic stack of stable pairs ( c.f .
Example LaTeXMLRef ) .
Let LaTeXMLMath be an allowable family of stable pairs of degree LaTeXMLMath .
LaTeXMLMath is a smoothable family if for every geometric point LaTeXMLMath the induced deformation of LaTeXMLMath is smoothable .
Let LaTeXMLMath be the category of noetherian schemes over LaTeXMLMath .
We define a groupoid LaTeXMLMath as follows : LaTeXMLEquation .
LaTeXMLMath is a separated and proper Deligne–Mumford stack .
Thus LaTeXMLMath gives a compactification LaTeXMLMath .
The proof that LaTeXMLMath is a Deligne–Mumford stack is given in Section LaTeXMLRef .
We prove that LaTeXMLMath is separated and proper in Theorems LaTeXMLRef and LaTeXMLRef below .
We first give the proof of Theorem LaTeXMLRef .
In fact we prove the following more precise result .
Let LaTeXMLMath be a stable pair of degree LaTeXMLMath .
Then for all LaTeXMLMath , LaTeXMLMath if LaTeXMLMath and LaTeXMLMath if LaTeXMLMath .
LaTeXMLMath is slc and LaTeXMLMath is LaTeXMLMath -Cartier by Definiton LaTeXMLRef , ( 1 ) and ( 2 ) .
Thus LaTeXMLMath is slc , and LaTeXMLMath at the strictly slc singularities of LaTeXMLMath .
Now LaTeXMLMath and moreover LaTeXMLMath if LaTeXMLMath , so the bound is clear at stricly slc points of LaTeXMLMath .
Since LaTeXMLMath has a LaTeXMLMath -Gorenstein smoothing , the slt singularities are of 3 types ( compare Theorem LaTeXMLRef ) : LaTeXMLMath where LaTeXMLMath .
LaTeXMLMath where LaTeXMLMath .
LaTeXMLMath .
We have LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath in cases 1 , 2 , and 3 .
We now use the two properties LaTeXMLMath is slc and LaTeXMLMath is Cartier to bound LaTeXMLMath and LaTeXMLMath in cases ( 1 ) and ( 2 ) .
In case ( 1 ) , we first remark that LaTeXMLMath and LaTeXMLMath by Lemma LaTeXMLRef , using the fact that LaTeXMLMath smoothes to LaTeXMLMath .
Consider the local smooth cover LaTeXMLMath of LaTeXMLMath at LaTeXMLMath , let LaTeXMLMath denote the inverse image of LaTeXMLMath , say LaTeXMLMath .
Then LaTeXMLMath since LaTeXMLMath is log canonical at LaTeXMLMath .
Let LaTeXMLMath be the local equation of LaTeXMLMath at LaTeXMLMath and LaTeXMLMath be a monomial of minimal multiplicity .
Then LaTeXMLMath , and LaTeXMLMath , using LaTeXMLMath Cartier and LaTeXMLMath .
Thus in particular LaTeXMLMath .
Suppose LaTeXMLMath , then LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
It follows that if LaTeXMLMath we have LaTeXMLMath , and if LaTeXMLMath we have LaTeXMLMath .
This proves the bounds in case ( 1 ) .
In case ( 2 ) , let LaTeXMLMath be the local analytic component LaTeXMLMath of LaTeXMLMath at LaTeXMLMath , let LaTeXMLMath be the double curve , LaTeXMLMath .
Then LaTeXMLMath is log canonical at LaTeXMLMath .
Let LaTeXMLMath be the smooth cover LaTeXMLMath of LaTeXMLMath , let LaTeXMLMath denote the inverse images of LaTeXMLMath .
By adjunction LaTeXMLMath is log canonical , equivalently LaTeXMLMath is reduced .
Write LaTeXMLMath .
Then LaTeXMLMath where LaTeXMLMath .
We have LaTeXMLMath ( respectively LaTeXMLMath ) if LaTeXMLMath ( respectively LaTeXMLMath ) , using LaTeXMLMath Cartier ( respectively LaTeXMLMath Cartier ) .
It follows that LaTeXMLMath if LaTeXMLMath , LaTeXMLMath if LaTeXMLMath .
∎ We next show that LaTeXMLMath is separated .
This is almost immediate from the general theory of moduli of surfaces .
I ’ ve included my own proof of a foundational result ( Lemma LaTeXMLRef ) because I could not find a proof in the literature .
Let LaTeXMLMath be a normal variety , flat over LaTeXMLMath .
Let LaTeXMLMath be an effective Weil divisor , flat over LaTeXMLMath .
Let LaTeXMLMath be a projective resolution of LaTeXMLMath , write LaTeXMLMath for the strict transform of LaTeXMLMath .
Assume that the exceptional locus is a divisor , write LaTeXMLMath for the sum of the exceptional divisors dominating LaTeXMLMath .
We say LaTeXMLMath is a semistable log-resolution of LaTeXMLMath if LaTeXMLMath is reduced and LaTeXMLMath is a simple normal crossing divisor .
LaTeXMLMath is separated We use the valuative criterion .
So , suppose we are given LaTeXMLMath and LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath .
We need to show that LaTeXMLMath .
Note that we may assume that LaTeXMLMath and LaTeXMLMath is smooth for LaTeXMLMath and LaTeXMLMath , since the open subset of LaTeXMLMath of pairs LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is smooth is dense .
After base change , we may assume that LaTeXMLMath .
Possibly after base change , we can construct a common semistable log resolution LaTeXMLMath of LaTeXMLMath and LaTeXMLMath that is an isomorphism over LaTeXMLMath .
We now claim that we can reconstruct the two families as the LaTeXMLMath canonical model of LaTeXMLMath for LaTeXMLMath .
So LaTeXMLMath and LaTeXMLMath as required .
To prove the claim , we just need to verify that LaTeXMLMath is log canonical and relatively ample for LaTeXMLMath .
We can check ampleness on the central fibre .
We have LaTeXMLMath by adjunction .
This is ample since LaTeXMLMath is ample for some LaTeXMLMath and LaTeXMLMath .
Now , LaTeXMLMath is slc for some LaTeXMLMath by the definition of LaTeXMLMath , and the same is true for LaTeXMLMath .
It follows that LaTeXMLMath is log canonical by Lemma LaTeXMLRef .
This completes the proof .
∎ Let LaTeXMLMath be a flat family of surfaces , LaTeXMLMath a LaTeXMLMath -divisor that is flat over LaTeXMLMath .
Suppose LaTeXMLMath is normal and LaTeXMLMath is LaTeXMLMath -Cartier .
Suppose also that LaTeXMLMath admits a semistable log-resolution .
Then LaTeXMLMath is log canonical iff LaTeXMLMath is slc .
If LaTeXMLMath is log canonical , then trivially LaTeXMLMath is slc by adjunction .
We now prove the converse .
We may work locally at LaTeXMLMath .
Take a semistable log resolution LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath canonical model over LaTeXMLMath .
Then LaTeXMLMath is an isomorphism over the locus where LaTeXMLMath is log canonical .
LaTeXMLMath is log canonical at any codimension LaTeXMLMath point of LaTeXMLMath .
For the implication LaTeXMLMath slc LaTeXMLMath LaTeXMLMath log canonical is easy in the case LaTeXMLMath .
So no exceptional divisor of LaTeXMLMath has centre a codimension LaTeXMLMath point of LaTeXMLMath .
Now let LaTeXMLMath be the normalisation of a component of LaTeXMLMath , LaTeXMLMath the inverse image of the double curve of LaTeXMLMath on LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLMath be the normalisation of the strict transform of this component in LaTeXMLMath , LaTeXMLMath the inverse image of the double curve of LaTeXMLMath on LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Write LaTeXMLMath for the induced map .
We have LaTeXMLEquation where the primes denote strict transforms and the LaTeXMLMath are the LaTeXMLMath -exceptional divisors .
Then LaTeXMLMath for all LaTeXMLMath because LaTeXMLMath is slc .
Now LaTeXMLMath since there are no LaTeXMLMath -exceptional divisors with centre a codimension LaTeXMLMath point of LaTeXMLMath .
So , we have LaTeXMLEquation where LaTeXMLMath for all LaTeXMLMath since LaTeXMLMath is relatively ample .
If LaTeXMLMath for some LaTeXMLMath , we have LaTeXMLMath , a contradiction .
So , in particular , for each component LaTeXMLMath of LaTeXMLMath , with strict transform LaTeXMLMath in LaTeXMLMath , the map LaTeXMLMath does not contract any component of the double curve of LaTeXMLMath on LaTeXMLMath .
It follows that there are no LaTeXMLMath -exceptional divisors over LaTeXMLMath ( recall that every such divisor has centre LaTeXMLMath ) .
Moreover , for each component LaTeXMLMath of LaTeXMLMath , LaTeXMLMath .
Hence LaTeXMLMath , so LaTeXMLMath has no exceptional divisors .
Then LaTeXMLMath and LaTeXMLMath LaTeXMLMath -ample implies LaTeXMLMath is an isomorphism , so LaTeXMLMath is log canonical as required .
∎ The Lemma above is similar to LaTeXMLCite , p. 325 , Theorem 5.1 ( a ) .
This is an inversion of adjunction type result ( compare LaTeXMLCite , p. 174 , Theorem 5.50 ) , but we can ’ t use the general theorem to prove our Lemma .
We now give the proof that LaTeXMLMath is proper .
The main steps of the argument are in Theorem LaTeXMLRef and Theorem LaTeXMLRef .
These theorems motivate the definition of a stable pair above .
First complete LaTeXMLMath to a family LaTeXMLMath .
Possibly after base change ( which we suppress in our notation ) we can take a semistable log-resolution LaTeXMLMath of LaTeXMLMath which is an isomorphism over LaTeXMLMath .
Then LaTeXMLMath is dlt .
Run a LaTeXMLMath MMP over LaTeXMLMath .
Let LaTeXMLMath be the end product .
Then LaTeXMLMath must be relatively nef , since it is zero on the generic fibre .
Now it follows that LaTeXMLMath .
For we can write LaTeXMLMath , where the LaTeXMLMath are the components of LaTeXMLMath , since LaTeXMLMath is zero on the generic fibre .
Without loss of generality we may assume LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , using the relation LaTeXMLMath .
Restricting to LaTeXMLMath , we see that LaTeXMLMath relatively nef implies that LaTeXMLMath for all LaTeXMLMath such that LaTeXMLMath .
Since LaTeXMLMath is connected , repeating the argument we obtain LaTeXMLMath for all LaTeXMLMath , so LaTeXMLMath as claimed .
Compare Lemma LaTeXMLRef , ( 1 ) .
We have LaTeXMLMath dlt and LaTeXMLMath is LaTeXMLMath -factorial , so LaTeXMLMath is dlt .
We now run a LaTeXMLMath MMP over LaTeXMLMath and let LaTeXMLMath be the end product .
Then LaTeXMLMath is a del Pezzo fibration , since the generic fibre is a del Pezzo surface ( namely LaTeXMLMath ) .
Now , LaTeXMLMath and LaTeXMLMath LaTeXMLMath -factorial gives LaTeXMLMath irreducible — for we have an exact sequence LaTeXMLEquation by Lemma LaTeXMLRef , where the LaTeXMLMath are the irreducible components of LaTeXMLMath .
The dlt property of LaTeXMLMath now gives that LaTeXMLMath is normal and log terminal .
Since LaTeXMLMath is dlt and LaTeXMLMath , we have LaTeXMLMath log canonical and LaTeXMLMath .
We obtain LaTeXMLMath is log canonical and LaTeXMLMath by adjunction .
The sharpening in the case LaTeXMLMath is clear .
Finally , since LaTeXMLMath is LaTeXMLMath -Cartier and LaTeXMLMath , LaTeXMLMath is LaTeXMLMath -Cartier .
∎ First complete LaTeXMLMath to a family LaTeXMLMath .
By Theorem LaTeXMLRef and its proof , we may assume that LaTeXMLMath is log canonical , and LaTeXMLMath .
After base change , take a semistable log resolution LaTeXMLMath which is an isomorphism over LaTeXMLMath .
Now run a LaTeXMLMath MMP over LaTeXMLMath .
Here LaTeXMLMath is chosen such that for LaTeXMLMath an exceptional divisor of LaTeXMLMath , if LaTeXMLMath then LaTeXMLMath .
Let LaTeXMLMath be the end product .
By the choice of LaTeXMLMath , we have LaTeXMLMath .
Now take the LaTeXMLMath canonical model over LaTeXMLMath , denote this LaTeXMLMath .
Then LaTeXMLMath is log canonical and relatively ample over LaTeXMLMath , and LaTeXMLMath since LaTeXMLMath .
We obtain LaTeXMLMath is slc and ample and LaTeXMLMath by adjunction .
The sharpening in the case LaTeXMLMath is obvious .
Finally since LaTeXMLMath and LaTeXMLMath is LaTeXMLMath -Cartier , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
∎ LaTeXMLMath is proper We use the valuative criterion of properness .
Taking LaTeXMLMath , we need to show that , possibly after base change , there exists an extension LaTeXMLMath .
We may assume that LaTeXMLMath and LaTeXMLMath is smooth as in the proof of separatedness above .
Applying Theorem LaTeXMLRef we obtain our result .
∎ We now work towards a classification of the surfaces LaTeXMLMath that occur in stable pairs LaTeXMLMath .
We collect the relevant results from the above in the proposition below .
Let LaTeXMLMath be a stable pair .
Then LaTeXMLMath is ample , LaTeXMLMath is slc , and LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing .
LaTeXMLMath is ample by Definition LaTeXMLRef , ( 1 ) and ( 2 ) .
There exists a LaTeXMLMath -Gorenstein smoothing LaTeXMLMath of LaTeXMLMath by Definition LaTeXMLRef , ( 3 ) .
Finally , LaTeXMLMath is slc , and LaTeXMLMath is LaTeXMLMath -Cartier , so LaTeXMLMath is slc .
∎ Let LaTeXMLMath be a surface with normal crossing singularities in codimension LaTeXMLMath .
Let LaTeXMLMath denote the double curve of LaTeXMLMath .
Write LaTeXMLMath for the decomposition of LaTeXMLMath into its irreducible components .
Let LaTeXMLMath .
Let LaTeXMLMath be the normalisation of LaTeXMLMath .
Write LaTeXMLMath for the inverse image of LaTeXMLMath .
Then LaTeXMLMath , where LaTeXMLMath is the normalisation and LaTeXMLMath is the inverse image of LaTeXMLMath .
First we give a rough classification of the components LaTeXMLMath .
Then we describe how to glue these components together to recover LaTeXMLMath .
( LaTeXMLCite p119 Theorem 4.15 ) Let LaTeXMLMath be a log canonical pair , where LaTeXMLMath is a surface and LaTeXMLMath is an effective Weil divisor .
Assume LaTeXMLMath .
Then we have the following cases : ( a ) LaTeXMLMath , ( a , r ) =1 .
( b ) LaTeXMLMath , ( a , r ) =1 .
( c ) LaTeXMLMath , ( a , r ) =1 , where the LaTeXMLMath -action is etale in codimension LaTeXMLMath and interchanges LaTeXMLMath and LaTeXMLMath .
We will denote cases ( a ) , ( b ) and ( c ) by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath respectively .
The LaTeXMLMath stands for dihedral — these singularities generalise the dihedral Du Val singularities .
Let LaTeXMLMath be a proper log canonical pair with LaTeXMLMath ample .
Then LaTeXMLMath belongs to one of the following types : LaTeXMLMath .
Then LaTeXMLMath has at most one strictly log canonical singularity .
LaTeXMLMath and LaTeXMLMath log terminal .
Then LaTeXMLMath has singularities LaTeXMLMath at LaTeXMLMath , with LaTeXMLMath .
LaTeXMLMath , where the two components intersect in a node of LaTeXMLMath .
Then LaTeXMLMath has a singularity of type LaTeXMLMath at the node of LaTeXMLMath , and at most one other singularity of type LaTeXMLMath on each component of LaTeXMLMath .
Moreover LaTeXMLMath has log terminal singularities away from LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath has a singularity of type LaTeXMLMath .
Then LaTeXMLMath has at most one other singularity of type LaTeXMLMath at LaTeXMLMath and LaTeXMLMath has log terminal singularities away from LaTeXMLMath .
To prove the theorem , we need the connectedness theorem of Shokurov : ( LaTeXMLCite p174 Corollary 5.49 ) Let LaTeXMLMath be a normal , proper variety , LaTeXMLMath a Weil divisor , and suppose that LaTeXMLMath is nef .
Then the locus of log canonical singularities , i.e. , LaTeXMLMath taken over all LaTeXMLMath with discrepancy LaTeXMLMath , is connected .
First , by Theorem LaTeXMLRef , either LaTeXMLMath and there is at most one strictly log canonical singularity on LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath is connected , and LaTeXMLMath is log terminal away from LaTeXMLMath .
Now let LaTeXMLMath be a component of LaTeXMLMath .
We have LaTeXMLMath is ample , so LaTeXMLMath .
Expanding the left hand side , LaTeXMLEquation .
Now LaTeXMLMath and LaTeXMLMath , thus LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
We know that LaTeXMLMath is log canonical , so we have the classification given in Theorem LaTeXMLRef of the singularities of LaTeXMLMath at LaTeXMLMath .
We calculate that each point of type ( a ) , ( b ) , ( c ) on LaTeXMLMath contributes LaTeXMLMath , LaTeXMLMath , LaTeXMLMath to LaTeXMLMath respectively .
The theorem now follows easily .
∎ We now want to give the classification of the surfaces LaTeXMLMath , by glueing together components as above .
We first state the classification of nonnormal slc surface singularities .
LaTeXMLMath is 2-to-1 .
Let LaTeXMLMath be a component , write LaTeXMLMath for the inverse image in LaTeXMLMath .
Then either LaTeXMLMath has two components mapping birationally to LaTeXMLMath , or LaTeXMLMath has one component mapping 2-to-1 to LaTeXMLMath .
In the latter case we say LaTeXMLMath is obtained by folding LaTeXMLMath .
If we are working locally at LaTeXMLMath and we are in the latter case , let LaTeXMLMath .
We say LaTeXMLMath is pinched at LaTeXMLMath .
( see LaTeXMLCite , Section 4 ) Let LaTeXMLMath be a slc singularity , assume that LaTeXMLMath is non-normal .
Then we have the following cases : LaTeXMLMath , where LaTeXMLMath .
LaTeXMLMath , where , for LaTeXMLMath a generator of LaTeXMLMath , we have LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath where LaTeXMLMath odd , LaTeXMLMath .
LaTeXMLMath is a degenerate cusp : LaTeXMLMath , with LaTeXMLMath of type LaTeXMLMath , glued to LaTeXMLMath along one component of LaTeXMLMath and LaTeXMLMath along the other component , so that we have a cycle of components .
LaTeXMLMath is a LaTeXMLMath -quotient of a degenerate cusp : LaTeXMLMath , where for LaTeXMLMath , LaTeXMLMath is of type LaTeXMLMath and is glued to LaTeXMLMath along one component of LaTeXMLMath and LaTeXMLMath along the other component .
For LaTeXMLMath or LaTeXMLMath there are two cases .
Either LaTeXMLMath is of type LaTeXMLMath and LaTeXMLMath is glued to LaTeXMLMath along the single component of LaTeXMLMath .
Or LaTeXMLMath is of type LaTeXMLMath , LaTeXMLMath is glued to LaTeXMLMath along one component of LaTeXMLMath and the other component is pinched at LaTeXMLMath .
Similarly for LaTeXMLMath .
So we have a chain of components .
( see LaTeXMLCite , p. 326 , Remark 5.2 ) The following cases admit a LaTeXMLMath -Gorenstein smoothing : ( 1 ) LaTeXMLMath .
( 2 ) None .
( 3 ) LaTeXMLMath .
( 4 ) All .
( 5 ) Unknown .
Let LaTeXMLMath be a slc proper surface with LaTeXMLMath ample .
Then LaTeXMLMath belongs to one of the following types ( we use the notation of Theorem LaTeXMLRef to describe the components ) : LaTeXMLMath normal .
LaTeXMLMath is a surface of type I. LaTeXMLMath , slt .
LaTeXMLMath are surfaces of type II , glued along LaTeXMLMath .
LaTeXMLMath irreducible , non-normal , slt .
LaTeXMLMath is a surface of type II , LaTeXMLMath is formed by folding LaTeXMLMath .
LaTeXMLMath has a degenerate cusp .
LaTeXMLMath is a union of components of type III which are glued together to form a cycle .
More precisely , LaTeXMLMath , where LaTeXMLMath is glued to LaTeXMLMath and LaTeXMLMath along the two components of LaTeXMLMath for all LaTeXMLMath , in such a way that the nodes of the curves LaTeXMLMath all coincide — this point is the degenerate cusp .
LaTeXMLMath has a LaTeXMLMath quotient of a degenerate cusp .
LaTeXMLMath has some components of type III glued together to form a chain .
At each end we glue on either a component of type IV or a component of type III with one component of LaTeXMLMath folded .
The nodes of the curves LaTeXMLMath and the dihedral singularities on the components of type IV all coincide — this point is the LaTeXMLMath -quotient of a degenerate cusp .
Here , the singularities of type LaTeXMLMath on LaTeXMLMath are glued together together in pairs LaTeXMLMath and LaTeXMLMath to give a singularity of type LaTeXMLMath on LaTeXMLMath .
The classification in the Theorem is immediate from the classification of the components LaTeXMLMath in Theorem LaTeXMLRef using the classification of slc singularities in Theorem LaTeXMLRef .
∎ We remark that a surface of type B* does not admit a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath — see Theorem LaTeXMLRef .
Notation as in theorem LaTeXMLRef .
If LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing , we have the following additional conditions : The normal , log terminal singularities of LaTeXMLMath are cyclic quotient singularities of the form LaTeXMLMath .
The singularities of type LaTeXMLMath on LaTeXMLMath are glued together together in pairs LaTeXMLMath and LaTeXMLMath to give a singularity of type LaTeXMLMath on LaTeXMLMath .
If LaTeXMLMath is pinched at LaTeXMLMath , then either LaTeXMLMath is of type LaTeXMLMath or LaTeXMLMath is smooth .
( 1 ) is the well known classification of smoothable log terminal surface singularities ( LaTeXMLCite , p. 313 Propn 3.10 ) .
Conditions ( 2 ) and ( 3 ) follow from Remark LaTeXMLRef .
∎ If LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath , then in ( 1 ) we have LaTeXMLMath and LaTeXMLMath .
See Lemma LaTeXMLRef .
In this section we prove that all the components of LaTeXMLMath are rational unless LaTeXMLMath is an elliptic cone .
Let LaTeXMLMath be a slc proper surface with LaTeXMLMath ample .
Let LaTeXMLMath be a component .
Let LaTeXMLMath be the minimal resolution of LaTeXMLMath .
Write LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath be a minimal model of LaTeXMLMath .
Note that LaTeXMLMath is birationally ruled ( see below ) , so LaTeXMLMath is either ruled or LaTeXMLMath .
Write LaTeXMLMath .
Notation as above .
LaTeXMLMath is an effective LaTeXMLMath -divisor with components of multiplicity LaTeXMLMath .
LaTeXMLMath is nef and big , and is zero exactly on LaTeXMLMath .
LaTeXMLMath is nef and big .
Write LaTeXMLMath where the LaTeXMLMath are the exceptional divisors of LaTeXMLMath , and LaTeXMLMath is the strict transform of LaTeXMLMath .
Then LaTeXMLMath .
Here LaTeXMLMath for all LaTeXMLMath since LaTeXMLMath is log canonical .
LaTeXMLMath is minimal , so LaTeXMLMath is LaTeXMLMath -nef , hence also LaTeXMLMath is LaTeXMLMath -nef .
It follows that LaTeXMLMath for all LaTeXMLMath .
This proves ( 1 ) .
( 2 ) is immediate since LaTeXMLMath by definition and LaTeXMLMath is ample .
( 3 ) now follows since LaTeXMLMath ∎ Either LaTeXMLMath is rational , or LaTeXMLMath is an elliptic cone and LaTeXMLMath .
First , we claim LaTeXMLMath is birationally ruled .
For LaTeXMLMath is nef and big , so LaTeXMLMath for all LaTeXMLMath .
LaTeXMLMath is effective , so LaTeXMLMath for all LaTeXMLMath .
It follows that LaTeXMLMath is birationally ruled .
We may assume that LaTeXMLMath is not rational .
Then LaTeXMLMath is ruled over a curve of positive genus , let LaTeXMLMath be a ruling and let LaTeXMLMath denote the composite LaTeXMLMath .
Suppose that there is no horizontal component of LaTeXMLMath .
Thus LaTeXMLMath is a sum of fibres .
Now LaTeXMLMath nef and big implies that LaTeXMLMath is nef and big .
Then LaTeXMLMath by Serre duality and Kodaira vanishing , a contradiction .
Thus there is an irrational component of LaTeXMLMath .
By Theorem LaTeXMLRef we know that LaTeXMLMath has only rational components , so by the classification of log canonical singularities it follows that LaTeXMLMath has a simple elliptic singularity .
Let LaTeXMLMath denote the LaTeXMLMath -exceptional elliptic curve on LaTeXMLMath .
Then LaTeXMLMath has multiplicity LaTeXMLMath in LaTeXMLMath and LaTeXMLMath is horizontal .
Now LaTeXMLMath is big so LaTeXMLMath for LaTeXMLMath a fibre of the ruling , thus LaTeXMLMath , LaTeXMLMath is a section .
Next we claim that LaTeXMLMath is in fact ruled .
Suppose not , then there exists a degenerate fibre .
Let LaTeXMLMath be a component meeting LaTeXMLMath .
Then LaTeXMLMath is not contained in LaTeXMLMath .
We have LaTeXMLMath , with equality iff LaTeXMLMath is contracted by LaTeXMLMath .
But also LaTeXMLEquation with equality only if LaTeXMLMath is a LaTeXMLMath curve .
Thus LaTeXMLMath is a LaTeXMLMath curve and is contracted by LaTeXMLMath , a contradiction since LaTeXMLMath is minimal .
So LaTeXMLMath is ruled over an elliptic curve .
LaTeXMLMath is obtained from LaTeXMLMath by contracting the negative section and so LaTeXMLMath is an elliptic cone .
Finally LaTeXMLMath by Shokurov ’ s connectedness theorem ( compare Theorem LaTeXMLRef ) so LaTeXMLMath .
∎ We prove bounds on the values LaTeXMLMath for LaTeXMLMath a component of LaTeXMLMath , using the existence of a smoothing to LaTeXMLMath .
We use these in Theorem LaTeXMLRef and Theorem LaTeXMLRef to give necessary and sufficient conditions for surfaces of types A and B to be smoothable to LaTeXMLMath .
For types C and D we do not have necessary and sufficient criteria for smoothability as yet .
However the bounds we give here substantially simplify the explicit calculations we do for LaTeXMLMath in Section LaTeXMLRef ( note types C and D only occur if LaTeXMLMath — see Theorem LaTeXMLRef ) .
Let LaTeXMLMath be a slc proper surface with LaTeXMLMath ample , and LaTeXMLMath a smoothing with LaTeXMLMath .
( 1 ) LaTeXMLMath , with equality only if LaTeXMLMath is LaTeXMLMath -factorial .
Here LaTeXMLMath number of components of LaTeXMLMath and LaTeXMLMath number of components of LaTeXMLMath , not counting components obtained by folding a component of LaTeXMLMath .
( 2 ) If LaTeXMLMath , LaTeXMLMath .
( 3 ) If LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be a slc proper surface with LaTeXMLMath ample , and LaTeXMLMath a LaTeXMLMath -Gorenstein smoothing with LaTeXMLMath .
In the cases enumerated in Theorem LaTeXMLRef , we have the following bounds for LaTeXMLMath .
LaTeXMLMath .
Either LaTeXMLMath , or LaTeXMLMath and LaTeXMLMath is LaTeXMLMath -factorial .
LaTeXMLMath for all LaTeXMLMath , and LaTeXMLMath , equal only if LaTeXMLMath is LaTeXMLMath -factorial .
LaTeXMLMath for LaTeXMLMath a middle component , LaTeXMLMath for LaTeXMLMath an end component , and LaTeXMLMath , equal only if LaTeXMLMath is LaTeXMLMath -factorial .
Case B* does not occur by Theorem LaTeXMLRef .
Let LaTeXMLMath be a proper surface , normal crossing in codimension LaTeXMLMath , and LaTeXMLMath a smoothing with LaTeXMLMath .
( 1 ) We have an exact sequence LaTeXMLEquation ( 2 ) Assume in addition that LaTeXMLMath is projective .
Then LaTeXMLMath is an isomorphism .
( 1 ) Certainly LaTeXMLEquation is exact , so we just need to show the first map is injective .
Suppose LaTeXMLMath , then using LaTeXMLMath , we can replace this with a relation LaTeXMLMath , where , without loss of generality , LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Then LaTeXMLMath is effective and linearly equivalent to LaTeXMLMath .
So LaTeXMLMath since LaTeXMLMath is proper .
Thus LaTeXMLMath for all LaTeXMLMath such that LaTeXMLMath .
Since LaTeXMLMath is connected , repeating the argument we obtain LaTeXMLMath for all LaTeXMLMath , as required .
( 2 ) Suppose LaTeXMLMath is a Cartier divisor which is numerically equivalent to LaTeXMLMath .
Then LaTeXMLMath , thus LaTeXMLMath , some LaTeXMLMath .
Now a similar argument to the above shows LaTeXMLMath as required .
∎ Let LaTeXMLMath be a slc proper surface with LaTeXMLMath ample , and LaTeXMLMath a smoothing with LaTeXMLMath .
( 1 ) LaTeXMLMath and LaTeXMLMath are isomorphisms .
( 2 ) LaTeXMLMath is exact .
Dually , LaTeXMLMath is exact .
Here LaTeXMLMath is defined as follows : if LaTeXMLMath is a component of LaTeXMLMath obtained by identifying two components LaTeXMLMath and LaTeXMLMath of LaTeXMLMath , we have LaTeXMLMath , where we ’ re using the decomposition LaTeXMLMath , and the nonzero entries are in the LaTeXMLMath th and LaTeXMLMath th positions .
If LaTeXMLMath is a component of LaTeXMLMath obtained by folding a component of LaTeXMLMath , then LaTeXMLMath .
( 1 ) First we show that LaTeXMLMath is an isomorphism .
From the exponential sequence for LaTeXMLMath we obtain the long exact cohomology sequence LaTeXMLEquation .
We have LaTeXMLMath by Serre duality .
Thus LaTeXMLMath since LaTeXMLMath is ample .
Now LaTeXMLMath gives LaTeXMLMath .
So we obtain that LaTeXMLMath is an isomorphism .
Now , since LaTeXMLMath , cohomology and base change ( LaTeXMLCite , p. 290 , Theorem 12.11 ) gives LaTeXMLMath , thus LaTeXMLMath since LaTeXMLMath is affine .
Using the exponential sequence for LaTeXMLMath we obtain that LaTeXMLMath is an isomorphism .
Finally , LaTeXMLMath is an isomorphism since LaTeXMLMath is a homotopy retract of LaTeXMLMath , so LaTeXMLMath is an isomorphism as claimed .
Now , since LaTeXMLMath is surjective , LaTeXMLMath is surjective and LaTeXMLMath is injective .
But LaTeXMLMath is clearly surjective since LaTeXMLMath is the only closed fibre of LaTeXMLMath .
Hence LaTeXMLMath and LaTeXMLMath are isomorphisms .
( 2 ) First , we show that we have isomorphisms LaTeXMLMath , LaTeXMLMath and dually LaTeXMLMath , LaTeXMLMath .
Well , first from the proof of ( 1 ) above we have LaTeXMLMath , and by Lemma LaTeXMLRef ( 2 ) we have LaTeXMLMath , so we obtain LaTeXMLMath .
This gives LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath be a component , let LaTeXMLMath be a resolution .
We may assume that LaTeXMLMath is not normal , otherwise we are done by the above .
Then LaTeXMLMath is rational by Theorem LaTeXMLRef .
Thus LaTeXMLMath , it follows that LaTeXMLMath .
We have LaTeXMLMath .
For LaTeXMLMath since LaTeXMLMath is ample , and LaTeXMLMath since LaTeXMLMath is rational ( using Leray ) .
So LaTeXMLMath using the exponential sequence .
Thus LaTeXMLMath as desired .
It remains to show that LaTeXMLEquation is exact , where the map LaTeXMLMath is as in the statement of the lemma , and we work with LaTeXMLMath coefficients .
We use the Mayer-Vietoris sequence inductively here , separating off one component at a time .
First we separate the double curves LaTeXMLMath where the two branches of LaTeXMLMath at LaTeXMLMath belong to the same component .
In this case , LaTeXMLMath is homotopy equivalent to LaTeXMLMath , where LaTeXMLMath is LaTeXMLMath with the two branches at LaTeXMLMath separated , LaTeXMLMath is LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath are glued along LaTeXMLMath , where LaTeXMLMath is obtained using the 2-to-1 map LaTeXMLMath .
Note that LaTeXMLMath is either ( a ) LaTeXMLMath , 2-to-1 , or ( b ) LaTeXMLMath where the two components are joined at a node and each maps isomorphically onto LaTeXMLMath .
Then LaTeXMLEquation is exact , using LaTeXMLMath .
In case ( a ) we have LaTeXMLEquation thus LaTeXMLMath is an isomorphism .
In case ( b ) , writing LaTeXMLMath , we have LaTeXMLEquation thus LaTeXMLEquation is exact .
Next we separate the components of LaTeXMLMath .
Let LaTeXMLMath be a component ( assumed normal ) , and write LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation is exact , using LaTeXMLMath ( recall LaTeXMLMath is either LaTeXMLMath or LaTeXMLMath ) .
Now , using these steps repeatedly , we obtain our result .
∎ ( 1 ) We have LaTeXMLEquation exact , where LaTeXMLMath maps to zero if LaTeXMLMath is obtained by folding LaTeXMLMath .
Now LaTeXMLMath , LaTeXMLEquation and LaTeXMLMath .
Thus LaTeXMLMath , with equality iff LaTeXMLMath is LaTeXMLMath -factorial .
So , using the exact sequence above , we obtain LaTeXMLMath , equal only if LaTeXMLMath is LaTeXMLMath -factorial .
( 2 ) The maps LaTeXMLMath and LaTeXMLMath are dual .
Let LaTeXMLMath be the component of LaTeXMLMath meeting LaTeXMLMath , let LaTeXMLMath denote the intersection .
By the exact sequence of Lemma LaTeXMLRef , ( 2 ) , we find that LaTeXMLMath is injective ( recall that LaTeXMLMath ) .
For , if LaTeXMLMath is in the kernel , LaTeXMLMath for some LaTeXMLMath .
Intersecting with a relatively ample divisor on LaTeXMLMath , we find that LaTeXMLMath , so LaTeXMLMath .
Now using the description of LaTeXMLMath in Lemma LaTeXMLRef we see that LaTeXMLMath .
For locally at LaTeXMLMath , LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath ( where LaTeXMLMath is a divisor flat over LaTeXMLMath , of degree LaTeXMLMath on LaTeXMLMath ) .
So we obtain LaTeXMLMath as desired .
( 3 ) Similarly , in this case we find that LaTeXMLMath , and LaTeXMLMath , so we obtain LaTeXMLMath .
∎ All this follows from Theorem LaTeXMLRef except for the claim that LaTeXMLMath for LaTeXMLMath a middle component of a surface of type LaTeXMLMath .
To prove this , observe that in this case LaTeXMLMath is injective .
For write LaTeXMLMath , where LaTeXMLMath is glued to LaTeXMLMath by glueing LaTeXMLMath to LaTeXMLMath .
Suppose LaTeXMLMath is in the kernel of LaTeXMLMath .
By the exact sequence of Lemma LaTeXMLRef , ( 2 ) , we have LaTeXMLEquation for some LaTeXMLMath .
Thus LaTeXMLMath .
Working inductively , we obtain LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath as claimed .
We now conclude as in the proof of Theorem LaTeXMLRef , ( 3 ) .
∎ A surface of type B* ( see Theorem LaTeXMLRef ) does not admit a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Suppose LaTeXMLMath is a counter example , let LaTeXMLMath be a LaTeXMLMath -Gorenstein smoothing with LaTeXMLMath .
We have LaTeXMLMath by Theorem LaTeXMLRef , ( 1 ) .
We have LaTeXMLMath since LaTeXMLMath is LaTeXMLMath -Cartier .
Thus LaTeXMLMath , so LaTeXMLMath , using LaTeXMLMath ample and LaTeXMLMath .
So LaTeXMLMath .
Let LaTeXMLMath be the minimal resolution of LaTeXMLMath .
Then LaTeXMLMath is rational by Theorem LaTeXMLRef .
Applying Noether ’ s formula we obtain LaTeXMLMath .
So the resolution LaTeXMLMath has strictly decreased LaTeXMLMath .
However , we calculate below that the only possible singularities on LaTeXMLMath will increase LaTeXMLMath when we take the minimal resolution , so we have a contradiction .
LaTeXMLMath has singularities of type LaTeXMLMath and LaTeXMLMath , with the latter cases occurring in pairs LaTeXMLMath and LaTeXMLMath , by Theorem LaTeXMLRef .
Now , given a cyclic quotient singularity LaTeXMLMath , let LaTeXMLMath be the expansion of LaTeXMLMath as a Hirzebruch continued fraction .
The geometric interpretation of this is that the minimal resolution of the singularity has exceptional locus a chain of smooth rational curves with self-intersections LaTeXMLMath .
Then on taking the minimal resolution of the singularity , the change in LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath denotes the inverse of LaTeXMLMath modulo LaTeXMLMath ( LaTeXMLCite , p. 111 ) .
We have LaTeXMLMath in the case LaTeXMLMath ( LaTeXMLCite , p. 112 ) .
Finally , we calcuate LaTeXMLMath in the case of a pair of singularities LaTeXMLMath .
To see this , note that if we write LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath .
∎ Manetti has classified normal log terminal degenerations of LaTeXMLMath in LaTeXMLCite — we will refer to such surfaces as Manetti surfaces .
We state the basic result below .
( compare LaTeXMLCite , p. 90 , Main Theorem ) Suppose LaTeXMLMath is a normal log terminal proper surface with LaTeXMLMath ample .
Then LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath iff LaTeXMLMath has singularities of type LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath .
We now classify the normal log canonical degenerations of LaTeXMLMath .
Let LaTeXMLMath be a normal log canonical proper surface with LaTeXMLMath ample .
Then LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath iff LaTeXMLMath is a Manetti surface or LaTeXMLMath is an elliptic cone of degree LaTeXMLMath .
If LaTeXMLMath is a smoothing of a normal proper surface LaTeXMLMath to LaTeXMLMath , then LaTeXMLMath is projective and LaTeXMLMath -Gorenstein , and LaTeXMLMath is relatively ample .
The projectivity is proved in LaTeXMLCite , p. 95 , Theorem 4 , the rest follows since LaTeXMLMath ( by Lemma LaTeXMLRef ) .
Let LaTeXMLMath be an slc proper surface .
Suppose LaTeXMLMath has a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Then every normal log terminal singularity of LaTeXMLMath is a cyclic quotient singularity of type LaTeXMLMath , where LaTeXMLMath .
Moreover LaTeXMLMath .
First , we know that every normal log terminal singularity of LaTeXMLMath is of the form LaTeXMLMath , since we assume there exists a LaTeXMLMath -Gorenstein smoothing ( compare Theorem LaTeXMLRef ( 1 ) ) .
We need to show that LaTeXMLMath .
We sketch the proof here , for details see LaTeXMLCite , p. 103 , Propn 13 ( i ) and Remark 6 .
We compute that the Milnor fibre LaTeXMLMath of a smoothing of a singularity of type LaTeXMLMath has LaTeXMLMath and negative definite intersection product .
Now , since LaTeXMLMath has positive definite intersection product , it follows that LaTeXMLMath , so LaTeXMLMath as required .
Finally , we show LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -Gorenstein smoothing of LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath .
Then locally at LaTeXMLMath , we have LaTeXMLMath , generated by LaTeXMLMath ( see LaTeXMLCite , p. 313 , Propn 3.10 , and LaTeXMLCite , p. 135 , Propn 2.2.7 ) .
But LaTeXMLMath locally at LaTeXMLMath , where LaTeXMLMath is a divisor , flat over LaTeXMLMath , that restricts to a hyperplane section on LaTeXMLMath .
Thus LaTeXMLMath .
For another proof using the Milnor fibre , see LaTeXMLCite , p. 105 , Theorem 15 ( ii ) .
∎ We call singularities of the form LaTeXMLMath singularities of class LaTeXMLMath .
We call singularities of the form LaTeXMLMath singularities of class LaTeXMLMath .
Suppose given a normal log canonical del Pezzo surface LaTeXMLMath which admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
We may assume LaTeXMLMath is strictly log canonical , otherwise LaTeXMLMath is a Manetti surface .
Let LaTeXMLMath be the minimal resolution of LaTeXMLMath .
If LaTeXMLMath is not rational , then LaTeXMLMath is an elliptic cone by Theorem LaTeXMLRef .
Then LaTeXMLMath gives that LaTeXMLMath is an elliptic cone of degree LaTeXMLMath .
So we may assume LaTeXMLMath is rational .
The Leray spectral sequence gives an exact sequence LaTeXMLEquation .
Now LaTeXMLMath since LaTeXMLMath is ample , and LaTeXMLMath since LaTeXMLMath is rational .
So LaTeXMLMath , LaTeXMLMath has rational singularities .
We can now use a result of Manetti ( LaTeXMLCite , p. 95 , Theorem 4 , and p. 100 , Theorem 11 ) : Let LaTeXMLMath be a birational morphism , with LaTeXMLMath maximal ( so LaTeXMLMath is an isomorphism over the negative section LaTeXMLMath of LaTeXMLMath ) .
Let LaTeXMLMath denote the birational ruling so obtained .
Then the exceptional locus of LaTeXMLMath is the strict transform LaTeXMLMath of LaTeXMLMath together with the irreducible components of the degenerate fibres of LaTeXMLMath of self-intersection LaTeXMLMath .
In particular , LaTeXMLMath .
Moreover every degenerate fibre contains a unique LaTeXMLMath curve .
We quickly sketch the proof of this .
First , since LaTeXMLMath smoothes to LaTeXMLMath , we have LaTeXMLMath , and LaTeXMLMath since LaTeXMLMath is minimal .
Manetti deduces there is no horizontal curve LaTeXMLMath on LaTeXMLMath with LaTeXMLMath except possibly LaTeXMLMath .
Now since LaTeXMLMath , it follows that the exceptional locus of LaTeXMLMath is LaTeXMLMath together with all the components of the degenerate fibres of self intersection LaTeXMLMath , and every degenerate fibre has a unique LaTeXMLMath curve .
There are two types of rational strictly log canonical surface singularities — namely a LaTeXMLMath quotient of a cusp and a quotient of a simple elliptic singularity .
Consider the minimal resolutions of these singularities .
In each case the exceptional locus is a union of smooth rational curves .
For a LaTeXMLMath quotient of a cusp , the exceptional locus consists of a chain of curves with two LaTeXMLMath curves off each end component of the chain .
For a quotient of a simple elliptic singularity , the exceptional locus consists of a curve with three chains of curves off it .
We now analyse how these could possibly fit into the minimal resolution LaTeXMLMath of LaTeXMLMath .
Consider the minimal model program yielding LaTeXMLMath in the neighbourhood of a given fibre of LaTeXMLMath .
At each stage we contract a LaTeXMLMath curve , meeting at most LaTeXMLMath components of the fibre , and disjoint from LaTeXMLMath ( since LaTeXMLMath is an isomorphism over LaTeXMLMath ) .
We know that LaTeXMLEquation .
This set decomposes into the exceptional loci of the minimal resolutions of one log canonical rational singularity and some LaTeXMLMath singularities .
First concentrate on the log canonical singularity ; let LaTeXMLMath denote the exceptional locus of its minimal resolution .
Then LaTeXMLMath contains a curve LaTeXMLMath which meets LaTeXMLMath other components of LaTeXMLMath — we call such a curve a fork of LaTeXMLMath .
Suppose LaTeXMLMath is a degenerate fibre of LaTeXMLMath containing a fork LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath denote the component meeting LaTeXMLMath ( this is the strict transform of the corresponding fibre LaTeXMLMath of LaTeXMLMath ) .
Then we have a decomposition LaTeXMLMath , where LaTeXMLMath is the unique -1 curve in LaTeXMLMath .
LaTeXMLMath is a string of curves , with one end component meeting LaTeXMLMath , LaTeXMLMath contracts to a LaTeXMLMath singularity ( or is empty ) .
LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are nonempty configurations of curves meeting LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath respectively .
Then in the MMP LaTeXMLMath , we contract LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath in that order .
( Note : LaTeXMLMath and LaTeXMLMath are nonempty because LaTeXMLMath is a fork .
One might think LaTeXMLMath could be empty since LaTeXMLMath contains LaTeXMLMath , but in that case LaTeXMLMath and must be contracted before we can contract LaTeXMLMath , a contradiction ) .
Next suppose that LaTeXMLMath is a degenerate fibre of LaTeXMLMath that does not contain a fork of LaTeXMLMath .
Then we have a decomposition LaTeXMLMath , where LaTeXMLMath is the unique -1 curve in LaTeXMLMath .
LaTeXMLMath is a string of curves , with one end component meeting LaTeXMLMath , LaTeXMLMath contracts to a LaTeXMLMath singularity ( or is empty ) .
LaTeXMLMath is a non-empty string of curves meeting LaTeXMLMath and with one end component meeting LaTeXMLMath .
We now analyse these two cases for each of the two types of singularity .
We call them fibre types I and II .
First suppose LaTeXMLMath has a LaTeXMLMath quotient of a cusp singularity .
So LaTeXMLMath where LaTeXMLMath is a chain of LaTeXMLMath ’ s , and LaTeXMLMath ( respectively LaTeXMLMath ) are LaTeXMLMath -curves meeting LaTeXMLMath ( respectively LaTeXMLMath ) .
Suppose LaTeXMLMath is contained in a degenerate fibre LaTeXMLMath .
Then as above we can write LaTeXMLMath where without loss of generality LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath .
Note that LaTeXMLMath can not contain the other fork LaTeXMLMath of LaTeXMLMath , since then LaTeXMLMath contradicting the description above .
We contract LaTeXMLMath first .
We deduce that the curves in the string LaTeXMLMath have self-intersections LaTeXMLMath .
Thus LaTeXMLMath contracts to a LaTeXMLMath singularity , where LaTeXMLMath is the length of the string .
But this is never a LaTeXMLMath singularity ( since LaTeXMLMath ) , a contradiction .
So LaTeXMLMath is empty .
We can now calculate that the curves in the string LaTeXMLMath have self-intersections LaTeXMLMath if LaTeXMLMath .
Then LaTeXMLMath , a contradiction .
Hence LaTeXMLMath .
Next suppose LaTeXMLMath is not contained in a degenerate fibre .
Then LaTeXMLMath is horizontal , hence LaTeXMLMath .
Then it follows that we have a fibre of type II with LaTeXMLMath , a -2 curve .
We deduce that LaTeXMLMath is a single -2 curve .
But then LaTeXMLMath contracts to a LaTeXMLMath singularity , which is not LaTeXMLMath , a contradiction .
Combining , we deduce that LaTeXMLMath , and we have two fibres of the form LaTeXMLMath as above , and LaTeXMLMath .
There are no further degenerate fibres .
It only remains to calculate LaTeXMLMath .
We use LaTeXMLMath to deduce LaTeXMLMath .
I claim that the surface LaTeXMLMath constructed above does not admit a LaTeXMLMath -Gorenstein smoothing .
Let LaTeXMLMath be the index one cover of LaTeXMLMath at the singular point .
Then LaTeXMLMath has a cusp singularity and the exceptional locus of the minimal resolution LaTeXMLMath is a cycle of rational curves of self-intersections LaTeXMLMath .
Suppose LaTeXMLMath has a LaTeXMLMath -Gorenstein smoothing , then , taking the canonical cover of the smoothing at the singular point we obtain a smoothing of LaTeXMLMath .
Let LaTeXMLMath denote the Milnor fibre of the smoothing of LaTeXMLMath .
Consider the intersection product on LaTeXMLMath , write LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are the number of zero , positive and negative eigenvalues of the intersection form .
Since LaTeXMLMath is normal and Gorenstein , we have LaTeXMLCite LaTeXMLEquation .
In our case we calculate LaTeXMLMath , a contradiction .
So LaTeXMLMath is not smoothable , hence LaTeXMLMath does not have a LaTeXMLMath -Gorenstein smoothing ( not even locally ) .
Now suppose LaTeXMLMath has a quotient of a simple elliptic singularity .
So LaTeXMLMath where LaTeXMLMath , LaTeXMLMath is a chain of smooth rational curves and LaTeXMLMath meets LaTeXMLMath , for LaTeXMLMath and LaTeXMLMath .
We first give a partial classification of these singularities .
We can contract the chains LaTeXMLMath to obtain a partial resolution LaTeXMLMath .
Write LaTeXMLMath for the image of LaTeXMLMath under LaTeXMLMath .
Then the chains LaTeXMLMath contract to singularities of type LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be the indices of these singularities , then LaTeXMLMath ( because LaTeXMLMath is assumed to be strictly log canonical — the condition is equivalent to LaTeXMLMath ) .
Thus LaTeXMLMath or LaTeXMLMath after reordering .
In particular , we see that each chain LaTeXMLMath is either a single LaTeXMLMath of self-intersection LaTeXMLMath , or a chain of LaTeXMLMath LaTeXMLMath ’ s of self-intersection LaTeXMLMath .
We claim that the fork LaTeXMLMath of LaTeXMLMath can not be contained in a fibre LaTeXMLMath .
By the classification above , its enough to show that LaTeXMLMath , since this then forces LaTeXMLMath , a contradiction .
Write LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLEquation .
Since LaTeXMLMath is an isomorphism over LaTeXMLMath , we have LaTeXMLMath because LaTeXMLMath is LaTeXMLMath -exceptional .
So LaTeXMLMath , writing LaTeXMLMath for a fibre of LaTeXMLMath .
Here LaTeXMLMath where LaTeXMLMath is the multiplicity of LaTeXMLMath in LaTeXMLMath .
Now LaTeXMLMath since LaTeXMLMath is log canonical and LaTeXMLMath is minimal .
Hence LaTeXMLEquation so LaTeXMLMath as required .
Thus LaTeXMLMath is horizontal , LaTeXMLMath and we have 3 degenerate fibres of type II .
In each case LaTeXMLMath is a single curve of self-intersection LaTeXMLMath or a string of LaTeXMLMath LaTeXMLMath -curves .
If the fibre LaTeXMLMath is a string , we deduce that LaTeXMLMath is a string of LaTeXMLMath LaTeXMLMath -curves or a single curve of self-intersection LaTeXMLMath respectively .
Now , since LaTeXMLMath contracts to a LaTeXMLMath singularity , we deduce LaTeXMLMath is a single LaTeXMLMath -curve and LaTeXMLMath .
If LaTeXMLMath is not a string , we find that LaTeXMLMath is a string of three LaTeXMLMath -curves , LaTeXMLMath meets the middle component , and LaTeXMLMath is empty , hence again LaTeXMLMath .
So LaTeXMLMath for all LaTeXMLMath , contradicting the classification above .
It remains to show that an elliptic cone of degree LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
We prove this in Lemma LaTeXMLRef below .
∎ Let LaTeXMLMath be an elliptic cone of degree LaTeXMLMath .
Then LaTeXMLMath has a smoothing to LaTeXMLMath .
Given an elliptic cone LaTeXMLMath of degree LaTeXMLMath , write LaTeXMLMath for the minimal resolution of LaTeXMLMath .
Then LaTeXMLMath is a ruled surface over an elliptic curve LaTeXMLMath , i.e. , LaTeXMLMath , where LaTeXMLMath is a line bundle on LaTeXMLMath of degree LaTeXMLMath .
We claim that LaTeXMLMath is determined up to isomorphism by its section LaTeXMLMath .
For LaTeXMLMath acts transitively on LaTeXMLMath , and given a line bundle LaTeXMLMath of degree LaTeXMLMath we have LaTeXMLMath for some LaTeXMLMath ( c.f .
LaTeXMLCite , p. 337 , Exercise 4.6 ( b ) ) , thus LaTeXMLMath acts transitively on the line bundles of degree LaTeXMLMath .
Our claim follows .
Let LaTeXMLMath be the spectrum of a DVR , and write LaTeXMLMath .
Let LaTeXMLMath be an elliptic curve in the special fibre .
Let LaTeXMLMath be the blowup of LaTeXMLMath in LaTeXMLMath .
Then the special fibre LaTeXMLMath consists of the strict transform LaTeXMLMath of LaTeXMLMath together with a ruled surface LaTeXMLMath of degree LaTeXMLMath over the elliptic curve LaTeXMLMath .
We contract LaTeXMLMath to obtain a family LaTeXMLMath which is a smoothing of an elliptic cone of degree LaTeXMLMath over LaTeXMLMath to LaTeXMLMath .
∎ Note that any smoothing of LaTeXMLMath is LaTeXMLMath -Gorenstein since LaTeXMLMath is Cartier .
The aim of this section is to define the notion of a relative Weil divisor and explain the conditions ‘ LaTeXMLMath and LaTeXMLMath commute with base change ’ in the definition ( LaTeXMLRef ) of an allowable family of stable pairs .
See also Section LaTeXMLRef .
We first recall some of Kollár ’ s theory of push forward and base change for open immersions .
These results are proved in LaTeXMLCite ( unpublished ) , I at least provide the statements here .
Let LaTeXMLMath be a morphism of schemes .
Let LaTeXMLMath be an open subscheme , LaTeXMLMath , and LaTeXMLMath a subscheme proper over LaTeXMLMath .
Let LaTeXMLMath be a coherent sheaf on LaTeXMLMath which is flat over LaTeXMLMath .
Given a morphism LaTeXMLMath , write LaTeXMLMath , LaTeXMLMath etc. , LaTeXMLMath for the induced morphism , LaTeXMLMath .
We say that the push forward of LaTeXMLMath commutes with LaTeXMLMath if the natural map LaTeXMLMath is an isomorphism in a neighbourhood of LaTeXMLMath .
We say that the push forward of LaTeXMLMath commutes with arbitrary base change if the push forward of LaTeXMLMath commutes with any LaTeXMLMath .
In our applications , LaTeXMLMath contains all closed points of LaTeXMLMath , so an isomorphism in a neighbourhood of LaTeXMLMath is a global isomorphism .
More specifically , we are only interested in the following special case : LaTeXMLMath is an invertible sheaf , LaTeXMLMath is a family of CM reduced surfaces , LaTeXMLMath is finite for every LaTeXMLMath , and either LaTeXMLMath is proper and LaTeXMLMath or LaTeXMLMath is local and LaTeXMLMath .
Then LaTeXMLMath is a ‘ divisorial sheaf ’ as defined below .
We write ‘ LaTeXMLMath commutes with base change ’ to mean the push forward of LaTeXMLMath commutes with arbitrary base change in this case ( since the choice of LaTeXMLMath is immaterial by Lemma LaTeXMLRef ( 1 ) ) .
Notation as above .
The push forward of LaTeXMLMath commutes with any flat LaTeXMLMath Let LaTeXMLMath be faithfully flat .
Then the push forward of LaTeXMLMath commutes with arbitrary base change iff the push forward of LaTeXMLMath commutes with arbitrary base change .
Assume that the push forward of LaTeXMLMath commutes with arbitrary base change and let LaTeXMLMath be a morphism , then the push forward of LaTeXMLMath commutes with arbitrary base change .
The following are equivalent : The push forward of LaTeXMLMath commutes with arbitrary base change .
The push forward of LaTeXMLMath commutes with every LaTeXMLMath where LaTeXMLMath is a closed point .
The natural morphism LaTeXMLMath is surjective for every LaTeXMLMath where LaTeXMLMath is a closed point .
If the push forward of LaTeXMLMath commutes with arbitrary base change then LaTeXMLMath is flat over LaTeXMLMath .
Notation as above .
Assume that LaTeXMLMath is local , LaTeXMLMath satisfies Serre ’ s condition LaTeXMLMath , and that LaTeXMLMath is coherent .
Then the following are equivalent : The push forward of LaTeXMLMath commutes with arbitrary base change .
The push forward of LaTeXMLMath commutes with LaTeXMLMath .
For every local Artinian subscheme LaTeXMLMath the push forward of LaTeXMLMath commutes with LaTeXMLMath .
Notation as above .
Assume that LaTeXMLMath is Noetherian , LaTeXMLMath is projective , and that LaTeXMLMath is LaTeXMLMath and LaTeXMLMath is coherent for all LaTeXMLMath .
Then there exists a locally closed stratification LaTeXMLMath , such that if LaTeXMLMath is a reduced scheme and LaTeXMLMath is morphism , then the push forward of LaTeXMLMath commutes with arbitrary base change iff LaTeXMLMath factors through LaTeXMLMath .
The assumption that LaTeXMLMath is projective is necessary for general LaTeXMLMath .
However , if we assume that LaTeXMLMath is the spectrum of a complete local ring , then the conclusion holds for arbitrary LaTeXMLMath .
We would like to remove the requirement that LaTeXMLMath is a reduced scheme — we do not know if this is possible .
We now define the notion of a relative Weil divisor for a family of CM reduced surfaces LaTeXMLMath over an arbitrary base LaTeXMLMath .
This is a generalisation of Mumford ’ s notion of a relative Cartier divisor ( LaTeXMLCite , Lecture 10 ) .
Let LaTeXMLMath be a family of CM reduced surfaces .
We say a codimension LaTeXMLMath closed subscheme LaTeXMLMath of LaTeXMLMath is a relative effective Weil divisor if there exists an open subscheme LaTeXMLMath and an effective Cartier divisor LaTeXMLMath on LaTeXMLMath , flat over LaTeXMLMath , such that LaTeXMLMath is finite for each LaTeXMLMath .
LaTeXMLMath , the scheme theoretic closure of LaTeXMLMath in LaTeXMLMath .
We define a relative Weil divisor to be a formal difference LaTeXMLMath of relative effective Weil divisors .
Given a relative Weil divisor LaTeXMLMath , we define an associated sheaf LaTeXMLMath , where LaTeXMLMath is the open subscheme where LaTeXMLMath and LaTeXMLMath are Cartier , and LaTeXMLMath is the invertible sheaf corresponding to the Cartier divisor LaTeXMLMath as usual .
We say that LaTeXMLMath is Cartier if LaTeXMLMath is invertible .
Given LaTeXMLMath , we define the pullback LaTeXMLMath of a relative effective Weil divisor LaTeXMLMath to LaTeXMLMath via LaTeXMLMath .
We define the pullback of a general relative Weil divisor by linearity .
We say a coherent sheaf LaTeXMLMath on LaTeXMLMath is a divisorial sheaf if LaTeXMLMath for some relative Weil divisor LaTeXMLMath .
Equivalently , there exists an open subscheme LaTeXMLMath such that LaTeXMLMath is invertible , LaTeXMLMath is finite for each LaTeXMLMath and LaTeXMLMath .
Given a divisorial sheaf LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath denote the sheaf LaTeXMLMath ( corresponds to multiplication of the divisor by LaTeXMLMath ) .
The assumption that LaTeXMLMath is flat over LaTeXMLMath is equivalent to the following : for all LaTeXMLMath , LaTeXMLMath does not contain any component of LaTeXMLMath ( LaTeXMLCite , Lecture 10 , p. 72 ) .
Given a relative Weil divisor LaTeXMLMath on LaTeXMLMath , the sheaf LaTeXMLMath is coherent ( using Remark LaTeXMLRef below ) .
If LaTeXMLMath is a divisorial sheaf and LaTeXMLMath is any open subscheme such that LaTeXMLMath is finite for each LaTeXMLMath then LaTeXMLMath ( using Lemma LaTeXMLRef ( 1 ) ) .
Note that LaTeXMLMath is not the same as LaTeXMLMath in general , and moreover LaTeXMLMath is not necessarily flat over LaTeXMLMath .
See Lemma LaTeXMLRef and Example LaTeXMLRef below .
Our next result is a technical lemma which , given a family LaTeXMLMath and a sheaf LaTeXMLMath on LaTeXMLMath , flat over LaTeXMLMath , relates the LaTeXMLMath property for the fibres LaTeXMLMath of LaTeXMLMath to a relative LaTeXMLMath -type property for LaTeXMLMath .
Note that it is not true that if every fibre LaTeXMLMath is LaTeXMLMath then the sheaf LaTeXMLMath is LaTeXMLMath — we can easily construct a counter example where the base LaTeXMLMath is not LaTeXMLMath .
Let LaTeXMLMath be a family of CM reduced surfaces and LaTeXMLMath a coherent sheaf on LaTeXMLMath which is flat over LaTeXMLMath .
Suppose that for each closed point LaTeXMLMath the sheaf LaTeXMLMath on LaTeXMLMath satisfies Serre ’ s condition LaTeXMLMath .
Let LaTeXMLMath be an open subscheme such that the set LaTeXMLMath is finite for each LaTeXMLMath .
Then the natural map LaTeXMLMath is an isomorphism .
Suppose that for each closed point LaTeXMLMath the sheaf LaTeXMLMath on LaTeXMLMath is invertible in codimension LaTeXMLMath .
Then there exists LaTeXMLMath such that the set LaTeXMLMath is finite for each LaTeXMLMath and LaTeXMLMath is invertible .
In particular , if LaTeXMLMath is invertible in codimension LaTeXMLMath and LaTeXMLMath for each closed point LaTeXMLMath , then LaTeXMLMath is a divisorial sheaf .
( 1 ) Write LaTeXMLMath .
We work locally at a closed point LaTeXMLMath , say LaTeXMLMath .
Then LaTeXMLMath is a closed subscheme with support LaTeXMLMath .
The sheaf LaTeXMLMath is LaTeXMLMath by assumption , so there exists a regular sequence LaTeXMLMath for LaTeXMLMath at LaTeXMLMath .
Replacing LaTeXMLMath by LaTeXMLMath if necessary , we may assume that LaTeXMLMath .
Now lift LaTeXMLMath to LaTeXMLMath , then LaTeXMLMath is a regular sequence for LaTeXMLMath at LaTeXMLMath ( LaTeXMLCite , p. 177 , Corollary of Theorem 22.5 ) .
Equivalently , we have an exact sequence LaTeXMLEquation .
Consider the natural map LaTeXMLMath , write LaTeXMLMath for the kernel and LaTeXMLMath for the cokernel .
LaTeXMLMath and LaTeXMLMath have support contained in the set LaTeXMLMath , so any given element of LaTeXMLMath or LaTeXMLMath is annihilated by some power of LaTeXMLMath .
So , if LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath , so in particular LaTeXMLMath , contradicting the exact sequence above .
Similiarly if LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath .
Again using the exact sequence above , since LaTeXMLMath we obtain LaTeXMLMath for some LaTeXMLMath , it follows that LaTeXMLMath , a contradiction .
Thus LaTeXMLMath , so the map LaTeXMLMath is an isomorphism as claimed .
( 2 ) If LaTeXMLMath is invertible at LaTeXMLMath then , working locally at LaTeXMLMath , lifting an isomorphism LaTeXMLMath we obtain a surjection LaTeXMLMath , by Nakayama ’ s Lemma .
Now , by flatness of LaTeXMLMath , it follows that LaTeXMLMath is an isomorphism , so LaTeXMLMath is invertible at LaTeXMLMath .
∎ In particular , if LaTeXMLMath is a family of CM reduced surfaces , and LaTeXMLMath is an open inclusion such that LaTeXMLMath is finite for each LaTeXMLMath , then LaTeXMLMath .
The sheaf LaTeXMLMath is divisorial for a family LaTeXMLMath of slc surfaces .
For LaTeXMLMath is flat over LaTeXMLMath , and , for each LaTeXMLMath , the natural map LaTeXMLMath is an isomorphism and LaTeXMLMath is invertible in codimension LaTeXMLMath and LaTeXMLMath .
We write LaTeXMLMath for a relative Weil divisor such that LaTeXMLMath .
Let LaTeXMLMath be a family of CM reduced surfaces , LaTeXMLMath a relative effective Weil divisor .
Then LaTeXMLMath for all LaTeXMLMath iff LaTeXMLMath commutes with base change .
Moreover , in this case LaTeXMLMath is flat over LaTeXMLMath .
We may assume that LaTeXMLMath and LaTeXMLMath are affine , write LaTeXMLMath , LaTeXMLMath .
We have a commutative diagram : LaTeXMLEquation .
Thus LaTeXMLMath iff LaTeXMLMath is surjective , by the snake lemma .
This proves the first part using Lemma LaTeXMLRef ( 3 ) .
Moreover , we see that in this case LaTeXMLMath for all LaTeXMLMath , hence LaTeXMLMath is flat over LaTeXMLMath as required .
∎ Let LaTeXMLMath be a family of surfaces over the spectrum of a DVR LaTeXMLMath with generic fibre LaTeXMLMath and special fibre LaTeXMLMath ( this can be realised as a family of scrolls in a projective space ) .
We can contract the negative section LaTeXMLMath of LaTeXMLMath to obtain a family LaTeXMLMath .
Let LaTeXMLMath be the image of LaTeXMLMath , then LaTeXMLMath is not LaTeXMLMath -Cartier at LaTeXMLMath .
However the special fibre LaTeXMLMath has LaTeXMLMath Cartier .
Let LaTeXMLMath be a relative Weil divisor , then necessarily LaTeXMLMath , let LaTeXMLMath be the restriction of the divisor LaTeXMLMath to the special fibre .
Then the natural map LaTeXMLMath is not surjective .
For otherwise LaTeXMLMath is invertible by Nakayama ’ s Lemma , a contradiction .
Thus the scheme theoretic fibre LaTeXMLMath has an embedded point at LaTeXMLMath , in particular LaTeXMLMath .
Let LaTeXMLMath be a family of CM reduced surfaces and LaTeXMLMath a codimension 1 closed subscheme , flat over LaTeXMLMath .
Then for LaTeXMLMath we can define the restriction LaTeXMLMath of LaTeXMLMath to the fibre LaTeXMLMath ( we take the double dual of the ideal sheaf of the fibre LaTeXMLMath as the ideal sheaf of LaTeXMLMath ) .
Consider the locus LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath ( equivalently LaTeXMLMath is LaTeXMLMath ) .
Since LaTeXMLMath is flat over LaTeXMLMath , this locus is open .
Finally , let LaTeXMLMath be the open locus of points LaTeXMLMath such that LaTeXMLMath is Cartier in codimension LaTeXMLMath .
Then LaTeXMLMath is a relative Weil divisor on LaTeXMLMath by Lemma LaTeXMLRef .
We prove below some foundational results concerning LaTeXMLMath -Cartier relative Weil divisors .
Let LaTeXMLMath be a family of CM reduced surfaces , with LaTeXMLMath local .
Let LaTeXMLMath be a relative Weil divisor .
Suppose that LaTeXMLMath is Cartier .
Then LaTeXMLMath commutes with base change if and only if LaTeXMLMath is Cartier .
If LaTeXMLMath commutes with base change then LaTeXMLMath is an isomorphism , and LaTeXMLMath is invertible by assumption .
Since LaTeXMLMath is coherent , it follows that LaTeXMLMath is invertible by Nakayama ’ s Lemma and the flatness of LaTeXMLMath ( using Lemma LaTeXMLRef ) .
Conversely , suppose LaTeXMLMath is Cartier .
By Lemma LaTeXMLRef ( 2 ) , we need only show that LaTeXMLMath is an isomorphism .
Both sides are invertible by assumption and the map is an isomorphism in codimension LaTeXMLMath , thus it is an isomorphism since LaTeXMLMath is LaTeXMLMath .
∎ Let LaTeXMLMath be a family of CM reduced surfaces with LaTeXMLMath local , and LaTeXMLMath a LaTeXMLMath -Cartier relative Weil divisor on LaTeXMLMath of index LaTeXMLMath .
We define a LaTeXMLMath quotient LaTeXMLMath as follows : LaTeXMLEquation where the multiplication is given by fixing an isomorphism LaTeXMLMath .
Let LaTeXMLMath be the locus where LaTeXMLMath is Cartier and LaTeXMLMath the inverse image in LaTeXMLMath .
We have LaTeXMLMath is etale .
The relative Weil divisor LaTeXMLMath is Cartier .
LaTeXMLMath .
Conversely , any LaTeXMLMath quotient LaTeXMLMath satisfying these criteria is of the form above .
We refer to such a cover as a cyclic cover of LaTeXMLMath defined by LaTeXMLMath , or , in the case LaTeXMLMath , an index one cover of LaTeXMLMath .
The construction is unique up to the choice of an element of LaTeXMLMath .
In particular if LaTeXMLMath is a local analytic germ , the construction is unique .
This is a straightforward generalisation of the usual cyclic covering trick LaTeXMLCite .
∎ Note that LaTeXMLMath is a single point ( using LaTeXMLMath ) .
Let LaTeXMLMath be a family of CM reduced surfaces , with LaTeXMLMath local , and LaTeXMLMath a relative Weil divisor on LaTeXMLMath .
Suppose LaTeXMLMath is LaTeXMLMath -Cartier .
Then LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath Cartier gives LaTeXMLMath Cartier , so LaTeXMLMath .
We claim that LaTeXMLMath .
We may work locally at LaTeXMLMath .
First suppose that LaTeXMLMath where LaTeXMLMath is a local Artinian LaTeXMLMath -algebra .
Let LaTeXMLMath be a cyclic cover defined by LaTeXMLMath .
Let LaTeXMLMath be the locus where LaTeXMLMath is Cartier ( then LaTeXMLMath is the open subscheme of LaTeXMLMath with underlying space LaTeXMLMath ) and let LaTeXMLMath be the inverse image in LaTeXMLMath .
We have LaTeXMLMath .
Thus LaTeXMLMath connected implies LaTeXMLMath is connected .
Hence LaTeXMLMath is connected , since LaTeXMLMath and LaTeXMLMath have the same underlying space .
But LaTeXMLEquation thus the cover LaTeXMLEquation of LaTeXMLMath is connected , it follows that the index LaTeXMLMath of LaTeXMLMath is equal to LaTeXMLMath .
The general case now follows using Lemma LaTeXMLRef and Lemma LaTeXMLRef — we have LaTeXMLMath Cartier for every Artinian subscheme LaTeXMLMath , hence LaTeXMLMath is Cartier , so LaTeXMLMath .
∎ The aim of this section is to give necessary conditions for a groupoid LaTeXMLMath over LaTeXMLMath to admit algebraic , everywhere versal deformations ( Theorem LaTeXMLRef ) .
This theory was developed by Artin in LaTeXMLCite , starting from the work of Schlessinger LaTeXMLCite .
In Section LaTeXMLRef we use our result to prove that LaTeXMLMath is an algebraic stack .
Let LaTeXMLMath be the category of noetherian LaTeXMLMath -algebras .
We say a morphism LaTeXMLMath in LaTeXMLMath is an extension if it is surjective , we say it is an infinitesimal extension if it has nilpotent kernel .
Let LaTeXMLMath be a groupoid over LaTeXMLMath .
We write LaTeXMLMath for the fibre over LaTeXMLMath .
Given LaTeXMLMath in LaTeXMLMath and LaTeXMLMath , we write LaTeXMLMath for the groupoid of maps LaTeXMLMath in LaTeXMLMath lying over LaTeXMLMath .
We write LaTeXMLMath , LaTeXMLMath for the isomorphism classes of LaTeXMLMath , LaTeXMLMath , these define functors LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be a groupoid over LaTeXMLMath .
We say that LaTeXMLMath is limit preserving if the natural functor LaTeXMLEquation is an equivalence of categories for every direct system LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath , let LaTeXMLMath be an extension , and LaTeXMLMath an infinitesimal extension with kernel LaTeXMLMath , where LaTeXMLMath is a finite LaTeXMLMath -module ( i.e. , writing LaTeXMLMath for the kernel of LaTeXMLMath , we have LaTeXMLMath in LaTeXMLMath ) .
We write LaTeXMLMath for the trivial extension of LaTeXMLMath by LaTeXMLMath , namely the LaTeXMLMath -module LaTeXMLMath with multiplication given by LaTeXMLMath .
We define conditions ( S1 ) ( a ) , ( S1 ) ( b ) and ( S2 ) for a groupoid LaTeXMLMath over LaTeXMLMath as follows : Let LaTeXMLEquation be a diagram in LaTeXMLMath , where LaTeXMLMath is as in Notation LaTeXMLRef .
Assume that the composed map LaTeXMLMath is surjective .
Let LaTeXMLMath .
Then the canonical map LaTeXMLEquation is surjective .
Let LaTeXMLMath be surjective , and let LaTeXMLMath be a finite LaTeXMLMath -module .
Let LaTeXMLMath have direct image LaTeXMLMath .
Then the canonical map LaTeXMLEquation is bijective .
If LaTeXMLMath satisfies ( S1 ) ( b ) , LaTeXMLMath has a natural LaTeXMLMath -module structure , and the underlying additive group acts on LaTeXMLMath LaTeXMLCite .
Write LaTeXMLMath .
LaTeXMLMath is a finite LaTeXMLMath -module .
By an obstruction theory LaTeXMLMath for LaTeXMLMath we mean the following data : For each infinitesimal extension LaTeXMLMath and element LaTeXMLMath , a functor LaTeXMLEquation .
For each LaTeXMLMath as in Notation LaTeXMLRef and LaTeXMLMath , an element LaTeXMLMath which is zero iff LaTeXMLMath .
The data is required to be functorial .
That is , given a morphism of extensions LaTeXMLEquation where LaTeXMLMath is surjective and LaTeXMLMath , LaTeXMLMath are finite LaTeXMLMath -modules , and given LaTeXMLMath with direct image LaTeXMLMath , we have a natural map LaTeXMLMath determined by LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
We also require that the data is linear in LaTeXMLMath , i.e. , given LaTeXMLMath and LaTeXMLMath finite LaTeXMLMath -modules , the map LaTeXMLEquation is an LaTeXMLMath -module homomorphism .
We say LaTeXMLMath is algebraic if it is of finite type over LaTeXMLMath .
We say LaTeXMLMath is algebraic if LaTeXMLMath is algebraic .
We state some further conditions we require for LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath are compatible with etale localisation , i.e. , given LaTeXMLMath etale , LaTeXMLMath with direct image LaTeXMLMath etc. , we have LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLMath is compatible with completions , i.e. , for LaTeXMLMath a maximal ideal of LaTeXMLMath , we have LaTeXMLEquation .
Constructibility : For LaTeXMLMath reduced , there is an open dense set of points LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation .
The lifting property for LaTeXMLMath is the following : Given LaTeXMLMath and LaTeXMLMath in LaTeXMLMath such that the direct image LaTeXMLMath of LaTeXMLMath in LaTeXMLMath is surjective , there exists LaTeXMLMath such that LaTeXMLMath .
An element LaTeXMLMath is formally versal at LaTeXMLMath if the lifting property holds whenever LaTeXMLMath is a finite length extension of the residue field LaTeXMLMath .
We say LaTeXMLMath is formally smooth over LaTeXMLMath if the lifting property holds whenever LaTeXMLMath is an infinitesimal extension .
Let LaTeXMLMath be a limit preserving groupoid over LaTeXMLMath .
Assume that we are given an obstruction theory LaTeXMLMath for LaTeXMLMath .
Suppose that LaTeXMLMath satisfies ( S1 ) ( a ) , ( b ) and ( S2 ) and that LaTeXMLMath and LaTeXMLMath satisfy the conditions of LaTeXMLRef for algebraic LaTeXMLMath , LaTeXMLMath .
Suppose also that , if LaTeXMLMath is a complete local ring in LaTeXMLMath , the map LaTeXMLEquation has dense image .
Then , given LaTeXMLMath there exists an algebraic ring LaTeXMLMath with a closed point LaTeXMLMath and LaTeXMLMath , formally smooth over LaTeXMLMath , such that LaTeXMLMath .
This follows from LaTeXMLCite , p. 170 , Corollary 3.2 and p. 175 , Theorem 4.4 .
∎ We define a workable theory of LaTeXMLMath -Gorenstein deformations of slc surfaces as suggested in LaTeXMLCite — we say LaTeXMLMath is LaTeXMLMath -Gorenstein if LaTeXMLMath commutes with base change for all LaTeXMLMath ( Definition LaTeXMLRef ) .
In particular , for LaTeXMLMath , the family LaTeXMLMath is LaTeXMLMath -Gorenstein by definition .
This theory is used in Section LaTeXMLRef to show that LaTeXMLMath is an algebraic stack using the methods of Artin .
We first review some standard deformation theory .
We consider families of schemes LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath is a noetherian scheme , flat over LaTeXMLMath , which is either of finite type over LaTeXMLMath or affine , and is separated for the LaTeXMLMath -adic topology for each LaTeXMLMath .
Given an infinitesimal extension LaTeXMLMath in LaTeXMLMath , write LaTeXMLMath for the set of deformations of LaTeXMLMath over LaTeXMLMath .
Given a family LaTeXMLMath extending LaTeXMLMath , write LaTeXMLMath for the group of automorphisms of LaTeXMLMath over LaTeXMLMath which restrict to the identity on LaTeXMLMath .
Given LaTeXMLMath in LaTeXMLMath , we also write LaTeXMLMath for LaTeXMLMath .
We want to allow the fibres to be local , in particular we can not assume that LaTeXMLMath is of finite type .
We need to insist that LaTeXMLMath is separated for the LaTeXMLMath -adic topology for each LaTeXMLMath to ensure that , e.g. , there are no empty fibres of LaTeXMLMath ( c.f .
LaTeXMLCite , p. 21 ) .
Let LaTeXMLMath and let LaTeXMLMath be a family of schemes over LaTeXMLMath as in Notation LaTeXMLRef .
Let LaTeXMLMath be a cotangent complex for LaTeXMLMath , in the derived category of coherent sheaves on LaTeXMLMath ( LaTeXMLCite , p. 44 , Definition 2.1.3 ) .
Given a coherent sheaf LaTeXMLMath on LaTeXMLMath , define LaTeXMLMath .
Define LaTeXMLMath and LaTeXMLMath .
Here , given a complex of sheaves , we use LaTeXMLMath to denote the cohomology sheaves and LaTeXMLMath to denote the hypercohomology groups .
If LaTeXMLMath is of finite type we may assume that the sheaves LaTeXMLMath are coherent , so in particular the LaTeXMLMath are coherent .
We only need the cases LaTeXMLMath for some finite LaTeXMLMath -module LaTeXMLMath .
There is a local-to-global spectral sequence relating the LaTeXMLMath and the LaTeXMLMath : LaTeXMLEquation .
This is the usual hypercohomology spectral sequence , see LaTeXMLCite , p. 445 .
Let LaTeXMLMath be a family of schemes and LaTeXMLMath a coherent sheaf on LaTeXMLMath .
Then LaTeXMLMath .
LaTeXMLMath is supported on the locus where LaTeXMLMath is not a smooth morphism .
LaTeXMLMath is supported on the locus where LaTeXMLMath is not an lci morphism .
See LaTeXMLCite , 2.3 , Theorem 3.1.5 and Corollary 3.2.2 .
∎ Notation as in LaTeXMLRef .
Let LaTeXMLMath be a family of schemes and LaTeXMLMath a family extending LaTeXMLMath .
There exists a canonical element LaTeXMLMath such that LaTeXMLMath iff LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath is a principal homogeneous space under LaTeXMLMath .
Given LaTeXMLMath extending LaTeXMLMath , LaTeXMLMath is naturally isomorphic to LaTeXMLMath .
For the local case , we use LaTeXMLCite , p. 66 , Theorem 4.3.3 , and p. 50 , 2.3.2 ( which shows that LaTeXMLMath for each LaTeXMLMath ) .
The global case follows formally .
∎ We now develop a LaTeXMLMath -Gorenstein deformation theory for slc surfaces .
We first state the most obvious definition of a LaTeXMLMath -Gorenstein deformation ( we refer to this as ‘ weakly LaTeXMLMath -Gorenstein ’ ) , and then refine our definition .
Let LaTeXMLMath be a family of slc surfaces .
We say LaTeXMLMath is weakly LaTeXMLMath - Gorenstein if the relative Weil divisor LaTeXMLMath is LaTeXMLMath -Cartier .
This is the definition suggested in LaTeXMLCite , p. 185 , Remark 6.27 and is equivalent to the definition used in LaTeXMLCite .
However , it is not well understood over an Artinian base , in particular there is no known obstruction theory for weakly LaTeXMLMath -Gorenstein deformations .
To remedy this we make the following definition .
Let LaTeXMLMath be a family of slc surfaces .
We say LaTeXMLMath is LaTeXMLMath - Gorenstein if LaTeXMLMath commutes with base change for all LaTeXMLMath .
Given a LaTeXMLMath -Gorenstein family of slc surfaces LaTeXMLMath and an infinitesimal extension LaTeXMLMath in LaTeXMLMath , write LaTeXMLMath for the set of LaTeXMLMath -Gorenstein deformations of LaTeXMLMath over LaTeXMLMath .
Note that LaTeXMLMath -Gorenstein implies weakly LaTeXMLMath -Gorenstein by Lemma LaTeXMLRef .
Equivalently , we require that LaTeXMLMath commutes with base change for LaTeXMLMath , where LaTeXMLMath is Cartier for each geometric fibre LaTeXMLMath .
For , by Lemma LaTeXMLRef , if LaTeXMLMath commutes with base change then LaTeXMLMath invertible , thus locally on LaTeXMLMath we have LaTeXMLMath .
This definition was given in LaTeXMLCite , p. 260 , Definition 5.2 .
Geometrically , the LaTeXMLMath -Gorenstein deformations of a local slc surface LaTeXMLMath are precisely those deformations which lift to deformations of the index one cover LaTeXMLMath .
This is made more precise in Proposition LaTeXMLRef below .
This description enables us to prove an analogous result to Theorem LaTeXMLRef for LaTeXMLMath -Gorenstein deformations ( Theorem LaTeXMLRef ) .
In particular , we define an obstruction theory for LaTeXMLMath -Gorenstein deformations .
An outline of this theory was given in a preprint of Hassett ( c.f .
LaTeXMLCite ) .
We also show that , for a smoothing LaTeXMLMath of an slc surface over the spectrum of a DVR , LaTeXMLMath is LaTeXMLMath -Gorenstein iff it is weakly LaTeXMLMath -Gorenstein .
Given LaTeXMLMath a LaTeXMLMath -Gorenstein family of slc surfaces , with LaTeXMLMath and LaTeXMLMath local , fix an index one cover LaTeXMLMath .
This is a LaTeXMLMath quotient , where LaTeXMLMath .
Then LaTeXMLMath is flat , and for any LaTeXMLMath in LaTeXMLMath , LaTeXMLMath is an index one cover of LaTeXMLMath .
Let LaTeXMLMath be a local extension in LaTeXMLMath , and consider the set of families of slc surfaces LaTeXMLMath extending LaTeXMLMath .
We have an action of LaTeXMLMath on this set coming from the LaTeXMLMath action on LaTeXMLMath , let LaTeXMLMath be an invariant element .
Then LaTeXMLMath is a LaTeXMLMath -Gorenstein family of slc surfaces extending LaTeXMLMath and LaTeXMLMath is an index one cover .
In particular , if LaTeXMLMath is an infinitesimal extension , we have a natural isomorphism LaTeXMLEquation .
LaTeXMLEquation with inverse LaTeXMLMath where LaTeXMLMath is the unique index one cover extending LaTeXMLMath .
We are given a family LaTeXMLMath such that LaTeXMLMath commutes with base change for all LaTeXMLMath .
Then in particular LaTeXMLMath is LaTeXMLMath -Cartier and LaTeXMLMath by Lemma LaTeXMLRef .
Let LaTeXMLMath be an index one cover .
Then LaTeXMLMath is flat over LaTeXMLMath , because LaTeXMLEquation and LaTeXMLMath is flat over LaTeXMLMath for each LaTeXMLMath by Lemma LaTeXMLRef .
Given LaTeXMLMath in LaTeXMLMath , the isomorphisms LaTeXMLMath show that LaTeXMLMath is an index one cover of LaTeXMLMath .
Given LaTeXMLMath a LaTeXMLMath invariant extension of LaTeXMLMath we write LaTeXMLMath , then LaTeXMLMath is a flat family extending LaTeXMLMath .
We claim that the quotient LaTeXMLMath is an index one cover of LaTeXMLMath .
First , LaTeXMLMath is invertible , and LaTeXMLMath commutes with base change , thus LaTeXMLMath is invertible .
Thus LaTeXMLMath is Cartier .
In particular LaTeXMLMath is LaTeXMLMath -Cartier , so LaTeXMLMath by Lemma LaTeXMLRef .
Let LaTeXMLMath denote the open subscheme of LaTeXMLMath where LaTeXMLMath is invertible , and write LaTeXMLMath for the corresponding open subscheme of LaTeXMLMath .
Then LaTeXMLMath is an etale LaTeXMLMath quotient ( because the map is etale over LaTeXMLMath ) .
Finally LaTeXMLMath by Lemma LaTeXMLRef ( 1 ) .
So LaTeXMLMath is an index one cover , by Proposition LaTeXMLRef .
Moreover , we claim LaTeXMLMath commutes with base change for all LaTeXMLMath .
First LaTeXMLMath commutes with base change since it is invertible , using Lemma LaTeXMLRef .
Also , by the above , LaTeXMLMath is an index one cover of LaTeXMLMath , and LaTeXMLMath is an index one cover LaTeXMLMath of LaTeXMLMath by assumption .
It follows that LaTeXMLMath is an isomorphism for LaTeXMLMath .
Now LaTeXMLMath commutes with base change for all LaTeXMLMath by assumption , thus LaTeXMLMath commutes with base change for LaTeXMLMath using Lemma LaTeXMLRef ( 2 ) .
So LaTeXMLMath commutes with base change for all LaTeXMLMath by Remark LaTeXMLRef .
Finally , if LaTeXMLMath is a infinitesimal extension , there is a unique index one cover of LaTeXMLMath extending LaTeXMLMath — for such an index one cover LaTeXMLMath is determined by the choice of an isomorphism LaTeXMLMath extending a given isomorphism LaTeXMLMath and thus by a unit LaTeXMLMath .
Moreover multiplying LaTeXMLMath by LaTeXMLMath for some LaTeXMLMath does not change the isomorphism type of the extension LaTeXMLMath of LaTeXMLMath .
But we can always take LaTeXMLMath th roots in LaTeXMLMath since LaTeXMLMath is infinitesimal , thus LaTeXMLMath is uniquely determined as claimed .
The last part of the Proposition follows .
∎ Let LaTeXMLMath be the spectrum of a DVR with generic point LaTeXMLMath .
Let LaTeXMLMath be a weakly LaTeXMLMath -Gorenstein family of slc surfaces such that LaTeXMLMath is canonical .
Then LaTeXMLMath is LaTeXMLMath -Gorenstein .
We may work locally at LaTeXMLMath .
Let LaTeXMLMath be an index one cover .
It is enough to show that the map LaTeXMLMath of the special fibres is an index one cover of LaTeXMLMath , by Proposition LaTeXMLRef .
Now LaTeXMLMath extends to an index one cover LaTeXMLMath , we need to show that LaTeXMLMath is LaTeXMLMath to deduce LaTeXMLMath .
First , LaTeXMLMath slc and LaTeXMLMath canonical implies that LaTeXMLMath is canonical .
For there exists a finite base change LaTeXMLMath such that LaTeXMLMath admits a semistable resolution , then LaTeXMLMath is canonical by Lemma LaTeXMLRef .
Using LaTeXMLCite , p. 310 , Lemma 3.3 we deduce that LaTeXMLMath is canonical .
Now LaTeXMLMath canonical implies that LaTeXMLMath is canonical , and canonical singularities are rational so in particular CM .
Hence LaTeXMLMath is CM since LaTeXMLMath ( where LaTeXMLMath is a uniformising parameter ) .
This completes the proof .
∎ Let LaTeXMLMath and let LaTeXMLMath be a LaTeXMLMath -Gorenstein family of slc surfaces .
Let LaTeXMLMath be a coherent sheaf on LaTeXMLMath .
We define a complex LaTeXMLMath in the derived category of coherent sheaves on LaTeXMLMath as follows : Let LaTeXMLMath be a local index one cover of LaTeXMLMath , a LaTeXMLMath quotient , where LaTeXMLMath is the local index of LaTeXMLMath .
Let LaTeXMLMath be a cotangent complex for LaTeXMLMath .
Define LaTeXMLMath locally .
Now define LaTeXMLMath and LaTeXMLMath .
Note that , since the functor LaTeXMLMath , LaTeXMLMath is exact , we have LaTeXMLEquation locally .
We have a local-to-global spectral sequence LaTeXMLEquation as above .
Let LaTeXMLMath , let LaTeXMLMath be a LaTeXMLMath -Gorenstein family of slc surfaces , and LaTeXMLMath , some finite LaTeXMLMath -module LaTeXMLMath .
Then LaTeXMLMath .
Let LaTeXMLMath be the inclusion of the locus where LaTeXMLMath is invertible .
Let LaTeXMLMath be a local index one cover , a LaTeXMLMath quotient say , and write LaTeXMLMath for the inverse image of LaTeXMLMath , then LaTeXMLMath is etale .
We have LaTeXMLMath and LaTeXMLMath using Lemma LaTeXMLRef , thus LaTeXMLMath .
Similiarly , LaTeXMLMath , so , using LaTeXMLMath , we find LaTeXMLMath .
But LaTeXMLMath and LaTeXMLMath agree on LaTeXMLMath , so we obtain our result .
∎ Notation as in LaTeXMLRef .
Let LaTeXMLMath be a LaTeXMLMath -Gorenstein family of slc surfaces and LaTeXMLMath a LaTeXMLMath -Gorenstein family of slc surfaces extending LaTeXMLMath .
There exists a canonical element LaTeXMLMath such that LaTeXMLMath iff LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath is a principal homogeneous space under LaTeXMLMath .
Given a LaTeXMLMath -Gorenstein family LaTeXMLMath extending LaTeXMLMath , LaTeXMLMath is naturally isomorphic to LaTeXMLMath .
For the local case we use Theorem LaTeXMLRef together with Proposition LaTeXMLRef .
We may assume LaTeXMLMath is a local ring .
Given a LaTeXMLMath -Gorenstein family of slc surfaces LaTeXMLMath extending LaTeXMLMath , take an index one cover LaTeXMLMath ( a LaTeXMLMath quotient , say ) .
This is flat over LaTeXMLMath and restricts to an index one cover LaTeXMLMath by Proposition LaTeXMLRef .
Now , by Theorem LaTeXMLRef ( 1 ) , there is a canonical element LaTeXMLMath such that LaTeXMLMath iff LaTeXMLMath .
By functoriality of the obstruction map , we have LaTeXMLMath using Remark LaTeXMLRef .
We define LaTeXMLMath .
In order to show property ( 1 ) holds , we just need to verify that if LaTeXMLMath then LaTeXMLMath , for by Proposition LaTeXMLRef we have a bijection LaTeXMLMath — we use Lemma LaTeXMLRef below .
Property ( 2 ) follows immediately from Proposition LaTeXMLRef and Theorem LaTeXMLRef ( 2 ) since LaTeXMLMath .
Finally , ( 3 ) is a special case of Theorem LaTeXMLRef ( 3 ) .
The global case now follows formally exactly as for ordinary deformation theory using the spectral sequence of Remark LaTeXMLRef ( we use Lemma LaTeXMLRef to identify LaTeXMLMath with LaTeXMLMath , the sheaf of infinitesimal automorphisms ) .
∎ Let LaTeXMLMath be an infinitesimal extension in LaTeXMLMath with LaTeXMLMath local and LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath be a family of local schemes , LaTeXMLMath a family extending LaTeXMLMath , and LaTeXMLMath a family of closed subschemes .
Suppose given a LaTeXMLMath action on LaTeXMLMath which extends to a LaTeXMLMath action on LaTeXMLMath .
Then , if there exists an extension LaTeXMLMath of LaTeXMLMath , there exists a LaTeXMLMath -invariant extension .
Write LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLMath be an ideal defining a closed subscheme LaTeXMLMath ( not necessarily flat over LaTeXMLMath ) extending LaTeXMLMath .
Then LaTeXMLMath is flat iff LaTeXMLMath for some liftings LaTeXMLMath of LaTeXMLMath and every relation LaTeXMLMath , LaTeXMLMath , between the LaTeXMLMath lifts to a relation LaTeXMLMath , LaTeXMLMath , between the LaTeXMLMath ( c.f .
LaTeXMLCite ) .
In our case , pick generators LaTeXMLMath of LaTeXMLMath which are LaTeXMLMath eigenfunctions ( using LaTeXMLCite , p. 219 , Lemma 7.29 ) .
Pick LaTeXMLMath eigenfunctions LaTeXMLMath lifting LaTeXMLMath .
Let LaTeXMLEquation be exact .
We define a LaTeXMLMath action on LaTeXMLMath ( in the obvious way ) such that LaTeXMLMath is LaTeXMLMath equivariant , and hence obtain a LaTeXMLMath action on the module of relations LaTeXMLMath .
We have a well defined map LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath are some lifts of the LaTeXMLMath .
Here we first regard LaTeXMLMath as an element of LaTeXMLMath and then take its image in LaTeXMLMath .
Then LaTeXMLMath iff LaTeXMLMath is flat over LaTeXMLMath , by the criterion above .
Thus there exists a flat extension LaTeXMLMath of LaTeXMLMath iff LaTeXMLEquation — for this is the condition that we can change the LaTeXMLMath so that LaTeXMLMath becomes the zero map .
Now , by construction , LaTeXMLMath is a LaTeXMLMath invariant element of LaTeXMLMath , so equivalently we require that LaTeXMLEquation .
In this case we can replace the LaTeXMLMath by LaTeXMLMath eigenfunctions LaTeXMLMath to obtain a LaTeXMLMath invariant flat extension LaTeXMLMath .
This completes the proof .
∎ The aim of this section is to prove that the groupoid LaTeXMLMath defined in Section LaTeXMLRef is an algebraic stack .
We use the theory of Artin LaTeXMLCite reviewed in Section LaTeXMLRef .
We do not work directly with LaTeXMLMath , instead we define a related groupoid LaTeXMLMath which ( roughly ) has the same definition as LaTeXMLMath except that we drop the smoothability assumption .
We show that LaTeXMLMath is an algebraic stack ( not necessarily proper in general ) and then obtain LaTeXMLMath as a closed substack .
We first verify that LaTeXMLMath satisfies the conditions of Theorem LaTeXMLRef , and hence obtain local patches of the stack .
The bulk of the work is giving a concrete description of LaTeXMLMath and constructing an obstruction theory LaTeXMLMath ( Theorem LaTeXMLRef ) .
Given the local patches it ’ s easy to show that LaTeXMLMath is an algebraic stack ( we just need to show that LaTeXMLMath is ‘ relatively representable ’ ) .
One might think that we could construct LaTeXMLMath as a quotient of some locally closed subscheme of a Hilbert scheme of pairs LaTeXMLMath .
However , we do not know that the base change conditions for LaTeXMLMath are locally closed ( c.f .
Theorem LaTeXMLRef and Remark LaTeXMLRef ) .
Thus we can not obtain LaTeXMLMath in this way .
Let LaTeXMLMath be a proper connected surface .
Let LaTeXMLMath be an effective Weil divisor on LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath is a quasistable pair of degree LaTeXMLMath if There exists LaTeXMLMath such that LaTeXMLMath is slc and ample .
LaTeXMLMath , and moreover LaTeXMLMath if LaTeXMLMath .
LaTeXMLMath if LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
A stable pair is quasistable .
We just need to show that the smoothability property for a stable pair LaTeXMLMath of degree LaTeXMLMath gives LaTeXMLMath if LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
First note that trivially LaTeXMLMath if LaTeXMLMath admits a smoothing to LaTeXMLMath .
If LaTeXMLMath the only possible singularities of LaTeXMLMath are of local analytic types LaTeXMLMath where LaTeXMLMath and LaTeXMLMath ( using Theorem LaTeXMLRef , Theorem LaTeXMLRef and Lemma LaTeXMLRef ) .
In the second case LaTeXMLMath shows LaTeXMLMath .
Thus LaTeXMLMath .
We now prove LaTeXMLMath .
First note that LaTeXMLMath LaTeXMLMath by Serre duality , and LaTeXMLMath is ample .
We have an exact sequence LaTeXMLEquation where LaTeXMLMath is the normalisation of LaTeXMLMath , and LaTeXMLMath is the normalisation of the double curve LaTeXMLMath of LaTeXMLMath .
We obtain a short exact sequence LaTeXMLEquation where LaTeXMLMath .
In particular , since LaTeXMLMath is ample , we have LaTeXMLMath .
Thus the long exact sequence of cohomology associated to the short exact sequence above gives LaTeXMLMath LaTeXMLMath .
If LaTeXMLMath is klt , then LaTeXMLMath by Kodaira vanishing .
Otherwise , LaTeXMLMath is an elliptic cone by Theorem LaTeXMLRef and Theorem LaTeXMLRef , and LaTeXMLMath , LaTeXMLMath .
An easy calculation shows that LaTeXMLMath in this case .
∎ We say that LaTeXMLMath is a family of quasistable pairs of degree LaTeXMLMath over LaTeXMLMath if LaTeXMLMath is a flat family over LaTeXMLMath , LaTeXMLMath is a relative Weil divisor over LaTeXMLMath , and for every geometric point LaTeXMLMath of LaTeXMLMath , the fibre LaTeXMLMath is a quasistable pair of degree LaTeXMLMath .
We say that LaTeXMLMath is an allowable family if LaTeXMLMath and LaTeXMLMath commute with base change for all LaTeXMLMath .
We define a groupoid LaTeXMLMath over LaTeXMLMath as follows : For LaTeXMLMath let LaTeXMLEquation .
We also regard LaTeXMLMath as a groupoid over the category LaTeXMLMath of noetherian LaTeXMLMath -algebras without further comment .
LaTeXMLMath is a subgroupoid of LaTeXMLMath This is immediate from Lemma LaTeXMLRef — LaTeXMLMath is the subgroupoid of smoothable families .
∎ In what follows , we suppress the degree LaTeXMLMath to simplify our notation .
Let LaTeXMLMath be the spectrum of a DVR with generic point LaTeXMLMath .
A family LaTeXMLMath of quasistable pairs such that LaTeXMLMath is canonical is allowable iff LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
The ’ only if ’ part follows from Lemma LaTeXMLRef .
So , suppose given a family LaTeXMLMath of quasistable pairs such that LaTeXMLMath is canonical and LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -Cartier .
Then LaTeXMLMath is a LaTeXMLMath -Gorenstein family by Proposition LaTeXMLRef .
Let LaTeXMLMath be an index one cover , with special fibre LaTeXMLMath an index one cover of LaTeXMLMath .
Since LaTeXMLMath is LaTeXMLMath -Cartier , and LaTeXMLMath is Cartier ( using LaTeXMLMath and LaTeXMLMath ) , we have LaTeXMLMath Cartier , using Lemma LaTeXMLRef .
It follows that LaTeXMLMath locally , some LaTeXMLMath , hence LaTeXMLMath commutes with base change for all LaTeXMLMath .
Thus LaTeXMLMath is allowable as required .
∎ For LaTeXMLMath , LaTeXMLMath iff the following conditions are satisfied : LaTeXMLMath is a LaTeXMLMath -Gorenstein family of slc surfaces .
LaTeXMLMath is a codimension LaTeXMLMath closed subscheme , flat over LaTeXMLMath .
LaTeXMLMath is a quasistable pair of degree LaTeXMLMath for every geometric point LaTeXMLMath of LaTeXMLMath .
LaTeXMLMath for every geometric point LaTeXMLMath of LaTeXMLMath .
Here LaTeXMLMath is the scheme theoretic fibre of LaTeXMLMath over LaTeXMLMath , and LaTeXMLMath is the restriction of LaTeXMLMath to the fibre LaTeXMLMath defined by taking the double dual of the ideal sheaf of LaTeXMLMath .
Given LaTeXMLMath , LaTeXMLMath is flat over LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath by Lemma LaTeXMLRef .
The other conditions are satisfied by the definition of LaTeXMLMath .
Conversely let LaTeXMLMath satisfy the conditions ( 1 ) to ( 4 ) .
We claim that LaTeXMLMath .
Note that LaTeXMLMath is a relative Weil divisor ( c.f .
Example LaTeXMLRef ) .
It is enough to show that LaTeXMLMath commutes with base change for all LaTeXMLMath .
By Lemma LaTeXMLRef , we may assume that LaTeXMLMath is a local Artinian ring .
So , by induction , it is enough to show the following : Let LaTeXMLMath be a local ring with residue field LaTeXMLMath and LaTeXMLMath a small extension ( i.e. , the kernel LaTeXMLMath of LaTeXMLMath is annihilated by the maximal ideal of LaTeXMLMath ) .
Suppose given LaTeXMLMath and LaTeXMLMath extending LaTeXMLMath which satisfies conditions ( 1 ) to ( 4 ) .
Then LaTeXMLMath .
We work locally on LaTeXMLMath .
Let LaTeXMLMath be an index one cover ( a LaTeXMLMath quotient , say ) , LaTeXMLMath the index one cover obtained by restriction to LaTeXMLMath , and LaTeXMLMath .
Given an LaTeXMLMath invariant extension LaTeXMLMath of LaTeXMLMath we obtain an extension LaTeXMLMath of LaTeXMLMath .
We claim that every extension LaTeXMLMath of LaTeXMLMath occurs in this way .
Let LaTeXMLMath be the special fibre of LaTeXMLMath .
Then LaTeXMLMath is an index one cover , and since LaTeXMLMath and LaTeXMLMath we have LaTeXMLMath Cartier .
Thus LaTeXMLMath has unobstructed embedded deformations ( locally ) and the extensions LaTeXMLMath of LaTeXMLMath form a prinicipal homogeneous space under LaTeXMLMath ( using Theorem LaTeXMLRef ) .
There exists a LaTeXMLMath invariant extension by Lemma LaTeXMLRef , hence also LaTeXMLMath has unobstructed embedded deformations ( locally ) and the extensions LaTeXMLMath of LaTeXMLMath form a prinicipal homogeneous space under LaTeXMLMath .
To prove our claim , we just need to identify the sheaves LaTeXMLMath and LaTeXMLMath .
We have the exact sequence LaTeXMLEquation applying the functor LaTeXMLMath we obtain LaTeXMLEquation .
Now LaTeXMLMath , since LaTeXMLMath is invertible in codimension LaTeXMLMath and LaTeXMLMath .
Hence we have a short exact sequence LaTeXMLEquation and similiarly LaTeXMLEquation .
Applying the exact functor LaTeXMLMath to the last exact sequence we obtain LaTeXMLEquation thus LaTeXMLMath as required .
So , given an extension LaTeXMLMath of LaTeXMLMath , it is obtained from an extension LaTeXMLMath of LaTeXMLMath by taking the LaTeXMLMath -quotient .
In particular , LaTeXMLMath is Cartier .
Thus locally LaTeXMLMath for some LaTeXMLMath , hence LaTeXMLMath commutes with base change for all LaTeXMLMath as required .
∎ Let LaTeXMLMath and LaTeXMLMath .
The forgetful map of functors ( of infinitesimal extensions of LaTeXMLMath ) LaTeXMLEquation is smooth .
Given an infinitesimal extension LaTeXMLMath in LaTeXMLMath and a LaTeXMLMath -Gorenstein family LaTeXMLMath extending LaTeXMLMath , we need to show that there exists an extension LaTeXMLMath of LaTeXMLMath ( using Lemma LaTeXMLRef ) .
By the proof of Lemma LaTeXMLRef there are no local obstructions .
We may assume that LaTeXMLMath is local and LaTeXMLMath is a small extension .
Let LaTeXMLMath have residue field LaTeXMLMath and write LaTeXMLMath for the special fibre of LaTeXMLMath .
Then the obstruction to extending LaTeXMLMath to some LaTeXMLMath lies in LaTeXMLMath by Theorem LaTeXMLRef ( 1 ) .
Moreover , since there are no local obstructions , using the local-to-global spectral sequence for LaTeXMLMath we see that the obstruction lies in LaTeXMLMath .
We have an exact sequence LaTeXMLEquation ( see the proof of Lemma LaTeXMLRef ) .
Now LaTeXMLMath by assumption , and LaTeXMLMath since LaTeXMLMath is ample , thus LaTeXMLMath .
So the obstruction is zero , and there is an extension LaTeXMLMath of LaTeXMLMath as required .
∎ Let LaTeXMLMath be an infinitesimal extension in LaTeXMLMath , LaTeXMLMath a family of schemes over LaTeXMLMath , LaTeXMLMath a family of closed subschemes , and LaTeXMLMath a family extending LaTeXMLMath .
Write LaTeXMLMath for the set of embedded deformations of LaTeXMLMath over LaTeXMLMath inside LaTeXMLMath .
Write LaTeXMLMath for the quotient of LaTeXMLMath by the action of LaTeXMLMath .
Given LaTeXMLMath extending LaTeXMLMath , write LaTeXMLMath for the group of automorphisms of LaTeXMLMath over LaTeXMLMath which restrict to the identity on LaTeXMLMath .
Given LaTeXMLMath in LaTeXMLMath , we also write LaTeXMLMath for LaTeXMLMath , similiarly for LaTeXMLMath and LaTeXMLMath .
Notation as in LaTeXMLRef .
Let LaTeXMLMath be a family of schemes , LaTeXMLMath a family extending LaTeXMLMath , and LaTeXMLMath a family extending LaTeXMLMath .
Let LaTeXMLMath be a family of closed subschemes , and LaTeXMLMath a family of closed subschemes extending LaTeXMLMath .
There is a canonical element LaTeXMLMath such that LaTeXMLMath iff LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath is a principal homogeneous space under LaTeXMLMath .
We have a natural map LaTeXMLEquation which identifies the action of LaTeXMLMath on LaTeXMLMath in terms of the action of LaTeXMLMath .
If LaTeXMLMath , it is a principal homogeneous space under LaTeXMLMath .
Given a family of closed subschemes LaTeXMLMath extending LaTeXMLMath , LaTeXMLMath is naturally isomorphic to LaTeXMLMath .
For ( 1 ) and ( 2 ) see LaTeXMLCite , p. 28 , Proposition 2.5 ( in fact only the case LaTeXMLMath , LaTeXMLMath local is treated in LaTeXMLCite , but the same argument proves the general case ) .
We now prove ( 3 ) .
If LaTeXMLMath , it is a principal homogeneous space under LaTeXMLMath by ( 2 ) .
Now LaTeXMLMath acts on LaTeXMLMath , so we obtain a homomorphism LaTeXMLEquation .
In the case LaTeXMLMath , LaTeXMLMath , LaTeXMLMath we obtain the map LaTeXMLMath above .
We have a natural isomorphism LaTeXMLMath by Theorem LaTeXMLRef ( 3 ) , which is compatible with the maps to LaTeXMLMath .
The result now follows .
∎ Given LaTeXMLMath , LaTeXMLMath a finite LaTeXMLMath -module and LaTeXMLMath , define LaTeXMLEquation and LaTeXMLEquation .
Notation as Definition LaTeXMLRef .
The sets LaTeXMLMath and LaTeXMLMath have natural LaTeXMLMath -module structures , and , writing LaTeXMLMath , we have an exact sequence of LaTeXMLMath -modules LaTeXMLEquation .
LaTeXMLEquation We have LaTeXMLMath and LaTeXMLMath , so we have a natural map LaTeXMLEquation with kernel LaTeXMLMath and cokernel LaTeXMLMath ( compare Theorem LaTeXMLRef ( 3 ) ) .
In particular , LaTeXMLMath is an LaTeXMLMath -module , moreover LaTeXMLMath is an LaTeXMLMath -module since the functor LaTeXMLMath satisfies Artin ’ s criterion S1 ( b ) .
Now LaTeXMLMath LaTeXMLMath and LaTeXMLMath so we have a natural map LaTeXMLEquation with kernel LaTeXMLMath by Lemma LaTeXMLRef .
Finally , LaTeXMLMath is surjective by Theorem LaTeXMLRef .
∎ Notation as in LaTeXMLRef .
Let LaTeXMLMath and LaTeXMLMath .
We have LaTeXMLMath iff LaTeXMLMath , where LaTeXMLMath is the canonical element constructed above .
If LaTeXMLMath , LaTeXMLMath is a principal homogeneous space under LaTeXMLMath .
Given LaTeXMLMath , LaTeXMLMath is naturally isomorphic to LaTeXMLMath .
Property ( 1 ) is immediate from Theorem LaTeXMLRef and Theorem LaTeXMLRef — because the forgetful map LaTeXMLMath is smooth , the obstruction theory for LaTeXMLMath gives an obstruction theory for LaTeXMLMath .
We have a natural map LaTeXMLMath which is surjective by Theorem LaTeXMLRef .
Assuming LaTeXMLMath , LaTeXMLMath is a principal homogeneous space under LaTeXMLMath by Theorem LaTeXMLRef ( 2 ) .
Moreover , given LaTeXMLMath , LaTeXMLMath is the set LaTeXMLMath .
This is a principal homogeneous space under LaTeXMLMath , where LaTeXMLEquation as in Theorem LaTeXMLRef ( 3 ) .
Now , the set LaTeXMLMath has a natural action of LaTeXMLMath , since the functor LaTeXMLMath satisfies Artin ’ s criterion S1 ( b ) .
This action is compatible with the actions just described via the exact sequence of Lemma LaTeXMLRef , it follows that LaTeXMLMath is a principal homogeneous space under LaTeXMLMath as required .
Finally , property ( 3 ) is a special case of Theorem LaTeXMLRef ( 3 ) .
∎ We can now identify the functor LaTeXMLMath and an obstruction theory LaTeXMLMath for our groupoid LaTeXMLMath .
Given LaTeXMLMath and a finite LaTeXMLMath -module LaTeXMLMath we have LaTeXMLEquation using Theorem LaTeXMLRef ( 2 ) .
Next , given an extension LaTeXMLMath in LaTeXMLMath , LaTeXMLMath and a finite LaTeXMLMath -module LaTeXMLMath , define LaTeXMLEquation .
Given an extension LaTeXMLMath of LaTeXMLMath by LaTeXMLMath , define LaTeXMLEquation .
Then , by Theorem LaTeXMLRef ( 1 ) , these data give an obstruction theory for LaTeXMLMath .
We are now ready to prove the existence of ( algebraic ) formally smooth deformations of LaTeXMLMath — roughly , these provide the local patches of an algebraic stack .
Given LaTeXMLMath , there exists an algebraic ring LaTeXMLMath with a closed point LaTeXMLMath and LaTeXMLMath , formally smooth over LaTeXMLMath , such that LaTeXMLMath .
The theorem is obtained by applying Theorem LaTeXMLRef to our functor LaTeXMLMath .
We verify the conditions of the theorem in Lemmas LaTeXMLRef , LaTeXMLRef , LaTeXMLRef , and LaTeXMLRef below .
∎ Let LaTeXMLMath , LaTeXMLMath then LaTeXMLMath defines a projective embedding LaTeXMLMath for LaTeXMLMath sufficiently large and divisible .
We have that LaTeXMLMath is LaTeXMLMath -Cartier and relatively ample using Lemma LaTeXMLRef and the base change property for LaTeXMLMath , LaTeXMLMath .
So , taking a sufficiently large and divisible multiple of LaTeXMLMath , we obtain a projective embedding of LaTeXMLMath .
∎ Note that it is not necessarily true that there exists LaTeXMLMath such that for every LaTeXMLMath , the sheaf LaTeXMLMath defines a projective embedding of LaTeXMLMath .
The point is that LaTeXMLMath defines a stack which is only locally of finite type , i.e. , we may require infinitely many patches .
However , if we restrict ourselves to smoothable pairs , i.e. , if we consider the stack LaTeXMLMath , we can show that there is such an LaTeXMLMath ( using the bound on the index provided by Theorem LaTeXMLRef ) , and deduce that LaTeXMLMath is of finite type ( Theorem LaTeXMLRef ) .
Given LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , the functor LaTeXMLEquation .
LaTeXMLEquation is represented by a quasiprojective scheme LaTeXMLMath .
We have a canonical polarisations LaTeXMLMath and LaTeXMLMath on LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath , and so by LaTeXMLCite we know that the functor LaTeXMLEquation .
LaTeXMLEquation is represented by a quasiprojective scheme LaTeXMLMath .
It is then easy to construct LaTeXMLMath as a locally closed subscheme of LaTeXMLMath .
∎ ( Open loci results ) Let LaTeXMLMath be a projective family of surfaces and LaTeXMLMath a codimension LaTeXMLMath closed subscheme , flat over LaTeXMLMath .
Let LaTeXMLMath be the locus of points LaTeXMLMath such that LaTeXMLMath is CM , reduced and Gorenstein in codimension LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath is Cartier in codimension LaTeXMLMath Then LaTeXMLMath is open , LaTeXMLMath is a relative Weil divisor on LaTeXMLMath , and LaTeXMLMath corresponds to a relative Weil divisor LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be a projective family of CM reduced surfaces , Gorenstein in codimension LaTeXMLMath , and LaTeXMLMath a relative effective Weil divisor on LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath commute with base change for all LaTeXMLMath .
Then the locus LaTeXMLMath where the geometric fibres of LaTeXMLMath are quasistable of degree LaTeXMLMath is open .
( 1 ) The locus where the fibres LaTeXMLMath are CM is open by LaTeXMLCite , p. 177 , Corollary to Theorem 22.5 .
LaTeXMLMath is reduced iff it is regular in codimension LaTeXMLMath — this is an open condition .
Since LaTeXMLMath commutes with base change , the requirement that LaTeXMLMath is Gorenstein in codimension LaTeXMLMath is open .
The condition LaTeXMLMath is equivalent to requiring that the sheaf LaTeXMLMath is LaTeXMLMath , which is an open condition , again by LaTeXMLCite ( note that LaTeXMLMath is flat over LaTeXMLMath ) .
Assuming this is satisfied , the natural map LaTeXMLMath is an isomorphism , thus the condition LaTeXMLMath Cartier in codimension LaTeXMLMath is open .
Hence LaTeXMLMath is open as required .
Using Lemma LaTeXMLRef we deduce that LaTeXMLMath and LaTeXMLMath are relative Weil divisors .
( 2 ) It is enough to show the following : Let LaTeXMLMath be the spectrum of a DVR with generic point LaTeXMLMath and closed point LaTeXMLMath .
Let LaTeXMLMath be a family of pairs such that LaTeXMLMath and LaTeXMLMath commute with base change for all LaTeXMLMath , and such that the special fibre LaTeXMLMath is quasistable of degree LaTeXMLMath .
Then LaTeXMLMath is quasistable of degree LaTeXMLMath .
Clearly LaTeXMLMath is ample , and it is also slc by Lemma LaTeXMLRef .
We are given LaTeXMLMath , we claim that this implies LaTeXMLMath — the essential point here is that LaTeXMLMath , so LaTeXMLMath is discrete .
To prove the claim , observe that LaTeXMLMath is Cartier by Lemma LaTeXMLRef and Lemma LaTeXMLRef , and the restriction map LaTeXMLMath is an isomorphism ( c.f .
Proof of Lemma LaTeXMLRef ( 1 ) ) .
Thus LaTeXMLMath , and restricting to the generic fibre we obtain our result .
We similiarly obtain LaTeXMLMath in the case LaTeXMLMath .
If LaTeXMLMath then we are given LaTeXMLMath , now LaTeXMLMath using Lemma LaTeXMLRef , it follows that LaTeXMLMath .
Given LaTeXMLMath , it follows that LaTeXMLMath by semicontinuity ( using the base change property for LaTeXMLMath , note that LaTeXMLMath is flat over LaTeXMLMath by Lemma LaTeXMLRef ) .
Finally , LaTeXMLMath is trivially an open condition .
This completes the proof .
∎ LaTeXMLMath is limit preserving .
Let LaTeXMLMath be a direct system in LaTeXMLMath such that the limit LaTeXMLMath lies in LaTeXMLMath .
Suppose given LaTeXMLMath , we need to show that this is obtained from some LaTeXMLMath by pullback .
We know that LaTeXMLMath is projective , fix an embedding LaTeXMLMath .
Then LaTeXMLMath is obtained by pullback from some projective flat family LaTeXMLMath for some LaTeXMLMath ( since LaTeXMLMath is of finite type , where LaTeXMLMath denotes the Hilbert polynomial of the fibres of LaTeXMLMath ) .
By Lemma LaTeXMLRef we have that LaTeXMLMath is a codimension LaTeXMLMath closed subscheme which is flat over LaTeXMLMath .
It follows ( since LaTeXMLMath is of finite type , where LaTeXMLMath denotes the Hilbert polynomial of the fibres of LaTeXMLMath ) that there exists LaTeXMLMath such that LaTeXMLMath , and LaTeXMLMath , such that LaTeXMLMath is obtained from LaTeXMLMath by pullback to LaTeXMLMath .
By Lemma LaTeXMLRef ( 1 ) , we may also assume that each fibre LaTeXMLMath is CM , reduced and Gorenstein in codimension LaTeXMLMath , and that LaTeXMLMath and LaTeXMLMath are relative Weil divisors .
We now analyse the push forward and base change conditions .
Let LaTeXMLMath be a coherent sheaf on an open subset LaTeXMLMath , flat over LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath the corresponding objects obtained by pullback to LaTeXMLMath .
We are interested in the cases LaTeXMLMath where LaTeXMLMath is the locus where LaTeXMLMath is Cartier .
LaTeXMLMath , where LaTeXMLMath is the locus where LaTeXMLMath is Cartier .
Assume that the push forward of LaTeXMLMath commutes with base change , and that LaTeXMLMath is coherent for each LaTeXMLMath .
We work locally at LaTeXMLMath , say LaTeXMLMath .
The natural map LaTeXMLMath is surjective , so pick a finite set of elements of LaTeXMLMath which generate LaTeXMLMath over LaTeXMLMath ( recall LaTeXMLMath is assumed to be coherent ) .
Since LaTeXMLMath , these are defined over some LaTeXMLMath where LaTeXMLMath .
Write LaTeXMLMath , LaTeXMLMath for the objects obtained by pullback from LaTeXMLMath to LaTeXMLMath , and say LaTeXMLMath .
Then , by construction , we have that LaTeXMLMath is surjective , hence by Lemma LaTeXMLRef ( 3 ) the push forward of LaTeXMLMath commutes with base change in a neighbourhood of LaTeXMLMath , which we may assume is LaTeXMLMath .
As in Remark LaTeXMLRef , we only need to consider a finite number of sheaves LaTeXMLMath .
Thus there exists LaTeXMLMath such that LaTeXMLMath and , writing LaTeXMLMath for the pullback of LaTeXMLMath , the sheaves LaTeXMLMath and LaTeXMLMath commute with base change for all LaTeXMLMath .
Finally , by Lemma LaTeXMLRef ( 2 ) we may assume that every geometric fibre of LaTeXMLMath is quasistable .
Then LaTeXMLMath is an element of LaTeXMLMath .
This completes the proof of the lemma .
∎ F satisfies conditions ( S1 ) ( a ) , ( b ) and ( S2 ) .
Let LaTeXMLEquation be a diagram in LaTeXMLMath as in the statement of condition ( S1 ) ( a ) .
Write LaTeXMLMath , then LaTeXMLMath is an infinitesimal extension of LaTeXMLMath .
We need to show that LaTeXMLMath is surjective for LaTeXMLMath .
So , let LaTeXMLMath and LaTeXMLMath be families which extend some LaTeXMLMath .
Define LaTeXMLMath as follows : let LaTeXMLMath ( where LaTeXMLMath denotes the underlying topological spaces ) , and LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath is a flat family of pairs extending LaTeXMLMath and LaTeXMLMath ( to prove flatness , use LaTeXMLCite p. 216 Lemma 3.4 ) .
Thus , by Lemma LaTeXMLRef , we only need to verify that LaTeXMLMath is a LaTeXMLMath -Gorenstein family to obtain LaTeXMLMath as required .
To see this , take an index one cover LaTeXMLMath , this gives an index one cover LaTeXMLMath on restriction to LaTeXMLMath by Proposition LaTeXMLRef , extend this to an index one cover LaTeXMLMath .
Define LaTeXMLMath by LaTeXMLMath as above , then LaTeXMLMath is an index one cover extending LaTeXMLMath and thus LaTeXMLMath is a LaTeXMLMath -Gorenstein family by Proposition LaTeXMLRef as required .
Suppose given LaTeXMLMath , an extension LaTeXMLMath in LaTeXMLMath , a finite LaTeXMLMath -module LaTeXMLMath , and LaTeXMLMath .
The condition ( S1 ) ( b ) states that the natural map LaTeXMLEquation is an isomorphism .
By Theorem LaTeXMLRef ( 2 ) , each side is naturally identified with LaTeXMLMath , so the map is an isomorphism as required .
Finally , the finiteness condition ( S2 ) for LaTeXMLMath is obvious from the construction of the module LaTeXMLMath using the properness of LaTeXMLMath .
∎ For LaTeXMLMath a complete local ring in LaTeXMLMath , the map LaTeXMLEquation is bijective .
Write LaTeXMLMath .
An element of LaTeXMLMath is a sequence of compatible families LaTeXMLMath .
This defines a pair of formal schemes LaTeXMLMath , where LaTeXMLMath denotes the formal spectrum .
We have a line bundle LaTeXMLMath on LaTeXMLMath such that the restriction to the special fibre is ample , e.g .
LaTeXMLMath where LaTeXMLMath .
By Grothendieck ’ s Existence Theorem ( LaTeXMLCite , III.5.4.5 ) , LaTeXMLMath is the completion of a proper pair LaTeXMLMath along the fibre LaTeXMLMath .
We claim that LaTeXMLMath .
First observe that LaTeXMLMath and LaTeXMLMath commute with base change for all LaTeXMLMath by Lemma LaTeXMLRef ( 3 ) .
Then , by Lemma LaTeXMLRef , the set of points LaTeXMLEquation is open .
But it contains the closed point by assumption , hence it is the whole of LaTeXMLMath .
Thus LaTeXMLMath as claimed , so the map LaTeXMLMath is surjective .
Now suppose given LaTeXMLMath and LaTeXMLMath which give the same element of LaTeXMLMath , i.e. , we have compatible isomorphisms LaTeXMLEquation for each LaTeXMLMath .
Equivalently , using Lemma LaTeXMLRef , we have compatible maps LaTeXMLEquation and thus a map LaTeXMLEquation so there is an isomorphism LaTeXMLMath over LaTeXMLMath extending the LaTeXMLMath .
Thus the map LaTeXMLMath is injective .
∎ LaTeXMLMath and LaTeXMLMath satisfy the conditions LaTeXMLRef for algebraic LaTeXMLMath .
( 1 ) Let LaTeXMLMath in LaTeXMLMath be etale ( in fact we only require LaTeXMLMath flat ) .
Given a flat family of schemes LaTeXMLMath and a coherent sheaf LaTeXMLMath on LaTeXMLMath we have natural isomorphisms LaTeXMLEquation by LaTeXMLCite , p. 50 , 2.3.2 , using LaTeXMLMath flat .
Given a LaTeXMLMath -Gorenstein family of slc surfaces LaTeXMLMath and a coherent sheaf LaTeXMLMath on LaTeXMLMath , let LaTeXMLMath be a local index one cover ( a LaTeXMLMath quotient , say ) , then LaTeXMLEquation by Remark LaTeXMLRef .
Thus we obtain LaTeXMLEquation by applying LaTeXMLMath to the natural isomorphism above with LaTeXMLMath .
By LaTeXMLCite , p. 255 , Proposition 9.3 we have natural isomorphisms LaTeXMLEquation for any quasi-coherent sheaf LaTeXMLMath since LaTeXMLMath is flat .
Thus LaTeXMLEquation for each LaTeXMLMath , using the spectral sequence of Remark LaTeXMLRef and flatness of LaTeXMLMath .
Also , given coherent sheaves LaTeXMLMath , LaTeXMLMath on LaTeXMLMath , we have LaTeXMLEquation since LaTeXMLMath is flat ( LaTeXMLCite , p. 52 , Theorem 7.11 ) .
Now suppose given LaTeXMLMath an extension in LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath a finite LaTeXMLMath -module .
Since LaTeXMLEquation we see that LaTeXMLMath commutes with etale localisation by ( a ) above .
Using the exact sequence of Lemma LaTeXMLRef together with ( a ) and ( b ) above we find that LaTeXMLMath commutes with etale localisation .
( 2 ) We need to show that , given LaTeXMLMath , LaTeXMLMath a finite LaTeXMLMath -module , and LaTeXMLMath a maximal ideal , we have LaTeXMLEquation .
By ( 1 ) , we may assume that LaTeXMLMath , then we need to show LaTeXMLEquation that is , LaTeXMLEquation .
This is a result in the style of Lemma LaTeXMLRef , and is proved in the same way .
( 3 ) We may assume that LaTeXMLMath is an integral domain , since we are given that LaTeXMLMath is reduced , and LaTeXMLMath and LaTeXMLMath commute with etale localisation .
We first show that , for LaTeXMLMath integral , and LaTeXMLMath a family of schemes over LaTeXMLMath , there is an open affine subset LaTeXMLMath such that if LaTeXMLMath is a morphism in LaTeXMLMath which factors through LaTeXMLMath , then the natural maps LaTeXMLEquation are isomorphisms .
Recall the construction of the LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath is a cotangent complex for LaTeXMLMath , and LaTeXMLMath is obtained via the spectral sequence LaTeXMLEquation .
If LaTeXMLMath is a cotangent complex for LaTeXMLMath , then given LaTeXMLMath , LaTeXMLMath is a cotangent complex for LaTeXMLMath ( LaTeXMLCite , p. 47 , 2.2.1 ( c ) ) .
Recall that , for a coherent sheaf LaTeXMLMath on LaTeXMLMath ( of finite type ) , where LaTeXMLMath integral , there is a non-empty open affine subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath is flat over LaTeXMLMath ( LaTeXMLCite , Lecture 8 , p. 57 ) .
It follows that , given coherent sheaves LaTeXMLMath on LaTeXMLMath , there exists an open affine subset LaTeXMLMath such that the natural map LaTeXMLEquation is an isomorphism whenever LaTeXMLMath factors through LaTeXMLMath ( compare the proof of LaTeXMLCite , p. 52 , Theorem 7.11 ) .
Thus , in our case , there is an open affine subset LaTeXMLMath such that the natural maps LaTeXMLEquation are isomorphisms for all LaTeXMLMath and all LaTeXMLMath .
Similiarly , shrinking LaTeXMLMath if necessary , we may assume the natural maps LaTeXMLEquation are isomorphisms for all LaTeXMLMath and all LaTeXMLMath .
So combining , we have isomorphisms LaTeXMLEquation for all LaTeXMLMath and all LaTeXMLMath .
Next , we may assume that the natural maps LaTeXMLEquation are isomorphisms for all LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , by the theory of cohomology and base change for projective morphisms ( see e.g .
LaTeXMLCite , III.12 , in particular p. 288 , Corollary 12.9 ) , thus LaTeXMLEquation .
Finally , we may assume that we obtain LaTeXMLEquation using the local-to-global spectral sequence , by imposing certain flatness requirements .
The same argument shows that the LaTeXMLMath commute with base changes LaTeXMLMath for some LaTeXMLMath , where LaTeXMLMath is a family of slc surfaces .
We use the local identification LaTeXMLEquation where LaTeXMLMath is a local index one cover which is a LaTeXMLMath quotient .
We use the exact sequence of Lemma LaTeXMLRef to show that LaTeXMLMath commutes with base changes LaTeXMLMath for some LaTeXMLMath .
We may assume that the modules LaTeXMLMath , LaTeXMLMath and LaTeXMLMath commute with base changes LaTeXMLMath factoring through some LaTeXMLMath by the above , and if we assume some flatness conditions we obtain that LaTeXMLMath commutes with base changes LaTeXMLMath using the 5-lemma .
Thus , given LaTeXMLMath integral , LaTeXMLMath a finite LaTeXMLMath -module , and LaTeXMLMath an extension , LaTeXMLMath and LaTeXMLMath commute with base changes LaTeXMLMath for some open affine subset LaTeXMLMath .
This completes the proof .
∎ LaTeXMLMath is a Deligne-Mumford algebraic stack , locally of finite type over LaTeXMLMath .
First , it is clear that LaTeXMLMath defines a stack .
The only non-trivial point is that etale descent data for LaTeXMLMath are effective — this follows since for each LaTeXMLMath and LaTeXMLMath we have a canonical polarisation LaTeXMLMath for some LaTeXMLMath .
By Theorem LaTeXMLRef , we can construct a scheme LaTeXMLMath locally of finite type over LaTeXMLMath and LaTeXMLMath such that the morphism of stacks LaTeXMLMath is smooth and surjective .
The construction is as follows : For every LaTeXMLMath , let LaTeXMLMath be an algebraic ring with a closed point LaTeXMLMath and LaTeXMLMath , formally smooth over LaTeXMLMath , such that LaTeXMLMath .
Let LaTeXMLMath be the disjoint union of the schemes LaTeXMLMath , and LaTeXMLMath the union of the families LaTeXMLMath .
We have that the diagonal LaTeXMLMath is representable , quasicompact and separated , i.e. , for every LaTeXMLMath and every pair of elements LaTeXMLMath , LaTeXMLMath , the functor LaTeXMLEquation .
LaTeXMLEquation is represented by a scheme which is quasicompact and separated over LaTeXMLMath .
For LaTeXMLMath is represented by a scheme LaTeXMLMath , quasiprojective over LaTeXMLMath , by Lemma LaTeXMLRef .
Thus LaTeXMLMath is an algebraic stack , locally of finite type over LaTeXMLMath .
Finally , we claim that LaTeXMLMath is a Deligne-Mumford algebraic stack , i.e. , there exists a scheme LaTeXMLMath locally of finite type over LaTeXMLMath and LaTeXMLMath such that the morphism of stacks LaTeXMLMath is etale and surjective .
By LaTeXMLCite , p. 104 , Theorem 4.21 , it is enough to show that the diagonal map LaTeXMLMath is unramified , i.e. , for every LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , the scheme LaTeXMLMath is unramified over LaTeXMLMath .
We may assume that LaTeXMLMath , LaTeXMLMath algebraically closed .
We know that LaTeXMLMath is finite for LaTeXMLMath by LaTeXMLCite , since LaTeXMLMath is slc and ample .
It follows that LaTeXMLMath has no infinitesimal automorphisms ( since LaTeXMLMath ) and hence LaTeXMLMath is unramified as required .
∎ LaTeXMLMath is a Deligne-Mumford algebraic stack , of finite type over LaTeXMLMath .
It is immediate that LaTeXMLMath is a Deligne-Mumford algebraic stack , locally of finite type over LaTeXMLMath by Theorem LaTeXMLRef and Definition LaTeXMLRef .
It remains to show that it is of finite type .
By Theorem LaTeXMLRef there exists LaTeXMLMath such that LaTeXMLMath is Cartier for every stable pair LaTeXMLMath of degree LaTeXMLMath .
By smoothability , the Hilbert polynomials of LaTeXMLMath and LaTeXMLMath with respect to the polarisation LaTeXMLMath are the same as the Hilbert polynomials of LaTeXMLMath and LaTeXMLMath a curve of degree LaTeXMLMath with respect to the polarisation LaTeXMLMath .
Also , by LaTeXMLCite , Theorem 2.1.2 , there exists LaTeXMLMath such that LaTeXMLMath is very ample and has no higher cohomology for all LaTeXMLMath .
Thus there is a Hilbert scheme LaTeXMLMath of pairs LaTeXMLMath such that every stable pair of degree LaTeXMLMath occurs in the universal family .
Now , let LaTeXMLMath be a local affine patch of LaTeXMLMath with universal family LaTeXMLMath .
Pick an projective embedding LaTeXMLMath defined by LaTeXMLMath .
We obtain a map LaTeXMLMath , and moreover taking all possible such embeddings we obtain a map LaTeXMLEquation .
We perform this construction everywhere locally on LaTeXMLMath .
The ( set-theoretic ) image of the maps LaTeXMLMath is the set LaTeXMLMath of points LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath is a stable pair of degree LaTeXMLMath embedded by LaTeXMLMath .
We claim that LaTeXMLMath is a locally closed subset of LaTeXMLMath and moreover that the images of the maps LaTeXMLMath give an open cover of LaTeXMLMath .
Assuming this for the moment , we deduce that there exists a finite open subcover of LaTeXMLMath , and hence a finite affine open cover of LaTeXMLMath .
Thus LaTeXMLMath is of finite type as required .
We now prove our claim above .
We first apply Lemma LaTeXMLRef ( 1 ) to deduce that there exists an open locus LaTeXMLMath containing LaTeXMLMath such that for all LaTeXMLMath , LaTeXMLMath is CM , reduced and Gorenstein in codimension LaTeXMLMath , and LaTeXMLMath is Cartier in codimension LaTeXMLMath .
Moreover , writing LaTeXMLMath for the universal family , LaTeXMLMath and LaTeXMLMath are relative Weil divisors on LaTeXMLMath .
Next , we apply Theorem LaTeXMLRef , to deduce that there exists a locally closed subset LaTeXMLMath containing LaTeXMLMath such that , writing LaTeXMLMath for the universal family , LaTeXMLMath and LaTeXMLMath commute with base changes LaTeXMLMath with LaTeXMLMath reduced , for all LaTeXMLMath .
There now exists an open subset LaTeXMLMath containing LaTeXMLMath such that LaTeXMLMath is quasistable for all LaTeXMLMath by Lemma LaTeXMLRef ( 2 ) .
Let LaTeXMLMath be the closure in LaTeXMLMath of the locus where LaTeXMLMath , then LaTeXMLMath contains LaTeXMLMath and LaTeXMLMath is a stable pair of degree LaTeXMLMath for all LaTeXMLMath .
Finally , LaTeXMLMath is the locus where LaTeXMLMath is defined by LaTeXMLMath — this is an locally closed condition .
Thus LaTeXMLMath is a locally closed subset of LaTeXMLMath .
Now let LaTeXMLMath be a local patch of LaTeXMLMath with universal family LaTeXMLMath and LaTeXMLMath a map as constructed above , we claim that the image of LaTeXMLMath is an open subset of LaTeXMLMath .
Let LaTeXMLMath be the spectrum of a DVR with closed point LaTeXMLMath and generic point LaTeXMLMath , and LaTeXMLMath a morphism ( where LaTeXMLMath is given its reduced structure ) such that LaTeXMLMath lies in the image of LaTeXMLMath .
It is enough to show that LaTeXMLMath also lies in the image of LaTeXMLMath .
Writing LaTeXMLMath for the pullback of the universal family of LaTeXMLMath to LaTeXMLMath , we have LaTeXMLMath ( note that the base change conditions are satisfied by construction ) .
We deduce our result by the versality of LaTeXMLMath .
∎ We provide a characterisation of the slc del Pezzo surfaces of type B which have a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Suppose LaTeXMLMath is a surface of type B .
Then LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath iff LaTeXMLMath has singularities of types LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
Moreover there are at most two non-normal singularities with index LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath or LaTeXMLMath .
We use the LaTeXMLMath -Gorenstein deformation theory developed in Section LaTeXMLRef .
Write LaTeXMLMath .
We use the short hand LaTeXMLMath , and similiarly LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
The following theorem describes how to calculate the LaTeXMLMath .
Let LaTeXMLMath be a scheme over LaTeXMLMath .
We have LaTeXMLMath , the tangent sheaf of LaTeXMLMath .
LaTeXMLMath can be calculated locally from an embedding LaTeXMLMath of LaTeXMLMath in an affine space via the exact sequence LaTeXMLMath .
In particular LaTeXMLMath if LaTeXMLMath is smooth .
LaTeXMLMath if LaTeXMLMath is a local complete intersection .
All this follows from Theorem LaTeXMLRef apart from the exact sequence of ( 2 ) — see LaTeXMLCite , p. 51 , 2.3.6 .
∎ We also recall the following results from Section LaTeXMLRef — if LaTeXMLMath is a local index one cover of LaTeXMLMath , a LaTeXMLMath quotient say , then LaTeXMLEquation and moreover LaTeXMLMath .
Let LaTeXMLMath be a surface with two normal irreducible components LaTeXMLMath , LaTeXMLMath meeting in a smooth double curve LaTeXMLMath .
Suppose LaTeXMLMath has only singularities of the form LaTeXMLMath at LaTeXMLMath .
Then , in a neighbourhood of LaTeXMLMath , LaTeXMLEquation .
Here LaTeXMLMath is the double curve on LaTeXMLMath , and we calculate LaTeXMLMath by moving LaTeXMLMath on LaTeXMLMath , and restricting to LaTeXMLMath — we obtain a LaTeXMLMath -divisor on LaTeXMLMath which is well defined modulo linear equivalence .
We have that LaTeXMLMath is a LaTeXMLMath -divisor on LaTeXMLMath .
In particular , LaTeXMLMath is a line bundle on LaTeXMLMath of degree LaTeXMLMath .
Let LaTeXMLMath , we first work locally analytically at LaTeXMLMath .
Suppose that LaTeXMLMath is normal crossing at LaTeXMLMath , i.e. , LaTeXMLEquation .
Then , using the exact sequence LaTeXMLMath , we deduce that LaTeXMLMath is a line bundle on LaTeXMLMath .
Moreover we obtain a natural isomorphism LaTeXMLMath — for we have LaTeXMLEquation .
Suppose now that LaTeXMLMath has a singularity LaTeXMLMath at LaTeXMLMath .
Let LaTeXMLMath be the canonical cover , then LaTeXMLMath is normal crossing .
Write LaTeXMLMath , LaTeXMLMath for the inverse images of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath for the inverse images of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath on LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
We have a natural isomorphism LaTeXMLMath , and LaTeXMLMath .
Patching together this local information , we obtain that globally LaTeXMLMath , where LaTeXMLMath is defined as detailed above .
∎ Suppose LaTeXMLMath is a surface of type B which satisfies the conditions ( 1 ) , ( 2 ) , ( 3 ) of Theorem LaTeXMLRef .
Then we have LaTeXMLMath .
Let LaTeXMLMath for LaTeXMLMath and LaTeXMLMath be the minimal resolutions of the components of LaTeXMLMath .
Then LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath by Noether ’ s formula , and LaTeXMLEquation where the LaTeXMLMath are the indices of the non-Gorenstein singularities of LaTeXMLMath at LaTeXMLMath ( see the proof of Theorem LaTeXMLRef ) .
Thus LaTeXMLEquation .
We also have LaTeXMLEquation and LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
Solving for LaTeXMLMath we obtain LaTeXMLMath .
∎ Suppose LaTeXMLMath is a surface of type B which satisfies the conditions ( 1 ) , ( 2 ) , ( 3 ) of Theorem LaTeXMLRef .
Then LaTeXMLMath .
We have an exact sequence LaTeXMLEquation .
Apply the functor LaTeXMLMath — we obtain an exact sequence LaTeXMLEquation .
Thus we have an inclusion LaTeXMLMath with cokernel supported on LaTeXMLMath .
It follows that LaTeXMLMath is surjective .
So it is enough to show that LaTeXMLMath for LaTeXMLMath .
Write LaTeXMLMath for a component LaTeXMLMath .
By Serre duality , LaTeXMLEquation .
We claim that LaTeXMLMath has a nonzero global section .
Assuming this , we have LaTeXMLEquation .
Now , letting LaTeXMLMath be the minimal resolution , we have LaTeXMLMath since LaTeXMLMath has only quotient singularities ( LaTeXMLCite , Lemma 1.11 ) .
Thus LaTeXMLMath , since LaTeXMLMath is rational by Theorem LaTeXMLRef .
So LaTeXMLMath as required .
It remains to show that LaTeXMLMath has a nonzero global section .
We have an exact sequence LaTeXMLEquation .
Now LaTeXMLMath by Serre duality and Kodaira vanishing ( recall that LaTeXMLMath is log terminal and LaTeXMLMath is ample ) .
So we are done if LaTeXMLMath has a nonzero global section .
A local calculation shows that LaTeXMLMath , where LaTeXMLMath is the sum of the singular points of LaTeXMLMath lying on LaTeXMLMath .
Now LaTeXMLMath is isomorphic to LaTeXMLMath , and there are at most LaTeXMLMath singular points of LaTeXMLMath on LaTeXMLMath by assumption , thus LaTeXMLMath and LaTeXMLMath has a nonzero global section as required .
∎ Let LaTeXMLMath be a surface of type B which admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Then there are at most two non-normal points of LaTeXMLMath of index greater than LaTeXMLMath .
For a smoothable surface LaTeXMLMath of type B , the non-Gorenstein singularities at LaTeXMLMath are of type LaTeXMLMath .
Moreover , if we have LaTeXMLMath such singularities with indices LaTeXMLMath then LaTeXMLMath , in particular LaTeXMLMath .
We show that if LaTeXMLMath smoothes to LaTeXMLMath , then LaTeXMLMath .
We assume LaTeXMLMath and obtain a contradiction .
Let LaTeXMLMath be a LaTeXMLMath -Gorenstein smoothing of LaTeXMLMath over the spectrum LaTeXMLMath of a DVR with generic point LaTeXMLMath , such that LaTeXMLMath .
Write LaTeXMLMath , then there are two cases : LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is LaTeXMLMath -factorial .
LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is not LaTeXMLMath -factorial .
Here we use Corollary LaTeXMLRef , also LaTeXMLMath by Lemma LaTeXMLRef ( 1 ) and we can calculate LaTeXMLMath using Lemma LaTeXMLRef ( 2 ) .
Suppose first LaTeXMLMath is of type ( 1 ) .
We claim that we can contract the divisor LaTeXMLMath to obtain a relative minimal model LaTeXMLMath .
We have LaTeXMLMath canonical by Lemma LaTeXMLRef , so we can use the relative MMP theory .
It ’ s enough to show that the double curve LaTeXMLMath of LaTeXMLMath generates an extremal ray in LaTeXMLMath ( note that LaTeXMLMath is relatively ample so certainly LaTeXMLMath ) .
Let LaTeXMLMath have singularity LaTeXMLMath generically along LaTeXMLMath .
Then LaTeXMLEquation .
Now LaTeXMLMath since LaTeXMLMath , thus LaTeXMLMath and LaTeXMLMath generates an extremal ray on LaTeXMLMath .
By the exact sequence LaTeXMLEquation ( c.f .
Lemma LaTeXMLRef ) , we see that LaTeXMLMath generates an extremal ray on LaTeXMLMath .
Hence we can contract LaTeXMLMath to obtain a LaTeXMLMath -factorial family of surfaces LaTeXMLMath with generic fibre LaTeXMLMath .
The special fibre LaTeXMLMath is obtained from LaTeXMLMath by contracting the double curve LaTeXMLMath .
Thus LaTeXMLMath has a log-terminal singularity whose minimal resolution has exceptional locus a tree of rational curves with one fork .
But this singularity is not smoothable , a contradiction .
Now suppose that LaTeXMLMath is of type ( 2 ) .
Then LaTeXMLMath is not LaTeXMLMath -factorial .
Thus we have a point LaTeXMLMath such that , locally analytically at LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
Working locally at LaTeXMLMath , we construct a small partial resolution LaTeXMLMath .
Assume first that LaTeXMLMath .
Let LaTeXMLMath be the blowup of LaTeXMLMath .
Writing LaTeXMLMath and LaTeXMLMath , we have two affine pieces of LaTeXMLMath as follows : LaTeXMLEquation .
LaTeXMLEquation Thus LaTeXMLMath is normal crossing at the strict transform LaTeXMLMath of LaTeXMLMath and LaTeXMLMath has a LaTeXMLMath singularity along LaTeXMLMath .
The only possible singularity of LaTeXMLMath away from LaTeXMLMath is a LaTeXMLMath singularity at LaTeXMLMath .
Write LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath for the strict transforms .
Then LaTeXMLMath is an isomorphism , and LaTeXMLMath contracts a LaTeXMLMath to the smooth point LaTeXMLMath .
In the case LaTeXMLMath we obtain LaTeXMLMath as the LaTeXMLMath quotient of the construction above .
Now work globally on LaTeXMLMath .
Note that LaTeXMLMath is slt , LaTeXMLMath is canonical , and LaTeXMLMath .
We first claim that LaTeXMLMath is projective .
For it is easy to see that the special fibre LaTeXMLMath is projective , and the restriction map LaTeXMLMath is an isomorphism ( using LaTeXMLMath nef and big , c.f .
proof of Lemma LaTeXMLRef ( 1 ) ) .
Moreover LaTeXMLMath is LaTeXMLMath factorial ( c.f .
Corollary LaTeXMLRef ) .
We now obtain a contradiction as for case ( 1 ) above .
∎ First suppose that LaTeXMLMath is a surface of type B that admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
The condition ( 1 ) on the singularities of LaTeXMLMath is satisified by Theorem LaTeXMLRef , Lemma LaTeXMLRef and Lemma LaTeXMLRef .
Let LaTeXMLMath be a LaTeXMLMath -Gorenstein smoothing of LaTeXMLMath , where LaTeXMLMath is the spectrum of a DVR with generic point LaTeXMLMath , such that LaTeXMLMath .
We have LaTeXMLMath since LaTeXMLMath is LaTeXMLMath -Cartier , so ( 2 ) holds .
Finally , ( 3 ) is satisfied by Corollary LaTeXMLRef .
Thus the conditions are necessary .
Now let LaTeXMLMath be a surface of type B which satisfies the conditions ( 1 ) , ( 2 ) and ( 3 ) .
We first construct a first order LaTeXMLMath -Gorenstein deformation LaTeXMLMath of LaTeXMLMath , where LaTeXMLMath .
We then show that this extends to a LaTeXMLMath -Gorenstein smoothing of LaTeXMLMath over a DVR .
Recall that there is a natural isomorphism LaTeXMLMath .
Now , by the local-to-global spectral sequence , we have an exact sequence LaTeXMLEquation .
By Lemma LaTeXMLRef we have LaTeXMLMath , so LaTeXMLEquation is exact .
We next specify an element of LaTeXMLMath , and describe the local first order deformations of LaTeXMLMath it determines .
We can then lift this to an element of LaTeXMLMath defining a global LaTeXMLMath -Gorenstein first order deformation LaTeXMLMath of LaTeXMLMath .
The sheaf LaTeXMLMath is supported on the singular locus of LaTeXMLMath .
So , to define a global section of LaTeXMLMath , we have to define a section in a neighbourhood of each connected component of the singular locus of LaTeXMLMath .
An isolated singularity LaTeXMLMath of LaTeXMLMath is of the form LaTeXMLMath by assumption .
Write LaTeXMLMath for the local index one cover of LaTeXMLMath , then LaTeXMLEquation .
We define a local first order LaTeXMLMath -Gorenstein deformation of LaTeXMLMath as follows : LaTeXMLEquation .
This corresponds to a local section of LaTeXMLMath .
Now consider LaTeXMLMath , the locus of non-isolated singularities of LaTeXMLMath .
In a neighbourhood of LaTeXMLMath , LaTeXMLMath is a line bundle on LaTeXMLMath of degree LaTeXMLMath by Lemma LaTeXMLRef .
Moreover LaTeXMLMath and LaTeXMLMath by Lemma LaTeXMLRef .
So there exists a section LaTeXMLMath of LaTeXMLMath at LaTeXMLMath , with reduced divisor of zeroes missing the non-Gorenstein points of LaTeXMLMath , i.e. , LaTeXMLEquation where LaTeXMLMath are distinct points of index one on LaTeXMLMath .
Then LaTeXMLMath defines local first order deformations of LaTeXMLMath at the points LaTeXMLMath of LaTeXMLMath of the following forms .
If LaTeXMLMath and LaTeXMLMath at LaTeXMLMath , some LaTeXMLMath , then we have LaTeXMLEquation for some unit LaTeXMLMath .
If LaTeXMLMath for some LaTeXMLMath then LaTeXMLMath at LaTeXMLMath and we have LaTeXMLEquation for some unit LaTeXMLMath .
Observe that LaTeXMLMath has unobstructed LaTeXMLMath -Gorenstein deformations .
For the local index one covers of LaTeXMLMath are local complete intersections , so LaTeXMLMath ( thus LaTeXMLMath has unobstructed LaTeXMLMath -Gorenstein deformations locally ) .
Also , we have LaTeXMLMath , since LaTeXMLMath is an invertible sheaf on LaTeXMLMath of non-negative degree , and LaTeXMLMath by Lemma LaTeXMLRef .
Thus LaTeXMLMath by the local-to-global spectral sequence , so LaTeXMLMath has unobstructed LaTeXMLMath -Gorenstein deformations as claimed .
Hence we can extend the deformation LaTeXMLMath to a sequence of compatible LaTeXMLMath -Gorenstein deformations LaTeXMLMath , where LaTeXMLMath .
They determine a formal scheme LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath is the completion of LaTeXMLMath , and LaTeXMLMath denotes the formal spectrum .
By the Grothendieck Existence Theorem , LaTeXMLMath is the completion of a projective LaTeXMLMath -Gorenstein family LaTeXMLMath ( c.f .
Proof of Lemma LaTeXMLRef ) .
We claim that this is a smoothing of LaTeXMLMath .
For , from the explicit descriptions of the local first order deformations above , we see that LaTeXMLMath has only singularities of types LaTeXMLMath and LaTeXMLMath , hence LaTeXMLMath is smooth over the generic point LaTeXMLMath of LaTeXMLMath as required .
The geometric generic fibre LaTeXMLMath is a smooth del Pezzo surface with LaTeXMLMath , hence LaTeXMLMath .
Thus LaTeXMLMath is a LaTeXMLMath -Gorenstein smoothing of LaTeXMLMath to LaTeXMLMath as required .
∎ We include below an important result we have proved along the way .
Let LaTeXMLMath be an slt del Pezzo surface that admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Then LaTeXMLMath has unobstructed LaTeXMLMath -Gorenstein deformations .
LaTeXMLMath is either a Manetti surface or of type B .
We proved the result in the type B case in the proof of Theorem LaTeXMLRef .
If LaTeXMLMath is a Manetti surface then the local index one covers of LaTeXMLMath are local complete intersections so LaTeXMLMath , LaTeXMLMath since LaTeXMLMath has LaTeXMLMath dimensional support , and LaTeXMLMath by LaTeXMLCite , p. 113 , Proof of Theorem 21 .
Thus LaTeXMLMath , so LaTeXMLMath has unobstructed LaTeXMLMath -Gorenstein deformations as required .
∎ We state and prove two major simplifications we obtain in the case LaTeXMLMath .
First , if LaTeXMLMath is a stable pair of degree LaTeXMLMath then LaTeXMLMath is slt .
Second , the stack LaTeXMLMath is smooth .
Let LaTeXMLMath be a stable pair of degree LaTeXMLMath .
Suppose LaTeXMLMath .
Then LaTeXMLMath is slt .
So LaTeXMLMath is either a Manetti surface ( that is , normal , with log terminal singularities ) or of type B .
In particular LaTeXMLMath has LaTeXMLMath or LaTeXMLMath components .
Write LaTeXMLMath for a component of LaTeXMLMath , and let LaTeXMLMath denote LaTeXMLMath .
We need to show that LaTeXMLMath is log terminal .
First we show that there are no singularities of types LaTeXMLMath or LaTeXMLMath .
For suppose LaTeXMLMath is a component of LaTeXMLMath which passes through such a point .
Then there is at most one other singularity of LaTeXMLMath on LaTeXMLMath , of type LaTeXMLMath .
We calculate LaTeXMLMath ( we allow LaTeXMLMath — this is the case where there are no other singularities on LaTeXMLMath ) .
Now LaTeXMLMath , so LaTeXMLMath .
But LaTeXMLMath misses the strictly log canonical point of LaTeXMLMath , since LaTeXMLMath is slc , thus LaTeXMLMath is Cartier near LaTeXMLMath .
So LaTeXMLMath , a contradiction .
It only remains to show that no normal strictly log canonical singularities can occur .
Otherwise , LaTeXMLMath is a normal surface , so is an elliptic cone by Theorem LaTeXMLRef .
Let LaTeXMLMath be a ruling , LaTeXMLMath the minimal resolution of LaTeXMLMath and LaTeXMLMath the exceptional curve .
Thus LaTeXMLMath is a ruled surface over an elliptic curve , LaTeXMLMath is the negative section , and the strict transform LaTeXMLMath of LaTeXMLMath is a ruling .
We calculate LaTeXMLMath .
Now LaTeXMLMath implies LaTeXMLMath .
But LaTeXMLMath misses the singularity of LaTeXMLMath since LaTeXMLMath is log canonical , so LaTeXMLMath is Cartier .
Thus LaTeXMLMath , a contradiction .
∎ In particular we have a classification of the surfaces LaTeXMLMath occurring by Theorem LaTeXMLRef and Theorem LaTeXMLRef .
If LaTeXMLMath the situation is much more complicated .
For example in the case LaTeXMLMath the maximum number of components of a surface LaTeXMLMath is LaTeXMLMath .
See Example LaTeXMLRef .
LaTeXMLMath is smooth for LaTeXMLMath .
First , by Theorem LaTeXMLRef and Theorem LaTeXMLRef above , for LaTeXMLMath , LaTeXMLMath has unobstructed LaTeXMLMath -Gorenstein deformations .
Second , by Theorem LaTeXMLRef , the map of functors LaTeXMLMath is smooth .
Note that LaTeXMLMath in this case , i.e. , every LaTeXMLMath -Gorenstein deformation of LaTeXMLMath is smoothable .
Thus LaTeXMLMath is smooth as required .
∎ LaTeXMLMath is not smooth if LaTeXMLMath — see Example LaTeXMLRef .
We set out below the complete classification of stable pairs of degrees 4 and 5 .
In degree 6 we give a complete list of candidates for the surfaces that occur , however we have yet to establish which surfaces of types C and D are smoothable .
I explain how the classification was obtained in Section LaTeXMLRef .
Given a weighted projective space LaTeXMLMath , let LaTeXMLMath denote a section of LaTeXMLMath .
We also write LaTeXMLMath to denote a general section of LaTeXMLMath .
Given a rational ruled surface LaTeXMLMath , let LaTeXMLMath and LaTeXMLMath denote a fibre and the negative section respectively .
When we refer to the singularities of LaTeXMLMath below , we mean the points of LaTeXMLMath where LaTeXMLMath is not normal crossing , i.e. , where we do not have LaTeXMLMath normal crossing and LaTeXMLMath normal crossing .
Surfaces LaTeXMLMath : LaTeXMLEquation .
Singularities of LaTeXMLMath : LaTeXMLEquation .
Surfaces LaTeXMLMath : LaTeXMLEquation .
Singularities of LaTeXMLMath : LaTeXMLEquation .
We have the following cases for the surfaces LaTeXMLMath : LaTeXMLEquation .
In this case we only have a partial solution – we give a complete list of candidates , but we have yet to establish which are smoothable .
We give below a list of possible components LaTeXMLMath of the surfaces LaTeXMLMath , i.e. , LaTeXMLMath is a component of the normalisation of LaTeXMLMath and LaTeXMLMath is the inverse image of the double curve .
We then glue these together , following the instructions in Theorem LaTeXMLRef and Theorem LaTeXMLRef , to recover LaTeXMLMath .
We have the following constraints : LaTeXMLMath .
Equivalently , LaTeXMLMath , where LaTeXMLMath runs over the components of LaTeXMLMath .
LaTeXMLMath in case C , LaTeXMLMath in case D , where LaTeXMLMath is the number of components of LaTeXMLMath — see Corollary LaTeXMLRef .
Recall that the components LaTeXMLMath are of two types III and IV .
We tabulate these separately .
For a singularity of type LaTeXMLMath , the exceptional locus of the minimal resolution consists of a chain LaTeXMLMath of curves together with two -2 curves meeting LaTeXMLMath .
The strict transform of LaTeXMLMath meets LaTeXMLMath .
We use the sequence LaTeXMLMath of self-intersections to describe the LaTeXMLMath singularities occurring below .
Type III : LaTeXMLEquation .
Type IV : LaTeXMLEquation .
In almost all cases the parameter is just the weight of the surface LaTeXMLMath , namely the maximal LaTeXMLMath such that there exists a birational morphism LaTeXMLMath , where LaTeXMLMath is the minimal resolution of LaTeXMLMath .
This is only false when LaTeXMLMath in type III cases 1,2,10 and 12 – then the weight is undefined in case 1 ( because LaTeXMLMath ) and equals LaTeXMLMath in the other cases .
We describe the components of type III concretely below , using toric language .
Note that we can also give an explicit description of the type IV cases — we describe LaTeXMLMath by expressing its minimal resolution LaTeXMLMath as a blowup of LaTeXMLMath .
This is omitted here .
Let LaTeXMLMath denote the weighted blowup of a smooth point of a surface LaTeXMLMath with weights LaTeXMLMath with respect to some local analytic coordinates .
Unless otherwise stated , we assume that the point and choice of coordinates is general .
We write LaTeXMLMath for the exceptional curve and use primes to denote strict transforms of curves .
If we refer to LaTeXMLMath , it is assumed that LaTeXMLMath passes through the centre LaTeXMLMath of the blowup , e.g .
if LaTeXMLMath , LaTeXMLMath denotes the strict transform of a general section of LaTeXMLMath through LaTeXMLMath .
Let LaTeXMLMath LaTeXMLMath denote the surface obtained from LaTeXMLMath in the following way : First perform a sequence of two blowups away from the negative section to obtain a degenerate fibre which is a chain of curves of self intersections LaTeXMLMath .
Then contract the two -2 curves .
Thus LaTeXMLMath has one double fibre with two LaTeXMLMath singularities on it , and a negative section with square LaTeXMLMath .
We let LaTeXMLMath and LaTeXMLMath denote the negative section and fibre as usual .
We write LaTeXMLMath for the double fibre with its reduced structure .
Type III ( toric descriptions ) : LaTeXMLEquation .
We aim to classify the surfaces LaTeXMLMath for LaTeXMLMath .
We provide a brief overview of our method below .
We first obtain a list of candidates satisfying certain combinatorial conditions .
We then check which of these candidates actually occur — the main point is to establish the smoothability of LaTeXMLMath .
This last step is incomplete in the case LaTeXMLMath .
Let LaTeXMLMath be a stable pair of degree LaTeXMLMath and LaTeXMLMath a component of LaTeXMLMath .
Let LaTeXMLMath be the minimal resolution of LaTeXMLMath and define a LaTeXMLMath -divisor LaTeXMLMath on LaTeXMLMath by LaTeXMLMath , LaTeXMLMath ( as in Notation LaTeXMLRef ) .
We first give a list of candidates for the pairs LaTeXMLMath .
We then glue these together as instructed in Theorem LaTeXMLRef to obtain our list of candidates for LaTeXMLMath .
We do not work directly with the pairs LaTeXMLMath — instead we consider the pairs LaTeXMLMath .
Note that it is immediate to recover LaTeXMLMath from LaTeXMLMath — we just contract the components of LaTeXMLMath where LaTeXMLMath is zero .
For LaTeXMLMath is ample and LaTeXMLMath by definition , moreover LaTeXMLMath by Lemma LaTeXMLRef below .
Note that we may assume LaTeXMLMath is rational , since otherwise LaTeXMLMath is an elliptic cone and LaTeXMLMath .
We begin our classification of the LaTeXMLMath by giving a list of possible singularities .
We deduce the possible configurations of rational curves making up the LaTeXMLMath -divisor LaTeXMLMath , together with multiplicities .
We obtain a finite list of possible slt singularities on LaTeXMLMath by using the conditions LaTeXMLMath and LaTeXMLMath slc to bound the index of LaTeXMLMath ( c.f .
the proof of Theorem LaTeXMLRef ) .
We are content to note that the strictly log canonical singularities of LaTeXMLMath are of two types LaTeXMLMath and LaTeXMLMath .
The corresponding configurations of components of LaTeXMLMath are easily understood — working locally over the singular point LaTeXMLMath we have : LaTeXMLMath : LaTeXMLMath , where LaTeXMLMath is a chain of rational curves , and LaTeXMLMath and LaTeXMLMath intersect LaTeXMLMath and LaTeXMLMath respectively .
LaTeXMLMath : LaTeXMLMath , where LaTeXMLMath is a chain of rational curves , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -curves meeting LaTeXMLMath , and LaTeXMLMath intersects LaTeXMLMath .
Note that , if LaTeXMLMath , LaTeXMLMath is log terminal by Theorem LaTeXMLRef , so these cases do not occur .
Assume LaTeXMLMath is rational and LaTeXMLMath , then there exists a birational morphism LaTeXMLMath , where LaTeXMLMath — fix one such morphism LaTeXMLMath , with LaTeXMLMath maximal .
Then LaTeXMLMath is an isomorphism over the negative section LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath denote the composite LaTeXMLMath .
We next analyse all the ways that we can fit the possible LaTeXMLMath configurations into a surface LaTeXMLMath with a birational ruling LaTeXMLMath as above .
We require that LaTeXMLMath is nef and big , and positive outside LaTeXMLMath — this imposes strong restrictions on curves in degenerate fibres which are not contained in LaTeXMLMath ( see Lemma LaTeXMLRef ) .
Also , we have LaTeXMLMath if LaTeXMLMath , where LaTeXMLMath denotes the strict transform of LaTeXMLMath under LaTeXMLMath .
We obtain the candidates for LaTeXMLMath and LaTeXMLMath in this way in the Proof of Theorem LaTeXMLRef .
We omit the derivation of the list of candidates for LaTeXMLMath .
It is rather different in style , since LaTeXMLMath may have strictly log canonical singularities .
In particular we do not obtain a finite list of possible singularities as our first step .
We begin by giving a list of possible degenerate fibres of LaTeXMLMath .
Then , with some work , we obtain a complete list of candidates .
In each case LaTeXMLMath is a collection of components of fibres together with LaTeXMLMath and at most one other horizontal component .
With the notation as above , the LaTeXMLMath -divisor LaTeXMLMath is effective and LaTeXMLMath .
We have LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath by definition ( compare the proof of Proposition LaTeXMLRef ) .
We need to show LaTeXMLMath for all LaTeXMLMath .
If LaTeXMLMath for some LaTeXMLMath , then LaTeXMLMath for all LaTeXMLMath in the same connected component of LaTeXMLMath using LaTeXMLMath LaTeXMLMath -nef .
It follows that this connected component contracts to a canonical singularity of LaTeXMLMath .
This must be a Du Val singularity of LaTeXMLMath with LaTeXMLMath locally .
But then we have a normal log-terminal singularity on LaTeXMLMath which is not of type LaTeXMLMath , a contradiction .
∎ Let LaTeXMLMath be a stable pair of degree LaTeXMLMath or LaTeXMLMath .
For LaTeXMLMath we have d=4 : LaTeXMLMath or LaTeXMLMath .
d=5 : LaTeXMLMath or LaTeXMLMath .
d=6 : LaTeXMLMath or LaTeXMLMath .
See the Proof of Theorem LaTeXMLRef — a finer analysis in the cases LaTeXMLMath and LaTeXMLMath gives our result .
∎ Let LaTeXMLMath or LaTeXMLMath .
Then the connected components of LaTeXMLMath are chains of smooth rational curves .
We have the following possibilities for the multiplicities and self-intersections of the components : LaTeXMLMath LaTeXMLEquation .
LaTeXMLMath LaTeXMLEquation .
Here we write LaTeXMLMath to denote a smooth rational curve LaTeXMLMath on LaTeXMLMath passing through a singularity of type LaTeXMLMath , such that locally analytically LaTeXMLMath .
X is slt for LaTeXMLMath by Theorem LaTeXMLRef , thus LaTeXMLMath is log terminal .
Recall that the log terminal singularities of LaTeXMLMath are of types LaTeXMLMath and LaTeXMLMath .
Lemma LaTeXMLRef states which values of the indices LaTeXMLMath and LaTeXMLMath are possible .
It only remains to determine what combinations of singularities can lie on LaTeXMLMath for LaTeXMLMath .
Suppose LaTeXMLMath , and let LaTeXMLMath have singularities of indices LaTeXMLMath at LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath ( compare Theorem LaTeXMLRef ( II ) ) .
Moreover , LaTeXMLMath is 3-divisible by Lemma LaTeXMLRef below .
Using the restrictions on the indices in Lemma LaTeXMLRef , we obtain the solutions LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath , LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath for LaTeXMLMath .
Combining our results we obtain the lists of possible connected components of LaTeXMLMath above .
∎ Suppose LaTeXMLMath .
Let LaTeXMLMath be a curve .
Then LaTeXMLMath is divisible by 3 ( i.e .
writing LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , we have LaTeXMLMath ) .
Equivalently , let LaTeXMLMath be a curve that is not LaTeXMLMath -exceptional , then LaTeXMLMath is divisible by 3 .
We use the relation LaTeXMLMath .
Thus LaTeXMLMath , so it ’ s enough to show that the index of LaTeXMLMath is not 3-divisible .
See the proof of Lemma LaTeXMLRef .
∎ Suppose LaTeXMLMath is a curve contained in a degenerate fibre of LaTeXMLMath which is not a component of LaTeXMLMath .
Then LaTeXMLMath is a -1 curve and LaTeXMLMath .
We know LaTeXMLMath is not LaTeXMLMath -exceptional by Lemma LaTeXMLRef , hence LaTeXMLMath .
But LaTeXMLMath .
Hence LaTeXMLMath and LaTeXMLMath as claimed .
∎ We refer to such a curve LaTeXMLMath as a LaTeXMLMath -curve Suppose LaTeXMLMath is a LaTeXMLMath -curve .
We list the possible intersections of LaTeXMLMath with LaTeXMLMath .
LaTeXMLMath : LaTeXMLMath intersects one component of LaTeXMLMath of multiplicity LaTeXMLMath .
LaTeXMLMath : LaTeXMLMath intersects one component of LaTeXMLMath of multiplicity LaTeXMLMath , or two components of multiplicity LaTeXMLMath , or two components of multiplicities LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath : LaTeXMLMath intersects one component of LaTeXMLMath of multiplicity LaTeXMLMath .
All intersections are transverse and at a single point , except possibly the LaTeXMLMath case for LaTeXMLMath — here the two components of LaTeXMLMath may coincide , and then further the two points of intersection may coincide to yield a point of contact of order LaTeXMLMath .
This is immediate from the index calculations of Lemma LaTeXMLRef together with Lemma LaTeXMLRef and Lemma LaTeXMLRef .
∎ The stable pairs of degrees LaTeXMLMath and LaTeXMLMath are as described in Section LaTeXMLRef .
Let LaTeXMLMath be a stable pair of degree LaTeXMLMath , where LaTeXMLMath or LaTeXMLMath .
Let LaTeXMLMath be a component of LaTeXMLMath .
Then LaTeXMLMath is rational since LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath , since LaTeXMLMath is ample and 3-divisible ( again using LaTeXMLMath ) .
So assume LaTeXMLMath , and choose LaTeXMLMath as in Notation LaTeXMLRef .
We classify the pairs LaTeXMLMath , and hence the pairs LaTeXMLMath , and finally glue these together to form the surfaces LaTeXMLMath .
Each degenerate fibre of the birational ruling LaTeXMLMath consists of some irreducible components of LaTeXMLMath and some LaTeXMLMath -curves .
We have a list of possible connected components of LaTeXMLMath in Lemma LaTeXMLRef , and Corollary LaTeXMLRef describes how LaTeXMLMath -curves and irreducible components of LaTeXMLMath may intersect .
Note also that two LaTeXMLMath -curves LaTeXMLMath in a degenerate fibre LaTeXMLMath do not intersect unless LaTeXMLMath .
We classify the LaTeXMLMath by calculating all the ways we can piece together connected components of LaTeXMLMath and LaTeXMLMath -curves to form a net of curves which is a union of fibres and horizontal curves on some surface LaTeXMLMath with a birational ruling LaTeXMLMath .
So , suppose given some LaTeXMLMath , and assume LaTeXMLMath ( otherwise LaTeXMLMath is smooth , so LaTeXMLMath ) .
Let LaTeXMLMath be a connected component of LaTeXMLMath .
We consider the cases of Lemma LaTeXMLRef .
We first treat the case LaTeXMLMath .
In case ( 4 ) LaTeXMLMath can not intersect a LaTeXMLMath -curve .
Thus if LaTeXMLMath intersects a degenerate fibre , it must contain the whole degenerate fibre .
But by inspection LaTeXMLMath can not contain a degenerate fibre , so LaTeXMLMath does not intersect any degenerate fibre .
It follows that there are no degenerate fibres .
But we know there are at least two curves of negative self-intersection , a contradiction .
So case ( 4 ) does not occur .
It now follows that case ( 5 ) can not occur .
For a LaTeXMLMath with singularities at LaTeXMLMath corresponding to case ( 5 ) must be glued to a LaTeXMLMath with singularities at LaTeXMLMath corresponding to case ( 4 ) in order to form a smoothable surface LaTeXMLMath .
In case ( 3 ) , let LaTeXMLMath be the components of LaTeXMLMath , ordered as in Lemma LaTeXMLRef .
Only LaTeXMLMath can intersect a LaTeXMLMath -curve and LaTeXMLMath can not contain a degenerate fibre .
Suppose that a component of LaTeXMLMath is contained in a degenerate fibre LaTeXMLMath .
Then LaTeXMLMath consists of LaTeXMLMath together with some LaTeXMLMath -curves meeting LaTeXMLMath , and possibly LaTeXMLMath or LaTeXMLMath .
The only possibility is LaTeXMLMath where LaTeXMLMath are LaTeXMLMath -curves .
Then LaTeXMLMath and LaTeXMLMath must be horizontal , but LaTeXMLMath can not intersect LaTeXMLMath , a contradiction .
So every component of LaTeXMLMath is horizontal .
In particular there can be no degenerate fibres ( because LaTeXMLMath can not intersect a LaTeXMLMath -curve ) , a contradiction .
Thus case ( 3 ) does not occur .
In case ( 1 ) LaTeXMLMath can not intersect a LaTeXMLMath -curve .
So there are no degenerate fibres ( compare ( 4 ) above ) .
Thus LaTeXMLMath , so LaTeXMLMath .
In case ( 2 ) again LaTeXMLMath can not intersect a LaTeXMLMath -curve .
Thus LaTeXMLMath is ruled , LaTeXMLMath , and so LaTeXMLMath .
Finally LaTeXMLMath , since LaTeXMLMath is ample and 3-divisible .
This completes the classification of the LaTeXMLMath for LaTeXMLMath .
Now let LaTeXMLMath .
In case ( 6 ) LaTeXMLMath can not intersect a LaTeXMLMath -curve .
We obtain a contradiction as for LaTeXMLMath , ( 4 ) .
So ( 6 ) does not occur .
It follows that case ( 7 ) can not occur , as for LaTeXMLMath , ( 5 ) .
For a LaTeXMLMath with singularities at LaTeXMLMath corresponding to case ( 7 ) must be glued to a LaTeXMLMath with singularities at LaTeXMLMath corresponding to case ( 6 ) to form a smoothable surface LaTeXMLMath .
In case ( 4 ) , let LaTeXMLMath be the components of LaTeXMLMath , ordered as above .
Only LaTeXMLMath can intersect a LaTeXMLMath -curve , and a LaTeXMLMath -curve intersecting LaTeXMLMath does not intersect any other irreducible components of LaTeXMLMath .
We proceed as in LaTeXMLMath , ( 3 ) to obtain a contradiction , thus ( 4 ) does not occur .
In case ( 5 ) , LaTeXMLMath can not intersect a LaTeXMLMath -curve .
Thus LaTeXMLMath is ruled , LaTeXMLMath and LaTeXMLMath .
The curve LaTeXMLMath can not be horizontal using LaTeXMLMath ample and 3-divisible , thus LaTeXMLMath is a fibre and LaTeXMLMath .
In case ( 3 ) , write LaTeXMLMath for the components of LaTeXMLMath , ordered as above .
Then only LaTeXMLMath and LaTeXMLMath can intersect a LaTeXMLMath -curve .
Now LaTeXMLMath can not contain a fibre , thus we see that LaTeXMLMath has components in at most one fibre , and then this fibre contains LaTeXMLMath or LaTeXMLMath .
If every component of LaTeXMLMath were horizontal , then LaTeXMLMath would intersect a LaTeXMLMath -curve in a degenerate fibre , a contradiction .
Thus LaTeXMLMath has components in a unique degenerate fibre LaTeXMLMath .
We classify the possible fibres LaTeXMLMath above .
We have LaTeXMLMath because LaTeXMLMath intersects LaTeXMLMath , it follows that LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath consists of LaTeXMLMath , some LaTeXMLMath -curves ( meeting LaTeXMLMath ) and possibly LaTeXMLMath — but such a configuration is never a fibre , a contradiction .
Hence LaTeXMLMath .
If there is a LaTeXMLMath -curve LaTeXMLMath meeting LaTeXMLMath , then LaTeXMLMath .
Otherwise , we have some LaTeXMLMath -curves meeting LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath are two such curves then LaTeXMLMath , a contradiction .
Thus there is exactly one LaTeXMLMath -curve , LaTeXMLMath say , meeting LaTeXMLMath .
Then LaTeXMLMath intersects another irreducible component LaTeXMLMath of LaTeXMLMath which has multiplicity LaTeXMLMath or LaTeXMLMath ( by Corollary LaTeXMLRef ) and hence self-intersection LaTeXMLMath or LaTeXMLMath by Lemma LaTeXMLRef .
We have LaTeXMLMath .
If LaTeXMLMath we find LaTeXMLMath , a contradiction .
Thus LaTeXMLMath , we deduce LaTeXMLMath .
Note that LaTeXMLMath is a connected component of LaTeXMLMath of type ( 1 ) .
So , we have two possible types of degenerate fibre LaTeXMLMath — either LaTeXMLMath where LaTeXMLMath meets LaTeXMLMath , or LaTeXMLMath , a chain of curves , where LaTeXMLMath .
In each case LaTeXMLMath is horizontal — it follows that there are no more degenerate fibres , since LaTeXMLMath can not intersect a LaTeXMLMath -curve .
We see that LaTeXMLMath is obtained from LaTeXMLMath by a sequence of blowups , and LaTeXMLMath .
Also LaTeXMLMath so LaTeXMLMath .
A graded ring calculation shows that LaTeXMLMath in the first case .
In the second case we see LaTeXMLMath by toric methods .
Case ( 8 ) is very similiar : writing LaTeXMLMath for the components of LaTeXMLMath , we obtain the same possible degenerate fibres LaTeXMLMath as above , LaTeXMLMath horizontal , and LaTeXMLMath a fibre .
LaTeXMLMath is obtained from LaTeXMLMath by a sequence of blowups , and LaTeXMLMath .
A graded ring calculation shows that LaTeXMLMath in the first case , where LaTeXMLMath .
In the second case we have LaTeXMLMath , where LaTeXMLMath .
Finally , for cases ( 1 ) and ( 2 ) , we may assume that LaTeXMLMath has index LaTeXMLMath , because I have already classified the LaTeXMLMath with an index LaTeXMLMath singularity above .
Then the same calculation as for LaTeXMLMath gives LaTeXMLMath or LaTeXMLMath .
This completes the classification of the LaTeXMLMath for LaTeXMLMath .
We calculate that each surface LaTeXMLMath has LaTeXMLMath , and LaTeXMLMath is a log del Pezzo surface .
We now glue the LaTeXMLMath together to obtain the surfaces LaTeXMLMath .
The normal surfaces LaTeXMLMath are the surfaces LaTeXMLMath with LaTeXMLMath .
The non-normal surfaces LaTeXMLMath are obtained by glueing components LaTeXMLMath , LaTeXMLMath along LaTeXMLMath , LaTeXMLMath so that each LaTeXMLMath singularity on LaTeXMLMath is glued to a LaTeXMLMath singularity on LaTeXMLMath to give a singularity of type LaTeXMLMath on LaTeXMLMath .
We also require that LaTeXMLMath , equivalently LaTeXMLMath ( in fact this is automatic in our cases LaTeXMLMath and LaTeXMLMath ) .
Finally , we conclude that each surface LaTeXMLMath constructed as above occurs in a stable pair LaTeXMLMath of degree LaTeXMLMath or LaTeXMLMath .
We need to show that there exists a divisor LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath is slc — this is easy to check .
Moreover , we require that LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath is LaTeXMLMath -Cartier — this follows from Theorem LaTeXMLRef , using the existence of a LaTeXMLMath -Gorenstein smoothing LaTeXMLMath ( Theorem LaTeXMLRef and Theorem LaTeXMLRef ) .
∎ We first construct explicit smoothings of some slc del Pezzo surfaces of type C. We thus obtain examples of stable pairs of degree LaTeXMLMath — in particular , we obtain an example where the surface has LaTeXMLMath components .
We next give an example to show that LaTeXMLMath is not smooth if LaTeXMLMath ( Example LaTeXMLRef ) .
Our final example shows that the relative smoothability assumption LaTeXMLRef in the definition of LaTeXMLMath is necessary if LaTeXMLMath .
More specifically , we construct an slc del Pezzo surface LaTeXMLMath which admits LaTeXMLMath -Gorenstein smoothings to both LaTeXMLMath and a surface which is not smoothable .
Let LaTeXMLMath be a cycle of smooth rational curves , i.e. , LaTeXMLMath , where LaTeXMLMath for all LaTeXMLMath , LaTeXMLMath is nodal , and the dual graph of LaTeXMLMath is a cycle .
Let LaTeXMLMath be the spectrum of a complete DVR with generic point LaTeXMLMath .
Then there exists a smoothing LaTeXMLMath of LaTeXMLMath to an elliptic curve .
Let LaTeXMLMath be a relatively ample line bundle on LaTeXMLMath , write LaTeXMLMath .
Define LaTeXMLMath .
The special fibre LaTeXMLMath is a LaTeXMLMath -bundle over LaTeXMLMath with components LaTeXMLMath ruled over LaTeXMLMath .
The generic fibre is a ruled surface of degree LaTeXMLMath over an elliptic curve .
Contracting the negative sections , we obtain a family LaTeXMLMath with special fibre LaTeXMLMath ( an slc del Pezzo surface of type C ) and generic fibre an elliptic cone of degree LaTeXMLMath .
If we fix LaTeXMLMath ( equivalently LaTeXMLMath ) , we deduce that LaTeXMLMath is smoothable , since an elliptic cone of degree LaTeXMLMath is smoothable by Lemma LaTeXMLRef .
We generalise Construction LaTeXMLRef .
Let LaTeXMLMath and LaTeXMLMath be as above .
Let LaTeXMLMath be a sheaf on LaTeXMLMath which is invertible in codimension LaTeXMLMath and LaTeXMLMath .
Locally at a node LaTeXMLMath , we have LaTeXMLEquation .
Write LaTeXMLMath for the local universal cover LaTeXMLMath .
Then locally LaTeXMLMath , where the subscript LaTeXMLMath is used to denote the eigensubsheaf where a generator LaTeXMLMath acts as LaTeXMLMath .
In particular , LaTeXMLMath is invertible locally .
Assume that LaTeXMLMath is relatively ample , write LaTeXMLMath .
Define LaTeXMLEquation where we write LaTeXMLMath for the double dual of the LaTeXMLMath th symmetric power of a sheaf .
Where LaTeXMLMath is invertible , LaTeXMLMath is a LaTeXMLMath -bundle and thus easily understood .
Let LaTeXMLMath be a node where LaTeXMLMath is not invertible .
Locally at LaTeXMLMath , with notation as above , we have LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath have weights LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath with respect to the LaTeXMLMath action , and LaTeXMLMath and LaTeXMLMath have weight LaTeXMLMath with respect to the grading of the ring .
So , locally over LaTeXMLMath , we have two affine pieces of LaTeXMLMath as follows : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath , write LaTeXMLMath , LaTeXMLMath .
Then the surface LaTeXMLMath has singularities LaTeXMLMath and LaTeXMLMath , and the 3-fold LaTeXMLMath has a LaTeXMLMath singularity generically along the double curve .
We can now give a global description of LaTeXMLMath .
The surface LaTeXMLMath has components LaTeXMLMath which are fibred over the components LaTeXMLMath of LaTeXMLMath .
The fibres of LaTeXMLMath are smooth rational curves ( when given their reduced structure ) .
The double curve LaTeXMLMath is the sum of the ( reduced ) fibres of LaTeXMLMath over the nodes of LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be a node of LaTeXMLMath on LaTeXMLMath where LaTeXMLMath is not invertible .
With notation as above , the fibre of LaTeXMLMath over LaTeXMLMath has multiplicity LaTeXMLMath , and LaTeXMLMath has singularities LaTeXMLMath and LaTeXMLMath at the fibre .
The bundle LaTeXMLMath has a negative section of self-intersection LaTeXMLMath .
We obtain LaTeXMLMath by glueing the components LaTeXMLMath along the fibres over the nodes of LaTeXMLMath .
The singularities of type LaTeXMLMath on the components glue to give singularities of type LaTeXMLMath on LaTeXMLMath , elsewhere LaTeXMLMath is normal crossing .
The negative sections of LaTeXMLMath glue to give a section of LaTeXMLMath .
The family LaTeXMLMath is a LaTeXMLMath -Gorenstein smoothing of LaTeXMLMath , with generic fibre a ruled surface over an elliptic curve of degree LaTeXMLMath .
We now contract the negative sections .
We obtain a LaTeXMLMath -Gorenstein family LaTeXMLMath with special fibre LaTeXMLMath an slc del Pezzo surface of type C and generic fibre an elliptic cone of degree LaTeXMLMath .
Again , assuming LaTeXMLMath , we deduce that LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Let LaTeXMLMath be a surface of type C. Assume that LaTeXMLMath has slt singularities of types LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath .
For every component LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is toric , i.e. , LaTeXMLMath is a toric surface and LaTeXMLMath is a sum of toric strata of codimension LaTeXMLMath , and LaTeXMLMath .
Then LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
Note that conditions ( 1 ) and ( 2 ) are necessary .
Let LaTeXMLMath be a component of LaTeXMLMath .
Then there is a unique toric blowup LaTeXMLMath which blows up the node of LaTeXMLMath such that the strict transforms of the components of LaTeXMLMath give fibres of a fibration LaTeXMLMath .
For , if LaTeXMLMath is given by the fan defined by the vectors LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath correspond to the components of LaTeXMLMath , the map LaTeXMLMath corresponds to the subdivision of the fan obtained by adding the vector LaTeXMLMath .
Write LaTeXMLMath for the strict transform of LaTeXMLMath .
Glueing the LaTeXMLMath together so that the negative sections form a cycle , we obtain a partial resolution LaTeXMLMath .
We claim that , for a suitable choice of LaTeXMLMath and LaTeXMLMath in Construction LaTeXMLRef , we can recover LaTeXMLMath as the special fibre of LaTeXMLMath .
Hence we obtain a smoothing of LaTeXMLMath as above .
To prove the claim , note that LaTeXMLMath is uniquely determined by the following data — the self-intersections LaTeXMLMath of the negative sections of the components LaTeXMLMath , and the LaTeXMLMath singularities at the double curves .
Let LaTeXMLMath be any smoothing of LaTeXMLMath .
Possibly after base change , we can choose LaTeXMLMath locally at the nodes of LaTeXMLMath to obtain the required singularities ( compare Construction LaTeXMLRef ) .
These local choices extend to a global sheaf LaTeXMLMath .
We require that LaTeXMLMath in order to obtain the correct self-intersections of the negative sections — we can achieve this by twisting LaTeXMLMath by a suitable line bundle .
∎ Every candidate surface LaTeXMLMath for LaTeXMLMath ( see LaTeXMLRef ) of type C with components of Picard number LaTeXMLMath does indeed occur in a stable pair LaTeXMLMath of degree LaTeXMLMath .
For , by Proposition LaTeXMLRef , LaTeXMLMath admits a LaTeXMLMath -Gorenstein smoothing to LaTeXMLMath .
It only remains to show that there exists a divisor LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath is a stable pair of degree LaTeXMLMath .
Equivalently , we require LaTeXMLMath such that LaTeXMLMath is slc for some LaTeXMLMath .
Let LaTeXMLMath be the partial resolution as in the Proof of Proposition LaTeXMLRef , and let LaTeXMLMath be the fibration .
Let LaTeXMLMath be a general double section of LaTeXMLMath which is disjoint from the negative section .
Then LaTeXMLMath has the required properties .
Note that smoothability of the pair LaTeXMLMath follows by Theorem LaTeXMLRef .
In particular , we obtain an example of a stable pair LaTeXMLMath of degree 6 such that the surface LaTeXMLMath has LaTeXMLMath components .
We construct LaTeXMLMath by glueing 18 components LaTeXMLMath together to form a surface of type C. At the degenerate cusp all the components of LaTeXMLMath are smooth , and LaTeXMLMath has 9 slt singularities of type LaTeXMLMath .
We note that LaTeXMLMath is the maximum number of components of a surface LaTeXMLMath occurring in a stable pair of degree LaTeXMLMath .
For , by Lemma LaTeXMLRef we have LaTeXMLMath Cartier , thus for each component LaTeXMLMath of LaTeXMLMath we have LaTeXMLMath , in particular LaTeXMLMath .
Since LaTeXMLMath , there are at most LaTeXMLMath components as required .
The deformation space LaTeXMLMath of an elliptic cone LaTeXMLMath of degree LaTeXMLMath consists of LaTeXMLMath smooth LaTeXMLMath -dimensional components , together with an embedded component at the origin .
The components of LaTeXMLMath meet as transversely as possible , i.e. , LaTeXMLMath is isomorphic to the collection of coordinate axes in LaTeXMLMath .
Each component corresponds to a smoothing of LaTeXMLMath to LaTeXMLMath as constructed in Lemma LaTeXMLRef .
Write LaTeXMLMath for the section of LaTeXMLMath .
There is an action of the LaTeXMLMath -torsion of LaTeXMLMath on LaTeXMLMath which permutes the components transitively , the LaTeXMLMath -torsion elements induce automorphisms of each component .
For details see LaTeXMLCite , p. 220 , Example 4.5 .
Now , we can add a divisor LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath is a stable pair of degree LaTeXMLMath , for any LaTeXMLMath such that LaTeXMLMath .
We describe the singularities of LaTeXMLMath at the point LaTeXMLMath .
By Theorem LaTeXMLRef , LaTeXMLMath is smooth over the closed subscheme of LaTeXMLMath corresponding to LaTeXMLMath -Gorenstein smoothable deformations .
First , since LaTeXMLMath is Cartier , every deformation is LaTeXMLMath -Gorenstein .
Second , the smoothable deformations correspond ( by definition ) to the scheme theoretic closure in LaTeXMLMath of the locus where the geometric fibres are isomorphic to LaTeXMLMath .
In our case this is exactly LaTeXMLMath by the explicit description above .
Thus LaTeXMLMath is smooth over LaTeXMLMath .
In particular , LaTeXMLMath is not smooth at LaTeXMLMath .
We define an slc del Pezzo surface LaTeXMLMath of type C as follows : The normalisation LaTeXMLMath has two components LaTeXMLMath and LaTeXMLMath , where we write LaTeXMLMath for the fibre and LaTeXMLMath for the negative section of a rational ruled surface LaTeXMLMath .
LaTeXMLMath is glued to LaTeXMLMath by identifying the negative section of LaTeXMLMath with a fibre of LaTeXMLMath and vice versa .
Thus LaTeXMLMath has a degenerate cusp , we remark that locally analytically at this point we have LaTeXMLEquation .
We claim that LaTeXMLMath smoothes to LaTeXMLMath .
We construct an explicit smoothing as follows : Let LaTeXMLMath be the spectrum of a DVR , let LaTeXMLMath .
Let LaTeXMLMath denote the blowup of LaTeXMLMath with centre a smooth conic LaTeXMLMath in the special fibre LaTeXMLMath .
Then the special fibre LaTeXMLMath consists of the strict transform LaTeXMLMath of LaTeXMLMath together with the exceptional divisor LaTeXMLMath , glued along the conic LaTeXMLMath and the negative section LaTeXMLMath .
Let LaTeXMLMath denote the blowup of LaTeXMLMath with centre a fibre LaTeXMLMath of LaTeXMLMath .
Then LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is the exceptional divisor , glued to the fibre LaTeXMLMath of LaTeXMLMath along its negative section , and glued to the LaTeXMLMath -curve of LaTeXMLMath along a fibre .
We calculate that LaTeXMLMath is ample on LaTeXMLMath and LaTeXMLMath , and defines the fibering contraction LaTeXMLMath of LaTeXMLMath .
Thus some multiple of LaTeXMLMath defines a divisorial contraction LaTeXMLMath which contracts LaTeXMLMath to a curve .
We observe that the special fibre LaTeXMLMath is our surface LaTeXMLMath , so LaTeXMLMath is a smoothing of LaTeXMLMath as required .
We next construct a partial smoothing of LaTeXMLMath to a rigid normal crossing surface .
We consider the components of LaTeXMLMath separately .
First , the component LaTeXMLMath has a deformation LaTeXMLMath with generic fibre LaTeXMLMath .
The family LaTeXMLMath can be obtained as a deformation of scrolls in some LaTeXMLMath .
Second , the component LaTeXMLMath has a deformation LaTeXMLMath with generic fibre LaTeXMLMath , where LaTeXMLMath is a section of LaTeXMLMath disjoint from LaTeXMLMath .
Here LaTeXMLMath is the trivial deformation .
Then , possibly after base change , we can glue LaTeXMLMath and LaTeXMLMath along LaTeXMLMath and LaTeXMLMath to obtain a partial smoothing LaTeXMLMath of LaTeXMLMath .
The generic fibre is the normal crossing surface LaTeXMLMath obtained by glueing LaTeXMLMath and LaTeXMLMath .
The surface LaTeXMLMath is rigid , in particular it is not smoothable .
To see this , note that , by Lemma LaTeXMLRef , we have LaTeXMLMath .
So LaTeXMLMath , thus all deformations of LaTeXMLMath are locally trivial .
It is easy to see that LaTeXMLMath has no locally trivial deformations ( by considering the components separately ) , hence LaTeXMLMath is rigid as claimed .
We can add a divisor LaTeXMLMath such that LaTeXMLMath is a stable pair of degree LaTeXMLMath , for any LaTeXMLMath such that LaTeXMLMath .
Then , working locally at LaTeXMLMath , the stack LaTeXMLMath is smooth over the closed subscheme LaTeXMLMath of LaTeXMLMath corresponding to smoothable deformations .
We want to emphasise that , set-theoretically , LaTeXMLMath is strictly smaller than LaTeXMLMath .
For , by our construction above , there exists an irreducible component of LaTeXMLMath whose generic point corresponds to a surface LaTeXMLMath that is not smoothable .
Thus this component is not contained in LaTeXMLMath .
In particular , we see that the relative smoothability condition ( LaTeXMLRef ) for LaTeXMLMath is necessary .
For , let LaTeXMLMath be a deformation of LaTeXMLMath over the spectrum of a complete DVR such that LaTeXMLMath has generic fibre LaTeXMLMath .
Write LaTeXMLMath for the groupoid obtained by dropping the relative smoothability assumption in the definition of LaTeXMLMath .
Then LaTeXMLMath ( since the generic fibre is not smoothable ) , but LaTeXMLMath for any Artinian thickening LaTeXMLMath of LaTeXMLMath .
It follows that there does not exist a versal deformation LaTeXMLMath of LaTeXMLMath , thus LaTeXMLMath is not an algebraic stack .
Let LaTeXMLMath , where LaTeXMLMath is a smooth compact hypersurface in LaTeXMLMath and LaTeXMLMath denotes the Lebesgue measure on LaTeXMLMath .
Let LaTeXMLMath .
If the hypersurface LaTeXMLMath has non-vanishing Gaussian curvature , then LaTeXMLEquation for LaTeXMLMath .
Moreover , the result is sharp .
See LaTeXMLCite , LaTeXMLCite .
If the hypersurface LaTeXMLMath is convex and the order of contact with every tangent line is finite , the optimal exponents for the inequality LaTeXMLMath are known in LaTeXMLMath , ( see LaTeXMLCite ) , and in any dimension in the range LaTeXMLMath , ( see LaTeXMLCite ) .
More precisely , the result in the range LaTeXMLMath is the following .
Let LaTeXMLMath be a smooth convex compact finite type hypersurface , in the sense that the order of contact with every tangent line is finite .
Then for LaTeXMLMath , the following condition is necessary and sufficient for the maximal inequality LaTeXMLMath .
LaTeXMLEquation for every tangent hyperplane LaTeXMLMath not passing through the origin , where LaTeXMLMath denotes the distance from a point LaTeXMLMath to the tangent hyperplane LaTeXMLMath .
In fact , the condition LaTeXMLMath is a necessary condition for any smooth compact hypersurface in LaTeXMLMath .
See LaTeXMLCite , Theorem 2 .
In this paper we shall consider convex radial hypersurfaces of the form LaTeXMLEquation where LaTeXMLMath is a ball centered at the origin , LaTeXMLMath , LaTeXMLMath is convex , LaTeXMLMath , LaTeXMLMath increasing , LaTeXMLMath , and LaTeXMLMath is allowed to vanish of infinite order .
If LaTeXMLMath does vanish of infinite order , the condition LaTeXMLMath can not hold for any LaTeXMLMath .
Since the condition LaTeXMLMath is necessary by Theorem 1 above , our only hope is to look for an inequality of the form LaTeXMLEquation where LaTeXMLMath is an Orlicz space , near LaTeXMLMath , associated to a Young function LaTeXMLMath , with the norm given by LaTeXMLEquation .
The following result was proved in LaTeXMLCite .
Let LaTeXMLMath be as in LaTeXMLMath with LaTeXMLMath .
Assume that for each LaTeXMLMath LaTeXMLEquation .
Put LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath let LaTeXMLMath be a non-decreasing function such that LaTeXMLMath if LaTeXMLMath is sufficiently large , LaTeXMLMath if LaTeXMLMath , and LaTeXMLMath if LaTeXMLMath .
Let LaTeXMLMath .
Then for every LaTeXMLMath there exists a constant LaTeXMLMath such that the estimate LaTeXMLMath holds .
The examples show ( see LaTeXMLCite , Example 3.3 ) that Theorem 2 is sharp for some surfaces , for example if LaTeXMLMath , LaTeXMLMath , but not for others , for example if LaTeXMLMath .
In this paper we shall give a set of simple sufficient conditions for the inequality LaTeXMLMath for some classes of Orlicz functions LaTeXMLMath .
We will show that our result is sharp for a wide class of both finite type and infinite type LaTeXMLMath ’ s .
Acknowledgements : The author wishes to thank Jim Wright for teaching him the technique needed to prove the three dimensional case of Lemma 5 below .
Assume that LaTeXMLMath is a Young function such that LaTeXMLMath , where LaTeXMLMath is a non-decreasing function such that LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath .
Assume that there exist constants LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath such that LaTeXMLEquation and for every LaTeXMLMath LaTeXMLEquation .
Our main reason for making these assumptions about LaTeXMLMath is the following generalization of the Marcienkiewicz interpolation theorem due to Bak .
See LaTeXMLCite , Lemma 1.1 .
Let LaTeXMLMath .
Suppose that the operator LaTeXMLMath is simultaneously weak type LaTeXMLMath and LaTeXMLMath , namely there exist constants LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation Suppose that LaTeXMLMath satisfies the assumptions above .
Then there exists a constant LaTeXMLMath depending only on LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
Lemma 3 has the following interesting consequence .
Let LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is a smooth cutoff function , and let LaTeXMLMath denote the same operator with LaTeXMLMath localized to the interval LaTeXMLMath .
It was proved in LaTeXMLCite that LaTeXMLMath , LaTeXMLMath , with norm LaTeXMLMath .
Let LaTeXMLMath .
It follows by Lemma 3 that LaTeXMLMath if LaTeXMLMath and LaTeXMLMath .
Our main results are the following .
Let LaTeXMLMath be as in LaTeXMLMath .
Let LaTeXMLMath .
Suppose that LaTeXMLMath satisfies the conditions LaTeXMLMath and LaTeXMLMath above .
Suppose that LaTeXMLMath .
Then the estimate LaTeXMLMath holds if LaTeXMLEquation .
The main technical result involved in the proof of Theorem 4 is the following version of the standard stationary phase estimates .
Let LaTeXMLMath .
Let LaTeXMLEquation with LaTeXMLMath , where LaTeXMLMath is as in LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath is independent of LaTeXMLMath and LaTeXMLMath .
Moreover , if LaTeXMLMath is replaced by LaTeXMLMath then the estimate LaTeXMLMath still holds with LaTeXMLMath on the right-hand side replaced by LaTeXMLMath .
The main technical result used in the proof of Theorem 2 is the following .
See LaTeXMLCite , Theorem 2.1 .
Let LaTeXMLMath be a non-negative function that is compactly supported in the interval LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath be as in LaTeXMLMath where LaTeXMLMath satisfies the condition of Theorem 2 .
Let LaTeXMLMath denote LaTeXMLMath in Lemma 5 with LaTeXMLMath in place of the characteristic function of the annulus LaTeXMLMath .
Then for every multi-index LaTeXMLMath with LaTeXMLMath there exists a constant LaTeXMLMath independent of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath such that LaTeXMLEquation where LaTeXMLMath if LaTeXMLMath , and LaTeXMLMath if LaTeXMLMath .
The point is that even though a higher dimensional analog of Lemma 6 may be difficult to obtain , we get around the problem by using Lemma 5 .
We have to settle for the uniform decay of order LaTeXMLMath instead of LaTeXMLMath , but this is enough in dimension LaTeXMLMath as we shall see below .
The idea is , roughly speaking , the following .
We are trying to prove LaTeXMLMath estimates for maximal operators associated to radial convex surfaces .
If the surface is infinitely flat , then Theorem 2 in LaTeXMLCite implies that LaTeXMLMath estimates are not possible for LaTeXMLMath .
So we are looking for LaTeXMLMath estimates where LaTeXMLMath is very close to LaTeXMLMath , so interpolating between LaTeXMLMath and LaTeXMLMath in the right way should do the trick .
However , in order to obtain LaTeXMLMath boundedness of the maximal operator , we only need decay LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath , so we should be alright .
If LaTeXMLMath a bit more integration by parts will be required .
The rest of the paper is organized as follows .
In the next section we shall prove Theorem 4 assuming Lemma 5 .
In the following section we shall prove Lemma 5 .
In the final section of the paper we shall discuss the sharpness of Theorem 4 and give some examples .
Let LaTeXMLMath , where LaTeXMLMath is a smooth cutoff function supported in LaTeXMLMath , such that LaTeXMLMath .
Let LaTeXMLMath .
Making a change of variables we see that LaTeXMLEquation where LaTeXMLEquation .
We shall prove that LaTeXMLEquation .
By interpolating with the trivial estimate LaTeXMLMath using Lemma 3 , we shall conclude that LaTeXMLEquation .
Since the LaTeXMLMath norms of LaTeXMLMath and LaTeXMLMath are reciprocals of each other , it follows that LaTeXMLMath if LaTeXMLEquation .
So it remains to prove LaTeXMLMath .
The proof follows from the standard Sobolev imbedding theorem type argument .
See for example LaTeXMLCite .
We shall use the following version which follows from the proof of Theorem 15 in LaTeXMLCite .
See also , for example , LaTeXMLCite , LaTeXMLCite .
Suppose that LaTeXMLMath is the Lebesgue measure on the hypersurface LaTeXMLMath supported in an ellipsoid with eccentricities LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath .
Suppose that LaTeXMLEquation .
LaTeXMLEquation for some LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLEquation .
Application of Lemma 7 immediately yields LaTeXMLMath since by Lemma 5 LaTeXMLMath is a universal constant and LaTeXMLMath .
This completes the proof of Theorem 4 .
We must show that LaTeXMLEquation with LaTeXMLMath independent of LaTeXMLMath and LaTeXMLMath .
Our plan is as follows .
We will first show that if either LaTeXMLMath , or LaTeXMLMath , then LaTeXMLMath .
If LaTeXMLMath , we will show that LaTeXMLMath .
This will complete the proof since LaTeXMLMath if LaTeXMLMath .
Going into polar coordinates and applying stationary phase , we get LaTeXMLEquation .
Since the Gaussian curvature on LaTeXMLMath does not vanish , it is a classical result that LaTeXMLEquation .
It follows that LaTeXMLMath if either LaTeXMLMath or LaTeXMLMath .
If LaTeXMLMath , let LaTeXMLMath .
Since LaTeXMLMath is convex , it follows that LaTeXMLMath .
Since LaTeXMLMath , it follows by the Van Der Corput Lemma that the expression in LaTeXMLMath is bounded by LaTeXMLMath .
The estimate for LaTeXMLMath follows in the same way if we observe that the derivative with respect to LaTeXMLMath brings down a factor of LaTeXMLMath , and LaTeXMLMath .
This completes the proof of Lemma 5 if LaTeXMLMath .
To prove the three dimensional case we go into polar coordinates , integrate in the angular variables and use the well known asymptotics for the Fourier transform of the Lebesgue measure on the circle to obtain LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is a symbol of order LaTeXMLMath , LaTeXMLMath is as above , and LaTeXMLMath .
Let LaTeXMLEquation so the integral in LaTeXMLMath becomes LaTeXMLEquation .
Integrating by parts we get LaTeXMLEquation .
Let LaTeXMLMath be defined by the relation LaTeXMLMath .
We have LaTeXMLMath .
If LaTeXMLMath this quantity is bounded below by LaTeXMLMath and the Van der Corput lemma gives the decay LaTeXMLMath for LaTeXMLMath .
Using the fact LaTeXMLMath is a symbol of order LaTeXMLMath we see that LaTeXMLMath is bounded by LaTeXMLMath , LaTeXMLMath large .
This handles the case LaTeXMLMath and LaTeXMLMath .
On the other hand , LaTeXMLMath .
Split up the integral that defines LaTeXMLMath into two pieces : LaTeXMLMath and LaTeXMLMath .
The second integral was just handled above .
In the first integral LaTeXMLMath .
The Van der Corput Lemma yields decay LaTeXMLMath .
Taking the properties of the symbol LaTeXMLMath into account , as before , we get the decay LaTeXMLMath .
This takes care of the case LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath and the Van der Corput lemma yields the decay LaTeXMLMath for LaTeXMLMath .
This completes the proof of the three dimensional case .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Theorem 4 yields boundedness for LaTeXMLMath .
This is sharp by Theorem 1 .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then Theorem 4 yields boundedness for LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
Then Theorem 4 tells us that the maximal operator is bounded if LaTeXMLMath .
Testing LaTeXMLMath against LaTeXMLMath , where LaTeXMLMath is the characteristic function of the ball of radius LaTeXMLMath centered at the origin , shows that this result is sharp .
The same procedure establishes sharpness of the estimate given in Example 2 .
In fact , testing LaTeXMLMath against LaTeXMLMath shows that the summation condition of Theorem 4 is pretty close to being sharp .
It is not hard to see that , at least up to a LaTeXMLMath factor , LaTeXMLMath bounded on LaTeXMLMath implies that LaTeXMLEquation .
This would literally follow , without the log factor , from the proof of Theorem 2 of LaTeXMLCite if we assumed , in addition , that LaTeXMLMath , for every LaTeXMLMath .
The condition LaTeXMLMath is equivalent ( after making a change of variables and going into polar coordinates ) to LaTeXMLEquation .
The expression LaTeXMLMath is equivalent to the summation condition of Theorem 4 if LaTeXMLMath does not vanish to infinite order .
If LaTeXMLMath vanishes to infinite order , the two condition are still often equivalent , as in the Example 3 above .
It would be interesting to extend the results of this paper to a more general class of hypersurfaces .
For example , one could consider hypersurfaces of the form LaTeXMLMath where LaTeXMLMath is as above and LaTeXMLMath is a smooth convex finite type function .
Some recent results ( see e.g .
LaTeXMLCite , LaTeXMLCite , and LaTeXMLCite ) suggest that such an analysis should be possible .
We shall address this issue in a subsequent paper ( LaTeXMLCite .
More generally , a bigger challange would be to consider a hypersurface of the form LaTeXMLMath , where LaTeXMLMath is a smooth function of LaTeXMLMath variables that vanishes of infinite order at the origin .
At the moment , obtaining sharp Orlicz estimates , even in the case where the determinant of the Hessian matrix of LaTeXMLMath only vanishes at the origin , does not seem accessible .
We investigate when the fundamental group of the smooth part of a LaTeXMLMath surface or Enriques surface with Du Val singularities , is finite .
As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold .
This result supports Conjecture A below , while Conjecture A ( or alternatively the rational-connectedness conjecture in [ KoMiMo ] which is still open when the dimension is at least 4 ) would imply that every log terminal Fano variety has a finite fundamental group .
Introduction We work over the complex numbers field LaTeXMLMath .
In this note , we consider the topological fundamental group LaTeXMLMath of the smooth part LaTeXMLMath of a normal projective variety LaTeXMLMath .
In general , it is difficult to calculate such groups .
Even in surface case , we still do not know whether there is a plane curve LaTeXMLMath such that the group LaTeXMLMath is non-residually finite ; we note also that only in 1993 , D. Toledo constructed the first example of compact complex algebraic variety with non-residually finite fundamental group , which answered a question of J. P. Serre .
In the present paper , the algebraic variety LaTeXMLMath is assumed to be either a LaTeXMLMath surface , or an Enriques surface or a Q -Fano 3-fold , which has at worst log terminal singularities .
We will see from Theorem 3 and its proof that LaTeXMLMath of these three different objects are closely inter-related .
First , let LaTeXMLMath be a LaTeXMLMath surface with at worst Du Val singularities ( which is certainly log terminal ; see [ Ka ] ) .
Then LaTeXMLMath is still simply connected ( cf .
[ Ko1 , Theorem 7.8 ] ) .
By [ Ni1 , Theorem 1 ] , the number LaTeXMLMath is bounded by 16 , and if LaTeXMLMath then LaTeXMLMath is infinite ( cf .
Remark 1.4 ) .
Recently , Barth [ B1 ] has extended this result in the following way : if each point in LaTeXMLMath is of Dynkin type LaTeXMLMath ( LaTeXMLMath ) then LaTeXMLMath and in the case LaTeXMLMath , LaTeXMLMath is infinite .
Our Theorem 1 below also implies that the condition LaTeXMLMath ( resp .
LaTeXMLMath ) in the result of Nikulin ( resp .
Barth ) is actually necessary and sufficient for LaTeXMLMath to be infinite ( see Theorem 1 below for the precise statement ) .
A similar result is obtained for the fundamental group LaTeXMLMath of the smooth part of an Enriques surface with at worst Du Val singularities ( Theorem 2 ) .
In contrast with the K3 case , LaTeXMLMath may not be abelian and may not be LaTeXMLMath -elementary in the abelian case .
One motivation behind this note is Theorem 3 below in connection with the study of higher dimensional geometry and an attempt to solve the conjectures below .
In what follows , a normal variety LaTeXMLMath with at worst log terminal singularities is Q -Fano if , by definition , the anti-canonical divisor LaTeXMLMath is Q -Cartier and ample .
Conjecture A .
Let LaTeXMLMath be a Q -Fano LaTeXMLMath -fold .
Then the topological fundamental group LaTeXMLMath of the smooth part LaTeXMLMath of LaTeXMLMath is finite .
Conjecture B .
Let LaTeXMLMath be a Q -Fano LaTeXMLMath -fold .
Then the topological fundamental group LaTeXMLMath is finite .
Conjecture C. Let LaTeXMLMath be a Q -Fano LaTeXMLMath -fold .
Then LaTeXMLMath is rational-connected .
Here LaTeXMLMath is rational-connected , if any two general points of LaTeXMLMath can be connected by a single irreducible rational curve .
Clearly , Conjecture A implies Conjecture B. Conjecture A was proposed in [ Z1 ] and was answered in affirmative when the Fano index of LaTeXMLMath is greater than LaTeXMLMath .
When LaTeXMLMath , Conjecture A was proved to be true in [ GZ1,2 ] or [ Z2 ] ( see [ FKL ] and [ KM ] for new proofs ; see also [ Z3 ] ) .
Conjecture C implies Conjecture B [ C , Ko1 ] .
Conjecture C has been proved when LaTeXMLMath [ C , KoMiMo ] , but it is still open when LaTeXMLMath .
Our Theorem 3 below is a support towards Conjecture A .
Now we state our Theorem 1 .
Let LaTeXMLMath be a LaTeXMLMath surface with Du Val singularities .
Let LaTeXMLMath be a minimal resolution , LaTeXMLMath the reduced exceptional divisor of LaTeXMLMath and LaTeXMLMath the sublattice of LaTeXMLMath generated by the cohomology classes of irreducible components of LaTeXMLMath .
The universal covering map LaTeXMLMath can be extended to a morphism ( LaTeXMLMath ) LaTeXMLMath such that LaTeXMLMath ; indeed , if LaTeXMLMath is finite , then LaTeXMLMath is the normalization of LaTeXMLMath in the function field LaTeXMLMath ; if LaTeXMLMath is infinite , LaTeXMLMath is given in Theorem 1 ( 3 ) .
Theorem 1 .
Let LaTeXMLMath be a prime number and LaTeXMLMath a LaTeXMLMath surface with LaTeXMLMath ( LaTeXMLMath ) singularities of type LaTeXMLMath i.e. , type LaTeXMLMath and no other singularities .
Then one of the LaTeXMLMath rows in Table LaTeXMLMath occurs ; each of these LaTeXMLMath rows is realized by a concrete example .
Table LaTeXMLMath shows precisely the topological fundamental group LaTeXMLMath and LaTeXMLMath ; in particular , we have : ( 1 ) LaTeXMLMath ; if LaTeXMLMath then LaTeXMLMath .
( 2 ) Suppose that LaTeXMLMath is finite .
Then LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath LaTeXMLMath a compactification of the universal cover of LaTeXMLMath is a LaTeXMLMath surface with at worst several type LaTeXMLMath singularities .
( 3 ) Suppose that LaTeXMLMath is infinite .
Then LaTeXMLMath or LaTeXMLMath , and there is a LaTeXMLMath -Galois cover LaTeXMLMath unramified over LaTeXMLMath such that LaTeXMLMath ( LaTeXMLMath a lattice LaTeXMLMath is an abelian surface .
Hence we have an exact sequence : LaTeXMLEquation .
The composition LaTeXMLMath of the natural morphisms LaTeXMLMath , restricted over LaTeXMLMath , is the universal covering map of LaTeXMLMath .
Our next theorem utilizes Theorem 1 but needs some lattice-theoretical arguments to determine the group structure of the fundamental group .
Theorem 2 .
Let LaTeXMLMath be a prime number .
Let LaTeXMLMath be an Enriques surface containing a configuration of smooth rational curves of Dynkin type LaTeXMLMath ( the direct sum of LaTeXMLMath copies of LaTeXMLMath ) , and let LaTeXMLMath be the surface with these LaTeXMLMath curves on LaTeXMLMath removed .
Then one of the LaTeXMLMath rows in Table LaTeXMLMath occurs ; in particular , LaTeXMLMath is soluble and it is infinite if and only if LaTeXMLMath .
Though the realization of the LaTeXMLMath rows in Table LaTeXMLMath are unknown yet , each of the remaining LaTeXMLMath rows in Table LaTeXMLMath is realized by a concrete example .
The following is an application of Theorems 1 and 2 and a partial answer to Conjecture A above .
Theorem 3 .
Let LaTeXMLMath be a prime number .
Let LaTeXMLMath be a Fano LaTeXMLMath -fold with a Cartier divisor LaTeXMLMath such that LaTeXMLMath is linearly equivalent to zero for LaTeXMLMath or LaTeXMLMath .
Suppose that a member LaTeXMLMath of LaTeXMLMath is irreducible normal and has LaTeXMLMath singularities of type LaTeXMLMath and no other singularities .
Then the fundamental group LaTeXMLMath of the smooth part LaTeXMLMath of LaTeXMLMath is the image of a group in Table LaTeXMLMath or LaTeXMLMath .
In particular , LaTeXMLMath is soluble ; and if LaTeXMLMath , then LaTeXMLMath for some LaTeXMLMath .
Remark 4 .
( 1 ) On a Q -Fano 3-fold LaTeXMLMath , a relation LaTeXMLMath with LaTeXMLMath a Cartier divisor occurs when LaTeXMLMath has Fano index 1 and Cartier index LaTeXMLMath .
It is conjectured that in this situation LaTeXMLMath .
This conjecture is confirmed by T. Sano [ Sa ] under the stronger condition that LaTeXMLMath has at worst terminal cyclic quotient singularities .
On the other hand , a result of Minagawa [ Mi ] shows that any terminal Q -Fano 3-fold of Fano index 1 can be deformed to a Q -Fano 3-fold of Cartier index 1 , 2 .
( 2 ) By Ambro [ A , Main Theorem ] , a general member of LaTeXMLMath is normal irreducible and has at worst log terminal singularities ; so LaTeXMLMath has at worst Du Val or type LaTeXMLMath or type LaTeXMLMath singularities since LaTeXMLMath ( cf .
the proof of Theorem 3 ) , whence the condition on Sing LaTeXMLMath in Theorem 3 is quite reasonable .
By the proof in §4 , we always have a surjective homomorphism LaTeXMLMath , where LaTeXMLMath Sing LaTeXMLMath .
In [ SZ ] , a sufficient condition for LaTeXMLMath to be finite is given when LaTeXMLMath .
( 3 ) The author has not been able to construct an example of LaTeXMLMath in Theorem 3 satisfying LaTeXMLMath or LaTeXMLMath .
Theorems 1 , 2 and 3 are proved respectively in §2 , §3 and §4 .
Acknowledgment .
Part of this work was done when the second author was visiting University of Missouri and University of Michigan in 1999 .
He would like to thank both institutions for their hospitality .
This work was finalized during the first author ’ s visit to Singapore under the Sprint programme of the Department of Mathematics .
He thanks NUS for the financial support and the hospitality .
Both authors are very thankful to Professor I. Shimada for the interest in this paper who actually found an alternative shorter proof of the Table 1 using computer .
They also thank Professor R. V. Gurjar for his crucial help in proving Theorem 3 .
This work was partially financed by an Academic Research Fund of National University of Singapore .
§1 .
The LaTeXMLMath case with LaTeXMLMath 1.1 .
We will frequently and implicitly use the following observation : Let LaTeXMLMath be as in Theorem 1 .
Then there is a Galois LaTeXMLMath -cover LaTeXMLMath ramified exactly over a LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath if and only if there is a relation LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is a Cartier divisor and LaTeXMLMath is an effective Cartier divisor with support equal to LaTeXMLMath and coefficients in LaTeXMLMath coprime to LaTeXMLMath .
Indeed , assuming the above equivalent conditions , one note that LaTeXMLMath is a linear chain of LaTeXMLMath -curves and can check that LaTeXMLMath for some integer LaTeXMLMath coprime to LaTeXMLMath .
Moreover , one has LaTeXMLMath with LaTeXMLMath the LaTeXMLMath -image of LaTeXMLMath , and LaTeXMLMath is given by : LaTeXMLEquation .
Lemma .
( 1 ) Let LaTeXMLMath and let LaTeXMLMath be a minimal resolution of singular points not in LaTeXMLMath .
Then LaTeXMLMath , where LaTeXMLMath .
( 2 ) One has LaTeXMLMath for some LaTeXMLMath , where for a sublattice LaTeXMLMath of LaTeXMLMath , we denote by LaTeXMLMath its primitive closure .
( 3 ) LaTeXMLMath is primitive in LaTeXMLMath LaTeXMLMath LaTeXMLMath is a perfect group LaTeXMLMath LaTeXMLMath does not include any LaTeXMLMath -divisible subset ( cf .
LaTeXMLMath below ) .
Proof .
( 1 ) follows from [ Ko1 , Theorem 7.8 ] since LaTeXMLMath has at worst log terminal singularities .
The first isomorphism in ( 2 ) follows from the proof of [ X , Lemma 2 ] , while the second follows from the assumption on LaTeXMLMath .
( 3 ) is a consequence of ( 1 ) and 1.1 .
Definition and Remark 1.2 .
Let LaTeXMLMath be as in Theorem LaTeXMLMath .
A subset LaTeXMLMath of LaTeXMLMath is LaTeXMLMath - divisible if there is a Galois LaTeXMLMath -cover LaTeXMLMath ramified exactly over LaTeXMLMath ( cf .
1.1 ) .
When LaTeXMLMath , LaTeXMLMath is LaTeXMLMath -divisible if and only if LaTeXMLMath is LaTeXMLMath -divisible in the lattice LaTeXMLMath ( see also 3.8 ) .
Lemma 1.3 .
( cf .
[ Ni1 , Lemma 3 ] ) Let LaTeXMLMath be as in Theorem LaTeXMLMath .
Suppose that there is a Galois LaTeXMLMath -cover LaTeXMLMath , ramified exactly over LaTeXMLMath LaTeXMLMath i.e. , LaTeXMLMath is LaTeXMLMath -divisible LaTeXMLMath .
Then LaTeXMLMath fits one of the following cases : LaTeXMLEquation .
Moreover , if LaTeXMLMath , then LaTeXMLMath is an abelian surface and hence LaTeXMLEquation if LaTeXMLMath fits one of the remaining LaTeXMLMath cases , then LaTeXMLMath is a ( smooth ) LaTeXMLMath surface and hence LaTeXMLMath .
Proof .
By the assumption , for each singular point LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is a single point and LaTeXMLMath is smooth .
Now LaTeXMLMath implies that LaTeXMLMath , whence LaTeXMLMath is either abelian with Euler number LaTeXMLMath or LaTeXMLMath with LaTeXMLMath .
The lemma follows from the calculation ( noting that K3 surfaces are simply connected ) : LaTeXMLEquation .
Remark 1.4 .
There is a converse to Lemma 1.3 by [ Ni1 ] and [ B1 ] .
Suppose that LaTeXMLMath is a LaTeXMLMath surface with LaTeXMLMath where LaTeXMLMath ( resp .
LaTeXMLMath where LaTeXMLMath ) .
Then LaTeXMLMath ( resp .
LaTeXMLMath ) and Sing LaTeXMLMath is LaTeXMLMath -divisible with LaTeXMLMath ( resp .
LaTeXMLMath ) ; so there is a Galois LaTeXMLMath -cover LaTeXMLMath unramified over LaTeXMLMath so that LaTeXMLMath is an abelian surface .
In particular , LaTeXMLMath is infinite soluble and all assertions in Theorem 1 ( 3 ) hold .
When LaTeXMLMath , the covering involution of LaTeXMLMath coincides with LaTeXMLMath .
Lemma 1.5 .
Let LaTeXMLMath be as in Theorem LaTeXMLMath with LaTeXMLMath .
Suppose that LaTeXMLMath are two distinct LaTeXMLMath -divisible LaTeXMLMath -point subsets of LaTeXMLMath .
Then either LaTeXMLMath and LaTeXMLMath , or LaTeXMLMath and LaTeXMLMath .
Proof .
By 1.1 , one has LaTeXMLMath , where LaTeXMLMath .
Then LaTeXMLMath .
Now the lemma follows from 1.1 and Lemma 1.3 ( noting that LaTeXMLMath always holds by Remark 1.4 ) .
Lemma 1.6 .
Let LaTeXMLMath be as in Theorem LaTeXMLMath with LaTeXMLMath .
( 1 ) Suppose that LaTeXMLMath .
Then there are LaTeXMLMath -divisible LaTeXMLMath -point subsets LaTeXMLMath of LaTeXMLMath with LaTeXMLMath .
( 2 ) Suppose that LaTeXMLMath .
Then there is a LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath includes only one LaTeXMLMath -divisible subset LaTeXMLMath .
( 3 ) Suppose that LaTeXMLMath .
Then there is an LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath does not include any LaTeXMLMath -divisible subset .
Proof .
In view of Lemma 1.5 , for ( 1 ) , it suffices to show that LaTeXMLMath includes two distinct LaTeXMLMath -divisible ( 8-point ) subsets .
By the proof of [ Ni1 , Lemma 4 ] or [ B1 , Lemma 2 ] , LaTeXMLMath includes a LaTeXMLMath -divisible 8-point subset LaTeXMLMath .
The same reasoning shows that any 12-point set consisting of 7 points in LaTeXMLMath and the LaTeXMLMath singular points of LaTeXMLMath not in LaTeXMLMath , includes a LaTeXMLMath -divisible 8-point subset LaTeXMLMath ( LaTeXMLMath ) .
( 1 ) is proved .
For ( 2 ) , applying ( 1 ) , we get LaTeXMLMath -divisible 8-point subsets LaTeXMLMath of LaTeXMLMath with LaTeXMLMath .
Take two singular points LaTeXMLMath of LaTeXMLMath not in LaTeXMLMath , and one point LaTeXMLMath in LaTeXMLMath but not in LaTeXMLMath .
Applying Lemma 1.5 , we see that we can take LaTeXMLMath as LaTeXMLMath , for LaTeXMLMath or LaTeXMLMath .
For ( 3 ) , we let LaTeXMLMath be any subset of LaTeXMLMath in ( 2 ) containing not more than 7 points of LaTeXMLMath .
1.7 .
Let LaTeXMLMath be an abelian surface with LaTeXMLMath the involution LaTeXMLMath .
Denote by LaTeXMLMath the set of the 16 LaTeXMLMath -fixed points , which is a subgroup of LaTeXMLMath consisting of the 2-torsion points .
One can regard LaTeXMLMath as a LaTeXMLMath -dimensional vector space over the field LaTeXMLMath .
The quotient LaTeXMLMath is a LaTeXMLMath surface with 16 singularities LaTeXMLMath of Dynkin type LaTeXMLMath dominated by the points in LaTeXMLMath .
The bijection LaTeXMLMath defines on the latter a LaTeXMLMath -dimensional LaTeXMLMath -vector space structure .
One sees easily that LaTeXMLMath and LaTeXMLMath is generated by the involution LaTeXMLMath and LaTeXMLMath .
Suppose that LaTeXMLMath is a LaTeXMLMath -divisible 8-point subset of LaTeXMLMath and LaTeXMLMath the corresponding LaTeXMLMath -cover ramified exactly over LaTeXMLMath .
Then each singular point of LaTeXMLMath not in LaTeXMLMath splits into two type LaTeXMLMath singularities of LaTeXMLMath and these 16 points form the singular locus LaTeXMLMath .
So LaTeXMLMath with LaTeXMLMath an abelian surface ( Remark 1.4 ) , and LaTeXMLMath is generated by the involution LaTeXMLMath and LaTeXMLMath .
The covering LaTeXMLMath induces LaTeXMLMath .
One can verify that LaTeXMLMath is an index-2 sublattice of LaTeXMLMath .
This way , we obtain a commutative diagram : LaTeXMLEquation .
Note that LaTeXMLMath ( and hence LaTeXMLMath ) is a rank-3 linear map between LaTeXMLMath -vector spaces of dimension 4 .
Lemma 1.8 .
Let LaTeXMLMath be a Kummer surface .
Then we have : ( 1 ) An LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath is LaTeXMLMath -divisible if and only if it is an affine hyperplane of the LaTeXMLMath -vector space LaTeXMLMath .
( 2 ) For both LaTeXMLMath , there is a LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath is equal to LaTeXMLMath ( resp .
LaTeXMLMath ) when LaTeXMLMath ( resp .
LaTeXMLMath ) , where LaTeXMLMath is a minimal resolution of singularities not in LaTeXMLMath and LaTeXMLMath is the smooth part of LaTeXMLMath .
Proof .
( 1 ) follows from [ Ni1 , Cor .
5 and Remark 1 ] .
( 2 ) Let LaTeXMLMath be LaTeXMLMath -divisible 8-point subsets of LaTeXMLMath with LaTeXMLMath ( Lemma 1.6 ) .
As in 1.7 , let LaTeXMLMath be the double cover ramified exactly over LaTeXMLMath .
Then one can verify that LaTeXMLMath LaTeXMLMath is an affine hyperplane of LaTeXMLMath and hence the group LaTeXMLMath equals LaTeXMLMath ( see the proof of Lemma 1.3 ) .
The covering map LaTeXMLMath implies that this group is an index-2 subgroup of LaTeXMLMath where LaTeXMLMath .
Hence the group LaTeXMLMath has order 4 ; since LaTeXMLMath has no type LaTeXMLMath singularity , this group equals LaTeXMLMath ( cf .
[ X , Theorem 3 ] ) .
One has LaTeXMLMath by Lemma 1.1 .
By Lemma 1.6 , we can find a 12-point subset LaTeXMLMath of LaTeXMLMath so that LaTeXMLMath contains only one LaTeXMLMath -divisible 8-point subset LaTeXMLMath .
As in 1.7 , let LaTeXMLMath be the double cover ramified exactly over LaTeXMLMath .
Let LaTeXMLMath be a minimal resolution with LaTeXMLMath , a disjoint union of 16 smooth rational curves .
The covering map LaTeXMLMath implies that LaTeXMLMath has the ( trivial ) group in ( iii ) below as an index-2 subgroup .
So ( 2 ) is reduced to the proof of the claim below .
Claim 1.8.1 .
( i ) The 8-point set LaTeXMLMath is not an affine hyperplane of LaTeXMLMath , and hence does not include any LaTeXMLMath -divisible set .
( ii ) The fundamental group of LaTeXMLMath with the 8 points in ( 1 ) removed , is trivial .
If the first assertion of the claim is false , then the 8-point set would be an affine hyperplane and hence its LaTeXMLMath -image is contained in an affine hyperplane LaTeXMLMath of LaTeXMLMath which has to consist of the 4 points LaTeXMLMath and 4 points in LaTeXMLMath , and LaTeXMLMath would include two distinct LaTeXMLMath -divisible subsets LaTeXMLMath , a contradiction .
By ( i ) and Lemma 1.1 , the group in ( ii ) is perfect .
Moreover , this group is soluble and hence trivial because it is the image of LaTeXMLMath while the latter is soluble [ Remark 1.4 ] .
This proves the claim and also the lemma .
Lemma 1.9 .
Let LaTeXMLMath be as in Theorem LaTeXMLMath with LaTeXMLMath .
Suppose that LaTeXMLMath is primitive in LaTeXMLMath ( this is true if LaTeXMLMath ; see Lemmas LaTeXMLMath and LaTeXMLMath ) .
Then LaTeXMLMath and LaTeXMLMath ( each LaTeXMLMath is realizable ) .
Proof .
By the proof of [ Ni1 , Lemma 4 ] or [ B1 , Lemma 3 ] , the primitivity of LaTeXMLMath implies that LaTeXMLMath .
¿From [ Ni2 , Theorem 1.14.4 ] and its remark , one deduces that there is a unique primitive embedding of LaTeXMLMath into the K3 lattice .
Now by the connectivity theorem [ Ni3 , Theorem 2.10 ] , we are reduced to show the lemma for any particular LaTeXMLMath satisfying the same condition of the lemma .
Let LaTeXMLMath be a ( smooth ) Kummer surface with 16 disjoint smooth rational curves .
By Lemmas 1.1 and 1.6 , among these 16 , there are 11 curves LaTeXMLMath ( LaTeXMLMath ) such that if LaTeXMLMath is the contraction of LaTeXMLMath ( LaTeXMLMath ) then LaTeXMLMath does not include any LaTeXMLMath -divisible subsets .
Thus LaTeXMLMath as in the proof of Claim 1.8.1 .
The lemma is proved .
Proposition 1.10 .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath be as in Theorem LaTeXMLMath with LaTeXMLMath .
( 1 ) If LaTeXMLMath , then LaTeXMLMath equals LaTeXMLMath or LaTeXMLMath ( both are realizable ; cf .
Lemma LaTeXMLMath ) .
( 2 ) Suppose that LaTeXMLMath and LaTeXMLMath is non-primitive in LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath ( each LaTeXMLMath is realizable ) .
( 3 ) If LaTeXMLMath , then LaTeXMLMath equals LaTeXMLMath .
( 4 ) If LaTeXMLMath , then LaTeXMLMath equals LaTeXMLMath .
( 5 ) If LaTeXMLMath , then LaTeXMLMath equals LaTeXMLMath .
Proof .
( 1 ) By the proof of [ Ni1 , Lemma 4 ] or [ B1 , Lemma 3 ] , LaTeXMLMath is not primitive .
So there is a double cover LaTeXMLMath ramified exactly at an 8-point subset LaTeXMLMath of LaTeXMLMath .
One has LaTeXMLMath , consisting of 8 singular points of type LaTeXMLMath .
If LaTeXMLMath is not LaTeXMLMath -divisible , then the condition in Lemma 1.9 is satisfied ( Lemma 1.1 ) , whence LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is LaTeXMLMath -divisible then LaTeXMLMath as in Lemma 1.8 .
( 2 ) follows from Lemma 1.3 and the arguments in ( 3 ) .
For the realization of each LaTeXMLMath , we let LaTeXMLMath be any affine hyperplane of LaTeXMLMath ( cf .
1.7 ) and LaTeXMLMath a minimal resolution of any LaTeXMLMath points not in LaTeXMLMath .
( 3 ) By Lemma 1.6 , there are two 8-point subsets LaTeXMLMath of LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be the double cover ramified exactly over LaTeXMLMath .
Then LaTeXMLMath is a LaTeXMLMath -divisible 8-point subset of LaTeXMLMath ; to see this , we apply 1.1 , pull back the relation on LaTeXMLMath arising from the LaTeXMLMath -divisible set LaTeXMLMath to a relation on a minimal resolution of LaTeXMLMath and apply 1.1 again .
Note that LaTeXMLMath consists of 10 points of type LaTeXMLMath .
Let LaTeXMLMath be the double cover ramified exactly over LaTeXMLMath .
Then LaTeXMLMath consists of 4 points of type LaTeXMLMath .
So LaTeXMLMath by Lemma 1.9 .
Thus LaTeXMLMath so that LaTeXMLMath .
Since LaTeXMLMath has at worst type LaTeXMLMath singularities , LaTeXMLMath ( cf .
[ X , Theorem 3 ] ) .
( 4 ) LaTeXMLMath implies that LaTeXMLMath with LaTeXMLMath ( Lemmas 1.6 and 1.9 ) .
Let LaTeXMLMath be the double cover ramified exactly over LaTeXMLMath .
Set LaTeXMLMath ( LaTeXMLMath ) .
Then LaTeXMLMath .
As in ( 3 ) , LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -divisible .
Hence LaTeXMLMath as in the proof of Lemma 1.8 .
So LaTeXMLMath by the same reasoning as in ( 3 ) .
( 5 ) Let LaTeXMLMath be a LaTeXMLMath -divisible 8-point subset of LaTeXMLMath ( Lemma 1.9 ) and let LaTeXMLMath be the double cover ramified exactly over LaTeXMLMath .
Then LaTeXMLMath consists of exactly 14 points of type LaTeXMLMath .
Now ( 5 ) follows from ( 4 ) and the reasoning in ( 3 ) .
§2 .
The LaTeXMLMath case with LaTeXMLMath We shall prove Theorem 1 at the end of the section .
We treat first the case LaTeXMLMath .
Let us start with : Example 2.1 .
For each LaTeXMLMath , we shall construct an example of LaTeXMLMath satisfying the conditions of Theorem 1 with LaTeXMLMath and LaTeXMLMath ; in particular , LaTeXMLMath is primitive in LaTeXMLMath .
It suffices to do for LaTeXMLMath .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and trivial Mordell Weil group LaTeXMLMath .
This is No.39 in [ MP , the Table ] or No.91 in [ SZ , Table 2 ] .
Clearly , LaTeXMLMath together with some fibre components form a divisor LaTeXMLMath of Dynkin type LaTeXMLMath .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
By [ No , Lemma 1.5 ] , if one lets LaTeXMLMath be a general fibre , then one has an exact sequence : LaTeXMLEquation .
Note that the first homomorphism above factors through LaTeXMLMath ( LaTeXMLMath ) where LaTeXMLMath is a fibre of type LaTeXMLMath .
Hence LaTeXMLMath is cyclic .
Since the group LaTeXMLMath is trivial and all fibres are reducible , the components of LaTeXMLMath form a partial LaTeXMLMath -basis of LaTeXMLMath and hence LaTeXMLMath is primitive in LaTeXMLMath .
Thus the group LaTeXMLMath is perfect ( Lemma 1.1 ) ; so it is trivial .
Example 2.2 .
Here is an example of LaTeXMLMath satisfying Theorem 1 with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and the Mordell Weil group LaTeXMLMath .
This is No.108 in [ MP , the Table ] or No.8 in [ SZ , Table 2 ] .
Write the 6 singular fibres as ( in natural ordering ) LaTeXMLEquation so that LaTeXMLMath meets components with index 0 .
Let LaTeXMLMath be a generator of the group LaTeXMLMath .
By the height pairing in [ Sh ] , one can verify that ( after relabeling ) LaTeXMLMath meets LaTeXMLMath and LaTeXMLMath meets LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLMath , which is of Dynkin type LaTeXMLMath .
Expressing LaTeXMLMath as a LaTeXMLMath -combination of LaTeXMLMath ( a general fibre ) and fibre components of index LaTeXMLMath , we get : LaTeXMLEquation .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
Denote by LaTeXMLMath the image on LaTeXMLMath of LaTeXMLMath .
Then LaTeXMLMath .
Let LaTeXMLMath be the canonical Galois LaTeXMLMath -cover unramified over LaTeXMLMath .
Note that LaTeXMLMath consists of 6 points of type LaTeXMLMath ( the preimages of LaTeXMLMath , LaTeXMLMath ) .
Let LaTeXMLMath be a minimal resolution with LaTeXMLMath the exceptional divisor .
Then the preimage on LaTeXMLMath of LaTeXMLMath is a disjoint union of LaTeXMLMath so that LaTeXMLMath .
If LaTeXMLMath is LaTeXMLMath -divisible , then as in 1.1 , we get a relation : LaTeXMLMath .
Intersecting this with LaTeXMLMath we see that LaTeXMLMath meets at least two components of LaTeXMLMath , a contradiction .
Hence LaTeXMLMath does not include any LaTeXMLMath -divisible subset ( cf .
Lemma 1.3 ) and hence LaTeXMLMath by Proposition 2.5 ( 1 ) below .
Thus LaTeXMLMath .
2.3 .
Let LaTeXMLMath be an abelian surface with an order-3 symplectic automorphism LaTeXMLMath so that LaTeXMLMath is a 9-point set .
Such an example is shown in [ BL ] .
Then LaTeXMLMath is a LaTeXMLMath surface with 9 singularities of type LaTeXMLMath .
Lemma 2.4 .
Let LaTeXMLMath be as in LaTeXMLMath .
In the following , we let LaTeXMLMath be a minimal resolution of singularities not in LaTeXMLMath and LaTeXMLMath .
( 1 ) For each LaTeXMLMath , there is a LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath .
( 2 ) There is an LaTeXMLMath -point subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath .
Proof .
By [ B1 , Claim 2 in §4 ] : “ each pair of points lie on a unique line ” , which means that each 7-point subset of LaTeXMLMath includes a unique LaTeXMLMath -divisible subset 6-point subset LaTeXMLMath .
Let LaTeXMLMath ( resp .
LaTeXMLMath with LaTeXMLMath a singular point of LaTeXMLMath not in LaTeXMLMath ) when LaTeXMLMath ( resp .
LaTeXMLMath ) .
Let LaTeXMLMath be the Galois LaTeXMLMath -cover ramified exactly over LaTeXMLMath .
Now LaTeXMLMath consists of LaTeXMLMath points of type LaTeXMLMath and hence does not include any LaTeXMLMath -divisible subsets ( Lemma 1.3 ) .
So the group LaTeXMLMath is perfect ( Lemma 1.1 ) .
We have also LaTeXMLMath .
Now the group LaTeXMLMath is trivial because it is also soluble being the subgroup of LaTeXMLMath , while the latter is the image of the soluble group LaTeXMLMath [ Remark 1.4 ] .
This proves ( 1 ) .
( 2 ) Let LaTeXMLMath be an 8-point subset of LaTeXMLMath including two 6-point subsets LaTeXMLMath with LaTeXMLMath ( in notation of [ B1 ] , the lines determined by LaTeXMLMath have a unique common point ) .
Now ( 2 ) is similar to Lemma 1.8 ( 2 ) or Proposition 1.10 ( 3 ) .
Proposition 2.5 .
Let LaTeXMLMath be as in Theorem LaTeXMLMath with LaTeXMLMath .
( 1 ) Suppose that LaTeXMLMath is primitive in LaTeXMLMath , i.e. , LaTeXMLMath does not include any LaTeXMLMath -divisible subset ( this is true if LaTeXMLMath ) .
Then LaTeXMLMath and LaTeXMLMath ( each LaTeXMLMath is realizable by Example LaTeXMLMath ) .
( 2 ) Suppose that LaTeXMLMath and LaTeXMLMath is non-primitive in LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath ( both LaTeXMLMath are realizable by Lemma LaTeXMLMath ) .
( 3 ) Suppose that LaTeXMLMath .
Then LaTeXMLMath equals LaTeXMLMath , or LaTeXMLMath ( both groups are realizable by Example LaTeXMLMath and Lemma LaTeXMLMath ) .
Proof .
( 1 ) As in [ B1 , Lemma 3 ] , the primitivity of LaTeXMLMath implies that LaTeXMLMath .
Now as in Lemma 1.9 , we are reduced to show LaTeXMLMath for a particular LaTeXMLMath satisfying Theorem 1 with LaTeXMLMath and LaTeXMLMath .
So just let LaTeXMLMath be the one constructed in Example 2.1 , and ( 1 ) is proved .
( 2 ) By 1.1 and Lemma 1.3 , one has LaTeXMLMath , and there is a LaTeXMLMath -divisible 6-point subset LaTeXMLMath of LaTeXMLMath and a corresponding Galois LaTeXMLMath -cover LaTeXMLMath ramified exactly over LaTeXMLMath .
Now LaTeXMLMath consists of LaTeXMLMath points of type LaTeXMLMath and hence does not include any LaTeXMLMath -divisible subsets ( Lemma 1.3 ) .
Thus LaTeXMLMath by ( 1 ) , whence LaTeXMLMath .
( 3 ) This is similar to Proposition 1.10 ( applying ( 1 ) ) .
Next we consider the case LaTeXMLMath .
We begin with examples .
Example 2.6 .
( 1 ) For each LaTeXMLMath , we construct an example LaTeXMLMath satisfying the conditions in Theorem 1 with LaTeXMLMath and LaTeXMLMath ; in particular , LaTeXMLMath is primitive in LaTeXMLMath .
It suffices to construct an LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and trivial Mordell Weil group LaTeXMLMath .
This is No.64 in [ MP , the Table ] or No.9 in [ SZ , Table 2 ] .
Clearly , some fibre components form a divisor LaTeXMLMath of Dynkin type LaTeXMLMath .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
Then as in Example 2.1 , one has LaTeXMLMath .
( 2 ) For each LaTeXMLMath , we construct an example LaTeXMLMath satisfying the conditions in Theorem 1 with LaTeXMLMath and LaTeXMLMath ; in particular , LaTeXMLMath is primitive in LaTeXMLMath .
It suffices to construct an LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and trivial Mordell Weil group .
This is No.29 in [ MP , the Table ] or No.41 in [ SZ , Table 2 ] .
Clearly , LaTeXMLMath and some fibre components form a divisor LaTeXMLMath of Dynkin type LaTeXMLMath .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
Then as in Example 2.1 , one has LaTeXMLMath .
Example 2.7 .
( 1 ) We construct an example LaTeXMLMath satisfying the conditions in Theorem 1 with LaTeXMLMath and LaTeXMLMath .
Also see Remark 3.3 for another construction .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and the Mordell Weil group LaTeXMLMath .
This is No.9 in [ MP , the Table ] or No.54 in [ SZ , Table 2 ] .
Write the type LaTeXMLMath singular fibres as ( in natural ordering ) LaTeXMLEquation so that LaTeXMLMath meets LaTeXMLMath .
Then a generator LaTeXMLMath of the group LaTeXMLMath meets LaTeXMLMath after relabeling .
Let LaTeXMLMath .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
As in Example 2.2 , we can verify that LaTeXMLEquation and proceed as there to obtain LaTeXMLMath .
( 2 ) We construct an example LaTeXMLMath satisfying the conditions in Theorem 1 with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and the Mordell Weil group LaTeXMLMath .
This is No.30 in [ MP , the Table ] or No.13 in [ SZ , Table 2 ] .
Write the type LaTeXMLMath singular fibres as ( in natural ordering ) LaTeXMLEquation so that LaTeXMLMath meets LaTeXMLMath .
Then a generator LaTeXMLMath of the group LaTeXMLMath meets LaTeXMLMath after relabeling .
Let LaTeXMLMath .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
As in Example 2.2 , we can verify that LaTeXMLEquation and proceed as there to obtain LaTeXMLMath .
Example 2.8 .
For each LaTeXMLMath , we construct a LaTeXMLMath surface LaTeXMLMath so that LaTeXMLMath has a type LaTeXMLMath singularity as its only singularity and LaTeXMLMath .
It suffices to construct an LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be an elliptic LaTeXMLMath surface with a section LaTeXMLMath , singular fibres of type LaTeXMLMath and trivial Mordell Weil group .
This is No.1 in [ MP , the Table ] or No.112 in [ SZ , Table 2 ] .
Clearly , some fibre components form a divisor LaTeXMLMath of Dynkin type LaTeXMLMath .
Let LaTeXMLMath be the contraction of LaTeXMLMath .
Then as in Example 2.1 , one has LaTeXMLMath .
Proposition 2.9 .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath be as in Theorem LaTeXMLMath .
Then we have : ( 1 ) LaTeXMLMath ; if LaTeXMLMath then LaTeXMLMath ; if LaTeXMLMath then LaTeXMLMath ; if LaTeXMLMath then LaTeXMLMath .
( 2 ) Suppose that LaTeXMLMath and LaTeXMLMath is primitive in LaTeXMLMath ( this is true when LaTeXMLMath and LaTeXMLMath , or LaTeXMLMath and LaTeXMLMath ) .
Then LaTeXMLMath ( all LaTeXMLMath are realizable by Example LaTeXMLMath ) .
( 3 ) Suppose that LaTeXMLMath and LaTeXMLMath is non-primitive in LaTeXMLMath .
Then LaTeXMLMath equals LaTeXMLMath or LaTeXMLMath , and LaTeXMLMath ( both cases are realizable by Example LaTeXMLMath ) .
( 4 ) Suppose that LaTeXMLMath .
Then LaTeXMLMath ( all prime numbers LaTeXMLMath are realizable by Example LaTeXMLMath ) .
Proof .
( 1 ) follows from the calculation LaTeXMLMath .
As in Lemma 1.9 , the assertions ( 2 ) and ( 4 ) need to be verified only for a particular LaTeXMLMath in Example 2.6 or 2.8 , and hence are true .
( 3 ) By Lemma 1.1 , Sing LaTeXMLMath is LaTeXMLMath -divisible .
So ( 3 ) follows from Lemma 1.3 .
Now Theorem 1 in the introduction is a consequence of Remark 1.4 , Lemma 1.9 and Propositions 1.10 , 2.5 and 2.9 .
§3 .
The fundamental group of an open Enriques surface We shall prove Theorem 2 in the section .
Let LaTeXMLMath be an Enriques surface .
The second cohomology group LaTeXMLMath is isomorphic to LaTeXMLMath , where the torsion is the canonical class LaTeXMLMath .
The free part LaTeXMLMath admits a canonical structure of a lattice which is even , unimodular and of signature LaTeXMLMath and hence isomorphic to LaTeXMLMath , where LaTeXMLMath is the unimodular hyperbolic lattice of signature LaTeXMLMath , and LaTeXMLMath the negative definite lattice associated with the Dynkin diagram of type LaTeXMLMath .
Assume that LaTeXMLMath contains a configuration of smooth rational curves of Dynkin type LaTeXMLMath , where LaTeXMLMath is a prime .
Then the pair LaTeXMLMath is one of the following : LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Conversely , for each pair LaTeXMLMath in the above list , by considering various ellipic fibrations one can prove the existence of an Enriques surface with LaTeXMLMath singularities of type LaTeXMLMath ( see [ CD ] ) .
Suppose that an Enriques surface LaTeXMLMath contains a configuration of rational curves of Dynkin type LaTeXMLMath .
We fix the following notation : LaTeXMLMath = the open Enriques surface obtained by deleting those LaTeXMLMath rational curves from LaTeXMLMath .
LaTeXMLMath = the inverse of LaTeXMLMath in the LaTeXMLMath cover LaTeXMLMath of LaTeXMLMath .
Lemma 3.1 .
If LaTeXMLMath , then LaTeXMLMath .
Proof .
In this case LaTeXMLMath corresponds to the case LaTeXMLMath in Table 1 , so it is simply connected .
Lemma 3.2 .
( 1 ) If LaTeXMLMath , then LaTeXMLMath .
( 2 ) If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , or the dihedral group LaTeXMLMath of order LaTeXMLMath .
The first case occurs if the LaTeXMLMath on the LaTeXMLMath cover of LaTeXMLMath is primitive ; the second if the LaTeXMLMath on LaTeXMLMath is non-primitive ; the third if the LaTeXMLMath on LaTeXMLMath is primitive , while the LaTeXMLMath on the LaTeXMLMath cover is non-primitive .
All three cases occur .
( See Examples LaTeXMLMath below . )
Proof .
( 1 ) This case follows immediately from Table 1 .
( 2 ) Since LaTeXMLMath is an extension of LaTeXMLMath by LaTeXMLMath , we see from Table 1 that LaTeXMLMath , LaTeXMLMath , or the dihedral group LaTeXMLMath of order 10 .
The second group contains a normal subgroup of index 5 , and occurs as LaTeXMLMath only if there is a Galois covering of LaTeXMLMath of degree 5 , unramified over LaTeXMLMath .
The third group contains no normal subgroup of index 5 .
Lemma 3.3 .
Let LaTeXMLMath be an odd prime .
Let LaTeXMLMath be a divisor on an Enriques surface LaTeXMLMath .
Suppose that Pic LaTeXMLMath contains a subgroup LaTeXMLMath of finite index coprime to LaTeXMLMath , and that the intersection number of LaTeXMLMath with any element of LaTeXMLMath is a multiple of LaTeXMLMath .
Then LaTeXMLMath is LaTeXMLMath -divisible in Pic LaTeXMLMath .
Proof .
This follows from the unimodularity of Pic LaTeXMLMath / ( torsion ) and the LaTeXMLMath -divisibility of the 2-torsion LaTeXMLMath .
Examples 3.4 .
( 3.4.1 ) The case with LaTeXMLMath and LaTeXMLMath Let LaTeXMLMath be the Example IV from [ Kon ] ; this is one of the 7 families of Enriques surfaces with finite automorphisms .
There are 20 smooth rational curves LaTeXMLMath on LaTeXMLMath .
( See Figure 4.4 in [ Kon ] . )
Take 8 curves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath on LaTeXMLMath , which form a configuration of Dynkin type LaTeXMLMath .
These are irreducible components of an elliptic pencil of type LaTeXMLMath .
We claim that the divisor LaTeXMLEquation is 5-divisible in Pic LaTeXMLMath .
To see this , first note that LaTeXMLMath intersects with any of the 20 curves LaTeXMLMath in a multiple of 5 points .
Next , consider an elliptic pencil of type LaTeXMLMath together with a double section to infer that among the 20 curves are there 10 curves which generate a sublattice isomorphic to LaTeXMLEquation a sublattice of index LaTeXMLMath of the unimodular lattice Pic LaTeXMLMath / ( torsion ) .
Now apply Lemma 3.3 .
( 3.4.2 ) The case with LaTeXMLMath and LaTeXMLMath Let LaTeXMLMath be the same surface as in Example ( 3.4.1 ) .
Take 8 curves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath on LaTeXMLMath , which form a configuration of Dynkin type LaTeXMLMath .
These are irreducible components of the same elliptic pencil of type LaTeXMLMath as above .
The corresponding 16 curves on the LaTeXMLMath -cover of LaTeXMLMath form a configuration of Dynkin type LaTeXMLMath , and can be found in Figure 4.3 in [ Kon ] .
It is checked that for any mod 5 nontrivial integral linear combination of the 16 curves can one find a smooth rational curve which intersects the combination in a non-multiple of 5 points .
So , the LaTeXMLMath is primitive .
( 3.4.3 ) The case with LaTeXMLMath and LaTeXMLMath Let LaTeXMLMath be the Example I from [ Kon ] ( see also [ D ] ) .
There are 12 smooth rational curves LaTeXMLMath on LaTeXMLMath .
We give the dual graph below for the readers ’ convenience .
Figure 3.4.3 Take 8 curves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath on LaTeXMLMath , which form a configuration of Dynkin type LaTeXMLMath .
These are irreducible components of an elliptic pencil of type LaTeXMLMath .
By intersecting with LaTeXMLMath and LaTeXMLMath , we see easily that the LaTeXMLMath is primitive .
On the other hand , the corresponding 16 curves LaTeXMLMath on the LaTeXMLMath -cover of LaTeXMLMath form a 5-divisible configuration of Dynkin type LaTeXMLMath .
To see this , note that the 16 curves are irreducible components of an elliptic pencil of type LaTeXMLMath , so that the divisor LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation is clearly 5-divisible .
Lemma 3.5 .
Let LaTeXMLMath be an Enriques surface .
( 1 ) If LaTeXMLMath has an elliptic pencil with a singular fibre of type LaTeXMLMath , then the configuration of Dynkin type LaTeXMLMath consisting of non-central components of this fibre is primitive , while the corresponding LaTeXMLMath curves on the LaTeXMLMath -cover of LaTeXMLMath form a LaTeXMLMath -divisible configuration of Dynkin type LaTeXMLMath .
( 2 ) Any configuration of smooth rational curves on LaTeXMLMath of Dynkin type LaTeXMLMath contains exactly one LaTeXMLMath -divisible sub-configuration of Dynkin type LaTeXMLMath .
Proof .
( 1 ) Write the singular fibre as LaTeXMLEquation .
The 6 curves LaTeXMLMath , … , LaTeXMLMath form a non-primitive LaTeXMLMath if and only if the divisor LaTeXMLEquation is LaTeXMLMath -divisible , if and only if a general fibre is LaTeXMLMath -divisible , which is impossible , because no elliptic pencil on an Enriques surface has a triple fibre .
On the other hand , the corresponding 12 curves LaTeXMLMath on the LaTeXMLMath -cover form a LaTeXMLMath -divisible configuration of Dynkin type LaTeXMLMath , as the 12 curves are irreducible components of an elliptic pencil of type LaTeXMLMath , and hence the divisor on the LaTeXMLMath -cover LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation is clearly LaTeXMLMath -divisible .
( 2 ) Let LaTeXMLMath be the sublattice of the unimodular lattice Pic LaTeXMLMath / ( torsion ) generated by the given 8 curves of Dynkin type LaTeXMLMath .
Let LaTeXMLMath be its primitive closure .
Since the discriminant group of LaTeXMLMath is a 3-elementary group with 4 generators and the orthogonal complement LaTeXMLMath has rank 2 , LaTeXMLMath must have order 3 or LaTeXMLMath .
In other words , LaTeXMLMath is not primitive and contains exactly one or four LaTeXMLMath -divisible sub-configurations of Dynkin type LaTeXMLMath .
The second possibility can be ruled out by the following claim and ( 1 ) .
Claim 3.5.1 .
Any configuration of smooth rational curves on LaTeXMLMath of Dynkin type LaTeXMLMath is equivalent , by a composition of reflections in a smooth rational curve , to a configuration of the same type consisting of irreducible components of an elliptic pencil of type LaTeXMLMath or LaTeXMLMath .
To prove the claim , observe that LaTeXMLMath det LaTeXMLMath is a perfect square , so that we can find an isotropic element of LaTeXMLMath , and hence a primitive isotropic element LaTeXMLMath of Pic LaTeXMLMath which is orthogonal to the 8 curves .
The divisor LaTeXMLMath consists of an elliptic configuration LaTeXMLMath and , possibly , trees of smooth rational curves , say , LaTeXMLMath .
These trees may contain some of the 8 curves .
Let LaTeXMLMath be the composition of reflections in a smooth rational curve LaTeXMLMath which maps LaTeXMLMath to LaTeXMLMath .
Then LaTeXMLMath maps the 8 curves to 8 smooth rational curves which are irreducible components of the elliptic pencil LaTeXMLMath .
( A reflection is , in general , not even an effective isometry , but in our case LaTeXMLMath has the desired property . )
Finally , It is easy to check that if an elliptic pencil on an Enriques surface contains 8 smooth rational curves of Dynkin type LaTeXMLMath , then it must be of type LaTeXMLMath or LaTeXMLMath .
Lemma 3.6 .
( 1 ) If LaTeXMLMath , or LaTeXMLMath , then LaTeXMLMath .
( 2 ) If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , or the symmetry group LaTeXMLMath of order LaTeXMLMath .
The first case occurs if the LaTeXMLMath on the LaTeXMLMath cover of LaTeXMLMath is primitive ; the second if the LaTeXMLMath on LaTeXMLMath is non-primitive ; the third if the LaTeXMLMath on LaTeXMLMath is primitive , while the LaTeXMLMath on the LaTeXMLMath cover is non-primitive .
All three cases occur .
( See Examples LaTeXMLMath below . )
( 3 ) If LaTeXMLMath , then LaTeXMLMath , or LaTeXMLMath .
The first case occurs if the LaTeXMLMath on the LaTeXMLMath cover of LaTeXMLMath contains only one LaTeXMLMath -divisible LaTeXMLMath ; the second if the LaTeXMLMath on the LaTeXMLMath cover is a union of two LaTeXMLMath -divisible LaTeXMLMath .
The second case is supported by an example .
( See Example LaTeXMLMath below . )
Proof .
( 1 ) These two cases follow from Table 1 .
( 2 ) From Table 1 , we see that LaTeXMLMath is an extension of LaTeXMLMath or LaTeXMLMath by LaTeXMLMath and hence is isomorphic to LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
( 3 ) From Table 1 , we see that LaTeXMLMath is an extension of LaTeXMLMath , or LaTeXMLMath , by LaTeXMLMath .
There are 5 possibilities : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , where the last group is the nonabelian group of order LaTeXMLMath , LaTeXMLEquation .
By Lemma 3.5 ( 2 ) , the LaTeXMLMath is non-primitive , so LaTeXMLMath has a normal subgroup of index 3 .
This rules out the second and fifth possibilities .
Note that the third group has 4 normal subgroups of index 3 .
The third case occurs if and only if the LaTeXMLMath on LaTeXMLMath contains four different LaTeXMLMath -divisible LaTeXMLMath , if and only if the LaTeXMLMath on LaTeXMLMath is of index LaTeXMLMath in its primitive closure in Pic LaTeXMLMath / ( torsion ) .
This is impossible again by Lemma 3.5 ( 2 ) .
Example 3.7 .
( 3.7.1 ) The case with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the Example II from [ Kon ] .
There are 12 smooth rational curves LaTeXMLMath on LaTeXMLMath .
We give the dual graph below for the readers ’ convenience .
Figure 3.7.1 Take 6 curves LaTeXMLMath on LaTeXMLMath , which form a configuration of Dynkin type LaTeXMLMath , and let LaTeXMLMath be the surface with these 6 curves removed from LaTeXMLMath .
On the LaTeXMLMath cover of LaTeXMLMath we have 12 curves LaTeXMLEquation which form a configuration of Dynkin type LaTeXMLMath .
We claim that this LaTeXMLMath is primitive , whence LaTeXMLMath by Lemma 3.6 .
Suppose that there is an integral linear combination of the 12 curves LaTeXMLEquation which is LaTeXMLMath -divisible in the Picard lattice of the LaTeXMLMath cover .
Intersecting LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , we see that modulo 3 LaTeXMLEquation .
Thus LaTeXMLMath .
Similarly , intersecting LaTeXMLMath with LaTeXMLMath , we see that all coefficients of LaTeXMLMath are 0 modulo 3 .
This proves the claim .
( 3.7.2 ) The case with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the Example V from [ Kon ] .
There are 20 smooth rational curves LaTeXMLMath on LaTeXMLMath ; see Figure 5.5 in [ Kon ] .
Take 6 curves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , on LaTeXMLMath , which form a configuration of Dynkin type LaTeXMLMath .
We claim that the divisor LaTeXMLEquation is LaTeXMLMath -divisible in Pic LaTeXMLMath .
To see this , first note that LaTeXMLMath intersects with any of the 20 curves LaTeXMLMath in a multiple of 3 points .
Next , consider the elliptic pencil LaTeXMLMath , which is of type LaTeXMLMath .
Its irreducible components together with a double section LaTeXMLMath generate a sublattice isomorphic to LaTeXMLEquation a sublattice of index LaTeXMLMath of the unimodular lattice Pic LaTeXMLMath / ( torsion ) .
Now apply Lemma 3.3 .
( 3.7.3 ) The case with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be an Enriques surface with an elliptic pencil containing a singular fibre of type LaTeXMLMath .
Take the 6 curves of Dynkin type LaTeXMLMath out of this fibre .
Then the result follows from Lemma 3.5 ( 1 ) .
( 3.7.4 ) The case with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the Example V from [ Kon ] .
There are 20 smooth rational curves LaTeXMLMath on LaTeXMLMath ; see Figure 5.5 in [ Kon ] .
Take 8 curves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , on LaTeXMLMath , which form a configuration of Dynkin type LaTeXMLMath .
These are irreducible components of an elliptic pencil of type LaTeXMLMath .
We have proved in Example 3.7.2 that the divisor LaTeXMLEquation is LaTeXMLMath -divisible in Pic LaTeXMLMath .
On the other hand , by Lemma 3.5 ( 1 ) , the 6 curves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath form a primitive configuration of Dynkin type LaTeXMLMath , whose pull back on the LaTeXMLMath -cover form a LaTeXMLMath -divisible configuration of Dynkin type LaTeXMLMath .
Definition 3.8 .
Let LaTeXMLMath be an Enriques surface with a configuration of Dynkin type LaTeXMLMath , i.e .
mutually disjoint LaTeXMLMath smooth rational curves .
The configuration is called LaTeXMLMath -divisible LaTeXMLMath -point set if the sum of the LaTeXMLMath curves is equal to LaTeXMLMath for an integral divisor LaTeXMLMath on LaTeXMLMath ; since LaTeXMLMath is the only torsion element in Pic ( LaTeXMLMath ) and since LaTeXMLMath , there are exactly two double covers of LaTeXMLMath both branched exactly at these LaTeXMLMath curves .
Let LaTeXMLMath be the LaTeXMLMath cover of LaTeXMLMath .
Then the pull back on LaTeXMLMath of a Dynkin type LaTeXMLMath configuration on LaTeXMLMath , is of Dynkin type LaTeXMLMath .
Hence a configuration of Dynkin type LaTeXMLMath is LaTeXMLMath -divisible only if LaTeXMLMath , or 8 .
Note also that the pull back on LaTeXMLMath of LaTeXMLMath on LaTeXMLMath is LaTeXMLMath -divisible if and only if the LaTeXMLMath is congruent to 0 or LaTeXMLMath modulo 2 in Pic ( LaTeXMLMath ) .
Let LaTeXMLMath and LaTeXMLMath be distinct LaTeXMLMath -divisible LaTeXMLMath -point sets on an Enriques surface .
Then LaTeXMLMath , or 2 .
If LaTeXMLMath , then the symmetric difference LaTeXMLMath is also a LaTeXMLMath -divisible 4-point set .
Lemma 3.9 .
( 1 ) If LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , then LaTeXMLMath .
( 2 ) If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
The first case occurs if the LaTeXMLMath on the LaTeXMLMath cover of LaTeXMLMath is primitive ; the second if the LaTeXMLMath on LaTeXMLMath is LaTeXMLMath -divisible ; the third if the LaTeXMLMath on LaTeXMLMath is primitive , while the LaTeXMLMath on the LaTeXMLMath cover is LaTeXMLMath -divisible .
( 3 ) If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
The first case occurs if the LaTeXMLMath on the LaTeXMLMath cover of LaTeXMLMath is primitive ; the second if the LaTeXMLMath on LaTeXMLMath contains a LaTeXMLMath -divisible LaTeXMLMath -point subset ; the third if the LaTeXMLMath on LaTeXMLMath is primitive , while the LaTeXMLMath on the LaTeXMLMath cover contains a LaTeXMLMath -divisible LaTeXMLMath -point subset .
( 4 ) If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
The first case occurs if the LaTeXMLMath on LaTeXMLMath contains no LaTeXMLMath -divisible LaTeXMLMath -point subset ; the second if the LaTeXMLMath on the LaTeXMLMath cover of LaTeXMLMath contains only one LaTeXMLMath -divisible LaTeXMLMath -point subset and the LaTeXMLMath on LaTeXMLMath contains a LaTeXMLMath -divisible LaTeXMLMath -point subset ; the third if the LaTeXMLMath on the LaTeXMLMath cover is a union of two LaTeXMLMath -divisible LaTeXMLMath -point subsets and the LaTeXMLMath on LaTeXMLMath contains only one LaTeXMLMath -divisible LaTeXMLMath -point subset ; the fourth if the LaTeXMLMath on LaTeXMLMath is a union of two LaTeXMLMath -divisible LaTeXMLMath -point subsets .
( 5 ) If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , where LaTeXMLEquation .
The first case occurs if the LaTeXMLMath on LaTeXMLMath is a union of three LaTeXMLMath -divisible LaTeXMLMath -point subsets ; the second if the LaTeXMLMath on LaTeXMLMath is a union of one LaTeXMLMath and two LaTeXMLMath -divisible LaTeXMLMath -point subsets ; the third if the LaTeXMLMath on LaTeXMLMath contains only one LaTeXMLMath -divisible LaTeXMLMath -point subset .
( 6 ) If LaTeXMLMath , then LaTeXMLMath .
All cases are supported by examples except the case with LaTeXMLMath and LaTeXMLMath .
( See Examples 3.12 . )
Proof .
( 1 ) and ( 6 ) follow from Table 1 .
( 2 ) and ( 3 ) also follow from Table 1 .
Note that if a subconfiguration of Dynkin type LaTeXMLMath on LaTeXMLMath is LaTeXMLMath -divisible , i.e .
LaTeXMLMath is linearly equivalent to LaTeXMLMath for some LaTeXMLMath Pic ( LaTeXMLMath ) , then both LaTeXMLMath and LaTeXMLMath determine Galois double covers of LaTeXMLMath , which correspond to two of the three normal subgroups of LaTeXMLMath of index 2 .
( 4 ) From Table 1 , we see that LaTeXMLMath is an extension of LaTeXMLMath by LaTeXMLMath or LaTeXMLMath .
There are 5 possibilities : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , or the dihedral group LaTeXMLMath of order 8 .
The last can be ruled out by observing that if LaTeXMLMath , then LaTeXMLMath must have an odd number of normal subgroups isomorphic to LaTeXMLMath , while LaTeXMLMath has exactly two such subgroups .
( 5 ) In this case , LaTeXMLMath is an extension of LaTeXMLMath by LaTeXMLMath .
There are 4 possibilities : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
The last can be ruled out by noting that a double cover of LaTeXMLMath branched along the union of four out of the seven curves is again an Enriques surface ( with 4 points blown up ) and that , by ( 4 ) , no open Enriques surface with LaTeXMLMath can have LaTeXMLMath as its fundamental group .
Note that LaTeXMLMath ( resp .
LaTeXMLMath ) has exactly 7 ( resp .
3 ) normal subgroups of index 2 .
The group LaTeXMLMath has 15 normal subgroups of index 2 , and hence occurs as LaTeXMLMath only if the LaTeXMLMath contains 7 different LaTeXMLMath -divisible 4-point subsets .
This condition is equivalent to that the LaTeXMLMath is a union LaTeXMLMath of three LaTeXMLMath -divisible 4-point subsets LaTeXMLMath , LaTeXMLMath , where the seven LaTeXMLMath -divisible 4-point subsets are LaTeXMLMath and LaTeXMLMath .
3.10 .
Let LaTeXMLMath be the Kummer surface LaTeXMLMath , where LaTeXMLMath is an elliptic curve with fundamental period LaTeXMLMath .
Let LaTeXMLMath be the 2-torsion point LaTeXMLMath and consider the following involution of LaTeXMLMath LaTeXMLEquation .
Then LaTeXMLMath induces a fixed point free involution LaTeXMLMath on LaTeXMLMath and the quotient surface LaTeXMLMath is an Enriques surface .
On LaTeXMLMath we have 12 smooth rational curves coming from the sixteen 2-torsion points , ( a 2-torsion ) LaTeXMLMath , and LaTeXMLMath ( a 2-torsion ) .
Their dual graph is given in Figure 3.10 .
Figure 3.10 There contained in the graph are 16 different configurations of type LaTeXMLMath , half of them giving elliptic pencils on LaTeXMLMath and the other half corresponding to half elliptic pencils .
We may assume that LaTeXMLMath is an elliptic pencil .
Then LaTeXMLMath is also an elliptic pencil .
Modulo 2 in Pic ( LaTeXMLMath ) there are many congruences .
To raise a few , we have the following : ( 1 ) LaTeXMLMath mod 2 in Pic ( LaTeXMLMath ) .
( 2 ) LaTeXMLMath mod 2 .
( 3 ) LaTeXMLMath mod 2 .
3.11 .
If LaTeXMLMath , then the special Enriques surface LaTeXMLMath has additional 8 smooth rational curves , LaTeXMLMath [ Kon , Example III ] .
Their dual graph is [ Kon , Fig 3.5 , p. 212 ] , but we will use LaTeXMLMath instead of LaTeXMLMath in [ Kon ] .
Examples 3.12 .
( 3.12.1 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 4 curves , LaTeXMLMath , and LaTeXMLMath .
Then the sum of the 8 curves on LaTeXMLMath has intersection number 1 with LaTeXMLMath .
Here we denote by LaTeXMLMath the inverse on LaTeXMLMath of LaTeXMLMath .
( 3.12.2 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 4 curves , LaTeXMLMath , and LaTeXMLMath .
These form a LaTeXMLMath -divisible 4-point set ( 3.10 . ( 2 ) ) .
( 3.12.3 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 4 curves , LaTeXMLMath , and LaTeXMLMath .
Use 3.10 . ( 3 ) .
( 3.12.4 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 5 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.5 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 5 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.6 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 5 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.7 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 6 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.8 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 6 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.9 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 6 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.10 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 6 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.11 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 7 curves , LaTeXMLMath , and LaTeXMLMath .
( 3.12.12 ) LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
Take the 7 curves , LaTeXMLMath , and LaTeXMLMath .
Combining results in this section , we conclude Theorem 2 .
§4 .
The proof of Theorem 3 We now prove Theorem 3 .
Claim 1 .
H is either a LaTeXMLMath or an Enriques surface with at worst Du Val singularities of type LaTeXMLMath .
Note that LaTeXMLMath .
So we have only to show that LaTeXMLMath ( and hence the irregularity of the resolution of LaTeXMLMath also vanishes because LaTeXMLMath has only rational singularities ) .
Consider the exact sequence : LaTeXMLEquation .
This induces a long exact sequence of cohomologies .
Now the Kawamata-Viehweg vanishing theorem implies that LaTeXMLMath , whence LaTeXMLMath LaTeXMLMath .
This proves Claim 1 .
Embed LaTeXMLMath in a projective space and let LaTeXMLMath be a general hyperplane on LaTeXMLMath such that LaTeXMLMath Sing LaTeXMLMath and LaTeXMLMath is a smooth irreducible curve on LaTeXMLMath , whence LaTeXMLMath is smooth along this curve because LaTeXMLMath is Cartier .
This is possible because the normal surface LaTeXMLMath has only finitely many singular points .
By the result of Hamm-Le in [ HL , Theorem 1.1.3 ] , one has LaTeXMLMath Sing LaTeXMLMath .
Claim 2 .
The natural homomorphism LaTeXMLMath Sing V ) LaTeXMLMath Sing LaTeXMLMath is surjective .
Let LaTeXMLMath be the minimal resolution .
By the assumption , LaTeXMLMath is away from Sing LaTeXMLMath , and hence the pull back on LaTeXMLMath , denoted also by LaTeXMLMath , of LaTeXMLMath is still smooth irreducible and also nef and big .
Note that LaTeXMLMath Sing LaTeXMLMath Sing LaTeXMLMath because LaTeXMLMath is Cartier .
Hence LaTeXMLMath Sing LaTeXMLMath by the choice of LaTeXMLMath ; similarly , LaTeXMLMath Sing LaTeXMLMath Sing LaTeXMLMath .
By [ No , Cor .
2.3 and the proof of Cor .
2.4B ] , we obtain the surjectivity of the homomorphism LaTeXMLMath , where LaTeXMLMath is the inverse of LaTeXMLMath and the latter equality comes from the observation that LaTeXMLMath is the minimal resolution of singular points in ( Sing LaTeXMLMath and the fact that every singular point on LaTeXMLMath is log terminal because so is LaTeXMLMath and the generality of LaTeXMLMath [ Ko1 , Theorem 7.8 ] .
This proves Claim 2 .
Combining Claim 2 with the equality preceding it , we get a surjective homomorphism LaTeXMLMath Sing LaTeXMLMath .
Since the above map factors through LaTeXMLMath Sing LaTeXMLMath , the latter map is also surjective .
On the other hand , LaTeXMLMath Sing LaTeXMLMath Sing LaTeXMLMath , whence we have an inclusion LaTeXMLMath Sing LaTeXMLMath Sing LaTeXMLMath and its induced surjective homomorphism LaTeXMLMath Sing LaTeXMLMath .
This , combined with the early sujective map in the preceding paragraph , produces a surjective homomorphism LaTeXMLMath .
This , together with Claim 1 and Theorems 1 and 2 , implies Theorem 3 .
Added in proof .
After the paper was submitted , we learnt that Conjecture B has been proved by S. Takayama under even weaker condition [ Ta ] , though Conjecture A is still open .
References [ A ] F. Ambro , Ladders on Fano varieties , alg-geom/ 9710005 .
[ B1 ] W. Barth , LaTeXMLMath surfaces with nine cusps , alg-geom/ 9709031 .
[ B2 ] W. Barth , On the classification of LaTeXMLMath surfaces with nine cusps , math.AG/ 9805082 .
[ BL ] C. Birkenhake and H. Lange , A family of abelian surfaces and curves of genus four , Manuscr .
Math .
85 ( 1994 ) , 393–407 .
[ C ] F. Campana , On twistor spaces of the class LaTeXMLMath , J. Diff .
Geom .
33 ( 1991 ) , 541–549 .
[ CD ] F. Cossec and I. Dolgachev , Enriques surfaces I , Progress in Math .
Vol .
76 ( 1989 ) , Birkhauser .
[ D ] I. Dolgachev , On automorphisms of Enriques surfaces , Invent .
Math .
76 ( 1984 ) , 163-177 .
[ FKL ] A. Fujiki , R. Kobayashi and S. Lu , On the fundamental group of certain open normal surfaces , Saitama Math .
J .
11 ( 1993 ) , 15–20 .
[ GZ 1 , 2 ] R. V. Gurjar and D. -Q. Zhang , LaTeXMLMath of smooth points of a log del Pezzo surface is finite , I ; II , J .
Math .
Sci .
Univ .
Tokyo 1 ( 1994 ) , 137–180 ; 2 ( 1995 ) , 165–196 .
[ GM ] M. Goresky and R. MacPherson , Stratified Morse theory , Springer , 1988 .
[ HL ] H. A. Hamm and D. T. Le , Lefschetz theorems on quasi-projective varieties , Bull .
Soc .
math .
France , 113 ( 1985 ) , 123-142 .
[ Ka ] Y. Kawamata , The cone of curves of algebraic varieties .
Ann .
Math .
119 ( 1984 ) , 603–633 .
[ KM ] S. Keel and J. McKernan , Rational curves on quasi-projective varieties , Mem .
Amer .
Math .
Soc .
140 ( 1999 ) .
[ Kol1 ] J. Kollar , Shafarevich maps and plurigenera of algebraic varieties , Invent .
Math .
113 ( 1993 ) , 177–215 .
[ Kol2 ] J. Kollar , Shafarevich maps and automorphic forms , M. B. Porter Lectures at Rice Univ. , Princeton Univ .
Press .
[ Ke ] J. Keum , Note on elliptic LaTeXMLMath surfaces , Trans .
A.M.S .
352 ( 2000 ) , 2077-2086 .
[ KoMiMo ] J. Kollar , Y. Miyaoka and S. Mori , Rationally connected varieties , J. Alg .
Geom .
1 ( 1992 ) , 429–448 .
[ Kon ] S. Kondo , Enriques surfaces with finite automorphism groups , Japan J .
Math .
12 ( 1986 ) , 191-282 .
[ Mi ] T. Minagawa , Deformations of Q -Calabi-Yau 3-folds and Q -Fano 3-folds of Fano index 1 , math.AG/ 9905106 .
[ MP ] R. Miranda and U. Persson , Mordell-Weil groups of extremal elliptic LaTeXMLMath surfaces , in : Problems in the theory of surfaces and their classification ( Cortona , 1988 ) , Sympo .
Math .
XXXII , Acad .
Press , London , 1991 , pp .
167–192 .
[ Mu ] S. Mukai , Finite groups of automorphisms of LaTeXMLMath surfaces and the Mathieu group , Invent .
Math .
94 ( 1988 ) , 183–221 .
[ Ni1 ] V. V. Nikulin , On Kummer surfaces , Math .
USSR Izv .
9 ( 1975 ) , 261–275 .
[ Ni2 ] V. V. Nikulin , Integral symmetric bilinear forms and some of their applications , Math .
USSR Izv .
14 ( 1980 ) , 103–167 .
[ Ni3 ] V. V. Nikulin , Finite automorphism groups of Kahler LaTeXMLMath surfaces , Trans .
Moscow Math .
Soc .
38 ( 1980 ) , 71–135 .
[ No ] M. V. Nori , Zariski ’ s conjecture and related problems , Ann .
Sci .
Ecole Norm .
Sup .
16 ( 1983 ) , 305–344 .
[ SZ ] I. Shimada and D. -Q. Zhang , Classification of extremal elliptic LaTeXMLMath surfaces and fundamental groups of open LaTeXMLMath surfaces , Nagoya Math .
J. to appear .
[ Sa ] T. Sano , On classification of non-Gorenstein Q -Fano 3-folds of Fano index 1 , J .
Math .
Soc .
Japan , 47 ( 1995 ) , 369-380 .
[ Sh ] T. Shioda , On the Mordell-Weil lattices , Comment .
Math .
Univ .
Sancti pauli , 39 ( 1990 ) , 211–240 .
[ Ta ] S. Takayama , Simple connectedness of weak Fano varieties , J. Alg .
Geom .
9 ( 2000 ) , 403–407 .
[ T ] D. Toledo , Projective varieties with non-residually finite fundamental group , Inst .
Hautes Etudes Sci .
Publ .
Math .
77 ( 1993 ) , 103–119 .
[ X ] G. Xiao , Galois covers between LaTeXMLMath surfaces , Ann .
Inst .
Fourier ( Grenoble ) , 46 ( 1996 ) , 73–88 .
[ Z1 ] D. -Q. Zhang , The fundamental group of the smooth part of a log Fano variety , Osaka J .
Math .
32 ( 1995 ) , 637–644 .
[ Z2 ] D. -Q. Zhang , Algebraic surfaces with nef and big anti-canonical divisor , Math .
Proc .
Cambridge Philos .
Soc .
117 ( 1995 ) , 161–163 .
[ Z3 ] D. -Q. Zhang , Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts , Trans .
Amer .
Math .
Soc .
348 ( 1996 ) , 4175–4184 .
J. Keum Korea Institute for Advanced Study 207-43 Cheongryangri-dong , Dongdaemun-gu Seoul 130-012 , Korea e-mail : jhkeumkias.re.kr D. -Q. Zhang Department of Mathematics , National University of Singapore 2 Science Drive 2 , Singapore 117543 Republic of Singapore e-mail : matzdqmath.nus.edu.sg Figures 3.4.3 , 3.7.1 and 3.10 are respectively Figures 1.4 , 2.4 and 3.5 ( with LaTeXMLMath there replaced by LaTeXMLMath here ) in [ Kon ] .
In Table 1 below , we write LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be the minimal resolution with LaTeXMLMath .
LaTeXMLMath or LaTeXMLMath is a 2-divisible configuration of LaTeXMLMath ’ s of Dynkin type LaTeXMLMath .
LaTeXMLMath or LaTeXMLMath is a 3-divisible configuration of LaTeXMLMath ’ s of Dynkin type LaTeXMLMath .
Table 1 No.1 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.2 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.3 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.4 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is smooth .
No.5 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.6 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is smooth .
No.7 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is smooth .
No.8 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.9 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.10 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.11 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.12 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is smooth .
No.13 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.14 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.15 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is smooth .
No.16 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.17 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is smooth .
No.18 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
In Table 2 below , LaTeXMLMath is the LaTeXMLMath cover of the Enriques surface LaTeXMLMath .
LaTeXMLMath is LaTeXMLMath minus a configuration LaTeXMLMath of LaTeXMLMath ’ s of Dynkin type LaTeXMLMath .
LaTeXMLMath is the inverse on LaTeXMLMath of LaTeXMLMath ; so LaTeXMLMath has Dynkin type LaTeXMLMath .
LaTeXMLMath or LaTeXMLMath ( resp .
LaTeXMLMath or LaTeXMLMath ) is a 2-divisible configuration of LaTeXMLMath ’ s of Dynkin type LaTeXMLMath ( resp .
LaTeXMLMath ) on LaTeXMLMath ( resp .
on LaTeXMLMath ) .
LaTeXMLMath or LaTeXMLMath ( resp .
LaTeXMLMath ) is a 3-divisible configuration of LaTeXMLMath ’ s of Dynkin type LaTeXMLMath ( resp .
LaTeXMLMath ) on LaTeXMLMath ( resp .
on LaTeXMLMath ) .
LaTeXMLMath is a LaTeXMLMath on LaTeXMLMath .
LaTeXMLMath is the symmetric group on 3 letters .
LaTeXMLMath is the dihedral group of order 10 .
We do not know if No.14 or No.20 in Table 2 is realizable .
Table 2 No.1 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.2 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.3 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.4 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.5 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.6 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.7 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.8 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.9 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath .
No.10 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.11 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.12 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.13 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.14 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.15 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.16 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.17 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.18 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.19 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.20 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath .
No.21 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath contains only one LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath .
No.22 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.23 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
No.24 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.25 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is non-primitive in LaTeXMLMath ; LaTeXMLMath .
No.26 : LaTeXMLMath ; LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath is primitive in LaTeXMLMath ; LaTeXMLMath .
In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces ( possibly cyclic coverings of LaTeXMLMath ) , branched over knots .
When the base space is a 3-sphere , we prove that the associated branching set is a two-bridge knot of genus one , and we determine its type .
Moreover , a geometric cyclic presentation for the fundamental groups of these manifolds is obtained in several interesting cases , including the ones corresponding to the branched cyclic coverings of LaTeXMLMath .
Mathematics Subject Classification 2000 : Primary 57M12 , 57R65 ; Secondary 20F05 , 57M05 , 57M25 .
Keywords : Takahashi manifolds , branched cyclic coverings , cyclically presented groups , geometric presentations of groups , Dehn surgery .
Takahashi manifolds are closed orientable 3-manifolds introduced in LaTeXMLCite by Dehn surgery on LaTeXMLMath , with rational coefficients , along the LaTeXMLMath -component link LaTeXMLMath depicted in Figure LaTeXMLRef .
These manifolds have been intensively studied in LaTeXMLCite , LaTeXMLCite , and LaTeXMLCite .
In the latter two papers , a nice topological characterization of all Takahashi manifolds as two-fold coverings of LaTeXMLMath , branched over the closure of certain rational 3-string braids , is given .
A Takahashi manifold is called periodic when the surgery coefficients have the same cyclic symmetry of order LaTeXMLMath of the link LaTeXMLMath , i.e .
the coefficients are LaTeXMLMath and LaTeXMLMath alternately .
Several important classes of 3-manifolds , such as ( fractional ) Fibonacci manifolds LaTeXMLCite and Sieradsky manifolds LaTeXMLCite , represent notable examples of periodic Takahashi manifolds .
In this paper we show that each periodic Takahashi manifold is an LaTeXMLMath -fold cyclic covering of the connected sum of two lens spaces , branched over a knot .
This knot arises from a component of the Borromean rings , by performing a surgery with coefficients LaTeXMLMath and LaTeXMLMath along the other two components .
For particular values of the surgery coefficients ( including the classes of manifolds cited above ) , the periodic Takahashi manifolds turn out to be LaTeXMLMath -fold cyclic coverings of LaTeXMLMath , branched over two-bridge knots of genus one LaTeXMLCite .
For the characterization of two-bridge knots of genus one , see LaTeXMLCite .
, whose parameters are obtained using Kirby-Rolfsen calculus LaTeXMLCite ( compare the analogous result of LaTeXMLCite , obtained by a different approach ) .
Observe that in LaTeXMLCite a characterization of all periodic Takahashi manifolds as LaTeXMLMath -fold cyclic coverings of LaTeXMLMath , branched over the closure of certain rational 3-string braids , is presented , but the result is incorrect , as we show in Remark 1 .
For many interesting periodic Takahashi manifolds - including the ones corresponding to branched cyclic coverings of LaTeXMLMath - a cyclic presentation for the fundamental group is provided and proved to be geometric , i.e .
arising from a Heegaard diagram , or , equivalently , from a canonical spine LaTeXMLCite .
We denote by LaTeXMLMath the Takahashi manifold obtained by Dehn surgery on LaTeXMLMath along the LaTeXMLMath -component link LaTeXMLMath of Figure LaTeXMLRef , with surgery coefficients LaTeXMLMath respectively , cyclically associated to the components of LaTeXMLMath .
A Takahashi manifold is periodic when LaTeXMLMath and LaTeXMLMath , for every LaTeXMLMath .
Denote by LaTeXMLMath the periodic Takahashi manifold LaTeXMLMath .
From now on , without loss of generality , we can always suppose that : LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Moreover , if LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , we shall denote by LaTeXMLMath the lens space of type LaTeXMLMath .
As usual , LaTeXMLMath is homeomorphic to LaTeXMLMath and LaTeXMLMath is homeomorphic to LaTeXMLMath , for all LaTeXMLMath ( including LaTeXMLMath ) .
Notice that LaTeXMLMath is the Fractional Fibonacci manifold LaTeXMLMath defined in LaTeXMLCite and , in particular , LaTeXMLMath is the Fibonacci manifold LaTeXMLMath studied in LaTeXMLCite .
Moreover , LaTeXMLMath is the Sieradsky manifold LaTeXMLMath introduced in LaTeXMLCite and studied in LaTeXMLCite .
Because of the symmetries of LaTeXMLMath , the homeomorphisms LaTeXMLEquation can easily be obtained for all LaTeXMLMath and LaTeXMLMath .
A balanced presentation of the fundamental group of every Takahashi manifold is given in LaTeXMLCite , and in LaTeXMLCite it is shown that this presentation is geometric , i.e .
it arises from a Heegaard diagram ( or , equivalently , from a canonical spine ) .
As a consequence , LaTeXMLMath admits the following geometric presentation with LaTeXMLMath generators and LaTeXMLMath relators : LaTeXMLEquation where the subscripts are mod LaTeXMLMath .
When LaTeXMLMath , we can easily get a cyclic presentation LaTeXMLCite with LaTeXMLMath generators : LaTeXMLMath .
LaTeXMLEquation where the subscripts are mod LaTeXMLMath .
For all LaTeXMLMath and LaTeXMLMath , the cyclic presentation ( 1 ) of LaTeXMLMath is geometric .
Proof .
If LaTeXMLMath then LaTeXMLMath is homeomorphic to the connected sum of LaTeXMLMath copies of LaTeXMLMath , and therefore the statement is straightforward .
If LaTeXMLMath , the presentation becomes LaTeXMLEquation .
Figure LaTeXMLRef shows an RR-system which induces ( LaTeXMLMath ) , and so , by LaTeXMLCite , this presentation is geometric .
If LaTeXMLMath , the presentation becomes LaTeXMLEquation .
Therefore , if we replace LaTeXMLMath with LaTeXMLMath , Figure LaTeXMLRef also gives an RR-system inducing ( LaTeXMLMath ) .
Since the link LaTeXMLMath is a two-component trivial link , we immediately get the following results : For all LaTeXMLMath , the manifold LaTeXMLMath is homeomorphic to the connected sum of lens spaces LaTeXMLMath .
In particular , LaTeXMLMath is homeomorphic to the lens space LaTeXMLMath and LaTeXMLMath is homeomorphic to LaTeXMLMath .
Proof .
LaTeXMLMath is obtained by Dehn surgery on LaTeXMLMath , with coefficients LaTeXMLMath and LaTeXMLMath , along the trivial link with two components LaTeXMLMath .
Now we prove the main result of the paper : For all LaTeXMLMath and LaTeXMLMath , the periodic Takahashi manifold LaTeXMLMath is the LaTeXMLMath -fold cyclic covering of the connected sum of lens spaces LaTeXMLMath , branched over a knot LaTeXMLMath which does not depend on LaTeXMLMath .
Moreover , LaTeXMLMath arises from a component of the Borromean rings , by performing a surgery with coefficients LaTeXMLMath and LaTeXMLMath along the other two components .
Proof .
Both the link LaTeXMLMath and the surgery coefficients defining LaTeXMLMath are invariant with respect to the rotation LaTeXMLMath of LaTeXMLMath , which sends the LaTeXMLMath -th component of LaTeXMLMath onto the LaTeXMLMath -th component ( mod LaTeXMLMath ) .
Let LaTeXMLMath be the cyclic group of order LaTeXMLMath generated by LaTeXMLMath .
Observe that the fixed-point set of the action of LaTeXMLMath on LaTeXMLMath is a trivial knot disjoint from LaTeXMLMath .
Therefore , we have an action of LaTeXMLMath on LaTeXMLMath , with a knot LaTeXMLMath as fixed-point set .
The quotient LaTeXMLMath is precisely the manifold LaTeXMLMath , which is homeomorphic to LaTeXMLMath by Lemma LaTeXMLRef , and LaTeXMLMath is obviously a knot LaTeXMLMath , which only depends on LaTeXMLMath and LaTeXMLMath .
Moreover , LaTeXMLMath is the Borromean rings , as showed in Figure LaTeXMLRef .
This proves the statement .
We can give another description of the branching set LaTeXMLMath , as the inverse image of a trivial knot in a certain two-fold branched covering .
Denote by LaTeXMLMath the link depicted in Figure LaTeXMLRef .
It is composed by the closure of the rational 3-string braid LaTeXMLMath , which is the connected sum of the two-bridge knots or links LaTeXMLMath and LaTeXMLMath , and by a trivial knot .
Moreover , denote : ( i ) by LaTeXMLMath the orbifold from the proof of Theorem LaTeXMLRef , whose underlying space is LaTeXMLMath and whose singular set is the knot LaTeXMLMath , with index LaTeXMLMath ; ( ii ) by LaTeXMLMath the orbifold whose underlying space is LaTeXMLMath and whose singular set is the closure of the rational 3-string braid LaTeXMLMath , with index LaTeXMLMath ; and ( iii ) by LaTeXMLMath the orbifold whose underlying space is LaTeXMLMath and whose singular set is the link LaTeXMLMath , with index LaTeXMLMath and LaTeXMLMath as pointed out in Figure LaTeXMLRef .
Assuming the previous notations , the following commutative diagram holds for each periodic Takahashi manifold .
Proof .
The link LaTeXMLMath admits an invertible involution LaTeXMLMath , whose axis intersects each component in two points ( see the dashed line of Figure LaTeXMLRef ) , and the rotation symmetry LaTeXMLMath of order LaTeXMLMath which was discussed in Theorem LaTeXMLRef .
These symmetries induce symmetries ( also denoted by LaTeXMLMath and LaTeXMLMath ) on the periodic Takahashi manifold LaTeXMLMath , such that LaTeXMLMath .
We have LaTeXMLMath ( see LaTeXMLCite and LaTeXMLCite ) and LaTeXMLMath ( see Theorem LaTeXMLRef ) .
It is immediate to see that LaTeXMLMath induces a symmetry ( also denoted by LaTeXMLMath ) on the orbifold LaTeXMLMath , and LaTeXMLMath is the orbifold LaTeXMLMath .
As we see from Figure LaTeXMLRef , LaTeXMLMath induces a strongly invertible involution ( also denoted by LaTeXMLMath ) on the link LaTeXMLMath .
Using the Montesinos algorithm we see that LaTeXMLMath .
This concludes the proof .
As a consequence , the branching set LaTeXMLMath of Theorem LaTeXMLRef can be obtained as the inverse image of the trivial component of LaTeXMLMath in the two-fold branched covering LaTeXMLMath .
From Theorem LaTeXMLRef we can get the following result , which has already been obtained in LaTeXMLCite by a different technique .
For all LaTeXMLMath and LaTeXMLMath , the periodic Takahashi manifold LaTeXMLMath is the LaTeXMLMath -fold cyclic covering of LaTeXMLMath , branched over the two-bridge knot of genus one LaTeXMLMath .
Proof .
From Theorem LaTeXMLRef , LaTeXMLMath is the LaTeXMLMath -fold cyclic covering of LaTeXMLMath , branched over a knot LaTeXMLMath which does not depend on LaTeXMLMath .
By isotopy and Kirby-Rolfsen moves it is easy to obtain ( see Figure LaTeXMLRef ) a diagram of LaTeXMLMath , which is a Conway ’ s normal form of type LaTeXMLMath .
This proves the statement .
Proposition LaTeXMLRef covers the results of LaTeXMLCite , LaTeXMLCite and LaTeXMLCite concerning LaTeXMLMath -fold branched cyclic coverings of two-bridge knots .
Moreover , for all LaTeXMLMath , the periodic Takahashi manifold LaTeXMLMath is homeomorphic to the Lins-Mandel manifold LaTeXMLMath LaTeXMLCite , the Minkus manifold LaTeXMLMath LaTeXMLCite and the Dunwoody manifold LaTeXMLMath LaTeXMLCite .
Moreover , observe that all cyclic coverings of two-bridge knots of genus one are periodic Takahashi manifolds .
Remark 1 The results of Corollaries 8 , 9 and 11 of LaTeXMLCite , concerning periodic Takahashi manifolds as LaTeXMLMath -fold cyclic branched coverings of the closure of certain ( rational ) 3-string braids , are incorrect .
This is evident from the following counterexamples .
If LaTeXMLMath and LaTeXMLMath then the first homology group of the 3-fold cyclic branched covering of the closure of the 3-string braid LaTeXMLMath has order LaTeXMLMath , but LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath then the first homology group of the 4-fold cyclic branched covering of the closure of the rational 3-string braid LaTeXMLMath has order LaTeXMLMath , but LaTeXMLMath .
Note that the corollaries are valid if LaTeXMLMath .
The following conjecture is naturally suggested by the previous results .
Conjecture Let LaTeXMLMath be fixed .
Then , for all LaTeXMLMath , the periodic Takahashi manifolds LaTeXMLMath are LaTeXMLMath -fold cyclic coverings of LaTeXMLMath , branched over a knot which does not depend on LaTeXMLMath , if and only if LaTeXMLMath .
Added in revision - The referee pointed out that it is possible to prove the conjecture for “ almost all cases ” by using the hyperbolic Dehn surgery theorem and the shortest geodesic arguments by Kojima LaTeXMLCite .
Acknowledgement The author wishes to thank the referee for his valuable suggestions to improve this paper and Massimo Ferri and Andrei Vesnin for the useful discussions on the topics .
MICHELE MULAZZANI , Department of Mathematics , University of Bologna , I-40127 Bologna , ITALY , and C.I.R.A.M. , Bologna , ITALY .
E-mail : mulazza @ dm.unibo.it
This contribution is the first in a series of three : it reports on the construction of ( a fine sheaf of ) diffeomorphism invariant Colombeau algebras LaTeXMLMath on open sets of Eucildean space ( LaTeXMLCite ) , which completes earlier approaches ( LaTeXMLCite ) .
Part II and III will show , among others , the way to an intrinsic definition of Colombeau algebras on manifolds which , locally , reproduces the algebra ( s ) LaTeXMLMath .
Key words .
Algebras of generalized functions , Colombeau algebras , diffeomorphism invariance Mathematics Subject Classification ( 2000 ) .
Primary 46F30 ; Secondary 46E50 , 35D05 , 26E15 .
Since its introduction in LaTeXMLCite the question of the functor property of Colombeau ’ s construction was at hand as a crucial one : let LaTeXMLMath be a diffeomorphism between open subsets of LaTeXMLMath , is it possible to extend the operation LaTeXMLMath on smooth distributions on LaTeXMLMath to an operation LaTeXMLMath on the Colombeau algebra such that LaTeXMLMath and LaTeXMLMath are satisfied or—to phrase it differently—is it possible to achieve a diffeomorphism invariant construction of full Colombeau algebras ?
( In this work we shall focus on full algebras distinguished by the existence of a canonical embedding of distributions and henceforth omit the term “ full ” .
By contrast the so called special ( or simplified ) algebras with their elements basically depending on a real regularization parameter LaTeXMLMath do not allow for a canonical embedding of distributions .
However , by their relative ease of construction and the fact that diffeomorphism invariance of the basic definitions is automatically satisfied they provide a flexible tool to model singularities in an nonlinear context : global analysis in this setting has been investigated in LaTeXMLCite . )
To begin with we briefly recall Colombeau ’ s construction as given in LaTeXMLCite and shall refer to the corresponding algebra as the “ elementary ” one .
Let LaTeXMLMath and for LaTeXMLMath set LaTeXMLMath .
Then we define : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation The “ elementary ” Colombeau algebra of generalized functions on LaTeXMLMath finally is defined as the quotient space LaTeXMLEquation .
LaTeXMLMath is a fine sheaf of differential algebras .
Smooth functions are embedded into LaTeXMLMath by the “ constant ” embedding LaTeXMLMath , i.e. , LaTeXMLMath , turning LaTeXMLMath into a faithful subalgebra .
Distributions , on the other hand , are embedded via ( anti ) convolution with the LaTeXMLMath ’ s , i.e .
LaTeXMLMath with LaTeXMLEquation .
It is one of the fundamental properties of Colombeau algebras that LaTeXMLMath .
Now , let LaTeXMLMath denote a diffeomorphism as above .
Accepting LaTeXMLMath and ( LaTeXMLRef ) as the explicit forms of LaTeXMLMath resp .
LaTeXMLMath , the basic requirement LaTeXMLMath amounts to LaTeXMLEquation ( where LaTeXMLMath ) , which in turn , enforces LaTeXMLEquation as the definition of the action of the diffeomorphism on a representative LaTeXMLMath of a generalized function in a Colombeau algebra on LaTeXMLMath .
Colombeau and Meril in their paper LaTeXMLCite ( using earlier ideas of LaTeXMLCite ) made the first decisive steps to incorporate formula ( LaTeXMLRef ) into the construction of a Colombeau algebra which they claimed to be diffeomorphism invariant .
Before discussing their work in some more detail let us introduce some terminology ( cf .
LaTeXMLCite , Section 9 ) which eases the understanding of the definitions to be given below .
Every Colombeau algebra is constructed as a quotient space of moderate modulo negligible sequences ( nets ) of ( smooth ) functions LaTeXMLMath belonging to some basic space usually denoted by LaTeXMLMath ( plus some superscript to distinguish the algebras to be constructed ) .
The respective properties of moderateness and negligibility are then defined by inserting scaled test objects ( e.g .
LaTeXMLMath with LaTeXMLMath resp .
LaTeXMLMath as above ) into LaTeXMLMath and analyzing the asymptotic behavior of the latter on these “ paths ” as the scaling parameter LaTeXMLMath tends to zero ( and consequently LaTeXMLMath weakly ) : we shall refer to the respective processes as testing for moderateness resp .
negligibility .
In this terminology diffeomorphism invariance of a Colombeau algebra is ensured by diffeomorphism invariance of the respective tests , of course including diffeomorphism invariance of the respective class of ( scaled ) test objects .
As opposed to this testing procedure the elements of the algebra themselves do not depend in any way from LaTeXMLMath .
We regard this distinction as fundamental clarifying several misinterpretations in the literature and call it the policy of “ separating definitions from testing. ” In LaTeXMLCite , Section 3 there was given a blueprint collecting all the definitions and theorems necessary for the construction of a Colombeau algebra .
In the following we shall use this collection as a guiding line in discussing the various variants of the algebra proposed in the literature beginning with the one of Colombeau and Meril LaTeXMLCite .
There are basically three modifications introduced by the authors of LaTeXMLCite distinguishing their construction—which we call LaTeXMLMath —from LaTeXMLMath , namely : Smooth dependence of LaTeXMLMath on LaTeXMLMath in place of arbitrary dependence .
Dependence of test objects on LaTeXMLMath , i.e. , bounded paths LaTeXMLMath .
Asymptotically vanishing moments ( see below ) of test objects as compared to the stronger condition LaTeXMLMath for all LaTeXMLMath ( which is the naive analog of LaTeXMLMath in the case of LaTeXMLMath ) .
Condition ( i ) is necessary to guarantee smoothness of LaTeXMLMath with respect to LaTeXMLMath ( cf .
transformation ( LaTeXMLRef ) ) .
However , the technical prize to pay here is the use of calculus in infinite dimensional spaces : Colombeau and Meril in particular used the concept of Silva-differentiability LaTeXMLCite .
However , instead of giving the proofs they rather “ invited the reader to admit ” ( LaTeXMLCite , p. 362 ) the respective smoothness properties .
Change ( ii ) together with ( iii ) obviously was introduced to obtain a diffeomorphism invariant analog of the vanishing moment conditions defined above .
More precisely , define LaTeXMLMath to be the set of all smooth , bounded paths LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation It may now be shown ( LaTeXMLCite , §3 ) that these moment conditions indeed are invariant under the action of a diffeomorphism .
Colombeau and Meril chose their basic space to be LaTeXMLMath .
Note that this definition is not in accordance with the policy of “ separating definitions from testing ” as propagated above .
Moreover , their definition of the objects constituting the Colombeau algebra was not unambiguous .
However , following the interpretations of LaTeXMLCite and LaTeXMLCite , the testing process in LaTeXMLCite is defined by inserting test objects of the form LaTeXMLMath into the first slot of LaTeXMLMath .
More precisely , LaTeXMLEquation .
LaTeXMLEquation Finally the Colombeau-Meril algebra on LaTeXMLMath is defined as the quotient space LaTeXMLEquation .
Using these definitions , all the main properties of LaTeXMLMath carry over to LaTeXMLMath , with almost identical proofs .
Indeed , boundedness of the paths LaTeXMLMath in LaTeXMLMath assures similar estimates as in the case of single functions LaTeXMLMath .
Unfortunately , in addition to the ambiguities mentioned above the class of test objects as defined by Colombeau and Meril still is not preserved under the action of a diffeomorphism .
Nevertheless , despite these defects ( which , apparently , went unnoticed by nearly all workers in the field ) their construction was quoted and used many times ( among others LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
It was only in 1998 that J. Jelínek in LaTeXMLCite pointed out the error in LaTeXMLCite by giving a ( rather simple ) counter example which we shall discuss in a moment .
In the same paper , he presented another version of the theory which avoided ( some of ) the shortcomings of LaTeXMLCite and has to be considered as the second decisive step towards a diffeomorphism invariant version of a Colombeau algebra .
Taking a closer look on the nature of test objects as used by Colombeau and Meril , from ( LaTeXMLRef ) we see that the action of a diffeomorphism LaTeXMLMath introduces an additional LaTeXMLMath -dependence in the first slot of LaTeXMLMath .
This in turn may be exploited by giving an example of a function in LaTeXMLMath which is constant in LaTeXMLMath ( hence the estimates of the derivatives follow trivially ) but whose LaTeXMLMath -transform depending on LaTeXMLMath fails to be moderate .
More precisely , set LaTeXMLMath .
Then according to LaTeXMLRef we have LaTeXMLEquation .
We next discuss in some detail the algebra proposed by J. Jelínek in LaTeXMLCite which we shall call LaTeXMLMath LaTeXMLMath ” obviously stands for diffeomorphism invariant .
In fact Jelínek ’ s construction comes very close to diffeomorphism invariance although the last gaps were only closed in LaTeXMLCite , however , without the necessity to change the main definitions .
.
Analogously to the previous construction we start by listing the main features distinguishing LaTeXMLMath from LaTeXMLMath .
( Smooth ) dependence of test objects also on LaTeXMLMath .
In testing for moderateness test objects may take arbitrary values in LaTeXMLMath , independently of any moment condition .
While in the light of Jelínek ’ s counterexample ( i ) is compelling there seems to be no apparent necessity for ( ii ) .
Apparently ( ii ) widens the range of test objects which in turn reduces LaTeXMLMath resp .
LaTeXMLMath in size .
Yet is has to be admitted that by this reduction no generalized function of interest , neither for the further development of the theory nor in applications is lost .
For the construction of a diffeomorphism invariant Colombeau algebra omitting ( ii ) see Part II resp .
LaTeXMLCite .
Here , however , we focus on LaTeXMLMath which we regard as the standard diffeomorphism invariant algebra .
While Colombeau in his “ Elementary Introduction ” LaTeXMLCite chose to embed distributions via convolution with a mollifier , i.e. , ( cf .
also ( LaTeXMLRef ) above ) LaTeXMLEquation .
Jelínek ( following in fact earlier ideas of Colombeau presented in LaTeXMLCite ) decided to embed distributions by letting them act on the test function , i.e. , LaTeXMLEquation .
Since both embeddings are simply related by a translation , i.e. , LaTeXMLMath with LaTeXMLEquation .
LaTeXMLEquation they give rise to equivalent descriptions of virtually every Colombeau algebra , which we call C- resp .
J-formalism .
In LaTeXMLCite , Section 5 a translation formalism allowing to change from one setting to the other at any place of the construction was established and used in turn to clarify subtle questions of infinite-dimensional calculus .
Before giving the actual definitions of LaTeXMLMath we briefly comment on these issues .
Jelínek uses LaTeXMLCite as main reference while the presentation of LaTeXMLCite and LaTeXMLCite is based upon the more convenient calculus of LaTeXMLCite .
The basic idea of the latter is that a map LaTeXMLMath between locally convex spaces is smooth if it transports smooth curves in LaTeXMLMath to smooth curves in LaTeXMLMath , where the notion of smooth curves is straightforward ( via limits of difference quotients ) .
This notion of smoothness in general is weaker than Silva-differentiability but coincides with the latter on all the spaces used in the construction of Colombeau algebras .
Moreover , it displays the following decisive advantage in applications to partial differential equations : if one is to construct a generalized solution to a nonlinear singular equation this is done componentwise , i.e. , for fixed LaTeXMLMath .
Smoothness of the respective solution in LaTeXMLMath is then guaranteed already by classical theorems on smooth dependence of solutions on parameters .
We now give a brief description of Jelínek ’ s algebra LaTeXMLMath : contrary to the original presentation using the LaTeXMLMath -formalism for its better familiarity ( however , omitting the superscript LaTeXMLMath from now on ) .
For a comparison of the respective features of the two formalisms we refer to the table in LaTeXMLCite , Section 5 .
Apart from closing a gap in the construction of LaTeXMLCite the presentation in LaTeXMLCite supplies those parts of the resp .
arguments which have not been included in LaTeXMLCite .
This applies , in particular , to the questions of smoothness and stability of LaTeXMLMath , LaTeXMLMath w.r.t .
differentiation and the fact that transformed test objects are not defined on the whole of LaTeXMLMath in general .
Forced by the choice of the embedding ( LaTeXMLRef ) we define the basic space to be LaTeXMLEquation .
Partial derivatives on LaTeXMLMath —which will become the derivatives in the algebra—in the C-formalism are simply defined by LaTeXMLEquation .
Recall that test objects have to depend on LaTeXMLMath and LaTeXMLMath , in particular are chosen to be smooth , bounded paths LaTeXMLMath ( resp .
LaTeXMLMath ) .
Denoting their space by LaTeXMLMath we are able to formulate the tests for moderateness and negligibility .
Definition .
LaTeXMLMath is called moderate if LaTeXMLMath LaTeXMLEquation .
The set of all moderate elements LaTeXMLMath will be denoted by LaTeXMLMath .
LaTeXMLMath is called negligible if LaTeXMLMath LaTeXMLEquation .
The set of all negligible elements LaTeXMLMath will be denoted by LaTeXMLMath .
The key ingredients in proving diffeomorphism invariance as well as stability with respect to derivatives ( i.e. , that the LaTeXMLMath -derivative of a moderate resp .
negligible function again is moderate resp .
negligible ; this becomes a peculiar issue due to the additional LaTeXMLMath -dependence of LaTeXMLMath ) are several equivalent formulations of the tests given above .
To settle the question of stability w.r.t .
differentiation Jelínek introduced an alternate , yet equivalent form of tests involving differentials of LaTeXMLMath with respect to the test function-slot denoted by LaTeXMLMath .
( LaTeXMLCite , Th .
17 , resp .
Th .
18 , LaTeXMLMath ) .
We only formulate the respective test for moderateness ( the case of negligibility being analogous ) and refer to the original for the ingenious proofs .
We presume that the author was completely aware of the role Ths .
17 and 18 had to play in this respect yet for some reasons he decided not to address this issue .
Theorem .
LaTeXMLMath is a member of LaTeXMLMath if and only if the following condition is satisfied : LaTeXMLEquation .
LaTeXMLEquation uniformly for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Here LaTeXMLMath denotes the tangent space of LaTeXMLMath and the operator LaTeXMLMath is derived from LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation Finally to prove that LaTeXMLMath is moderate if LaTeXMLMath was , observe that LaTeXMLMath .
Then the claim follows from LaTeXMLEquation .
We now turn to the central issue of diffeomorphism invariance .
First we present a heuristical calculation which clearly shows which path has to be pursued .
Suppose we want to prove moderateness of LaTeXMLMath .
Given LaTeXMLMath then we would have to estimate LaTeXMLEquation .
Hence we would need LaTeXMLMath to pass a test for moderateness w.r.t .
test objects of the form ( denoting by LaTeXMLMath the projection to the fist component ) LaTeXMLEquation .
But unfortunately LaTeXMLMath since it is only defined if LaTeXMLMath , whereas we want LaTeXMLMath to be a test function on the whole of LaTeXMLMath .
However , LaTeXMLMath belongs to a class of test objects providing an apparently weaker , yet , as it finally turns out , equivalent test .
More precisely , from LaTeXMLCite , Th .
10.5 we have that LaTeXMLMath is moderate if and only if it fulfills the following condition ( Z ) Condition ( Z ) is one of 6 tests proved to be equivalent in LaTeXMLCite , Theorem 10.5 called Theorem A–Z there .
Note , however , that this neither indicates that the authors of LaTeXMLCite originally intended to give 26 equivalent tests , nor that ( Z ) for some mysterious reason was considered to be the ultimate condition ; rather it was invented during a train ride returning from a workshop in Novi Sad to Vienna and “ Z ” simply stands for “ Zug ” which is the German word for train .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath and for LaTeXMLMath the following holds : For each LaTeXMLMath there exists LaTeXMLMath and a subset LaTeXMLMath of LaTeXMLMath which is open in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath is smooth on LaTeXMLMath , and for all LaTeXMLMath , LaTeXMLMath is bounded in LaTeXMLMath .
Now diffeomorphism invariance of the notion of moderateness is established by the following Theorem .
Let LaTeXMLMath be a diffeomorphism and LaTeXMLMath .
Define LaTeXMLMath ) by LaTeXMLEquation .
For LaTeXMLMath , set LaTeXMLEquation .
Then LaTeXMLMath satisfies the requirements specified for test objects in condition ( Z ) .
In some more detail assume LaTeXMLMath to be moderate .
We show that LaTeXMLMath passes the test used in Definition LaTeXMLRef ( i ) .
Indeed given LaTeXMLMath , LaTeXMLMath and LaTeXMLMath define LaTeXMLMath as in the preceding theorem .
Then by the chain rule LaTeXMLEquation where each LaTeXMLMath is bounded on LaTeXMLMath .
Since LaTeXMLMath satisfies condition ( Z ) the claim follows .
However , matters become more complicated in the case of negligibility .
First note that the resp .
test objects take values in LaTeXMLMath ( LaTeXMLMath ) which is not a diffeomorphism invariant property .
The way out is provided by the re-introduction of asymptotically vanishing moments ( cf .
LaTeXMLRef ) into the theory by building up ( another ) equivalent test using this notion .
Indeed Jelínek ( LaTeXMLCite , 18 ( 4 LaTeXMLMath ) ) has formulated such a condition which , however , unfortunately is not equivalent to the notion of negligibility as defined above .
Moreover , this condition is so strong that even LaTeXMLMath , hence the property of LaTeXMLMath being an algebra homomorphism on LaTeXMLMath is lost .
However , in LaTeXMLCite , Section 7 this flaw was removed , namely by demanding also all derivatives of the test objects to have asymptotically vanishing moments .
More precisely we say that a test object LaTeXMLMath is of type LaTeXMLMath if on ( a given ) LaTeXMLMath LaTeXMLEquation .
Then we have Theorem .
LaTeXMLMath is negligible if and only if LaTeXMLMath LaTeXMLMath LaTeXMLMath such that LaTeXMLMath of type LaTeXMLMath : LaTeXMLEquation .
Diffeomorphism invariance of the notion of negligibility is now established by the above theorem in conjunction with an analog of Theorem LaTeXMLRef , as well as an analog of condition ( Z ) above .
Finally we define our main object of desire .
Definition .
The diffeomorphism invariant Colombeau algebra on LaTeXMLMath is defined as the quotient LaTeXMLEquation .
Summing up we have constructed a differential algebra LaTeXMLMath in a diffeomorphism invariant way , in particular allowing for a diffeomorphism invariant embedding of distributions .
Moreover , LaTeXMLMath ( as usual ) is a fine sheaf of differential algebras ( LaTeXMLCite , Section 8 ) .
We finally turn to the issue of commutativity of the embedding with partial derivatives in the algebra .
This will guarantee that LaTeXMLMath indeed possesses all the favorable properties of a Colombeau algebra .
To this end it is useful to change to the J-formalism .
Recall from ( LaTeXMLRef ) that derivatives in the C-formalism are just given by partial derivatives .
Using the translation formalism of LaTeXMLCite we derive that in the J-formalism , i.e. , on LaTeXMLMath we have LaTeXMLEquation .
We now see immediately that if LaTeXMLMath then LaTeXMLMath ( cf .
( LaTeXMLRef ) ) is independent of LaTeXMLMath hence the second term in ( LaTeXMLRef ) vanishes .
Moreover , since LaTeXMLMath is linear in LaTeXMLMath , LaTeXMLMath which is exactly the LaTeXMLMath -image of LaTeXMLMath .
We prove that in the core model below LaTeXMLMath the following holds true .
Let LaTeXMLMath .
There is then a sequence LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath do we have that LaTeXMLMath is stationary in LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath for all limit ordinals LaTeXMLMath and for all LaTeXMLMath do we have that LaTeXMLMath is mutually stationary if and only if the range of LaTeXMLMath is finite .
More on mutual stationarity Preliminary version Ralf Schindler Institut für Formale Logik , Universität Wien , 1090 Wien , Austria rds @ logic.univie.ac.at http : //www.logic.univie.ac.at/ LaTeXMLMath rds/ Let LaTeXMLMath be a non-empty set of regular cardinals .
Recall that LaTeXMLMath is called mutually stationary if and only if for each large enough regular cardinal LaTeXMLMath and for each model LaTeXMLMath with universe LaTeXMLMath there is some LaTeXMLMath such that for all LaTeXMLMath , LaTeXMLMath .
We refer the reader to LaTeXMLCite ( in particular , to LaTeXMLCite ) .
It is an open problem to decide whether there is a model of set theory in which LaTeXMLMath must be mutually stationary provided each individual LaTeXMLMath is stationary .
Foreman and Magidor have shown that in LaTeXMLMath there is some LaTeXMLMath such that each LaTeXMLMath is stationary , but LaTeXMLMath is not mutually stationary ( cf .
LaTeXMLCite ) .
The purpose of this paper is to extend their result LaTeXMLCite to the core model below LaTeXMLMath which we introduced in LaTeXMLCite .
We shall need the following lemma , which is essentially due to Baumgartner LaTeXMLCite .
Let LaTeXMLMath , and let LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath be non-empty sets of successor cardinals such that LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLMath be mutually stationary for all LaTeXMLMath and such that LaTeXMLMath .
Then LaTeXMLMath is mutually stationary as well .
Proof .
It certainly suffices to prove LaTeXMLRef for LaTeXMLMath .
Let LaTeXMLMath be the cardinal predecessor of LaTeXMLMath .
Let LaTeXMLMath be a model expanding LaTeXMLMath for some large regular LaTeXMLMath .
As LaTeXMLMath is mutually stationary we may pick some LaTeXMLMath such that LaTeXMLMath and for all LaTeXMLMath , LaTeXMLMath .
Pick LaTeXMLMath such that for all LaTeXMLMath , LaTeXMLMath is cofinal in LaTeXMLMath .
Expand LaTeXMLMath by LaTeXMLMath to get LaTeXMLMath .
As LaTeXMLMath is mutually stationary we may pick some LaTeXMLMath such that LaTeXMLMath and for all LaTeXMLMath , LaTeXMLMath .
However , due to the presence of LaTeXMLMath , we ’ ll also have that for all LaTeXMLMath , LaTeXMLMath .
As LaTeXMLMath , we are done .
LaTeXMLMath Lemma LaTeXMLRef We can now state and prove our main result .
Our proof will closely follow LaTeXMLCite to a certain extent .
Suppose that LaTeXMLMath , where LaTeXMLMath is the core model below LaTeXMLMath from LaTeXMLCite .
Let LaTeXMLMath .
There is then a sequence LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath do we have that LaTeXMLMath is stationary in LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath for all limit ordinals LaTeXMLMath and for all LaTeXMLMath do we have that LaTeXMLMath is mutually stationary if and only if the range of LaTeXMLMath is finite .
Proof .
We commence by defining the sequence LaTeXMLMath .
If LaTeXMLMath is a singular ordinal , then we let LaTeXMLMath be the lexicographically least tuple LaTeXMLMath such that LaTeXMLEquation is cofinal in LaTeXMLMath .
For an ordinal LaTeXMLMath and a natural number LaTeXMLMath we define LaTeXMLEquation .
We are now going to show that LaTeXMLMath witnesses the truth of Theorem LaTeXMLRef .
We shall first prove that if LaTeXMLMath is a limit ordinal and LaTeXMLMath is such that LaTeXMLMath is finite then LaTeXMLMath is mutually stationary .
By Lemma LaTeXMLRef it will be enough if we prove this under the assumption that LaTeXMLMath for some LaTeXMLMath .
Let LaTeXMLMath be a model with universe LaTeXMLMath , where LaTeXMLMath is a large regular cardinal .
Let LaTeXMLMath be least such that LaTeXMLMath , and let LaTeXMLMath be the LaTeXMLMath ordinal LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLEquation .
One can then argue as for LaTeXMLCite that for every LaTeXMLMath , LaTeXMLMath .
The only thing to notice that the use of the condensation lemma for LaTeXMLMath can be replaced by LaTeXMLCite in a straightforward way .
We shall now prove that if LaTeXMLMath is a limit ordinal and LaTeXMLMath is such that LaTeXMLMath is mutually stationary then LaTeXMLMath is finite .
Let LaTeXMLMath be a large regular cardinal , and let LaTeXMLEquation be such that for all LaTeXMLMath do we have that LaTeXMLMath .
In particular LaTeXMLMath for each such LaTeXMLMath .
Let LaTeXMLEquation be such that LaTeXMLMath is transitive , and let LaTeXMLMath .
Let LaTeXMLMath , and let LaTeXMLMath be least such that there is some LaTeXMLMath .
Note that LaTeXMLMath , because otherwise LaTeXMLMath would be an extender which collapses the cardinal LaTeXMLMath .
Let LaTeXMLMath denote the ( padded ) coiteration of LaTeXMLMath with LaTeXMLMath .
We ’ ll have that LaTeXMLMath , and that LaTeXMLMath .
We first want to show that LaTeXMLMath is trivial ( i.e. , that LaTeXMLMath doesn ’ t move in the comparison with LaTeXMLMath ) , and that if LaTeXMLMath is least such that LaTeXMLMath then LaTeXMLMath is set-sized and LaTeXMLMath .
Claim 1 .
Let LaTeXMLMath be such that LaTeXMLMath .
Then for no LaTeXMLMath do we have that LaTeXMLMath and LaTeXMLMath .
Proof .
Otherwise LaTeXMLMath would be an extender which collapses the cardinal LaTeXMLMath .
LaTeXMLMath Claim 1 Claim 2 .
LaTeXMLMath is trivial .
Proof .
Assume not .
Let LaTeXMLMath , where LaTeXMLMath is least in LaTeXMLMath , and let LaTeXMLMath .
Let LaTeXMLMath be minimal with LaTeXMLMath .
We let LaTeXMLMath enumerate the cardinals of LaTeXMLMath in the half-open interval LaTeXMLMath , and we let LaTeXMLMath for LaTeXMLMath .
For each LaTeXMLMath we let LaTeXMLMath be the least LaTeXMLMath such that LaTeXMLMath , and we let LaTeXMLMath be the largest initial segment LaTeXMLMath of LaTeXMLMath such that all bounded subsets of LaTeXMLMath which are in LaTeXMLMath are in LaTeXMLMath as well .
By Claim 1 we shall have that LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
Moreover , LaTeXMLMath is sound above LaTeXMLMath .
We let LaTeXMLMath denote the phalanx LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath be the padded iteration trees arising from the comparison of LaTeXMLMath with LaTeXMLMath .
We understand that LaTeXMLMath either has a last ill-founded model , or else that LaTeXMLMath and LaTeXMLMath are lined up .
The following says that it is the latter which will hold .
Subclaim 1 .
LaTeXMLMath is coiterable with LaTeXMLMath .
Proof .
Suppose that LaTeXMLMath has a last ill-founded model .
Let LaTeXMLMath be such that LaTeXMLMath is regular and large enough , LaTeXMLMath is countable and transitive , and LaTeXMLMath .
Then LaTeXMLMath witnesses that LaTeXMLMath is not iterable .
Fix LaTeXMLMath for a moment .
Let LaTeXMLEquation where LaTeXMLMath is such that LaTeXMLMath .
Let LaTeXMLMath denote the canonical embedding from LaTeXMLMath into LaTeXMLMath .
Notice that by LaTeXMLMath it is clear that LaTeXMLMath is not cofinal in LaTeXMLMath .
Hence if LaTeXMLMath denotes the canonical embedding from LaTeXMLMath into LaTeXMLMath then LaTeXMLMath is the critical point of LaTeXMLMath .
Unfortunately , LaTeXMLMath might not be premouse but rather a proto-mouse ; this will in fact be the case if LaTeXMLMath has a top extender with critical point LaTeXMLMath and LaTeXMLMath , as then LaTeXMLMath is discontinuous at LaTeXMLMath .
Let us therefore define an object LaTeXMLMath as follows .
We set LaTeXMLMath if LaTeXMLMath is a premouse .
Otherwise , if LaTeXMLMath is the critical point of the top extender of LaTeXMLMath , we let LaTeXMLEquation where LaTeXMLMath is the top extender fragment of LaTeXMLMath , LaTeXMLMath is maximal with LaTeXMLEquation and LaTeXMLMath is such that LaTeXMLMath .
Now because LaTeXMLMath and LaTeXMLMath are both sound above LaTeXMLMath we can apply LaTeXMLCite ( or an elaboration of it in case that LaTeXMLMath is a proto-mouse ) and deduce that LaTeXMLMath .
But as LaTeXMLMath , this means that LaTeXMLMath .
Therefore LaTeXMLMath is an initial segment of LaTeXMLMath .
Now if LaTeXMLMath denotes the phalanx LaTeXMLEquation then , due to the existence of the family of maps LaTeXMLEquation we know that LaTeXMLMath can not be iterable , as LaTeXMLMath is not iterable .
However , any iteration of LaTeXMLMath can be construed as an iteration of LaTeXMLMath , and thus in turn of LaTeXMLMath .
But LaTeXMLMath is iterable .
Contradiction !
LaTeXMLMath Subclaim 1 Now notice that LaTeXMLMath is trivial , LaTeXMLMath , and that LaTeXMLMath will be the first extender used in LaTeXMLMath .
By LaTeXMLCite , we ’ ll then LaTeXMLMath is below LaTeXMLMath .
If the assumption that LaTeXMLMath is below LaTeXMLMath is dropped then at the time of writing I don ’ t see how to prove the iterability of the phalanx needed to verify Claim 2. in fact have that no extender from LaTeXMLMath will be applied to ( an inital segment of ) the last model of the phalanx LaTeXMLMath , LaTeXMLMath .
We may therefore finally argue as in the proof of LaTeXMLCite to derive a contradiction .
LaTeXMLMath Claim 2 Now let LaTeXMLMath enumerate the cardinals of LaTeXMLMath in the half-open interval LaTeXMLMath , and let LaTeXMLMath for LaTeXMLMath .
For each LaTeXMLMath we let LaTeXMLMath be the least LaTeXMLMath such that LaTeXMLMath , we let LaTeXMLMath be the largest initial segment LaTeXMLMath of LaTeXMLMath such that all bounded subsets of LaTeXMLMath which are in LaTeXMLMath are in LaTeXMLMath as well , and we let LaTeXMLMath be the LaTeXMLMath such that LaTeXMLEquation .
Notice that LaTeXMLMath will always be defined by Claims 1 and 2 .
The following is easy to verify .
Claim 3 .
LaTeXMLMath is finite .
We now let for each LaTeXMLMath , LaTeXMLEquation .
Let LaTeXMLMath denote the canonical embedding from LaTeXMLMath into LaTeXMLMath .
Unfortunately , again even if it is well-founded , LaTeXMLMath might not be premouse but rather a proto-mouse ; this will in fact be the case if LaTeXMLMath has a top extender with critical point LaTeXMLMath and LaTeXMLMath , as then LaTeXMLMath is discontinuous at LaTeXMLMath .
Let us therefore define , inductively for LaTeXMLMath , objects LaTeXMLMath as follows .
We understand that we let the construction break down as soon as one of the models defined is ill-founded .
Simultaneously to their definition we notice that LaTeXMLMath , and we define LaTeXMLMath as the least LaTeXMLMath such that LaTeXMLMath .
We set LaTeXMLMath if LaTeXMLMath is a premouse .
Otherwise , if LaTeXMLMath is the critical point of the top extender of LaTeXMLMath , we let LaTeXMLEquation where LaTeXMLMath is the top extender fragment of LaTeXMLMath .
It is straightforward to see that if all the LaTeXMLMath are well-defined then LaTeXMLMath , so that by Claim 3 we immediately get : Claim 4 .
LaTeXMLMath is finite .
We now aim to prove that all the LaTeXMLMath are well-defined .
In fact : Claim 5 .
For each LaTeXMLMath do we have that LaTeXMLMath .
Proof .
Fix LaTeXMLMath .
We assume that LaTeXMLMath is a premouse , leaving the rest as an exercise for the reader .
Let LaTeXMLMath .
By standard arguments in oder to show Claim 5 it will certainly be enough to prove the following .
Subclaim 2 .
LaTeXMLMath is iterable .
Proof .
Let LaTeXMLMath be such that LaTeXMLMath is regular and large enough , LaTeXMLMath is countable and transitive , and LaTeXMLMath .
It suffices to prove that LaTeXMLMath can be embedded into an iterable phalanx .
Let LaTeXMLMath .
Let LaTeXMLEquation and let LaTeXMLMath be the canonical embedding from LaTeXMLMath into LaTeXMLMath .
As LaTeXMLMath , the critical point of LaTeXMLMath will be LaTeXMLMath .
Using LaTeXMLCite we then get as above that in fact LaTeXMLMath .
We claim that LaTeXMLEquation defines a sufficiently elementary embedding from LaTeXMLMath into LaTeXMLMath .
Here , LaTeXMLMath and LaTeXMLMath is a LaTeXMLMath good function with parameters from LaTeXMLMath .
The reason for this is that for appropriate LaTeXMLMath and LaTeXMLMath do we have that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This now implies that we my embedded LaTeXMLMath into the phalanx LaTeXMLEquation .
However , LaTeXMLMath , and hence this latter phalanx is iterable .
LaTeXMLMath Subclaim 2 LaTeXMLMath Claim 5 We have shown that if LaTeXMLMath , say LaTeXMLMath , then LaTeXMLMath .
By Claim 4 this establishes the Theorem .
LaTeXMLMath Theorem LaTeXMLRef Many questions remain open .
Can Theorem LaTeXMLRef be extended to the core model of LaTeXMLCite ?
Or can Theorem LaTeXMLRef even be extended to fine structural models which do not know how to fully iterate themselves ?
Finally : Can one use methods provided by the current paper to get a reasonable lower bound for the consistency strength of the assumption that LaTeXMLMath must be mutually stationary provided every individual LaTeXMLMath is stationary ?
Let LaTeXMLMath be the symmetric operator given by the restriction of LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath is a self-adjoint operator on the Hilbert space LaTeXMLMath and LaTeXMLMath is a linear dense set which is closed with respect to the graph norm on LaTeXMLMath , the operator domain of LaTeXMLMath .
We show that any self-adjoint extension LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath can be additively decomposed by the sum LaTeXMLMath , where both the operators LaTeXMLMath and LaTeXMLMath take values in the strong dual of LaTeXMLMath .
The operator LaTeXMLMath is the closed extension of LaTeXMLMath to the whole LaTeXMLMath whereas LaTeXMLMath is explicitly written in terms of a ( abstract ) boundary condition depending on LaTeXMLMath and on the extension parameter LaTeXMLMath , a self-adjoint operator on an auxiliary Hilbert space isomorphic ( as a set ) to the deficiency spaces of LaTeXMLMath .
The explicit connection with both Kreĭn ’ s resolvent formula and von Neumann ’ s theory of self-adjoint extensions is given .
2328 Given a self-adjoint operator LaTeXMLMath , let LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath is a dense linear subspace which is closed with respect to the graph norm .
Then LaTeXMLMath is a closed , densely defined , symmetric operator .
Since LaTeXMLMath , LaTeXMLMath is not essentially self-adjoint , as LaTeXMLMath is a non-trivial extension of LaTeXMLMath , and , by the famed von Neumann ’ s formulae LaTeXMLCite , we know that LaTeXMLMath has an infinite family of self-adjoint extensions LaTeXMLMath parametrized by the unitary maps LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath , where LaTeXMLMath Kernel LaTeXMLMath denotes the deficiency spaces .
In section 2 we define a family LaTeXMLMath of extensions of LaTeXMLMath by means of a Kreĭn-like formula i.e .
by explicitly giving its resolvent LaTeXMLMath ( see Theorem 2.1 ) .
By using the approach developed in LaTeXMLCite , we describe the domain of LaTeXMLMath in terms of the boundary condition LaTeXMLMath , where LaTeXMLMath is a surjective continuos linear mapping with Kernel LaTeXMLMath , LaTeXMLMath is self-adjoint and LaTeXMLMath is a Hilbert space isomorphic ( as a set ) to LaTeXMLMath .
In section 3 we use the resolvent LaTeXMLMath given in Theorem 2.1 to re-write LaTeXMLMath in a more appealing way as a sum LaTeXMLMath where both LaTeXMLMath and LaTeXMLMath take values in the strong dual ( with respect to the graph norm ) of LaTeXMLMath ( see Theorem 3.1 ) ; LaTeXMLMath is nothing else that the closed extension of LaTeXMLMath to the whole Hilbert space LaTeXMLMath and LaTeXMLMath is explicitly given in terms of the maps LaTeXMLMath and LaTeXMLMath giving the boundary conditions .
This result gives an extension , and a rephrasing in terms of boundary conditions , of the results obtained in LaTeXMLCite ( and references therein , in particular LaTeXMLCite ) , where LaTeXMLMath is strictly positive and LaTeXMLMath is closed in LaTeXMLMath ( see Remark 3.5 ) .
As regards boundary conditions the reader is also refered to LaTeXMLCite , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath the kernel of the evaluation map along a regular submanifold , and to LaTeXMLCite , where LaTeXMLMath is an arbitrary injective self-adjoint operator .
Successively , is section 4 , we study the connection of the self-adjoint extensions defined in the previuos sections with the ones given by von Neumann ’ s theory LaTeXMLCite .
We prove ( see Theorem 4.1 ) that the operator LaTeXMLMath defined in Theorem 3.4 , of which the self-adjoint LaTeXMLMath is a restriction , coincides with LaTeXMLMath ; moreover we explicitly define a map on self-adjoint operators LaTeXMLMath to unitary operators LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath denotes the von Neumann ’ s extension corresponding to LaTeXMLMath .
Such correspondence is then explicitly inverted ( see Theorem 4.3 ) .
This shows ( see Corollary 4.4 ) that LaTeXMLMath coincides with a self-adjoint extension LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath if and only if the boundary condition LaTeXMLMath holds for some self-adjoint operator LaTeXMLMath .
In section 5 we conclude with some examples both in the case of finite and infinite deficiency indices .
Example 5.1 ( also see Remark 4.2 ) shows that , in the case dim LaTeXMLMath , our results reproduce the theory of finite rank perturbations as given in LaTeXMLCite , §3.1 , and thus they can be viewed as an extension of such a theory to the infinite rank case .
In example 5.2 we give two examples in the infinite rank case : infinitely many point interaction in three dimensions and singular perturbations , supported on LaTeXMLMath -sets with LaTeXMLMath , of traslation invariant pseudo-differential operators with domain the Sobolev space LaTeXMLMath .
Given a Banach space LaTeXMLMath we denote by LaTeXMLMath its strong dual .
LaTeXMLMath denotes the space of linear operators from the Banach space LaTeXMLMath to the Banach space LaTeXMLMath ; LaTeXMLMath .
LaTeXMLMath denotes the Banach space of bounded , everywhere defined , linear operators on the Banach space LaTeXMLMath to the Banach space LaTeXMLMath ; LaTeXMLMath .
Given LaTeXMLMath densely defined , the closed operator LaTeXMLMath is the adjoint of LaTeXMLMath i.e .
LaTeXMLEquation .
If LaTeXMLMath is a complex Hilbert space with scalar product ( conjugate-linear with respect to the first variable ) LaTeXMLMath , then LaTeXMLMath denotes the conjugate-linear isomorphism defined by LaTeXMLEquation .
The Hilbert adjoint LaTeXMLMath of the densely defined linear operator LaTeXMLMath is defined as LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath denote Fourier transform and convolution respectively .
LaTeXMLMath , LaTeXMLMath , is the usual scale of Sobolev-Hilbert spaces , i.e .
LaTeXMLMath is the space of tempered distributions with a Fourier transform which is square integrable with respect to the measure with density LaTeXMLMath .
Given the Hilbert space LaTeXMLMath with scalar product LaTeXMLMath ( we denote by LaTeXMLMath the corresponding norm and put LaTeXMLMath ) , let LaTeXMLMath be a self-adjoint operator and let LaTeXMLMath be a linear dense set which is closed with respect to the graph norm on LaTeXMLMath .
We denote by LaTeXMLMath the Hilbert space given by the set LaTeXMLMath equipped with the scalar product LaTeXMLMath leading to the graph norm , i.e .
LaTeXMLEquation .
We remark that in the sequel we will avoid to identify LaTeXMLMath with its dual .
Indeed we will use the duality map induced by the scalar product on LaTeXMLMath ( see the next section for the details ) .
Being LaTeXMLMath closed we have LaTeXMLMath and we can then consider the orthogonal projection LaTeXMLMath .
From now on , since this gives advantages in concrete applications where usually a variant of LaTeXMLMath is what is known in advance , more generally we will consider a linear map LaTeXMLEquation where LaTeXMLMath is a Hilbert space with scalar product LaTeXMLMath and corrsponding norm LaTeXMLMath , such that LaTeXMLEquation and LaTeXMLEquation the bar denoting here the closure in LaTeXMLMath .
We put LaTeXMLEquation .
By ( 2.1 ) one has LaTeXMLMath so that LaTeXMLEquation .
Regarding ( 2.2 ) we have the following Hypothesis ( 2.2 ) is equivalent to LaTeXMLEquation when one uses the embedding of LaTeXMLMath into LaTeXMLMath given by the map LaTeXMLMath .
Defining as usual the annihilator of LaTeXMLMath by LaTeXMLEquation one has that denseness of LaTeXMLMath is equivalent to LaTeXMLEquation .
Since LaTeXMLMath the proof is concluded if the range of LaTeXMLMath is closed .
This follows from the closed range theorem since the range of LaTeXMLMath is closed by the surjectivity hypothesis .
∎ Being LaTeXMLMath the resolvent set of LaTeXMLMath , we define LaTeXMLMath , LaTeXMLMath , by LaTeXMLEquation and we then introduce , for any LaTeXMLMath , the two linear operators LaTeXMLMath and LaTeXMLMath by LaTeXMLEquation .
By ( 2.2 ) one has LaTeXMLEquation and , as an immediate consequence of the first resolvent identity for LaTeXMLMath ( see LaTeXMLCite , Lemma 2.1 ) LaTeXMLEquation .
These relations imply LaTeXMLEquation and LaTeXMLEquation .
By LaTeXMLCite ( combining Theorem 2.1 , Proposition 2.1 , Lemma 2.2 , Remarks 2.10 , 2.12 and 2.13 ) one then obtains the following Given LaTeXMLMath define LaTeXMLEquation and , given then any self-adjoint operator LaTeXMLMath , define LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
Then LaTeXMLMath is the resolvent of the self-adjoint extension of LaTeXMLMath defined by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Here we just give the main steps of the proof refering to LaTeXMLCite , §2 , for the details .
One starts writing the presumed resolvent of an extension LaTeXMLMath of LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath has to be determined .
Self-adjointness requires LaTeXMLMath or , equivalently , LaTeXMLEquation .
Therefore posing LaTeXMLMath , where LaTeXMLMath , ( 2.6 ) is equivalent to LaTeXMLEquation .
The resolvent identity LaTeXMLEquation is then equivalent to LaTeXMLEquation .
Suppose now that there exist a ( necessarily closed ) operator LaTeXMLEquation and an open set LaTeXMLMath , invariant with respect to complex conjugation , such that LaTeXMLEquation .
Then ( 2.9 ) forces LaTeXMLMath to satisfy the relation LaTeXMLEquation whereas ( 2.7 ) , at least in the case LaTeXMLMath is densely defined , and has a bounded inverse given by LaTeXMLMath as we are pretending , is equivalent to LaTeXMLEquation .
By LaTeXMLCite , Lemma 2.2 , for any self-adjoint LaTeXMLMath , the linear operator LaTeXMLEquation satisfies ( 2.10 ) , ( 2.11 ) and , by LaTeXMLCite , Proposition 2.1 , has a bounded inverse for any LaTeXMLMath ( at this point hypothesis ( 2.1 ) is used ) .
Therefore ( see the proof of Theorem 2.1 in LaTeXMLCite ) LaTeXMLEquation is the resolvent of a self-adjoint operator LaTeXMLMath ( here hypotheses ( 2.2 ) is needed ) .
For any LaTeXMLMath one has LaTeXMLEquation .
LaTeXMLEquation the definition of LaTeXMLMath being LaTeXMLMath -independent thanks to resolvent identity ( 2.8 ) .
Being LaTeXMLMath injective , ( 2.3 ) and ( 2.5 ) imply LaTeXMLEquation and so the definition LaTeXMLEquation is LaTeXMLMath -independent .
Therefore any LaTeXMLMath can be equivalently re-written as LaTeXMLEquation where LaTeXMLMath and LaTeXMLEquation .
This implies , for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Conversely any LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , admits the decomposition LaTeXMLMath , where LaTeXMLEquation .
Note that LaTeXMLMath by ( 2.5 ) and LaTeXMLMath .
∎ The results quoted in the previous theorem are consequences of an alternative version of Kreĭn ’ s resolvent formula .
The original one was obtained in LaTeXMLCite , LaTeXMLCite , LaTeXMLCite for the cases where dim LaTeXMLMath , dim LaTeXMLMath , dim LaTeXMLMath respectively ; also see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite for more recent formulations .
In standard Kreĭn ’ s formula ( usually written with LaTeXMLMath ) the main ingredient is the orthogonal projection LaTeXMLMath whereas we used , exploiting the a priori knowledge of the self-adjoint operator LaTeXMLMath , the map LaTeXMLMath , which plays the role of the orthogonal projection LaTeXMLMath .
Thus the knowledge of LaTeXMLMath is not needed .
The version given in LaTeXMLCite allows LaTeXMLMath to be not surjective and LaTeXMLMath can be a Banach space ; the use of the map LaTeXMLMath simplifies the exposition and makes easier to work out concrete applications .
Indeed , as we already said , frequently what is explicitely known is the map LaTeXMLMath and LaTeXMLMath is then simply defined as its kernel : see the many examples in LaTeXMLCite where LaTeXMLMath is the trace ( restriction ) map along some null subset of LaTeXMLMath and LaTeXMLMath is a ( pseudo- ) differential operator .
Moreover this approach allows a natural formulation in terms of the boundary condition LaTeXMLMath .
Note that , since LaTeXMLMath if and only if LaTeXMLMath , once the reference point LaTeXMLMath has been chosen , the decomposition LaTeXMLMath of a generic element LaTeXMLMath of LaTeXMLMath by a regular part LaTeXMLMath and a singular one LaTeXMLMath is univocal .
As regards the definition of LaTeXMLMath , the one given in the theorem above is not the only possible definition of the operator LaTeXMLMath .
Any other not necessarily bounded , densely defined operator satisfying LaTeXMLEquation .
LaTeXMLEquation and such that LaTeXMLMath is boundedly invertible would suffice ; moreover hypothesis ( 2.1 ) is not necessary ( see LaTeXMLCite , Theorem 2.1 ) ; note that , once LaTeXMLMath is given , LaTeXMLMath univocally defines LaTeXMLMath and hence LaTeXMLMath itself .
For alternative choices of LaTeXMLMath we refer to LaTeXMLCite ; also see LaTeXMLCite where it is shown how , under the hypotheses Kernel LaTeXMLMath and LaTeXMLMath , it is always possible to take LaTeXMLMath in Theorem 2.1 ( at the expense of having then LaTeXMLMath in the completion of LaTeXMLMath with respect to the norm LaTeXMLMath ) .
However we remark that any different choice ( either of LaTeXMLMath or of the operator LaTeXMLMath itself ) does not change the family of extensions as a whole .
In the case LaTeXMLMath has a non-empty real resolvent set , by LaTeXMLCite , Remark 2.7 , if in Theorem 2.1 one consider only the sub-family of extensions in which the LaTeXMLMath ’ s have bounded inverses , then one can take LaTeXMLMath .
More generally one can take LaTeXMLMath independently of the invertibility of LaTeXMLMath ; however this could give rise to implicit conditions ( related to the location of the spectrum of LaTeXMLMath ) on the choice of LaTeXMLMath .
We define the pre-Hilbert space LaTeXMLMath as the set LaTeXMLMath equipped with the scalar product LaTeXMLEquation .
We denote then by LaTeXMLMath the Hilbert space given by the completion of LaTeXMLMath .
We will avoid to identify LaTeXMLMath and LaTeXMLMath with their duals ; indeed , see Lemma 3.1 below , we will identify LaTeXMLMath with LaTeXMLMath .
As usual LaTeXMLMath will be treated as a ( dense ) subspace of LaTeXMLMath by means of the canonical embedding LaTeXMLEquation which associates to LaTeXMLMath the set of all the Cauchy sequences converging to LaTeXMLMath .
Considering also the canonical embedding ( with dense range ) LaTeXMLEquation we can then define the conjugate linear operator LaTeXMLEquation as the unique bounded extension of LaTeXMLEquation .
Analogously we define the conjugate linear operator LaTeXMLEquation as the unique bounded extension of LaTeXMLEquation .
These definitions immediately lead to the following One has LaTeXMLEquation so that LaTeXMLEquation .
We will denote by LaTeXMLEquation the pairing between LaTeXMLMath and LaTeXMLMath .
It is nothing else that the extension of the scalar product of LaTeXMLMath , being LaTeXMLEquation .
We consider now the linear operator LaTeXMLEquation .
Since LaTeXMLEquation the operator LaTeXMLMath has an unique extension LaTeXMLEquation .
Let LaTeXMLMath be the adjoint of the linear operator LaTeXMLMath when viewed as an element of LaTeXMLMath .
Then one has LaTeXMLEquation .
Being LaTeXMLMath injective , by continuity and density the thesis follows from the identity LaTeXMLEquation ∎ If we use the symbol LaTeXMLMath to denote the linear operator LaTeXMLMath when we consider it as an element of LaTeXMLMath , and if we use LaTeXMLMath as a substitute of LaTeXMLMath , then by Lemma 3.2 and a slight abuse of notations we can write LaTeXMLEquation .
By the same abuse of notations we define LaTeXMLMath by LaTeXMLEquation .
Now we can reformulate Theorem 2.1 in terms of additive perturbations : Define LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
Then the linear operator LaTeXMLMath is LaTeXMLMath -valued and coincides with LaTeXMLMath when restricted to LaTeXMLMath , i.e .
when a boundary condition of the kind LaTeXMLMath holds for some self-adjoint operator LaTeXMLMath .
Therefore , posing LaTeXMLMath , one has LaTeXMLEquation and , in the case LaTeXMLMath has a bounded inverse , LaTeXMLEquation where LaTeXMLEquation .
By the definition of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath one has , for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation The proof is then concluded by Theorem 2.1 .
∎ In the case LaTeXMLMath and LaTeXMLMath is boundedly invertible , by Theorem 2.1 and Remark 2.4 ( taking LaTeXMLMath ) one can define LaTeXMLMath either by LaTeXMLMath or , equivalently , by LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
Since , for any LaTeXMLMath , one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation the self-adjoint extension LaTeXMLMath could be defined directly in terms of LaTeXMLMath by LaTeXMLEquation .
This reproduces the formulae appearing in LaTeXMLCite , Lemma 2.3 , where however no additive representaion of the extension LaTeXMLMath is given , and in LaTeXMLCite where an additive representaion is obtained only when LaTeXMLMath is closed in LaTeXMLMath .
In this section we explore the connection between the results given in the previous sections and von Neumann ’ s theory of self-adjoint extensions LaTeXMLCite .
Such a theory ( see e.g .
LaTeXMLCite , §13 , for a very compact exposition ) tells us that LaTeXMLEquation the direct sum decomposition being orthogonal with respect to the graph inner product of LaTeXMLMath ; any self-adjoint extension LaTeXMLMath of LaTeXMLMath is then obtained by restricting LaTeXMLMath to a subspace of the kind LaTeXMLMath , where LaTeXMLMath is unitary .
For simplicity in the next theorem we will consider only the case LaTeXMLMath and we put LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath as defined in Theorem 3.4 .
Then LaTeXMLEquation .
The linear operator LaTeXMLEquation is a continuos bijection which becomes unitary when one puts on LaTeXMLMath the scalar product LaTeXMLEquation .
The linear operator LaTeXMLEquation is unitary and the corresponding von Neumann ’ s extension LaTeXMLMath coincides with the self-adjoint operator LaTeXMLMath defined in Theorems 2.1 and 3.4 .
By the definition of LaTeXMLMath one has LaTeXMLEquation and so , since LaTeXMLEquation and LaTeXMLEquation in conclusion there follows LaTeXMLEquation if and only if Range LaTeXMLMath is closed .
By the closed range theorem Range LaTeXMLMath is closed if and only if the Range LaTeXMLMath is closed , and this is equivalent to the range of LaTeXMLMath being closed .
Being LaTeXMLMath surjective , LaTeXMLMath is injective with a closed range and so LaTeXMLEquation is a bijection .
By von Neumann ’ s theory we know that any LaTeXMLMath can be univocally decomposed as LaTeXMLEquation i.e .
LaTeXMLEquation .
The above decomposition can be then rearranged as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By ( 2.4 ) one has LaTeXMLEquation .
Since the scalar product of LaTeXMLMath can be equivalently written as LaTeXMLEquation one has LaTeXMLEquation .
This implies , since Range LaTeXMLMath is closed , LaTeXMLEquation .
Thus , being LaTeXMLMath , the vector LaTeXMLEquation is a generic element of LaTeXMLMath and we have shown that LaTeXMLMath .
It is then straighforward to check that LaTeXMLMath .
By ( 4.1 ) one has LaTeXMLEquation .
This implies LaTeXMLEquation thus LaTeXMLMath is isometric if and only if LaTeXMLEquation where LaTeXMLMath .
By using the identities LaTeXMLMath and LaTeXMLEquation one has LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and so LaTeXMLMath is an isometry .
By again using identity ( 4.2 ) one can check that LaTeXMLMath has an inverse defined by LaTeXMLEquation .
Thus LaTeXMLMath is unitary .
Let us now take LaTeXMLMath .
Then LaTeXMLEquation and so LaTeXMLMath and LaTeXMLEquation ∎ Note that when LaTeXMLMath is bounded , in the previuos theorem one can re-write the unitary LaTeXMLMath as LaTeXMLEquation .
Being LaTeXMLMath always bounded when dim LaTeXMLMath , the previous theorem gives an analogue of Theorem 3.1.2 in LaTeXMLCite avoiding however the use of an admissible matrix LaTeXMLMath ( see LaTeXMLCite , definition 3.1.2 ) .
The previous theorem has the following converse : Let LaTeXMLMath be a self-adjoint extension of LaTeXMLMath as given by von Neumann ’ s theory .
Suppose that LaTeXMLMath and let LaTeXMLMath be the Cayley transform of LaTeXMLMath .
Then the set LaTeXMLEquation is dense , LaTeXMLEquation is self-adjoint and the corresponding self-adjoint operator LaTeXMLMath , defined in Theorems 2.1 and 3.4 , coincides with LaTeXMLMath .
By ( 4.1 ) one has LaTeXMLEquation .
Thus , by inverting the relation LaTeXMLMath given in the previous theorem , one obtains LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLMath if and only if LaTeXMLMath ( see e.g .
LaTeXMLCite , Lemma 1 ) , the range of LaTeXMLMath is dense and thus LaTeXMLMath is densely defined as LaTeXMLMath is a continuos bijection .
By ( 4.1 ) one has LaTeXMLEquation and so , since LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is self-adjoint if and only if LaTeXMLEquation .
Such an equality is then an immediate conseguence of the unitarity of both LaTeXMLMath and LaTeXMLMath .
∎ LaTeXMLMath as defined in Theorem 3.4 coincides with a self-adjoint extension LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath if and only if the boundary condition LaTeXMLMath holds for some self-adjoint operator LaTeXMLMath .
Finite rank perturbations .
Suppose dim LaTeXMLMath , so that LaTeXMLMath and LaTeXMLMath .
Then necessarily LaTeXMLEquation with LaTeXMLMath .
Hypotheses ( 2.1 ) and ( 2.2 ) correspond to LaTeXMLEquation and LaTeXMLEquation .
Considering then an Hermitean invertible matrix LaTeXMLMath with inverse LaTeXMLMath , by Theorem 3.4 one can define the self-adjoint operator LaTeXMLEquation with LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation According to Theorem 2.1 its resolvent is given by LaTeXMLEquation where LaTeXMLEquation .
The operator LaTeXMLMath above coincides with a generic finite rank perturbation of the self-adjoint operator LaTeXMLMath as defined in LaTeXMLCite , §3.1 .
In order to realize that the resolvent written above ( in the case LaTeXMLMath ) is the same given there , the identity LaTeXMLEquation has to be used .
The previous construction can be applied to the case of so-called point interactions in three dimensions ( see LaTeXMLCite and references therein ) .
Since in example 5.2 below we will consider the case of infinitely many point interactions , here we just treat the simplest situation in which only one point interection ( placed at the origin ) is present .
In this case we take LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Therefore LaTeXMLMath is simply the evaluation map at the origin LaTeXMLEquation and we have the family of self-adjoint operators LaTeXMLMath , LaTeXMLMath , defined as ( we take LaTeXMLMath ) LaTeXMLEquation on the domain LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
This reproduces the family given in LaTeXMLCite , §1.5.1 , and coincides with the family LaTeXMLMath given in LaTeXMLCite , §I.1.1 , when one takes LaTeXMLMath .
The case LaTeXMLMath can be then recovered by directly using Theorem 3.4 in the case LaTeXMLMath .
Infinite rank perturbations .
Suppose dim LaTeXMLMath .
Then ( we suppose LaTeXMLMath is separable ) LaTeXMLMath , LaTeXMLMath and necessarily LaTeXMLEquation with LaTeXMLMath .
The generalization of the finite rank case to this situation is then evident .
As concrete example one can consider infinitely many point interactions in three dimensions by taking LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath as before and an infinite and countable set LaTeXMLMath such that LaTeXMLEquation .
Defining then LaTeXMLMath , by LaTeXMLCite ( see page 172 ) one has LaTeXMLEquation where LaTeXMLEquation and hypotheses ( 2.1 ) and ( 2.2 ) are an immediate conseguence of the discreteness of LaTeXMLMath ( see LaTeXMLCite , example 3.4 ) .
By Theoren 3.4 , given any invertible infinite Hermitean matrix LaTeXMLMath with a bounded inverse LaTeXMLMath , one can then define the family of self-adjoint operators LaTeXMLEquation on the domain LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
When LaTeXMLEquation the self-adjoint extension LaTeXMLMath coincides with the operator LaTeXMLMath given in LaTeXMLCite , §III.1.1 ( also see LaTeXMLCite , example 3.4 ) .
In more general situations where the set LaTeXMLMath is not discrete the use of the unitary isomorphism LaTeXMLMath given no advantages and , how the following example shows , it is better to work with LaTeXMLMath itself .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , where the self-adjoint pseudo-differential operator LaTeXMLMath is defined by LaTeXMLEquation with LaTeXMLMath is a real-valued function such that LaTeXMLEquation .
We want now to define the self-adjoint extensions of the restriction of LaTeXMLMath to functions vanishing on a LaTeXMLMath -set , with LaTeXMLMath .
A Borel set LaTeXMLMath is called a LaTeXMLMath -set , LaTeXMLMath , if LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath -dimensional Hausdorff measure and LaTeXMLMath is the closed LaTeXMLMath -dimensional ball of radius LaTeXMLMath centered at the point LaTeXMLMath ( see LaTeXMLCite , §1.1 , chap .
VIII ) .
Examples of LaTeXMLMath -sets are LaTeXMLMath -dimensional Lipschitz submanifolds and ( when LaTeXMLMath is not an integer ) self-similar fractals of Hausdorff dimension LaTeXMLMath ( see LaTeXMLCite , chap .
II , example 2 ) .
We take as the linear operator LaTeXMLMath the unique continuous surjective ( thus ( 2.1 ) holds true ) map LaTeXMLEquation such that , for LaTeXMLMath -a.e .
LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath denotes the LaTeXMLMath -dimensional Lebesgue measure of LaTeXMLMath .
We refer to LaTeXMLCite , Theorems 1 and 3 , chap .
VII , for the existence of the map LaTeXMLMath ; obviously it coincides with the usual evaluation along LaTeXMLMath when restricted to smooth functions .
The definition of the Besov-like space LaTeXMLMath is quite involved and we will not reproduce it here ( see LaTeXMLCite , §2.1 , chap .
V ) .
However , in the case LaTeXMLMath ( i.e .
LaTeXMLMath ) , LaTeXMLMath can be alternatively defined ( see LaTeXMLCite , §1.1 , chap .
V ) as the Hilbert space of LaTeXMLMath having finite norm LaTeXMLEquation where LaTeXMLMath denotes the restriction of the LaTeXMLMath -dimensional Hausdorff measure LaTeXMLMath to the set LaTeXMLMath .
The adjoint map LaTeXMLMath gives rive , for any LaTeXMLMath , to the signed measure LaTeXMLMath defined by LaTeXMLEquation .
Since LaTeXMLMath has support given by the closure of LaTeXMLMath , hypothesis ( 2.2 ) is always verified when the closure of LaTeXMLMath has zero Lebesgue measure .
Defining then LaTeXMLEquation one has LaTeXMLEquation .
Therefore , given any self-adjoint LaTeXMLMath , one has the family of self-adjoint extensions LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation ( see LaTeXMLCite , example 3.6 , LaTeXMLCite , §4 , for alternative definitions ) .
When LaTeXMLMath is a compact Riemannian manifold , LaTeXMLMath the Laplace-Beltrami operator , one has LaTeXMLEquation and LaTeXMLEquation where , for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLMath denoting the volume element of LaTeXMLMath .
Therefore in this case , when LaTeXMLMath ( i.e .
LaTeXMLMath ) , taking LaTeXMLMath , LaTeXMLMath , one can define the self-adjoint extension LaTeXMLEquation and so the construction given here generalizes the examples given in LaTeXMLCite and LaTeXMLCite .
Also see LaTeXMLCite , example 14 , for an alternative definition .
Let LaTeXMLMath be a holomorphic foliation of general type on LaTeXMLMath which admits a rational first integral .
We provide bounds for the degree of the first integral of LaTeXMLMath just in function of the degree , the birational invariants of LaTeXMLMath and the geometric genus of a generic leaf .
Similar bounds for invariant algebraic curves are also obtained and examples are given showing the necessity of the hypothesis .
Instituto de Matemática Pura e Aplicada , IMPA , Estrada Dona Castorina , 110 Jardim Botânico , 22460-320 - Rio de Janeiro , RJ , Brasil .
email : jvp @ impa.br In LaTeXMLCite , Poincaré studied the following problem : ” Is it possible to decide if an algebraic differential equation in two variables is algebraically integrable ? ” .
In the modern terminology the question above can rephrased as : ” Is it possible to decide if a holomorphic foliation LaTeXMLMath on the complex projective plane LaTeXMLMath admits a rational first integral ? ” .
Poincaré observed that in order to solve this problem is sufficient to find a bound for the degree of the generic leaf of LaTeXMLMath .
For a modern version of some of Poincaré ’ s results on this direction see LaTeXMLCite .
Since 1991 paper of Cerveau and Lins Neto LaTeXMLCite , this problem has received the attention of many mathematicians .
The interested reader should consult the papers by Carnicer LaTeXMLCite , Soares LaTeXMLCite , Brunella–Mendes LaTeXMLCite , Esteves LaTeXMLCite and Zamora LaTeXMLCite to have some idea of these recent developments .
Recently , see LaTeXMLCite , Lins Neto has constructed families of foliations with fixed degree and local analytic type of the singularities where foliations with rational first integral of arbitrarily large degree appear .
This shows that it is impossible to bound the degree of the first integral in function of local information of the singularities .
In this work we investigate the problem of bounding the degree of the first integral putting in evidence , for the first time in this problem , the Kodaira dimension of the foliation .
This concept has been introduced independently by L.G .
Mendes LaTeXMLCite and M. McQuillan LaTeXMLCite .
As in the birational theory of algebraic surface the Kodaira dimension measures the abundance of sections in the canonical bundle and its powers .
For algebraic surfaces the canonical bundle is nothing more than the bundle of holomorphic LaTeXMLMath –forms .
In the foliated case it will be the bundle of LaTeXMLMath -forms defined over the leaves of the foliation .
Formally , if we denote the canonical bundle of a holomorphic foliation LaTeXMLMath by LaTeXMLMath then the Kodaira dimension of LaTeXMLMath is defined as LaTeXMLEquation and the sequence of integers LaTeXMLMath is the plurigenera of LaTeXMLMath .
The possible values for the Kodaira dimension of a holomorphic foliation are LaTeXMLMath or LaTeXMLMath .
When LaTeXMLMath , LaTeXMLMath is , up to bimeromorphic maps and ramified coverings , generated by a global holomorphic vector field .
If LaTeXMLMath then LaTeXMLMath is a Riccati foliation or a turbulent foliation or some particular fibration .
In the case LaTeXMLMath has negative Kodaira dimension McQuillan has conjectured that LaTeXMLMath is a rational fibration or a Hilbert modular foliation .
Finally , when LaTeXMLMath is a foliation with LaTeXMLMath we say that LaTeXMLMath is a foliation of general type .
For more details see LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
Our objective is to investigate the Poincaré problem for the foliations of general type .
In this direction , our first result is : Let LaTeXMLMath be a holomorphic foliation of general type on LaTeXMLMath .
Suppose that LaTeXMLMath admits a meromorphic first integral .
Then there exists a bound on the degree of the first integral depending only on the degree and the plurigenera of LaTeXMLMath and on the geometric genus of the generic leaf .
After some extra work we are able to extend the previous result to bound the degree of invariant curves of foliations which do not necessarily admit a rational first integral .
More precisely we prove : Let LaTeXMLMath be a holomorphic foliation of general type on LaTeXMLMath .
Suppose that LaTeXMLMath admits an invariant algebraic curve LaTeXMLMath .
Then there exists a bound for the degree of LaTeXMLMath depending only on the degree and the plurigenera of LaTeXMLMath and on the geometric genus of LaTeXMLMath .
Next we discuss some examples showing that the geometric genus has to appear as a parameter for the bound , and that for foliations of Kodaira dimension distinct from two it is impossible to bound the degree of invariant curves just in function of the degree of the foliation , even if the genus of the curve is fixed .
In this section we recall some basic definitions used in this work .
Central to our exposition are the concepts of plurigenera and Kodaira dimension for holomorphic foliations .
These concepts were introduced in this context independently by L. G. Mendes and M. McQuillan .
For more information on the subject see LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
A holomorphic foliation LaTeXMLMath on a compact complex surface LaTeXMLMath is given by an open covering LaTeXMLMath and holomorphic vector fields LaTeXMLMath over each LaTeXMLMath such that whenever the intersection of LaTeXMLMath and LaTeXMLMath is non–empty there exists an invertible holomorphic function LaTeXMLMath satisfying LaTeXMLMath .
The collection LaTeXMLMath defines a holomorphic line–bundle LaTeXMLMath , called the tangent bundle of LaTeXMLMath .
The dual of LaTeXMLMath is the cotangent bundle LaTeXMLMath , also called the canonical bundle LaTeXMLMath .
Let LaTeXMLMath be a foliation on the complex surface LaTeXMLMath .
The plurigenera of LaTeXMLMath is defined as LaTeXMLEquation .
Recall that a reduced foliation LaTeXMLMath is a foliation such that every singularity LaTeXMLMath is reduced in Seidenberg ’ s sense , i.e. , for every vector field LaTeXMLMath generating LaTeXMLMath in a neighboorhoud of a singular point LaTeXMLMath , the eigenvalues of the linear part of LaTeXMLMath are not both zero and their quotient , when defined , is not a positive rational number .
Let LaTeXMLMath be a foliation on the complex surface LaTeXMLMath , and LaTeXMLMath any reduced foliation bimeromorphically equivalent to LaTeXMLMath .
The Kodaira dimension of LaTeXMLMath is given by LaTeXMLEquation .
When the foliation has Kodaira dimension LaTeXMLMath we say that the foliation is of general type .
It follows from the next proposition that the Kodaira dimension is well defined and is a bimeromorphic invariant of LaTeXMLMath , for a proof see LaTeXMLCite .
Let LaTeXMLMath be a reduced foliation on the algebraic surface LaTeXMLMath and LaTeXMLMath be a reduced foliation on the algebraic surface LaTeXMLMath .
If there exists a birational map LaTeXMLMath sending LaTeXMLMath to LaTeXMLMath , then LaTeXMLMath , for every nonnegative integer LaTeXMLMath .
Consequently LaTeXMLMath .
Proof of Theorem LaTeXMLRef : Let LaTeXMLMath be the minimal resolution of LaTeXMLMath .
Denote by LaTeXMLMath the reduced foliation LaTeXMLMath .
Since LaTeXMLMath admits a meromorphic first integral and LaTeXMLMath is free of dicritical singularities we have that LaTeXMLMath is a fibration .
Let LaTeXMLMath be a generic fiber of LaTeXMLMath .
Since LaTeXMLMath is of general type we can suppose that the genus LaTeXMLMath of LaTeXMLMath is at least LaTeXMLMath .
Riemman-Roch Theorem implies that LaTeXMLEquation if LaTeXMLMath is at least LaTeXMLMath .
Take LaTeXMLMath to be first non-negative integer that satisfies LaTeXMLMath .
From the choice of LaTeXMLMath , the restriction map LaTeXMLEquation has non empty kernel .
In other words there exists a global holomorphic section of LaTeXMLMath that vanishes identically on LaTeXMLMath .
Since LaTeXMLMath is a morphism , for every holomorphic section LaTeXMLMath of LaTeXMLMath we have that LaTeXMLMath is a holomorphic section of LaTeXMLMath .
Hence LaTeXMLEquation one can conclude that the generic leaf of LaTeXMLMath has degree at most LaTeXMLMath .
The bound obtained in the theorem above seems to depend on an infinite numbers of invariants of LaTeXMLMath , LaTeXMLMath for every positive integer LaTeXMLMath .
Now we introduce a new bimeromorphic invariant , the height of a holomorphic foliation , which can be used to substitute the plurigenera as a parameter for the bound .
Let LaTeXMLMath be a holomorphic foliation on an compact complex surface of non-negative Kodaira dimension LaTeXMLMath .
As usual let LaTeXMLMath be any resolution of LaTeXMLMath .
We define the height of LaTeXMLMath , LaTeXMLMath , to be the first positive integer LaTeXMLMath such that LaTeXMLMath has LaTeXMLMath algebraically independent holomorphic sections .
Let LaTeXMLMath be a reduced holomorphic foliation of general type on the compact complex surface LaTeXMLMath .
If the height of LaTeXMLMath is LaTeXMLMath then LaTeXMLMath .
proof : Let LaTeXMLMath be a vector space generated by three algebraically independent global holomorphic sections of LaTeXMLMath .
If we denote these sections by LaTeXMLMath and LaTeXMLMath and consider the morphism LaTeXMLMath , LaTeXMLEquation we obtain that LaTeXMLMath .
Therefore we have that LaTeXMLEquation and the lemma follows .
Let LaTeXMLMath be a foliation of general type on LaTeXMLMath .
If LaTeXMLMath admits a meromorphic first integral then there exists a bound for the degree of the first integral depending only on the degree and the height of LaTeXMLMath and the geometric genus of the generic leaf .
proof : Follows easily from lemma LaTeXMLRef and the proof of theorem LaTeXMLRef .
A positive answer to the following conjecture would imply that the bound obtained in theorem LaTeXMLRef would depend just on the degree of LaTeXMLMath and the geometric genus of the generic leaf .
If LaTeXMLMath is a holomorphic foliation of general type on LaTeXMLMath then there exists a bound for the height of LaTeXMLMath depending only on the degree of LaTeXMLMath .
In order to extend Theorem LaTeXMLRef to bound degree of an invariant curve LaTeXMLMath we must control the vanishing order of LaTeXMLMath along LaTeXMLMath .
Let LaTeXMLMath be a reduced singularity of a holomorphic foliation LaTeXMLMath and LaTeXMLMath be a local smooth separatrix .
If LaTeXMLMath in a neighboorhoud of LaTeXMLMath is generated by a holomorphic vector field LaTeXMLMath then the vanishing order of LaTeXMLMath along LaTeXMLMath at LaTeXMLMath is given by the Poincaré-Hopf index of LaTeXMLMath at LaTeXMLMath .
We will use the notation LaTeXMLMath .
In more concrete terms , since LaTeXMLMath is smooth at LaTeXMLMath in a suitable coordinate system we can write LaTeXMLEquation and we set LaTeXMLMath as LaTeXMLMath .
Note that this index is a particular case of the Gomez-Mont–Seade–Verjovski index which is defined for separatrices with arbitrary singularities .
Let LaTeXMLMath be a reduced holomorphic foliation on a surface LaTeXMLMath and LaTeXMLMath a smooth invariant curve .
Then LaTeXMLMath LaTeXMLMath .
proof : To prove item LaTeXMLMath first consider an open covering LaTeXMLMath of LaTeXMLMath by Stein open sets .
Over each LaTeXMLMath of the covering take a holomorphic vector field LaTeXMLMath generating LaTeXMLMath .
Suppose that we have at most one singularity of LaTeXMLMath over each LaTeXMLMath .
If the restriction of LaTeXMLMath to LaTeXMLMath does not have any singularity then we can interpret LaTeXMLMath as a local generator of LaTeXMLMath , i.e. , we have a canonical isomorphism between LaTeXMLMath and LaTeXMLMath .
When there exists a LaTeXMLMath such that LaTeXMLMath then the restriction of LaTeXMLMath to LaTeXMLMath gives a section of LaTeXMLMath vanishing at LaTeXMLMath with order LaTeXMLMath , i.e. , we have a canonical isomorphism between LaTeXMLMath and LaTeXMLMath .
Glueing the local canonical isomorphisms we obtain LaTeXMLEquation and item LaTeXMLMath follows by taking the dual .
Item LaTeXMLMath follows from item LaTeXMLMath and the long exact sequence in cohomology associated to LaTeXMLEquation where LaTeXMLMath is the effective divisor given by LaTeXMLEquation .
If LaTeXMLMath is a holomorphic foliation on LaTeXMLMath we are going to say that LaTeXMLMath is the safe resolution of LaTeXMLMath if it is obtained by taking the minimal resolution of LaTeXMLMath and after that blowing-up each singularity once .
We do that in order to guarantee that every irreducible curve invariant by LaTeXMLMath is smooth .
Let LaTeXMLMath be a holomorphic foliation on LaTeXMLMath and LaTeXMLMath an invariant algebraic curve .
Let LaTeXMLMath be the safe resolution of LaTeXMLMath and LaTeXMLMath the strict transform of LaTeXMLMath .
Then there exists a bound for LaTeXMLMath depending only on LaTeXMLMath .
proof : Let ’ s say that a singularity is quasi-reduced if it has Milnor number LaTeXMLMath or if it is a reduced saddle-node .
First suppose that every singularity of LaTeXMLMath is quasi–reduced .
If LaTeXMLMath is a reduced singularity then we have at most two branches of LaTeXMLMath passing through LaTeXMLMath .
After doing a blow-up at LaTeXMLMath the contribution of the singularities infinitely near LaTeXMLMath to LaTeXMLMath will be at most two .
When LaTeXMLMath is a dicritical singularity than we can have infinitely many branches of LaTeXMLMath passing through LaTeXMLMath .
Although after resolving LaTeXMLMath it is not hard to see that at most two branches of the strict transform of LaTeXMLMath have singularities infinitely near LaTeXMLMath .
Again the contribution of the singularities infinitely near LaTeXMLMath to LaTeXMLMath is at most two .
If LaTeXMLMath is a reduced saddle-node then the strong separatrix will contribute with one to LaTeXMLMath and the weak separatrix , if exists , with at most LaTeXMLMath .
Since the number of singularities of LaTeXMLMath is bounded by LaTeXMLMath , where LaTeXMLMath , the existence of a bound for LaTeXMLMath in this particular case is proved .
To prove the general case one has just to observe two facts .
The first fact is that there exists a positive integer LaTeXMLMath , depending on the degree of LaTeXMLMath , such that with LaTeXMLMath blow-ups we can always obtain a foliation with all singularities quasi–reduced .
The second fact is that a weak separatrix of a reduced saddle-node that appears after at most LaTeXMLMath blow-ups will have bounded contribution to LaTeXMLMath .
Proof of Theorem LaTeXMLRef : The proof is completely similar to the proof of Theorem LaTeXMLRef .
The only difference is that we have to use lemma LaTeXMLRef and lemma LaTeXMLRef to guarantee the existence of the positive integer LaTeXMLMath such that LaTeXMLEquation .
Again we are able to substitute the plurigenera of LaTeXMLMath by the height of LaTeXMLMath and obtain the following corollary .
Let LaTeXMLMath be a foliation of general type on LaTeXMLMath .
If LaTeXMLMath admits an invariant algebraic curve LaTeXMLMath then there exists a bound for the degree of LaTeXMLMath depending only on the degree and the height of LaTeXMLMath and the geometric genus of LaTeXMLMath .
Let LaTeXMLMath the family of holomorphic foliations on LaTeXMLMath defined , in an affine chart , by LaTeXMLEquation .
Write LaTeXMLMath as LaTeXMLMath .
If LaTeXMLMath is positive then we have a first integral of degree LaTeXMLMath , otherwise the first integral has degree LaTeXMLMath .
Since the resolution of these foliations are rational fibrations , LaTeXMLMath .
∎ Let LaTeXMLMath the family of holomorphic foliations on family of holomorphic foliations on LaTeXMLMath defined by LaTeXMLEquation .
These foliations , constructed by Lins Neto in LaTeXMLCite , are foliations of degree LaTeXMLMath on LaTeXMLMath with singularities with fixed local analytic type and with first integrals of arbitrarily large degree .
Lins Neto also observed that whenever we have a first integral in this family then the generic leaf has geometric genus LaTeXMLMath .
McQuillan , see LaTeXMLCite or LaTeXMLCite , showed that Lins Neto examples up to a three-fold covering and birational transformations are linear vector fields on a torus and that the Kodaira dimension of each particular example is zero .
∎ The Gauss hypergeometric equation LaTeXMLEquation whenever LaTeXMLMath , admits as general solution in the neighboorhoud of zero the function ( see LaTeXMLCite ) LaTeXMLEquation where LaTeXMLMath are arbitrary constants to be determined by the boundary conditions and LaTeXMLEquation .
Here LaTeXMLMath stands for LaTeXMLMath .
The classical change of variable LaTeXMLMath , see LaTeXMLCite p. 104 , associates a Riccati foliation to any second order differential equation .
In this new coordinate the foliation induced by Gauss hypergeometric equation can be written as LaTeXMLEquation .
If LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is arbitrary then the foliation induce by ( LaTeXMLRef ) does not admit a rational first integral and has an invariant rational curve of degree LaTeXMLMath defined by the polynomial LaTeXMLEquation .
Since these foliations have Kodaira dimension one , it is impossible to bound the degree of rational curves ( geometric genus LaTeXMLMath ) for this class of foliations .
∎ By pulling back the family presented in example LaTeXMLRef Lins Neto constructed holomorphic foliations of general type on LaTeXMLMath , again with singularities with fixed local analytic type and with first integrals of arbitrarily large degree .
Let LaTeXMLMath the family of holomorphic foliations on family of holomorphic foliations on LaTeXMLMath defined by LaTeXMLEquation .
It is shown in section LaTeXMLMath of LaTeXMLCite that if we take the pull-back of LaTeXMLMath by the morphism LaTeXMLMath given in homogeneous coordinates by LaTeXMLMath , then the induced family of foliations , denoted by LaTeXMLMath will have the following properties .
the foliations in the family have degree LaTeXMLMath ; There is a finite set of parameters LaTeXMLMath such that the restricted family LaTeXMLMath has non degenerated singularities of fixed analytic type and between these singularities there are exactly LaTeXMLMath dicritical singularities ; There exists a countable and dense set of parameters LaTeXMLMath , such that for any LaTeXMLMath the foliation LaTeXMLMath has a rational first integral of degree LaTeXMLMath , satisfying the properties that for any LaTeXMLMath the sets LaTeXMLMath and LaTeXMLEquation are finite .
Suppose that LaTeXMLMath is sufficiently large .
If LaTeXMLMath is a foliation in the family then there exists a polynomial of degree LaTeXMLMath that vanishes on all dicritical singularities of LaTeXMLMath .
Hence the product of this polynomial with any linear homogeneous polynomial will lift to a section of the canonical sheaf of the resolved foliation .
This is sufficient to assure that the foliation is of general type .
∎ We finally remark that the bounds obtained for the degree of invariant algebraic curves on LaTeXMLMath can be easily extended to any compact complex surface LaTeXMLMath with Picard group isomorphic to LaTeXMLMath .
The author wants to thank C. Camacho , for many suggestions on the style of the exposition , and L. G. Mendes for many helpfull discussions and explanations about his work on the Kodaira dimension for holomorphic foliations .
The author is supported by FAPERJ .
In this paper we define and investigate LaTeXMLMath –homology cobordism invariants of LaTeXMLMath –homology 3–spheres which turn out to be related to classical invariants of knots .
As an application we show that many lens spaces have infinite order in the LaTeXMLMath –homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given LaTeXMLMath –homology sphere .
We also give some new examples of 3–manifolds which can not be obtained by integral surgery on a knot .
In recent years , gauge theoretical tools and new results in 4–dimensional topology have successfully been used to study the structure of the integral homology cobordism group .
It is for instance a consequence of Donaldson ’ s famous theorem about the intersection forms of smooth 4–manifolds that the Poincaré homology sphere LaTeXMLMath has infinite order in this group , and M. Furuta found a family of Brieskorn spheres which generates a subgroup of infinite rank LaTeXMLCite .
Recently N. Saveliev LaTeXMLCite showed , using the LaTeXMLMath –invariant introduced in LaTeXMLCite , that a Brieskorn sphere with non–trivial Rokhlin invariant has infinite order in the integral homology cobordism group .
However , many 3–manifolds arising naturally in knot theory , for instance double coverings of the 3–sphere branched along knots , are not integral homology spheres , but still LaTeXMLMath –homology spheres .
As in the case of integral homology spheres , the set of LaTeXMLMath –homology spheres modulo the LaTeXMLMath –homology cobordism relation forms a group , the so called LaTeXMLMath –homology cobordism group LaTeXMLMath .
To study this group , we introduce , based on Furuta ’ s result on the intersection forms of smooth 4–dimensional spin manifolds LaTeXMLCite , two invariants of LaTeXMLMath –homology spheres which turn out to be in fact invariants of the cobordism class ( see Theorem LaTeXMLRef ) .
Exploiting that these invariants are closely related to classical knot invariants like signature and slice genus , we prove estimates for them in the case of lens spaces , which enables us to exhibit many examples of lens spaces which have infinite order in the LaTeXMLMath –homology cobordism group , and we determine the slice genera of certain Montesinos knots .
A second relation between knot theory and cobordism classes of LaTeXMLMath –homology spheres is provided by the simple fact that surgery along a knot with odd framing produces a LaTeXMLMath –homology sphere .
In this case our invariants can again be related to the slice genus of the knot .
Using this , we prove a lower bound for the slice genera of knots on which integral surgery yields a given LaTeXMLMath –homology sphere .
We also give new examples of 3–manifolds which can not be obtained by integral surgery on a knot .
Recall that a closed connected and oriented 3–manifold LaTeXMLMath is called a LaTeXMLMath –homology sphere if LaTeXMLMath .
Two LaTeXMLMath –homology spheres LaTeXMLMath are called LaTeXMLMath –homology cobordant if there exists a smooth 4–dimensional manifold LaTeXMLMath with LaTeXMLMath such that the inclusions LaTeXMLMath induce isomorphisms LaTeXMLMath .
The set of LaTeXMLMath –homology cobordism classes of LaTeXMLMath –homology spheres forms a group , the so called LaTeXMLMath –homology cobordism group , which we denote by LaTeXMLMath .
Addition in this group is given by taking the connected sum and the zero element is the equivalence class of the 3–sphere .
If LaTeXMLMath is a LaTeXMLMath –homology sphere , there is a Rokhlin invariant LaTeXMLMath which is defined to be the residue class of the signature of any ( smooth ) spin 4–manifold with boundary LaTeXMLMath .
It is easy to see that the Rokhlin invariant of a LaTeXMLMath –homology sphere is always even .
The Rokhlin invariant defines a homomorphism LaTeXMLEquation with image LaTeXMLMath ( the Rokhlin invariant of LaTeXMLMath is two ) .
Let LaTeXMLMath be a LaTeXMLMath –homology sphere .
We define LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath runs over all smooth spin 4–manifolds with boundary LaTeXMLMath .
Surprisingly enough this simple definition actually yields bordism invariants which can be estimated if the manifold in question is a double covering of the 3–sphere branched along some knot .
The following theorem summarizes some properties of the invariants LaTeXMLMath and LaTeXMLMath .
The invariants LaTeXMLMath and LaTeXMLMath are finite numbers .
Moreover LaTeXMLMath where equality occurs if and only if LaTeXMLMath and LaTeXMLMath .
For every LaTeXMLMath –homology sphere LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are LaTeXMLMath –homology spheres , then LaTeXMLEquation .
LaTeXMLEquation If LaTeXMLMath and LaTeXMLMath are LaTeXMLMath –homology cobordant then we have LaTeXMLMath and LaTeXMLMath .
In particular LaTeXMLMath if LaTeXMLMath is the boundary of a LaTeXMLMath –acyclic 4–manifold .
If LaTeXMLMath or LaTeXMLMath then LaTeXMLMath has infinite order in the group LaTeXMLMath .
The same is true if LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is a double covering of the 3–sphere branched along a knot LaTeXMLMath , then LaTeXMLEquation where LaTeXMLMath denotes the slice genus of LaTeXMLMath and LaTeXMLMath its signature .
For the proof of statement LaTeXMLRef , let LaTeXMLMath be a LaTeXMLMath –homology sphere .
Pick a spin manifold LaTeXMLMath with boundary LaTeXMLMath .
Now suppose we are given another spin manifold LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath .
As LaTeXMLMath is a LaTeXMLMath –homology sphere , LaTeXMLMath is spin and LaTeXMLMath .
Hence LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we used Furuta ’ s Theorem LaTeXMLCite in the third line .
Since LaTeXMLMath does not depend on LaTeXMLMath we can conclude that LaTeXMLMath is finite and bounded from above by LaTeXMLMath .
Since this is true for every LaTeXMLMath we also obtain that LaTeXMLMath and the finiteness of LaTeXMLMath .
Now suppose we are given a LaTeXMLMath –homology sphere LaTeXMLMath such that LaTeXMLMath .
Then we can find spin 4–manifolds LaTeXMLMath and LaTeXMLMath with LaTeXMLMath such that LaTeXMLEquation .
Now consider the spin manifold LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , the above equality implies that LaTeXMLMath .
By Furuta ’ s Theorem this is only possible if LaTeXMLMath , and we obtain that LaTeXMLMath .
Hence we have LaTeXMLMath and LaTeXMLMath as claimed .
Now let us proof property LaTeXMLRef .
Pick a spin manifold LaTeXMLMath with boundary LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLMath bounds LaTeXMLMath , and we obtain that LaTeXMLEquation .
Hence we have LaTeXMLMath .
If we choose a spin manifold LaTeXMLMath with boundary LaTeXMLMath such that LaTeXMLMath , then LaTeXMLMath bounds LaTeXMLMath and we have LaTeXMLEquation .
These two inequalities imply the desired result .
As to assertion LaTeXMLRef first note that – thanks to property LaTeXMLRef – it suffices to prove this for LaTeXMLMath .
Choose spin manifolds LaTeXMLMath and LaTeXMLMath with boundaries LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLMath is spin with boundary LaTeXMLMath and LaTeXMLMath .
To prove assertion LaTeXMLRef note that again we only have to prove the required property for LaTeXMLMath .
The condition LaTeXMLMath implies that there exists a spin 4–manifold LaTeXMLMath with boundary LaTeXMLMath such that LaTeXMLMath .
Pick manifolds LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Consider the spin manifolds LaTeXMLEquation .
LaTeXMLEquation Then LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Similarly LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
So we obtain LaTeXMLEquation .
LaTeXMLEquation and the claim follows .
To prove the second part of the assertion we only have to prove that LaTeXMLMath .
Since LaTeXMLMath we have LaTeXMLMath and LaTeXMLMath .
By statement LaTeXMLRef , we obtain LaTeXMLEquation and therefore LaTeXMLMath .
The first part of statement LaTeXMLRef is an immediate consequence of LaTeXMLRef and LaTeXMLRef .
As to the second part , the assumption LaTeXMLMath implies that there exists a spin manifold LaTeXMLMath with boundary LaTeXMLMath such that LaTeXMLMath .
Now suppose there is some LaTeXMLMath such that LaTeXMLMath is the boundary of a LaTeXMLMath –acyclic manifold LaTeXMLMath .
Consider the spin manifold LaTeXMLEquation .
Then LaTeXMLMath which , by Furuta ’ s Theorem , implies that LaTeXMLMath .
But LaTeXMLMath , hence we obtain that LaTeXMLMath , in contradiction to LaTeXMLMath .
To prove property LaTeXMLRef , note that given a surface LaTeXMLMath in LaTeXMLMath with boundary LaTeXMLMath , there is a double covering LaTeXMLMath branched along LaTeXMLMath with boundary LaTeXMLMath .
The manifold LaTeXMLMath is spin , and it is well known that LaTeXMLMath whereas LaTeXMLMath ( see LaTeXMLCite for a proof ) .
This implies the claimed inequalities .
∎ Observe that the invariants LaTeXMLMath and LaTeXMLMath are both not additive .
In fact , if one of them were additive , then property LaTeXMLRef would imply that LaTeXMLMath for all LaTeXMLMath –homology spheres LaTeXMLMath .
As there are clearly LaTeXMLMath –homology spheres with non–zero Rokhlin invariant – for instance LaTeXMLMath – this is impossible by property LaTeXMLRef .
We also note that the invariants LaTeXMLMath and LaTeXMLMath are in general not integers ( however they are always multiples of LaTeXMLMath ) and that they are related by the formula LaTeXMLEquation to the Rokhlin invariant .
In particular LaTeXMLMath is always an even integer .
Suppose we are given positive integers LaTeXMLMath such that exactly one of these three numbers is even , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath denote the weighted graph shown in figure LaTeXMLRef and consider the 4–manifold LaTeXMLMath obtained by plumbing according to LaTeXMLMath .
It is not hard to check that the determinant of LaTeXMLMath is – up to sign – the number LaTeXMLMath and is therefore odd , hence the boundary LaTeXMLMath is a LaTeXMLMath –homology sphere .
As the signature of LaTeXMLMath is LaTeXMLMath and the rank is LaTeXMLMath , we obtain that LaTeXMLEquation .
Now it has been proved in LaTeXMLCite and LaTeXMLCite that the bilinear form LaTeXMLMath can be realized by a collection of LaTeXMLMath –curves in a K3–surface , and a regular neighborhood of such a configuration of spheres is diffeomorphic to LaTeXMLMath , hence we have an embedding LaTeXMLMath .
Let LaTeXMLMath denote the closure of the complement .
Then LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , and we obtain LaTeXMLEquation .
Now we claim that LaTeXMLMath .
In fact , if LaTeXMLMath , this follows from Theorem LaTeXMLRef , as in this case LaTeXMLMath .
Our conditions on LaTeXMLMath exclude the case that LaTeXMLMath , so the only remaining case we have to check is LaTeXMLMath .
But then we already know that LaTeXMLMath , and therefore we can again conclude that LaTeXMLMath , which proves our claim .
As the difference LaTeXMLMath is always an even integer , this discussion shows that the above estimates for LaTeXMLMath and LaTeXMLMath are sharp , i.e .
LaTeXMLEquation .
Note that LaTeXMLMath is minus the Brieskorn sphere LaTeXMLMath , hence we can conclude that LaTeXMLMath .
As LaTeXMLMath , we obtain that LaTeXMLMath has infinite order in the group LaTeXMLMath although it bounds a rational ball LaTeXMLCite .
This also follows from the results in LaTeXMLCite .
As we have seen in Example LaTeXMLRef , the Brieskorn sphere LaTeXMLMath is an element of infinite order in the kernel of the natural map LaTeXMLMath , where the latter group is defined in the obvious way .
Also note that this map is not onto , as the Rokhlin invariant of every LaTeXMLMath –homology sphere is even .
A similar argument shows that LaTeXMLMath is not onto .
It is conceivable that this map is also not one–to–one , and it would be interesting to find explicit examples of integral homology spheres which are not the boundary of a LaTeXMLMath –acyclic 4–manifold but are bounded by some LaTeXMLMath –acyclic manifold .
We remark that Furuta ’ s arguments given in LaTeXMLCite actually show that the family LaTeXMLMath of Brieskorn spheres generates a subgroup of infinite order in LaTeXMLMath ( this has been used by H. Endo LaTeXMLCite to prove that the corresponding family of Pretzel knots spans an infinite dimensional subspace in the smooth knot concordance group ) , so the restriction of LaTeXMLMath to this subgroup is one–to–one .
Suppose we are given positive integers LaTeXMLMath such that exactly one of these three numbers is even , LaTeXMLMath and LaTeXMLMath .
Then the Montesinos knot LaTeXMLMath has slice genus and unknotting number LaTeXMLMath .
Let LaTeXMLMath denote the knot LaTeXMLMath .
The double covering LaTeXMLMath of LaTeXMLMath branched along this knot is LaTeXMLMath , hence LaTeXMLEquation by Example LaTeXMLRef .
It is not hard to verify that LaTeXMLMath .
By Theorem LaTeXMLRef we obtain that LaTeXMLEquation so LaTeXMLMath .
Now the knot LaTeXMLMath is build up from the rational tangles LaTeXMLMath and LaTeXMLMath .
The rational tangle LaTeXMLMath can be changed to the trivial tangle by LaTeXMLMath crossing changes if LaTeXMLMath is even and by LaTeXMLMath crossing changes if LaTeXMLMath is odd , so we obtain that LaTeXMLMath .
Of course LaTeXMLMath , and therefore we can conclude that LaTeXMLMath .
∎ Note that the Montesinos knots in Corollary LaTeXMLRef are examples of knots where the inequality LaTeXMLMath from LaTeXMLCite is not sharp .
In some sense the last statement of Theorem LaTeXMLRef can be seen as a refinement of this inequality involving the cobordism class of the double branched covering .
Suppose that LaTeXMLMath is an integral homology sphere with non–zero Rokhlin invariant which is the result of rational surgery on a torus knot .
Then LaTeXMLMath .
This follows from LaTeXMLCite where Saveliev shows that certain Brieskorn spheres bound spin 4–manifolds LaTeXMLMath with intersection forms LaTeXMLMath , LaTeXMLMath and uses this to prove that they have infinite order in the homology cobordism group .
For these homology spheres LaTeXMLMath .
As A. Casson and J. Harer demonstrated in LaTeXMLCite , all lens spaces of the form LaTeXMLMath with coprime numbers LaTeXMLMath and LaTeXMLMath where LaTeXMLMath is odd bound LaTeXMLMath –acyclic 4–manifolds , hence they are zero in the group LaTeXMLMath .
In this section , we prove estimates for the invariants LaTeXMLMath and LaTeXMLMath of lens spaces and use them to find many examples of lens spaces which have in fact infinite order in the LaTeXMLMath –homology cobordism group .
In particular we will obtain a complete list of those lens spaces LaTeXMLMath with odd LaTeXMLMath whose cobordism classes have infinite order .
It is well known that a lens space LaTeXMLMath is a double covering of the 3–sphere branched along the two–bridge link LaTeXMLMath , see for instance LaTeXMLCite .
To be able to apply statement LaTeXMLRef in Theorem LaTeXMLRef , we have to compute respectively to estimate the signature and the slice genus of such a link .
Let us start by fixing some notations .
Assume that we are given a sequence LaTeXMLMath of LaTeXMLMath non–zero integers .
Then we define the link LaTeXMLMath to be the 4–plat obtained by closing the 3–string braid LaTeXMLMath as pictured in figure LaTeXMLRef ( here we use the convention that the generators of the braid group have positive crossings if the two strings involved have parallel orientations ) .
The link LaTeXMLMath is a knot if and only if LaTeXMLMath is odd , otherwise it is a two component link .
Assume that we are given coprime integers LaTeXMLMath such that LaTeXMLMath is odd .
Then an admissible continuous fractions decomposition is a presentation of LaTeXMLMath as a continuous fraction LaTeXMLEquation where the LaTeXMLMath and LaTeXMLMath are integers such that LaTeXMLMath for all LaTeXMLMath .
Observe that such an admissible continuous fractions decomposition exists for every pair LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
It is also well known LaTeXMLCite that the 4–plat LaTeXMLMath is nothing else than the 2–bridge link LaTeXMLMath .
This description of 2–bridge links turns out to be particularly useful for computing the signature and slice genus of such a link .
The following fact can be found in LaTeXMLCite .
Assume that LaTeXMLMath are coprime integers and that LaTeXMLMath is odd .
Pick an admissible continuous fractions expansion LaTeXMLEquation .
Then the signature of the two–bridge link LaTeXMLMath is given by LaTeXMLEquation .
To find an upper bound for the slice genus of a two–bridge knot we will use the following general observation .
Suppose that LaTeXMLMath and LaTeXMLMath are links such that LaTeXMLMath is obtained from LaTeXMLMath by LaTeXMLMath positive and LaTeXMLMath negative crossing changes .
Then LaTeXMLEquation .
The trace of a homotopy given by the crossing changes is a union of immersed annuli LaTeXMLMath in LaTeXMLMath with LaTeXMLMath positive and LaTeXMLMath negative self–intersection points connecting LaTeXMLMath and LaTeXMLMath ( see LaTeXMLCite ) .
Pick a connected surface LaTeXMLMath with boundary LaTeXMLMath such that LaTeXMLMath .
By gluing this surface with LaTeXMLMath we obtain a connected immersed surface LaTeXMLMath having boundary LaTeXMLMath whose genus is LaTeXMLMath and which has LaTeXMLMath positive and LaTeXMLMath negative self intersection points .
Now let us assume that LaTeXMLMath .
Since we can join two self–intersection points of opposite signs by a handle we can construct a surface bounding LaTeXMLMath which has genus LaTeXMLMath and LaTeXMLMath positive self–intersection points .
Replacing the remaining self–intersections points by LaTeXMLMath handles , we end up with an embedded surface with boundary LaTeXMLMath which has genus LaTeXMLMath and the claim follows .
In the case that LaTeXMLMath a similar argument applies .
∎ Suppose that we have a sequence LaTeXMLMath of non–zero integers such that LaTeXMLMath is odd .
Define numbers LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation Then the slice genus LaTeXMLMath of the knot LaTeXMLMath is bounded from above by LaTeXMLEquation .
Suppose that we have an index LaTeXMLMath such that LaTeXMLMath .
If LaTeXMLMath is odd , we can deform the knot LaTeXMLMath into the knot defined by the sequence LaTeXMLMath by performing LaTeXMLMath negative crossing changes .
If LaTeXMLMath is even , we can do LaTeXMLMath negative crossing changes to obtain LaTeXMLMath .
Repeating this for every index LaTeXMLMath for which LaTeXMLMath is positive , we eventually obtain a knot LaTeXMLMath for which LaTeXMLMath if LaTeXMLMath and LaTeXMLMath otherwise after having performed LaTeXMLEquation negative crossing changes .
A similar reduction can be done if LaTeXMLMath .
In this case we can do LaTeXMLMath respectively LaTeXMLMath positive crossing changes , depending on whether LaTeXMLMath is even or odd .
So we see that after performing LaTeXMLEquation additional positive crossing changes , we end up with a link LaTeXMLMath where LaTeXMLMath and LaTeXMLMath if and only if LaTeXMLMath is even .
Observe that the knot LaTeXMLMath is the boundary of an obvious Seifert surface which has genus LaTeXMLMath .
Now assume that LaTeXMLMath , i.e .
LaTeXMLMath .
By Proposition LaTeXMLRef we can conclude that LaTeXMLEquation .
LaTeXMLEquation If we have LaTeXMLMath , we can use the same argument to obtain the lower bound LaTeXMLEquation .
LaTeXMLEquation Since of course either LaTeXMLMath or LaTeXMLMath the claimed inequality follows .
∎ Suppose that LaTeXMLMath are coprime odd numbers and that we are given a continuous fractions decomposition LaTeXMLEquation with non–zero integers LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLEquation .
LaTeXMLEquation Then LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation It is well known that the lens space LaTeXMLMath is a double covering of LaTeXMLMath , branched along the two–bridge knot LaTeXMLMath , see for instance LaTeXMLCite .
By Lemma LaTeXMLRef , there exists a surface LaTeXMLMath with boundary LaTeXMLMath which has genus LaTeXMLEquation .
Using the expression for the signature of LaTeXMLMath derived in Lemma LaTeXMLRef , Theorem LaTeXMLRef now immediately yields the desired result .
∎ As to the orientation of lens spaces , we are using the convention from LaTeXMLCite that the oriented lens space LaTeXMLMath for LaTeXMLMath is the double covering of the 3–sphere branched along the two–bridge link LaTeXMLMath .
With this choice of orientations , the covering of the left–handed trefoil knot is LaTeXMLMath .
Note that this is in accordance with the convention used in LaTeXMLCite , where the lens space LaTeXMLMath is defined to be the result of rational surgery along the unknot with framing LaTeXMLMath ( see LaTeXMLCite , Exercise 6.3.5 ) .
By computing the terms appearing in the statement of Proposition LaTeXMLRef we can find many examples of lens spaces which have infinite order in the LaTeXMLMath –homology cobordism group .
The following table shows – up to orientation – all the lens spaces whose first homology groups have odd order less or equal than 13 together with the estimates for LaTeXMLMath and LaTeXMLMath provided by Proposition LaTeXMLRef , the continuous fractions decomposition used for the computation and the order of the lens space in LaTeXMLMath as far as it is known ( note that the lens spaces LaTeXMLMath and LaTeXMLMath have orientation reversing diffeomorphisms , the fact that LaTeXMLMath is the boundary of a LaTeXMLMath –acyclic manifold is proved in LaTeXMLCite ) ) .
Suppose that LaTeXMLMath are coprime odd numbers and that LaTeXMLMath has a continuous fractions decomposition LaTeXMLEquation such that LaTeXMLMath for all LaTeXMLMath .
Then the lens space LaTeXMLMath has infinite order in the LaTeXMLMath –homology cobordism group .
Suppose LaTeXMLMath is some positive integer .
Then the lens space LaTeXMLMath has infinite order in LaTeXMLMath .
In fact , first note that LaTeXMLMath and LaTeXMLMath are odd and coprime ( if some prime LaTeXMLMath divides LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath , in contradiction to LaTeXMLMath ) .
A continuous fractions decomposition is given by LaTeXMLEquation .
Since all the coefficients are positive Corollary LaTeXMLRef applies and the claim follows .
As the lens spaces LaTeXMLMath and LaTeXMLMath are diffeomorphic if and only if LaTeXMLMath , we obtain in particular that there is an infinite family of lens spaces which have infinite order in the LaTeXMLMath –homology cobordism group .
We have seen that a LaTeXMLMath –homology sphere obtained by integral surgery on the unknot – namely a lens space LaTeXMLMath – has infinite order in the LaTeXMLMath –homology cobordism group unless it is an integral homology sphere , in which case it is LaTeXMLMath –homology cobordant to LaTeXMLMath .
It turns out that the same is true if the knot is slice , unless the framing is LaTeXMLMath .
In fact , suppose that LaTeXMLMath is obtained by doing surgery with odd framing LaTeXMLMath on a slice knot LaTeXMLMath .
Without loss of generality we can assume that LaTeXMLMath .
Let LaTeXMLMath be the trace of the surgery , i.e .
LaTeXMLMath is obtained from LaTeXMLMath by attaching a 2–handle along LaTeXMLMath with framing LaTeXMLMath .
Then an embedded disk in the 4–ball with boundary LaTeXMLMath and the core of the 2–handle can be glued together to given an embedded sphere LaTeXMLMath with self–intersection number LaTeXMLMath which generates LaTeXMLMath .
Let LaTeXMLMath be the manifold obtained from LaTeXMLMath by removing a tubular neighborhood of LaTeXMLMath .
As the boundary of such a neighborhood is a lens space LaTeXMLMath , we have LaTeXMLMath .
Now it is easy to see that LaTeXMLMath is LaTeXMLMath –acyclic , hence LaTeXMLMath is LaTeXMLMath –homology cobordant to LaTeXMLMath .
This implies that LaTeXMLMath has infinite order in LaTeXMLMath if LaTeXMLMath .
Note that in the case that LaTeXMLMath , LaTeXMLMath is even a LaTeXMLMath –acyclic manifold and as LaTeXMLMath we obtain that the integral homology sphere LaTeXMLMath is the boundary of a LaTeXMLMath –acyclic manifold .
We close this section with the remark that the above computations for lens spaces can be generalized to prove bounds for LaTeXMLMath and LaTeXMLMath if LaTeXMLMath is a Seifert fibred LaTeXMLMath –homology sphere , note that these spaces are double coverings of the 3–sphere branched along a Montesinos link .
As the arguments and calculations are very similar to the computations in the ( special ) case of lens spaces we only state the result for one class of examples .
Let LaTeXMLMath be a positive integer and consider the Seifert fibred space LaTeXMLMath , which is the double coverings of LaTeXMLMath branched along the Montesinos knot LaTeXMLMath .
The obvious Seifert surface for this knot is obtained from Seifert surfaces for the two–bridge knots LaTeXMLMath and LaTeXMLMath by adding three bands , one of them having LaTeXMLMath half twists .
Using this and the results about the signature and the slice genus of a two–bridge knot which we obtained in this section one easily derives that LaTeXMLMath and LaTeXMLMath .
Note that , as for every knot , we have the inequality LaTeXMLMath , so our estimate for the slice genus happens to be sharp .
Now we can conclude that LaTeXMLMath .
By Theorem LaTeXMLRef this shows that LaTeXMLMath has infinite order in LaTeXMLMath .
Also note that the order of the first homology of LaTeXMLMath is linear in LaTeXMLMath , so LaTeXMLMath if LaTeXMLMath , and we obtain an infinite family of Seifert fibred spaces which have infinite order in LaTeXMLMath .
As indicated in the introduction , a second main source of LaTeXMLMath –homology spheres is surgery on knots with odd integral framings .
In this section , we shall see how one can obtain information on the invariants LaTeXMLMath and LaTeXMLMath if a LaTeXMLMath –homology sphere LaTeXMLMath is described by integral surgery on a knot .
Combined with other methods for calculating these invariants , this provides a lower bound for the slice genus of such a knot .
We also give examples of 3–manifolds which can not be obtained by integral surgery on a knot .
Suppose we are given a knot LaTeXMLMath in the 3–sphere .
Then surgery with odd framing on this knot yields a LaTeXMLMath –homology sphere LaTeXMLMath .
First let us derive a relation between the Arf invariant of the knot , the framing and the Rokhlin invariant LaTeXMLMath of LaTeXMLMath .
For this purpose recall that if LaTeXMLMath is a simply connected 4–manifold whose boundary is a LaTeXMLMath –homology sphere and LaTeXMLMath is a closed embedded characteristic surface , we have a well defined Arf invariant LaTeXMLMath which does only depend of the homology class of the surface and the Rokhlin invariant of the boundary ( as one can see by gluing with a simply connected spin manifold with boundary LaTeXMLMath , note that the condition that LaTeXMLMath is a LaTeXMLMath –homology sphere implies that LaTeXMLMath is still characteristic in the resulting closed 4–manifold ) .
Let LaTeXMLMath be a knot and assume that LaTeXMLMath is some odd number .
Let LaTeXMLMath denote the LaTeXMLMath –homology sphere which is the result of surgery along LaTeXMLMath with framing LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLEquation .
Let LaTeXMLMath denote the simply connected 4–manifold which is obtained from LaTeXMLMath by adding a handle along LaTeXMLMath with framing LaTeXMLMath .
Then LaTeXMLMath is a LaTeXMLMath –homology sphere , in fact LaTeXMLMath .
Furthermore LaTeXMLMath .
As the boundary of LaTeXMLMath is a LaTeXMLMath –homology sphere , the LaTeXMLMath –intersection form is non–degenerate and hence the fact that LaTeXMLMath implies that LaTeXMLMath is not spin , i.e .
LaTeXMLMath is the non–zero element of LaTeXMLMath .
Pick a Seifert surface LaTeXMLMath for LaTeXMLMath and let LaTeXMLMath denote the surface which is obtained by gluing LaTeXMLMath with the core of the 2–handle .
Then LaTeXMLMath , and therefore LaTeXMLMath .
Let LaTeXMLMath denote the homology class of LaTeXMLMath and denote the Poincaré duality map LaTeXMLMath by LaTeXMLMath .
Then the fact that the self–intersection of LaTeXMLMath is odd implies that the image of LaTeXMLMath under the restriction LaTeXMLMath is the non–zero element , i.e .
we have LaTeXMLEquation .
Now pick a spin manifold LaTeXMLMath with boundary LaTeXMLMath and consider the closed manifold LaTeXMLMath .
The fact that LaTeXMLMath is a LaTeXMLMath –homology sphere implies that LaTeXMLMath .
As LaTeXMLMath is spin the surface LaTeXMLMath represents LaTeXMLMath , i.e .
LaTeXMLMath is characteristic .
Therefore we can conclude that LaTeXMLMath .
By Novikov additivity , LaTeXMLMath .
By definition of the Rokhlin invariant we also have that LaTeXMLMath , and therefore we obtain that LaTeXMLMath .
By LaTeXMLCite , we obtain LaTeXMLEquation for the Arf invariant of the surface LaTeXMLMath .
However it is also known LaTeXMLCite that LaTeXMLMath , and the proof of the proposition is complete .
∎ Assume that a LaTeXMLMath –homology sphere LaTeXMLMath is obtained by integral surgery on a knot .
Then LaTeXMLEquation .
Here the sign is minus if the framing is positive , otherwise it is plus .
Suppose that LaTeXMLMath is obtained by surgery on a knot LaTeXMLMath with framing LaTeXMLMath .
First let us consider the case that LaTeXMLMath is positive ( note that LaTeXMLMath must be odd ) .
Then LaTeXMLMath , and by Proposition LaTeXMLRef , we have that LaTeXMLMath .
If LaTeXMLMath is negative , we obtain LaTeXMLMath , and multiplying this by minus one gives the desired result .
∎ If a lens space LaTeXMLMath can be obtained by surgery on a knot , then LaTeXMLMath or LaTeXMLMath is a square modulo p by Proposition 1 in LaTeXMLCite .
It is interesting that if LaTeXMLMath is an odd prime , this criterion is actually eqivalent to the congruence of Corollary LaTeXMLRef , which can therefore be seen as a generalisation of the result for lens spaces in LaTeXMLCite .
In fact , assume that LaTeXMLMath is a square mod LaTeXMLMath ( we can restrict ourselves to this case after possibly reversing the orientation ) , i.e .
LaTeXMLMath .
By LaTeXMLCite , p. 137 , we have LaTeXMLEquation where LaTeXMLMath is a Dedekind sum .
It is also known LaTeXMLCite that LaTeXMLMath ( the sign depending on orientation conventions ) , note that , as LaTeXMLMath is odd , the Rokhlin invariant is even and therefore LaTeXMLMath is an integer .
Multiplying equation ( LaTeXMLRef ) by 2p we therefore obtain LaTeXMLEquation .
But LaTeXMLMath is an odd integer , hence LaTeXMLMath , and we end up with LaTeXMLEquation which is the prediction made by Corollary LaTeXMLRef .
A similar calculation shows that if the congruence of Corollary LaTeXMLRef holds for a lens space LaTeXMLMath , one of the Jacobi symbols LaTeXMLMath and LaTeXMLMath must be one , hence , as LaTeXMLMath is a prime , LaTeXMLMath or LaTeXMLMath is a square modulo LaTeXMLMath .
In some cases Corollary LaTeXMLRef can be used to show that certain 3–manifolds are not the result of integral surgery on a knot ( although they can of course be obtained by integral surgery along a link ) or to determine the sign of the framing .
Let us consider an example of a connected sum where each summand is the result of surgery on a knot but the sum is not .
Let LaTeXMLMath .
Note that the first homology of LaTeXMLMath is cyclic of order 21 .
Clearly both summands can be obtained by surgery on knots , namely by LaTeXMLMath respectively LaTeXMLMath surgery on the unknot .
However LaTeXMLMath , and as LaTeXMLMath we can conclude that LaTeXMLMath is not the result of integral surgery on a knot .
Note that , as LaTeXMLMath can of course be obtained by integral surgery along the trivial two–component link , its surgery number as defined in LaTeXMLCite is two .
The same argument shows that all manifolds of the form LaTeXMLMath have surgery number two .
Suppose that LaTeXMLMath .
The lens space LaTeXMLMath is the result of surgery with framing LaTeXMLMath on the unknot .
However Corollary LaTeXMLRef shows that this manifold can not be obtained by surgery on a knot with positive framing .
In fact , LaTeXMLMath , LaTeXMLMath , and if LaTeXMLMath , then LaTeXMLMath .
As LaTeXMLMath is not a square modulo 5 , Proposition 1 in LaTeXMLCite implies that LaTeXMLMath is not the result of surgery on a knot .
In LaTeXMLCite , D. Auckly gave another argument for this fact which can be generalized to the following Assume that LaTeXMLMath is a knot , LaTeXMLMath , which can be unknotted using only one crossing change , and suppose that neither LaTeXMLMath nor LaTeXMLMath is a square modulo LaTeXMLMath .
Let LaTeXMLMath denote the double covering of the 3–sphere branched along LaTeXMLMath .
Then LaTeXMLMath is the result of rational surgery on a knot but can not be obtained by integral surgery on a knot .
Suppose for a moment that LaTeXMLMath can be obtained as the result of integral surgery on some knot .
The trace of this surgery is a simply connected 4–manifold LaTeXMLMath with second homology LaTeXMLMath .
If LaTeXMLMath denotes the framing of the surgery the intersection form on LaTeXMLMath is LaTeXMLMath times the standard form on LaTeXMLMath .
Using this one easily sees that the linking form on LaTeXMLMath maps the generator of LaTeXMLMath to LaTeXMLMath , see for instance LaTeXMLCite for a proof .
Note that LaTeXMLMath .
Now it has been pointed out in LaTeXMLCite that – as a consequence of the fact that the unknotting number is one – the linking form on LaTeXMLMath takes precisely the values LaTeXMLMath in LaTeXMLMath for some sign LaTeXMLMath , in particular LaTeXMLMath is in the image .
Hence there exist integers LaTeXMLMath such that LaTeXMLEquation .
Multiplying this by LaTeXMLMath leads to LaTeXMLMath in contradiction to our assumption .
∎ Let LaTeXMLMath denote the Montesinos knot LaTeXMLMath which is denoted by LaTeXMLMath in LaTeXMLCite .
The double covering LaTeXMLMath of LaTeXMLMath branched along LaTeXMLMath is then a Seifert fibred space LaTeXMLMath , note that this 3–manifold is actually irreducible .
One easily sees that LaTeXMLMath and LaTeXMLMath ( the reason being that the rational tangle LaTeXMLMath can be changed to the trivial tangle by one crossing change ) .
As LaTeXMLMath is not a square modulo 15 , Proposition LaTeXMLRef applies and we obtain that LaTeXMLMath can not be obtained by integral surgery on a knot although it is the result of rational Dehn surgery on some knot in LaTeXMLMath .
Note that LaTeXMLMath and LaTeXMLMath , hence LaTeXMLMath , so Corollary LaTeXMLRef does not yield this result .
If a LaTeXMLMath –homology sphere is the result of surgery on a link then the trace of this surgery will be a natural choice for a 4–manifold bounded by LaTeXMLMath .
Unfortunately this trace will in general not be spin , but one can always do surgery on an embedded surface to obtain a spin manifold .
Suppose that LaTeXMLMath is a simply connected 4–manifold whose boundary is a LaTeXMLMath –homology sphere and that LaTeXMLMath is a closed characteristic surface having non–zero self–intersection number .
Let LaTeXMLMath denote the sign of LaTeXMLMath .
Then there exists a simply connected spin 4–manifold LaTeXMLMath with LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation where we think of the Arf invariant LaTeXMLMath as an element of LaTeXMLMath .
Clearly we can restrict ourselves to the case that the self–intersection number of LaTeXMLMath is positive , the case of negative self–intersection number then follows by reversing the orientation .
Let us first consider the case that the Arf invariant LaTeXMLMath is zero .
Let LaTeXMLMath denote the manifold which is obtained from LaTeXMLMath by attaching LaTeXMLMath copies of LaTeXMLMath and consider the surface LaTeXMLMath given by gluing LaTeXMLMath with the exceptional divisors .
Then LaTeXMLMath , and clearly LaTeXMLMath .
As explained in LaTeXMLCite we can now construct a sphere LaTeXMLMath , where LaTeXMLMath is obtained from LaTeXMLMath by attaching LaTeXMLMath copies of LaTeXMLMath such that LaTeXMLMath and such that LaTeXMLMath is still characteristic .
Blowing down this sphere gives a spin 4–manifold LaTeXMLMath as required .
In the case that LaTeXMLMath , consider LaTeXMLMath and the surface LaTeXMLMath obtained by gluing LaTeXMLMath with a torus representing 3 times the generator of LaTeXMLMath .
Then LaTeXMLMath and the claim follows from what we just proved .
∎ Suppose that a LaTeXMLMath –homology sphere is obtained by integral surgery with framing LaTeXMLMath on a knot LaTeXMLMath .
Let LaTeXMLMath denote the sign of the framing and let LaTeXMLEquation ( note that , according to Proposition LaTeXMLRef , the number in parentheses is actually a multiple of 8 ) .
Finally let LaTeXMLMath and let LaTeXMLMath denote the slice genus of the knot LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLEquation .
By assumption , LaTeXMLMath is obtained by surgery on a knot LaTeXMLMath .
The trace of this surgery is a 4–dimensional handlebody LaTeXMLMath with boundary LaTeXMLMath which is the result of attaching a 2–handle along LaTeXMLMath to LaTeXMLMath .
Consequently LaTeXMLMath and LaTeXMLMath .
By Proposition LaTeXMLRef , LaTeXMLMath , and clearly LaTeXMLMath , i.e .
LaTeXMLMath .
Now pick a surface LaTeXMLMath with boundary LaTeXMLMath such that LaTeXMLMath and let LaTeXMLMath denote a Seifert surface for LaTeXMLMath .
Then the surfaces LaTeXMLMath and LaTeXMLMath formed by gluing the core of the 2–handle with LaTeXMLMath respectively LaTeXMLMath clearly have the same homology class LaTeXMLMath .
As in the proof of Proposition LaTeXMLRef , this homology class is characteristic , and LaTeXMLMath .
It is known that the Arf invariant of the surface LaTeXMLMath obtained by gluing LaTeXMLMath and the core of the 2–handle is LaTeXMLMath , and as LaTeXMLMath , the same is true for LaTeXMLMath Now we can apply Lemma LaTeXMLRef to LaTeXMLMath to obtain a spin 4–manifold LaTeXMLMath with boundary LaTeXMLMath and the claimed estimates follow from the definitions of the invariants LaTeXMLMath and LaTeXMLMath .
∎ A similar estimate can be derived in the more general case that the LaTeXMLMath –homology sphere LaTeXMLMath is obtained by integral surgery on a link which has a characteristic knot ( which one can always assume after sliding handles ) , such a characteristic knot will again define a characteristic surface to which one can apply Lemma LaTeXMLRef .
We are now ready to use Proposition LaTeXMLRef , combined with the information obtained from Proposition LaTeXMLRef , to derive a lower bound for the slice genus of a knot on which integral surgery can be performed to obtain a given LaTeXMLMath –homology sphere LaTeXMLMath .
Note that if LaTeXMLMath is the result of surgery of a knot , then we can , by Corollary LaTeXMLRef , choose the orientation of LaTeXMLMath such that LaTeXMLMath , therefore we state our result only in this case .
Assume that LaTeXMLMath is a LaTeXMLMath –homology sphere and that LaTeXMLMath and LaTeXMLMath Let LaTeXMLEquation were we think of LaTeXMLMath as an element of LaTeXMLMath .
If LaTeXMLMath can be obtained by integral surgery on a knot LaTeXMLMath , then LaTeXMLEquation .
Let LaTeXMLMath denote the framing of the knot .
Of course LaTeXMLMath .
Once we can show that LaTeXMLMath must be positive the claimed inequality follows from Proposition LaTeXMLRef .
So let us assume that LaTeXMLMath .
Then LaTeXMLMath , and by Proposition LaTeXMLRef , we have LaTeXMLEquation .
However combining this with the assumption LaTeXMLMath shows that LaTeXMLMath .
But this is only possible if LaTeXMLMath , which contradicts our assumptions .
∎ Assume that LaTeXMLMath is an even number and consider the lens space LaTeXMLMath .
Note that this lens space is – up to orientation – LaTeXMLMath with LaTeXMLMath , LaTeXMLMath .
As LaTeXMLMath and LaTeXMLMath are coprime , LaTeXMLMath can be obtained by surgery on a knot by Theorem 1 in LaTeXMLCite .
An admissible continuous fractions decomposition is given by LaTeXMLEquation .
Using this one can easily derive that LaTeXMLMath and LaTeXMLMath .
Hence Theorem LaTeXMLRef implies that the slice genus of any knot LaTeXMLMath on which integral surgery can be performed to obtain LaTeXMLMath must be at least LaTeXMLMath .
Of course the slice genus of a knot is at most its genus , and for every knot the crossing number is at least two times the genus ( otherwise Seifert ’ s algorithm would produce a Seifert surface of smaller genus ) , so we also obtain a lower bound for the crossing number .
The most obvious infinite family of lens spaces which can be obtained by integral surgery on a knot is given by the lens spaces LaTeXMLMath .
All these spaces can be obtained by surgery on a single knot , namely the unknot ( but of course with different framings ) .
Example LaTeXMLRef shows that this situation is not typical and that one actually has to use infinitely many knots ( even infinitely many concordance classes of knots ) to obtain all the lens spaces which are the result of integral surgery on a knot , i.e .
we have the For every natural number LaTeXMLMath there exists a lens space LaTeXMLMath which can be obtained by integral surgery on a knot such that every such knot has slice genus at least LaTeXMLMath .
Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra , then for each element of the semigroup a ( generalized ) projection operator exists that is defined on the entire Geometric Algebra .
A single fundamental theorem holds for all ( generalized ) projection operators .
This theorem makes previous projection operator formulas LaTeXMLCite equivalent to each other .
The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism ‘ grade involution ’ and the anti-automorphism ‘ inverse ’ on the semigroup of invertible versors .
This class of projection operators is studied in some detail as the natural generalization of the subspace projection operators .
Other generalized projection operators include projections onto any invertible element , or a weighted projection onto any element .
This last projection operator even implies a possible projection operator for the zero element .
We introduce a class of generalized projection operators on a Geometric Algebra LaTeXMLMath indexed by a nonempty set of generators , LaTeXMLMath , and two functions from LaTeXMLMath to LaTeXMLMath denoted : LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath is closed under the geometric product , LaTeXMLMath , and LaTeXMLMath .
Each class of projection operators includes a function LaTeXMLMath from LaTeXMLMath to LaTeXMLMath for each element LaTeXMLMath of LaTeXMLMath defined by LaTeXMLEquation .
The paper begins with the statement and proof of the Fundamental Theorem of Projection Operators and then examines projection operators for specific choices of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
The body of the paper relates the fundamental theorem to familiar projection operators and to novel projection operators .
The paper ends with a summary of future work .
The Fundamental Theorem of Projetion Operators ( FToPO ) states for any projection operators LaTeXMLMath and LaTeXMLMath LaTeXMLEquation .
Proof : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This theorem allows projection operators to be treated directly rather than as derivative objects .
This can help for applications like those in reference LaTeXMLCite where projection operators are used as fundamental objects of computation in Geometric Algebra .
The most familiar projection is to let the set LaTeXMLMath be the set of invertible versors ( the semigroup generated by the invertible blades ) , LaTeXMLMath be the grade involution , and LaTeXMLMath be the inverse operation .
If LaTeXMLMath is an invertible blade and LaTeXMLMath is a vector , LaTeXMLMath is the projection of LaTeXMLMath onto the subspace characterized by LaTeXMLMath .
Proof : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation So this class of projections is indeed familiar .
Here are seven formulas for the vector domain portion of the projection operators LaTeXMLMath and LaTeXMLMath of two invertible blades LaTeXMLMath and LaTeXMLMath from reference LaTeXMLCite .
LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Using the FToPO it is easy to see that 1 .
LaTeXMLMath 2 .
LaTeXMLMath … LaTeXMLMath 6 .
LaTeXMLMath 7 .
LaTeXMLMath 2. from the FToPO and since LaTeXMLMath .
The latter is clear since the reverse of an invertible blade is a nonzero scalar multiple of itself .
LaTeXMLMath 3. from the FToPO .
LaTeXMLMath 4. because for invertible blades LaTeXMLMath and LaTeXMLMath LaTeXMLMath therefore LaTeXMLMath .
LaTeXMLMath 5. from the FToPO .
LaTeXMLMath 6. from the FToPO and since LaTeXMLMath .
LaTeXMLMath 7. because for an invertible blade LaTeXMLMath , LaTeXMLMath .
The full generality of the FToPO interrelates the composition of projections onto blades with projections onto versors .
Projecting onto a versor is a new operation , but we will show a simple motivation , and that motivation will reproduce the familiar projection onto a blade when the versor in question is , in fact , a blade .
Formula ( LaTeXMLRef ) clearly shows that LaTeXMLMath is the average of two objects , namely LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is a versor then the second object is always a vector .
Specifically , let LaTeXMLMath be a versor .
Now define LaTeXMLMath and inductively define LaTeXMLMath then inductively it is clear that each LaTeXMLMath is a vector .
Expand LaTeXMLMath to get LaTeXMLEquation .
As a family of versors LaTeXMLMath approaches the blade LaTeXMLMath , the vector LaTeXMLMath becomes a vector whose rejection from LaTeXMLMath remains the same as LaTeXMLMath , while the projection of LaTeXMLMath swings around to become diametrically opposite the projection of LaTeXMLMath .
Thus , LaTeXMLMath smoothly becomes the projection of LaTeXMLMath onto LaTeXMLMath .
The extension of projection to versors was required to fully utilize the FToPO for blades and the interpretation of the projection of blades is truly an explanation of the older , more familiar projection onto subspaces .
However , there are also more formal extensions of the idea of projection , of which two are explored here .
The simplest formal extension of the familiar projection is to let the set LaTeXMLMath be the set of all invertible elements , LaTeXMLMath be the grade involution , and LaTeXMLMath be the inverse operation .
This clearly is just an enlargement of the domain , LaTeXMLMath , of objects that can be projected onto .
We show that the interpretation of the projection onto a nonversor LaTeXMLMath is problematic .
This is because if LaTeXMLMath is a vector for each vector LaTeXMLMath , it follows that LaTeXMLMath is a versor .
Assume that for each vector LaTeXMLMath , LaTeXMLMath is a vector .
Then isomorphically embed the problem into a nontrivial , nondegenerate Geometric Algebra using a LIFT as described in the appendix and define LaTeXMLMath .
Clearly LaTeXMLMath is a vector-valued linear function of a vector variable ( i.e .
LaTeXMLMath is a linear transformation ) .
Furthermore LaTeXMLMath is actually an orthogonal transformation of the enlarged vector space .
LaTeXMLEquation .
Since the enlarged space is nontrivial and nondegenerate reference LaTeXMLCite guarantees that there exists a nonzero versor LaTeXMLMath that performs the same transformation , i.e .
there exists a nonzero versor LaTeXMLMath such that LaTeXMLMath for each vector LaTeXMLMath .
A short computation now shows that LaTeXMLMath for each vector LaTeXMLMath , where LaTeXMLMath denotes the reverse of LaTeXMLMath .
Note that LaTeXMLMath is a scalar and fix an arbitrary vector LaTeXMLMath .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
So by Lemma ( LaTeXMLRef ) of the appendix , LaTeXMLMath is a scalar .
Now LaTeXMLMath and since LaTeXMLMath is a scalar multiple of LaTeXMLMath it is a versor too .
This means that projecting onto a nonversor , while defined , results in some vectors going to nonvectors .
The interpretation of such a transformation is an outstanding issue .
The most general nontrivial projection operator is to let the set LaTeXMLMath be the set , LaTeXMLMath of all elements , LaTeXMLMath be the grade involution , and LaTeXMLMath be the reverse .
As discussed in the previous section the interpretation of the projection onto nonversors is problematic .
If LaTeXMLMath is an invertible versor then LaTeXMLMath is proportional to LaTeXMLMath , so in the case where LaTeXMLMath is an invertible versor the two projections are not very different .
The inverse projection operation is the average , while the reverse projection operation is the weighted average of LaTeXMLMath and LaTeXMLMath .
The class of reverse projection operators is defined for all multivectors , so there is even a projection operator for the zero element , and in fact LaTeXMLMath .
Projection operators for noninvertible elements are actually quite interesting , for instance if an element , LaTeXMLMath , is idempotent ( LaTeXMLMath ) then LaTeXMLMath follows from the FToPO easily .
The uses for the reverse projection operator are still unknown .
When LaTeXMLMath the reverse projection operator is the same as the inverse projection operator .
When LaTeXMLMath then the reverse projection operator is a weighted average that depends on the scale of LaTeXMLMath .
So possibly the reverse projection operator has use as a statistical projection where the scale of LaTeXMLMath determines the certainty of the element , or possibly the actual operation on the elements will not be useful , but instead the algebraic properties of the projection operator itself will provide a meaningful ( and useful ) measure of the scale of a multivector .
This paper is a short introduction to a new class of projection operators in a Geometric Algebra .
Even without taking projections onto new elements the Fundamental Theorem of Projection Operators ( FToPO ) unifies and generalizes the standard identities of projections onto subspaces .
The outright generalizations of projection operators fall into three different potentially useful cases , each of which calls for application or interpretation .
The first case is the projection onto versors , which the author believes is the natural generalization of the projection onto blades .
The second case is the projection onto the zero element , which is simple enough that it can be appended to any other class of projection operators and preserve the FToPO , and can thereby introduce scale to projection operators .
The third case is the class of weighted projection operators , LaTeXMLMath , that are sensitive to the scale of their generators , LaTeXMLMath .
Each case is a call for further work .
The projection onto versors have a clear interpretation and only need applications demonstrate its worth .
The zero element was given a geometric interpretation in reference LaTeXMLCite , and that interpretation should be reconciled with the projection operator presented here .
The weighted projection operators need both a solid interpretation and applications and therefore will probably not be well understood for some time to come .
Since the class of reverse projection operators is a weighted projection operator that has a projection operator LaTeXMLMath for each element LaTeXMLMath of the Geometric Algebra , there is at least the hope that the reverse projection operators can help elucidate the geometric properties of arbitrary elements of a Geometric Algebra .
The appendix contains two results used in the earlier proofs .
As taken from LaTeXMLCite , a LIFT ( ‘ linear injective function ’ transformation ) from one Geometric Algebra to another Geometric Algebra is defined as a linear injective map that preserves the outer product and the scalars .
In more detail , given two geometric algebras , LaTeXMLMath and LaTeXMLMath , and a linear injective function , LaTeXMLMath , from the vectors of LaTeXMLMath to the vectors of LaTeXMLMath then LaTeXMLMath is a LIFT between the two algebras , where LaTeXMLMath is the outermorphism of LaTeXMLMath .
This paper uses a LIFT to isomorphically embed a degenerate Geometric Algebra into a nondegenerate , nontrivial geometric algebra .
If the Geometic Algebra is trivial it is just the scalars , and it is embedded into a Geometric Algebra over a one-dimensional Euclidean vector space .
If the algebra is degenerate then it is isomorphically embedded into a nondegenerate algebra as described in reference LaTeXMLCite .
Let LaTeXMLMath be a Geometric Algebra over a nondegenerate finite dimensional vector space LaTeXMLMath then LaTeXMLEquation .
Proof : Since LaTeXMLMath is nondegenerate it has an orthogonal basis of invertible vectors LaTeXMLMath .
The set LaTeXMLMath is a basis for LaTeXMLMath where LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath implies that LaTeXMLMath when LaTeXMLMath .
Since LaTeXMLMath for each LaTeXMLMath it is clear that LaTeXMLMath for all LaTeXMLMath , therefore LaTeXMLMath is a scalar .
We introduce a new construction of error-correcting codes from algebraic curves over finite fields .
Modular curves of genus LaTeXMLMath over a field of size LaTeXMLMath yield nonlinear codes more efficient than the linear Goppa codes obtained from the same curves .
These new codes now have the highest asymptotic transmission rates known for certain ranges of alphabet size and error rate .
Both the theory and possible practical use of these new record codes require the development of new tools .
On the theoretical side , establishing the transmission rate depends on an error estimate for a theorem of Schanuel applied to the function field of an asymptotically optimal curve .
On the computational side , actual use of the codes will hinge on the solution of new problems in the computational algebraic geometry of curves .
STOC ’ 01 , July 6-8 , 2001 , Hersonissos , Crete , Greece .
1 In this section we first review the construction and properties of Goppa codes , to put our work in its context .
We then define our new nonlinear codes and give lower bounds on their minimal distance .
We conclude this section by stating lower bounds on the size of our codes and comparing our codes ’ parameters with those of Goppa codes .
In the next section we prove the bounds claimed in the Introduction .
In the final section we discuss theoretical and computational questions raised by our construction , and show how to solve these problems for the nonlinear codes obtained from rational curves .
Fix a finite field LaTeXMLMath of LaTeXMLMath elements .
Let LaTeXMLMath be a projective , smooth , irreducible algebraic curve of genus LaTeXMLMath defined over LaTeXMLMath , with LaTeXMLMath rational points .
To any divisor LaTeXMLMath on LaTeXMLMath of degree LaTeXMLMath , Goppa ( LaTeXMLCite , see also LaTeXMLCite ) regards the space of sections of LaTeXMLMath as a linear LaTeXMLMath code with alphabet LaTeXMLMath , for some LaTeXMLMath ( because a nonzero section of LaTeXMLMath has at most LaTeXMLMath zeros ) and LaTeXMLMath ( by the Riemann-Roch theorem ) .
Thus the transmission rate LaTeXMLMath and the error-detection rate LaTeXMLMath of Goppa ’ s codes are related by LaTeXMLEquation .
This lower bound improves as LaTeXMLMath increases .
How large can LaTeXMLMath get as LaTeXMLMath ?
An upper bound is LaTeXMLEquation ( Drinfeld-Vlăduţ LaTeXMLCite ) .
We say a curve of genus LaTeXMLMath is “ asymptotically optimal ” if it has at least LaTeXMLMath rational points over LaTeXMLMath .
If LaTeXMLMath is even , i.e. , if LaTeXMLMath is an integer , then modular curves of various flavors — classical ( elliptic ) , Shimura , or Drinfeld — attain LaTeXMLEquation .
LaTeXMLCite , and are thus asymptotically optimal .
Therefore if LaTeXMLMath there exist arbitrarily long linear codes over LaTeXMLMath with LaTeXMLEquation and this is the best that can be obtained from ( LaTeXMLRef ) .
Once LaTeXMLMath , these codes improve on the Gilbert-Varshamov bound for suitable LaTeXMLMath .
Actual construction of these codes requires explicit equations for LaTeXMLMath .
The definitions of modular curves do not readily yield useful equations , but in recent years many families of modular curves have been given by LaTeXMLMath explicit equations in LaTeXMLMath variables , each equation of degree LaTeXMLMath .
See LaTeXMLCite for classical and Shimura curves , LaTeXMLCite for further Shimura curves , and LaTeXMLCite for Drinfeld modular curves .
LaTeXMLCite are in two variables but of degree exponential in LaTeXMLMath ; but they are easily put in an equivalent form of degree LaTeXMLMath by introducing LaTeXMLMath more variables .
Using the resulting codes for error-resistant communication also requires polynomial-time decoding of any word at distance LaTeXMLMath from a codeword ; this and more has also been recently accomplished LaTeXMLCite .
The Goppa codes generalize the Reed-Solomon codes , which are the special case where LaTeXMLMath is a projective line LaTeXMLMath ( so LaTeXMLMath ) .
In this special case , the Goppa code can be identified with the space of polynomials of degree at most LaTeXMLMath in one variable , interpreted as words by evaluation at each element of LaTeXMLMath .
LaTeXMLMath are extended Reed-Solomon codes of length LaTeXMLMath , with one coordinate for each element of LaTeXMLMath , and an additional coordinate for the leading coefficient , corresponding to evaluation at the point at infinity of LaTeXMLMath .
Our new idea is to replace these polynomials by rational functions of bounded degree , say degree LaTeXMLMath .
Since a rational function of degree LaTeXMLMath is determined by two polynomials of degree LaTeXMLMath , we expect that LaTeXMLMath will play a role comparable to half the degree of the divisor LaTeXMLMath used to construct a Goppa code .
The notions of a rational function and its degree extend to curves LaTeXMLMath of arbitrary genus .
Given LaTeXMLMath with LaTeXMLMath rational points , we thus define LaTeXMLMath for any LaTeXMLMath as follows : LaTeXMLMath consists of the rational functions LaTeXMLMath on LaTeXMLMath , defined over LaTeXMLMath , such that LaTeXMLMath .
To give LaTeXMLMath the structure of an error-correcting code , choose an enumeration LaTeXMLMath of the LaTeXMLMath -rational points of LaTeXMLMath , and identify LaTeXMLMath with the LaTeXMLMath -tuple LaTeXMLEquation of values of LaTeXMLMath at points of LaTeXMLMath .
Since LaTeXMLMath may have poles on some LaTeXMLMath , some LaTeXMLMath values may be LaTeXMLMath .
Thus the alphabet for our new code LaTeXMLMath is not a finite field but a set of size LaTeXMLMath , the projective line LaTeXMLMath over the finite field LaTeXMLMath .
In other words , we are identifying a function LaTeXMLMath with its graph as a map from LaTeXMLMath to LaTeXMLMath , just as a polynomial in the Reed-Solomon code was identified with its graph as a map from LaTeXMLMath to LaTeXMLMath .
It is readily seen ( Prop .
LaTeXMLRef below ) that if LaTeXMLMath are distinct rational functions of degrees LaTeXMLMath on LaTeXMLMath then LaTeXMLMath holds for at most LaTeXMLMath points LaTeXMLMath of LaTeXMLMath .
Therefore LaTeXMLMath has minimal distance at least LaTeXMLMath .
In particular , since we assume LaTeXMLMath , different functions of degree LaTeXMLMath yield different words in LaTeXMLMath .
More generally , let LaTeXMLMath be a divisor of degree zero on LaTeXMLMath .
For each LaTeXMLMath we define LaTeXMLMath to be the set of rational sections of degree LaTeXMLMath of the line bundle LaTeXMLMath associated to LaTeXMLMath .
That is , LaTeXMLMath consists of the zero function together with the nonzero rational functions LaTeXMLMath on LaTeXMLMath whose divisor LaTeXMLMath is of the form LaTeXMLMath for some divisor LaTeXMLMath whose positive and negative parts each have degree at most LaTeXMLMath .
To give LaTeXMLMath the structure of an error-correcting code , choose for each LaTeXMLMath -rational point LaTeXMLMath of LaTeXMLMath a rational function LaTeXMLMath whose divisor has the same order at LaTeXMLMath as LaTeXMLMath , and identify each LaTeXMLMath with the LaTeXMLMath -tuple LaTeXMLEquation .
Different choices of LaTeXMLMath yield isomorphic codes ( Lemma LaTeXMLRef below ) .
In particular , if LaTeXMLMath we recover our earlier definition of LaTeXMLMath by setting each LaTeXMLMath .
We shall see in Prop .
LaTeXMLRef that here , too , any two distinct rational sections of degrees LaTeXMLMath agree on at most LaTeXMLMath points , so LaTeXMLMath has minimal distance at least LaTeXMLMath , and LaTeXMLMath can be recovered uniquely from the LaTeXMLMath -tuple ( LaTeXMLRef ) .
Linearly equivalent divisors yield isomorphic codes ( Lemma LaTeXMLRef ) , so LaTeXMLMath can be regarded as a degree-zero divisor modulo linear equivalence , i.e. , as an element of the Jacobian LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath be the average size of LaTeXMLMath as LaTeXMLMath varies over LaTeXMLMath : LaTeXMLEquation .
We shall show ( Thm .
LaTeXMLRef ) that if LaTeXMLMath is an asymptotically optimal curve then , for each LaTeXMLEquation the estimate LaTeXMLEquation holds as long as LaTeXMLMath .
The threshold ( LaTeXMLRef ) is low enough to allow all ratios LaTeXMLMath for which the estimate ( LaTeXMLRef ) exceeds LaTeXMLMath .
In particular , if LaTeXMLMath , our codes have on average LaTeXMLEquation times as many words as the Goppa codes of the same length and designed minimal distance must have by Riemann-Roch .
With a somewhat longer argument we show ( Thm .
LaTeXMLRef ) that the same estimate holds for each individual LaTeXMLMath , but with a higher threshold LaTeXMLMath defined below ( equations LaTeXMLRef , LaTeXMLRef ) .
We can not simply conclude that our codes transmit asymptotically LaTeXMLMath more bits per letter than Goppa ’ s , because our alphabet size is larger by LaTeXMLMath than that of the Goppa codes .
A direct comparison would require Goppa codes over a field of LaTeXMLMath elements .
But it is rare that LaTeXMLMath and LaTeXMLMath are both prime powers ( one of them must be a power of LaTeXMLMath , the other a Mersenne or Fermat prime ) ; and they can never both be squares .
Nevertheless we claim that a fair comparison can be made , and shows our codes to be better in a range of parameters that includes all the Goppa codes that improve on Gilbert-Varshamov .
We base this claim on two observations .
First , if a code over an alphabet of LaTeXMLMath letters is as good as a Goppa code , its parameters should obey the relation obtained by extrapolating ( LaTeXMLRef ) to an alphabet of size LaTeXMLMath , that is , LaTeXMLEquation .
By ( LaTeXMLRef ) , our codes ’ parameters satisfy LaTeXMLEquation .
This improves on ( LaTeXMLRef ) as long as LaTeXMLEquation .
This condition holds for all LaTeXMLMath for which ( LaTeXMLRef ) is better than the Gilbert-Varshamov bound .
For a second approach , instead of extrapolating Goppa codes to alphabets of size LaTeXMLMath , we degrade our codes by artificially reducing the alphabet size to LaTeXMLMath .
To do this , we choose for each LaTeXMLMath a forbidden letter LaTeXMLMath , and consider only words LaTeXMLMath such that LaTeXMLMath for every LaTeXMLMath .
If the LaTeXMLMath are chosen independently at random from LaTeXMLMath , the expected number of such words LaTeXMLMath is LaTeXMLMath .
These words constitute a code of length LaTeXMLMath and minimal distance LaTeXMLMath over an alphabet of size LaTeXMLMath .
But by ( LaTeXMLRef ) the size of this code is within a subexponential factor LaTeXMLMath of LaTeXMLMath , the Riemann-Roch lower bound on the number of words in the Goppa code with the same alphabet size , length , and designed distance !
Since an average degradation of LaTeXMLMath is thus asymptotically as good as a Goppa code , we may justifiably claim that LaTeXMLMath itself is better than Goppa .
We establish the lower bound LaTeXMLMath on the minimal distance of LaTeXMLMath , the independence of LaTeXMLMath of the choice of LaTeXMLMath , and the isomorphism LaTeXMLMath when the degree-zero divisors LaTeXMLMath are linearly equivalent .
We then prove the asymptotic formula ( LaTeXMLRef ) for LaTeXMLMath , and indicate how to modify our analysis to estimate the size of individual codes LaTeXMLMath .
Let LaTeXMLMath be a divisor of degree LaTeXMLMath on a curve LaTeXMLMath over LaTeXMLMath , and suppose LaTeXMLMath are distinct sections of LaTeXMLMath of degrees LaTeXMLMath .
Then the words associated to LaTeXMLMath by ( LaTeXMLRef ) agree on at most LaTeXMLMath coordinates .
In particular , LaTeXMLMath has minimal distance at least LaTeXMLMath .
We may assume that the LaTeXMLMath are nonzero .
Let LaTeXMLMath be the divisors LaTeXMLMath , LaTeXMLMath .
These LaTeXMLMath are degree- LaTeXMLMath divisors whose positive and negative parts LaTeXMLMath each have degree LaTeXMLMath .
Set LaTeXMLMath , a nonzero rational function on LaTeXMLMath .
If LaTeXMLMath agree on the LaTeXMLMath -th coordinate then LaTeXMLMath is either a pole of both LaTeXMLMath and LaTeXMLMath or a zero of LaTeXMLMath .
Let LaTeXMLEquation and LaTeXMLMath .
Then the negative part of the degree-zero divisor LaTeXMLMath is bounded above by LaTeXMLMath , and thus has degree at most LaTeXMLMath .
Thus the positive part of LaTeXMLMath also has degree at most LaTeXMLMath .
Hence there are at most LaTeXMLMath choices of LaTeXMLMath for which LaTeXMLMath .
Since there are LaTeXMLMath common poles , we deduce that the words associated to LaTeXMLMath have at most LaTeXMLMath common coordinates , as claimed .
All choices of LaTeXMLMath in ( LaTeXMLRef ) yield equivalent codes .
Let LaTeXMLMath be any other choice , and set LaTeXMLMath .
Then LaTeXMLMath is a rational function on LaTeXMLMath with neither pole nor zero at LaTeXMLMath .
Thus using LaTeXMLMath instead of LaTeXMLMath in ( LaTeXMLRef ) multiplies the LaTeXMLMath -th coordinate of every word by the nonzero scalar LaTeXMLMath , for each LaTeXMLMath .
Since each coordinate is changed by a permutation of the alphabet LaTeXMLMath , an equivalent code results .
If LaTeXMLMath are linearly equivalent divisors of degree LaTeXMLMath then the codes LaTeXMLMath , LaTeXMLMath are isomorphic .
Let LaTeXMLMath be the divisor of the function LaTeXMLMath .
Then LaTeXMLMath is a rational section of degree LaTeXMLMath of LaTeXMLMath if and only if LaTeXMLMath is a rational section of degree LaTeXMLMath of LaTeXMLMath .
This identifies LaTeXMLMath and LaTeXMLMath as sets .
Having chosen LaTeXMLMath for LaTeXMLMath , we may choose LaTeXMLMath for LaTeXMLMath .
Then ( LaTeXMLRef ) gives the same coordinates for LaTeXMLMath as an element of LaTeXMLMath that LaTeXMLMath has as an element of LaTeXMLMath .
This identifies LaTeXMLMath and LaTeXMLMath as error-correcting codes .
Some remarks on automorphisms : for nonzero LaTeXMLMath we have an isomorphism LaTeXMLMath from LaTeXMLMath to itself .
Thus the multiplicative group LaTeXMLMath acts on LaTeXMLMath .
For general LaTeXMLMath we expect that this is the full automorphism group of LaTeXMLMath .
By comparison , the Goppa codes , being linear , have many more automorphisms : translation by any codeword , as well as scalar multiplication .
Like the Goppa codes , our LaTeXMLMath can inherit more symmetries from automorphisms of LaTeXMLMath and/or LaTeXMLMath .
Thus if LaTeXMLMath has an automorphism taking LaTeXMLMath to a divisor linearly equivalent to LaTeXMLMath then LaTeXMLMath inherits this automorphism by Lemma LaTeXMLRef .
In particular , every automorphism of LaTeXMLMath acts in LaTeXMLMath .
Likewise , if LaTeXMLMath can be defined over a subfield LaTeXMLMath of LaTeXMLMath then Gal LaTeXMLMath acts on LaTeXMLMath .
Finally , LaTeXMLMath also has automorphisms by the group LaTeXMLMath , which acts on LaTeXMLMath by fractional linear transformations .
Indeed , each LaTeXMLMath yields the automorphism LaTeXMLMath of LaTeXMLMath .
These automorphisms have no Goppa-code analogue .
This requires more work .
For instance , the functions in LaTeXMLMath can be regarded the elements of height LaTeXMLMath of the function field LaTeXMLMath .
By a function-field analogue of a theorem of Schanuel LaTeXMLCite , announced by Serre LaTeXMLCite and proved by DiPippo LaTeXMLCite and Wan LaTeXMLCite ( independently but in the same way ) , for any genus- LaTeXMLMath curve LaTeXMLMath over LaTeXMLMath the number of such elements is asymptotic to LaTeXMLEquation as LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath -function of the curve ( defined below ) .
We shall see later that LaTeXMLEquation if LaTeXMLMath is an asymptotically optimal curve .
The same formula can be obtained for the number of rational sections of LaTeXMLMath of degree at most LaTeXMLMath .
But we need formulas valid not for LaTeXMLMath but for LaTeXMLMath , and this requires explicit and sufficiently small error terms in the asymptotic formula ( LaTeXMLRef ) .
It is enough to count the elements of LaTeXMLMath , which are rational sections LaTeXMLMath of LaTeXMLMath of degree exactly LaTeXMLMath .
These are the functions whose divisors are of the form LaTeXMLMath where LaTeXMLMath are effective divisors of degree exactly LaTeXMLMath with disjoint supports .
Necessarily LaTeXMLMath is linearly equivalent to LaTeXMLMath .
Conversely , for each ordered pair LaTeXMLMath of degree- LaTeXMLMath effective divisors with disjoint supports such that LaTeXMLMath , there are LaTeXMLMath rational functions LaTeXMLMath whose divisor is LaTeXMLMath .
Thus LaTeXMLMath is LaTeXMLMath times the number of such ordered pairs LaTeXMLMath .
Averaging over LaTeXMLMath in LaTeXMLMath lets us ignore the condition LaTeXMLMath .
Now it is easy to count pairs LaTeXMLMath of effective divisors of degree LaTeXMLMath without the additional condition of disjoint supports : the count is LaTeXMLMath , where LaTeXMLMath is the number of effective divisors of degree LaTeXMLMath .
But each such pair LaTeXMLMath is uniquely LaTeXMLMath for some effective divisors LaTeXMLMath with the supports of LaTeXMLMath disjoint .
Thus LaTeXMLEquation where LaTeXMLMath , and for LaTeXMLMath we define LaTeXMLEquation which is the number of pairs LaTeXMLMath of effective divisors of degree LaTeXMLMath and disjoint supports .
The identity ( LaTeXMLRef ) states that the sequence LaTeXMLMath is the convolution of LaTeXMLMath with LaTeXMLMath .
Thus LaTeXMLEquation where LaTeXMLEquation .
This leads us to study the functions LaTeXMLMath .
Now LaTeXMLMath is closely related to the zeta function LaTeXMLMath of LaTeXMLMath , defined by LaTeXMLEquation .
Indeed LaTeXMLMath .
Define LaTeXMLEquation .
It is known that LaTeXMLMath , the L-function of LaTeXMLMath , is a polynomial of degree LaTeXMLMath in LaTeXMLMath , of the form LaTeXMLEquation where the LaTeXMLMath , the “ eigenvalues of Frobenius ” for LaTeXMLMath , are LaTeXMLMath conjugate pairs of complex numbers , all of absolute value LaTeXMLMath .
( This is the “ Riemann hypothesis ” for LaTeXMLMath , here a celebrated theorem of Weil . )
Hence LaTeXMLEquation .
This yields the exact formula LaTeXMLEquation .
It follows that LaTeXMLMath has a simple pole at LaTeXMLMath with residue LaTeXMLEquation and no other singularities except for simple poles at LaTeXMLMath and LaTeXMLMath .
Thus LaTeXMLMath has a simple pole at LaTeXMLMath with residue LaTeXMLEquation and no other poles with LaTeXMLMath , whence LaTeXMLEquation as LaTeXMLMath .
It is further known that LaTeXMLMath is given by the formula LaTeXMLEquation ( “ Dirichlet class number formula ” for function fields ) .
Hence LaTeXMLEquation so we have recovered ( LaTeXMLRef ) averaged over LaTeXMLMath .
Still , we need estimates on LaTeXMLMath for LaTeXMLMath , not as LaTeXMLMath .
To go further we use the distribution of the LaTeXMLMath on the circle LaTeXMLMath .
Let LaTeXMLMath be the argument of LaTeXMLMath : LaTeXMLEquation .
It is known that a family of curves LaTeXMLMath is asymptotically optimal if and only if LaTeXMLEquation for each nonzero integer LaTeXMLMath ( see for instance “ Remark 1 ” in LaTeXMLCite ) .
Thus if LaTeXMLMath is asymptotically optimal then for any continuous function LaTeXMLMath we have LaTeXMLEquation where the LaTeXMLMath are the Fourier coefficients of LaTeXMLMath : LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLEquation for LaTeXMLMath , we calculate LaTeXMLEquation for all LaTeXMLMath in LaTeXMLMath , uniformly in any half-plane LaTeXMLMath with LaTeXMLMath .
In particular , since LaTeXMLMath for our curves , we have LaTeXMLEquation as we claimed in ( LaTeXMLRef ) .
We can now prove : For LaTeXMLMath define LaTeXMLMath by LaTeXMLEquation .
Then LaTeXMLEquation .
We have LaTeXMLEquation for all LaTeXMLMath , with strict inequality if LaTeXMLMath .
If LaTeXMLMath is asymptotically optimal ( i.e. , if LaTeXMLMath varies in a family of curves of genus LaTeXMLMath with LaTeXMLMath rational points ) , and for each LaTeXMLMath we choose LaTeXMLMath with LaTeXMLMath , then LaTeXMLMath is given asymptotically by ( LaTeXMLRef ) .
We estimate the error in ( LaTeXMLRef ) using contour integration .
By ( LaTeXMLRef ) and the discussion around ( LaTeXMLRef ) we have LaTeXMLEquation for any LaTeXMLMath .
( In fact we obtain ( LaTeXMLRef ) for all LaTeXMLMath , but we shall soon need to assume LaTeXMLMath . )
On the circle LaTeXMLMath we have LaTeXMLEquation by ( LaTeXMLRef ) .
We estimate LaTeXMLMath by using another contour integral to express LaTeXMLMath in terms of LaTeXMLMath : For all LaTeXMLMath with LaTeXMLMath we have LaTeXMLEquation .
Consider first LaTeXMLMath with LaTeXMLMath .
For such LaTeXMLMath we obtain LaTeXMLEquation by integrating termwise the product of the absolutely convergent series ( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath .
For any LaTeXMLMath other than LaTeXMLMath , the integrand extends to a meromorphic function on LaTeXMLMath with simple poles at LaTeXMLMath and a multiple pole at LaTeXMLMath .
The contour in ( LaTeXMLRef ) encloses the poles LaTeXMLMath but not the poles LaTeXMLMath .
Thus analytic continuation gives LaTeXMLEquation for all LaTeXMLMath , for any contour that encloses LaTeXMLMath but not LaTeXMLMath .
Now when LaTeXMLMath the contour in ( LaTeXMLRef ) encloses LaTeXMLMath but not LaTeXMLMath .
Thus we can evaluate the contour integral in ( LaTeXMLRef ) by starting from ( LaTeXMLRef ) , adding the residue at LaTeXMLMath , and subtracting the residue at LaTeXMLMath .
The former residue is LaTeXMLMath , and the latter is LaTeXMLMath .
This proves ( LaTeXMLRef ) .
Thus ( LaTeXMLRef ) is LaTeXMLEquation .
LaTeXMLEquation We use ( LaTeXMLRef , LaTeXMLRef ) to estimate both parts of this .
For the single integral , we find LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus the single integral is LaTeXMLMath .
We shall show that the double integral is exponentially smaller than LaTeXMLMath ; this will prove ( LaTeXMLRef ) .
To estimate the integrand , let LaTeXMLMath , so LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath , so LaTeXMLEquation .
LaTeXMLEquation Thus our proof of ( LaTeXMLRef ) will be complete once we show LaTeXMLEquation or equivalently LaTeXMLEquation and this follows from the observation that LaTeXMLEquation .
It remains to prove ( LaTeXMLRef ) and to show that the “ main term ” in ( LaTeXMLRef ) is indeed exponentially larger than the “ error term ” as long as LaTeXMLMath .
By ( LaTeXMLRef ) , the main term is LaTeXMLEquation .
Thus strict inequality in the upper bound ( LaTeXMLRef ) is what we need to show that ( LaTeXMLRef ) exceeds the “ error term ” LaTeXMLMath .
The ratio between LaTeXMLMath and the claimed upper bound is LaTeXMLEquation .
Trying LaTeXMLMath we find that LaTeXMLEquation so the upper bound holds for all LaTeXMLMath .
Moreover the bound is strict if LaTeXMLMath is a decreasing function of LaTeXMLMath at LaTeXMLMath .
We calculate that the logarithmic derivative of LaTeXMLMath at LaTeXMLMath is LaTeXMLEquation .
This is negative once LaTeXMLMath , so Theorem LaTeXMLRef is proved .
LaTeXMLMath We showed above that LaTeXMLMath is LaTeXMLMath times the number of ordered pairs LaTeXMLMath of effective degree- LaTeXMLMath divisors with disjoint supports such that LaTeXMLMath .
Call this number LaTeXMLMath , so that the total count LaTeXMLMath introduced in ( LaTeXMLRef ) is LaTeXMLMath .
We expect that LaTeXMLMath is approximated by LaTeXMLMath if LaTeXMLMath is large enough .
To prove this we use a known device from analytic number theory : for each character LaTeXMLMath of the finite abelian group LaTeXMLMath , define LaTeXMLEquation .
This is the sum of LaTeXMLMath over all ordered pairs of effective divisors LaTeXMLMath of degree LaTeXMLMath with disjoint supports .
From the LaTeXMLMath we can recover LaTeXMLMath by the usual formula LaTeXMLEquation .
When LaTeXMLMath is the trivial character ( the character sending all of LaTeXMLMath to LaTeXMLMath ) , the sum LaTeXMLMath reduces to LaTeXMLMath ; we expect that the other LaTeXMLMath will be smaller .
As with LaTeXMLMath , we analyze the LaTeXMLMath by comparing them with LaTeXMLEquation the sum extending over all pairs of effective divisors LaTeXMLMath , whether disjointly supported or not .
Again , any such pair is uniquely LaTeXMLMath with LaTeXMLMath effective divisors such that LaTeXMLMath have disjoint supports ; and necessarily LaTeXMLMath .
Thus we have a convolution formula LaTeXMLEquation generalizing ( LaTeXMLRef ) .
We deduce that LaTeXMLEquation with LaTeXMLMath as above and LaTeXMLEquation .
We can factor LaTeXMLMath by writing LaTeXMLEquation .
Since LaTeXMLMath are not in general divisors of degree zero , this requires that LaTeXMLMath be extended from LaTeXMLMath to the group LaTeXMLMath of linear equivalence classes of divisors on LaTeXMLMath of arbitrary degree .
For each LaTeXMLMath , choose an arbitrary extension of LaTeXMLMath to a homomorphism from LaTeXMLMath to the unit circle .
[ For instance , fix a divisor LaTeXMLMath of degree LaTeXMLMath , and let LaTeXMLMath be an arbitrary complex number of norm LaTeXMLMath ; any such choice of LaTeXMLMath yields a unique extension of LaTeXMLMath to LaTeXMLMath . ]
Then LaTeXMLEquation where LaTeXMLMath is the sum of the values of LaTeXMLMath on effective divisors of degree LaTeXMLMath .
[ Changing LaTeXMLMath to LaTeXMLMath , for some LaTeXMLMath of norm LaTeXMLMath , multiplies LaTeXMLMath and LaTeXMLMath by LaTeXMLMath and LaTeXMLMath respectively , and thus does not change their product . ]
For a nontrivial character LaTeXMLMath we have LaTeXMLMath for all LaTeXMLMath , because by Riemann-Roch each degree- LaTeXMLMath class in LaTeXMLMath is represented the same number of times in the sum LaTeXMLMath .
LaTeXMLMath the formula LaTeXMLEquation holds not only on average over LaTeXMLMath ( this average estimate is ( LaTeXMLRef ) ) but also for each LaTeXMLMath .
We thus recover Schanuel ’ s theorem with a sharp error term .
But again our present application requires estimates for LaTeXMLMath , not LaTeXMLMath .
Thus LaTeXMLEquation holds not only on average over LaTeXMLEquation is a finite sum .
This sum , called the LaTeXMLMath -function associated to LaTeXMLMath , is again known to satisfy a Riemann hypothesis , which yields a factorization LaTeXMLEquation for some LaTeXMLMath all of absolute value LaTeXMLMath .
Unlike the eigenvalues of Frobenius LaTeXMLMath for LaTeXMLMath , the LaTeXMLMath are of unknown distribution even for an asymptotically optimal LaTeXMLMath .
Thus instead of asymptotic formulas for LaTeXMLEquation we get only an upper bound : LaTeXMLEquation for all LaTeXMLMath .
But an upper bound is all we need because LaTeXMLMath contributes only to the error terms LaTeXMLMath , LaTeXMLMath .
Since LaTeXMLMath is a polynomial , we need not worry about nonzero poles in the contour integral LaTeXMLEquation for LaTeXMLMath , which holds for all LaTeXMLMath .
Therefore LaTeXMLEquation .
Using contour integration about a circle of radius LaTeXMLMath to isolate the LaTeXMLMath term of ( LaTeXMLRef ) , we obtain LaTeXMLEquation for any positive LaTeXMLMath .
Minimizing this over LaTeXMLMath , summing over the LaTeXMLMath choices of LaTeXMLMath , and using our known estimates for LaTeXMLMath and LaTeXMLMath , we find : For LaTeXMLMath define LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath .
Then LaTeXMLEquation for every degree- LaTeXMLMath divisor LaTeXMLMath .
There exists a unique LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLMath for all LaTeXMLMath .
If LaTeXMLMath is asymptotically optimal , and for each LaTeXMLMath we choose LaTeXMLMath with LaTeXMLMath , then LaTeXMLMath is given asymptotically by LaTeXMLEquation .
Estimate ( LaTeXMLRef ) follows from ( LaTeXMLRef ) and the bound ( LaTeXMLRef ) on each term with LaTeXMLMath nontrivial , together with the facts LaTeXMLMath and LaTeXMLEquation ( see ( LaTeXMLRef , LaTeXMLRef ) ) .
For the remainder term to be exponentially smaller we must have LaTeXMLMath ( from Thm .
LaTeXMLRef ) and LaTeXMLEquation .
The ratio between the two sides is LaTeXMLEquation where again LaTeXMLMath .
For all LaTeXMLMath , the product ( LaTeXMLRef ) exceeds LaTeXMLMath .
For LaTeXMLMath the product clearly falls below LaTeXMLMath once LaTeXMLMath is large enough .
Thus ( LaTeXMLRef ) equals LaTeXMLMath for some LaTeXMLMath , with the minimum attained at some LaTeXMLMath ; since LaTeXMLMath is a decreasing function of LaTeXMLMath for that LaTeXMLMath , the inequality ( LaTeXMLRef ) holds for all LaTeXMLMath .
It is not hard to check that LaTeXMLMath — even the lower bound LaTeXMLMath on ( LaTeXMLRef ) suffices for this .
The claim ( LaTeXMLRef ) now follows from ( LaTeXMLRef ) and Thm .
LaTeXMLRef .
The following short table lists LaTeXMLMath rounded to four decimals for LaTeXMLMath and LaTeXMLMath a prime power LaTeXMLMath : LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath 4.3461 1.8541 1.1606 0.8348 0.5276 LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath 0.4440 0.3827 0.2990 0.2448 0.1919 Since the definition of LaTeXMLMath requires LaTeXMLMath , we must have LaTeXMLMath , so the threshold LaTeXMLMath is too high for LaTeXMLMath .
For these small LaTeXMLMath , we get information only about the average size LaTeXMLMath of the codes LaTeXMLMath with small LaTeXMLMath .
But it is only for LaTeXMLMath that any of the algebraic-geometry codes improve on Gilbert-Varshamov .
For LaTeXMLMath it turns out that LaTeXMLMath is larger than the maximal LaTeXMLMath for which LaTeXMLMath attains or exceeds the Gilbert-Varshamov bound .
For LaTeXMLMath , we find that LaTeXMLMath is within the range of codes whose average size LaTeXMLMath improves on Gilbert-Varshamov ; thus in each case we have a subrange in which each individual code LaTeXMLMath is known to be exponentially larger than the Gilbert-Varshamov bound .
As LaTeXMLMath increases , LaTeXMLMath , so this subrange of LaTeXMLMath values covers almost all of LaTeXMLMath .
A new construction of error-correcting codes automatically raises new decoding problems .
When the codes come from algebraic curves , these problems can be stated in terms of the geometry of the curves .
For example , for LaTeXMLMath , the problem of nearest-neighbor decoding is a special case of the following problem : Problem 1 .
Given : an algebraic curve LaTeXMLMath of genus LaTeXMLMath over a field LaTeXMLMath ; a list LaTeXMLMath of LaTeXMLMath -rational points of LaTeXMLMath ; an LaTeXMLMath -tuple LaTeXMLMath in LaTeXMLMath ; and integers LaTeXMLMath .
Find a rational function LaTeXMLMath of degree at most LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath with at most LaTeXMLMath exceptions , assuming that at least one such LaTeXMLMath exists .
Similarly for LaTeXMLMath : Problem 1 ’ .
Given : an algebraic curve LaTeXMLMath of genus LaTeXMLMath over a field LaTeXMLMath ; a divisor LaTeXMLMath of degree zero on LaTeXMLMath ; a list LaTeXMLMath of LaTeXMLMath -rational points of LaTeXMLMath , and functions LaTeXMLMath whose divisor has the same order at LaTeXMLMath as LaTeXMLMath ; an LaTeXMLMath -tuple LaTeXMLMath in LaTeXMLMath ; and integers LaTeXMLMath .
Find a rational section LaTeXMLMath of LaTeXMLMath of degree at most LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath with at most LaTeXMLMath exceptions , assuming that at least one such LaTeXMLMath exists .
By Prop .
LaTeXMLRef , if LaTeXMLMath then LaTeXMLMath is uniquely determined ; if LaTeXMLMath equals or exceeds LaTeXMLMath , but not by too much , one might still hope that there are few enough spurious LaTeXMLMath that “ list decoding ” ( that is , finding all possible LaTeXMLMath , not just one ) may be feasible as in LaTeXMLCite .
The special case LaTeXMLMath of Problem 1 or 1 ’ is the error detection or recognition problem : is a given word in the code ?
For a Goppa code , the recognition problem is readily solved in time polynomial in the length of the code : the code is linear , so recognition reduces to linear algebra .
But the new codes LaTeXMLMath are nonlinear , and an efficient error-detection algorithm is not obvious .
Another , possibly even more fundamental , difficulty is enumerating LaTeXMLMath .
To use LaTeXMLMath in any error-correcting application other than the highly unlikely application of transmitting the values of a low-degree rational section of LaTeXMLMath , one must have an efficient means of generating the LaTeXMLMath -th codeword as a function of LaTeXMLMath , and of inverting this function to recover the integer LaTeXMLMath transmitted .
For a linear code with a known basis , enumeration is no harder than recognition , but again the problem seems nontrivial for our nonlinear codes LaTeXMLMath .
It is not necessary to enumerate every codeword : if LaTeXMLMath , an efficiently computable and invertible injection from LaTeXMLMath to LaTeXMLMath would still let us use an LaTeXMLMath -word subcode of LaTeXMLMath for error-resistant communication .
But LaTeXMLMath must not be so much smaller than LaTeXMLMath as to reduce the asymptotic transmission rate .
Thus we ask : Problem 2 .
Find LaTeXMLMath and an injection LaTeXMLMath such that both LaTeXMLMath and the inverse function LaTeXMLMath are efficiently computable .
We show that both Problems 1 and 2 have polynomial-time solutions when LaTeXMLMath has genus zero .
( In that case , all degree-zero divisors are linearly equivalent , so Problems 1 and 1 ’ are equivalent . )
This does not directly address the issue of using LaTeXMLMath for error-resistant communications , because that application requires curves of large genus ; the most direct generalization of our solution to arbitrary LaTeXMLMath requires exhaustion over LaTeXMLMath and thus takes time exponential in the genus .
Nevertheless we have hope that our solutions can be adapted to the large-genus case , especially for Problems 1 and 1 ’ .
This is because we solve Problem 1 in genus zero by adapting a known algorithm for decoding Reed-Solomon codes .
Goppa codes are large-genus generalizations of Reed-Solomon codes , and can be decoded efficiently LaTeXMLCite .
It may be possible to combine ideas from these decoding algorithms and our genus-zero solution of Problem 1 to solve that Problem in general .
In the genus-zero case , all LaTeXMLMath with the same LaTeXMLMath are isomorphic .
Thus we may and shall assume LaTeXMLMath , and call the codes simply “ LaTeXMLMath ” , suppressing the subscript .
This LaTeXMLMath consists of rational functions in one variable LaTeXMLMath , evaluated at LaTeXMLMath ( one of which may be LaTeXMLMath ) .
A rational function LaTeXMLMath of degree LaTeXMLMath is a quotient LaTeXMLMath of relatively prime polynomials LaTeXMLMath in LaTeXMLMath of degree LaTeXMLMath : LaTeXMLEquation with the leading coefficients LaTeXMLMath not both zero .
A condition LaTeXMLMath is a homogeneous linear equation in the LaTeXMLMath coefficients LaTeXMLMath .
( If LaTeXMLMath the equation becomes LaTeXMLMath ; if LaTeXMLMath the equation is LaTeXMLMath if LaTeXMLMath is finite , LaTeXMLMath if LaTeXMLMath .
LaTeXMLMath , LaTeXMLMath that appear here and later can be avoided by using homogeneous coordinates on LaTeXMLMath and regarding LaTeXMLMath as the quotient of two degree- LaTeXMLMath homogeneous polynomials in two variables . )
Thus the recognition problem amounts to solving the LaTeXMLMath simultaneous linear equations coming from LaTeXMLMath , which we can do in time polynomial in LaTeXMLMath .
We claim that every nonzero solution is proportional to LaTeXMLMath and thus recovers the function LaTeXMLMath , as long as LaTeXMLMath — exactly the condition we imposed on LaTeXMLMath when we defined of LaTeXMLMath .
Indeed , suppose LaTeXMLMath is another solution , yielding another rational function LaTeXMLMath .
Then the polynomial LaTeXMLMath , of degree at most LaTeXMLMath , vanishes at all finite LaTeXMLMath , and its LaTeXMLMath coefficient vanishes if some LaTeXMLMath .
Thus LaTeXMLMath is identically zero , and LaTeXMLMath as claimed .
If LaTeXMLMath is of degree LaTeXMLMath , the same argument shows that the linear equations on LaTeXMLMath will have a solution space of dimension LaTeXMLMath , and any nonzero solution vector recovers LaTeXMLMath as LaTeXMLMath .
We have thus solved the genus-zero case of Problem 1 for LaTeXMLMath and LaTeXMLMath .
The same system of simultaneous linear equations with LaTeXMLMath replaced by LaTeXMLMath also solves the genus-zero case of Problem 1 for any LaTeXMLMath such that LaTeXMLMath — that is , for all LaTeXMLMath less than half the designed distance LaTeXMLMath of the code .
To see this , suppose LaTeXMLMath differs from the word LaTeXMLMath in at most LaTeXMLMath coordinates , and let LaTeXMLMath be an “ error-locating polynomial ” : a polynomial of degree at most LaTeXMLMath that vanishes at each finite LaTeXMLMath where LaTeXMLMath .
( If one of the errors is at LaTeXMLMath then LaTeXMLMath has degree at most LaTeXMLMath . )
Then the coefficients of the polynomials LaTeXMLMath and LaTeXMLMath satisfy the linear equations on the coefficients of polynomials of degree LaTeXMLMath whose quotient agrees with LaTeXMLMath at all LaTeXMLMath .
Any solution LaTeXMLMath of these equations yields polynomials LaTeXMLMath such that LaTeXMLMath , which now is a polynomial of degree LaTeXMLMath , vanishes at all finite LaTeXMLMath and has vanishing LaTeXMLMath coefficient if some LaTeXMLMath .
Again it follows that LaTeXMLMath identically and LaTeXMLMath .
Thus as claimed we can decode the codes LaTeXMLMath associated to LaTeXMLMath up to the error-correcting bound LaTeXMLMath .
In the genus-zero case the enumeration problem also has a polynomial-time solution , even without relaxing it to a large subset of LaTeXMLMath as in Problem 2 .
When LaTeXMLMath , the LaTeXMLMath -function of LaTeXMLMath is the constant LaTeXMLMath , so we know LaTeXMLMath exactly , and thus also LaTeXMLMath and LaTeXMLMath .
We calculate : LaTeXMLEquation .
LaTeXMLEquation whence LaTeXMLMath for LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLEquation ( so the asymptotic formula ( LaTeXMLRef ) is exact here !
This result , but not the simpler proof we give next , already occurs in LaTeXMLCite , as a special case of a formula for LaTeXMLMath depending only on the zeta function of LaTeXMLMath in the case that LaTeXMLMath is hyperelliptic . )
We next construct a bijection LaTeXMLMath from LaTeXMLMath to a finite field LaTeXMLMath containing LaTeXMLMath with degree LaTeXMLMath .
Since LaTeXMLMath is readily enumerated ( choose a basis for LaTeXMLMath as a vector space over its prime field ) , our bijection will yield a complete enumeration of LaTeXMLMath .
To construct LaTeXMLMath , fix LaTeXMLMath that generates LaTeXMLMath over LaTeXMLMath , and define LaTeXMLMath for all LaTeXMLMath .
Note that LaTeXMLMath can not be LaTeXMLMath , because the denominator of LaTeXMLMath has degree at most LaTeXMLMath , and thus can not vanish at LaTeXMLMath .
Moreover , LaTeXMLMath is an injection : if LaTeXMLMath are distinct rational functions of degree at most LaTeXMLMath we can not have LaTeXMLMath , because then LaTeXMLMath would be a root of a polynomial of degree at most LaTeXMLMath , and thus could not generate the field extension LaTeXMLMath .
Since LaTeXMLMath it follows that LaTeXMLMath is a bijection .
To invert LaTeXMLMath , we must express any LaTeXMLMath as LaTeXMLMath for some polynomials LaTeXMLMath of degrees LaTeXMLMath .
This , too , can be done by solving LaTeXMLMath simultaneous linear equations , and thus in time polynomial in LaTeXMLMath .
For instance , find the intersection of the two LaTeXMLMath -vector subspaces LaTeXMLEquation and LaTeXMLEquation of dimension LaTeXMLMath in LaTeXMLMath .
Note that the intersection has dimension at least LaTeXMLMath , and thus contains a nonzero vector .
This proves directly that the injection LaTeXMLMath is onto , and thus also completes an alternative proof of the formula ( LaTeXMLRef ) .
Remark : The algorithms in these section are polynomial-time but far from optimal .
The simultaneous linear equations that arise are of a special form that can be solved much more quickly by other methods such as fast gcd ’ s in LaTeXMLMath .
Our results also suggest at least three theoretical problems .
When LaTeXMLMath , it is known that Goppa ’ s code can be modified to improve on both Gilbert-Varshamov and ( LaTeXMLRef ) near the crossover points between these two lower bounds .
Problem 3 .
Does our construction of LaTeXMLMath admit similar improvements near the crossover points between ( LaTeXMLRef ) and the Gilbert-Varshamov bound for codes over an alphabet of LaTeXMLMath letters ?
A second problem is whether the thresholds LaTeXMLMath and LaTeXMLMath of Thms .
LaTeXMLRef and Thm .
LaTeXMLRef are best possible : Problem 4 .
Can the bounds LaTeXMLMath and LaTeXMLMath be reduced ?
In particular , can any of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath be replaced by a threshold LaTeXMLMath ?
If LaTeXMLMath can be pushed below LaTeXMLMath then ( LaTeXMLRef ) will yield a deterministic construction of arbitrarily long algebraic-geometry codes over a five-letter alphabet with LaTeXMLMath both bounded away from zero .
Note that by ( LaTeXMLRef ) Goppa codes do not do this when LaTeXMLMath .
For a five-letter alphabet , Thm .
LaTeXMLRef proves the existence of such codes , but does not let us specify one in time polynomial in LaTeXMLMath , because of the averaging over LaTeXMLMath .
We may thus ask : Problem 5 .
Is it possible to compute , in polynomial or random polynomial time , a choice of LaTeXMLMath that makes LaTeXMLMath at least as large as average , and thus with LaTeXMLMath both provably bounded away from zero ?
Finally , a more speculative kind of problem concerns our earlier observation that degrading LaTeXMLMath to a LaTeXMLMath -letter alphabet yields nonlinear codes with exactly the same LaTeXMLMath as Goppa codes .
Is this more than a coincidence ?
That is , Problem 6 .
Give a conceptual explanation for the factor LaTeXMLMath in ( LaTeXMLRef ) , and for the fact that it exactly cancels the degradation factor LaTeXMLMath .
Thanks to the Packard foundation for financial support , to Joel Rosenberg for a careful reading of an earlier draft , and to Stephen DiPippo for the references LaTeXMLCite .
We give a proof of Dyson ’ s Lemma for a product of smooth projective varieties of arbitrary dimension .
WEAK POSITIVITY AND DYSON ’ S LEMMA Markus Wessler By Liouville ’ s theorem , complex numbers which can be approximated by rational numbers very well are necessarily transcendent .
In other words , if LaTeXMLMath is algebraic of degree LaTeXMLMath , then for all LaTeXMLMath there are only finitely many rational numbers LaTeXMLMath such that LaTeXMLMath .
But since in the situation above an infinite sequence of rational numbers LaTeXMLMath could be constructed , satisfying LaTeXMLMath for all LaTeXMLMath , it was clear that there was some lower bound for the exponent .
It took about hundred years until Roth in 1955 could prove that replacing LaTeXMLMath by LaTeXMLMath was in fact the optimal bound .
In most approaches , auxiliary polynomials in two or more variables were constructed .
In his paper LaTeXMLCite , Dyson describes explicitly which properties these polynomials should have .
In order to replace the exponent LaTeXMLMath by LaTeXMLMath , he proved a statement about the existence of certain polynomials , which is known as Dyson ’ s Lemma today .
Fixing a certain number of points in the complex plain , Dyson considers polynomials in two variables of multidegree LaTeXMLMath with respect to a special kind of zero conditions in these points , called the index .
Clearly there exists some polynomial satisfying them , provided the number of conditions is bounded by LaTeXMLMath .
Now Dyson finds out that , increasing the number of conditions too much , it becomes impossible to find such a polynomial at all .
In other words , assuming the existence of such a polynomial , he shows that the number of conditions is necessarily bounded by LaTeXMLMath for some constant LaTeXMLMath depending on LaTeXMLMath and LaTeXMLMath .
Moreover , this constant tends to one when increasing the ratio LaTeXMLMath , which means that the number of conditions is asymptotically independent .
We want to reformulate this in terms of algebraic geometry .
The question should be the following .
If one compactifies the situation by considering LaTeXMLMath over LaTeXMLMath ( or over some algebraically closed field ) and if one identifies polynomials with sections of some special sheaf and encodes the zero conditions into some ideal sheaf , how can the existence of a special section be interpreted then ?
This immediately leads to the notion of weak positivity , and it is exactly what Esnault-Viehweg stated and proved in LaTeXMLCite , using positivity statements and vanishing theorems .
The next question arising is : how can this be generalized further ?
The case of a product of arbitrary curves has been treated by Nakamaye LaTeXMLCite , for example , using derivations .
The proof of Dyson ’ s Lemma in LaTeXMLCite , however , is based on positivity methods from algebraic geometry .
Trying to give a corresponding proof for a product of arbitrary curves , one is led to a much more general situation .
We shall , therefore , in section 5 give a proof for Dyson ’ s Lemma as it is stated below ( LaTeXMLRef ) and then , in section 6 , deduce from this the result for a product of curves , where a slightly stronger positivity statement can be obtained .
We now consider the following situation .
Let LaTeXMLMath be a product of smooth projective varieties , defined over an algebraically closed field LaTeXMLMath of characteristic zero .
For LaTeXMLMath , let us denote by LaTeXMLMath the dimension of LaTeXMLMath and by LaTeXMLMath the projection onto the LaTeXMLMath -th factor .
Let us fix very ample sheaves LaTeXMLMath on LaTeXMLMath and let us write LaTeXMLMath .
For every LaTeXMLMath -tuple LaTeXMLMath of non-negative integers let us denote by LaTeXMLMath the induced sheaf on LaTeXMLMath .
Let LaTeXMLMath be a finite set of LaTeXMLMath points on LaTeXMLMath with projections LaTeXMLMath onto the factors .
Let us fix positive rational numbers LaTeXMLMath and an LaTeXMLMath -tuple LaTeXMLMath of non-negative integers .
Let us assume that LaTeXMLMath .
For LaTeXMLMath we define LaTeXMLMath to be the ideal sheaf generated by LaTeXMLMath for LaTeXMLMath and LaTeXMLMath Finally , let us fix non-negative integers LaTeXMLMath such that LaTeXMLMath is globally generated and let us define LaTeXMLMath .
In this situation , we are going to prove : If LaTeXMLMath is effective , then LaTeXMLMath is weakly positive over a product open set , where LaTeXMLMath with LaTeXMLEquation and LaTeXMLMath for LaTeXMLMath .
Comparing this to the classical situation , we notice that the ideal sheaf LaTeXMLMath carries the index conditions , the effectivity of LaTeXMLMath corresponds to the existence of a polynomial of multidegree LaTeXMLMath satisfying these conditions , and the maximal number of possible conditions is encoded in LaTeXMLMath .
Let us recall the following positivity notions : Let LaTeXMLMath be a quasi-projective variety and let LaTeXMLMath be an open subset .
A locally free sheaf LaTeXMLMath on LaTeXMLMath is called weakly positive over LaTeXMLMath , if there exists an ample invertible sheaf LaTeXMLMath on LaTeXMLMath such that for all LaTeXMLMath the sheaf LaTeXMLMath is semi-ample over LaTeXMLMath , which means that for some LaTeXMLMath the sheaf LaTeXMLMath is globally generated over LaTeXMLMath .
If LaTeXMLMath is weakly positive over LaTeXMLMath , we call LaTeXMLMath weakly positive .
Obviously , we may replace LaTeXMLMath by arbitrary multiples .
An invertible sheaf LaTeXMLMath on LaTeXMLMath is called very ample with respect to LaTeXMLMath , if LaTeXMLMath is globally generated over LaTeXMLMath by sections of a finite dimensional subspace LaTeXMLMath and the natural map LaTeXMLMath defined by these sections is an embedding .
LaTeXMLMath is called ample with respect to LaTeXMLMath , if some power of LaTeXMLMath is very ample with respect to LaTeXMLMath .
Let LaTeXMLMath be a quasi-projective variety , let LaTeXMLMath be an invertible sheaf on LaTeXMLMath and let LaTeXMLMath be an open subset .
Then LaTeXMLMath is ample with respect to LaTeXMLMath if and only if there exists a blowing up LaTeXMLMath with centre outside LaTeXMLMath such that for some ample invertible sheaf LaTeXMLMath on LaTeXMLMath and some LaTeXMLMath we have an inclusion LaTeXMLMath , which is an isomorphism over LaTeXMLMath .
Proof : We only have to show the condition is necessary and to this end we may assume that LaTeXMLMath is very ample with respect to LaTeXMLMath .
Let LaTeXMLMath be the space of sections generating LaTeXMLMath over LaTeXMLMath .
Thus we have a rational map LaTeXMLMath , given by the map LaTeXMLMath , surjective over LaTeXMLMath .
If LaTeXMLMath is the image sheaf of this map , then we consider the blowing up LaTeXMLMath with respect to the ideal sheaf LaTeXMLMath .
By LaTeXMLCite , II.7.17.3 there exists a morphism LaTeXMLMath and an inclusion LaTeXMLMath .
Since LaTeXMLMath is not necessarily ample , we have to continue blowing up .
Let LaTeXMLMath be the image of LaTeXMLMath under LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLMath factorizes over LaTeXMLMath , where LaTeXMLMath is the projective bundle of LaTeXMLMath ( see LaTeXMLCite , II.7 ) .
We have natural maps LaTeXMLEquation .
Let LaTeXMLMath be the image sheaf of this composed map .
We consider the blowing up LaTeXMLMath of LaTeXMLMath , which is an ideal sheaf , since LaTeXMLMath is birational .
The sheaf LaTeXMLEquation is therefore invertible on LaTeXMLMath , and we obtain LaTeXMLMath as an invertible quotient of LaTeXMLMath .
By LaTeXMLCite , II.7.12 this corresponds to a morphism LaTeXMLMath , factorizing canonically over some morphism LaTeXMLMath .
Then LaTeXMLMath is the blowing up we need .
By construction we obtain an exceptional divisor LaTeXMLMath for LaTeXMLMath such that LaTeXMLMath is relatively ample .
Then there exists some LaTeXMLMath such that LaTeXMLEquation is ample .
This sheaf is contained in LaTeXMLMath , and this inclusion is an isomorphism over LaTeXMLMath .
LaTeXMLMath This immediately implies the compatibility of locally ample sheaves with finite morphisms : Let LaTeXMLMath be a morphism of normal quasi-projective varieties , let LaTeXMLMath be an open subset such that LaTeXMLMath is finite , and let LaTeXMLMath be an invertible sheaf on LaTeXMLMath .
If LaTeXMLMath is ample with respect to LaTeXMLMath , then LaTeXMLMath is ample with respect to LaTeXMLMath .
Proof : If LaTeXMLMath is finite , then by LaTeXMLRef we find a blowing up LaTeXMLMath with centre outside LaTeXMLMath , an ample sheaf LaTeXMLMath on LaTeXMLMath , some LaTeXMLMath and an inclusion LaTeXMLMath , being an isomorphism over LaTeXMLMath .
Let LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation be the fibre product .
Since LaTeXMLMath is finite , LaTeXMLMath is ample on LaTeXMLMath .
Moreover , we have an inclusion LaTeXMLEquation being an isomorphism over LaTeXMLMath , which by LaTeXMLRef implies that LaTeXMLMath is ample with respect to LaTeXMLMath .
For the general case we may , considering the Stein factorization of LaTeXMLMath , assume that LaTeXMLMath is birational and an isomorphism over LaTeXMLMath .
But then we are done , since LaTeXMLMath and hence the sections of LaTeXMLMath correspond to the sections of LaTeXMLMath .
LaTeXMLMath The definition of weak positivity is independent of the choice of the ample invertible sheaf LaTeXMLMath .
Moreover , we have : Let LaTeXMLMath a quasi-projective variety , LaTeXMLMath a locally free sheaf on LaTeXMLMath and LaTeXMLMath an open subset .
Then the following statements are equivalent : ( a ) LaTeXMLMath is weakly positive over LaTeXMLMath .
( b ) There exists an ample invertible sheaf LaTeXMLMath on LaTeXMLMath such that for all LaTeXMLMath the sheaf LaTeXMLMath is globally generated over LaTeXMLMath for some LaTeXMLMath .
( c ) For every ample invertible sheaf LaTeXMLMath on LaTeXMLMath and for all LaTeXMLMath , the sheaf LaTeXMLMath is globally generated over LaTeXMLMath for some LaTeXMLMath .
( d ) There exists an invertible sheaf LaTeXMLMath on LaTeXMLMath , ample with respect to LaTeXMLMath , such that for all LaTeXMLMath the sheaf LaTeXMLMath is globally generated over LaTeXMLMath for some LaTeXMLMath .
( e ) For every invertible sheaf LaTeXMLMath on LaTeXMLMath , ample with respect to LaTeXMLMath , and for all LaTeXMLMath , the sheaf LaTeXMLMath is globally generated over LaTeXMLMath for some LaTeXMLMath .
( f ) There exists an invertible sheaf LaTeXMLMath on LaTeXMLMath , such that for all LaTeXMLMath the sheaf LaTeXMLMath is globally generated over LaTeXMLMath for some LaTeXMLMath .
Proof : LaTeXMLMath is just the definition of weak positivity , and we obviously have LaTeXMLMath .
So it remains to show LaTeXMLMath .
Let LaTeXMLMath be ample with respect to LaTeXMLMath and let LaTeXMLMath .
By LaTeXMLRef we find a blowing up LaTeXMLMath , an ample sheaf LaTeXMLMath on LaTeXMLMath and some LaTeXMLMath with an inclusion LaTeXMLMath , being an isomorphism over LaTeXMLMath .
Let LaTeXMLMath and let us choose LaTeXMLMath such that LaTeXMLMath is globally generated .
Thus , for some LaTeXMLMath , we have a surjective map LaTeXMLMath .
By assumption , there exists some LaTeXMLMath such that , for some LaTeXMLMath , we find a map LaTeXMLEquation surjective over LaTeXMLMath .
Thus we obtain maps LaTeXMLEquation and hence LaTeXMLEquation all being surjective over LaTeXMLMath .
This induces a map LaTeXMLEquation surjective over LaTeXMLMath .
We may assume that the sheaf on the left hand side is globally generated , and , taking LaTeXMLMath , this implies LaTeXMLMath .
LaTeXMLMath We consider the following class of ideal sheaves : Let LaTeXMLMath be a normal quasi-projective variety and let LaTeXMLMath be an ideal sheaf on LaTeXMLMath .
Let LaTeXMLMath be a birational morphism such that LaTeXMLMath is normal and LaTeXMLMath is invertible .
1 .
We call LaTeXMLMath full , if the natural map LaTeXMLMath is an isomorphism .
2 .
If LaTeXMLMath is an invertible sheaf on LaTeXMLMath and LaTeXMLMath an open subset , then we call LaTeXMLMath weakly positive over LaTeXMLMath , if LaTeXMLMath is weakly positive over LaTeXMLMath .
3 .
If LaTeXMLMath is projective , then we call LaTeXMLMath numerically effective ( nef ) , if LaTeXMLMath is numerically effective on LaTeXMLMath .
Let LaTeXMLMath be a normal quasi-projective variety , let LaTeXMLMath be an invertible sheaf on LaTeXMLMath , let LaTeXMLMath be a full ideal sheaf on LaTeXMLMath and let LaTeXMLMath be an open subset .
( a ) LaTeXMLMath is weakly positive over LaTeXMLMath if and only if for every sheaf LaTeXMLMath , ample with respect to LaTeXMLMath , and for all LaTeXMLMath the sheaf LaTeXMLMath is globally generated over LaTeXMLMath for some LaTeXMLMath .
( b ) If LaTeXMLMath and LaTeXMLMath are invertible sheaves on LaTeXMLMath and if for every LaTeXMLMath there is some LaTeXMLMath such that LaTeXMLMath is weakly positive over LaTeXMLMath , then so is LaTeXMLMath .
Proof : LaTeXMLMath follows immediately from LaTeXMLMath and from LaTeXMLCite , 4.3 .
In order to show LaTeXMLMath , let us assume LaTeXMLMath is weakly positive over LaTeXMLMath .
We choose some birational morphism LaTeXMLMath such that LaTeXMLMath is normal , LaTeXMLMath is invertible on LaTeXMLMath and LaTeXMLMath is an isomorphism .
If LaTeXMLMath is ample with respect to LaTeXMLMath , then LaTeXMLMath is ample with respect to LaTeXMLMath by LaTeXMLRef .
If we choose LaTeXMLMath , then by LaTeXMLRef there exists some LaTeXMLMath such that LaTeXMLMath is globally generated over LaTeXMLMath .
Thus there exists a map LaTeXMLEquation surjective over LaTeXMLMath , and hence we obtain a map LaTeXMLEquation surjective over LaTeXMLMath , where the last equality holds because LaTeXMLMath is full .
Now we may assume that the sheaf on the left hand side is globally generated , which proves the necessity of the condition .
In order to show it is sufficient , too , let us choose again some birational morphismus LaTeXMLMath with the properties from above .
Let LaTeXMLMath and let LaTeXMLMath be ample with respect to LaTeXMLMath , hence LaTeXMLMath is ample with respect to LaTeXMLMath by LaTeXMLRef .
Then LaTeXMLMath is globally generated over LaTeXMLMath .
By definition there exists a map LaTeXMLEquation surjective over over LaTeXMLMath .
This implies the weak positivity of LaTeXMLMath over LaTeXMLMath and so by definition the weak posititity of LaTeXMLMath over LaTeXMLMath , which completes the proof of LaTeXMLMath .
LaTeXMLMath We shall need two more properties : Let LaTeXMLMath be a surjective morphism of normal quasi-projective varieties and let LaTeXMLMath be an open subset such that LaTeXMLMath is finite .
Let LaTeXMLMath be a locally free sheaf on LaTeXMLMath , let LaTeXMLMath be an invertible sheaf and let LaTeXMLMath be an ideal sheaf on LaTeXMLMath such that for every LaTeXMLMath the sheaf LaTeXMLMath is full and there exists a map LaTeXMLEquation surjective over LaTeXMLMath .
If in this situation LaTeXMLMath is weakly positive over LaTeXMLMath , then LaTeXMLMath is weakly positive over LaTeXMLMath .
Proof : Let LaTeXMLMath and let LaTeXMLMath be an ample invertible sheaf on LaTeXMLMath .
Since LaTeXMLMath is finite over LaTeXMLMath , LaTeXMLMath is ample over LaTeXMLMath by LaTeXMLRef .
If LaTeXMLMath is weakly positive over LaTeXMLMath , then by LaTeXMLRef there exists some LaTeXMLMath such that the sheaf LaTeXMLMath is globally generated over LaTeXMLMath .
Thus we obtain a map LaTeXMLEquation surjective over LaTeXMLMath and hence a map LaTeXMLEquation surjective over LaTeXMLMath .
By assumption and since LaTeXMLMath is full for every LaTeXMLMath , there exists a map LaTeXMLEquation surjective over LaTeXMLMath .
We may assume that the sheaf on the left hand side is globally generated , therefore LaTeXMLMath is globally generated over LaTeXMLMath which by LaTeXMLRef yields the weak positivity of LaTeXMLMath over LaTeXMLMath .
LaTeXMLMath Let LaTeXMLMath be a normal quasi-projective variety , let LaTeXMLMath be a locally free sheaf on LaTeXMLMath and let LaTeXMLMath be an open subset .
Then the following statements are equivalent : ( a ) LaTeXMLMath is weakly positive over LaTeXMLMath .
( b ) There exists some LaTeXMLMath and some sheaf LaTeXMLMath on LaTeXMLMath , ample with respect to LaTeXMLMath , such that for every morphism LaTeXMLMath , finite over LaTeXMLMath , with LaTeXMLMath for some LaTeXMLMath and some invertible sheaf LaTeXMLMath on LaTeXMLMath , the sheaf LaTeXMLMath is weakly positive over LaTeXMLMath .
( c ) There exists some LaTeXMLMath , some invertible sheaf LaTeXMLMath on LaTeXMLMath and for every LaTeXMLMath a morphism LaTeXMLMath , finite over LaTeXMLMath , such that LaTeXMLMath for some invertible sheaf LaTeXMLMath on LaTeXMLMath and such that LaTeXMLMath is weakly positive over LaTeXMLMath .
Proof : We only have to show LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath be an ample invertible sheaf on LaTeXMLMath such that LaTeXMLMath is ample , too .
By assumption there exists a morphism LaTeXMLMath , finite over LaTeXMLMath , such that LaTeXMLMath for some invertible sheaf LaTeXMLMath on LaTeXMLMath .
By LaTeXMLCite , 2.1 we may assume that in addition LaTeXMLMath for some invertible sheaf LaTeXMLMath on LaTeXMLMath .
Since LaTeXMLMath is weakly positive over LaTeXMLMath , so is LaTeXMLMath .
Hence , for some LaTeXMLMath we obtain a map LaTeXMLEquation .
LaTeXMLEquation surjective over LaTeXMLMath and thus a map LaTeXMLEquation surjective over LaTeXMLMath .
We may assume that the sheaf on the left hand side is globally generated and we find that LaTeXMLMath is globally generated over LaTeXMLMath .
Choosing LaTeXMLMath , we are done .
LaTeXMLMath Finally , let us recall the Fujita-Kawamata Positivity Theorem in a slightly modified version .
A proof due to Kollár can be found in LaTeXMLCite , 2.41 .
Let LaTeXMLMath be a projective surjective morphism of smooth quasi-projective varieties and let LaTeXMLMath be an open subset such that LaTeXMLMath is smooth and LaTeXMLMath is locally free over LaTeXMLMath .
Then LaTeXMLMath is weakly positive over LaTeXMLMath .
In the situation of LaTeXMLRef , let LaTeXMLMath be an invertible sheaf on LaTeXMLMath and let LaTeXMLMath be an effective normal crossing divisor on LaTeXMLMath such that LaTeXMLMath is semi-ample .
Then LaTeXMLMath is weakly positive over LaTeXMLMath .
Proof : For LaTeXMLMath sufficiently large , we have LaTeXMLMath , where LaTeXMLMath is a smooth divisor and LaTeXMLMath is a normal crossing divisor , and on the other hand we have LaTeXMLMath .
We thus may assume that LaTeXMLMath , and the statement follows from considering the cyclic covering according to this situation ( see LaTeXMLCite , 2.43 ) .
LaTeXMLMath Let LaTeXMLMath be a normal variety with at most rational singularities and let LaTeXMLMath be an effective Cartier divisor on LaTeXMLMath .
Let LaTeXMLMath be a blowing up such that both LaTeXMLMath is smooth and LaTeXMLMath has normal crossings .
For every LaTeXMLMath let us define LaTeXMLEquation .
Due to LaTeXMLCite , 5.10 , this definition does not depend on the chosen blowing up .
Moreover , we define LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLMath for LaTeXMLMath sufficiently large , so we define LaTeXMLEquation .
If in addition LaTeXMLMath is projective and LaTeXMLMath is an effective invertible sheaf on LaTeXMLMath , then let LaTeXMLEquation .
Immediately from the definition we see that , given any birational morphism LaTeXMLMath of varieties and any effective Cartier divisor LaTeXMLMath on LaTeXMLMath , we may choose a blowing up LaTeXMLMath in such a way that LaTeXMLMath is smooth and LaTeXMLMath is a normal crossing divisor on LaTeXMLMath and that in addition LaTeXMLMath factors over LaTeXMLMath , hence LaTeXMLEquation .
If LaTeXMLMath is a finite surjective morphism of smooth varieties , LaTeXMLMath an effective Cartier divisor on LaTeXMLMath and LaTeXMLMath , then there exists a map LaTeXMLEquation .
If LaTeXMLMath is the open subset such that LaTeXMLMath is étale , then this map is an isomorphism over LaTeXMLMath .
Proof : Let LaTeXMLMath be a blowing up such that LaTeXMLMath is smooth and LaTeXMLMath is a normal crossing divisor .
Let LaTeXMLMath be a desingularization of the fibre product LaTeXMLMath such that we have the following diagram : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We may assume that LaTeXMLMath is a normal crossing divisor on LaTeXMLMath .
We have LaTeXMLEquation .
Considering integral parts of LaTeXMLMath -divisors , there exists an injective map LaTeXMLEquation which is an isomorphism over LaTeXMLMath , and in this case even the multiplicities of the components do not change .
We obtain a map LaTeXMLEquation .
Since LaTeXMLMath is flat by LaTeXMLCite , III .
Exercise 9.3 , we obtain by flat base change ( LaTeXMLCite , III.9.3 ) a map LaTeXMLEquation being an isomorphism over LaTeXMLMath .
LaTeXMLMath We shall need the following fact which generalizes LaTeXMLCite , 5.21 : Let LaTeXMLMath be smooth projective varieties and let LaTeXMLMath denote their product .
For LaTeXMLMath let us consider effective divisors LaTeXMLMath and effective invertible sheaves LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath denote the induced divisor on LaTeXMLMath and let LaTeXMLMath denote the induced invertible sheaf on LaTeXMLMath .
Then we have ( a ) LaTeXMLMath .
( b ) LaTeXMLMath .
Proof : We may restrict ourselves to the case of two factors .
In order to show LaTeXMLMath , we may assume LaTeXMLMath .
Let us start with the case where LaTeXMLMath is a normal crossing divisor on LaTeXMLMath .
Let LaTeXMLMath be the second projection .
For LaTeXMLMath one has LaTeXMLMath for all fibres of the second projection , and by LaTeXMLCite , 5.18 one has LaTeXMLEquation for LaTeXMLMath .
In particular this holds for LaTeXMLMath , and since in this case the sheaf on the right hand side is equal to LaTeXMLMath , we have LaTeXMLMath .
Let now LaTeXMLMath be an arbitrary effective divisor on LaTeXMLMath and LaTeXMLMath .
We consider a blowing up LaTeXMLMath such that LaTeXMLMath is smooth and LaTeXMLMath is a normal crossing divisor on LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
For LaTeXMLMath on LaTeXMLMath we use the first case and the compatibility of relatively canonical sheaf with base change to obtain LaTeXMLEquation .
Using LaTeXMLRef and flat base change ( LaTeXMLCite , III.9.3 ) , this yields LaTeXMLEquation .
For LaTeXMLMath the right hand side is nothing but LaTeXMLMath , and we obtain LaTeXMLMath for this case , too .
To show that LaTeXMLMath is a lower bound , too , it suffices to show that for an open subset LaTeXMLMath one has LaTeXMLMath .
We thus may assume that LaTeXMLMath , and in the above calculation may choose LaTeXMLMath .
Then , since LaTeXMLMath , we obtain that LaTeXMLEquation is not an isomorphism , so LaTeXMLMath .
For LaTeXMLMath we may assume LaTeXMLMath .
Let us choose effective divisors LaTeXMLMath on LaTeXMLMath and LaTeXMLMath on LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath is a section of LaTeXMLMath , and by LaTeXMLMath we obtain LaTeXMLEquation .
For the other direction we consider the projections LaTeXMLMath and LaTeXMLMath .
One has LaTeXMLMath for every fibre LaTeXMLMath and LaTeXMLMath for every fibre LaTeXMLMath .
Let LaTeXMLMath be an effective divisor with LaTeXMLMath .
By LaTeXMLCite , 5.19 , the support of LaTeXMLMath is of the form LaTeXMLMath for some closed subvariety LaTeXMLMath and the support of LaTeXMLMath is of the form LaTeXMLMath for some closed subvariety LaTeXMLMath .
But the support of LaTeXMLMath is contained in the support of LaTeXMLMath , which yields the vanishing of LaTeXMLMath or in other words LaTeXMLMath .
LaTeXMLMath The usual vanishing and positivity theorems ( see LaTeXMLCite , for example ) can be extended to the situation described above : Let LaTeXMLMath be a normal projective variety with at most rational singularities , let LaTeXMLMath be an invertible sheaf on LaTeXMLMath , and let LaTeXMLMath be an effective Cartier divisor on LaTeXMLMath and LaTeXMLMath .
( a ) If LaTeXMLMath is nef and big , then for all LaTeXMLMath one has LaTeXMLEquation ( b ) If LaTeXMLMath is semi-ample and LaTeXMLMath is effective for some effective Cartier divisor LaTeXMLMath on LaTeXMLMath and some LaTeXMLMath , then for all LaTeXMLMath the map LaTeXMLEquation is surjective .
( c ) Let LaTeXMLMath be a projective surjective morphism on a smooth quasi-projective variety LaTeXMLMath and LaTeXMLMath the open subset such that LaTeXMLMath is locally free over LaTeXMLMath and LaTeXMLMath is smooth .
If LaTeXMLMath is semi-ample , then LaTeXMLMath is weakly positive over LaTeXMLMath .
Proof : Let LaTeXMLMath a blowing up such that LaTeXMLMath is a normal crossing divisor and LaTeXMLMath is smooth .
Writing LaTeXMLMath , the sheaf LaTeXMLMath is nef and big on LaTeXMLMath , and writing LaTeXMLMath , the sheaf LaTeXMLMath is effective .
We obtain LaTeXMLMath and LaTeXMLMath from the corresponding vanishing theorems for integral parts of LaTeXMLMath -divisors which hold on LaTeXMLMath ( see LaTeXMLCite , 2.28 and 2.33 , respectively ) .
Finally , LaTeXMLMath follows from LaTeXMLRef : If LaTeXMLMath is the induced map , then LaTeXMLEquation is weakly positive over LaTeXMLMath .
LaTeXMLMath The following two statements correspond to LaTeXMLCite , 6.5 and 6.4 , respectively .
Let LaTeXMLMath and LaTeXMLMath be smooth quasi-projective varieties and let LaTeXMLMath be a projective surjective morphism of relative dimension LaTeXMLMath .
Let us consider the open subset LaTeXMLMath such that LaTeXMLMath is smooth .
Let LaTeXMLMath be an invertible sheaf on LaTeXMLMath , let LaTeXMLMath be an effective divisor on LaTeXMLMath and let LaTeXMLMath be an open subset such that LaTeXMLMath is relatively semi-ample over LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLMath be an invertible sheaf on LaTeXMLMath such that LaTeXMLMath is ample with respect to LaTeXMLMath and such that LaTeXMLMath for all LaTeXMLMath ( here LaTeXMLMath denotes the Seshadri index , see LaTeXMLCite , for example ) .
Then there exists a nonempty open subset LaTeXMLMath such that LaTeXMLEquation over LaTeXMLMath is surjective .
Proof : By LaTeXMLCite , 5.10. we may choose an open subset LaTeXMLMath where LaTeXMLMath for all LaTeXMLMath .
( In fact , considering some blowing up LaTeXMLMath such that LaTeXMLMath is a normal crossing divisor , we may take LaTeXMLMath as the set of all LaTeXMLMath such that LaTeXMLMath is a normal crossing divisor . )
Let LaTeXMLMath be in general position and let us fix some LaTeXMLMath .
From now on , the index LaTeXMLMath means restriction to the fibre LaTeXMLMath .
We consider the blowing up LaTeXMLMath of the fibre LaTeXMLMath in LaTeXMLMath and we denote by LaTeXMLMath the exceptional divisor of LaTeXMLMath .
Then , by LaTeXMLCite , II .
Exercise 8.5 , we obtain that LaTeXMLMath .
Moreover , let LaTeXMLMath be a blowing up such that LaTeXMLMath is a normal crossing divisor and LaTeXMLMath is smooth .
We may assume that LaTeXMLMath factors as LaTeXMLMath for some morphism LaTeXMLMath .
We write LaTeXMLMath , where LaTeXMLMath is the part of the exceptional divisor with LaTeXMLMath .
We now have to prove that the sheaf LaTeXMLMath is globally generated over LaTeXMLMath .
We are obviously allowed to replace LaTeXMLMath and LaTeXMLMath by multiples and , hence , may assume that LaTeXMLMath is relatively globally generated over LaTeXMLMath .
Considering the subsheaf of LaTeXMLMath which is globally generated and replacing LaTeXMLMath by a blowing up with centre outside of LaTeXMLMath , making this sheaf invertible , we may assume that LaTeXMLMath is relatively globally generated , hence relatively numerically effective .
By assumption , LaTeXMLMath and LaTeXMLMath are nef , and LaTeXMLMath is nef and big , hence LaTeXMLMath is nef and big .
By LaTeXMLCite , 5.22 we obtain a surjection LaTeXMLEquation where LaTeXMLMath denotes the cokernel of LaTeXMLEquation .
By assumption , the support of LaTeXMLMath is contained in { z } .
Assuming LaTeXMLMath , we obtain LaTeXMLEquation and hence But this means LaTeXMLEquation so LaTeXMLMath contains LaTeXMLMath with a multiplicity of at least LaTeXMLMath , which contradicts the definition of LaTeXMLMath .
So LaTeXMLMath is concentrated in the point LaTeXMLMath , and we may apply LaTeXMLCite , 2.11 to obtain that LaTeXMLMath is globally generated over LaTeXMLMath .
LaTeXMLMath Keeping the assumptions from LaTeXMLRef , let LaTeXMLMath and LaTeXMLMath be invertible sheaves on LaTeXMLMath such that LaTeXMLMath is semi-ample over LaTeXMLMath , LaTeXMLMath is relatively semi-ample over LaTeXMLMath and LaTeXMLMath is relatively numerically effective .
Let us assume moreover that LaTeXMLMath and that , for some LaTeXMLMath , there exists a map LaTeXMLMath , surjective over LaTeXMLMath .
Then there exists a nonempty open subset LaTeXMLMath such that LaTeXMLMath is weakly positive over LaTeXMLMath .
Proof : Let us start with the case where LaTeXMLMath is flat .
Taking LaTeXMLMath , we see that LaTeXMLMath is relatively semi-ample over LaTeXMLMath .
Using LaTeXMLRef and since LaTeXMLMath , we obtain maps LaTeXMLEquation surjective over LaTeXMLMath for some nonempty open subset LaTeXMLMath .
If LaTeXMLMath is an ample invertible sheaf on LaTeXMLMath , then the sheaf LaTeXMLEquation is globally generated for LaTeXMLMath sufficiently large , and with LaTeXMLMath we obtain that LaTeXMLEquation is globally generated , too .
Since by LaTeXMLCite , 2.16 quotient sheaves inherit weak positivity , we obtain that LaTeXMLEquation is weakly positive over LaTeXMLMath .
By assumption we have a map LaTeXMLEquation surjective over LaTeXMLMath , which yields , using LaTeXMLCite , 2.16 again , the weak positivity of LaTeXMLMath over LaTeXMLMath , where LaTeXMLMath is sufficiently large .
This gives sense to the following definition : LaTeXMLEquation .
We have LaTeXMLMath , hence there exists a nonempty open subset LaTeXMLMath such that LaTeXMLMath is weakly positive over LaTeXMLMath .
Proof of LaTeXMLRef : LaTeXMLMath is ample with respect to LaTeXMLMath .
So by LaTeXMLRef there exists some LaTeXMLMath such that LaTeXMLEquation and hence LaTeXMLEquation is globally generated over LaTeXMLMath .
We may assume that LaTeXMLMath is globally generated over LaTeXMLMath .
Taking LaTeXMLEquation we obtain that LaTeXMLEquation for some general section LaTeXMLMath .
Since LaTeXMLMath is flat , we may assume that LaTeXMLEquation is locally free over LaTeXMLMath , and hence , by LaTeXMLRef , weakly positive over LaTeXMLMath .
By LaTeXMLCite , 4.3 we obtain that LaTeXMLEquation is weakly positive over LaTeXMLMath , too .
The sheaves LaTeXMLMath and LaTeXMLMath satisfy the conditions of LaTeXMLRef .
Moreover , making LaTeXMLMath larger if required , we may assume LaTeXMLEquation .
Thus we obtain that the composed map is surjective over LaTeXMLMath , since LaTeXMLMath .
Using LaTeXMLCite , 2.16 , we obtain that LaTeXMLMath is weakly positive over LaTeXMLMath , and so is LaTeXMLEquation .
Again by LaTeXMLCite , 2.16 we obtain the weak positivity of the quotient sheaf LaTeXMLEquation over LaTeXMLMath .
But by definition of LaTeXMLMath we obtain LaTeXMLEquation and so LaTeXMLMath .
LaTeXMLMath Now we have to get rid of the twist LaTeXMLMath , which we manage to do by LaTeXMLRef and the following fact : In the situation above , let LaTeXMLMath be a finite morphism , ramified over some divisor LaTeXMLMath , where LaTeXMLMath intersects the divisor LaTeXMLMath in codimension LaTeXMLMath .
Let LaTeXMLMath denote the fibre product .
Then , after replacing LaTeXMLMath by the complement of a closed subvariety of codimension LaTeXMLMath if necessary , the assumptions made in LaTeXMLRef hold true for the induced morphism LaTeXMLMath as well .
Proof : By assumption we may assume that LaTeXMLMath is even empty and that the induced morphism LaTeXMLMath is smooth over LaTeXMLMath .
Since moreover LaTeXMLMath is smooth over LaTeXMLMath , we obtain that LaTeXMLMath is smooth and so in particular LaTeXMLMath .
The compatibility of the relation LaTeXMLMath with the covering holds by LaTeXMLCite , 4.3 .
It remains to show that the relation LaTeXMLMath still holds .
But taking LaTeXMLMath and LaTeXMLMath , we may assume , after making LaTeXMLMath smaller if necessary , that LaTeXMLMath is empty and the relation then follows from LaTeXMLRef .
LaTeXMLMath We now choose for every LaTeXMLMath a finite morphism LaTeXMLMath , which satisfies the properties of LaTeXMLRef .
In addition we may , by LaTeXMLCite , 2.1 , assume that LaTeXMLMath for some invertible sheaf LaTeXMLMath on LaTeXMLMath .
This induces , for every LaTeXMLMath , a morphism LaTeXMLMath such that LaTeXMLMath for some invertible sheaf LaTeXMLMath on LaTeXMLMath .
By LaTeXMLRef , the bound LaTeXMLMath holds for LaTeXMLMath as well .
By LaTeXMLRef this implies , taking LaTeXMLMath , the weak positivity of LaTeXMLMath over LaTeXMLMath , which proves LaTeXMLRef for the flat case .
Now let us sketch how to reduce to the flat case if LaTeXMLMath is not flat .
Let LaTeXMLMath be the flat locus .
Let LaTeXMLMath denote the Hilbert scheme parametrizing the flat subvarieties of LaTeXMLMath ( see LaTeXMLCite , p. 42 , for example ) and inducing , by its universal property , a rational map LaTeXMLMath .
Extending this by LaTeXMLCite , II.7.17.3 to a morphism LaTeXMLMath , we obtain a factorization of the inclusion LaTeXMLMath over some birational map LaTeXMLMath .
By LaTeXMLCite , II.9.8.1 , the morphism LaTeXMLMath corresponds to a flat morphism LaTeXMLMath , where LaTeXMLMath turns out to be a component of the fibre product LaTeXMLMath .
Let LaTeXMLMath be a desingularization such that the preimage of the singular locus of LaTeXMLMath is a divisor and let LaTeXMLMath and LaTeXMLMath denote the induced maps .
Then , replacing the morphism LaTeXMLMath in LaTeXMLRef by LaTeXMLEquation where LaTeXMLMath is the maximal divisor in LaTeXMLMath such that LaTeXMLMath , the assumptions from LaTeXMLRef hold true for the induced sheaves and divisors as well .
To prove LaTeXMLRef , the essential fact is the existence of a morphism LaTeXMLMath or , correspondingly , LaTeXMLMath .
But this follows from duality of finite morphisms ( see LaTeXMLCite , III .
Exercise 6.10 ) and from the fact that LaTeXMLMath or , correspondingly , LaTeXMLMath , which can be proved using methods from LaTeXMLCite , III.6 and 7 .
By LaTeXMLCite , III.10.2 , we find that LaTeXMLMath , being an equidimensional morphism of smooth varieties outside LaTeXMLMath , is flat outside LaTeXMLMath , which by LaTeXMLRef and LaTeXMLRef for the flat case implies the weak positivity of LaTeXMLMath over LaTeXMLMath for an open subset LaTeXMLMath .
We may assume that LaTeXMLMath is of the form LaTeXMLMath for an open subset LaTeXMLMath .
But then LaTeXMLMath is weakly positive over LaTeXMLMath by LaTeXMLCite , 4.3 , which completes the proof of LaTeXMLRef .
LaTeXMLMath Following the notations from the introduction , we shall construct a covering which simplifies the index conditions defining the ideal sheaf LaTeXMLMath .
Let LaTeXMLMath .
For every point LaTeXMLMath we choose exactly LaTeXMLMath smooth hyperplane sections LaTeXMLMath of LaTeXMLMath in general position through LaTeXMLMath .
( If LaTeXMLMath we just add another arbitrary point . )
So LaTeXMLMath is an isolated point , where the hyperplane sections LaTeXMLMath intersect transversally .
We denote by LaTeXMLEquation the corresponding normal crossing divisor on LaTeXMLMath , where the sum runs over LaTeXMLMath and LaTeXMLMath .
So we have LaTeXMLEquation .
Now by LaTeXMLCite , Theorem 17 , there exists a smooth projective variety LaTeXMLMath and a finite morphism LaTeXMLMath such that LaTeXMLMath is a normal crossing divisor on LaTeXMLMath and LaTeXMLMath ramifies exactly over LaTeXMLMath , where , taking LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath we have : LaTeXMLEquation .
The coverings LaTeXMLMath constructed in this way for LaTeXMLMath induce a covering LaTeXMLEquation which is étale over LaTeXMLMath .
In the situation of LaTeXMLRef we consider the sheaf LaTeXMLEquation which is full .
Let LaTeXMLMath be a birational morphism such that LaTeXMLMath is smooth and LaTeXMLMath is invertible on LaTeXMLMath and let us fix a blowing up LaTeXMLMath such that LaTeXMLMath is invertible .
Let us assume moreover that there exists a morphism LaTeXMLMath making the diagram LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation commutative .
Let us denote by LaTeXMLMath the induced morphism .
Then for all LaTeXMLMath the trace map induces a surjective map LaTeXMLMath , and moreover , the ideal sheaf LaTeXMLMath is full .
Proof : Keeping notations simple , we restrict ourselves to the case LaTeXMLMath .
For the general case one has to consider the LaTeXMLMath -th powers .
We shall first show that the image of LaTeXMLMath under the trace map LaTeXMLMath is contained in LaTeXMLMath .
This statement is local , and we may , after fixing a point LaTeXMLMath and some point LaTeXMLMath mapping to LaTeXMLMath and keeping all the notations , replace LaTeXMLMath and LaTeXMLMath by the corresponding local rings .
Let LaTeXMLMath and LaTeXMLMath denote the local equations of LaTeXMLMath in the point LaTeXMLMath and of LaTeXMLMath in a point LaTeXMLMath mapping to LaTeXMLMath , respectively .
Thus for LaTeXMLMath , the ideal sheaves LaTeXMLMath and LaTeXMLMath are generated by expressions of the form LaTeXMLEquation where LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath respectively .
It remains to show that the image of LaTeXMLMath under the trace map is contained in LaTeXMLMath .
The coverings LaTeXMLMath are determined by the ramifications LaTeXMLMath in LaTeXMLMath , and we may now assume that LaTeXMLMath is a Galois covering with Galois group LaTeXMLMath .
The trace map then is nothing but the sum over all conjugates of LaTeXMLMath .
Now LaTeXMLMath is invariant under LaTeXMLMath , so the image of LaTeXMLMath under the trace map in LaTeXMLMath is generated by the images of the LaTeXMLMath -invariant elements LaTeXMLEquation where LaTeXMLMath are units for LaTeXMLMath .
But the LaTeXMLMath -invariance of such expressions just means the LaTeXMLMath -invariance of the single summands , and for such a summand , given by LaTeXMLMath , this just means that LaTeXMLMath for LaTeXMLMath and certain LaTeXMLMath .
So the image of LaTeXMLMath under the trace map in LaTeXMLMath is generated by expressions of the form LaTeXMLEquation where LaTeXMLMath .
This means that LaTeXMLMath so the image of LaTeXMLMath under the trace map is contained in LaTeXMLMath .
So the trace map induces a surjective map LaTeXMLMath .
Since LaTeXMLMath is full , we have LaTeXMLMath , and we obtain a surjective map LaTeXMLMath .
Finally , we observe that the inclusion LaTeXMLMath splits , and we obtain a surjective map LaTeXMLMath , which factors over LaTeXMLMath , as we have just seen .
So LaTeXMLMath is surjective and LaTeXMLMath is full .
LaTeXMLMath We shall now give the proof for LaTeXMLRef .
First we need some more notations .
An open subset LaTeXMLMath is called open subset of type LaTeXMLMath , if for LaTeXMLMath there exist open subsets LaTeXMLMath and an open subset LaTeXMLMath such that LaTeXMLMath is of the form LaTeXMLMath .
A subset of type LaTeXMLMath is called a product open set .
As one immediately sees , LaTeXMLRef follows from inductively applying the following If in the situation of LaTeXMLRef the sheaf LaTeXMLMath is weakly positive over some open set LaTeXMLMath of type LaTeXMLMath , then the sheaf LaTeXMLMath is weakly positive over some open set LaTeXMLMath of type LaTeXMLMath , where LaTeXMLMath with LaTeXMLEquation and LaTeXMLMath for LaTeXMLMath .
So we shall prove this .
To this end let LaTeXMLMath .
By LaTeXMLRef there exists some LaTeXMLMath such that LaTeXMLMath is globally generated over LaTeXMLMath , where LaTeXMLMath with LaTeXMLMath for LaTeXMLMath and LaTeXMLMath and where LaTeXMLMath for LaTeXMLMath and LaTeXMLMath are open subsets .
We notice that the ordering of the LaTeXMLMath is preserved ; we have LaTeXMLMath .
We choose a general section LaTeXMLMath of LaTeXMLMath and observe : There exists a product open subset LaTeXMLMath such that LaTeXMLEquation .
Moreover , by LaTeXMLRef we may assume that LaTeXMLMath is étale , and so by LaTeXMLRef we have LaTeXMLMath .
Proof : We consider the projection LaTeXMLMath onto the first LaTeXMLMath factors of the product , whose fibres are isomorphic to LaTeXMLMath .
One has LaTeXMLMath for an open subset LaTeXMLMath .
Let LaTeXMLMath , in other words , the fibre LaTeXMLMath intersects the open subset LaTeXMLMath .
Then we obtain LaTeXMLEquation .
By LaTeXMLCite , 5.11 we have LaTeXMLEquation for LaTeXMLMath .
Now by LaTeXMLRef and since the LaTeXMLMath are ordered , we obtain LaTeXMLEquation which , by LaTeXMLCite , 5.14 , yields LaTeXMLMath , where LaTeXMLMath is a neighbourhood of the fibre LaTeXMLMath .
We can say more precisely that LaTeXMLMath and thus is a product open subset .
LaTeXMLMath In order to apply LaTeXMLRef to our situation , we shall now check whether the assumptions are fulfilled .
For LaTeXMLMath let LaTeXMLMath be the coverings constructed in LaTeXMLRef , which ramify exactly in LaTeXMLMath , and let LaTeXMLMath .
Let LaTeXMLMath and let us write LaTeXMLMath and LaTeXMLMath .
In the situation described above , we have LaTeXMLEquation .
Proof of LaTeXMLRef : We consider the fibre product LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and notice that LaTeXMLMath ramifies exactly in LaTeXMLMath .
Compatibility of relatively canonical sheaves with base change yields LaTeXMLEquation and this holds true after adding divisors , which are not contained in the ramification locus , hence we have LaTeXMLEquation .
LaTeXMLEquation Considering on the other hand the fibre product LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation then LaTeXMLMath ramifies exactly in LaTeXMLMath , and the Riemann Hurwitz Formula implies LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus we get LaTeXMLEquation .
LaTeXMLEquation and so LaTeXMLEquation .
LaTeXMLEquation This proves LaTeXMLRef .
LaTeXMLMath Now LaTeXMLMath is the blowing up of the ideal sheaf LaTeXMLMath , given by powers of maximal ideals , so we have LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the exceptional divisor of LaTeXMLMath .
By choice of LaTeXMLMath there exists a surjective map LaTeXMLEquation so we obtain a surjective map LaTeXMLEquation where the last sheaf coincides with LaTeXMLMath over LaTeXMLMath .
Hence , there exists a map LaTeXMLEquation surjective over LaTeXMLMath .
Now for LaTeXMLMath and taking LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we may apply LaTeXMLRef to the induced morphism LaTeXMLMath and obtain that LaTeXMLMath is weakly positive over LaTeXMLEquation where LaTeXMLMath is a non-empty subset and LaTeXMLMath with LaTeXMLEquation .
Now we may assume that LaTeXMLMath is of the form LaTeXMLMath , where LaTeXMLMath is an open subset of type LaTeXMLMath .
By LaTeXMLRef , for every LaTeXMLMath there exists a surjective map LaTeXMLEquation and LaTeXMLMath is full for every LaTeXMLMath .
So taking LaTeXMLEquation and since LaTeXMLMath is finite , the assumptions of LaTeXMLRef are fulfilled , and we obtain the weak positivity of LaTeXMLMath over LaTeXMLMath .
Now we have LaTeXMLEquation .
Hence by LaTeXMLRef we obtain the weak positivity of LaTeXMLEquation over the open set LaTeXMLMath of type LaTeXMLMath , which completes the proof of LaTeXMLRef .
LaTeXMLMath As explained in the introduction , the case of curves is , in some sense , the most “ natural ” one .
Let us hence evaluate LaTeXMLRef for the special situation of a product LaTeXMLMath of smooth projective curves of genera LaTeXMLMath ( which , indeed , was the motivation for this paper ) .
Using notations from above and taking LaTeXMLMath , hence LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , we obtain LaTeXMLMath .
Near a point LaTeXMLMath , sections of this sheaf have an expansion LaTeXMLEquation for local parameters LaTeXMLMath in LaTeXMLMath .
Moreover , we can define in the usual way the index LaTeXMLMath of such a section LaTeXMLMath in the point LaTeXMLMath by LaTeXMLEquation .
The ideal sheaves LaTeXMLMath turn out to be generated by global sections LaTeXMLMath satisfying LaTeXMLMath .
In this situation we obtain from LaTeXMLRef the following result : If LaTeXMLMath is effective , then LaTeXMLMath is weakly positive over a product open set in LaTeXMLMath , where LaTeXMLMath with LaTeXMLEquation and LaTeXMLMath for LaTeXMLMath .
One can do slightly better here .
Making a very mild assumption on the position of the points and modifying the ideal sheaves a little , one can replace the notion of weak positivity over some product open set by the notion of numerical effectivity , hence by a global positivity statement .
We shall , however , not prove this here , since it can be achieved by exactly the same arguments used in LaTeXMLCite , 5 .
Markus Wessler , Universität Kassel , Fachbereich 17 , Mathematik und Informatik Heinrich-Plett-Str .
40 , D-34132 Kassel e-Mail : wessler @ mathematik.uni-kassel.de
hep-th/0104254 hep-th/0104254 D-Branes on Noncompact Calabi-Yau Manifolds : K-Theory and Monodromy D-Branes on Noncompact Calabi-Yau Manifolds : K-Theory and Monodromy Xenia de la Ossa , Bogdan Florea and Harald Skarke Mathematical Institute , University of Oxford , 24-29 St. Giles ’ , Oxford OX1 3LB , England We study D-branes on smooth noncompact toric Calabi-Yau manifolds that are resolutions of abelian orbifold singularities .
Such a space has a distinguished basis LaTeXMLMath for the compactly supported K-theory .
Using local mirror symmetry we demonstrate that the LaTeXMLMath have simple transformation properties under monodromy ; in particular , they are the objects that generate monodromy around the principal component of the discriminant locus .
One of our examples , the toric resolution of LaTeXMLMath , is a three parameter model for which we are able to give an explicit solution of the GKZ system .
April 2001 1 .
Introduction The simplest manifestation of mirror symmetry is an exchange of the Hodge numbers LaTeXMLMath and LaTeXMLMath of a Calabi-Yau LaTeXMLMath -fold LaTeXMLMath and its mirror LaTeXMLMath .
Interpreted naively , this would seem to imply identifications between LaTeXMLMath -cycles on LaTeXMLMath and holomorphic cycles on LaTeXMLMath .
This leads to the following puzzle .
Monodromy in the complex structure moduli space of LaTeXMLMath can take LaTeXMLMath -cycles to arbitrary other LaTeXMLMath -cycles , so this would lead to the counterintuitive picture of mixing cycles of arbitrary even dimension on LaTeXMLMath .
Mathematically this puzzle is resolved by Kontsevich ’ s conjecture [ 1 ] that the relevant objects on LaTeXMLMath are the elements of the bounded derived category LaTeXMLMath of coherent sheaves .
In terms of physics , we now have the following intuitive picture .
We should not think of a cycle as a geometric object per se , but as something that a D-brane can wrap .
A D-brane corresponds to a cycle with a vector bundle on it only in a semiclassical limit .
In a more general construction a D-brane can be obtained from higher dimensional branes and anti-branes , leading to an interpretation in terms of K-theory [ 2 ] that is consistent with Kontsevich ’ s approach .
Monodromy in the complexified Kähler moduli space of Calabi-Yau manifolds has been the object of recent studies both by mathematicians [ 3 ] - [ 8 ] and by physicists [ 9 ] - [ 20 ] .
One particular approach [ 13 ] [ 15 ] [ 16 ] [ 17 ] uses well known results on McKay correspondence [ 21 ] - [ 28 ] to obtain a special basis for the K-theory on LaTeXMLMath .
These authors study noncompact toric Calabi-Yau manifolds that are resolutions of singularities of the type LaTeXMLMath ( or more general Calabi-Yau singularities in [ 19 ] ) with a single exceptional divisor , mainly in order to describe compact Calabi-Yau manifolds as hypersurfaces in the exceptional divisor .
In this work we study D-branes on non-compact toric Calabi-Yau manifolds in their own right , with the aim of getting a better understanding of what the fundamental D-brane degrees of freedom are and how they behave under monodromy .
We show how to construct a distinguished basis for the compactly supported K-theory with a number of remarkable properties , the most striking being the fact that the elements of this basis seem to generate the monodromy around the principal component of the discriminant locus in the same way as the structure sheaf LaTeXMLMath does in the compact case .
We consider cases with more than one exceptional divisor , and we test the applicability of the above statements beyond the realm of McKay correspondence .
We do not have general proofs for our statements , but we demonstrate their validity in various examples with the help of local mirror symmetry .
The outline of this paper is as follows .
In the next section we present necessary material on toric varieties , their Mori and Kähler cones , and the secondary fan .
In section 3 we introduce local mirror symmetry , toric moduli spaces and the GKZ system .
While the material in these sections is known , its presentation relying on the holomorphic quotient approach to toric varieties may be useful ; besides , it serves to establish notations and to introduce some of our examples .
Section 4 is the core of this paper .
There we discuss K-theory and known results related to McKay correspondence and proceed to define the distinguished generators LaTeXMLMath of the compactly supported K-theory .
We find that these generators are the ones that are responsible for monodromy around the principal component of the discriminant locus .
In section 5 we demonstrate that our methods work in cases that are more complicated than examples of the type LaTeXMLMath .
We consider the case of LaTeXMLMath and find that it is possible to solve the corresponding GKZ system , with results that agree precisely with our assertions .
2 .
Toric Calabi-Yau manifolds We start with presenting some general considerations on non-compact toric Calabi-Yau manifolds and their Kähler and Mori cones that will be useful later .
The results obtained here are standard [ 29 ] - [ 32 ] , but our derivations from basic facts in toric geometry are possibly simpler than what can be found in the literature .
The data of a LaTeXMLMath -dimensional toric variety LaTeXMLMath can be specified in terms of a fan LaTeXMLMath in a lattice LaTeXMLMath isomorphic to LaTeXMLMath .
LaTeXMLMath is smooth whenever each of the LaTeXMLMath -dimensional cones in LaTeXMLMath is generated over LaTeXMLMath by exactly LaTeXMLMath lattice vectors that generate LaTeXMLMath over ZZ .
We will only consider this case .
Perhaps the simplest way of describing LaTeXMLMath is as follows : Assume that there are LaTeXMLMath one dimensional cones in LaTeXMLMath generated by lattice vectors LaTeXMLMath .
Assign a homogeneous variable LaTeXMLMath to each of the LaTeXMLMath and a multiplicative equivalence relation among the LaTeXMLMath , LaTeXMLEquation with LaTeXMLMath for any linear relation LaTeXMLMath among the generators LaTeXMLMath .
The LaTeXMLMath can be normalized to be integers without common divisor ; in the context of a gauged linear sigma model they are the charges with respect to the LaTeXMLMath fields .
The number of independent relations of the type ( 2.1 ) is LaTeXMLMath .
Define a subset LaTeXMLMath of LaTeXMLMath as the set of all LaTeXMLMath -tuples of LaTeXMLMath with the following property : If LaTeXMLMath vanishes for all LaTeXMLMath , then all LaTeXMLMath with LaTeXMLMath belong to the same cone .
Then LaTeXMLMath is LaTeXMLMath , where the division by LaTeXMLMath is implemented by taking equivalence classes with respect to the multiplicative relations ( 2.1 ) .
Every one dimensional cone generated by LaTeXMLMath corresponds in a natural way to the divisor LaTeXMLMath determined by LaTeXMLMath .
Similarly , an LaTeXMLMath dimensional cone spanned by LaTeXMLMath determines the codimension LaTeXMLMath subspace LaTeXMLMath of LaTeXMLMath .
Monomials of the type LaTeXMLMath are sections of line bundles LaTeXMLMath .
If we denote by LaTeXMLMath the lattice dual to LaTeXMLMath and by LaTeXMLMath the pairing between LaTeXMLMath and LaTeXMLMath , it is easily checked that monomials of the form LaTeXMLMath with LaTeXMLMath are meromorphic functions ( i.e. , invariant under ( 2.1 ) ) on LaTeXMLMath .
This implies the linear equivalence relations LaTeXMLEquation .
Conversely , if a divisor of the form LaTeXMLMath belongs to the trivial class , then there exists an LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
A calculation similar to the way the canonical divisor of LaTeXMLMath is determined shows that the canonical divisor of LaTeXMLMath is given by LaTeXMLMath .
Thus LaTeXMLMath is Calabi-Yau if and only if LaTeXMLMath is trivial , i.e .
if and only if there exists an LaTeXMLMath such that LaTeXMLMath for every LaTeXMLMath .
Therefore the LaTeXMLMath must all lie in the same affine hyperplane .
We will make use of this fact by drawing toric diagrams in dimension LaTeXMLMath that display only the endpoints of the LaTeXMLMath .
We will be interested in the Kähler moduli space of LaTeXMLMath .
The dual of the Kähler cone is the Mori cone spanned by effective curves .
Toric curves are determined by LaTeXMLMath -dimensional cones LaTeXMLMath in LaTeXMLMath .
If a curve is compact , the corresponding cone is the boundary between two LaTeXMLMath -dimensional cones LaTeXMLMath , LaTeXMLMath .
If we denote the integer generators of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath by LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , respectively ( remember that we are assuming that our cones are simplicial and their generators generate LaTeXMLMath ) , we find that LaTeXMLMath must lie in the intersection of the hyperplane of LaTeXMLMath with LaTeXMLMath and so there exists a unique linear relation of the form LaTeXMLMath with LaTeXMLMath and all LaTeXMLMath integer .
We will now argue that the LaTeXMLMath are actually the intersection numbers between the curve LaTeXMLMath determined by LaTeXMLMath and the toric divisors LaTeXMLMath .
Our general rules imply that intersection numbers between LaTeXMLMath different toric divisors are 1 or 0 depending on whether these divisors form a cone in LaTeXMLMath .
This implies LaTeXMLMath , LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
For calculating LaTeXMLMath with LaTeXMLMath we have to use linear equivalence relations of the type ( 2.2 ) .
To calculate LaTeXMLMath we may choose LaTeXMLMath to fulfill LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation i.e .
LaTeXMLMath where ‘ LaTeXMLMath ’ stands for LaTeXMLMath with LaTeXMLMath which do not intersect LaTeXMLMath .
Thus we find that LaTeXMLMath .
As our choice of LaTeXMLMath among the LaTeXMLMath with LaTeXMLMath was arbitrary , we have indeed shown that LaTeXMLMath for any LaTeXMLMath .
A set of generators for the Mori cone is then given by all those curves LaTeXMLMath whose LaTeXMLMath can not be written as nonnegative linear combinations of the other LaTeXMLMath .
The matrix LaTeXMLMath whose lines are the LaTeXMLMath of the Mori cone generators has the following remarkable properties : Any Matrix LaTeXMLMath consisting of LaTeXMLMath independent ( linear combinations of ) lines of LaTeXMLMath serves as a ‘ charge matrix ’ for the relations ( 2.1 ) .
If the Mori cone is simplicial , we just have LaTeXMLMath .
This will be the case in most of our examples , so we will not distinguish between LaTeXMLMath and LaTeXMLMath in these cases .
Any column of LaTeXMLMath is associated with a toric divisor LaTeXMLMath .
If a linear combination LaTeXMLMath of column vectors of LaTeXMLMath vanishes , then the corresponding divisor LaTeXMLMath has vanishing intersection with any effective curve , i.e .
it is trivial .
Therefore a diagram displaying the column vectors of LaTeXMLMath or LaTeXMLMath encodes the linear equivalence relations among the toric divisors LaTeXMLMath .
We may interpret these vectors as one dimensional cones of a fan , the so called ‘ secondary fan ’ of LaTeXMLMath .
Note , however , that two distinct but linearly equivalent toric divisors correspond to the same vector in the secondary fan .
As the entries of LaTeXMLMath are the intersections between the generators of the Mori cone and the divisors , the Kähler cone of LaTeXMLMath is determined by those LaTeXMLMath such that the corresponding linear combinations of the columns of LaTeXMLMath only have nonnegative entries .
We should stress that our analysis was in terms of a single fixed triangulation .
If we allow several distinct triangulations , the Mori cone vectors of any of them will lead to correct charge matrices LaTeXMLMath but the Kähler condition will depend on which combinations of the charge vectors correspond to the Mori cone , i.e .
on the choice of triangulation .
In this way several regions of a secondary fan constructed from some charge matrix LaTeXMLMath can correspond to different ‘ geometric phases ’ in the sense of [ 33 ] [ 34 ] .
We will now present some of the examples that we are going to use in this paper .
Example 1 : The toric resolution of LaTeXMLMath : We have toric divisors LaTeXMLMath corresponding to vectors LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath are non-compact and correspond to the coordinates of the original LaTeXMLMath on which LaTeXMLMath acts by LaTeXMLMath with LaTeXMLMath .
All other LaTeXMLMath are compact and are nothing but the effective curves .
The Mori cone vectors are determined by LaTeXMLMath , leading to LaTeXMLEquation .
Upon dropping the first and the last column , this becomes LaTeXMLMath , where LaTeXMLMath is the Cartan matrix of LaTeXMLMath .
Thus the generators of the Kähler cone , corresponding to linear combinations of the LaTeXMLMath that turn the columns of LaTeXMLMath into unit vectors , are given by LaTeXMLMath or , alternatively , by LaTeXMLEquation .
Example 2 : The toric resolution of LaTeXMLMath : The resolution of a singular space of the type LaTeXMLMath , where LaTeXMLMath acts on the coordinates of LaTeXMLMath by LaTeXMLEquation can be represented torically by vectors LaTeXMLMath subject to the single relation LaTeXMLMath ; the LaTeXMLMath lattice is just the lattice generated by the LaTeXMLMath .
The first LaTeXMLMath vectors LaTeXMLMath correspond to the original coordinates LaTeXMLMath whereas LaTeXMLMath corresponds to the single exceptional divisor LaTeXMLMath isomorphic to LaTeXMLMath .
The Mori cone is determined by the single relation , leading to LaTeXMLEquation .
Example 3 : The toric resolution of LaTeXMLMath : We first consider a singular space of the type LaTeXMLMath , where LaTeXMLMath acts on the coordinates of LaTeXMLMath by LaTeXMLEquation .
As a toric variety LaTeXMLMath is determined by three vectors LaTeXMLEquation in a lattice LaTeXMLMath isomorphic to LaTeXMLMath , the singularity resulting from the fact that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath generate only a sublattice of LaTeXMLMath .
A complete crepant ( i.e. , canonical class preserving ) toric resolution LaTeXMLMath is obtained by adding two further rays LaTeXMLEquation and triangulating the resulting diagram ( this triangulation is unique in the present case ) .
Fig .
1 : The resolution of LaTeXMLMath .
Fig .
1 : The resolution of LaTeXMLMath .
The resulting fan , with the redundant third coordinate suppressed , is shown in fig .
1 .
The structure of the resolution is easily read off from this diagram : We have two exceptional divisors LaTeXMLMath and LaTeXMLMath corresponding to LaTeXMLMath and LaTeXMLMath , respectively .
The star fans of LaTeXMLMath and LaTeXMLMath tell us that LaTeXMLMath is a LaTeXMLMath and LaTeXMLMath is a Hirzebruch surface LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath intersect along a curve LaTeXMLMath which is a hyperplane of the LaTeXMLMath and at the same time the negative section of LaTeXMLMath .
We denote by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath the noncompact toric divisors corresponding to the vertices LaTeXMLMath , LaTeXMLMath and LaTeXMLMath respectively ( i.e. , the zero loci of the coordinates of our original LaTeXMLMath ) .
Intersection numbers can be calculated by using the linear equivalences LaTeXMLEquation and the fact that three distinct toric divisors have an intersection number of 1 if they belong to the same cone and 0 otherwise .
As LaTeXMLMath is linearly equivalent to LaTeXMLMath we omit expressions involving LaTeXMLMath in the following .
Intersections of divisors are well defined whenever they involve at least one of LaTeXMLMath and LaTeXMLMath .
Triple intersections are given by LaTeXMLEquation and the vanishing of LaTeXMLMath .
Intersections of two distinct divisors are determined by LaTeXMLEquation where LaTeXMLMath is the fibre of the LaTeXMLMath .
The self-intersections of LaTeXMLMath and LaTeXMLMath are LaTeXMLEquation .
We also have : LaTeXMLEquation .
This implies that LaTeXMLMath and LaTeXMLMath form mutually dual bases of the Mori cone and the Kähler cone of LaTeXMLMath .
In terms of codimension one ( here , two dimensional ) cones LaTeXMLMath and the linear relations between the rays in the two cones of maximal dimension that contain LaTeXMLMath , we obtain the following linear relations among the vectors LaTeXMLMath of the fan : LaTeXMLEquation .
As a check on our intersection numbers , we observe that indeed LaTeXMLMath .
Fig .
2 : The secondary fan of the resolution of LaTeXMLMath .
Fig .
2 : The secondary fan of the resolution of LaTeXMLMath .
If we consider the matrix whose lines are the generators ( 2.17 ) of the Mori cone and draw the rays corresponding to the columns of this matrix , we obtain the secondary fan for LaTeXMLMath as shown in fig .
2 .
The linear relations among the vectors in this fan encode the linear equivalences ( 2.12 ) among the divisors .
3 .
Local mirror symmetry In our study of D-brane states we will have to address issues that involve quantum geometry .
A standard tool for this problem is the use of mirror symmetry .
In particular , classical periods in the mirror geometry get mapped to quantum corrected expressions related to the middle cohomology of the original space .
In the non-compact case one has to use local mirror symmetry .
For our applications of this subject we have relied mainly on [ 35 ] and we refer to this paper for further references .
The authors of [ 35 ] consider decompactifications of Calabi-Yau hypersurfaces in toric varieties such that the volumes of certain cycles remain compact .
They show that in the decompactification limit these cycles lead to differential equations that are identical with the GKZ differential systems of a lower dimensional geometry .
We will assume that this remains true even for cases where the non-compact Calabi-Yau geometry can not be identified with a limiting case of a compact Calabi-Yau hypersurface .
The local mirror of a LaTeXMLMath -dimensional noncompact Calabi-Yau geometry is determined by interpreting the diagram of the hyperplane containing the end points of the LaTeXMLMath now as a polytope LaTeXMLMath in a LaTeXMLMath -dimensional lattice LaTeXMLMath .
A polytope corresponds to a line bundle LaTeXMLMath over a toric variety LaTeXMLMath by the following construction : Fix any point in LaTeXMLMath to be the origin .
Describe the facets of LaTeXMLMath by equations LaTeXMLMath , where LaTeXMLMath , the lattice dual to LaTeXMLMath and fix the sign ambiguity about LaTeXMLMath in such a way that LaTeXMLMath is nonnegative for points LaTeXMLMath of LaTeXMLMath .
Choose LaTeXMLMath to be a toric variety whose one dimensional rays are the LaTeXMLMath corresponding to a variable LaTeXMLMath as in the previous section .
To every point LaTeXMLMath assign the monomial LaTeXMLMath .
Then LaTeXMLMath is the bundle whose sections are determined by polynomials of the type LaTeXMLEquation .
The ‘ local mirror ’ LaTeXMLMath of LaTeXMLMath is defined to be the vanishing locus of a section ( 3.1 ) of LaTeXMLMath .
In the present context we can give an alternative description of the LaTeXMLMath : We have LaTeXMLMath and may choose coordinates such that LaTeXMLMath .
Then we can write the affine function LaTeXMLMath as a linear function of the form LaTeXMLMath with LaTeXMLMath ( it is easy to check that the LaTeXMLMath are the elements of LaTeXMLMath dual to the LaTeXMLMath -dimensional cones at the boundary of the support of LaTeXMLMath ) .
Obviously the complex structure moduli space of LaTeXMLMath is parametrized by the LaTeXMLMath .
It is important to note , however , that different sets of LaTeXMLMath need not correspond to different complex structures .
In particular , a scaling LaTeXMLMath does not amount to a change in the complex structure but leads to a redefinition of the LaTeXMLMath , implying the equivalences LaTeXMLEquation for any LaTeXMLMath .
Given identifications of this type it is natural to seek a description in terms of toric geometry .
If we interpret the exponents of the LaTeXMLMath ’ s as linear relations among vectors LaTeXMLMath in a toric diagram and notice that the LaTeXMLMath generate LaTeXMLMath ( at least over the rational numbers ) , we find that the LaTeXMLMath fulfill LaTeXMLEquation for any LaTeXMLMath .
These are just the relations among the vectors of the secondary fan which encodes , as we saw , the linear equivalence relations ( 2.2 ) of the divisors LaTeXMLMath corresponding to the LaTeXMLMath .
There are some subtleties , however : As we saw in the previous section , it is possible that two distinct ( but linearly equivalent ) toric divisors lead to the same vector in the secondary fan .
We will show how to interpret this in the context of the examples .
Besides , it is possible that there are identifications in the moduli space that do not come from rescalings of the type LaTeXMLMath and hence have a structure different from ( 3.2 ) .
If this occurs , the toric variety associated with the secondary fan is called the ‘ simplified moduli space ’ LaTeXMLMath .
Depending on whether we have extra identifications or not , the toric variety corresponding to the secondary fan is a compactification of LaTeXMLMath ( the moduli space of all smooth local mirror hypersurfaces ) or a covering space of a compactification of LaTeXMLMath .
LaTeXMLMath will degenerate over various loci in LaTeXMLMath where LaTeXMLMath can be solved for all LaTeXMLMath without violating the conditions on which LaTeXMLMath are allowed to vanish simultaneously .
Some of these loci may just be toric divisors , but usually there is also at least one connected piece given by a polynomial equation in the LaTeXMLMath to which we will refer as the primary or principal component of the discriminant locus .
If we want to relate the mirror geometry to the original one , we have to find a region in the moduli space where quantum corrections are strongly suppressed .
This is the case for the deep interior of the Kähler cone , the so called large volume limit , which is dual to the large complex structure limit .
As we saw in section 2 , the Kähler cone can be determined by writing any divisor as a linear combination of toric divisors and demanding that the corresponding linear combination of columns of the matrix LaTeXMLMath contain only nonnegative entries .
If the resulting generators do not belong to the secondary fan , we have to blow up the moduli space in order to be able to change to the large complex structure variables .
In those cases where the Mori cone is simplicial we can draw the secondary fan by displaying the columns of LaTeXMLMath and the generators of the Kähler cone will be nothing but the unit vectors .
If we then write the linear relations among the vectors in the secondary fan in such a way that we express every vector in terms of the unit vectors and use the corresponding rules ( 2.1 ) to set all variables except the large complex structure variables LaTeXMLMath We hope that no confusion arises from the fact that we use the same symbol LaTeXMLMath for the coordinates of LaTeXMLMath and the large complex structure variables .
LaTeXMLMath to 1 , we find that the LaTeXMLMath can be expressed as LaTeXMLEquation .
Note that we do not include a sign here ( compare with e.g .
[ 36 ] ) .
If LaTeXMLMath is the resolution of an orbifold singularity of the type LaTeXMLMath there is another distinguished coordinate patch in the moduli space containing the orbifold locus where all LaTeXMLMath except the ones corresponding to the coordinates of the LaTeXMLMath are set to zero .
At this point the conformal field theory is expected to acquire a quantum symmetry .
We find that the moduli space in this case always has a singularity that looks locally like LaTeXMLMath .
The GKZ differential operators are calculated by using the following recipe : For every linear relation LaTeXMLMath , where LaTeXMLMath corresponds to any curve in the Mori cone ( see [ 35 ] ) we define a differential operator in terms of the LaTeXMLMath , LaTeXMLEquation .
Assume that we work in a specific coordinate patch given by some LaTeXMLMath .
In order to transform ( 3.5 ) to a system involving the LaTeXMLMath we can rewrite it in terms of operators LaTeXMLMath , commute all LaTeXMLMath to the left using LaTeXMLMath and then express the LaTeXMLMath as LaTeXMLMath with LaTeXMLMath .
We stress that the solutions of the GKZ system are not the periods on LaTeXMLMath but rather the logarithmic integrals of the periods .
While the periods are finite and non-vanishing on the moduli space wherever LaTeXMLMath is non-degenerate , the GKZ solutions have extra singularities at the zero loci of moduli space coordinates coming from the logarithmic integration .
The GKZ solutions are multivalued and undergo monodromy transformations around codimension one loci where they are not holomorphic .
We will be interested mainly in monodromies around the large complex structure divisors LaTeXMLMath and around the principal component of the discriminant locus .
In addition , there is the possibility of a non-trivial transformation ( ‘ orbifold monodromy ’ , which , strictly speaking , is not a monodromy ) if the moduli space looks locally like LaTeXMLMath .
We will now show how these concepts can be applied to our examples .
Example 1 : The mirror geometry of LaTeXMLMath : Here LaTeXMLMath is LaTeXMLMath and the polynomial is given by LaTeXMLEquation so the hypersurface LaTeXMLMath is just a collection of LaTeXMLMath points in LaTeXMLMath .
A ‘ singularity ’ of LaTeXMLMath occurs whenever two or more of these points coincide .
The secondary fan is determined by the columns of ( 2.5 ) .
For LaTeXMLMath we have to blow up the moduli space in order to have a coordinate patch described by the large complex structure variables LaTeXMLMath ( with LaTeXMLMath ) .
The GKZ operators corresponding to the Mori cone generators , LaTeXMLEquation become LaTeXMLEquation with LaTeXMLEquation .
We note that the space of solutions of ( 3.8 ) is too large unless we introduce further operators corresponding to linear combinations of the Mori cone generators .
The case of LaTeXMLMath allows for an explicit solution [ 36 ] : Here we have LaTeXMLEquation and LaTeXMLMath has a basis of solutions of the form LaTeXMLEquation .
Special points in the moduli space are the large complex structure limit LaTeXMLMath , the analog of the primary component of the discriminant locus at LaTeXMLMath , and the orbifold point at LaTeXMLMath where we introduce a new coordinate LaTeXMLMath by LaTeXMLMath .
We find the following transformation properties upon taking loops around these points : LaTeXMLEquation .
Example 2 : The local mirror geometry LaTeXMLMath of the resolution LaTeXMLMath of LaTeXMLMath is just the mirror geometry of a compact Calabi-Yau manifold realised as a degree LaTeXMLMath hypersurface in LaTeXMLMath , i.e .
LaTeXMLMath is a degree LaTeXMLMath hypersurface LaTeXMLEquation in LaTeXMLMath .
The GKZ operator LaTeXMLMath becomes LaTeXMLEquation in terms of the large complex structure variable LaTeXMLMath .
Example 3 : The mirror geometry of LaTeXMLMath , a genus two Riemann surface : Fig .
3 : The fan for LaTeXMLMath .
Fig .
3 : The fan for LaTeXMLMath .
Here LaTeXMLMath is LaTeXMLMath with the LaTeXMLMath acting on the homogeneous coordinates of LaTeXMLMath as LaTeXMLMath .
The polynomial corresponding to fig .
1 is given by LaTeXMLEquation where we have chosen the subscripts of the LaTeXMLMath to correspond to those of the LaTeXMLMath in fig .
1 .
The local mirror of LaTeXMLMath is given by the vanishing locus of ( 3.15 ) in LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath has fixed points whenever two of the three LaTeXMLMath vanish .
The vanishing locus of ( 3.15 ) passes through one of these fixed points if and only if one of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath vanishes .
Thus the generic hypersurface misses the fixed points .
A quintic polynomial in LaTeXMLMath defines , by a standard calculation , a Riemann surface of Euler number LaTeXMLMath .
As the LaTeXMLMath acts without fixed points on this surface , the Euler number is divided by 5 , showing that the local mirror geometry is that of a Riemann surface LaTeXMLMath with LaTeXMLMath , i.e .
genus LaTeXMLMath .
Scalings LaTeXMLMath imply the equivalences LaTeXMLEquation .
If we naively interpret the exponents of the LaTeXMLMath as a charge matrix LaTeXMLEquation with entries LaTeXMLMath and try to find a fan with rays LaTeXMLMath fulfilling LaTeXMLMath for LaTeXMLMath , we find that we have to take LaTeXMLMath , meaning that we should not distinguish between LaTeXMLMath and LaTeXMLMath .
This can be explained by the fact that taking LaTeXMLMath implies that we can multiply LaTeXMLMath with any nonzero number provided we divide LaTeXMLMath by the same number without affecting the other LaTeXMLMath , i.e .
as long as LaTeXMLMath and LaTeXMLMath are nonzero the complex structure of LaTeXMLMath depends only on LaTeXMLMath .
This is even true if one of LaTeXMLMath becomes zero , since an exchange of LaTeXMLMath and LaTeXMLMath can be compensated by exchanging LaTeXMLMath with LaTeXMLMath which does not affect the complex structure .
Thus we can consistently drop the third line and the third column of ( 3.17 ) to obtain a matrix LaTeXMLEquation .
This is just the matrix of linear relations for the secondary fan of fig .
2 .
The corresponding compact toric variety LaTeXMLEquation with LaTeXMLMath as in ( 3.16 ) is closely related to the moduli space LaTeXMLMath of smooth hypersurfaces of the type ( 3.15 ) : Smoothness implies LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , so LaTeXMLEquation i.e .
LaTeXMLMath is a compactification of LaTeXMLMath ( other sensible compactifications correspond to omitting LaTeXMLMath or LaTeXMLMath from fig .
2 ) .
LaTeXMLMath is contained in the single coordinate patch of LaTeXMLMath defined by the cone spanned by LaTeXMLMath and LaTeXMLMath .
In this patch we can parametrize the hypersurface as the vanishing locus of LaTeXMLEquation .
Having set LaTeXMLMath , LaTeXMLMath and LaTeXMLMath to one has used up most of the freedom coming from ( 3.16 ) , the remaining relation being LaTeXMLEquation .
As we just noticed , LaTeXMLMath becomes singular along the divisors LaTeXMLMath and LaTeXMLMath of LaTeXMLMath .
The remaining singularities can be found by looking for values of LaTeXMLMath , LaTeXMLMath where LaTeXMLMath for LaTeXMLMath can be solved by some LaTeXMLMath .
This results in the equation LaTeXMLEquation for the primary component of the discriminant locus .
We note that while LaTeXMLMath contains some of the singular loci , it misses others such as LaTeXMLMath , LaTeXMLMath and any points with three or four of the LaTeXMLMath vanishing .
The divisor LaTeXMLMath in LaTeXMLMath corresponds to two one dimensional loci LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
Our main concern with the moduli space has to do with the study of monodromies .
Thus we want to know what happens when we move around singularities at codimension one rather than what happens when we hit them .
For example , the monodromy around LaTeXMLMath depends only on nonvanishing values of LaTeXMLMath and not on how we interpret the locus LaTeXMLMath .
Therefore LaTeXMLMath is sufficient for our purposes .
Fig .
4 : The moduli space of the resolution of LaTeXMLMath .
Fig .
4 : The moduli space of the resolution of LaTeXMLMath .
A schematic representation of LaTeXMLMath is given in fig .
4 , with the the toric divisors shown as straight lines and the primary component of the discriminant locus indicated by curved lines .
The locus LaTeXMLMath is tangent to the discriminant locus at their point of intersection LaTeXMLMath whereas LaTeXMLMath corresponds to a transverse intersection with LaTeXMLMath .
At LaTeXMLMath ( i.e. , LaTeXMLMath ) the moduli space has a singularity where the Riemann surface remains smooth ; in addition there are LaTeXMLMath and LaTeXMLMath singularities at LaTeXMLMath and LaTeXMLMath , respectively .
As we saw above , there is a distinguished coordinate patch in LaTeXMLMath which contains all loci where LaTeXMLMath is smooth .
Now we want to study another distinguished set of coordinates corresponding to the ‘ large complex structure limit ’ .
We remember that the Kähler cone of LaTeXMLMath ( the resolution of LaTeXMLMath ) was spanned by LaTeXMLMath and LaTeXMLMath corresponding to the vectors LaTeXMLMath and LaTeXMLMath in the secondary fan ( fig .
2 ) respectively .
The large radius limit of LaTeXMLMath corresponds to the deep interior of the Kähler cone , so by local mirror symmetry the large complex structure limit is determined by the LaTeXMLMath coordinate patch in LaTeXMLMath given by LaTeXMLEquation .
In terms of LaTeXMLMath , LaTeXMLMath the principal component of the discriminant locus is determined by LaTeXMLEquation .
The GKZ system can be determined and solved with the methods described above .
There are five independent solutions , as expected , which are described in appendix A .
4 .
D-branes and tautological bundles We want to find out about the D-brane vacuum states in type II string theory on LaTeXMLMath .
The mathematical structure that captures the largest number of properties of brane states is , at present knowledge , the bounded derived category LaTeXMLMath of coherent sheaves on LaTeXMLMath [ 37 ] [ 38 ] ( but see the remarks in [ 39 ] [ 40 ] ) .
While we will make several remarks concerning LaTeXMLMath , we will work mainly with the somewhat coarser ( but easier to handle ) concepts of K-theory .
Let LaTeXMLMath be the Grothendieck group of coherent sheaves on LaTeXMLMath .
We expect compact brane states on a non-compact space LaTeXMLMath to correspond to classes of the compactly supported K-theory group LaTeXMLMath .
Using the duality between LaTeXMLMath and LaTeXMLMath we can determine a basis for LaTeXMLMath by first finding a basis for LaTeXMLMath .
Let us consider the situation where LaTeXMLMath is a smooth crepant resolution of a singularity of the type LaTeXMLMath , where LaTeXMLMath is a finite subgroup of LaTeXMLMath .
Since LaTeXMLMath is smooth , LaTeXMLMath is generated by vector bundles ( see e.g .
[ 41 ] ) .
Moreover , if LaTeXMLMath is a crepant resolution of an abelian singularity , LaTeXMLMath is in fact generated by LaTeXMLMath line bundles , where LaTeXMLMath is the order of LaTeXMLMath ( at least for LaTeXMLMath ) [ 25 ] .
Thus , for finding a basis for the group LaTeXMLMath related to fractional branes it is convenient to first determine a set LaTeXMLMath ( LaTeXMLMath ) of line bundles whose K-theory classes generate LaTeXMLMath .
Clearly there is no choice for the LaTeXMLMath that should be preferred a priori .
Rather , there are two distinct constructions , each of which is related to McKay correspondence : 1 .
Mathematicians ’ construction [ 22 ] – [ 25 ] : There is a vector bundle LaTeXMLMath ( the ‘ tautological vector bundle ’ ) transforming in the regular representation of LaTeXMLMath whose decomposition into irreducibles gives the line bundles LaTeXMLMath .
In particular , the LaTeXMLMath are generated by their sections and the action of LaTeXMLMath on the sections determines a one-to-one correspondence between the LaTeXMLMath and the characters of the irreducible representations of LaTeXMLMath .
In the case of a resolution of LaTeXMLMath with some finite group LaTeXMLMath the first Chern classes LaTeXMLMath , LaTeXMLMath form a basis of LaTeXMLMath dual to the basis of LaTeXMLMath given by the homology classes of a basis of effective curves LaTeXMLMath in the resolution .
In the case of a singularity of the type LaTeXMLMath with LaTeXMLMath an abelian subgroup of LaTeXMLMath in general there exist several crepant resolutions and not for every resolution it is possible to define line bundles as above .
However , it was shown in [ 24 ] that there exists a distinguished crepant resolution , named LaTeXMLMath -Hilb , on which it is still possible to define the tautological line bundles ( see also [ 23 ] , [ 27 ] , [ 28 ] ) LaTeXMLMath We thank A. Craw for emphasizing the importance of choosing the G-Hilb resolution to us .
.
The advantage of this approach is that it is rigorously proven for LaTeXMLMath and LaTeXMLMath .
2 .
Physicists ’ constructions : The authors of [ 13 ] suggest to consider , in the style of [ 42 ] , the world-volume theory of D0-branes , which is a theory of LaTeXMLMath LaTeXMLMath gauge fields and LaTeXMLMath LaTeXMLMath matrices .
It is conjectured ( and shown in several examples ) that the vacua of such a theory in the different phases corresponding to different choices of Fayet-Iliopoulos parameters all lead to moduli spaces that are nothing but the geometric phases of the resolutions LaTeXMLMath of LaTeXMLMath .
Now repeat this construction with an extra field transforming in a specific one dimensional representation LaTeXMLMath of LaTeXMLMath .
It is conjectured that , independently of the phase , this should lead to a space that is the total space of a line bundle LaTeXMLMath over LaTeXMLMath , and that repeating this for all characters LaTeXMLMath should give a basis LaTeXMLMath of LaTeXMLMath .
However , this construction is extremely tedious to work with .
A different method for determining LaTeXMLMath based on the boundary chiral ring associated to a certain two dimensional gauge theory has been proposed in [ 17 ] .
The implications of this approach have been worked out for the case of a single exceptional divisor that is a weighted projective space LaTeXMLMath with Fermat weights [ 17 ] or a Grassmannian LaTeXMLMath P. Mayr informs us that this approach works in more general situations as well .
[ 19 ] .
In all examples we are aware of , the LaTeXMLMath have no sections .
The advantage of this approach is that it appears to lead to dual classes LaTeXMLMath whose interpretations in terms of D-branes are very well behaved .
Roughly , the resulting LaTeXMLMath can be summarized in the following way .
There is a set of divisor classes LaTeXMLMath containing all Kähler cone generators LaTeXMLMath and the trivial class LaTeXMLMath such that all LaTeXMLMath are nef , i.e .
have nonnegative intersection with any curve in the Mori cone .
If we denote by LaTeXMLMath the line bundles LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
In two dimensions the LaTeXMLMath are just the trivial class and the Kähler cone generators .
In higher dimensions we have to add extra divisor classes which are nonnegative integer linear combinations of the LaTeXMLMath .
For the LaTeXMLMath with G-Hilb and LaTeXMLMath the authors of [ 23 ] , [ 27 ] have given an explicit construction .
In terms of the language used in this paper this can be summarised in the following way .
Through the sections we can assign a character to any LaTeXMLMath .
It is also possible to assign characters to toric curves .
Such a curve LaTeXMLMath corresponds up to a sign to some LaTeXMLMath leading to a linear equivalence as in ( 2.2 ) .
By collecting expressions with the same sign this can be written as LaTeXMLMath where LaTeXMLMath , LaTeXMLMath are effective divisors corresponding to the same character .
We then assign this character to LaTeXMLMath and the corresponding line segment in the diagram , and find that all the characters obtainable in this way also occur in the list of characters corresponding to the LaTeXMLMath .
Then every interior point LaTeXMLMath of the toric diagram is of one of the following types : 1 .
There are three pairs of line segments with the same character meeting in LaTeXMLMath .
In this case we add nothing to the list of LaTeXMLMath ( the classes assigned by [ 23 ] , [ 27 ] in this case are already among the Kähler cone generators ) .
2 .
There are two pairs of line segments with characters LaTeXMLMath , LaTeXMLMath meeting in LaTeXMLMath ( and possibly an extra line segment ) .
Then add LaTeXMLMath to the list of LaTeXMLMath .
3 .
There are three line segments with the same character LaTeXMLMath .
In this case add LaTeXMLMath to the list of LaTeXMLMath .
It turns out that this procedure always leads to a one to one correspondence between the LaTeXMLMath and the character table of LaTeXMLMath through the action of LaTeXMLMath on the sections .
In many cases the LaTeXMLMath are the same in the mathematicians ’ and physicists ’ constructions , i.e .
LaTeXMLMath .
However , [ 17 ] seems to suggest partial resolutions in the case with a single interior point where the exceptional divisor is a weighted projective space .
We note that the G-Hilb resolution may be incompatible with such a resolution or any refinement of it , as the following example shows .
Fig .
5 : LaTeXMLMath -Hilb and partial resolution of LaTeXMLMath .
Fig .
5 : LaTeXMLMath -Hilb and partial resolution of LaTeXMLMath .
In fig .
5 we have displayed the G-Hilb resolution of LaTeXMLMath constructed according to the rules of [ 23 ] , [ 27 ] and the partial resolution by an exceptional divisor LaTeXMLMath .
Clearly the former can not be obtained as a refinement of the latter .
In the following we always follow the mathematicians ’ approach .
The next step in our construction of D-brane states is to find a basis for LaTeXMLMath that is dual to the basis of LaTeXMLMath defined in terms of line bundles LaTeXMLMath .
According to [ 25 ] , there is a pairing LaTeXMLMath between representatives LaTeXMLMath of LaTeXMLMath and LaTeXMLMath of LaTeXMLMath that can be evaluated in terms of Chern characters LaTeXMLEquation with LaTeXMLMath the localized Chern character LaTeXMLMath Let LaTeXMLMath be the embedding of a compact submanifold LaTeXMLMath in a noncompact manifold LaTeXMLMath .
Elements of the compactly supported K-theory can be represented by either coherent sheaves LaTeXMLMath on LaTeXMLMath or by their finite resolution by vector bundles on LaTeXMLMath , that is by complexes LaTeXMLMath of vector bundles on LaTeXMLMath which are exact off LaTeXMLMath and whose homology is precisely the push-forward of LaTeXMLMath to LaTeXMLMath [ 43 ] .
Then , the local Chern character is defined such that LaTeXMLMath [ 44 ] .
of the complex LaTeXMLMath and LaTeXMLMath the Todd class of LaTeXMLMath .
There is also a closely related pairing which will become important when we study monodromies .
It is defined as LaTeXMLEquation with LaTeXMLMath the line bundle ( or , more generally , the complex ) dual to LaTeXMLMath .
If we restrict LaTeXMLMath to LaTeXMLMath , these pairings become well defined under the exchange of LaTeXMLMath and LaTeXMLMath and we find that LaTeXMLMath is always symmetric whereas LaTeXMLMath is symmetric in even dimensions and skew in odd dimensions , as a consequence of the fact that LaTeXMLMath is even when LaTeXMLMath is trivial .
The generally accepted way of obtaining a basis for LaTeXMLMath is to choose classes dual to those given by the line bundles LaTeXMLMath with respect to LaTeXMLMath .
Following this convention , we define classes of LaTeXMLMath by demanding that their representatives LaTeXMLMath fulfill LaTeXMLMath .
Thus we obtain LaTeXMLMath dual to LaTeXMLMath and LaTeXMLMath dual to LaTeXMLMath with respect to LaTeXMLMath and note that the LaTeXMLMath are dual to the LaTeXMLMath and LaTeXMLMath are dual to the LaTeXMLMath with respect to LaTeXMLMath .
So far we have not been specific about the representatives LaTeXMLMath of the compactly supported K-theory .
In the spirit of [ 2 ] we may interpret them as bound states of LaTeXMLMath –filling branes .
In mathematical terms this amounts to specifying a complex of vector bundles on LaTeXMLMath that is exact outside a compact locus LaTeXMLMath .
It is not hard to check in every example that we may indeed represent every LaTeXMLMath as a formal linear combination of line bundles of the form LaTeXMLMath and that the LaTeXMLMath obtained from the line bundles LaTeXMLMath form a basis for all Chern characters with support on the compact toric cycles .
Alternatively , one may wish to consider ‘ pure ’ branes defined in terms of the structure sheaves of the independent lower dimensional compact holomorphic cycles .
Given the structure sheaves LaTeXMLMath where the LaTeXMLMath form a basis for all compact holomorphic cycles on LaTeXMLMath , applying LaTeXMLMath push-forwards for every cycle of codimension LaTeXMLMath leads to sheaves LaTeXMLMath on LaTeXMLMath .
In order to relate these objects to Chern characters on LaTeXMLMath we have to use the Grothendieck-Riemann-Roch theorem , LaTeXMLEquation for embeddings of the type LaTeXMLMath .
Writing LaTeXMLMath allows us to define the charge vectors LaTeXMLMath .
Alternatively we may calculate the charge vectors by first calculating LaTeXMLEquation and noticing that LaTeXMLMath implies LaTeXMLMath , i.e .
LaTeXMLMath .
We note that the compact holomorphic cycles generate the compact homology of LaTeXMLMath , so the number of LaTeXMLMath is equal to LaTeXMLMath which is just the number of LaTeXMLMath -dimensional cones in LaTeXMLMath [ 29 ] .
At the large volume limit the Mukai vector LaTeXMLMath determines the central charge LaTeXMLMath This formula occurs implicitly in [ 45 ] and explicitly in [ 13 ] ; see also the remarks in [ 17 ] .
LaTeXMLEquation of the brane configuration , where the LaTeXMLMath are generators of the Kähler cone .
In particular we obtain LaTeXMLEquation for the central charges of D0-branes and D2-branes wrapping ( with trivial bundle ) the generators LaTeXMLMath of the Mori cone dual to the LaTeXMLMath .
These are the objects related by local mirror symmetry to the solutions of the GKZ system at the large complex structure point .
More precisely we expect the exact central charge LaTeXMLMath to be a linear combination of the GKZ solutions such that LaTeXMLEquation .
If we demand that LaTeXMLMath measure the complexified Kähler class at the large Kähler limit we have to make the identification LaTeXMLEquation .
Note that this is different from the conventions usually adopted in the literature , but we find that this is precisely the identification that works .
Linearity implies that the central charge corresponding to any LaTeXMLMath is given in terms of the charge vector by LaTeXMLEquation .
Finally , we return to the subject of monodromy .
In [ 1 ] it was conjectured ( and pushed further in the work of [ 3 ] [ 4 ] ) that the monodromies around loci in the moduli space where the mirror LaTeXMLMath of a Calabi-Yau threefold LaTeXMLMath becomes singular induce autoequivalences of LaTeXMLMath , the bounded derived category of coherent sheaves on LaTeXMLMath .
Moreover , in the case of a Fano surface embedded in a Calabi-Yau threefold , a relationship of these autoequivalences of LaTeXMLMath with mutations of exceptional collections supported on the Fano surface was pointed out in [ 4 ] .
For our purposes we will view the various monodromies mainly as automorphisms of LaTeXMLMath .
However , in some examples we will identify the monodromy actions on the exceptional collections of coherent sheaves supported on the compact divisors .
As in the case of the local mirror geometry , we will be interested in the following three types of transformations : — Monodromy around large Kähler structure divisors in the moduli space , — Monodromy around the primary component of the discriminant locus , — ‘ Orbifold monodromy ’ in the case LaTeXMLMath .
Only the monodromy around a divisor LaTeXMLMath in the moduli space where the Kähler parameter LaTeXMLMath ( associated with the divisor class LaTeXMLMath in LaTeXMLMath ) becomes infinite allows for a classical analysis .
In this case we just take LaTeXMLMath in ( 4.5 ) .
Because of the multiplicativity of Chern characters , the fact that the Chern character of a line bundle is the exponential of its first Chern class and the form of ( 4.5 ) , this transforms the LaTeXMLMath by tensoring them with LaTeXMLMath .
By ( 4.1 ) , the LaTeXMLMath transform by tensoring with LaTeXMLMath .
According to the observations in [ 46 ] [ 9 ] , ‘ orbifold monodromy ’ should cyclically permute the LaTeXMLMath if LaTeXMLMath is a resolution of LaTeXMLMath .
For the primary component of the discriminant locus we have the following picture : In the case of a compact Calabi-Yau variety LaTeXMLMath it is conjectured ( see [ 1 ] [ 3 ] [ 4 ] [ 5 ] [ 20 ] ) that a sheaf LaTeXMLMath is subjected to a Fourier-Mukai transform whose kernel is the structure sheaf LaTeXMLMath , implying that the Chern character of LaTeXMLMath transforms as LaTeXMLEquation where LaTeXMLMath is the pairing ( 4.2 ) .
In our case of non-compact LaTeXMLMath this can not work because it would violate compact support conditions , but we make the following observation : In all of our examples we obtain expressions for LaTeXMLMath that allow us to choose LaTeXMLMath in such a way that its restriction LaTeXMLMath to any compact toric cycle LaTeXMLMath is equal to LaTeXMLMath .
For the case of a resolution LaTeXMLMath of an orbifold singularity this means that our expressions for LaTeXMLMath are consistent with taking LaTeXMLMath to be the push-forward of the restriction of LaTeXMLMath to LaTeXMLMath .
Wherever we have the possibility of comparison with the mirror geometry , we find that the monodromy around the primary component of the discriminant locus is given by LaTeXMLEquation .
More precisely , the following happens : For one parameter models the principal component is pointlike .
If we decompose the GKZ solutions into logarithms and holomorphic pieces at LaTeXMLMath , the principal component is at the boundary of the radius of convergence of the holomorphic pieces .
In this case we find that the monodromy is given precisely by ( 4.11 ) provided we choose the simplest anti-clockwise path , LaTeXMLMath and the identification ( 4.8 ) .
With more than one parameter the discriminant locus consists of several disjoint pieces in the LaTeXMLMath coordinate patch ( these pieces join in the other coordinate patches ) , and there is no unambiguous choice of component or path .
We find , however , that at every branch one of the LaTeXMLMath ( possibly transformed by large complex structure monodromy ) becomes massless .
This is consistent with the picture that when we take LaTeXMLMath along some non-trivial paths like the ones corresponding to ‘ orbifold monodromy ’ we turn it into one of the other LaTeXMLMath .
If LaTeXMLMath is even there is a simple consistency check : If we require ( 4.11 ) to respect the pairing LaTeXMLMath then it is easily checked that this is equivalent to LaTeXMLMath ( in odd dimensions the analogous condition LaTeXMLMath is fulfilled automatically because of the skew symmetry of LaTeXMLMath ) .
This is indeed true in all of our examples .
Example 1 : Resolution of LaTeXMLMath : The case of LaTeXMLMath with LaTeXMLMath any discrete subgroup of LaTeXMLMath is well understood by mathematicians in the context of McKay correspondence .
If LaTeXMLMath is abelian and resolved by the introduction of a set LaTeXMLMath of exceptional curves , and if LaTeXMLMath is a basis of divisor classes dual to LaTeXMLMath then the LaTeXMLMath are given by LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
By ( 2.6 ) sections of the LaTeXMLMath are given , for example , by LaTeXMLMath , so the action of LaTeXMLMath on these sections through LaTeXMLMath , LaTeXMLMath indeed reproduces the characters LaTeXMLMath of LaTeXMLMath .
Using ( 4.1 ) and denoting by LaTeXMLMath the class of a point , we find LaTeXMLEquation and therefore LaTeXMLMath .
The restriction of LaTeXMLMath to the union of the LaTeXMLMath is the same as LaTeXMLMath except for the LaTeXMLMath points of the form LaTeXMLMath where LaTeXMLMath has rank two .
Upon subtracting the LaTeXMLMath sheaves with support on these points we arrive at a class that matches LaTeXMLMath .
It is easily checked that LaTeXMLMath for all LaTeXMLMath .
The large volume central charges are given by LaTeXMLMath and LaTeXMLMath .
In the case of LaTeXMLMath this implies LaTeXMLMath and LaTeXMLMath and we see that the principal component and orbifold monodromies found in the mirror geometry are precisely the ones generated by ( 4.11 ) and permutations LaTeXMLMath , respectively .
Example 2 : For LaTeXMLMath with LaTeXMLMath the restrictions of the LaTeXMLMath to the exceptional divisor LaTeXMLMath are nothing but LaTeXMLMath , LaTeXMLMath .
The independent holomorphic cycles are of the form LaTeXMLMath with LaTeXMLMath and the LaTeXMLMath restrict to LaTeXMLMath .
This example has been previously considered in [ 47 ] [ 48 ] [ 17 ] .
We include it as further evidence that the LaTeXMLMath have the properties stated above .
Defining LaTeXMLEquation with LaTeXMLMath the hyperplane divisor , we find that LaTeXMLEquation .
With LaTeXMLMath this simple recursion is solved by LaTeXMLMath and we obtain LaTeXMLMath , implying LaTeXMLMath for any LaTeXMLMath , LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
This leads to the following expressions for the LaTeXMLMath : LaTeXMLEquation .
Again the restriction of LaTeXMLMath to any compact toric cycle is the same as the structure sheaf of that cycle .
Alternatively we may determine the LaTeXMLMath by the ansatz LaTeXMLMath .
With LaTeXMLEquation we get LaTeXMLMath which leads to LaTeXMLEquation .
LaTeXMLEquation Using LaTeXMLMath and the reciprocity of LaTeXMLMath and LaTeXMLMath we get LaTeXMLEquation as it should be .
For LaTeXMLMath we find LaTeXMLMath .
The corresponding GKZ system has been studied at various places in the literature , e.g .
in [ 36 ] and [ 9 ] .
In terms of solutions LaTeXMLEquation the rule LaTeXMLMath leads to LaTeXMLMath .
Comparing with [ 9 ] , we find that this is precisely the expression denoted there by LaTeXMLMath which vanishes at the discriminant point LaTeXMLMath .
Example 3 : The Kähler cone is generated by LaTeXMLMath and LaTeXMLMath corresponding to the characters LaTeXMLMath and LaTeXMLMath , respectively .
By applying the rules outlined above , we assign the character LaTeXMLMath to each of the three line segments meeting at LaTeXMLMath in fig .
1 and to the line segment between LaTeXMLMath and LaTeXMLMath , whereas the remaining two line segments ( from LaTeXMLMath to LaTeXMLMath and LaTeXMLMath ) correspond to LaTeXMLMath .
Thus we get LaTeXMLMath because of the three line segments with equal characters meeting at LaTeXMLMath and LaTeXMLMath because of the two pairs of line segments at LaTeXMLMath .
Altogether we get representatives LaTeXMLMath with LaTeXMLEquation for the bases of LaTeXMLMath , where we have chosen the labels such that sections of LaTeXMLMath transform as LaTeXMLMath under LaTeXMLMath .
Using ( 4.1 ) we find that the localized Chern characters of the basis of LaTeXMLMath are given by LaTeXMLEquation with LaTeXMLMath the class of a point .
Let us now consider the branes defined in terms of the structure sheaves LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath of the independent lower dimensional cycles .
Denoting by LaTeXMLMath the result of three succesive inclusion maps acting on LaTeXMLMath , etc. , we arrive with the help of the Grothendieck-Riemann-Roch theorem ( 4.3 ) at the following result : LaTeXMLEquation .
This allows us to determine the D-brane charges LaTeXMLMath with LaTeXMLMath the D0-brane charge , LaTeXMLMath D2-brane charges and LaTeXMLMath D4-brane charges of the LaTeXMLMath as LaTeXMLEquation .
In particular , this means that LaTeXMLMath .
Note how LaTeXMLMath has rank 1 on LaTeXMLMath and on LaTeXMLMath except on their intersection LaTeXMLMath , where it has rank 2 which is compensated by subctracting LaTeXMLMath .
At this point we would like to mention that we have performed a similar analysis for LaTeXMLMath with arbitrary odd LaTeXMLMath and an action of the type LaTeXMLMath .
In that case the resolution requires a LaTeXMLMath and LaTeXMLMath Hirzebruch surfaces and we again obtain LaTeXMLMath whose sections transform by the characters of LaTeXMLMath and an expression for LaTeXMLMath that reduces to the structure sheaf on every compact toric cycle .
Returning to LaTeXMLMath we now give an alternative description of the compactly supported K-theory classes in terms of non-trivial sheaves on the exceptional divisors .
Again with the help of the Grothendieck-Riemann-Roch theorem , we find that we may choose representatives LaTeXMLMath in terms of the following combinations of push-forwards of sheaves : LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath inclusion maps .
By LaTeXMLMath we denote a stable bundle on LaTeXMLMath with the Chern character given by LaTeXMLEquation where LaTeXMLMath is the class of a point on LaTeXMLMath .
Note that LaTeXMLMath is a foundation of the helix of exceptional bundles on LaTeXMLMath and that LaTeXMLMath is a regular exceptional pair on LaTeXMLMath ( see [ 49 ] ) .
I n terms of the pure brane basis LaTeXMLMath the large volume central charge is LaTeXMLEquation .
We will now discuss monodromy by assuming that the assertions made in this section are correct .
The comparison with the mirror geometry is rather technical and can be found in appendix A .
We want to find monodromy matrices acting on the charge vectors , LaTeXMLMath , such that LaTeXMLMath , where LaTeXMLMath is the monodromy transformed version of LaTeXMLMath .
The monodromy around the orbifold locus cyclically permutes the charge vectors ( 4.24 ) .
Therefore , we obtain : LaTeXMLEquation .
Also , we can easily compute the large radius limit monodromies , LaTeXMLMath and LaTeXMLMath .
On the sheaves defined on the exceptional divisors the actions of the monodromies come from tensoring with the restrictions of LaTeXMLMath and LaTeXMLMath .
Therefore , the large radius limit monodromy LaTeXMLMath acts as following : the exceptional collection LaTeXMLMath on LaTeXMLMath is mutated to another exceptional collection , LaTeXMLMath , while on LaTeXMLMath is given by the tensoring with LaTeXMLMath , therefore taking regular exceptional pairs into regular exceptional pairs .
The action of the monodromy LaTeXMLMath is represented by tensoring any sheaf supported on LaTeXMLMath with LaTeXMLMath , hence again transforming regular pairs into regular pairs , while leaving any sheaf supported on LaTeXMLMath invariant .
Using ( 4.11 ) it is possible to compute the action of the monodromy around the principal component of the discriminant on the generators of LaTeXMLMath : LaTeXMLMath with LaTeXMLMath are invariant under this transformation , but LaTeXMLMath .
With the help of ( 4.4 ) , we readily obtain the monodromy around the conifold locus : LaTeXMLEquation .
The conifold monodromy , although preserving exceptional collections , acts in a very different way on LaTeXMLMath .
For example , we have LaTeXMLMath and LaTeXMLMath , that is the D4-branes can ’ jump ’ from one exceptional divisor to another .
However , as remarked in [ 20 ] , this is not very surprising since autoequivalences of LaTeXMLMath need not preserve the D-branes .
5 .
Beyond LaTeXMLMath Up to now we have only considered cases of the type LaTeXMLMath with a single triangulation .
We now want to examine the range of validity of our statements regarding the LaTeXMLMath and monodromy .
We first present another example , the resolution of LaTeXMLMath , which is still an orbifold but has several interesting features : It is not of the simple LaTeXMLMath type , it allows for more than one triangulation , its resolution involves three new non-compact toric divisors but no compact toric divisor , and finally it is a three parameter model whose GKZ system can be solved explicitly .
We will be able to show explicitly that the LaTeXMLMath vanish at ( branches of ) the principal component of the discriminant locus and nowhere else .
Aspects of D-brane states on this model have been studied previously in e.g .
[ 50 ] [ 51 ] .
Finally we examine the possibility of extending our results to cases not of the McKay type .
We find that they still hold in many examples but not in general .
Example 4 : A toric resolution of LaTeXMLMath : A singular space of the type LaTeXMLMath where every non-trivial element of LaTeXMLMath acts by flipping the sign of two of the three coordinates of LaTeXMLMath can be resolved by introducing three additional non-compact divisors and three compact curves .
There are several distinct possibilities for choosing the curves .
Fig .
6 : LaTeXMLMath -Hilb resolution of LaTeXMLMath .
Fig .
6 : LaTeXMLMath -Hilb resolution of LaTeXMLMath .
We use the LaTeXMLMath -Hilb resolution depicted in fig .
6 .
The Mori cone is generated by the following vectors : LaTeXMLEquation .
The generators of the Kähler cone are the divisors LaTeXMLMath , LaTeXMLMath and LaTeXMLMath corresponding to the vanishing of the coordinates of LaTeXMLMath .
The mirror geometry is determined by LaTeXMLEquation .
The large complex structure coordinates LaTeXMLMath are LaTeXMLEquation and the orbifold coordinates are LaTeXMLEquation .
The principal component of the discriminant locus is determined by LaTeXMLEquation .
The simplest formulation of the GKZ system can be obtained by mixing large complex structure and orbifold coordinates .
We find the operator LaTeXMLEquation from the first Mori cone vector and the same operator with cyclically permuted indices for the other two Mori cone vectors .
This simply implies LaTeXMLEquation for any solution LaTeXMLMath , i.e .
there must be a basis of solutions depending only on at most one of the LaTeXMLMath .
The sums of two Mori cone vectors lead to operators of the type LaTeXMLEquation which upon using ( 5.4 ) and ( 5.7 ) implies LaTeXMLEquation .
The whole GKZ system has three solutions of the type LaTeXMLMath and , as always , a constant solution .
Upon returning to large complex structure variables , we obtain LaTeXMLEquation and the corresponding index-permuted expressions for LaTeXMLMath and LaTeXMLMath .
The divisors LaTeXMLMath determining the line bundles LaTeXMLMath are just LaTeXMLMath , LaTeXMLMath and LaTeXMLMath and we find LaTeXMLEquation where LaTeXMLMath is the compact curve at the intersection of LaTeXMLMath and LaTeXMLMath etc .
In terms of structure sheaves we can represent LaTeXMLMath as LaTeXMLMath .
Noticing that all three curves intersect in the same point , we find that we can again view LaTeXMLMath as the object whose restriction to any compact toric cycle is the structure sheaf of that cycle .
The central charges are determined by LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , leading to LaTeXMLEquation .
At the orbifold point LaTeXMLMath we have the following situation : The moduli space develops a LaTeXMLMath singularity .
Provided we make the right choice of sheets for the square roots and logarithms , we find LaTeXMLMath and thus LaTeXMLMath for any LaTeXMLMath .
The ‘ orbifold monodromy ’ LaTeXMLMath , LaTeXMLMath acts as LaTeXMLEquation and the other elements of the orbifold monodromy act in similar ways .
LaTeXMLMath can become massless only if LaTeXMLEquation .
We can rewrite this in the form LaTeXMLMath such that E LaTeXMLMath and E LaTeXMLMath are expressions that do not contain LaTeXMLMath .
Then a necessary condition for ( 5.14 ) to hold is LaTeXMLMath and we can proceed to eliminate the other square roots in the same way .
The result is an equation proportional to the square of the expression determining the principal component of the discriminant locus ( 5.5 ) .
Conversely , if we solve ( 5.5 ) , e.g .
by setting LaTeXMLMath , plug this into ( 5.10 ) and choose the right sheets , we find that LaTeXMLMath indeed vanishes .
The same type of analysis works for the other LaTeXMLMath .
At this point it is natural to ask whether the analog of LaTeXMLMath , i.e .
the sheaf that is equal to the structure sheaf upon restriction to any compact toric cycle but of rank zero away from these cycles , might lead to monodromies in cases that are not related to McKay correspondence .
It turns out that this is very often the case ( at least for sufficiently simple examples ) , but not true in general .
Fig .
7 : Examples not of the McKay type .
Fig .
7 : Examples not of the McKay type .
Three examples where this works are shown in fig .
7 .
The first of these is the resolution of a conifold singularity and exactly solvable .
The other two ( anticanonical line bundles over LaTeXMLMath and LaTeXMLMath , respectively ) are two parameter models that we treated with analyses similar to the ones used for example 3 ( LaTeXMLMath ) .
Fig .
8 : Symmetric triangulation of the half-hexagon .
Fig .
8 : Symmetric triangulation of the half-hexagon .
As a counterexample , consider the Calabi-Yau manifold depicted in fig .
8 , whose GKZ system is again solvable .
Here we find the following : If we choose as the line bundles LaTeXMLMath the ones determined by the generators of the Kähler cone , we still find that the corresponding LaTeXMLMath has the same restriction to compact toric cycles as LaTeXMLMath .
However , only two of the three generators of LaTeXMLMath become massless at the conifold locus ( these statements are true for any triangulation ) .
In particular , for the symmetric triangulation the vanishing locus of the central charge of LaTeXMLMath does not coincide with the conifold locus .
Acknowledgements We would like to thank Philip Candelas and Duiliu Diaconescu for very useful conversations .
Appendix A .
Comparison of GKZ solutions and K-theory results for LaTeXMLMath The GKZ operators LaTeXMLMath This GKZ system has also been studied in [ 52 ] .
corresponding to the Mori cone generators ( 2.17 ) are given by LaTeXMLEquation .
This can be turned into a system involving LaTeXMLMath , LaTeXMLMath by standard manipulations described above .
In this way we arrive at the following expressions in terms of LaTeXMLMath : LaTeXMLEquation .
Solutions to this system can be obtained by considering LaTeXMLEquation ( the coefficients of LaTeXMLMath , LaTeXMLMath in the LaTeXMLMath –functions are the entries of the Mori cone vectors ( 2.17 ) ) and its partial derivatives w.r.t the LaTeXMLMath at LaTeXMLMath .
We use : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath .
This yields the constant solution LaTeXMLMath and , with LaTeXMLMath for LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath where LaTeXMLMath LaTeXMLMath LaTeXMLMath and the summations are taken over those values of LaTeXMLMath , LaTeXMLMath where the arguments of all factorials are non-negative .
Of the three expressions obtained by taking second derivatives only the first one and the linear combination LaTeXMLMath of the other two actually solve the GKZ system ( A.2 ) .
We note that there is also a linear combination of third derivatives ( involving third powers of logarithms ) that is annihilated by both operators occurring in ( A.2 ) .
The reason is that this system is not yet complete as we have written it : In principle we should write down a GKZ operator for every curve in the Mori cone .
Taking as an additional charge vector the sum LaTeXMLMath of our Mori cone generators , we see that the triple-log solution is excluded .
Fig .
9 : The real part of the discriminant locus .
Fig .
9 : The real part of the discriminant locus .
We now want to study monodromies of the GKZ solutions around the loci where LaTeXMLMath degenerates .
This is an easy exercise for the divisors LaTeXMLMath , LaTeXMLMath where the monodromy is determined by LaTeXMLMath .
We find that the following set of solutions is well behaved ( i.e. , transforms by an LaTeXMLMath matrix ) under the monodromies LaTeXMLMath : LaTeXMLEquation .
In the large complex structure coordinate patch the discriminant locus consists of several different branches .
The slice through real LaTeXMLMath , LaTeXMLMath is shown in fig .
9 .
There are two branches with LaTeXMLMath .
The one with LaTeXMLMath is tangent to the axis LaTeXMLMath in LaTeXMLMath .
Parts of this branch are at the boundary of the domain of convergence of the LaTeXMLMath ’ s in such a way that there is convergence like LaTeXMLMath .
Through a numerical analysis we found that at this locus LaTeXMLEquation which corresponds to the vanishing of LaTeXMLMath if we make the identifications LaTeXMLMath , LaTeXMLMath .
This is equivalent to the vanishing of a LaTeXMLMath -monodromy transformed version of LaTeXMLMath with ( 4.8 ) .
Similarly we find at the other branch with LaTeXMLMath which intersects LaTeXMLMath at LaTeXMLMath that LaTeXMLEquation .
This corresponds to a LaTeXMLMath -monodromy transformed version of LaTeXMLMath vanishing .
The third branch , with LaTeXMLMath and LaTeXMLMath is beyond the region of convergence .
We have preliminary evidence that at this branch a LaTeXMLMath -monodromy transformed version of LaTeXMLMath becomes massless .
References [ 1 ] % [ 2 ] % [ 3 ] % [ 4 ] % [ 5 ] % [ 6 ] % [ 7 ] % [ 8 ] % [ 9 ] % [ 10 ] % [ 11 ] % [ 12 ] % [ 13 ] % [ 14 ] % [ 15 ] % [ 16 ] % [ 17 ] % [ 18 ] % [ 19 ] % [ 20 ] % [ 21 ] % [ 22 ] % [ 23 ] % [ 24 ] % [ 25 ] % [ 26 ] % [ 27 ] % [ 28 ] % [ 29 ] % [ 30 ] % [ 31 ] % [ 32 ] % [ 33 ] % [ 34 ] % [ 35 ] % [ 36 ] % [ 37 ] % [ 38 ] % [ 39 ] % [ 40 ] % [ 41 ] % [ 42 ] % [ 43 ] % [ 44 ] % [ 45 ] % [ 46 ] % [ 47 ] % [ 48 ] % [ 49 ] % [ 50 ] % [ 51 ] % [ 52 ] %
The exact vacuum expectation values of the second level descendent fields LaTeXMLMath in the Bullough-Dodd model are calculated .
By performing quantum group restrictions , we obtain LaTeXMLMath in the LaTeXMLMath , LaTeXMLMath and LaTeXMLMath perturbed minimal CFTs .
In particular , the exact expectation value LaTeXMLMath is found to be proportional to the square of the bulk free energy .
Expectation values of descendent fields in the Bullough-Dodd model and related perturbed conformal field theories P. Baseilhac e-mail : pb18 @ york.ac.uk and M. Stanishkov e-mail : marian @ mail.apctp.org , On leave of absence from INRNE , Sofia , Bulgaria LaTeXMLMath Department of Mathematics , University of York Heslington , York YO105DD , United Kingdom LaTeXMLMath Asia Pacific Center for Theoretical Physics , Seoul , 130-012 , Korea In a 2-D integrable quantum field theory ( QFT ) which can be realized as a conformal field theory ( CFT ) perturbed by some relevant operator , it is well-known that any correlation function of local fields LaTeXMLMath in the short-distance limit can be reduced down to one-point functions LaTeXMLMath by successive application of the operator product expansion ( OPE ) LaTeXMLCite .
These vacuum expectation values ( VEV ) s contain important information about the IR environment .
Together with the structure constants characterizing the UV limit of the QFT , they provide the UV behaviour of the correlation functions whereas the so-called form-factors characterize their IR behaviour .
Since three years important progress has been made concerning the evaluation of some VEVs in different integrable QFTs .
In ref .
LaTeXMLCite , an explicit expression for the VEVs of the exponential fields in the sinh-Gordon and sine-Gordon models was proposed .
In ref .
LaTeXMLCite it was shown that this result can be obtained using the “ reflection amplitude ” LaTeXMLCite of the Liouville field theory .
This method was also applied in the so-called Bullough-Dodd model with real and imaginary coupling .
In QFT involving more fields , the VEVs for a two-parameter family of integrable QFTs introduced and studied in LaTeXMLCite gave rise to the VEVs of local operators in parafermionic sine-Gordon models and in integrable perturbed LaTeXMLMath coset CFT LaTeXMLCite .
Also , the VEVs in simply-laced affine Toda field theories are known for a long time LaTeXMLCite and the case of non-simply laced dual pairs was recently studied in LaTeXMLCite .
However , the higher-order corrections to the short-distance expansion of two-point correlation functions involve the VEVs of the descendent fields .
This question was addressed in LaTeXMLCite .
There , the VEV of the descendent field LaTeXMLMath in the sinh-Gordon ( ShG ) and sine-Gordon ( SG ) - with LaTeXMLMath - models was calculated as well as in its related perturbed CFT , i.e .
LaTeXMLMath perturbation of minimal models .
From this result , the next-order correction of the two-point function in the scaling Lee-Yang model was computed LaTeXMLCite .
The purpose of this paper is to calculate the VEV of the simplest non-trivial descendent field in the Bullough-Dodd ( BD ) model which is generally described by the following action in the Euclidean space : LaTeXMLEquation .
Here , the parameters LaTeXMLMath and LaTeXMLMath are introduced , as the two operators do not renormalize in the same way , on the contrary to any simply-laced affine Toda field theory .
This model has attracted over the years a certain interest , in particular in connection with perturbed minimal models : LaTeXMLMath minimal CFT perturbed by the operators LaTeXMLMath , LaTeXMLMath or LaTeXMLMath can be obtained by a quantum group ( QG ) restriction of imaginary Bullough-Dodd model LaTeXMLCite with special values of the coupling .
We will use this property to deduce the VEV LaTeXMLMath in the following perturbed minimal models : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we denote respectively LaTeXMLMath , LaTeXMLMath and LaTeXMLMath as specific primary operators of the unperturbed minimal model LaTeXMLMath and the parameters LaTeXMLMath , LaTeXMLMath and LaTeXMLMath characterize the strength of the perturbation .
This paper is organized as follows .
In the next section we introduce the notations and write the short-distance expansion of the two-point correlation function which involves the VEV of the descendent field LaTeXMLMath in the BD model associated with the action ( LaTeXMLRef ) .
Using the method based on the “ reflection relations ” LaTeXMLCite we find a conjecture for this last quantity in Section 3 .
Whereas it exists an ambiguity for the solution of these functional equations , we choose the “ minimal ” one which is compatible with the “ resonance conditions ” ( see ref .
LaTeXMLCite for details ) .
In Section 4 we compare the semi-classical limit of the short distance expansion of the two-point function with the semi-classical approximation based on the action ( LaTeXMLRef ) .
In Section 5 we deduce the VEV LaTeXMLMath in the models ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Concluding remarks follow in the last section .
Similarly to the ShG model LaTeXMLCite , the BD model can be regarded as a relevant perturbation of a Gaussian CFT .
In this free field theory , the field is normalized such that : LaTeXMLEquation and we have the classical equation of motion : LaTeXMLEquation .
Instead of considering the action ( LaTeXMLRef ) we turn directly to the case of an imaginary coupling constant which is the most interesting for our purpose in Section 5 .
The perturbation is then relevant if LaTeXMLMath .
Although the model ( LaTeXMLRef ) for real coupling is very different from the one with imaginary coupling in its physical content ( this latter model contains solitons and breathers ) , there are good reasons to believe that the expectation values obtained in the real coupling case provide also the expectation values for the imaginary coupling .
The calculation of the VEVs in both cases ( LaTeXMLMath real or imaginary ) within the standard perturbation theory agree through the identification LaTeXMLMath LaTeXMLCite .
With this substitution in ( LaTeXMLRef ) , the general short distance OPE for two arbitrary primary fields LaTeXMLMath and LaTeXMLMath takes the form : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and the dots in each term stand for the contributions of the descendents of each field .
The different coefficients in eq .
( LaTeXMLRef ) are computable within the conformal perturbation theory ( CPT ) LaTeXMLCite .
We obtain : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where any function LaTeXMLMath admits a power series expansion : LaTeXMLEquation .
Each coeffficient in ( LaTeXMLRef ) is expressed in terms of Coulomb type integrals .
The corresponding leading terms are respectively given by : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where we introduced the integrals : LaTeXMLEquation .
LaTeXMLEquation Notice that LaTeXMLMath and that the first subleading term of the coefficient LaTeXMLMath is : LaTeXMLEquation .
The integrals LaTeXMLMath have been evaluated explicitly in LaTeXMLCite with the result : LaTeXMLEquation where the usual notation LaTeXMLMath is used .
As we already said in the introduction , the next sub-leading terms in ( LaTeXMLRef ) involve the descendent fields .
There are four independent second-level descendent fields in BD : LaTeXMLEquation .
LaTeXMLEquation Similarly to the SG ( or ShG ) case , using ( LaTeXMLRef ) it is easy to show that linear combinations of these descendent fields can be written in terms of total derivatives of local fields ( we refer the reader to LaTeXMLCite for details about these relations ) .
As a result , the VEVs of the composite fields ( LaTeXMLRef ) can all be expressed in terms of a single VEV , say : LaTeXMLEquation .
Let us make an important observation .
The second sub-leading terms in the OPE ( LaTeXMLRef ) appear to be the third order descendents of the primary fields .
Analogously to the previous discussion linear combinations of them can be expressed in terms of total derivatives of some local fields .
As before , all the corresponding VEVs can be expressed through LaTeXMLMath .
Unlike the SG case , it is non-vanishing due to the absence of a conserved charge of spin 3 in the BD model .
We will not enter in details about this VEV since its computation is not our purpose in this paper .
One can now write the short-distance expansion for the two-point function : LaTeXMLEquation by taking the expectation value of the r.h.s .
of the OPE ( LaTeXMLRef ) in the BD model with imaginary coupling .
Due to the previous discussion , the first non-vanishing contribution of the VEVs of lowest descendent fields in the r.h.s .
of the VEV of ( LaTeXMLRef ) correspond to the following terms : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation respectively .
These coefficients also admit expansion similar to eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) .
In particular we have : LaTeXMLEquation .
Finally , the short-distance ( LaTeXMLMath ) expansion of the two-point correlation function in the BD model with imaginary coupling writes : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where we defined LaTeXMLMath and LaTeXMLMath by the ratios : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLMath is the VEV of the exponential field in the BD model .
A closed analytic expression for this latter VEV has been proposed in ref .
LaTeXMLCite : LaTeXMLEquation where LaTeXMLEquation .
Its integral representation is well defined if : LaTeXMLEquation and obtained by analytic continuation outside this domain .
It is then straightforward to obtain the result associated with the action ( LaTeXMLRef ) i.e .
for real values of the coupling constant LaTeXMLMath which follows from the obvious substitutions : LaTeXMLEquation .
LaTeXMLEquation In the ( Gaussian ) free field theory , the composite fields ( LaTeXMLRef ) are spinless with scale dimension : LaTeXMLEquation .
For generic value of the coupling LaTeXMLMath some divergences arise in the VEVs of the fields ( LaTeXMLRef ) due to the perturbation in ( LaTeXMLRef ) with imaginary coupling .
They are generally cancelled if we add specific counterterms which contain spinless local fields with cutt-off dependent coefficients .
For LaTeXMLMath the perturbation becomes relevant and a finite number of lower scale dimension couterterms are then sufficient .
However , this procedure is regularization scheme dependent , i.e .
one can always add finite counterterms .
For generic values of LaTeXMLMath this ambiguity in the definition of the renormalized expression for the fields ( LaTeXMLRef ) can be eliminated by fixing their scale dimensions to be ( LaTeXMLRef ) .
It exists however a set of values of LaTeXMLMath for which the ambiguity still remains , but here we will not consider these isolated cases .
In the BD model with imaginary coupling , this situation arises if two fields , say LaTeXMLMath and LaTeXMLMath , satisfy the resonance condition : LaTeXMLEquation associated with the ambiguity : LaTeXMLEquation .
In this specific case one says that the renormalized field LaTeXMLMath has an LaTeXMLMath -th resonance LaTeXMLCite with the field LaTeXMLMath .
Due to the condition ( LaTeXMLRef ) and using ( LaTeXMLRef ) we find immediatly that a resonance can appear between the descendent field LaTeXMLMath and the following primary fields : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation If we now look at the expression ( LaTeXMLRef ) , we notice that the contribution ( LaTeXMLRef ) , brought by the second level descendent field , and that of any of the exponential fields in LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , have the same power behaviour in LaTeXMLMath ( LaTeXMLMath ) at short-distance for the corresponding values of LaTeXMLMath in ( LaTeXMLRef ) .
The integrals which appear in these contributions and their corresponding poles are , respectively : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By analogy with the SG ( or ShG ) model , one expects that the VEV ( LaTeXMLRef ) ( and similarly for the real coupling case ) exhibits , at least , the same poles in order that the divergent contributions compensate each other .
This last requirement leads for instance to the relations : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation These last conditions will be used in the next section to fix the normalization of the VEV ( LaTeXMLRef ) .
Let us now turn to the evaluation of ( LaTeXMLRef ) which plays an important role in the two-point function ( LaTeXMLRef ) .
The BD model ( LaTeXMLRef ) can be regarded as two different perturbations of the Liouville field theory LaTeXMLCite .
First , one can consider the Liouville action : LaTeXMLEquation .
The perturbation is then identified with LaTeXMLMath .
Alternatively , we can take : LaTeXMLEquation as the initial action and consider LaTeXMLMath as a perturbation .
Using the first picture , the holomorphic stress-energy tensor : LaTeXMLEquation ensures the local conformal invariance of the Liouville field theory ( LaTeXMLRef ) and similarly for the anti-holomorphic part .
The exponential fields LaTeXMLMath are spinless primary fields with conformal dimension : LaTeXMLEquation .
The property of reflection relations which relates operators with the same quantum numbers is a characteristic of the CFT .
Using the CPT framework , one expects that similar relations are also satisfied in the perturbed case ( LaTeXMLRef ) .
With the change LaTeXMLMath in ( LaTeXMLRef ) and using the second picture ( LaTeXMLRef ) , one assumes that the VEV of the exponential field LaTeXMLMath satisfies simultaneously the following two functional equations LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation with LaTeXMLEquation .
The functions LaTeXMLMath , LaTeXMLMath are called “ reflection amplitudes ” .
An exact expression for LaTeXMLMath was presented in LaTeXMLCite .
LaTeXMLMath is obtained from LaTeXMLMath by the substitutions LaTeXMLMath and LaTeXMLMath .
Under certain assumptions about the analytic properties of the VEV , the system ( LaTeXMLRef ) was solved and the VEV for these exponential fields was derived in LaTeXMLCite .
Let us denote the descendent fields : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are arbitrary strings and LaTeXMLMath , LaTeXMLMath are the standard Virasoro generators : LaTeXMLEquation .
The descendent fields ( LaTeXMLRef ) and the ones obtained after the reflection LaTeXMLMath possess the same quantum numbers .
Consequently , using the arguments of LaTeXMLCite based on the CPT framework , one also expects that their VEVs in the perturbed theory ( LaTeXMLRef ) satisfy the following “ reflection relation ” : LaTeXMLEquation .
However , it is more convenient to use the basis : LaTeXMLEquation .
The main reason is that in ( LaTeXMLRef ) the components LaTeXMLMath , LaTeXMLMath of the modified stress-tensor depend on LaTeXMLMath .
Using ( LaTeXMLRef ) one can always express ( LaTeXMLRef ) in the basis ( LaTeXMLRef ) .
For our purpose we will need the relation LaTeXMLCite : LaTeXMLEquation .
Furthermore , using ( LaTeXMLRef ) it implies : LaTeXMLEquation which leads to the following reflection relation : LaTeXMLEquation .
LaTeXMLEquation One can also consider the second picture ( LaTeXMLRef ) where the Liouville theory has coupling LaTeXMLMath instead of LaTeXMLMath and is perturbed by LaTeXMLMath .
If we define the analytic continuation of ( LaTeXMLRef ) : LaTeXMLEquation then the two different pictures provide us the following two functional relations : LaTeXMLEquation .
LaTeXMLEquation Notice that these equations are invariant with respect to the symmetry LaTeXMLMath with LaTeXMLMath in agreement with the well-known self-duality of the BD-model .
As was shown in the previous section , the solution of these functional equations should exhibits , at least , the poles ( LaTeXMLRef ) through the identification LaTeXMLMath and LaTeXMLMath .
Since the solution of ( LaTeXMLRef ) is defined up to a multiplication constant , we naturally choose to fix it by imposing eqs .
( LaTeXMLRef ) .
We find that the “ minimal ” solution which follows from these constraints is : LaTeXMLEquation where LaTeXMLMath is the “ deformed ” Coxeter number LaTeXMLCite .
Here we have used the exact relation between the parameters LaTeXMLMath and LaTeXMLMath in the action ( LaTeXMLRef ) and the mass of the particle LaTeXMLMath LaTeXMLCite : LaTeXMLEquation .
It is easy to see that for LaTeXMLMath and LaTeXMLMath , LaTeXMLMath possess poles located at : LaTeXMLEquation .
But as long as we consider LaTeXMLMath that satisfy ( LaTeXMLRef ) there remain Notice that for LaTeXMLMath one also has the pole LaTeXMLMath .
However , this one is nothing but obtained from the reflection LaTeXMLMath .
the expected poles ( LaTeXMLRef ) : LaTeXMLEquation .
It is well-known that the Bullough-Dodd model at the specific value of the coupling LaTeXMLMath and the sinh-Gordon model at LaTeXMLMath give an equivalent Lagrangian representation of the same QFT .
Then , as expected , one can check that ( LaTeXMLRef ) evaluated at LaTeXMLMath coincides exactly with the same quantity in the ShG model evaluated at LaTeXMLMath .
Accepting the conjecture ( LaTeXMLRef ) and using eq .
( LaTeXMLRef ) for LaTeXMLMath one can easily deduce for instance : LaTeXMLEquation where LaTeXMLEquation is the bulk free energy of the BD model LaTeXMLCite .
As we saw previously , the OPE proposed in eq .
( LaTeXMLRef ) plays a crucial role in the determination of the prefactor of the LaTeXMLMath dependent part of LaTeXMLMath , using eqs .
( LaTeXMLRef ) .
It is therefore important to check this expression , using for instance the semi-classical expansion .
In what follows , we will compare ( LaTeXMLRef ) with the semi-classical calculations based on the action ( 1.1 ) .
Let us consider ( LaTeXMLRef ) for LaTeXMLMath , LaTeXMLMath in the classical limit LaTeXMLMath .
Then the saddle-point evaluation of the functional integral based on the action ( 1.1 ) leads to the field configuration LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is a solution of the classical Bullough-Dodd equation : LaTeXMLEquation with the following asymptotic conditions : LaTeXMLEquation .
LaTeXMLEquation Here we denoted : LaTeXMLEquation and LaTeXMLMath is the MacDonald function .
Such a solution was considered in LaTeXMLCite .
Taking into account the above considerations , the two-point function takes the following form in the semi-classical limit : LaTeXMLEquation .
Following LaTeXMLCite it is not difficult to obtain the first few terms in the LaTeXMLMath expansion : LaTeXMLEquation .
We would like now to compare these results with the corresponding limit of ( LaTeXMLRef ) .
First , using the result for the exact VEV in the Bullough-Dodd model ( LaTeXMLRef ) proposed in LaTeXMLCite we obtain the following ratios : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Furthermore , the mass- LaTeXMLMath relation ( LaTeXMLRef ) proposed in LaTeXMLCite gives : LaTeXMLEquation whereas , using eq .
( LaTeXMLRef ) and LaTeXMLMath which can be deduced from the results of LaTeXMLCite , we have : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation If we now use the same notations as above ( LaTeXMLRef ) , the semi-classical limit of the expression ( LaTeXMLRef ) takes the following form : LaTeXMLEquation .
It is straightforward to check that this result agrees perfectly with ( LaTeXMLRef ) through ( LaTeXMLRef ) .
For imaginary value of the coupling LaTeXMLMath , LaTeXMLMath and LaTeXMLMath the action of the BD model ( LaTeXMLRef ) becomes complex .
Whereas it is not clear if it can be defined as a QFT , this model is known to be integrable and its LaTeXMLMath -matrix was constructed in LaTeXMLCite .
It is known that this model possess a quantum group symmetry LaTeXMLMath with deformation parameter LaTeXMLMath LaTeXMLCite .
An important role is played by one of its subalgebras LaTeXMLMath .
Following LaTeXMLCite ( see also LaTeXMLCite ) , we can restrict the Hilbert space of states of the complex BD model at special values of the coupling constant , more precisely when LaTeXMLMath is a root of unity , i.e .
for : LaTeXMLEquation relative prime integers , in which case the complex BD is identified with the perturbed minimal models ( LaTeXMLRef ) or ( LaTeXMLRef ) , respectively .
In the following , LaTeXMLMath will denote a primary field of the minimal model LaTeXMLMath .
In the first case , the exact relation between the parameters LaTeXMLMath in ( LaTeXMLRef ) and the mass of the fundamental kink LaTeXMLMath can be found in LaTeXMLCite with the result : LaTeXMLEquation .
Here we denote LaTeXMLEquation .
For unitary minimal models LaTeXMLMath which , for LaTeXMLMath , corresponds to the massive phase LaTeXMLCite .
Using the particle-breather identification : LaTeXMLEquation and eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) for imaginary coupling LaTeXMLMath and parameter LaTeXMLMath , it is straightforward to get the VEV in the model associated with the action ( LaTeXMLRef ) : LaTeXMLEquation with LaTeXMLEquation where we introduce the useful notation : LaTeXMLEquation .
Here LaTeXMLMath is one of the degenerate ground states of the QFT ( LaTeXMLRef ) ( see LaTeXMLCite for a detailed discussion of the vacuum structure of the model ) .
Taking LaTeXMLMath in ( LaTeXMLRef ) to be the identity operator , it is easy to get : LaTeXMLEquation .
A simple check consists to consider the scaling Lee-Yang model which corresponds to LaTeXMLMath , LaTeXMLMath i.e .
LaTeXMLMath in ( LaTeXMLRef ) .
As LaTeXMLMath for these values , we must obtain the result of LaTeXMLCite .
Using ( LaTeXMLRef ) the lightest mass in ( LaTeXMLRef ) is : LaTeXMLEquation .
Replacing this expression in ( LaTeXMLRef ) for LaTeXMLMath , LaTeXMLMath and ( LaTeXMLRef ) it is easy to see that the results are in perfect agreement with the ones of LaTeXMLCite .
In the second restriction LaTeXMLMath , which leads to the action ( LaTeXMLRef ) .
The exact relation between the parameter LaTeXMLMath and the mass of the fundamental kink LaTeXMLMath is in this case LaTeXMLCite : LaTeXMLEquation .
Along the same line as for the LaTeXMLMath perturbation we obtain the following expression for the VEV in the model associated with the action ( LaTeXMLRef ) : LaTeXMLEquation with LaTeXMLEquation where LaTeXMLMath is one of the degenerate ground states LaTeXMLCite of the QFT ( 1.3 ) .
The analog of the formula ( LaTeXMLRef ) is now : LaTeXMLEquation .
Another subalgebra of LaTeXMLMath is the subalgebra LaTeXMLMath .
One can again restrict the phase space of the complex BD with respect to this subalgebra for a special value of the coupling : LaTeXMLEquation relative prime integers .
Then , for this value of the coupling , the BD model is identified with the perturbed minimal model with the action ( LaTeXMLRef ) .
The exact relation between LaTeXMLMath and the mass LaTeXMLRef ) is not clearly understood .
However it is expected that it possesses particles and kinks similarly to the other models ( see for instance refs .
LaTeXMLCite for details ) .
The physical mass scale LaTeXMLMath is then associated with one of its particles LaTeXMLCite .
LaTeXMLMath is LaTeXMLCite : LaTeXMLEquation .
Here we keep the definition ( LaTeXMLRef ) .
In particular , the massive phase corresponds to : LaTeXMLEquation .
LaTeXMLEquation As for the two previous cases one would like to obtain the expectation values of the descendent fields for any primary operator LaTeXMLMath .
For LaTeXMLMath these fields are not invariant with respect to the subalgebra LaTeXMLMath on the contrary to LaTeXMLMath .
However , one expects that they only differ by a c-number coefficient characterizing the degenerate structure of the vacua LaTeXMLMath .
Taking the ratio of the VEV of the descendent field of LaTeXMLMath associated with the action ( LaTeXMLRef ) and the VEV of the primary field itself , one obtains : LaTeXMLEquation with LaTeXMLEquation .
In particular , we have : LaTeXMLEquation .
In conclusion , we proposed in this paper an exact expression for the VEV of the second level descendent of the exponential field LaTeXMLMath in the BD model .
The calculation is based on the so-called “ reflection relations ” which lead to a system of functional equations for this VEV .
While the solution is not unique , we chose the “ minimal solution ” obeing some residue conditions .
By performing a quantum group restriction in the case of complex BD model we found also the VEVs of the descendents of the primary fields in the perturbed minimal CFT models ( 1.2 ) , ( 1.3 ) , ( 1.4 ) .
It is rather interesting to notice that in eq .
( LaTeXMLRef ) , the exact VEV LaTeXMLMath is simply related to the VEV of the trace of the energy momentum tensor : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath as follows : LaTeXMLEquation .
This was already noticed in LaTeXMLCite for the ShG case .
We expect this property to be general , i.e .
to be confirmed for other integrable theories .
However , we have no proof yet of this phenomena .
We would like to notice two important differences between ShG and BD models .
First , in the LaTeXMLMath expansion of the two-point function the quantity LaTeXMLMath ( LaTeXMLRef ) comes with a coefficient of order LaTeXMLMath .
Therefore , it can not be checked directly in the semi-classical approximation , although the later is in agreement with the short distance expansion of the two-point function , thus giving a strong support to our conjecture ( LaTeXMLRef ) .
For a direct check one has to go beyond the classical limit and consider the first order in the perturbation theory based on the action ( 1.1 ) .
Another difference is the appearing of the third level descendents in the OPE of the exponential fields .
As a consequence , the following quantity appear in the short distance expansion of the two-point function : LaTeXMLEquation .
In contrast with LaTeXMLMath it is sensitive to the semi-classical expansion - it combines with the integral LaTeXMLMath in ( LaTeXMLRef ) in order to match the corresponding term coming from the semiclassical calculation based on the action ( 1.1 ) .
The function ( LaTeXMLRef ) can also be obtained using the “ reflection relations ” approach .
Using our results , one can deduce easily the next-to leading contributions in the short distance behaviour of the two-point functions of primary operators for different perturbed minimal models : for instance the Ising model in a magnetic field LaTeXMLCite , the tricritical Ising model perturbed by its leading energy density operator ( with conformal dimension LaTeXMLMath ) LaTeXMLCite or perturbed by its subleading magnetic operator ( with conformal dimension LaTeXMLMath ) , and so on .
Several models can be worked out along the same line , using the known results for the three-point functions of the CFT .
For instance , the super ShG model or , more generally , the parafermionic ShG model LaTeXMLCite .
We intend to discuss these various questions in a forthcoming publication LaTeXMLCite .
We are grateful to Al .
B. Zamolodchikov and particularly V.A .
Fateev for valuable discussions and interest in this work .
We thanks for the hospitality of LPM ( Montpellier ) where part of this work was done .
M.S .
aknowledges the Physics Department of Bologna University and APCTP , Seoul , for hospitality and financial support .
MS ’ s work is supported under contract KOSEF grant 1999-2-112-001-5 .
PB ’ s work is supported in part by the EU under contract ERBFMRX CT960012 and Marie Curie fellowship HPMF-CT-1999-00094 .
The review is devoted to topological global aspects of quantal description .
The treatment concentrates on quantizations of kinematical observables — generalized positions and momenta .
A broad class of quantum kinematics is rigorously constructed for systems , the configuration space of which is either a homogeneous space of a Lie group or a connected smooth finite-dimensional manifold without boundary .
The class also includes systems in an external gauge field for an Abelian or a compact gauge group .
Conditions for equivalence and irreducibility of generalized quantum kinematics are investigated with the aim of classification of possible quantizations .
Complete classification theorems are given in two special cases .
It is attempted to motivate the global approach based on a generalization of imprimitivity systems called quantum Borel kinematics .
These are classified by means of global invariants — quantum numbers of topological origin .
Selected examples are presented which demonstrate the richness of applications of Borel quantization .
The review aims to provide an introductory survey of the subject and to be sufficiently selfcontained as well , so that it can serve as a standard reference concerning Borel quantization for systems admitting localization on differentiable manifolds .
Quantization of Kinematics on Configuration Manifolds Reviews in Mathematical Physics , Vol .
13 , No .
4 ( 2001 ) , pp .
1–47 .
H.-D. Doebner Arnold Sommerfeld Institut für Mathematische Physik Technische Universität Clausthal Leibnizstr .
10 D-38678 Clausthal ( Fed .
Rep. Germany ) P. Šťovíček and J. Tolar Doppler Institute Faculty of Nuclear Sciences and Physical Engineering Czech Technical University Břehová 7 CZ-115 19 Prague 1 ( Czech Republic ) The successful development of quantum theory in this century shows convincingly that it provides perhaps the most universal language for the description of physical phenomena .
In quantum theory , as in any other physical theory , two fundamental aspects can be distinguished : the mathematical formalism and the physical interpretation .
At the basis of the most common mathematical formalism of quantum mechanics lies the notion of a complex separable Hilbert space LaTeXMLMath of , in general , infinite dimension .
Normed vectors in LaTeXMLMath correspond to pure states of a quantum system , whereas quantal observables are represented by self-adjoint operators in LaTeXMLMath .
However , only the rules of a physical interpretation enable one to use quantum theory for the description of physical systems .
The principal general rule is Born ’ s statistical interpretation of the wave function .
For each physical system , or at least for a certain class of them , it is further necessary to specify which operators in LaTeXMLMath are associated with physical observables measured by certain measuring devices .
This means in particular that at least the operators of kinematical observables ( position and momentum ) , and the dynamical evolution law of the system are to be specified .
An important tool for the derivation of quantum models are quantization methods .
The primary aim of quantization of a given classical system is to associate self-adjoint operators with classical observables .
As a rule , two main methods are used .
The first one is based on Bohr ’ s correspondence principle : the physical meaning of quantum operators is found by looking at their classical counterparts .
In this way non-relativistic quantum mechanics was formulated by quantization of classical Hamiltonian mechanics , quantum theory of electromagnetic field by quantization of the Maxwell theory , etc .
LaTeXMLCite .
The correspondence principle can , of course , be the leading rule for quantization , if the observables already existed in a classical form .
What should be done in the case of quantum observables without a classical analogue like the spin ?
Here the second method is often applicable , which uses invariance principles connected with the symmetries of the system .
By Noether ’ s theorem the operators corresponding to conserved quantities can be found as generators of some projective representation of the symmetry group in LaTeXMLMath .
As a far reaching application of this approach let us mention the relativistic quantum theory of elementary particles based on the irreducible unitary representations of the Poincaré group .
Both methods were used from the very first days of quantum theory , always taking into account specific physical properties of the systems considered .
The first method usually appears in non-relativistic quantum mechanics as canonical quantization LaTeXMLCite , for systems with the Euclidean configuration space LaTeXMLMath .
The position coordinates LaTeXMLMath and the canonically conjugate momenta LaTeXMLMath are quantized into self-adjoint position and momentum operators LaTeXMLMath , LaTeXMLMath ( in a separable Hilbert space LaTeXMLMath ) , satisfying canonical commutation relations .
This was originally discovered and mathematically formulated independently by W. Heisenberg and E. Schrödinger in 1925–26 .
The uniqueness of the mathematical formulation up to unitary equivalence was then guaranteed by the Stone-von Neumann Theorem .
Quantum mechanics on LaTeXMLMath became very soon a successful theory which has been able to correctly describe experimental findings in vast areas of quantum physics .
However , in some cases it was necessary to look for a formulation of quantum mechanics when the configuration space of a system was not Euclidean LaTeXMLCite .
For instance , in connection with the studies of rotational spectra of molecules and of deformed nuclei , quantum rotators were introduced as fundamental quantum models with configuration spaces LaTeXMLMath ( the circle ) , LaTeXMLMath ( the 2-sphere ) and LaTeXMLMath ( the rotation group ) .
The textbook treatment of spinning top models ( quantum mechanics of angular momentum ) presents a successful application of the approach via invariance principles .
There were also attempts to enforce canonical quantization in cases where global Cartesian coordinates do not exist on the configuration manifold LaTeXMLMath .
A formal quantization of generalized coordinates LaTeXMLMath and conjugate momenta LaTeXMLMath was suggested LaTeXMLCite on a manifold LaTeXMLMath with the Riemann structure ( metric tensor LaTeXMLMath with determinant LaTeXMLMath ) : LaTeXMLEquation .
Note that the additional term in LaTeXMLMath ’ s makes them symmetric operators in LaTeXMLMath with respect to the Riemann measure LaTeXMLMath on LaTeXMLMath .
The main difficulty encountered here is that operators ( 1.1 ) are not globally defined since , in general , LaTeXMLMath are only local coordinates .
It is therefore desirable to invent quantization methods which employ global geometric objects .
On several occasions the formalism of quantum mechanics in connection with non–trivial topology of the configuration manifold lead to new non–classical effects .
A deep and in its time not completely understood and recognized accomplishment in this direction was Dirac ’ s famous investigation LaTeXMLCite of a quantum charged particle ( charge LaTeXMLMath ) in the external magnetic field of a point–like magnetic monopole ( magnetic charge LaTeXMLMath ) .
If the singular Dirac monopole is placed at the origin of a Cartesian coordinate system in LaTeXMLMath , one deals in fact with quantum mechanics on a topologically non-trivial effective configuration manifold LaTeXMLMath ( the three-dimensional Euclidean space with the origin excluded ) .
Here the formalism of quantum mechanics in connection with non-trivial topology of the configuration manifold leads to an unexpected topological quantum effect originating from a peculiar behaviour of the phase of a wave function : Dirac discovered that a quantal description exists only under the condition that the dimensionless quantity LaTeXMLMath is an integer .
Another phenomenon of this kind was noticed in 1959 by Y. Aharonov and D. Bohm LaTeXMLCite .
The origin of the Aharonov–Bohm effect can be traced to a shift of the phase of wave function due to an external magnetic flux imposed on a charged particle .
Here the effective Aharonov–Bohm configuration space is LaTeXMLMath , the three–dimensional Euclidean space with a straight line excluded .
In both mentioned cases the topologies of the configuration spaces differ from the trivial topology of the Euclidean space and play decisive rôle in quantum theory .
These remarks about the early history of quantum mechanics clearly point to the need for a systematic development of global quantization methods .
For systems with sufficiently symmetric configuration or phase spaces , two modern approaches in the theory of group representations can be applied : 1 .
Mackey ’ s quantization on homogeneous configuration manifolds LaTeXMLMath LaTeXMLCite .
Essentially , it is equivalent to the construction of systems of imprimitivity for a ( locally compact , separable ) group LaTeXMLMath , based on LaTeXMLMath .
2 .
The method of coadjoint orbits which play the role of homogeneous phase spaces LaTeXMLCite .
In the case of configuration or phase manifolds without geometric symmetries , two programs of global quantization were suggested : 3 .
Borel quantization on configuration manifolds LaTeXMLCite which extends the notion of Schrödinger systems LaTeXMLCite .
4 .
Geometric quantization on symplectic phase manifolds LaTeXMLCite .
These methods have been elaborated to differing degrees of sophistication and have , in general , different classes of classical systems as their domains of applicability .
Borel quantization is built on configuration spaces and reflects the topology of LaTeXMLMath .
For physical applications it is important that it yields both important classification theorems and explicit relations for quantization of kinematical observables .
Like canonical quantization , it is a two step procedure .
In a first step the kinematics , i.e .
position and momentum observables on LaTeXMLMath , is quantized .
The time dependence is introduced in a second step with a quantum analogue of a second order Riemannian dynamics on LaTeXMLMath LaTeXMLCite .
In its most general form it leads to Doebner–Goldin non–linear Schrödinger equations LaTeXMLCite .
Concerning other quantization methods respecting global properties of configuration or phase spaces we should especially mention : 5 .
The Feynman path integral method ( it was used , e.g. , in LaTeXMLCite for LaTeXMLMath = SO ( 3 ) and in LaTeXMLCite for configuration spaces of identical particles ) .
6 .
Quantization by deformation of classical mechanics LaTeXMLCite .
7 .
Dirac quantization of systems with constraints in phase space LaTeXMLCite .
This review article is devoted to the mathematical exposition of quantum Borel kinematics .
This method yields quantizations of kinematics for systems admitting localization on connected smooth finite-dimensional configuration manifolds without boundary .
We restrict our consideration exclusively to paracompact manifolds which ( by Whitney ’ s embedding theorem ) can be regarded as submanifolds of LaTeXMLMath .
In Chap .
2 , the Hilbert space formalism of quantum mechanics , Wigner ’ s Theorem on symmetry transformations , and the notion of Mackey ’ s system of imprimitivity are briefly surveyed .
The notion of quantum Borel kinematics is introduced in Chap .
3 .
In Chap .
4 , a family of quantum Borel kinematics is constructed .
This geometrical construction of quantum kinematics ( Sect .
4.2 ) is based on the notion of a generalized system of imprimitivity for the family of one-parameter groups of diffeomorphisms ( Sect .
3.1 ) .
It represents a generalization of quantum Borel kinematics of Ref .
LaTeXMLCite , especially in admitting an external gauge field with an arbitrary Abelian or compact structure group LaTeXMLMath .
Thus the construction involves associated LaTeXMLMath -bundles with finite-dimensional fibres LaTeXMLMath .
Chap .
4 is also devoted to questions of unitary equivalence and irreducibility ( Sects .
4.5 and 4.6 ) of this class of quantum kinematics .
Important special case of the vanishing external field is treated in Sect .
4.7 .
The classification of quantum Borel kinematics can not be considered to be complete .
In Chap .
5 theorems are stated which fully characterize them as well as two cases of complete classifications — elementary quantum Borel kinematics ( Sect .
5.5 ) and quantum Borel LaTeXMLMath –kinematics of type 0 ( Sect .
5.6 ) .
In these cases it is shown that the first and the second singular homology groups of the configuration manifold LaTeXMLMath are involved and provide the necessary topological tools for classification of quantizations .
We have payed particular attention to a selection of proper examples which complement each chapter and demonstrate the richness of possible applications .
From these examples , we mention a new derivation of the Dirac quantization condition from rotational symmetry LaTeXMLCite ( Ex .
2.4 ) , a topological description of the Aharonov–Bohm effect ( Ex .
5.2 ) , classification of elementary quantum Borel kinematics on arbitrary two–dimensional compact orientable manifolds ( Ex .
5.3 ) as well as in the real projective space — topologically non–trivial part of the configuration space of the system of two identical particles LaTeXMLCite ( Ex .
5.4 ) .
In quantum mechanics , a separable Hilbert space LaTeXMLMath is associated with a quantum system we are going to describe .
States of the system are represented by von Neumann ’ s statistical ( density ) operators — bounded self–adjoint positive operators in LaTeXMLMath with unit trace .
The set of states LaTeXMLMath introduced in this way is convex ; its extremal points are called pure states .
The pure states are just the projectors on one–dimensional subspaces of LaTeXMLMath .
We assume LaTeXMLMath of the original state LaTeXMLMath .
There are several options for a description of a non–ideal localization — e.g .
with the use of positive operator–valued measures LaTeXMLCite , Ch .
3. with LaTeXMLCite that , to a measurement on the system , taking values in a set LaTeXMLMath endowed with a LaTeXMLMath -algebra LaTeXMLMath of measurable subsets , there corresponds a projection–valued measure LaTeXMLMath on LaTeXMLMath .
To any measurable set LaTeXMLMath , a projector LaTeXMLMath is related such that LaTeXMLMath and LaTeXMLMath , provided LaTeXMLMath for LaTeXMLMath .
If the system is in a state LaTeXMLMath , then the formula LaTeXMLEquation gives the probability that the result of the measurement belongs to the set LaTeXMLMath .
The map LaTeXMLMath is evidently a probability measure on LaTeXMLMath .
From the above considerations it is clear that in the quantal formalism crucial role is played by an orthocomplemented lattice of projectors onto subspaces of the Hilbert space LaTeXMLMath LaTeXMLCite .
This lattice will be denoted by LaTeXMLMath ; partial ordering of LaTeXMLMath is defined as follows : LaTeXMLMath if and only if LaTeXMLMath ; the complement : LaTeXMLMath .
Clearly , LaTeXMLMath if and only if the corresponding subspaces are in inclusion ; LaTeXMLMath projects on the orthogonal complement .
Important properties of the lattice of projectors are : For any countable set LaTeXMLMath of elements from LaTeXMLMath there exist LaTeXMLMath and LaTeXMLMath in LaTeXMLMath ; The element LaTeXMLMath is defined by the following properties : 1 ) LaTeXMLMath for all LaTeXMLMath ; 2 ) if LaTeXMLMath is any element of LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath , then LaTeXMLMath .
In an analogous fashion , the element LaTeXMLMath is defined by : 1 ) LaTeXMLMath for all LaTeXMLMath ; 2 ) if LaTeXMLMath is any element of LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath , then LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath there exists an element LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
This element is equal to LaTeXMLMath and is unique with this property .
An orthocomplemented lattice satisfying LaTeXMLMath is called a logic .
Let the configuration manifold — which we shall always denote by LaTeXMLMath — be a LaTeXMLMath –space of a symmetry group LaTeXMLMath .
This means that an action of LaTeXMLMath on LaTeXMLMath is given , i.e. , to each element LaTeXMLMath of LaTeXMLMath there corresponds a transformation LaTeXMLMath of LaTeXMLMath onto itself such that : LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath .
Some important assertions , in particular Mackey ’ s Imprimitivity Theorem , can be stated provided the group LaTeXMLMath is locally compact and separable ( i.e .
with countable basis of the topology ) .
In the following we shall restrict our considerations to the case when LaTeXMLMath is a finite-dimensional connected Lie group .
It is just these groups that very often appear in physical applications .
Let the manifold LaTeXMLMath be also connected and smooth , and the mapping LaTeXMLMath be infinitely differentiable ( LaTeXMLMath ) .
Now we would like to associate , to each symmetry transformation LaTeXMLMath of the configuration space LaTeXMLMath , a symmetry transformation of the quantum mechanical description .
To be specific , we introduce two notions .
Definition 2.1 .
An automorphism of a logic LaTeXMLMath is a one-to-one mapping LaTeXMLMath which satisfies LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Definition 2.2 .
A convex automorphism of the set of states LaTeXMLMath is a one-to-one mapping LaTeXMLMath with the following property : given positive real numbers LaTeXMLMath such that LaTeXMLMath , then LaTeXMLMath .
Now we can state the conditions on symmetry transformations of the quantum mechanical description in the following form : There exists a homomorphism LaTeXMLMath from the group LaTeXMLMath into the group of automorphisms of the logic LaTeXMLMath , LaTeXMLMath .
There exists a homomorphism LaTeXMLMath from the group LaTeXMLMath into the group of convex automorphisms of the set of states LaTeXMLMath , LaTeXMLMath .
The probability does not change under the symmetry transformations , i.e .
LaTeXMLEquation .
Given a projection–valued measure LaTeXMLMath on a set LaTeXMLMath ( endowed with a LaTeXMLMath -algebra of measurable subsets ) which corresponds to measurements on the system with values in LaTeXMLMath , there exists a homomorphism LaTeXMLMath from the group LaTeXMLMath into the group of measurable and one-to-one mappings from LaTeXMLMath onto itself .
Every LaTeXMLMath induces an automorphism of the LaTeXMLMath -algebra , LaTeXMLMath .
We demand LaTeXMLEquation .
Automorphisms of the logic and convex automorphisms of the set of states are described by Wigner ’ s theorem : Theorem 2.1 ( LaTeXMLCite , Chap .
VII.3 ) .
Let LaTeXMLMath be a separable infinite–dimensional Hilbert space .
Then : All automorphisms of the logic LaTeXMLMath are of the form LaTeXMLEquation where LaTeXMLMath is a fixed unitary or antiunitary operator in LaTeXMLMath .
Two such operators induce the same automorphism of the logic if and only if they differ by a phase factor .
All convex automorphisms of the set of states LaTeXMLMath are of the form LaTeXMLEquation where LaTeXMLMath is a fixed unitary or antiunitary operator in LaTeXMLMath .
Two such operators induce the same convex automorphism of the set of states if and only if they differ by a phase factor .
Theorem 2.1 and conditions ( a ) , ( b ) imply that to each action LaTeXMLMath a pair of operators LaTeXMLMath , LaTeXMLMath is associated , both being unitary or antiunitary .
To fulfil condition ( c ) , operators LaTeXMLMath , LaTeXMLMath may differ by a phase factor at most .
Hence these operators can be identified , LaTeXMLEquation .
Since we consider only connected Lie groups , all operators LaTeXMLMath will be unitary LaTeXMLMath of the unit element LaTeXMLMath there exists LaTeXMLMath to each LaTeXMLMath such that LaTeXMLMath ; it is well known that LaTeXMLMath is generated by the elements of LaTeXMLMath .
.
Let us denote by LaTeXMLMath the group of unitary operators in LaTeXMLMath with strong topology .
The centre LaTeXMLMath of this group consists of operators LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath denotes the compact Lie group of complex numbers of unit modulus .
The quotient group LaTeXMLEquation is called the projective group of the Hilbert space LaTeXMLMath .
The conditions ( a ) — ( c ) can be summarized in one requirement ( abc ) There exists a homomorphism LaTeXMLMath .
Moreover , we shall demand LaTeXMLMath to be measurable .
In this case LaTeXMLMath is even continuous ( LaTeXMLCite , Chap .
VIII.5 ) .
The condition ( abc ) can be reformulated with the use of the notion of a projective representation .
Let LaTeXMLMath be the canonical homomorphism .
For each given homomorphism LaTeXMLMath there exists a measurable mapping LaTeXMLMath .
The mapping LaTeXMLMath is called the projective representation of LaTeXMLMath .
Two projective representations LaTeXMLMath , LaTeXMLMath are called equivalent if there exists a measurable mapping LaTeXMLMath such that LaTeXMLMath .
The homomorphism LaTeXMLMath obviously determines the projective representation uniquely up to this equivalence .
Given a projective representation LaTeXMLMath , there exists a measurable mapping LaTeXMLEquation such that LaTeXMLEquation .
The factor LaTeXMLMath is called a multiplier of LaTeXMLMath ; by definition it fulfils LaTeXMLEquation .
Two multipliers LaTeXMLMath , LaTeXMLMath are equivalent if there exists a measurable mapping LaTeXMLMath such that LaTeXMLEquation .
By definition a multiplier is exact ( or trivial ) , if it is equivalent to 1 .
The set of all multipliers with pointwise multiplication forms an Abelian group ; trivial multipliers form its invariant subgroup .
The corresponding quotient group is referred to as the multiplier group for LaTeXMLMath ; we shall denote it by LaTeXMLMath .
LaTeXMLCite , Chap .
X .
The discussion of condition ( d ) of Sect .
2.2 was postponed to this section , since its analysis requires the description of a concrete measurement on the system .
For localizable systems the position measurements play a distinguished rôle .
Results of position measurements are points of the configuration space LaTeXMLMath , i.e .
LaTeXMLMath .
So it is natural to consider the LaTeXMLMath -algebra LaTeXMLMath of Borel subsets of LaTeXMLMath as the LaTeXMLMath -algebra of measurable sets LaTeXMLMath -algebra LaTeXMLMath is generated by open subsets of manifold LaTeXMLMath .
.
The starting point of Mackey ’ s quantization and of quantum Borel kinematics is the notion of localization of a quantum system on a configuration manifold LaTeXMLMath .
It is mathematically modeled by a projection–valued measure LaTeXMLMath mapping Borel subsets LaTeXMLMath of LaTeXMLMath ( LaTeXMLMath ) into projection operators LaTeXMLMath on a separable Hilbert space LaTeXMLMath subject to the usual axioms of localization .
For convenience , these axioms are given below : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation For a given subset LaTeXMLMath , the projection LaTeXMLMath corresponds to a measurement which determines whether the system is localized in LaTeXMLMath ; its eigenvalues 1 ( 0 ) correspond to situations when the system is found completely inside ( outside ) LaTeXMLMath , respectively .
According to ( d ) , each action LaTeXMLMath induces a Borel transformation of LaTeXMLMath onto itself .
As already mentioned , the other three conditions ( abc ) imply the existence of a projective representation LaTeXMLMath of LaTeXMLMath .
Hence ( 2.1 ) can be written in the form LaTeXMLEquation .
Definition 2.3 .
A pair LaTeXMLMath where LaTeXMLMath is a ( projective ) representation of a group LaTeXMLMath and LaTeXMLMath is a projection–valued measure on a LaTeXMLMath -space LaTeXMLMath , is called a ( projective ) system of imprimitivity for the group LaTeXMLMath , if ( 2.2 ) holds for all LaTeXMLMath , LaTeXMLMath .
Two projective systems of imprimitivity are equivalent if the corresponding projective representations are equivalent and if the projection–valued measures are equal .
Stronger results can be obtained if the symmetry group LaTeXMLMath of LaTeXMLMath is sufficiently rich .
More precisely , we shall turn our attention to homogeneous spaces .
By definition , LaTeXMLMath is a homogeneous G-space if LaTeXMLMath acts transitively on LaTeXMLMath , i.e. , to each pair of points LaTeXMLMath there exists a transformation LaTeXMLMath such that LaTeXMLMath .
Let us fix a point LaTeXMLMath .
The isotropy subgroup of LaTeXMLMath in LaTeXMLMath will be denoted by LaTeXMLMath .
It is well known that LaTeXMLMath is a closed Lie subgroup of the Lie group LaTeXMLMath .
The space LaTeXMLMath of left cosets LaTeXMLMath , LaTeXMLMath , endowed with factor topology , can be given a differentiable ( LaTeXMLMath ) structure , thus becoming a smooth manifold , and the mapping LaTeXMLMath induces a diffeomorphism of LaTeXMLMath onto LaTeXMLMath ( LaTeXMLCite , Chap .
II.3 , II.4 ) .
Having identified LaTeXMLMath with LaTeXMLMath , the group LaTeXMLMath acts on LaTeXMLMath in the natural way , LaTeXMLMath .
The quadruple LaTeXMLMath can be viewed as a principal fibre bundle .
Let us note here that the requirement on LaTeXMLMath to be connected is not very restrictive provided LaTeXMLMath is connected : Proposition 2.2 ( LaTeXMLCite , Chap .
II.4 ) .
Let LaTeXMLMath be a finite dimensional Lie group acting transitively on a connected smooth manifold LaTeXMLMath and let LaTeXMLMath be the connected component of unity in LaTeXMLMath .
Then LaTeXMLMath acts transitively on LaTeXMLMath , too .
As shown by Mackey LaTeXMLCite , the transitive systems of imprimitivity ( i.e .
based on homogeneous spaces LaTeXMLMath ) can be completely classified .
The notions of irreducibility , unitary equivalence , direct sum decomposition , etc. , can be taken over for the systems of imprimitivity in exact analogy with these notions for ( projective ) unitary representations ( LaTeXMLCite , Chap .
1.2 ) .
In order to investigate questions of irreducibility , direct sum decomposition , etc. , a commuting ring LaTeXMLMath is considered , which consists of all bounded operators in LaTeXMLMath commuting with LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath , LaTeXMLMath .
We have for instance the property that a system of imprimitivity LaTeXMLMath is irreducible if and only if the ring LaTeXMLMath consists of multiples of the unit operator LaTeXMLMath only ( Schur ’ s Lemma ) .
Now , following G.W .
Mackey , we are going to describe the canonical construction of transitive systems of imprimitivity .
Let LaTeXMLMath be a locally compact group satisfying the second axiom of countability , LaTeXMLMath its closed subgroup .
On the coset space LaTeXMLMath there exists a quasi–invariant measure defined on the LaTeXMLMath -algebra of Borel subsets .
A measure LaTeXMLMath on LaTeXMLMath is called quasi–invariant with respect to the action of LaTeXMLMath , if for all LaTeXMLMath the measures LaTeXMLMath and LaTeXMLMath are mutually absolutely continuous .
Moreover , all quasi–invariant LaTeXMLMath -finite measures on LaTeXMLMath are mutually absolutely continuous ( LaTeXMLCite , Chap .
VIII.4 ) .
We fix a measure LaTeXMLMath from this class .
Further , let LaTeXMLMath be a multiplier of LaTeXMLMath and let LaTeXMLMath be a projective unitary representation of LaTeXMLMath with multiplier LaTeXMLMath restricted to LaTeXMLMath in a separable Hilbert space LaTeXMLMath .
Then we construct the Hilbert space LaTeXMLMath as the space of vector–valued functions LaTeXMLMath satisfying LaTeXMLMath is a Borel function on LaTeXMLMath for all LaTeXMLMath ; LaTeXMLMath , LaTeXMLMath ; LaTeXMLMath , where LaTeXMLMath is the norm induced by the inner product LaTeXMLEquation the integral is well-defined since , because of b ) , the inner product LaTeXMLMath in LaTeXMLMath remains constant on the left cosets LaTeXMLMath .
Henceforth we shall identify two functions on LaTeXMLMath which are equal LaTeXMLMath -almost everywhere .
Then the projection–valued measure LaTeXMLMath on LaTeXMLMath is canonically defined by LaTeXMLEquation where LaTeXMLEquation .
The projective LaTeXMLMath -representation of LaTeXMLMath , LaTeXMLMath , is given by LaTeXMLEquation where LaTeXMLMath is the Radon–Nikodým derivative .
The pair LaTeXMLMath is called a canonical system of imprimitivity and its equivalence class does not depend on the choice of a quasi-invariant measure LaTeXMLMath .
Theorem 2.3 ( The Imprimitivity Theorem LaTeXMLCite ) .
Let LaTeXMLMath be a locally compact group satisfying the second axiom of countability , LaTeXMLMath its closed subgroup and LaTeXMLMath a multiplier of LaTeXMLMath .
Let a pair LaTeXMLMath be a projective system of imprimitivity for LaTeXMLMath based on LaTeXMLMath with multiplier LaTeXMLMath .
Then there exists an LaTeXMLMath -representation LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath is equivalent to the canonical system of imprimitivity LaTeXMLMath .
For any two LaTeXMLMath -representations LaTeXMLMath , LaTeXMLMath of the subgroup LaTeXMLMath the corresponding canonical systems of imprimitivity are equivalent if and only if LaTeXMLMath , LaTeXMLMath are equivalent .
The commuting rings LaTeXMLMath and LaTeXMLMath are isomorphic .
The Imprimitivity Theorem shows how to obtain all systems of imprimitivity up to unitary equivalence , provided the multiplier group LaTeXMLMath is known .
More facts about the multiplier group can be given in the case when LaTeXMLMath is a connected and simply connected Lie group .
Then every multiplier is equivalent to a multiplier of class LaTeXMLMath .
These multipliers can be expressed in the form LaTeXMLMath where LaTeXMLMath is called an infinitesimal multiplier .
Let us introduce a coboundary operator LaTeXMLMath on real skew–symmetric multilinear forms on the Lie algebra LaTeXMLMath via LaTeXMLEquation where LaTeXMLMath is any LaTeXMLMath -form .
Then the Abelian group of infinitesimal multipliers of LaTeXMLMath is isomorphic to the second cohomology group LaTeXMLMath .
Examples LaTeXMLMath is isomorphic to the additive group of real skew–symmetric 2–forms on LaTeXMLMath ; given a 2-form LaTeXMLMath , then LaTeXMLMath is a multiplier ; LaTeXMLMath .
LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath is a connected and simply connected semi–simple Lie group , then LaTeXMLMath .
We note that any connected Lie group LaTeXMLMath can be replaced by its ( connected and simply connected ) universal covering Lie group LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath is given by LaTeXMLMath where LaTeXMLMath is the covering homomorphism .
As Example 2.3 ( see Sect.2.5 ) will show , the transition to the covering group can lead to richer results with reasonable physical interpretation .
Remark .
Detailed descriptions of general foundations of quantum mechanics can be found in LaTeXMLCite , Chap .
VI and VII ; of systems of imprimitivity in LaTeXMLCite , Chap .
IX , LaTeXMLCite and LaTeXMLCite ; of multipliers in LaTeXMLCite , Chap .
X .
Let LaTeXMLMath be a connected Lie group and LaTeXMLMath a ( not necessarily homogeneous ) LaTeXMLMath –space .
A one–parameter subgroup of LaTeXMLMath is a one–dimensional Lie subgroup including its parametrization , LaTeXMLMath G LaTeXMLMath .
There is a one–to–one correspondence between elements LaTeXMLMath of a Lie algebra LaTeXMLMath and one–parameter subgroups LaTeXMLMath which can be expressed by LaTeXMLMath ( LaTeXMLCite , Chap .
I.6.4 ) .
This correspondence can be used to define the mapping LaTeXMLMath ; then one has LaTeXMLMath , and LaTeXMLMath is a local diffeomorphism at the unit element of LaTeXMLMath .
To each LaTeXMLMath there corresponds a one–parameter subgroup LaTeXMLMath and a flow on LaTeXMLMath , LaTeXMLMath ; the corresponding vector field on LaTeXMLMath will be denoted by LaTeXMLMath .
If LaTeXMLMath is the mapping LaTeXMLMath depending on LaTeXMLMath , then obviously LaTeXMLMath .
In the terminology of LaTeXMLCite , an infinitesimal action is the mapping LaTeXMLMath , where LaTeXMLMath denotes the infinite–dimensional Lie algebra of smooth vector fields on LaTeXMLMath .
Let LaTeXMLMath be the subgroup of ineffectively acting elements from LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is said to act effectively on LaTeXMLMath .
LaTeXMLMath is closed and normal , the factor group LaTeXMLMath is a Lie group acting effectively on LaTeXMLMath .
Manifold LaTeXMLMath can be considered as a LaTeXMLMath -space if the action is given by LaTeXMLMath .
In this way the ineffectively acting elements can be eliminated LaTeXMLMath discrete it does not seem reasonable to eliminate LaTeXMLMath in this way ; see Ex .
2.3. .
Theorem 2.4 ( LaTeXMLCite , Chap .
III.3.7 ) .
The infinitesimal action LaTeXMLMath is linear .
For all LaTeXMLMath one has LaTeXMLEquation .
Hence the image of this mapping is a finite–dimensional Lie subalgebra in LaTeXMLMath .
The kernel is LaTeXMLMath , the Lie algebra of the group LaTeXMLMath of ineffective elements from LaTeXMLMath .
The proof of the first part is based on a straightforward calculation ( LaTeXMLCite , Chap .
III.3.7 ) .
For the last assertion we observe that , if LaTeXMLMath and LaTeXMLMath is the corresponding flow , then LaTeXMLMath so LaTeXMLMath .
Now let us consider the opposite situation .
Suppose we are given a finite–dimensional Lie subalgebra LaTeXMLMath in LaTeXMLMath such that all vector fields from LaTeXMLMath are complete .
Let LaTeXMLMath be the Lie algebra with the same vector space as LaTeXMLMath but with a Lie bracket LaTeXMLMath .
If LaTeXMLMath is the connected and simply connected Lie group with Lie algebra LaTeXMLMath ( LaTeXMLMath is unique up to isomorphism ) , then according to LaTeXMLCite , Chap .
III.4.7 , Theorem 6 , to each LaTeXMLMath there exists an open neighbourhood LaTeXMLMath and a uniquely defined local action of LaTeXMLMath on LaTeXMLMath ( i.e. , an action defined only for elements from some neighbourhood LaTeXMLMath of the unity ) such that the associated infinitesimal action is identical with the mapping LaTeXMLMath .
The neighbourhood LaTeXMLMath can be chosen small enough for LaTeXMLMath to be a diffeomorphism on it .
For LaTeXMLMath , LaTeXMLMath — the corresponding flow , LaTeXMLMath , LaTeXMLMath sufficiently small , we have LaTeXMLMath .
Since all vector fields LaTeXMLMath are complete , LaTeXMLMath can be chosen independently of LaTeXMLMath .
In this way we obtain a local homomorphism from LaTeXMLMath into the group of diffeomorphisms of LaTeXMLMath .
Since LaTeXMLMath is connected and simply connected , the domain of the local homomorphism can be unambiguously extended to the whole LaTeXMLMath ( LaTeXMLCite , Chap .
II and VII ) .
Example 2.1 .
LaTeXMLMath , LaTeXMLMath – the group of translations .
Both the multiplier group LaTeXMLMath and the isotropy subgroup are trivial .
So in this case exactly one irreducible system of imprimitivity exists ( up to unitary equivalence ) .
Let LaTeXMLMath be a Hilbert space and LaTeXMLMath be a pair of self–adjoint operators in LaTeXMLMath satisfying the commutation relation LaTeXMLMath .
Then if LaTeXMLMath is the spectral projection–valued measure of LaTeXMLMath , and LaTeXMLMath , then the commutation relation is equivalent to the identity LaTeXMLEquation .
So the Imprimitivity Theorem implies the Stone–von Neumann theorem ( cf .
LaTeXMLCite , Chap .
2.5 ) .
Example 2.2 .
LaTeXMLMath , LaTeXMLMath – the group of translations .
To each element LaTeXMLMath there corresponds a vector field LaTeXMLMath .
The isotropy subgroup is trivial .
LaTeXMLMath is isomorphic to the group LaTeXMLMath under addition : if LaTeXMLMath , then LaTeXMLEquation is a multiplier ( LaTeXMLMath is an arbitrary fixed non-zero constant ) .
The inequivalent irreducible systems of imprimitivity LaTeXMLMath are labelled by LaTeXMLMath : the Hilbert space is LaTeXMLMath and we find LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath is a self–adjoint operator , LaTeXMLEquation .
LaTeXMLMath , i.e .
LaTeXMLMath , and LaTeXMLMath .
The real number LaTeXMLMath can be given physical meaning : a particle with electric charge LaTeXMLMath moves on the plane LaTeXMLMath in an external magnetic field which is perpendicular to the plane and has constant value LaTeXMLMath ( the sign reflects the orientation ) .
Example 2.3 .
LaTeXMLMath , LaTeXMLMath U ( 1 ) — the group of rotations of the circle LaTeXMLMath .
Both LaTeXMLMath and the isotropy subgroup are trivial ; hence there exists exactly one irreducible system of imprimitivity ( up to unitary equivalence ) .
Now let us replace LaTeXMLMath U ( 1 ) by its universal covering LaTeXMLMath .
Then LaTeXMLMath and the isotropy subgroup LaTeXMLMath .
The irreducible unitary representations of LaTeXMLMath are labelled by elements LaTeXMLMath : for LaTeXMLMath let LaTeXMLEquation if LaTeXMLMath , then LaTeXMLMath .
We shall describe the system of imprimitivity for given LaTeXMLMath .
The Hilbert space LaTeXMLMath consists of ( equivalence classes of ) functions LaTeXMLMath such that LaTeXMLEquation almost everywhere ; the inner product is defined by LaTeXMLEquation .
We have LaTeXMLEquation where LaTeXMLMath and LaTeXMLEquation is self–adjoint .
The mapping LaTeXMLEquation is unitary , LaTeXMLMath can be identified with LaTeXMLMath .
We find LaTeXMLEquation .
A possible physical interpretation is connected with the Aharonov–Bohm effect LaTeXMLCite : a particle in LaTeXMLMath with electric charge LaTeXMLMath is moving on the circle LaTeXMLMath , LaTeXMLMath , and external magnetic flux LaTeXMLMath is concentrated along the LaTeXMLMath -axis passing through the centre of the circle .
If LaTeXMLMath is an integer , then the two quantum kinematics with fluxes LaTeXMLMath lead to the same observable results ( e.g. , the same interference pattern ) .
Example 2.4 .
LaTeXMLMath , the symmetry group LaTeXMLMath is replaced by the quantum mechanical symmetry group LaTeXMLMath LaTeXMLCite acting on LaTeXMLMath in the usual way LaTeXMLEquation where LaTeXMLMath and points LaTeXMLMath ( LaTeXMLMath ) are identified with the matrices LaTeXMLMath ; LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are the Pauli spin matrices .
The isotropy subgroup LaTeXMLMath of the north pole LaTeXMLMath consists of the diagonal matrices LaTeXMLMath , LaTeXMLMath , hence LaTeXMLMath .
If LaTeXMLMath is parametrized by LaTeXMLMath , LaTeXMLMath , LaTeXMLEquation then the projection LaTeXMLMath is given by LaTeXMLEquation .
The quadruple LaTeXMLMath constitutes a non-trivial principal bundle known as the Hopf fibration .
We shall explicitly write local trivializations of this bundle on sets LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are the north and the south pole , respectively .
A local trivialization is determined by a selected smooth local section .
We choose ( in spherical coordinates LaTeXMLMath , LaTeXMLMath ) LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath is simple , connected and simply connected , its multiplier group is trivial .
The irreducible representations LaTeXMLMath of the isotropy subgroup LaTeXMLMath are labeled by integers LaTeXMLMath , LaTeXMLMath .
In order to write down explicit expressions for the operators LaTeXMLMath of generalized momenta , it is convenient to work in the complex line bundle associated ( via LaTeXMLMath ) with the principal bundle .
Then the Hilbert space LaTeXMLMath of the canonical system of imprimitivity corresponding to LaTeXMLMath consists of measurable sections LaTeXMLMath in the complex line bundle ; each section LaTeXMLMath can be identified with a pair of functions LaTeXMLMath , where LaTeXMLEquation for almost all LaTeXMLMath .
We choose LaTeXMLMath , LaTeXMLMath , LaTeXMLMath as basis of the Lie algebra LaTeXMLMath .
The element LaTeXMLMath induces the vector field LaTeXMLMath on LaTeXMLMath , LaTeXMLMath .
Further , the relation LaTeXMLEquation defines a self–adjoint operator LaTeXMLMath in LaTeXMLMath .
Operators LaTeXMLMath , LaTeXMLMath corresponding to vector fields LaTeXMLMath , LaTeXMLMath can be obtained by cyclic permutations .
If for LaTeXMLMath , LaTeXMLMath is a vector field and if LaTeXMLMath is a smooth local section , then the self–adjoint operator LaTeXMLMath is determined by a pair of operators LaTeXMLMath , LaTeXMLMath ; a straightforward calculation yields LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath ; 1-forms LaTeXMLMath are the localizations on sets LaTeXMLMath of the connection 1-form LaTeXMLMath in the associated bundle .
In spherical coordinates LaTeXMLEquation .
LaTeXMLEquation ( The constant LaTeXMLMath is again arbitrary , non-zero , but fixed . )
On the intersection LaTeXMLMath we find LaTeXMLEquation .
We put LaTeXMLEquation .
The situation may have the following physical interpretation : a particle with charge LaTeXMLMath is moving in the magnetic field of the Dirac monopole with magnetic charge LaTeXMLMath placed at the origin LaTeXMLMath in LaTeXMLMath , so on the sphere LaTeXMLMath there is the external magnetic field LaTeXMLMath , LaTeXMLMath .
The relation LaTeXMLMath , LaTeXMLMath , coincides with the Dirac quantization condition LaTeXMLCite .
Operators LaTeXMLMath are the well-known conserved total angular momentum operators for a charged particle moving in the Dirac monopole field .
A more detailed discussion of this example was given in LaTeXMLCite .
The first treatments of the magnetic monopole using a connection in a fibre bundle appeared in LaTeXMLCite .
Example 2.5 .
LaTeXMLMath , LaTeXMLMath is the group of orientation preserving affine transformations of LaTeXMLMath .
Having identified LaTeXMLMath with LaTeXMLMath , LaTeXMLMath acts on LaTeXMLMath according to LaTeXMLEquation .
The isotropy subgroup LaTeXMLMath of the origin LaTeXMLMath consists of all matrices with LaTeXMLMath .
The irreducible unitary representations of LaTeXMLMath are of the form LaTeXMLMath , LaTeXMLMath .
The Lie algebra LaTeXMLMath consists of all matrices LaTeXMLEquation .
An element LaTeXMLMath induces a vector field LaTeXMLMath .
Let us investigate LaTeXMLMath : LaTeXMLMath is connected and simply connected , and a general real skew–symmetric 2-form LaTeXMLMath on LaTeXMLMath is of the form LaTeXMLMath for some constant LaTeXMLMath ; but it is easily verified that LaTeXMLMath where LaTeXMLMath is a 1-form on LaTeXMLMath , so LaTeXMLMath is trivial ( see Sect .
2.3 ) .
Thus the irreducible systems of imprimitivity are labelled by LaTeXMLMath .
Explicitly , LaTeXMLMath , LaTeXMLMath acts via multiplication by indicator function LaTeXMLMath , and LaTeXMLEquation .
The self–adjoint generalized momentum operators LaTeXMLMath , LaTeXMLMath , defined by LaTeXMLEquation are of the form LaTeXMLEquation .
Generally , for a given smooth manifold LaTeXMLMath there is , a priori , no geometric symmetry group .
As indicated in LaTeXMLCite , the investigation of vector fields on LaTeXMLMath is a meaningful starting point .
We denote by LaTeXMLMath the Lie algebra of smooth vector fields on LaTeXMLMath , by LaTeXMLMath its subalgebra of compactly supported vector fields , by LaTeXMLMath the family of all complete vector fields , LaTeXMLMath .
The flow LaTeXMLMath of a complete vector field LaTeXMLMath represents a one–parameter group of diffeomorphisms LaTeXMLMath of LaTeXMLMath , also called a dynamical system on LaTeXMLMath .
And , vice versa , every dynamical system is a flow of some ( uniquely determined ) complete vector field LaTeXMLEquation .
The family of dynamical systems on LaTeXMLMath will be denoted by LaTeXMLMath .
The following theorem summarizes some well-known facts from differential geometry LaTeXMLCite , LaTeXMLCite .
Theorem 3.1 .
Let LaTeXMLMath be a diffeomorphism .
Then LaTeXMLMath , where LaTeXMLMath , is a Lie algebra isomorphism ; the restriction LaTeXMLMath is also a Lie algebra isomorphism ; LaTeXMLMath is a bijection .
The mapping LaTeXMLEquation is bijective and LaTeXMLMath .
For every LaTeXMLMath the manifold LaTeXMLMath becomes a LaTeXMLMath -space for the group LaTeXMLMath .
Attempting to generalize Mackey ’ s quantization ( Sects .
2.3 , 2.4 ) we require that there exist : a Hilbert space LaTeXMLMath , a projection–valued measure LaTeXMLMath on LaTeXMLMath and unitary representations LaTeXMLMath in LaTeXMLMath of the flows LaTeXMLMath such that LaTeXMLEquation where the objects LaTeXMLMath , LaTeXMLMath do not depend on the choice of LaTeXMLMath .
Equation ( 3.1 ) is just a generalization of ( 2.2 ) .
Geometric shifts of Borel sets LaTeXMLMath by flows LaTeXMLMath along complete vector fields LaTeXMLMath are represented in LaTeXMLMath by unitary operators LaTeXMLMath such that ( 3.1 ) holds .
Generalized momentum operators can then be introduced via Stone ’ s Theorem as ( essentially self–adjoint ) infinitesimal generators LaTeXMLMath of the one-parameter groups of unitary operators — shifts in LaTeXMLMath of the localized quantum system , LaTeXMLEquation .
The quantization of ‘ classical ’ Borel kinematics LaTeXMLMath thus requires LaTeXMLCite the imprimitivity condition ( 3.1 ) for the unitary representation of the flow of each complete vector field individually .
Then we can state Definition 3.1 .
Quantum Borel kinematics is a pair LaTeXMLMath , where LaTeXMLMath is a projection–valued measure on LaTeXMLMath in a separable Hilbert space LaTeXMLMath , and LaTeXMLMath associates with each LaTeXMLMath a homomorphism LaTeXMLMath such that the following conditions are satisfied : Equation ( 3.1 ) holds for all LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ; The mapping LaTeXMLMath from the Lie algebra LaTeXMLMath into the space of essentially self–adjoint operators with common invariant dense domain in LaTeXMLMath is a Lie algebra homomorphism : LaTeXMLEquation .
LaTeXMLEquation Locality condition .
If two flows LaTeXMLMath , LaTeXMLMath , after restriction on the set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , coincide , then the mappings LaTeXMLMath coincide on the domain LaTeXMLMath , where LaTeXMLMath is the subspace of LaTeXMLMath projected out by LaTeXMLMath .
If in 2 ) only linearity ( 3.2 ) is required , we shall call LaTeXMLMath a generalized system of imprimitivity for LaTeXMLMath .
LaTeXMLMath acts transitively on LaTeXMLMath ( we suppose LaTeXMLMath to be connected , without boundary ) : for any two points LaTeXMLMath there exists LaTeXMLMath , LaTeXMLMath such that LaTeXMLMath .
It will describe quantum Borel kinematics with external gauge field .
It follows from condition 3 ) that , if LaTeXMLMath is known for all LaTeXMLMath , then it is determined for all LaTeXMLMath ; 3 ) further implies that LaTeXMLMath are differential operators .
Condition 2 ) may sometimes be too restrictive since ( 3.3 ) excludes a non-vanishing external gauge field on M ; in this connection see LaTeXMLCite and also Ex .
2.2 , Ex .
2.4 and Sect .
4.7 .
The projection–valued measure LaTeXMLMath induces in a natural way a quantization LaTeXMLMath of classical ( smooth ) real functions LaTeXMLMath on configuration space ( e.g .
coordinate functions , potentials , etc . ) .
Not necessarily bounded , self–adjoint quantum position operators LaTeXMLMath are uniquely determined by their spectral decompositions LaTeXMLEquation where the spectral function LaTeXMLMath is given by the spectral measure LaTeXMLMath on subsets LaTeXMLMath of LaTeXMLMath .
Equation ( 3.1 ) is then replaced by LaTeXMLEquation where LaTeXMLMath , and implies a generalization of the Heisenberg commutation relations in terms of coordinate-independent objects LaTeXMLEquation .
It is assumed that operators LaTeXMLMath have a common invariant dense domain LaTeXMLMath in LaTeXMLMath .
If an obvious relation LaTeXMLEquation for all LaTeXMLMath is still added , then ( 3.6 ) , ( 3.5 ) and ( 3.3 ) define a Schrödinger system in the sense of LaTeXMLCite .
We can say that Borel quantization on a smooth configuration manifold LaTeXMLMath associates the generalized position LaTeXMLMath and momentum operators LaTeXMLMath with smooth functions LaTeXMLMath and smooth vector fields LaTeXMLMath , respectively .
These quantum kinematical observables on LaTeXMLMath are globally defined , hence Borel quantization incorporates the global structure of LaTeXMLMath .
Remark .
The natural infinite–dimensional Lie algebra structure ( 3.6 ) , ( 3.5 ) and ( 3.3 ) of quantum Borel kinematics should be compared with the non–relativistic local current algebra for a Schrödinger second quantized field over LaTeXMLMath studied in LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
There is an apparent algebraic correspondence of the local density operator LaTeXMLMath with LaTeXMLMath and the local current operator LaTeXMLMath with LaTeXMLMath .
In both cases the representations of the algebraic structures yield possible quantum kinematics .
However , in contrast to quantum Borel kinematics , where LaTeXMLMath is a multiplication operator , local current algebra is more general , since also representations where LaTeXMLMath is not equal to LaTeXMLMath are admitted .
The question of existence and uniqueness of a measure which is quasi-invariant with respect to all diffeomorphisms LaTeXMLMath for which LaTeXMLMath , is answered by Theorem 3.2 .
The family of quasi-invariant measures on LaTeXMLMath is non-empty and , moreover , all measures in this family are mutually equivalent and form a unique invariant measure class .
Lebesgue measure class .
After completion , those subsets in LaTeXMLMath which have measure zero are exactly measure zero sets in the sense of Lebesgue .
Proof : The fact that the family of sets of zero measure in the sense of Lebesgue is invariant under diffeomorphisms is well known LaTeXMLCite .
The existence part of the theorem can be seen as follows .
Having embedded LaTeXMLMath in LaTeXMLMath ( Whitney ’ s Theorem ) , we can consider a tubular neighborhood LaTeXMLMath of LaTeXMLMath in the normal bundle ( LaTeXMLCite , Chap .
2.3 ) .
Denoting by LaTeXMLMath the associated submersion , we can define LaTeXMLMath for LaTeXMLMath , where LaTeXMLMath denotes the Lebesgue measure in LaTeXMLMath ; then LaTeXMLMath is quasi-invariant .
The assertion about uniqueness of the invariant measure class for LaTeXMLMath follows from the fact that the family of diffeomorphisms LaTeXMLMath includes all translations and the assertion for the group of translations is known ( LaTeXMLCite , Chap .
II.3 ) .
In general , LaTeXMLMath can be covered by a countable family of open sets , each of which is diffeomorphic to LaTeXMLMath and so the assertion is true again .
LaTeXMLMath Measure LaTeXMLMath will be called differentiable if the mapping LaTeXMLMath is smooth for all LaTeXMLMath , LaTeXMLMath .
For instance , measure LaTeXMLMath used in the proof of Theorem 3.2 is differentiable .
Every manifold is locally orientable ; having fixed an orientation on an open set LaTeXMLMath ( dim LaTeXMLMath ) , then to every differentiable measure LaTeXMLMath exactly one LaTeXMLMath -form LaTeXMLMath exists on LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLMath for every positively oriented basis in LaTeXMLMath .
For each LaTeXMLMath we define a function LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation where the LaTeXMLMath -form LaTeXMLMath is defined by LaTeXMLEquation .
Equivalently , if LaTeXMLMath , then LaTeXMLEquation .
In local coordinates , if LaTeXMLMath , LaTeXMLMath , then LaTeXMLEquation .
The structure of projection–valued measures on LaTeXMLMath is well known .
According to LaTeXMLCite , Chap .
IX.4 , we have a canonical representation of a localized quantum system : Let LaTeXMLMath be a projection–valued measure on LaTeXMLMath in a separable Hilbert space LaTeXMLMath .
Then there exist two sequences LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , the first one consisting of Hilbert spaces , the other of measures on LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath , LaTeXMLMath are mutually singular LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
for LaTeXMLMath .
The projection–valued measure LaTeXMLMath is unitarily equivalent to the measure LaTeXMLMath which acts via multiplication by indicator functions of subsets in the Hilbert space LaTeXMLMath , where LaTeXMLMath are the Hilbert spaces of vector–valued functions from LaTeXMLMath to LaTeXMLMath , LaTeXMLMath .
The measures LaTeXMLMath are determined uniquely up to equivalence .
If only one LaTeXMLMath is non-zero , the projection–valued measure LaTeXMLMath is called homogeneous .
Due to the transitivity of actions of the family LaTeXMLMath the following theorem holds ( for details see LaTeXMLCite , Chap .
IX.5 , IX.6 ) : Theorem 3.3 .
If LaTeXMLMath is a generalized system of imprimitivity for LaTeXMLMath , then LaTeXMLMath is homogeneous .
The unique non-zero measure LaTeXMLMath belongs to the Lebesgue measure class on LaTeXMLMath .
Thus the canonical representation of a localized quantum system on LaTeXMLMath involves a smooth measure , i.e .
a measure induced by the Lebesgue measure of the coordinate charts .
An LaTeXMLMath –homogeneous localized quantum system of degree LaTeXMLMath can be interpreted as a quantum system with internal degrees of freedom ; a 1–homogeneous localized quantum system will be called elementary as its LaTeXMLMath ’ s are related to elementary spectral measures .
We shall need also Theorem 3.4 .
( LaTeXMLCite , Chap .
IX.2 ) .
Let LaTeXMLMath be a Borel measure on LaTeXMLMath , LaTeXMLMath a Hilbert space , LaTeXMLMath , LaTeXMLMath a projection–valued measure on LaTeXMLMath acting in LaTeXMLMath via multiplication by indicator functions .
Then any bounded operator LaTeXMLMath in LaTeXMLMath commuting with LaTeXMLMath for all LaTeXMLMath ( LaTeXMLMath ) is of the form LaTeXMLEquation where LaTeXMLMath is a Borel mapping from LaTeXMLMath into the space of bounded operators in LaTeXMLMath such that LaTeXMLMath .
Function LaTeXMLMath is determined by LaTeXMLMath uniquely on LaTeXMLMath modulo a set of LaTeXMLMath -measure zero .
Quantum Borel kinematics are rather diverse ( even for trivial configuration space LaTeXMLMath ) , as the following example LaTeXMLCite shows .
Let LaTeXMLMath with a fixed basis .
To every vector field LaTeXMLMath we relate a matrix–valued function LaTeXMLEquation .
It is straightforward to verify LaTeXMLEquation .
Let LaTeXMLMath be a skew-Hermitean representation of the Lie algebra LaTeXMLMath in a Hilbert space LaTeXMLMath .
We define operators LaTeXMLMath , LaTeXMLMath in LaTeXMLMath by LaTeXMLEquation .
Then using ( 3.7 ) and the identity LaTeXMLMath , the pair LaTeXMLMath can be shown to be a quantum Borel kinematic .
Choosing the representation LaTeXMLMath in LaTeXMLMath to be given by LaTeXMLEquation where LaTeXMLMath is a real constant , we obtain LaTeXMLEquation .
This is just an example of the divergence term which we shall encounter in Sect .
4.7 .
Remark .
Consider the surjective mapping LaTeXMLEquation where LaTeXMLMath is the unit LaTeXMLMath -matrix .
This mapping permits to associate with every representation LaTeXMLMath of LaTeXMLMath a representation LaTeXMLMath of LaTeXMLMath .
Then our mapping LaTeXMLMath is the infinitesimal form of a representation of the group of diffeomorphisms of LaTeXMLMath induced from LaTeXMLMath ( see LaTeXMLCite for LaTeXMLMath ) .
For general results concerning the “ divergence–like ” terms in the framework of systems of imprimitivity for the group of diffeomorphisms , see LaTeXMLCite .
In order to motivate our construction of quantum Borel kinematics with external field via generalized systems of imprimitivity for LaTeXMLMath , let us consider quantum kinematics on LaTeXMLMath , for a charged particle in an external magnetic field LaTeXMLMath .
Since LaTeXMLMath , the Poincaré lemma implies that there exists a vector potential LaTeXMLMath such that LaTeXMLMath .
The Hilbert space is LaTeXMLMath , and the Hamiltonian LaTeXMLEquation .
If another potential LaTeXMLMath , LaTeXMLMath , is chosen , then according to the Poincaré theorem there exists a real function LaTeXMLMath such that LaTeXMLMath The Hamiltonian LaTeXMLMath is transformed as LaTeXMLMath where LaTeXMLMath is the unitary mapping LaTeXMLEquation .
This common quantum mechanical scheme can be reformulated in geometric language LaTeXMLCite : LaTeXMLMath is the space of measurable sections in the trivial Hermitian complex line bundle LaTeXMLMath associated with the principal bundle LaTeXMLMath ; LaTeXMLEquation is a localized connection 1–form ; LaTeXMLEquation is the curvature 2–form ; LaTeXMLEquation is a transition function in the principal bundle relating two different trivializations .
In general , if a magnetic field is given by a closed 2–form LaTeXMLMath on manifold LaTeXMLMath , a vector potential 1-form LaTeXMLMath such that LaTeXMLMath need not exist .
However , following LaTeXMLCite , one can always define vector potentials LaTeXMLMath locally , i.e .
on open sets LaTeXMLMath ( diffeomorphic to LaTeXMLMath ) such that { LaTeXMLMath } is an open covering of LaTeXMLMath We require that vector potentials LaTeXMLMath be related on the intersection LaTeXMLMath by a gauge transformation LaTeXMLEquation where LaTeXMLMath is a transition function .
Then LaTeXMLMath In this way we shall construct a principal bundle with typical fibre LaTeXMLMath and connection { LaTeXMLMath } .
We choose a measure LaTeXMLMath from the Lebesgue measure class on LaTeXMLMath and four objects LaTeXMLMath where : LaTeXMLMath ( or shortly LaTeXMLMath ) is a principal bundle over LaTeXMLMath ; its typical fibre LaTeXMLMath is an Abelian or compact Lie group ; LaTeXMLMath is a connection in LaTeXMLMath , and LaTeXMLMath is a unitary representation of LaTeXMLMath in a finite-dimensional Hilbert space LaTeXMLMath with inner product LaTeXMLMath We construct a separable Hilbert space LaTeXMLMath like in Sect .
2.3 , consisting of vector–valued functions LaTeXMLMath such that LaTeXMLMath is a Borel function on LaTeXMLMath for all LaTeXMLMath LaTeXMLMath ( LaTeXMLMath is an equivariant function ) , LaTeXMLMath where LaTeXMLMath is the norm induced by the inner product LaTeXMLEquation .
The integral is well defined as the integrand remains constant on the fibres .
Two functions LaTeXMLMath are identified if they coincide almost everywhere.The projection–valued measure LaTeXMLMath on LaTeXMLMath is defined via multiplication by indicator functions : LaTeXMLEquation .
For LaTeXMLMath we define the unitary representation of the additive group LaTeXMLMath LaTeXMLEquation .
Here LaTeXMLMath denotes the horizontal lift of the flow LaTeXMLMath on LaTeXMLMath .
It is easily verified that the constructed pair LaTeXMLMath is a generalized system of imprimitivity for LaTeXMLMath in the sense of Definition 3.1 .
The linearity of mapping LaTeXMLMath required by this definition will be investigated in Sect .
4.3 ( see Eq .
( 4.7 ) ) .
The equivalence class of LaTeXMLMath does not depend on measure LaTeXMLMath ; if LaTeXMLMath is another measure from the Lebesgue class , then LaTeXMLMath is the desired unitary mapping .
For simplicity we shall suppose LaTeXMLMath to be differentiable .
We assumed LaTeXMLMath to be Abelian or compact in order to deal only with finite–dimensional unitary representations LaTeXMLMath of LaTeXMLMath .
For dim LaTeXMLMath we identify LaTeXMLMath with LaTeXMLMath endowed with the standard inner product .
We say that the pair LaTeXMLMath is a generalized system of imprimitivity specified by the quadruple LaTeXMLMath We note that , for two diffeomorphic manifolds , there is a one–to–one correspondence between the equivalence classes of generalized systems of imprimitivity constructed in this way .
The associated vector bundle LaTeXMLMath will be constructed in the standard way ( see e.g .
LaTeXMLCite ) .
We introduce equivalence relation LaTeXMLEquation on LaTeXMLMath and put LaTeXMLMath ; the projection is LaTeXMLMath Each fibre becomes an LaTeXMLMath -dimensional Hilbert space : LaTeXMLEquation .
LaTeXMLEquation The inner product is well–defined since LaTeXMLMath is unitary .
For the same reason we can relate a unitary mapping LaTeXMLMath to each LaTeXMLMath LaTeXMLEquation and we have LaTeXMLMath Let LaTeXMLMath be the Hilbert space of measurable sections in the associated vector bundle , having finite norm induced by the inner product LaTeXMLEquation .
We can define a unitary mapping LaTeXMLMath in a natural way : LaTeXMLEquation .
The definition of LaTeXMLMath is correct since LaTeXMLMath and LaTeXMLMath The inverse mapping LaTeXMLMath is given by LaTeXMLMath Having performed the unitary transform LaTeXMLMath we replace LaTeXMLMath by a generalized system of imprimitivity LaTeXMLMath in LaTeXMLMath .
We shall compute explicit expressions .
Clearly , LaTeXMLMath acts via multiplication by indicator functions LaTeXMLEquation .
In order to express LaTeXMLMath we must first describe induced connection in the associated vector bundle LaTeXMLMath .
Having Hermitian structure on the fibres , we consider only Hermitian connections on LaTeXMLMath i.e .
connections for which all linear isomorphisms LaTeXMLMath ( shortly LaTeXMLMath see below ) , which belong to curves LaTeXMLMath in LaTeXMLMath , are unitary .
As is well known , there is a one-to-one correspondence between Hermitian connections and Hermitian covariant derivatives in LaTeXMLMath A covariant derivative LaTeXMLMath acting on smooth sections , LaTeXMLMath , LaTeXMLMath is Hermitian , if it satisfies ( in addition to four conditions LaTeXMLCite defining the covariant derivative ) the identity LaTeXMLEquation .
Sec LaTeXMLMath denotes the linear space of smooth sections in LaTeXMLMath A connection LaTeXMLMath in LaTeXMLMath induces a Hermitian connection in LaTeXMLMath given a piecewise smooth curve LaTeXMLMath in LaTeXMLMath and its lift LaTeXMLMath in LaTeXMLMath then LaTeXMLMath , LaTeXMLMath is the desired unitary mapping .
Its definition does not depend on the starting point LaTeXMLMath since LaTeXMLMath and LaTeXMLMath is another horizontal lift of the curve LaTeXMLMath Now the Hermitian covariant derivative is defined by the limit LaTeXMLEquation .
We are now in the position to give explicit formula for LaTeXMLMath : LaTeXMLEquation .
Let LaTeXMLMath denote the linear space of smooth and compactly supported sections in LaTeXMLMath the subspace LaTeXMLMath is dense in LaTeXMLMath As before , a self–adjoint operator LaTeXMLMath is defined by LaTeXMLEquation using ( 4.6 ) , ( 4.5 ) and ( 3.6 ) we find LaTeXMLEquation .
Note that expression ( 4.7 ) implies that LaTeXMLMath is linear in LaTeXMLMath Using unitary representation LaTeXMLMath we can associate a principal bundle LaTeXMLMath to the principal bundle LaTeXMLMath The construction is similar to that for LaTeXMLMath .
On LaTeXMLMath we introduce the equivalence relation LaTeXMLMath , LaTeXMLMath , and put LaTeXMLMath , LaTeXMLMath .
Lie group U ( r ) acts on LaTeXMLMath via LaTeXMLMath , LaTeXMLMath .
The mapping LaTeXMLMath is a bundle homomorphism — we have LaTeXMLMath With the help of this homomorphism we can transform the connection LaTeXMLMath in LaTeXMLMath into a connection LaTeXMLMath in LaTeXMLMath as described in the following theorem .
Theorem 4.1 .
( LaTeXMLCite , Chap .
II.5 ; LaTeXMLCite ) .
Let LaTeXMLMath be a principal bundle homomorphism and LaTeXMLMath a connection in LaTeXMLMath Then there exists a unique connection LaTeXMLMath in LaTeXMLMath such that the tangent mapping LaTeXMLMath maps every horizontal subspace of connection LaTeXMLMath onto a horizontal subspace of LaTeXMLMath The generalized system of imprimitivity LaTeXMLMath specified by LaTeXMLMath id ) , with id : LaTeXMLMath being the identity mapping ( fundamental representation ) , is equivalent to LaTeXMLMath We describe the corresponding unitary mapping .
If LaTeXMLMath there is a unique vector LaTeXMLMath such that LaTeXMLMath Indeed , if LaTeXMLMath then LaTeXMLMath and LaTeXMLMath hence LaTeXMLMath for each LaTeXMLMath The mapping defined in this way is unitary and transforms LaTeXMLMath in LaTeXMLMath since LaTeXMLMath ; LaTeXMLMath is transformed in LaTeXMLMath since LaTeXMLMath preserves the connection .
We can again associate a vector bundle LaTeXMLMath to the principal bundle LaTeXMLMath using the fundamental representation id .
Both LaTeXMLMath and LaTeXMLMath have the same base space and the same typical fibre .
In fact , LaTeXMLMath and LaTeXMLMath can be identified by the mapping LaTeXMLEquation where the homomorphism LaTeXMLMath was described above .
LaTeXMLMath is well–defined since LaTeXMLEquation .
LaTeXMLMath is surjective because LaTeXMLMath for each LaTeXMLMath LaTeXMLMath is injective because , for each LaTeXMLMath the induced mapping LaTeXMLMath is unitary .
There is again a Hermitian covariant derivative LaTeXMLMath in LaTeXMLMath which corresponds to the Hermitian connection LaTeXMLMath in LaTeXMLMath .
We can briefly say that the following diagram commutes : For the corresponding generalized systems of imprimitivity we find the following commuting diagram : All these generalized systems of imprimitivity are mutually unitarily equivalent .
With the help of the identical representation id of U LaTeXMLMath in LaTeXMLMath we can associate a vector bundle LaTeXMLMath to every principal bundle LaTeXMLMath To each LaTeXMLMath there corresponds the unitary mapping LaTeXMLMath which we henceforth denote by the same letter LaTeXMLMath , namely LaTeXMLMath .
Conversely , let LaTeXMLMath be a unitary mapping .
Then LaTeXMLMath necessarily has the form LaTeXMLMath where LaTeXMLMath is a unitary mapping LaTeXMLMath Hence there exists LaTeXMLMath such that LaTeXMLMath .
Every unitary mapping LaTeXMLMath which is represented by unitary mapping LaTeXMLMath is equal to the composition LaTeXMLMath since LaTeXMLMath .
Further , two points LaTeXMLMath coincide as unitary mappings if and only if LaTeXMLMath So we can return from the vector bundle LaTeXMLMath back to the principal bundle LaTeXMLMath .
The fibre LaTeXMLMath over LaTeXMLMath consists of unitary mappings LaTeXMLMath and the structure group LaTeXMLMath acts on LaTeXMLMath by composition LaTeXMLMath .
Definition 4.1 .
Two principal bundles LaTeXMLMath , LaTeXMLMath over the same base space and with the same structure group are said to be isomorphic if there exists a diffeomorphism LaTeXMLMath fulfilling LaTeXMLMath LaTeXMLMath for all LaTeXMLMath Two Hermitian vector bundles LaTeXMLMath , LaTeXMLMath are said to be isomorphic if there exists a diffeomorphism LaTeXMLMath such that the restrictions LaTeXMLMath are unitary mappings for all LaTeXMLMath Lemma 4.1 .
Two principal bundles both with structure group U LaTeXMLMath are isomorphic if and only if the corresponding associated vector bundles ( with typical fibres LaTeXMLMath ) are isomorphic .
Proof .
If LaTeXMLMath is an isomorphism , then LaTeXMLMath is an isomorphism of the associated vector bundles .
Conversely , let LaTeXMLMath be a unitary mapping LaTeXMLMath then LaTeXMLMath is unitary mapping LaTeXMLMath and there exists a unique LaTeXMLMath such that LaTeXMLMath The mapping LaTeXMLMath is the desired isomorphism .
LaTeXMLMath As already shown , a connection in a principal bundle can be carried over to the associated vector bundle .
If LaTeXMLMath , LaTeXMLMath id , this procedure can be inverted .
Let us take a Hermitian connection in the vector bundle LaTeXMLMath associated with principal bundle LaTeXMLMath .
The connection relates a family of unitary mappings LaTeXMLMath to every piecewise smooth curve LaTeXMLMath on LaTeXMLMath If LaTeXMLMath , LaTeXMLMath then LaTeXMLMath will be the lift in LaTeXMLMath of the curve LaTeXMLMath with the initial point LaTeXMLMath .
We have LaTeXMLMath This lifting prescription determines a unique connection LaTeXMLMath in the principal bundle LaTeXMLMath This correspondence between the connections in LaTeXMLMath and the Hermitian connections in LaTeXMLMath is one–to–one .
Definition 4.2 .
Connections LaTeXMLMath , LaTeXMLMath in principal bundles LaTeXMLMath , LaTeXMLMath respectively , are said to be isomorphic , if there exists an isomorphism LaTeXMLMath which maps connection LaTeXMLMath in connection LaTeXMLMath Hermitian covariant derivatives LaTeXMLMath , LaTeXMLMath in Hermitian vector bundles LaTeXMLMath , LaTeXMLMath respectively , are said to be isomorphic if there exists an isomorphism LaTeXMLMath such that LaTeXMLMath more precisely , LaTeXMLEquation .
Lemma 4.2 .
Connections LaTeXMLMath in principal bundles LaTeXMLMath , LaTeXMLMath respectively , are isomorphic if and only if the corresponding Hermitian covariant derivatives in the associated vector bundles ( with typical fibres LaTeXMLMath ) are isomorphic .
Proof .
According to Lemma 4.1 , to every isomorphism LaTeXMLMath there exists an isomorphism LaTeXMLMath fulfilling LaTeXMLMath and conversely .
If LaTeXMLMath transforms LaTeXMLMath in LaTeXMLMath and if LaTeXMLMath is the lift in LaTeXMLMath of a curve LaTeXMLMath with starting point LaTeXMLMath then LaTeXMLMath is the lift in LaTeXMLMath of the same curve with starting point LaTeXMLMath We have : if LaTeXMLMath then LaTeXMLMath Using the last relation we obtain LaTeXMLMath Conversely , the relation LaTeXMLMath implies LaTeXMLMath and so LaTeXMLMath this means that the isomorphism LaTeXMLMath preserves the connection .
LaTeXMLMath Theorem 4.2 .
Two generalized systems of imprimitivity LaTeXMLMath specified by quadruples LaTeXMLMath , are unitarily equivalent , if and only if the corresponding covariant derivatives in the associated vector bundles are isomorphic .
Proof .
First we have the necessary condition that the Hilbert spaces of representations LaTeXMLMath should have the same dimension , say LaTeXMLMath .
According to the results of Sect .
4.3 we can equivalently take the generalized systems of imprimitivity LaTeXMLMath , constructed in the associated vector bundles and investigate their unitary equivalence .
Thus consider a unitary mapping LaTeXMLMath relating the two generalized systems of imprimitivity .
Let LaTeXMLMath denote the fibre bundle over LaTeXMLMath with each fibre over LaTeXMLMath consisting of unitary mappings LaTeXMLMath .
From the equality LaTeXMLMath it follows that LaTeXMLMath is induced by a measurable section LaTeXMLMath in the bundle LaTeXMLMath .
From the equality LaTeXMLMath and from relation ( 4.6 ) one deduces that LaTeXMLEquation holds for all LaTeXMLMath , all LaTeXMLMath and for almost all ( depending on LaTeXMLMath ) LaTeXMLMath .
Here LaTeXMLMath , LaTeXMLMath , are unitary mappings corresponding to the curve LaTeXMLMath .
We shall show that the measurable section LaTeXMLMath , after proper redefinition on a set of measure zero , is smooth .
It is sufficient to verify this assertion locally , i.e .
to investigate the case LaTeXMLMath .
Let us consider all constant vector fields on LaTeXMLMath , the corresponding flows and LaTeXMLMath .
Thus we have for each LaTeXMLMath and almost all LaTeXMLMath the equality LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
It follows from Fubini ’ s Theorem that , for almost all LaTeXMLMath , the equality ( 4.9 ) holds true for almost all LaTeXMLMath .
We need one such LaTeXMLMath .
Then we can say that for almost all LaTeXMLMath we have LaTeXMLEquation .
Here the right–hand side depends differentiably on LaTeXMLMath and this proves our assertion that the section LaTeXMLMath can be defined in smooth manner .
From the identity ( 4.8 ) LaTeXMLEquation now follows immediately .
The converse part of the proof is easier .
If covariant derivatives LaTeXMLMath , LaTeXMLMath are isomorphic , there exists a smooth section LaTeXMLMath in LaTeXMLMath which defines a unitary maping LaTeXMLMath .
This unitary mapping carries the generalized system of imprimitivity LaTeXMLMath over to LaTeXMLMath .
Then it suffices to notice that ( 4.10 ) implies ( 4.8 ) .
LaTeXMLMath Corollary 4.1 .
Two generalized systems of imprimitivity specified by quadruples LaTeXMLMath , LaTeXMLMath , are equivalent if and only if the connections LaTeXMLMath , LaTeXMLMath are isomorphic .
Notation .
Let us consider a principal bundle LaTeXMLMath with connection LaTeXMLMath , and fix a point LaTeXMLMath .
Then , for every piecewise smooth closed curve LaTeXMLMath on LaTeXMLMath with base point LaTeXMLMath there exists a unique element LaTeXMLMath such that LaTeXMLMath is a starting point and LaTeXMLMath the end point of the lift of curve LaTeXMLMath .
We denote this element by LaTeXMLMath .
Theorem 4.3 .
Let LaTeXMLMath be generalized systems of imprimitivity specified by quadruples LaTeXMLMath , LaTeXMLMath , and let points LaTeXMLMath , LaTeXMLMath , LaTeXMLMath satisfy LaTeXMLMath .
The generalized systems of imprimitivity LaTeXMLMath , LaTeXMLMath are equivalent if and only if there exists a unitary mapping LaTeXMLMath such that for all piecewise smooth closed curves LaTeXMLMath on LaTeXMLMath with the base point LaTeXMLMath , LaTeXMLEquation is valid .
Proof .
We start with two remarks .
Taking LaTeXMLMath instead of LaTeXMLMath , then LaTeXMLMath .
According to Sect .
4.3 , to a principal bundle LaTeXMLMath with connection LaTeXMLMath we can associate a principal bundle LaTeXMLMath with connection LaTeXMLMath ; there also exists a homomorphism LaTeXMLMath preserving the connection and fulfilling LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
The systems of imprimitivity for these two principal bundles are equivalent .
It follows from these two remarks that the choice of a point LaTeXMLMath , plays no role and , moreover , we can restrict our considerations to the case LaTeXMLMath , LaTeXMLMath id .
We are thus considering principal bundles LaTeXMLMath with connections LaTeXMLMath .
It suffices to show that condition ( 4.11 ) is valid if and only if connections LaTeXMLMath are isomorphic .
If LaTeXMLMath is an isomorphism preserving the connection and if we choose LaTeXMLMath , then LaTeXMLMath holds for every closed curve LaTeXMLMath with base point LaTeXMLMath .
Conversely , let LaTeXMLMath for some LaTeXMLMath .
After having substituted LaTeXMLMath for LaTeXMLMath we can suppose LaTeXMLMath .
We define a partial mapping LaTeXMLMath on some subset of LaTeXMLMath into LaTeXMLMath : if LaTeXMLMath , is a piecewise smooth curve in LaTeXMLMath with starting point LaTeXMLMath and if LaTeXMLMath are the lifts of this curve in LaTeXMLMath with starting points LaTeXMLMath , respectively , we put LaTeXMLMath .
Let us investigate the case when two curves LaTeXMLMath , have the same end point LaTeXMLMath .
Let LaTeXMLMath be the lifts of curves LaTeXMLMath with starting points LaTeXMLMath , LaTeXMLMath , respectively .
There exists a unique LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath , be a closed curve with base point LaTeXMLMath , coinciding with LaTeXMLMath for LaTeXMLMath and with LaTeXMLMath for LaTeXMLMath .
Then we find LaTeXMLMath and hence LaTeXMLMath .
Thus function LaTeXMLMath is well–defined .
The domain of LaTeXMLMath consists of all points in LaTeXMLMath which can be connected with LaTeXMLMath by a horizontal curve .
But this domain can be extended to the whole LaTeXMLMath by the relation LaTeXMLMath .
In this way LaTeXMLMath becomes an isomorphism of principal bundles LaTeXMLMath and , by construction , preserves the connection .
LaTeXMLMath Let us consider principal bundle LaTeXMLMath with connection LaTeXMLMath .
We associate a subgroup LaTeXMLMath of LaTeXMLMath to each point LaTeXMLMath consists of all LaTeXMLMath such that the points LaTeXMLMath lie on a common horizontal curve in LaTeXMLMath .
The group LaTeXMLMath is called the holonomy group of the connection LaTeXMLMath with the base point LaTeXMLMath .
It has following properties : If LaTeXMLMath can be connected with LaTeXMLMath by a horizontal curve , then LaTeXMLMath ; LaTeXMLMath .
So all holonomy groups are conjugate subgroups in LaTeXMLMath and we need not specify the base point .
The restricted holonomy group LaTeXMLMath is the subgroup of LaTeXMLMath corresponding to horizontal lifts of those closed curves which are homotopic to LaTeXMLMath .
The groups LaTeXMLMath – being subgroups of LaTeXMLMath – are topological groups .
Theorem 4.4 .
( LaTeXMLCite , Chap .
II.3 , LaTeXMLCite ) .
The restricted holonomy group LaTeXMLMath is a connected Lie group , and it coincides with the arcwise connected component of unity in LaTeXMLMath .
Moreover , the quotient LaTeXMLMath is finite or countable .
The holonomy group LaTeXMLMath itself need not be a Lie group .
But it can be equipped with a new topology which induces the original topology on LaTeXMLMath and the quotient group LaTeXMLMath is discrete .
In this topology one can verify that the inclusion LaTeXMLMath is a homomorphism of Lie groups .
Definition 4.3 .
We say that a structure group LaTeXMLMath of a principal bundle LaTeXMLMath is reducible to a Lie group LaTeXMLMath , if there exists a principal bundle LaTeXMLMath and homomorphism LaTeXMLMath such that LaTeXMLMath with LaTeXMLMath being an injective homomorphism of Lie groups .
Moreover , if bundles LaTeXMLMath are endowed with connections LaTeXMLMath , respectively , and the homomorphism LaTeXMLMath preserves connection , we say that connection LaTeXMLMath is reducible to connection LaTeXMLMath .
Theorem 4.5 .
Let a generalized system of imprimitivity LaTeXMLMath be specified by a quadruple LaTeXMLMath .
If connection LaTeXMLMath is reducible to a connection LaTeXMLMath on a principal bundle LaTeXMLMath and LaTeXMLMath denotes the representation of the Lie group LaTeXMLMath , then the generalized system of imprimitivity specified by the quadruple LaTeXMLMath is equivalent to LaTeXMLMath .
Proof .
The assertion can be proved by a method completely analogous to that used at the end of Sect .
4.3 .
The desired unitary mapping can be constructed in terms of the injective homomorphism LaTeXMLMath .
LaTeXMLMath Theorem 4.6 ( LaTeXMLCite , Chap .
II.6 , LaTeXMLCite ) .
Let LaTeXMLMath be a principal bundle , LaTeXMLMath the holonomy group of connection LaTeXMLMath in LaTeXMLMath .
Then the structure group LaTeXMLMath is reducible to a connection in the reduced principal bundle LaTeXMLMath , the holonomy group of which is identical with LaTeXMLMath .
Theorem 4.7 .
Let LaTeXMLMath be a generalized system of imprimitivity specified by a quadruple LaTeXMLMath be the holonomy group of connection LaTeXMLMath be the restriction of representation LaTeXMLMath to the subgroup LaTeXMLMath .
Then the commuting algebras LaTeXMLMath are isomorphic .
Proof .
In view of Theorems 4.5 and 4.6 it suffices to consider the case LaTeXMLMath .
Let LaTeXMLMath be a bounded operator LaTeXMLMath .
If LaTeXMLMath commutes with all LaTeXMLMath , then , according to Theorem 3.4 , it is of the form LaTeXMLMath , where LaTeXMLMath is some measurable mapping from LaTeXMLMath into the space of operators in LaTeXMLMath such that LaTeXMLMath is true on almost all fibres .
Moreover , if LaTeXMLMath commutes with all LaTeXMLMath , then LaTeXMLMath holds almost everywhere .
We shall show that function LaTeXMLMath can be considered smooth after a redefinition on a set of measure zero .
To show this local property , we can consider LaTeXMLMath .
Then for vectors LaTeXMLMath the mapping LaTeXMLMath remains constant ( almost everywhere ) on horizontal lifts of straight lines in LaTeXMLMath with directions LaTeXMLMath and passing through LaTeXMLMath for almost all LaTeXMLMath .
More precisely , if LaTeXMLMath , then for all LaTeXMLMath and for almost all LaTeXMLMath ( depending on LaTeXMLMath and LaTeXMLMath ) the equality LaTeXMLEquation is valid for each LaTeXMLMath .
Using the Fubini Theorem we find that for almost all LaTeXMLMath , ( 4.12 ) holds for all LaTeXMLMath and almost all LaTeXMLMath .
Let us fix LaTeXMLMath with this property ; we can assume LaTeXMLMath .
Then we construct an auxiliary section LaTeXMLMath in LaTeXMLMath .
For LaTeXMLMath we lift the curve LaTeXMLMath , in the given connection , choosing LaTeXMLMath for the starting point , and we put LaTeXMLMath to be equal to the final point of the lifted curve .
Let LaTeXMLMath .
Then for almost all LaTeXMLMath and all LaTeXMLMath holds .
This proves our assertion that LaTeXMLMath can be considered smooth .
Namely , the mapping LaTeXMLMath is an automorphism of the principal bundle LaTeXMLMath .
So let us suppose that function LaTeXMLMath is smooth and again use the equation LaTeXMLMath , now being valid for all LaTeXMLMath .
We find that the linear mapping LaTeXMLMath , if restricted to the horizontal subspace , is zero .
Hence LaTeXMLMath is constant on horizontal curves in LaTeXMLMath .
Since the holonomy group coincides with the structure group , arbitrary two points in LaTeXMLMath can be connected by a horizontal curve .
Thus LaTeXMLMath for all LaTeXMLMath and we have LaTeXMLMath .
In this way we have associated a unique operator LaTeXMLMath to each LaTeXMLMath .
Conversely , one can relate a unique LaTeXMLMath to each LaTeXMLMath by means of the relation LaTeXMLMath .
The one–to–one correspondence LaTeXMLMath is the desired isomorphism .
LaTeXMLMath Corollary 4.2 .
LaTeXMLMath is irreducible if and only if LaTeXMLMath is irreducible .
Let LaTeXMLMath denote the Lie algebra of the group LaTeXMLMath , u LaTeXMLMath the Lie algebra of the group U LaTeXMLMath ( consisting of skew–Hermitian LaTeXMLMath –matrices ) , and LaTeXMLMath the curvature 2–form of connection LaTeXMLMath .
LaTeXMLMath is a 2–form on LaTeXMLMath taking values in LaTeXMLMath such that LaTeXMLMath holds for all LaTeXMLMath , where LaTeXMLMath .
By composition with the representation LaTeXMLMath of LaTeXMLMath we obtain a 2–form LaTeXMLMath taking values in u LaTeXMLMath .
Under the homomorphism of principal bundles LaTeXMLMath ( see Sect .
4.3 ) LaTeXMLMath is mapped into the curvature form LaTeXMLMath of the connection LaTeXMLMath .
For a vector bundle LaTeXMLMath associated to LaTeXMLMath let End LaTeXMLMath denote a vector bundle over LaTeXMLMath , with fibres ( over LaTeXMLMath ) consisting of linear endomorphisms of fibres ( from LaTeXMLMath into LaTeXMLMath ) .
2–forms LaTeXMLMath on LaTeXMLMath taking values in LaTeXMLMath and satisfying LaTeXMLMath , are in one–to–one correspondence with 2–forms LaTeXMLMath on LaTeXMLMath taking skew–adjoint values in the space of sections LaTeXMLMath ; the correspondence is expressed by the relation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are horizontal lifts of LaTeXMLMath with respect to the connection LaTeXMLMath .
In this way to LaTeXMLMath a 2–form LaTeXMLMath is related and LaTeXMLEquation holds ( LaTeXMLCite , Chap .
III.5 ) .
LaTeXMLMath is the curvature of the covariant derivative LaTeXMLMath .
A simple calculation using ( 4.7 ) then leads to LaTeXMLEquation since LaTeXMLMath .
If LaTeXMLMath ( this is true if and only if LaTeXMLMath ) , we say that the external field vanishes on the manifold .
We see that the field vanishes if and only if LaTeXMLEquation see Sect .
3.2 , eq .
( 3.3 ) .
We recall the mapping LaTeXMLMath introduced in Sect .
4.5 .
If the field vanishes on the manifold , then the value LaTeXMLMath depends only on the homotopy class of the curve LaTeXMLMath .
In this manner we obtain a representation LaTeXMLMath of the fundamental group LaTeXMLMath of manifold LaTeXMLMath .
Let LaTeXMLMath be the universal covering of LaTeXMLMath .
Since LaTeXMLMath , there exists exactly one flat connection LaTeXMLMath on the principal bundle LaTeXMLMath .
Moreover , according to Theorem 4.3 ( Eq .
( 4.11 ) ) the generalized system of imprimitivity LaTeXMLMath specified by the quadruple LaTeXMLMath is equivalent to LaTeXMLMath .
Conversely , since the connection LaTeXMLMath is flat , the field on LaTeXMLMath will vanish for every generalized system of imprimitivity LaTeXMLMath , no matter which representation LaTeXMLMath of LaTeXMLMath is chosen .
So we arrive at a canonical form for generalized systems of imprimitivity ( or quantum Borel kinematics ) with vanishing field ( with flat connection ) .
This form was already studied in detail ( see LaTeXMLCite and references therein ) .
The results of Sections 4.5 and 4.6 imply in this case : Theorem 4.8 .
Two generalized systems of imprimitivity ( or two quantum Borel kinematics ) with vanishing field LaTeXMLMath , are equivalent if and only if the representations LaTeXMLMath , LaTeXMLMath are equivalent .
The commuting algebras LaTeXMLMath are isomorphic .
The generalized systems of imprimitivity described by ( 4.1 ) , ( 4.2 ) are not of the most general form .
Starting from a characterization of Mackey ’ s systems of imprimitivity in terms of cocycles ( LaTeXMLCite , Theorem .
9.11 ) , the following theorem was proved in LaTeXMLCite : Theorem 5.1 .
Any LaTeXMLMath –homogeneous generalized system of imprimitivity on LaTeXMLMath is unitarily equivalent to a canonical one LaTeXMLMath , with LaTeXMLMath for some smooth measure LaTeXMLMath on LaTeXMLMath , LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath , and LaTeXMLEquation for all LaTeXMLMath and all LaTeXMLMath .
Equivalence classes of LaTeXMLMath –homogeneous generalized systems of imprimitivity are in one–to–one correspondence with equivalence classes of cocycles LaTeXMLMath .
Here LaTeXMLMath is a cocycle of LaTeXMLMath relative to the Lebesgue measure class on LaTeXMLMath with values in U ( LaTeXMLMath ) , i.e .
a Borel measurable map LaTeXMLMath with LaTeXMLEquation .
LaTeXMLEquation for almost all LaTeXMLMath and almost all LaTeXMLMath Two cocycles LaTeXMLMath , LaTeXMLMath , are called equivalent ( cohomologous ) , if there is a Borel function LaTeXMLMath , such that for all LaTeXMLMath and LaTeXMLMath , LaTeXMLMath LaTeXMLEquation .
Unfortunately , the classification given in Theorem 5.1 is not easy to handle , since the calculation of cocycles is rather tedious .
To be more specific , one has to impose further conditions on the operators under consideration .
To gain further insight into the structure of the shift operators ( 5.1 ) , one can perform a formal calculation .
In particular , assume for the moment that the cocycles of the representation are smooth maps from LaTeXMLMath into U ( LaTeXMLMath ) .
Then , by formal differentiation of ( 5.1 ) with respect to LaTeXMLMath at LaTeXMLMath , an expression for the generalized momentum operator LaTeXMLMath is obtained , LaTeXMLEquation where LaTeXMLMath and LaTeXMLEquation .
The first two terms on the right hand side of ( 5.2 ) are linear in the vector field LaTeXMLMath .
Though the set of complete vector fields LaTeXMLMath is not a linear space — the sum of two complete vector fields may not be complete — it contains the “ large ” linear subset LaTeXMLMath of vector fields with compact support for which one can demand linearity ( or demand ‘ partial ’ linearity at least for all complete linear combinations of complete vector fields ) , cf .
Theorem 3.1 ) .
Thus as a first additional assumption on LaTeXMLMath we require ( 3.2 ) , i.e .
LaTeXMLMath to be linear in that case .
Using the formal expression ( 5.2 ) , the commutator of LaTeXMLMath and LaTeXMLMath is obviously obtained again in the form ( 3.5 ) .
Finally , for the commutator LaTeXMLMath we obtain LaTeXMLEquation where LaTeXMLEquation .
If LaTeXMLMath were a localized connection 1–form , ( 5.4 ) would represent the local definition of a curvature 2–form LaTeXMLMath of a LaTeXMLMath –bundle over LaTeXMLMath .
The Jacobi identity for generalized momenta would then give us precisely the Bianchi identity LaTeXMLMath , where LaTeXMLMath is the covariant differential defined by the connection .
In order to arrive at this point we had to assume differentiability of the shift operators ( 5.1 ) and of the functions in the domain of momentum operators .
Now there are different ways of defining differentiable structures and thus differentiability on the set LaTeXMLMath ; for a discussion of this point we refer to LaTeXMLCite .
On the other hand , we have already interpreted LaTeXMLMath as a curvature 2–form that is in general related to a connection on a LaTeXMLMath –bundle over LaTeXMLMath .
This line of reasoning leads us to the following definition LaTeXMLCite : Definition 5.1 .
Let LaTeXMLMath be a differentiable manifold , LaTeXMLMath an LaTeXMLMath -homogeneous generalized system of imprimitivity on LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath a differential 2–form on LaTeXMLMath with values in Hermitian operators on LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath denote the corresponding generalized position and momentum operators , respectively .
Then : 1 ) The quadruple LaTeXMLMath is called an ( LaTeXMLMath –compatible ) quantum Borel LaTeXMLMath –kinematics ( QBK LaTeXMLMath ) , if LaTeXMLMath is ( partially ) linear , satisfies ( 5.3 ) , ( 5.4 ) , and the common invariant domain LaTeXMLMath of LaTeXMLMath ’ s and LaTeXMLMath ’ s for LaTeXMLMath and LaTeXMLMath is dense in LaTeXMLMath .
It is local , if LaTeXMLMath is local , and elementary , if LaTeXMLMath .
2 ) A quantum Borel kinematics is called differentiable , if it is equivalent to LaTeXMLMath of standard form , constructed from the following ingredients : Lebesgue measure LaTeXMLMath on LaTeXMLMath ; Hermitian vector bundle LaTeXMLMath over LaTeXMLMath with fibres diffeomorphic to LaTeXMLMath equipped with Hermitian inner product LaTeXMLMath ; A 2–form LaTeXMLMath with skew–adjoint values in the endomorphism bundle LaTeXMLMath ; The Hilbert space LaTeXMLMath is realized as the Hilbert space LaTeXMLMath of sections of LaTeXMLMath , i.e .
( measurable ) mappings LaTeXMLMath such that LaTeXMLMath and with finite norm induced by the inner product LaTeXMLEquation .
The common invariant domain LaTeXMLMath for LaTeXMLMath , LaTeXMLMath contains the set LaTeXMLMath of smooth sections of LaTeXMLMath with compact support and LaTeXMLMath ; The position operators LaTeXMLMath have the usual form of the Schrödinger representation LaTeXMLEquation .
For local differentiable quantum Borel kinematics the formal calculations can be made precise .
According to Sect .
5.2 it only remains to derive the representation of generalized momenta LaTeXMLMath in a standard form .
This is the content of Theorem 5.2 ( LaTeXMLCite ) .
Let LaTeXMLMath be a local differentiable quantum Borel kinematics on LaTeXMLMath in a standard form .
Then there is a Hermitian connection LaTeXMLMath with curvature LaTeXMLMath on LaTeXMLMath , i.e .
a connection compatible with the inner product LaTeXMLEquation a covariantly constant self–adjoint section LaTeXMLMath of End LaTeXMLMath , the bundle of endomorphisms of LaTeXMLMath , such that for all LaTeXMLMath and all LaTeXMLMath LaTeXMLEquation .
Moreover , LaTeXMLMath is a curvature 2–form on LaTeXMLMath satisfying the Bianchi identity LaTeXMLEquation where LaTeXMLMath denotes the covariant differential defined by the connection LaTeXMLMath .
The canonical form of generalized momenta ( 5.2 ) shows that by imposing LaTeXMLMath –compatibility the quantum system on LaTeXMLMath is influenced by an external classical field on LaTeXMLMath .
In general , this field is defined by a curvature 2–form on LaTeXMLMath , and thus a curvature LaTeXMLMath on the associated U LaTeXMLMath –principal bundle LaTeXMLMath .
We could think of this LaTeXMLMath as a classical gauge ( Yang–Mills ) field .
The simplest example was provided in Sect .
4.1 for LaTeXMLMath , LaTeXMLMath , where the 2–form LaTeXMLMath was interpreted as a coupling constant times a magnetostatic field on LaTeXMLMath , with the Bianchi identity corresponding to the Maxwell equation LaTeXMLMath .
Up to a coupling constant the connection LaTeXMLMath generalizes the notion of a vector potential .
For LaTeXMLMath the global connection form LaTeXMLMath corresponds to the vector potential LaTeXMLMath and the 2–form LaTeXMLMath to the magnetic field LaTeXMLMath of Maxwell ’ s theory .
The canonical form given in Theorem 5.2 indicates that a classification of differentiable quantum Borel kinematics amounts to a classification of Hermitian LaTeXMLMath –bundles with connection over LaTeXMLMath and covariantly constant self–adjoint sections of the corresponding endomorphism bundle .
LaTeXMLMath always exists and is covariantly constant and self–adjoint .
This is the content of Theorem 5.3 .
Two local differentiable quantum Borel kinematics LaTeXMLMath , LaTeXMLMath , in canonical form of Theorem 5.2 are equivalent , if and only if there is a strong , unitary , and connection ( and thus curvature ) preserving bundle isomorphism LaTeXMLMath mapping LaTeXMLMath into each other , i.e .
LaTeXMLEquation .
Unfortunately , there are no general existence and classification theorems of Hermitian LaTeXMLMath –bundles with connection .
Looking back to Sect .
4 , there a rather big class of local differentiable QBK LaTeXMLMath ’ s is constructively defined , however with LaTeXMLMath .
Hence even in these cases , the additional classification of covariantly constant self–adjoint sections has to be found as well .
This last problem was solved only in certain special cases — elementary quantum Borel kinematics and type 0 or type U ( 1 ) QBK LaTeXMLMath ’ s — described in the following sections .
The problem of existence and classification of elementary , i.e .
LaTeXMLMath local differentiable quantum Borel kinematics in terms of global geometrical properties ( cohomology groups ) of the underlying manifold LaTeXMLMath was completely solved LaTeXMLCite .
It is based on a theorem LaTeXMLCite concerning existence and classification of complex line bundles with hermitian connection .
Theorem 5.4 Let LaTeXMLMath be a connected differentiable manifold and LaTeXMLMath be a closed 2–form on LaTeXMLMath with LaTeXMLMath .
Then there exists a complex line bundle LaTeXMLMath with hermitian connection LaTeXMLMath of curvature LaTeXMLMath if and only if LaTeXMLMath satisfies the integrality condition LaTeXMLEquation for all closed 2–surfaces LaTeXMLMath in LaTeXMLMath .
In terms of cohomology theory , the de Rham class of LaTeXMLMath has to be integral , LaTeXMLEquation .
Hence non–isomorphic equivalence classes of principal bundles over a manifold LaTeXMLMath with the structure group U ( 1 ) are labeled by elements of the second cohomology group LaTeXMLMath .
The Lie algebra of U ( 1 ) coincides with the imaginary axis LaTeXMLMath .
Since U ( 1 ) is Abelian , the vector bundle End LaTeXMLMath is trivial .
So the curvature LaTeXMLMath is a purely imaginary 2–form on LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is the curvature 2–form of connection LaTeXMLMath .
If we put LaTeXMLEquation .
LaTeXMLMath can be interpreted as the 2–form of external magnetic field on LaTeXMLMath .
For an arbitrary 2–cycle LaTeXMLMath of the singular homology on LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
This leads to the Dirac quantization condition on the magnetic field LaTeXMLEquation where LaTeXMLMath .
We may interpret this result that the 2–cycle LaTeXMLMath — besides the usual magnetic field satisfying LaTeXMLMath — encloses a Dirac magnetic monopole with quantized magnetic charge LaTeXMLMath .
Furthermore , the various inequivalent choices of LaTeXMLMath for fixed curvature LaTeXMLMath are parametrized by LaTeXMLEquation where LaTeXMLMath denotes the group of characters of the fundamental group of LaTeXMLMath .
We should emphasize that LaTeXMLMath classifies pairs of hermitian line bundles LaTeXMLMath and compatible connections LaTeXMLMath .
This implies that the curvature 2–forms of two equivalent complex line bundles with hermitian connection are identical .
The classification of complex line bundles LaTeXMLMath themselves — disregarding their connection — is given by elements of the Čech cohomology Ȟ LaTeXMLMath : two complex line bundles are equivalent if and only if their Chern classes in LaTeXMLMath coincide , i.e .
the curvature 2–forms admissible on these two bundles are in the same integral de Rham cohomology class of LaTeXMLMath .
Theorem 5.4 is the basis of a classification theorem that goes back to LaTeXMLCite for the flat ( LaTeXMLMath ) case and was extended to the case of external fields LaTeXMLMath by LaTeXMLCite : Theorem 5.5 .
The equivalence classes of elementary local differentiable quantum Borel kinematics are in one–to–one correspondence to elements of the set LaTeXMLEquation .
For the proof it remains to classify the inequivalent choices of covariantly constant self–adjoint sections LaTeXMLMath of End LaTeXMLMath .
For a line bundle the endomorphism bundle is actually trivial : As the transition functions LaTeXMLMath of LaTeXMLMath commute with complex numbers , the induced transition functions of End LaTeXMLMath become trivial , LaTeXMLMath , hence End LaTeXMLMath .
Thus the sections of this bundle correspond to complex functions on LaTeXMLMath .
Furthermore , the induced connection on LaTeXMLMath is the trivial connection on LaTeXMLMath given by the Lie derivative .
Thus covariantly constant self–adjoint sections LaTeXMLMath of End LaTeXMLMath are real multiples of the identity , LaTeXMLEquation .
Obviously , LaTeXMLMath is not changed under strong bundle isomorphisms LaTeXMLMath , so each value of LaTeXMLMath for a given Hermitian line bundle determines an inequivalent local differentiable quantum Borel kinematics .
LaTeXMLMath Finally let us note that elementary quantum Borel kinematics with LaTeXMLMath are , in terms of constructions of Sect .
4 , described by generalized systems of imprimitivity LaTeXMLMath specified by the quadruple LaTeXMLMath id ) .
The whole variety of quantizations could be read off the formula ( 5.5 ) .
In order to get a more transparent result we define a QBK LaTeXMLMath of type 0 LaTeXMLCite by LaTeXMLEquation .
Then we obtain an identical formula for LaTeXMLMath as in QBK LaTeXMLMath LaTeXMLCite : LaTeXMLEquation .
As proved in LaTeXMLCite , on every smooth manifold LaTeXMLMath there exists a differentiable QBK LaTeXMLMath of type LaTeXMLMath .
Let us note that for LaTeXMLMath , the type LaTeXMLMath QBK LaTeXMLMath ’ s classify all possible Borel quantizations LaTeXMLCite ; this is not the case , however , for LaTeXMLMath .
Finally , a complete classification of QBK LaTeXMLMath ’ s of type LaTeXMLMath was possible in the case of flat connection ( LaTeXMLMath ) LaTeXMLCite .
It turns out that it is essentially the question of the topology of LaTeXMLMath .
The corresponding investigations can be summarized in Theorem 5.6 The set of classes of unitarily equivalent QBK LaTeXMLMath ’ s of type LaTeXMLMath with flat connection on LaTeXMLMath can be bijectively mapped onto the set of pairs LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath denotes the isomorphism class of flat LaTeXMLMath -bundles over LaTeXMLMath .
Since there is a one-to-one correspondence between the isomorphism classes of flat LaTeXMLMath -bundles over LaTeXMLMath and flat U LaTeXMLMath -principal bundles over LaTeXMLMath , we can use Milnor ’ s Lemma LaTeXMLCite .
A U LaTeXMLMath -principal bundle over LaTeXMLMath admits a flat connection if and only if it is induced from the universal covering bundle of LaTeXMLMath by a homomorphism of the fundamental group LaTeXMLMath into U LaTeXMLMath .
Thus , disregarding the real constant LaTeXMLMath , the set of inequivalent quantizations of type LaTeXMLMath with LaTeXMLMath –valued wave functions is isomorphic to the set LaTeXMLEquation of ( the conjugacy classes of ) LaTeXMLMath -dimensional unitary representations of the fundamental group of LaTeXMLMath .
In the case LaTeXMLMath , i.e .
of quantizations with complex–valued wave functions , the topological part of the classification reduces to LaTeXMLMath , i.e .
to the set of one-dimensional unitary representations of LaTeXMLMath LaTeXMLCite .
LaTeXMLCite and in geometric quantization LaTeXMLCite .
Since the commutator subgroup LaTeXMLMath ( generated by elements LaTeXMLMath ) belongs to the kernels of all such one-dimensional representations and since the singular homology group LaTeXMLMath is isomorphic to LaTeXMLMath ( the Hurewicz isomorphism ) , inequivalent QBK LaTeXMLMath ’ s are labeled by elements of the character group of LaTeXMLMath .
Finally , let us describe the general structure of the Abelian group LaTeXMLMath for compact LaTeXMLMath .
It has a decomposition LaTeXMLMath , where the free Abelian group LaTeXMLMath is LaTeXMLMath ( LaTeXMLMath terms ) , with LaTeXMLMath being the first Betti number of LaTeXMLMath , and the torsion Abelian group is LaTeXMLMath with LaTeXMLMath being cyclic groups of orders LaTeXMLMath ( torsion coefficients ) such that LaTeXMLMath positive integer .
Thus the characters of LaTeXMLMath can be parametrized by LaTeXMLMath -tuples LaTeXMLEquation with the numbers LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , classifying inequivalent quantum Borel 1–kinematics on LaTeXMLMath .
It is remarkable that elementary quantum Borel kinematics with vanishing external magnetic field find application in quantum mechanics ( Aharonov-Bohm effect LaTeXMLCite ) .
According to Sect .
4.3 they can be labeled by one–dimensional unitary representations of the fundamental group LaTeXMLMath , and consequently , as already explained in Sect .
4.9 , inequivalent systems of imprimitivity with vanishing magnetic field are labeled by elements of the character group of LaTeXMLMath .
Quantum Borel kinematics in the case of trivial fibration LaTeXMLMath was studied in detail in LaTeXMLCite .
In this case a localized connection 1–form LaTeXMLMath can be defined globally on the whole manifold LaTeXMLMath ; the closed 1–form LaTeXMLMath represents the vector potential of the vanishing magnetic field .
A covariant derivative on LaTeXMLMath has the form LaTeXMLEquation .
Two such covariant derivatives LaTeXMLMath , LaTeXMLMath are isomorphic if and only if there exists a function LaTeXMLMath such that LaTeXMLEquation .
Following the terminology of LaTeXMLCite , we say that the 1–forms LaTeXMLMath , LaTeXMLMath are logarithmically cohomologous ; LaTeXMLMath is said to be logarithmically exact .
A 1–form LaTeXMLMath is logarithmically exact if and only if LaTeXMLEquation holds for all 1–cycles LaTeXMLMath of the singular homology .
If LaTeXMLMath is a periodic cycle , i.e .
LaTeXMLMath for some LaTeXMLMath and 2-cycle LaTeXMLMath , then LaTeXMLEquation because a logarithmically exact form is closed .
So it suffices to check independent , non–periodic 1–cycles .
Their number is LaTeXMLMath , the first Betti number of manifold LaTeXMLMath .
Now , quantum Borel kinematics on LaTeXMLMath in the case of trivial fibration and of vanishing magnetic field is determined by a closed 1-form LaTeXMLMath .
Its cohomology class is in turn — according to the de Rham Theorem — determined by LaTeXMLMath periods LaTeXMLEquation .
For each LaTeXMLMath , LaTeXMLMath represents an external magnetic flux outside the manifold LaTeXMLMath and passing through the LaTeXMLMath -th independent cycle LaTeXMLMath .
Two potentials LaTeXMLMath , LaTeXMLMath determine the same quantum Borel kinematics if LaTeXMLEquation .
We conclude that in the considered special case the family of all inequivalent quantum Borel kinematics can be parametrized by elements LaTeXMLEquation .
In the examples we concentrate on elementary quantum Borel kinematics .
Example 5.1 .
LaTeXMLMath .
Since LaTeXMLMath , only the trivial principal bundle exists over LaTeXMLMath .
Every smooth function LaTeXMLMath U ( 1 ) can be written in the form LaTeXMLMath with LaTeXMLMath being a smooth real function .
Two 1–forms are logarithmically cohomologous if and only if they are cohomologous .
Thus ( an equivalence class of ) quantum kinematics is obtained by taking ( a cohomology class of ) a closed 2–form LaTeXMLMath of magnetic field .
There exists exactly one equivalence class of quantum Borel kinematics ( i.e .
with vanishing magnetic field ) — the standard one with LaTeXMLMath .
Example 5.2 .
LaTeXMLMath is the 3-dimensional real space LaTeXMLMath with the LaTeXMLMath -axis excluded ( Aharonov-Bohm configuration ) .
Then LaTeXMLMath , and there exists only the trivial principal bundle over LaTeXMLMath .
There exists exactly one ( up to homology ) independent cycle in LaTeXMLMath and it is non-periodic , i.e .
LaTeXMLMath .
Inequivalent quantum Borel kinematics are labeled by LaTeXMLMath U ( 1 ) , LaTeXMLMath ; LaTeXMLMath denotes the magnetic flux supported by the excluded line LaTeXMLMath .
The corresponding vector potential 1–form LaTeXMLMath can be chosen in the form LaTeXMLEquation .
From the point of view of quantum mechanics a charged particle can not distinguish between the flux LaTeXMLMath , LaTeXMLMath , and the zero one .
In fact , this is the effect discovered by Y.Aharonov and D.Bohm LaTeXMLCite .
Let us note that the Aharonov–Bohm effect in the presence of two solenoids or LaTeXMLMath solenoids placed along a straight line was studied in LaTeXMLCite .
Example 5.3 .
LaTeXMLMath be a compact orientable surface .
It is known that , up to diffeomorphism , such LaTeXMLMath are classified by the Euler characteristic LaTeXMLMath , where LaTeXMLMath is the genus of LaTeXMLMath .
For given LaTeXMLMath , we denote LaTeXMLMath ; it is modeled by a 2-sphere with LaTeXMLMath handles ( LaTeXMLCite , Chap .
9.3 ) .
Since LaTeXMLMath there exist countably many principal bundles over LaTeXMLMath labeled by LaTeXMLMath According to our physical interpretation , LaTeXMLMath can by related to the magnetic charge of a Dirac monopole enclosed by the closed surface , LaTeXMLMath Further , LaTeXMLMath thus inequivalent quantum Borel kinematics ( with vanishing magnetic field ) are labeled by elements of the dual group LaTeXMLMath Each quantum Borel kinematics depends on LaTeXMLMath external magnetic fluxes , each handle carrying two of them .
On a 2–torus , generalized quantum kinematics was studied in LaTeXMLCite , quantum Borel kinematics in LaTeXMLCite .
Example 5.4 .
LaTeXMLMath is a real projective space , LaTeXMLMath We can identify : LaTeXMLEquation .
LaTeXMLEquation Hence there exist two inequivalent principal bundles over LaTeXMLMath and two inequivalent QBK ’ s in mutually inequivalent fibrations .
The QBK ’ s can be explicitly described in the following way : the Hilbert spaces LaTeXMLMath are chosen as subspaces in LaTeXMLMath ( with measure LaTeXMLMath invariant under the transformation LaTeXMLMath ) , LaTeXMLMath if and only if LaTeXMLMath the two inequivalent systems of imprimitivity ( LaTeXMLMath ) are defined for both signs + and – by the operators LaTeXMLEquation acting in LaTeXMLMath and LaTeXMLMath , respectively .
The real projective space LaTeXMLMath appears in quantum mechanics e.g .
as a ( topologically non–trivial ) part of the effective configuration space of two indistinguishable point–like particles localized in the LaTeXMLMath –dimensional Euclidean space LaTeXMLMath The two cases with signs + and – correspond in quantum mechanics to the cases of bosonic and fermionic statistics , respectively .
More details can be found in LaTeXMLCite ; see also LaTeXMLCite .
It should be stressed that the case LaTeXMLMath presents unexpected features : in LaTeXMLCite it was found for a system of two particles in two dimensions that there is a continuous family of quantizations describing new statistics which interpolate between fermions and bosons .
These anomalous or fractional statistics were later discovered independently by LaTeXMLCite and by LaTeXMLCite , who actually coined the term ‘ anyons ’ for the corresponding particles .
Topological quantum numbers 2 distinguishable particles in LaTeXMLMath 2 indistinguishable particles in LaTeXMLMath Particle on orientable surface of genus LaTeXMLMath LaTeXMLMath LaTeXMLMath ACKNOWLEDGEMENTS J. T. and P. Š. are grateful to Prof. Dr. H.D .
Doebner for the kind hospitality extended to them on various occasions at the Arnold Sommerfeld Institute in Clausthal .
The authors thank A. Bóna for typing the manuscript .
Partial support of the Grant Agency of Czech Republic ( contract No .
202/96/0218 ) is acknowledged .
The list of references is by no means complete and we apologize to the authors of papers which have not been included .
It was pointed out ( [ 5 ] ) that the space of hermitian triples is an analogy of the hermitian connection space .
Generalizing the Ashtekar — Isham procedure one can quantize the space of hermitian triples as well as the original one .
Here we add an example how this similarity can be exploited in a quantum theory of riemannian geometry .
Let LaTeXMLMath be an orientable compact smooth riemannian 4 - dimensional manifold .
Consider the space of all triples LaTeXMLEquation where LaTeXMLMath is a riemannian metric , LaTeXMLMath is a compatible almost complex structure and LaTeXMLMath is the corresponding almost kaehler form .
This space naturally splits with respect to a discrete parametrization such that every connected component can be described as follows LaTeXMLEquation where LaTeXMLMath is the canonical class of LaTeXMLMath .
All details of the local geometry can be found in [ 5 ] , [ 6 ] .
For the hermitian connection space it was constructed a quantization programme ( see f.e .
[ 2 ] ) , dealing with based loops on 3 - dimensional based manifold LaTeXMLMath .
One can generalize this procedure for the space of hermitian triples over the product LaTeXMLMath where for every loop LaTeXMLMath on LaTeXMLMath we have the corresponding 2 - dimensional torus LaTeXMLMath on LaTeXMLMath ( or if one uses graphs for the definition as in [ 1 ] then one gets a Rieman surface ) .
Thus one can arrange a complex bounded function on the space LaTeXMLMath associated with each loop LaTeXMLMath and then repeat the construction of the corresponding LaTeXMLMath - algebra .
As well one can introduce the notion of cylindrical functions and generate a regular measure on the compactified space LaTeXMLMath .
Extending this approach to the framework of quantum theory of riemannian geometry ( see [ 1 ] ) one can exploit ” duality ” between two spaces in some special abelian case .
In this case we take a 3 - dimensional orientable compact smooth riemannian manifold as the based manifold and a complex line bundle LaTeXMLMath .
The configuration space is the compactified space of abelian hermitian connection LaTeXMLMath ( in notations of [ 1 ] ) but one takes the product LaTeXMLEquation as the phase space of the system .
Here LaTeXMLMath is a subspace of the space LaTeXMLMath of all hermitian triples over the product manifold LaTeXMLMath consists of all triples invariant under all LaTeXMLMath - rotation .
The key point is that the product space LaTeXMLEquation is a Poisson manifold thus one can say that in some weak sense the space LaTeXMLMath is dual to LaTeXMLMath .
Using this Poisson structure one could come in the same way as it is proposed for the cotangent bundle LaTeXMLMath ( see section 3A in [ 1 ] ) .
Moreover since the quantizations of both spaces depend on the same loops LaTeXMLMath in LaTeXMLMath one can introduce the notions of cylindrical function simultaneously on both spaces and then gets well defined Poisson brackets on the space of cylindrical functions on the product space .
In this section we follow [ 2 ] .
Let LaTeXMLMath be a real smooth connected orientable 3 - manifold .
On the principal LaTeXMLMath - bundle one has the space of hermitian connections denoted as LaTeXMLMath with natural gauge group LaTeXMLMath action .
Fixing a point LaTeXMLMath consider the space of based loops denoted as LaTeXMLMath which admits a group structure .
A natural equivalence relation on LaTeXMLMath is given by the requirement LaTeXMLEquation for every connection LaTeXMLMath where LaTeXMLMath is the corresponding holonomy .
The collection of the equivalence classes is denoted as LaTeXMLMath .
Every element LaTeXMLMath defines a function on the quotient space LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath represents class LaTeXMLMath and LaTeXMLMath represents class LaTeXMLMath .
Due to LaTeXMLMath trace identities one establishes that the set of such functions is closed under multiplication thus one gets a LaTeXMLMath - algebra LaTeXMLMath , consisting of all finite linear combination of LaTeXMLMath with complex coefficients .
This one is a subalgebra in LaTeXMLMath - algebra of all complex valued bounded continuous functions on LaTeXMLMath and one can take the completion LaTeXMLMath of LaTeXMLMath under the supremum norm in the ambient LaTeXMLMath - algebra .
This one is called the Ashtekar — Isham LaTeXMLMath - algebra .
Briefly speaking the Gelfand spectrum of LaTeXMLMath is a compactification of the quotient space LaTeXMLMath ; the Hilbert space of the quantized theory is represented by LaTeXMLMath where LaTeXMLMath is a regular diffeomorphism invariant measure on the compactified quotient space .
The construction of such a measure is a crucial step in the programme .
The definition uses the notion of cylindrical functions — special functions which form a dense subset in LaTeXMLMath .
Namely for a number of loops LaTeXMLMath consider the following projection LaTeXMLEquation defined by the holonomies around the loops .
Then for every regular function on the target space one takes the corresponding lifting to the quotient space — it gives a cylindrical function on the last one .
As well one lifts the product Haar measure to LaTeXMLMath ; taking the limit one gets a regular LaTeXMLMath - invariant measure on the compactified space .
Moreover this measure corresponds to an invariant of framed knots in LaTeXMLMath ( see [ 2 ] , [ 3 ] ) .
Let us adopt this construction to the case of hermitian triples .
Instead of sufficiently wide generalization made in [ 5 ] now we ’ d like to consider only a special case .
Namely let LaTeXMLMath be as above ; consider the product manifold LaTeXMLEquation and let an orientation on LaTeXMLMath is fixed as well as on LaTeXMLMath .
Then we can consider the space LaTeXMLMath with a fixed topological datum LaTeXMLMath .
For every loop LaTeXMLMath one has the function LaTeXMLEquation defined as follows .
Let LaTeXMLMath be the torus in LaTeXMLMath corresponds to LaTeXMLMath : LaTeXMLEquation .
This torus inherits its own orientation from a parametrization of LaTeXMLMath .
We take LaTeXMLEquation where LaTeXMLMath equals to LaTeXMLMath if the own orientation of LaTeXMLMath is compatible to the fixed orientation on LaTeXMLMath and LaTeXMLMath otherwise .
It ’ s clear that LaTeXMLMath is a bounded continuous complex function on LaTeXMLMath .
Thus again we can make all steps getting the corresponding LaTeXMLMath - algebra LaTeXMLMath with Gelfand spectrum denoted as LaTeXMLMath .
But in what follows we need just a subspace in LaTeXMLMath corresponds to such triples LaTeXMLMath which are invariant under all LaTeXMLMath - rotations defined on LaTeXMLMath due to its product structure .
This subspace we denote as LaTeXMLMath .
It depends only on the based manifold LaTeXMLMath and can be described without any references to the product manifold LaTeXMLMath ( see [ 7 ] ) .
Again using the quantization procedure we get the corresponding compactification LaTeXMLMath .
As above we can define the notion of cylindrical functions on LaTeXMLMath .
The definition will be the same : for each set of loops LaTeXMLMath one has a projection LaTeXMLEquation .
In terms of this projection we again define the cylindrical functions and the corresponding regular measure .
The limit gives what we need .
All details can be found in [ 7 ] .
In a classical setup the hermitian connection space is the configuration space .
At the same time one takes the cotangent bundle LaTeXMLMath as the phase space .
It is endowed with natural Poisson brackets but this brackets can not be extended to a sufficiently wide quantization procedure in the Ashtekar — Isham framework ( see section 3 A in [ 1 ] ) .
We ’ d like to propose an appropriate way avoiding the difficulties in the abelian case .
We claim that in this case the space of hermitian triples is almost dual to the hermitian connection space over a 3 - dimensional based manifold .
Let LaTeXMLMath be as above 3 - dimensional based manifold .
Consider a complex line bundle LaTeXMLMath over LaTeXMLMath with a fixed hermitian structure .
Let us fix an appropriate element LaTeXMLMath corresponding to an almost complex structure on LaTeXMLMath .
We take the induced component LaTeXMLMath and consider the direct product LaTeXMLEquation .
The space LaTeXMLMath is a Poisson manifold .
( see [ 8 ] for the definition and properties of Poisson manifolds and [ 7 ] for detailed explanations ) .
The key point is that in each point LaTeXMLMath the tangent space is isomorphic to LaTeXMLEquation and this representation is locally trivial .
Thus for any point of LaTeXMLMath the tangent space is represented as LaTeXMLEquation ( we rescale all vectors from the first summand by LaTeXMLMath ) .
For a pair of smooth functions LaTeXMLMath let LaTeXMLMath be components belong to LaTeXMLMath - summands in the dual decomposition LaTeXMLEquation .
The Poisson structure is given by formula LaTeXMLEquation .
In local coordinate LaTeXMLMath where LaTeXMLMath belongs to the first summand in the decomposition above LaTeXMLMath to the second and LaTeXMLMath to the third the Poisson structure has absolutely classical form LaTeXMLEquation while the functions depend only on LaTeXMLMath form the center of the corresponding Poisson algebra .
Now we can quantize the picture using the same loops for both components in LaTeXMLMath .
We introduce the spaces of cylindrical functions for both LaTeXMLMath and LaTeXMLMath denoting the first one as LaTeXMLMath and the second one as LaTeXMLMath .
Then one has the big space LaTeXMLMath endowed with the product regular measure .
The corresponding limit gives us an universal space LaTeXMLMath on LaTeXMLMath which is the product of quantized spaces LaTeXMLEquation and we have the corresponding LaTeXMLMath - invariant measure on the product space .
The point is that on this space we have the extended Poisson brackets LaTeXMLEquation satisfying obviously the distinguishing properties namely LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath doesn ’ t depend on the second ” variable ” LaTeXMLMath being a pure function on LaTeXMLMath and LaTeXMLMath depends only on the second ” variable ” .
Consider the mixed situation when one takes a cylindrical function LaTeXMLMath and a cylindrical function LaTeXMLMath pairing these functions .
Let LaTeXMLMath is defined by a set of loops LaTeXMLMath and LaTeXMLMath uses a set LaTeXMLMath .
In the total space LaTeXMLMath we have a component corresponds to the union set LaTeXMLMath .
Since the Poisson structure defined above is a constant Poisson structure then the Poisson brackets LaTeXMLMath should belong to the component in LaTeXMLMath .
Moreover one can consider the second type functions defined on LaTeXMLMath as derivations of the original space LaTeXMLMath .
Really if we denote the corresponding LaTeXMLMath - invariant measure on the compactified LaTeXMLMath as LaTeXMLMath then the action LaTeXMLEquation is correctly defined and satisfies the Leibnitz rule thus it can be used in quantization constructions .
This action on cylindrical functions gives cylindrical functions again but changes a ” grading ” because LaTeXMLMath is defined via the extended loop set LaTeXMLMath rather then the original function LaTeXMLMath with the loop set LaTeXMLMath .
As in [ 1 ] the construction easily extends to the situation with graphs instead of loop sets used above .
The details are contained in [ 7 ] .
As it was pointed out in [ 5 ] one has a twisted version of the construction above .
In the case above our ingredients in the direct product LaTeXMLEquation are independent ” variables ” .
But one can twist the picture imposing for example the following condition .
Every hermitian triple LaTeXMLMath from LaTeXMLMath defines a hermitian structure on a real rank 2 subbundle of LaTeXMLMath ( see [ 5 ] , [ 7 ] ) .
So each point LaTeXMLMath gives a complex line bundle with a fixed hermitian structure .
Let us take the corresponding hermitian connection space LaTeXMLMath as the fiber getting globally topologically nontrivial bundle LaTeXMLEquation with LaTeXMLMath - fibers .
One could try to exploit a Poisson structure on LaTeXMLMath familiar with the original one described above .
But there exists an alternative way .
The point is that LaTeXMLMath looks like an even super symplectic manifold ( the definition and properties can be found in [ 4 ] ) .
The based space LaTeXMLMath has a symplectic form defined as follows .
Let us recall from above that the tangent space in each point is isomorphic to the direct sum LaTeXMLEquation .
Thus one has a constant symplectic form defined by LaTeXMLEquation .
The fiber of LaTeXMLMath is an affine space over a real vector space endowed with inner product ( over point LaTeXMLMath the riemannian metric LaTeXMLMath should be exploited for the definition ) .
For the even super symplectic structure it remains to take an appropriate affine connection on LaTeXMLMath .
Thus in the construction one can consider over LaTeXMLMath the corresponding super brackets instead of the classical one .
On the other hand the example has been discussed in the present article looks like too special .
The construction hasn ’ t been extended to a nonabelian case .
The crucial point for a virtual extension is that the case of hermitian triples is ” abelian ” in some sense .
One could expect that a nontrivial generalization of the example lies in the framework of lorentzian geometry when one considers triples consist of lorentzian metrics and ” compatible ” almost complex structures .
Anyway the formula of the Poisson structure LaTeXMLMath on the product space LaTeXMLMath can be easily generalized to the case when LaTeXMLMath is the connection space on LaTeXMLMath - bundle over LaTeXMLMath but this generalization doesn ’ t look like a gauge invariant one being the result of pure formal approach so we finish this example hoping that more geometric approach will be found in a future .
At the end I ’ d like to thank Korean Institute for Advanced Study ( Seoul ) for hospitality and good working condition .
A few years ago Foda , Quano , Kirillov and Warnaar proposed and proved various finite analogs of the celebrated Andrews-Gordon identities .
In this paper we use these polynomial identities along with the combinatorial techniques introduced in our recent paper to derive Garrett , Ismail , Stanton type formulas for two variants of the Andrews-Gordon identities .
In 1961 , Gordon LaTeXMLCite found a natural generalization of the Rogers-Ramanujan partition theorem .
( Gordon ) For all LaTeXMLMath , LaTeXMLMath , the partitions of LaTeXMLMath of the frequency form LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , LaTeXMLMath ( for all LaTeXMLMath ) are equinumerous with the partitions of LaTeXMLMath into parts not congruent to LaTeXMLMath or LaTeXMLMath modulo LaTeXMLMath .
Thirteen years later , Andrews LaTeXMLCite proposed and proved the following analytic counterpart to Gordon ’ s theorem : ( Andrews ) For all LaTeXMLMath as in Theorem LaTeXMLRef , and LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation We note that the last equality in ( LaTeXMLRef ) follows from Jacobi ’ s triple product identity LaTeXMLCite .
Subsequently , Bressoud LaTeXMLCite interpreted the l.h.s .
of ( LaTeXMLRef ) in terms of weighted lattice paths .
Bressoud path is made of three basic steps ( see Fig .
LaTeXMLRef ) : LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath .
To calculate the weight of Bressoud path we define the peak of a path as a vertex preceded by the NE step and followed by the SE step .
The height of a peak is its LaTeXMLMath -coordinate , the weight of a peak is its LaTeXMLMath -coordinate .
The weight LaTeXMLMath of path LaTeXMLMath is defined as the sum of the weights of its peaks .
In the above example ( Fig .
LaTeXMLRef ) , the path has five peaks : LaTeXMLMath and its weight is LaTeXMLMath .
The relative height of a peak LaTeXMLMath is the largest positive integer LaTeXMLMath , for which we can find two vertices on the path : LaTeXMLMath such that LaTeXMLMath and such that between these two vertices there are no peaks of height LaTeXMLMath and every peak of height LaTeXMLMath has weight LaTeXMLMath .
The peaks in the above example ( Fig .
LaTeXMLRef ) have relative heights LaTeXMLMath , respectively .
We can now state Bressoud ’ s result LaTeXMLCite .
( Bressoud ) For LaTeXMLMath as in Theorem LaTeXMLRef and LaTeXMLMath , LaTeXMLMath LaTeXMLEquation where LaTeXMLEquation and LaTeXMLMath denotes a collection of all Bressoud lattice paths that start at LaTeXMLMath and end at LaTeXMLMath , which have no peaks higher than LaTeXMLMath , and , in addition , the number of peaks of relative height LaTeXMLMath is LaTeXMLMath for LaTeXMLMath .
Actually , it is straightforward to refine Bressoud ’ s analysis to show that LaTeXMLEquation where LaTeXMLEquation and the LaTeXMLMath -binomial coefficients are defined as LaTeXMLEquation .
This leads to LaTeXMLEquation where for LaTeXMLMath , LaTeXMLMath LaTeXMLEquation and LaTeXMLEquation .
Here and in the following , summation over LaTeXMLMath is over all non-negative integer tuples LaTeXMLMath .
We note that one can extract identity ( LaTeXMLRef ) from Lemma 4 in LaTeXMLCite with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , if one recognizes that , in this case , the somewhat non-trivial conditions imposed on the peaks therein are equivalent to our concise statement that all paths end at LaTeXMLMath .
Making use of ( LaTeXMLRef ) , one easily derives the following recursion relations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and verifies the initial conditions LaTeXMLEquation where the Kronecker delta function LaTeXMLMath is defined , as usual , as LaTeXMLEquation .
We note that formulas ( LaTeXMLRef ) – ( LaTeXMLRef ) specify the polynomials LaTeXMLMath uniquely .
Next , for LaTeXMLMath , we define polynomials LaTeXMLMath as LaTeXMLEquation .
LaTeXMLEquation Employing the standard LaTeXMLMath -binomial recurrences LaTeXMLCite one finds that LaTeXMLEquation .
It is not difficult to check that LaTeXMLEquation .
LaTeXMLEquation and for LaTeXMLMath LaTeXMLEquation .
The formulas ( LaTeXMLRef ) – ( LaTeXMLRef ) and ( LaTeXMLRef ) – ( LaTeXMLRef ) imply that LaTeXMLEquation .
Indeed , both sides of ( LaTeXMLRef ) satisfy identical recurrences and initial conditions .
Combining ( LaTeXMLRef ) and ( LaTeXMLRef ) we arrive at ( Foda , Quano and Kirillov ) LaTeXMLEquation .
LaTeXMLEquation The above theorem was first proven in LaTeXMLCite and LaTeXMLCite in a somewhat different fashion .
Using the following limiting formulas LaTeXMLEquation .
LaTeXMLEquation it is easy to check that in the limit LaTeXMLMath , Theorem LaTeXMLRef reduces to Theorem LaTeXMLRef .
Recently , motivated by LaTeXMLCite , we investigated in LaTeXMLCite the following multisums LaTeXMLEquation with LaTeXMLMath , LaTeXMLMath .
The combinatorial analysis of ( LaTeXMLRef ) with LaTeXMLMath given in LaTeXMLCite can be upgraded to the more general case LaTeXMLMath as summarized in three steps below .
First step .
We observe that LaTeXMLEquation where the path LaTeXMLMath is obtained from the path LaTeXMLMath by moving LaTeXMLMath by LaTeXMLMath units to the left along the LaTeXMLMath -axis .
Next , using ( LaTeXMLRef ) and ( LaTeXMLRef ) we derive that LaTeXMLEquation .
LaTeXMLEquation Second step .
For LaTeXMLMath every path LaTeXMLMath consists of two pieces joined together at some point LaTeXMLMath with LaTeXMLMath .
The first piece belongs to LaTeXMLMath and the second one to LaTeXMLMath .
This observation is equivalent to LaTeXMLEquation .
Third step .
LaTeXMLEquation .
Combining ( LaTeXMLRef ) – ( LaTeXMLRef ) one obtains LaTeXMLEquation .
LaTeXMLEquation Finally , letting LaTeXMLMath tend to infinity , we find with the aid of ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) our first variant of the Andrews-Gordon identities LaTeXMLEquation .
Formula ( LaTeXMLRef ) is a special case of ( 3.21 ) in LaTeXMLCite with LaTeXMLMath .
The other cases there can be treated in a completely analogous manner .
Actually , neither the polynomial analogs ( LaTeXMLRef ) of ( LaTeXMLRef ) , nor the path interpretation ( LaTeXMLRef ) of the Andrews-Gordon identities are unique .
In particular , in LaTeXMLCite Warnaar considered the path space that is based on Gordon frequency conditions in Theorem LaTeXMLRef , with an additional constraint that LaTeXMLMath for LaTeXMLMath .
This led him to the new polynomial versions of the Andrews-Gordon identities .
In the next section of this paper we will make essential use of Warnaar ’ s analysis to investigate the following multisums LaTeXMLEquation .
The rest of this article is organized as follows .
In Section LaTeXMLRef , we briefly discuss LaTeXMLMath -multinomial coefficients and Warnaar ’ s terminating versions of the Andrews-Gordon identities .
In Section LaTeXMLRef , we review a particle interpretation of Gordon ’ s frequency conditions given in LaTeXMLCite .
In Section LaTeXMLRef , by following easy steps being similar to three steps above , we derive our main formulas for ( LaTeXMLRef ) and their finite analogs .
We conclude with a short description of prospects for future work opened by this investigation .
We start by recalling the binomial theorem LaTeXMLEquation where LaTeXMLMath is the usual binomial coefficient .
By analogy , we introduce multinomial coefficients LaTeXMLMath for LaTeXMLMath as the coefficients in the expansion LaTeXMLEquation .
Multiple use of ( LaTeXMLRef ) yields an explicit sum representation LaTeXMLEquation .
Building on the work of Andrews LaTeXMLCite , Schilling LaTeXMLCite and Warnaar LaTeXMLCite have introduced the following LaTeXMLMath -analogs of ( LaTeXMLRef ) LaTeXMLEquation for LaTeXMLMath .
We list some important properties of LaTeXMLMath -multinomials ( LaTeXMLRef ) , which have been proven in LaTeXMLCite and LaTeXMLCite .
Symmetries : LaTeXMLEquation and LaTeXMLEquation .
Recurrences : LaTeXMLEquation .
LaTeXMLMath -Deformed tautologies : LaTeXMLEquation where LaTeXMLMath and LaTeXMLEquation .
Limiting behavior : For LaTeXMLMath LaTeXMLEquation .
Special values : LaTeXMLEquation and LaTeXMLEquation .
Next , for each LaTeXMLMath we consider the ordered sequences of integers LaTeXMLMath subject to Gordon ’ s conditions LaTeXMLEquation .
Each of these Gordon sequences can be represented graphically by a lattice path as illustrated by the example shown in Fig .
LaTeXMLRef .
The weight LaTeXMLMath of Gordon path LaTeXMLMath described above is LaTeXMLEquation .
For LaTeXMLMath , LaTeXMLMath , LaTeXMLMath we perform a weighted path count with the help of the following polynomials LaTeXMLEquation where LaTeXMLMath denotes the space of all Gordon paths subject to ( LaTeXMLRef ) that start at LaTeXMLMath and end at LaTeXMLMath .
In LaTeXMLCite , Warnaar , building on the work of Andrews and Baxter LaTeXMLCite , showed in a recursive fashion that for LaTeXMLMath , LaTeXMLMath LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation More specifically , it was proven in LaTeXMLCite that both sides of ( LaTeXMLRef ) satisfy the same recurrences LaTeXMLEquation and the same initial conditions LaTeXMLEquation .
Also in LaTeXMLCite , a particle interpretation of Gordon paths was given and , as a result , another representation for LaTeXMLMath was obtained ; namely , LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLEquation .
Comparing ( LaTeXMLRef ) and ( LaTeXMLRef ) we arrive at the polynomial identities LaTeXMLEquation which in the limit LaTeXMLMath reduce to the Andrews-Gordon identities ( LaTeXMLRef ) .
It is important to realize that while ( LaTeXMLRef ) and ( LaTeXMLRef ) are identical in the limit LaTeXMLMath , these identities are substantially different for finite LaTeXMLMath .
It should be noted that ( LaTeXMLRef ) with LaTeXMLMath is a corollary of Bressoud ’ s Lemma 3 in LaTeXMLCite .
It appears that this Lemma , as stated , is true for LaTeXMLMath only .
For LaTeXMLMath , the generating function LaTeXMLMath therein should be corrected to LaTeXMLMath .
We begin our particle description of Gordon lattice paths by considering first paths in LaTeXMLMath .
Following LaTeXMLCite , we introduce a special kind of paths from which all other paths in LaTeXMLMath can be constructed .
These paths , termed the minimal paths in LaTeXMLCite , are shown in Fig .
LaTeXMLRef .
In a minimal path , each column with a non-zero height LaTeXMLMath LaTeXMLMath is interpreted as a particle of charge LaTeXMLMath .
Two adjacent particles in a minimal path are separated by a single empty column .
In order to construct an arbitrary non-minimal path in LaTeXMLMath out of one and only one minimal path , we need to introduce the rules of particle motion from left to right .
Fig .
LaTeXMLRef illustrates these rules in the simplest case of an isolated particle of charge LaTeXMLMath going from LaTeXMLMath to LaTeXMLMath .
The complete transition requires LaTeXMLMath elementary moves .
Next , we consider the motion of a particle of charge LaTeXMLMath through a path configuration shown in Fig .
LaTeXMLRef and Fig .
LaTeXMLRef ( a ) .
In case of the path configuration in Fig .
LaTeXMLRef , the particle of charge LaTeXMLMath can not make any further move to the right .
In the case shown in Fig .
LaTeXMLRef ( a ) , we can make LaTeXMLMath elementary moves to obtain the new path configuration in Fig .
LaTeXMLRef ( b ) .
What happens next depends on the height of the column at LaTeXMLMath , as illustrated in Fig .
LaTeXMLRef and Fig .
LaTeXMLRef .
In case of the path configuration in Fig .
LaTeXMLRef , the particle of charge LaTeXMLMath can not move any further .
In the case shown in Fig .
LaTeXMLRef ( a ) we can make LaTeXMLMath elementary moves to end up with the path configuration in Fig .
LaTeXMLRef ( b ) .
Ignoring the first column at LaTeXMLMath , we see that the last configuration is practically the same as the one in Fig .
LaTeXMLRef ( b ) with LaTeXMLMath replaced by LaTeXMLMath .
It means that we can keep on moving the particle of charge LaTeXMLMath according to the rules in Fig .
LaTeXMLRef –Fig .
LaTeXMLRef .
Actually , it is easy to modify the above discussion in order to deal with the general boundary conditions LaTeXMLMath , LaTeXMLMath .
All we need to do is to refine the notion of a minimal path as in Fig .
LaTeXMLRef .
Two adjacent particles in Fig .
LaTeXMLRef are separated by at most one empty column .
Two half-columns at LaTeXMLMath and LaTeXMLMath are not interpreted as particles .
In LaTeXMLCite , Warnaar proved that using rules of particle motion , one can obtain each non-minimal path from one and only one minimal path in a completely bijective fashion .
The particle content LaTeXMLMath of LaTeXMLMath can be determined by reducing LaTeXMLMath to its minimal image LaTeXMLMath .
Now , since the sum of heights LaTeXMLMath of the path LaTeXMLMath is the invariant of the motion we find by counting the heights of LaTeXMLMath that LaTeXMLEquation where LaTeXMLMath denotes the space of all Gordon paths of the particle content LaTeXMLMath , which start at LaTeXMLMath and end at LaTeXMLMath .
Equation ( LaTeXMLRef ) implies that LaTeXMLEquation where LaTeXMLMath is obtained from LaTeXMLMath by moving LaTeXMLMath by LaTeXMLMath units to the left along the LaTeXMLMath -axis .
Formula ( LaTeXMLRef ) will play an important role in the sequel .
We shall also require another result established in LaTeXMLCite : ( Warnaar ) LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Note that ( LaTeXMLRef ) is an immediate consequence of Theorem LaTeXMLRef .
Having collected the necessary background information , we can derive the second variant of the Andrews-Gordon identities by following three easy steps very similar to those taken to derive our first variant in Section LaTeXMLRef .
First step .
We generalize ( LaTeXMLRef ) as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Next , we sum over LaTeXMLMath to obtain LaTeXMLEquation .
LaTeXMLEquation Second step .
Here , the argument is exactly the same as the one used in deriving ( LaTeXMLRef ) .
We simply state the result LaTeXMLEquation .
Third step .
If the path LaTeXMLMath is obtained from the path LaTeXMLMath by reflecting LaTeXMLMath across the LaTeXMLMath -axis , then LaTeXMLEquation .
Hence , LaTeXMLEquation .
LaTeXMLEquation Next , combining ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) we find that LaTeXMLEquation .
LaTeXMLEquation It follows from Theorem LaTeXMLRef that LaTeXMLEquation .
Letting LaTeXMLMath tend to infinity in ( LaTeXMLRef ) , we obtain with the aid of ( LaTeXMLRef ) and ( LaTeXMLRef ) LaTeXMLEquation .
LaTeXMLEquation Finally , recalling ( LaTeXMLRef ) we arrive at the desired second variant LaTeXMLEquation .
LaTeXMLEquation Setting LaTeXMLMath in ( LaTeXMLRef ) we easily obtain LaTeXMLEquation which is essentially the same ( modulo misprint ) as identity ( 3.3 ) in LaTeXMLCite .
Comparing ( LaTeXMLRef ) and ( LaTeXMLRef ) with LaTeXMLMath , the structural similarities of these two formulas become obvious .
This resemblance strongly suggests that our two variants are special cases of a more general formula for multisums of the form LaTeXMLEquation with LaTeXMLMath .
The unifying formula requires generalized LaTeXMLMath -multinomials that depend on LaTeXMLMath finitization parameters .
Fortunately , such objects have already appeared in the literature LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
In LaTeXMLCite , the polynomials ( 5.2 ) were termed LaTeXMLMath -supernomial coefficients .
We notice that LaTeXMLEquation and LaTeXMLEquation .
Using ( LaTeXMLRef ) , ( LaTeXMLRef ) along with ( LaTeXMLRef ) and ( LaTeXMLRef ) with LaTeXMLMath , it is easy to guess that LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation The remarkable formula ( LaTeXMLRef ) and its finite analogs will be the subject of our next paper .
We would like to thank G.E .
Andrews and C. Krattenthaler for their interest .
We are grateful to A. Riese for his help in preparing the figures for this manuscript , and S.O .
Warnaar for many patient explanations of his work LaTeXMLCite and for his comments .
Computing the equivariant Euler characteristic of Zariski and étale sheaves on curves by Bernhard Köck Dedicated to Vic Snaith on the occasion of his 60th birthday Abstract .
We prove an equivariant Grothendieck-Ogg-Shafarevich formula .
This formula may be viewed as an étale analogue of well-known formulas for Zariski sheaves generalizing the classical Chevalley-Weil formula .
We give a new approach to those formulas ( first proved by Ellingsrud/Lønsted , Nakajima , Kani and Ksir ) which can also be applied in the étale case .
Mathematics Subject Classification 2000 .
14F20 ; 14L30 ; 14H30 .
Key words .
Equivariant Euler characteristic , étale cohomology , Grothendieck-Ogg-Shafarevich formula , conductor , Lefschetz formula , Riemann-Roch formula , Hurwitz formula .
This paper deals with the Riemann-Roch problem for equivariant Zariski sheaves and equivariant étale sheaves on smooth projective curves , i.e .
with the computation of their ( equivariant ! )
Euler characteristic .
In the case of Zariski sheaves we give a new , very natural and quick approach to generalizations of the classical Chevalley-Weil formula proved by Ellingsrud/Lønsted ( see LaTeXMLCite ) , Kani ( see LaTeXMLCite ) and Nakajima ( see LaTeXMLCite ) in the 1980s and we derive generalizations of a more recent result of Ksir ( see LaTeXMLCite ) .
In the case of étale sheaves we develop and prove an equivariant Grothendieck-Ogg-Shafarevich formula by imitating our new approach for Zariski sheaves .
Let LaTeXMLMath be a connected smooth projective curve over an algebraically closed field LaTeXMLMath and let LaTeXMLMath be a finite subgroup of LaTeXMLMath of order LaTeXMLMath .
We assume throughout this paper that the canonical projection LaTeXMLMath is tamely ramified .
Using the coherent Lefschetz fixed point formula ( see LaTeXMLCite or LaTeXMLCite or LaTeXMLCite ) in conjunction with the Riemann-Roch formula and Hurwitz formula , we prove the following theorem ( see Theorem 1.1 ) which effectively implies all known formulas ( see Corollaries 1.3 , 1.4 , 1.7 and 1.8 ) for the equivariant Euler characteristic LaTeXMLEquation of a locally free LaTeXMLMath -sheaf LaTeXMLMath on LaTeXMLMath , considered as an element of the Grothendieck group LaTeXMLMath of all LaTeXMLMath -representations of LaTeXMLMath .
Theorem .
LaTeXMLEquation .
Here , LaTeXMLMath denotes the rank of LaTeXMLMath , LaTeXMLMath the fibre of LaTeXMLMath at LaTeXMLMath and LaTeXMLMath the character of LaTeXMLMath which is given by the representation of LaTeXMLMath on the cotangent space LaTeXMLMath .
We now pass to the étale world .
We fix a prime LaTeXMLMath .
Let LaTeXMLMath be a constructible LaTeXMLMath -sheaf on the étale site LaTeXMLMath which carries a LaTeXMLMath -action compatible with the given LaTeXMLMath -action on LaTeXMLMath .
We are interested in computing the equivariant Euler characteristic LaTeXMLEquation considered as an element of the Grothendieck group LaTeXMLMath of LaTeXMLMath -representations of LaTeXMLMath .
In the extreme case that LaTeXMLMath is the trivial group , this problem is solved by the classical Grothendieck-Ogg-Shafarevich formula ( see Theorem 2.12 on p. 190 in LaTeXMLCite ) .
In the extreme case that LaTeXMLMath is the constant sheaf with trivial LaTeXMLMath -action , a satisfactory answer to this problem follows from Remark 2.9 on p. 187 in LaTeXMLCite ( see Remark 2.2 ( b ) ) .
By imitating our approach for Zariski sheaves we prove the following result for an arbitrary group LaTeXMLMath and arbitrary sheaf LaTeXMLMath ( see Theorem 2.1 ) : Theorem .
We assume that the characteristic of LaTeXMLMath does not divide LaTeXMLMath .
Then we have : LaTeXMLEquation .
Here , LaTeXMLMath denotes the genus of LaTeXMLMath , LaTeXMLMath the generic point of LaTeXMLMath , LaTeXMLMath the sum of the wild conductors of LaTeXMLMath , LaTeXMLMath the ramification index of the canonical projection LaTeXMLMath at LaTeXMLMath and LaTeXMLMath the augmentation representation of the decomposition group LaTeXMLMath at LaTeXMLMath .
As a corollary , we obtain that LaTeXMLMath is divisible by LaTeXMLMath ( see Corollary 2.4 ) .
Furthermore , this formula has the following simple shape , if LaTeXMLMath is étale ( see Remark 2.2 ( a ) ) : LaTeXMLEquation .
In fact , this formula is valid without the assumption that LaTeXMLMath does not divide LaTeXMLMath .
In particular , the ( non-equivariant ) Euler characteristic LaTeXMLMath is divisible by LaTeXMLMath .
Acknowledgments .
I would like to thank Igor Zhukov for raising the equivariant Riemann-Roch problem for étale sheaves and for drawing my attention to the paper LaTeXMLCite .
Furthermore , I would like to thank him and Victor Snaith for helpful and encouraging discussions , for reading carefully a preliminary version of this paper and for suggesting several corrections .
The object of this section is to give a new approach to generalizations of the classical Chevalley-Weil formula published by Ellingsrud/Lønsted , Nakajima and Kani and to derive generalizations of a comparatively simple formula recently published by Ksir .
Let LaTeXMLMath be a connected smooth projective curve over an algebraically closed field LaTeXMLMath and let LaTeXMLMath be a finite subgroup of LaTeXMLMath of order LaTeXMLMath .
We assume in this section that the canonical projection LaTeXMLMath is tamely ramified .
We denote the genus of LaTeXMLMath and LaTeXMLMath by LaTeXMLMath and LaTeXMLMath , respectively .
For any ( closed ) point LaTeXMLMath , let LaTeXMLMath denote the decomposition group and let LaTeXMLMath denote the ramification index of LaTeXMLMath at LaTeXMLMath .
It is well-known ( see Corollaire 1 of Proposition 7 , Chapitre IV on p. 75 of LaTeXMLCite ) that LaTeXMLMath is a cyclic group of order LaTeXMLMath and that LaTeXMLEquation where LaTeXMLMath denotes the character which is given by the action of LaTeXMLMath on the cotangent space LaTeXMLMath .
For LaTeXMLMath we set LaTeXMLMath where LaTeXMLMath .
We denote the Grothendieck group of all LaTeXMLMath -representations of LaTeXMLMath ( of finite dimension ) by LaTeXMLMath .
It is free with basis LaTeXMLMath where LaTeXMLMath denotes the set of isomorphism classes of irreducible LaTeXMLMath -representations of LaTeXMLMath .
Now , let LaTeXMLMath be a locally free LaTeXMLMath -module on LaTeXMLMath of rank LaTeXMLMath , i.e. , we have LaTeXMLMath -isomorphisms LaTeXMLMath , LaTeXMLMath , which satisfy the usual composition rules .
Then , the Zariski cohomology groups LaTeXMLMath , LaTeXMLMath , are LaTeXMLMath -representations of LaTeXMLMath .
Let LaTeXMLEquation denote the equivariant Euler characteristic of LaTeXMLMath with values in LaTeXMLMath .
For any LaTeXMLMath , we view the fibre LaTeXMLMath as a LaTeXMLMath -representation of LaTeXMLMath .
The following theorem computes the equivariant Euler characteristic LaTeXMLMath .
Theorem 1.1 .
We have in LaTeXMLMath : LaTeXMLEquation .
Proof .
By classical representation theory ( see Corollary ( 17.10 ) on p. 424 in LaTeXMLCite ) it suffices to show that the Brauer characters of both sides of formula ( 1 ) coincide .
For any LaTeXMLMath -representation LaTeXMLMath of LaTeXMLMath and for any LaTeXMLMath of order prime to LaTeXMLMath we write LaTeXMLMath for the value of the Brauer character of LaTeXMLMath at LaTeXMLMath .
Recall that LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , are the eigenvalues of the LaTeXMLMath -linear map LaTeXMLMath on LaTeXMLMath and LaTeXMLMath is the Teichmüller character from the group of invertible elements in LaTeXMLMath to the group of invertible elements in the quotient field LaTeXMLMath of the Witt ring LaTeXMLMath of LaTeXMLMath .
( We set LaTeXMLMath and LaTeXMLMath , if LaTeXMLMath . )
Let LaTeXMLMath such that LaTeXMLMath does not divide the order of LaTeXMLMath and let LaTeXMLMath denote the set of points in LaTeXMLMath fixed by LaTeXMLMath .
Then we have : LaTeXMLEquation .
For LaTeXMLMath we have LaTeXMLMath , so the character value of the right hand side of the formula in Theorem 1.1 at the place LaTeXMLMath equals LaTeXMLEquation .
By the Lefschetz fixed point formula ( see Example 3 in LaTeXMLCite or LaTeXMLCite or LaTeXMLCite ; here we use the assumption that LaTeXMLMath does not divide LaTeXMLMath ) , this equals the character value of the left hand side at the place LaTeXMLMath .
For LaTeXMLMath we have : LaTeXMLEquation by the Hurwitz formula ( see Corollary 2.4 on p. 301 in LaTeXMLCite ) .
Hence , the character value of the right hand side of the formula in Theorem 1.1 at the place LaTeXMLMath equals LaTeXMLEquation .
By the Riemann-Roch formula ( see §1 in Chapter IV of LaTeXMLCite and Exercise 6.11 on p. 149 in LaTeXMLCite ) , this equals the character value of the left hand side at the place LaTeXMLMath .
Thus , the proof of Theorem 1.1 is complete .
LaTeXMLMath Lemma 1.2 .
Let LaTeXMLMath and LaTeXMLMath an LaTeXMLMath -th root of unity .
Then we have : LaTeXMLEquation .
Proof .
LaTeXMLMath .
LaTeXMLMath Remark .
A generalization of Theorem 1.1 and the subsequent Corollary 1.4 to the so-called weakly ramified case can be found in LaTeXMLCite .
The following corollary is the main result of the paper LaTeXMLCite by Ellingsrud and Lønsted ; it computes the multiplicity of any irreducible representation LaTeXMLMath in the equivariant Euler characteristic LaTeXMLMath , if LaTeXMLMath does not divide LaTeXMLMath .
While the proof of Ellingsrud and Lønsted is based on the study of the cokernel of the natural embedding LaTeXMLMath , we derive it from Theorem 1.1 and hence from the Lefschetz fixed point formula .
Corollary 1.3 ( Formula ( 3.7 ) in LaTeXMLCite ) .
We assume that LaTeXMLMath does not divide LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath denote the multiplicity of LaTeXMLMath in LaTeXMLMath where LaTeXMLMath .
Then we have in LaTeXMLMath : LaTeXMLEquation .
Proof .
Let LaTeXMLMath denote the usual character pairing .
Then , for all LaTeXMLMath and LaTeXMLMath , we have : LaTeXMLEquation .
Hence we have : LaTeXMLEquation .
Thus , Corollary 1.3 follows from Theorem 1.1 .
LaTeXMLMath The following corollary is the main result of the paper LaTeXMLCite by Nakajima .
Part ( a ) of it has also been proved by Kani ( see Theorem 2 in LaTeXMLCite ) .
In addition to Theorem 1.1 we use the facts that the Euler characteristic LaTeXMLMath is an element of the Grothendieck group LaTeXMLMath of projective LaTeXMLMath -modules ( see Theorem 1 in LaTeXMLCite or Remark 1.5 ( a ) below ) and that the Cartan homomorphism LaTeXMLMath is injective .
The corollary expresses the Euler characteristic LaTeXMLMath as an integral linear combination of certain projective LaTeXMLMath -modules .
Our proof shortens the somewhat lengthy calculations in LaTeXMLCite .
Corollary 1.4 ( Theorem 2 in LaTeXMLCite ) .
( a ) There is a projective LaTeXMLMath -module LaTeXMLMath ( which is unique up to isomorphism ) such that LaTeXMLEquation ( b ) For any LaTeXMLMath , let LaTeXMLMath be given by the equation LaTeXMLEquation furthermore , for any LaTeXMLMath , we fix a point LaTeXMLMath with LaTeXMLMath .
Then we have : LaTeXMLEquation .
Proof .
( a ) Applying Theorem 1.1 to the sheaf LaTeXMLMath with trivial LaTeXMLMath -action , we obtain the following equality in LaTeXMLMath and hence in LaTeXMLMath : LaTeXMLEquation .
This equality shows that the class of the projective LaTeXMLMath -module LaTeXMLMath is divisible by LaTeXMLMath in LaTeXMLMath .
Writing the quotient as a linear combination of classes of indecomposable projective modules we see that the quotient is in fact the class of a projective LaTeXMLMath -module , say LaTeXMLMath .
This immediately implies part ( a ) .
( b ) We first prove the following congruence : LaTeXMLEquation .
For this , we may obviously assume that LaTeXMLMath .
We write LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath denote the sheaf of meromorphic functions on LaTeXMLMath , i.e. , the constant sheaf associated with the function field LaTeXMLMath of LaTeXMLMath .
Then LaTeXMLMath is a LaTeXMLMath -subsheaf of the constant sheaf LaTeXMLMath .
But LaTeXMLMath is isomorphic to LaTeXMLMath as a LaTeXMLMath -sheaf since the twisted group ring LaTeXMLMath is isomorphic to the ring LaTeXMLMath of LaTeXMLMath -matrices over the function field LaTeXMLMath by Galois theory and since there is ( up to isomorphism ) only one module of LaTeXMLMath -dimension LaTeXMLMath over LaTeXMLMath .
So , we may assume that LaTeXMLMath for some equivariant Weil divisor LaTeXMLMath .
Now it is easy to see that LaTeXMLMath mod LaTeXMLMath for all LaTeXMLMath .
So , for any LaTeXMLMath we have : LaTeXMLMath mod LaTeXMLMath .
Thus , the congruence above is proved .
Hence , by Theorem 1.1 , we have in LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath On the other hand , we have : LaTeXMLEquation .
Thus , Corollary 1.4 is proved .
LaTeXMLMath Remark 1.5 .
( a ) In order to prove that LaTeXMLMath is in LaTeXMLMath for all locally free LaTeXMLMath -modules LaTeXMLMath , it suffices to show that the element LaTeXMLMath is in LaTeXMLMath which in turn follows from the fact that LaTeXMLMath is in LaTeXMLMath for one invertible LaTeXMLMath -module LaTeXMLMath on LaTeXMLMath .
( Apply twice the formula in part ( b ) of Corollary 1.4 ) .
If LaTeXMLMath , a nice and short proof of this fact using equivariant cohomology can be found in Borne ’ s thesis ( see Corollaire 3.14 on p. 61 in LaTeXMLCite ) .
( b ) The equation LaTeXMLEquation occurring in the proof of part ( a ) may be considered as an equivariant version of the classical Hurwitz formula , see Théorème 3.16 on p. 62 in Borne ’ s thesis LaTeXMLCite .
He gives a proof of this formula and of part ( a ) of Corollary 1.4 which does not use the work of Nakajima or Kani either and whose main ingredient is the coherent Lefschetz fixed point formula as well .
The following example illustrates part ( a ) of Corollary 1.4 ; it has been proved in LaTeXMLCite directly using Hilbert 90 ( see Proposition 3.7 on p. 56 in LaTeXMLCite ) .
Example 1.6 .
We assume that LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLMath be a character .
We write LaTeXMLMath for some LaTeXMLMath ( for all LaTeXMLMath ) .
Then we have : LaTeXMLEquation .
The following corollary is a main result of the paper LaTeXMLCite by Kani ; it generalizes the classical Chevalley-Weil formula .
Corollary 1.7 ( Corollary of Theorem 2 in LaTeXMLCite ) .
Let LaTeXMLMath denote the sheaf of holomorphic differentials on LaTeXMLMath .
Then we have in LaTeXMLMath : LaTeXMLEquation .
Proof .
It is well-known that LaTeXMLMath and that LaTeXMLMath is LaTeXMLMath -isomorphic to LaTeXMLMath for all LaTeXMLMath .
Hence , by Theorem 1.1 , we have in LaTeXMLMath : LaTeXMLEquation .
Since LaTeXMLMath is isomorphic to the trivial representation LaTeXMLMath , this proves Corollary 1.7 .
LaTeXMLMath The following corollary generalizes a recently published result of Ksir ( see LaTeXMLCite ) .
While we derive it from the previous corollary , her proof is much more elementary .
To be more precise : While our proof is based on the Lefschetz fixed point formula ( see the proof of Theorem 1.1 ) , her proof uses only the Riemann-Roch and Hurwitz theorem and some elementary character theory .
However , her proof seems to work only in the case that not only the representation LaTeXMLMath , but all ( irreducible ) LaTeXMLMath -representations of LaTeXMLMath are rationally valued .
Here , we call a LaTeXMLMath -module rationally valued if its ( Brauer ) character takes only rational values .
For each point LaTeXMLMath , we fix a point LaTeXMLMath in the fibre LaTeXMLMath .
Corollary 1.8 .
We assume that LaTeXMLMath does not divide LaTeXMLMath .
Let LaTeXMLMath be a non-trivial rationally valued irreducible LaTeXMLMath -representation of LaTeXMLMath .
Then the multiplicity of LaTeXMLMath in the LaTeXMLMath -representation LaTeXMLMath is equal to LaTeXMLEquation .
Proof .
This follows from Corollary 1.7 , the following Proposition 1.9 and the well-known fact that the multiplicity of LaTeXMLMath in the regular representation LaTeXMLMath is equal to LaTeXMLMath .
Proposition 1.9 .
We assume that LaTeXMLMath does not divide LaTeXMLMath .
Let LaTeXMLMath be a rationally valued irreducible LaTeXMLMath -representation of LaTeXMLMath .
Then the multiplicity of LaTeXMLMath in LaTeXMLMath and in its dual LaTeXMLMath is equal to LaTeXMLEquation .
Proof .
As in the proof of Corollary 1.3 we write LaTeXMLMath for the usual character pairing .
Then we have : LaTeXMLEquation since over any point LaTeXMLMath there are precisely LaTeXMLMath points in the fibre LaTeXMLMath and since LaTeXMLMath is conjugate to LaTeXMLMath for any other point LaTeXMLMath in LaTeXMLMath .
This proves the Proposition for LaTeXMLMath .
The same argument applies to LaTeXMLMath .
LaTeXMLMath Lemma 1.10 .
Let LaTeXMLMath be a cyclic group of order LaTeXMLMath coprime to LaTeXMLMath , let LaTeXMLMath be a rationally valued LaTeXMLMath -module , and let LaTeXMLMath be a primitive character of LaTeXMLMath .
Then we have : LaTeXMLEquation .
Proof .
It obviously suffices to consider the case LaTeXMLMath .
Since LaTeXMLMath is rationally valued and LaTeXMLMath is abelian , the class LaTeXMLMath of LaTeXMLMath in LaTeXMLMath belongs to the image of the canonical homomorphism LaTeXMLMath , see the Corollary of Proposition 35 , §12.2 , on p. 93 in Serre ’ s book LaTeXMLCite .
By Exercise 13.1 on pp .
104-105 in LaTeXMLCite , the permutation representations LaTeXMLMath , LaTeXMLMath a subgroup of LaTeXMLMath , form a LaTeXMLMath -basis of LaTeXMLMath .
Since both sides of the formula in the Lemma are additive in LaTeXMLMath , it therefore suffices to prove the Lemma in the case LaTeXMLMath where LaTeXMLMath is any subgroup of LaTeXMLMath .
Let LaTeXMLMath denote the order of LaTeXMLMath .
Then we obviously have : LaTeXMLEquation .
Thus we obtain : LaTeXMLEquation as was to be shown .
LaTeXMLMath Similarly to the deduction of Corollary 1.8 from Corollary 1.7 we deduce the following corollary from Corollary 1.4 .
An alternative approach to the following corollary based on Ksir ’ s paper LaTeXMLCite and Borne ’ s thesis LaTeXMLCite ca be found in the recent preprint LaTeXMLCite by Ksir and Joyner .
Corollary 1.11 .
We assume that LaTeXMLMath does not divide LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -equivariant divisor on LaTeXMLMath and let LaTeXMLMath be a rationally valued irreducible LaTeXMLMath -representation of LaTeXMLMath .
Then the multiplicity of LaTeXMLMath in the Euler characteristic LaTeXMLMath is equal to LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are given by LaTeXMLMath for any LaTeXMLMath .
Proof .
By Corollary ( 1.4 ) ( b ) we have the congruence LaTeXMLEquation in LaTeXMLMath mod LaTeXMLMath .
By the Riemann-Roch theorem and the Riemann-Hurwitz formula this congruence becomes an equality in LaTeXMLMath after adding the term LaTeXMLMath on the right hand side .
Now using the Proposition 1.9 and Frobenius reciprocity we obtain Theorem 2 .
LaTeXMLMath Remark .
Note that the multiplicity LaTeXMLMath of the character LaTeXMLMath of the cyclic group LaTeXMLMath in the restricted representation LaTeXMLMath can easily be computed .
For instance , if LaTeXMLMath is a power of a prime , the following lemma can be applied .
Lemma 1.12 .
Let LaTeXMLMath be a cyclic group of prime power order LaTeXMLMath , let LaTeXMLMath be a rationally valued LaTeXMLMath -representation of LaTeXMLMath , let LaTeXMLMath be a primitive character and let LaTeXMLMath with LaTeXMLMath and LaTeXMLMath coprime to LaTeXMLMath .
Then the multiplicity LaTeXMLMath of LaTeXMLMath in LaTeXMLMath is equal to LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the ( unique ) subgroups of LaTeXMLMath of order LaTeXMLMath and LaTeXMLMath , respectively .
Proof .
Easy .
LaTeXMLMath The goal of this section is to prove an equivariant Grothendieck-Ogg-Shafarevich formula .
Let LaTeXMLMath be a connected smooth projective curve over an algebraically closed field LaTeXMLMath and let LaTeXMLMath be a finite subgroup of LaTeXMLMath of order LaTeXMLMath .
We assume in this section that LaTeXMLMath does not divide LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be defined as in §1 .
Furthermore , we denote the generic point of LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be a prime and let LaTeXMLMath be a constructible LaTeXMLMath -sheaf on LaTeXMLMath with LaTeXMLMath -action , i.e. , we have isomorphisms LaTeXMLMath , LaTeXMLMath , which satisfy the usual composition rules .
Then the étale cohomology groups LaTeXMLMath , LaTeXMLMath , are LaTeXMLMath -representations of LaTeXMLMath .
Let LaTeXMLEquation denote the equivariant Euler characteristic ; here , LaTeXMLMath is the Grothendieck group of LaTeXMLMath -representations of LaTeXMLMath ( of finite dimension ) .
Furthermore , let LaTeXMLEquation denote the sum of the wild conductors of LaTeXMLMath ( see p. 188 in LaTeXMLCite ) and let LaTeXMLMath denote the augmentation representation of LaTeXMLMath ( for LaTeXMLMath ) .
The following theorem may be viewed as an analogue of Theorem 1.1 ; it computes the equivariant Euler characteristic LaTeXMLMath .
Theorem 2.1 ( Equivariant Grothendieck-Ogg-Shafarevich formula ) .
We have in LaTeXMLMath : LaTeXMLEquation .
Remark 2.2 .
Let LaTeXMLMath for all LaTeXMLMath with LaTeXMLMath .
Then the formula above has the following shape : LaTeXMLEquation here , LaTeXMLMath is the sum of the conductors of LaTeXMLMath ( see p. 188 in LaTeXMLCite ) .
This formula becomes particularly simple in the following two extreme cases .
( a ) Let LaTeXMLMath be étale .
Then we have : LaTeXMLEquation .
If LaTeXMLMath is the trivial group , this is the classical Grothendieck-Ogg-Shafarevich formula ( see Theorem 2.12 on p. 190 in LaTeXMLCite ) .
In particular , we obtain the following formula for an arbitrary group LaTeXMLMath : LaTeXMLEquation ( note that LaTeXMLMath by the Hurwitz formula ) .
This formula remains valid , if we drop the assumption that LaTeXMLMath does not divide LaTeXMLMath ( see the proof below ) .
In particular we obtain that the ( non-equivariant ) Euler characteristic LaTeXMLMath is divisible by LaTeXMLMath .
Finally , the latter formula may be viewed as an analogue of Theorem 2.4 in LaTeXMLCite .
( b ) Let LaTeXMLMath be the constant sheaf LaTeXMLMath with trivial LaTeXMLMath -action .
Then we obtain the following formula : LaTeXMLEquation .
This is the LaTeXMLMath -version of the formula in Remark 2.9 on p. 187 in LaTeXMLCite .
( Note that the Artin character is the character of the augmentation representation since LaTeXMLMath does not divide LaTeXMLMath . )
It can be derived from the LaTeXMLMath -version by applying the decomposition homomorphism as in lines 6 through 9 on p. 191 in LaTeXMLCite .
Proof ( of Theorem 2.1 ) .
As in the proof of Theorem 1.1 we will show that the ( Brauer ) character values of both sides coincide for all LaTeXMLMath .
So let LaTeXMLMath .
Then we have : LaTeXMLEquation .
Hence , for LaTeXMLMath , the character value of the right hand side of the formula in Theorem 2.1 at the place LaTeXMLMath equals LaTeXMLMath .
By the Lefschetz fixed point formula ( see Theorem 2 in LaTeXMLCite or LaTeXMLCite ; here , we use that LaTeXMLMath does not divide LaTeXMLMath ) , this equals the character value of the left hand side .
For LaTeXMLMath , the character value of the right hand side at the place LaTeXMLMath is LaTeXMLEquation by the Hurwitz formula .
By the classical Grothendieck-Ogg-Shafarevich formula ( see Theorem 2.12 on p. 190 in LaTeXMLCite ) , this equals the character value of the left hand side at the place LaTeXMLMath .
Thus , the proof of Theorem 2.1 is complete .
LaTeXMLMath The following corollary may be viewed as the analogue of Corollary 1.3 ; it computes the multiplicity of any irreducible LaTeXMLMath -representation LaTeXMLMath of LaTeXMLMath in the equivariant Euler characteristic LaTeXMLMath .
We write LaTeXMLMath for the set of isomorphism classes of irreducible LaTeXMLMath -representations and set LaTeXMLMath for LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath denote the multiplicity of the trivial representation LaTeXMLMath in LaTeXMLMath where LaTeXMLMath .
Corollary 2.3 .
We assume that LaTeXMLMath .
Then we have in LaTeXMLMath : LaTeXMLEquation .
Proof .
Let LaTeXMLMath denote the symmetric bilinear form given by LaTeXMLMath for any LaTeXMLMath -representations LaTeXMLMath , LaTeXMLMath of LaTeXMLMath .
Then we obviously have : ( a ) LaTeXMLMath for any LaTeXMLMath -representations LaTeXMLMath , LaTeXMLMath , LaTeXMLMath of LaTeXMLMath .
( b ) LaTeXMLMath for all LaTeXMLMath .
( c ) LaTeXMLMath for any subgroup LaTeXMLMath of LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Hence we have for all LaTeXMLMath : LaTeXMLEquation .
Hence we have : LaTeXMLEquation .
Thus , Corollary 2.3 follows from Theorem 2.1 .
LaTeXMLMath The analogue of the element LaTeXMLMath occurring in Corollary 1.4 ( a ) is LaTeXMLEquation .
It is obviously an element of LaTeXMLMath .
More generally , LaTeXMLMath is an element of LaTeXMLMath , since LaTeXMLMath is isomorphic to LaTeXMLMath for any LaTeXMLMath with LaTeXMLMath .
Furthermore , LaTeXMLMath is an integer .
Thus , Theorem 2.1 implies the following corollary .
Corollary 2.4 .
The sum LaTeXMLMath of the wild conductors LaTeXMLMath , LaTeXMLMath , is divisible by LaTeXMLMath .
In particular , Theorem 2.1 expresses LaTeXMLMath as an integral linear combination of LaTeXMLMath -representations ; thus , the analogue of Corollary 1.4 is already built into Theorem 2.1 .
School of Mathematics University of Southampton Southampton SO17 1BJ United Kingdom e-mail : bk @ maths.soton.ac.uk .
Suppose that LaTeXMLMath is a symplectic manifold and that there exists a Liouville vector field LaTeXMLMath defined in a neighborhood of and transverse to LaTeXMLMath .
Then LaTeXMLMath induces a contact form LaTeXMLMath on LaTeXMLMath which determines the germ of LaTeXMLMath along LaTeXMLMath .
One should think of the contact manifold LaTeXMLMath as controlling the behavior of LaTeXMLMath “ at infinity ” .
If LaTeXMLMath points out of LaTeXMLMath along LaTeXMLMath then we call LaTeXMLMath a convex filling of LaTeXMLMath , and if LaTeXMLMath points into LaTeXMLMath along LaTeXMLMath then we call LaTeXMLMath a concave filling of LaTeXMLMath .
Much attention has been given in recent years to constructions of convex fillings of contact LaTeXMLMath -manifolds ; see LaTeXMLCite for example .
Etnyre and Honda LaTeXMLCite very recently began a careful investigation of concave fillings and proved that every compact , oriented , contact LaTeXMLMath -manifold has ( infinitely many ) concave fillings , but their proof depends on a result of Lisca and Matic LaTeXMLCite that every Stein surface embeds as a domain in a closed Kähler manifold .
This result is not explicitly constructive and in particular does not give handlebody decompositions of the concave fillings .
In this paper we present a method to explicitly construct , handle by handle , concave fillings of contact LaTeXMLMath -manifolds , without reference to the result of Lisca and Matic .
The main theorem we prove is : Every closed , oriented , contact LaTeXMLMath -manifold has a concave filling and , furthermore , a handlebody decomposition of the filling can be given explicitly in terms of the contact structure .
While the author was writing this paper Akbulut and Ozbagci LaTeXMLCite presented , using different techniques , a constructive proof of the fact that every Stein surface embeds in a closed symplectic LaTeXMLMath -manifold .
The reader will also find some common themes in an earlier paper by Akbulut and Ozbagci LaTeXMLCite .
The author would like to thank Emmanuel Giroux for helpful correspondence regarding his recent results , Ichiro Torisu for pointing out Giroux ’ s results to the author and the Nankai Institute of Mathematics for support and hospitality .
Henceforth in this paper , unless explicitly stated otherwise , we adopt the following conventions : All manifolds are compact and oriented , and all LaTeXMLMath -manifolds are closed .
All contact structures are positive , and all symplectic manifolds are oriented by their symplectic structures .
For the basic definitions in symplectic and contact topology the reader is referred to LaTeXMLCite and LaTeXMLCite .
We will build our concave fillings by glueing together symplectic cobordisms , which we now define .
Let LaTeXMLMath be a symplectic LaTeXMLMath -manifold and let LaTeXMLMath and LaTeXMLMath be two LaTeXMLMath -manifolds with LaTeXMLMath .
Suppose there exists a Liouville vector field LaTeXMLMath defined on a neighborhood of and transverse to LaTeXMLMath , pointing in along LaTeXMLMath and out along LaTeXMLMath , and let LaTeXMLMath be the induced contact structure on LaTeXMLMath .
In this situation , following LaTeXMLCite and LaTeXMLCite , we call LaTeXMLMath a symplectic cobordism from LaTeXMLMath to LaTeXMLMath and we indicate the existence of such a cobordism with the notation LaTeXMLMath .
Note that this relation is not reflexive , but it is transitive : If LaTeXMLMath is a symplectic cobordism from LaTeXMLMath to LaTeXMLMath , for LaTeXMLMath , then it is possible to glue LaTeXMLMath to LaTeXMLMath ( after attaching a symplectic collar on LaTeXMLMath and possibly multiplying LaTeXMLMath by a positive constant ) to form a symplectic cobordism from LaTeXMLMath to LaTeXMLMath .
A convex filling of LaTeXMLMath is thus a symplectic cobordism from LaTeXMLMath to LaTeXMLMath while a concave filling of LaTeXMLMath is a symplectic cobordism from LaTeXMLMath to LaTeXMLMath .
Given any contact LaTeXMLMath -manifold LaTeXMLMath , the most basic building block is a cobordism LaTeXMLMath from LaTeXMLMath to itself constructed as follows : Let LaTeXMLMath be any contact form for LaTeXMLMath and let LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath coordinate .
Then LaTeXMLMath is a symplectic form and LaTeXMLMath is a Liouville vector field inducing the contact form LaTeXMLMath on the graph of any function LaTeXMLMath , so for any two functions LaTeXMLMath on LaTeXMLMath we can construct a cobordism LaTeXMLMath .
Use of this building block will often be assumed without explicit mention , but it is essential for all of our constructions , and in particular for the fact that cobordisms can be glued together .
Our main construction depends on the recent discovery of close connections between contact structures and open book decompositions , culminating in the work of Giroux LaTeXMLCite .
To discuss open book decompositions , we begin with conventions regarding mapping class groups .
If LaTeXMLMath is a compact surface with boundary , we take the mapping class group of LaTeXMLMath to be the group of orientation-preserving self-homeomorphisms of LaTeXMLMath fixing LaTeXMLMath pointwise , modulo isotopies fixing LaTeXMLMath pointwise .
We denote this mapping class group by LaTeXMLMath .
We multiply elements of LaTeXMLMath in the same order in which we compose functions , and in fact we will generally blur the distinction between a homeomorphism LaTeXMLMath and its equivalence class LaTeXMLMath .
If LaTeXMLMath is another compact surface , there is a natural inclusion LaTeXMLMath given by extending LaTeXMLMath to be the identity on LaTeXMLMath .
By “ polar coordinates ” on a solid torus LaTeXMLMath we will mean coordinates LaTeXMLMath , where LaTeXMLMath are polar coordinates on LaTeXMLMath and LaTeXMLMath .
By polar coordinates near a link LaTeXMLMath we will mean polar coordinates on a neighborhood of each component of LaTeXMLMath such that LaTeXMLMath .
An open book decomposition of a LaTeXMLMath -manifold LaTeXMLMath is a non-empty link LaTeXMLMath and a fibration LaTeXMLMath such that near LaTeXMLMath we can find polar coordinates LaTeXMLMath with respect to which LaTeXMLMath .
The link LaTeXMLMath is called the “ binding ” and the union of a fiber with LaTeXMLMath is called a “ page ” .
From such a structure we get the following data : The topological type of a page , a compact , oriented surface LaTeXMLMath with LaTeXMLMath .
The monodromy LaTeXMLMath , the element of LaTeXMLMath represented by the return map for flow along any vector field LaTeXMLMath on LaTeXMLMath which is transverse to the pages and is meridinal near LaTeXMLMath ( i.e .
LaTeXMLMath in polar coordinates near LaTeXMLMath ) .
We will often refer to the pair LaTeXMLMath as the open book decomposition , and the manifold with this open book decomposition as LaTeXMLMath .
We identify LaTeXMLMath with LaTeXMLMath where LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath and all LaTeXMLMath .
The pages are LaTeXMLMath and the binding is LaTeXMLMath .
Notice that LaTeXMLMath .
A contact vector field for a contact structure LaTeXMLMath is a vector field LaTeXMLMath such that flow along LaTeXMLMath preserves LaTeXMLMath .
A Reeb vector field for LaTeXMLMath is a contact vector field which is transverse to LaTeXMLMath .
Every Reeb vector field LaTeXMLMath arises from a contact form LaTeXMLMath for LaTeXMLMath as the unique vector field satisfying LaTeXMLMath and LaTeXMLMath .
Let us say that a vector field LaTeXMLMath on LaTeXMLMath respects a given open book decomposition of LaTeXMLMath if LaTeXMLMath is transverse to the pages , tangent to the binding , and satisfies the following orientation condition : LaTeXMLMath co-orients and hence orients each page , and also orients the binding , and we require that this orientation of the binding agree with its orientation as the boundary of a page .
A contact structure LaTeXMLMath on LaTeXMLMath is supported by a given open book decomposition if there exists a Reeb vector field for LaTeXMLMath which respects the open book decomposition The first result below is due , in various versions , to Giroux LaTeXMLCite , Torisu LaTeXMLCite , and Thurston and Winkelnkemper LaTeXMLCite Every open book decomposition of a LaTeXMLMath -manifold LaTeXMLMath supports a unique ( up to isotopy ) contact structure on LaTeXMLMath .
Thurston and Winkelnkemper proved existence without mentioning uniqueness .
Torisu proved existence and uniqueness , but using a different relationship between the contact structure and the open book decomposition defined in terms of convex Heegard splittings .
We will discuss Torisu ’ s version in section LaTeXMLRef .
Finally Giroux pointed out a simple argument for the uniqueness .
This result becomes most powerful when coupled with the following theorem .
Every contact structure on a LaTeXMLMath -manifold LaTeXMLMath is supported by an open book decomposition of LaTeXMLMath .
Given an open book decomposition LaTeXMLMath , we will refer to the unique supported contact structure on LaTeXMLMath as LaTeXMLMath , and we will abbreviate with the notation LaTeXMLMath .
If LaTeXMLMath is a disk and LaTeXMLMath then LaTeXMLMath and LaTeXMLMath is the standard contact structure LaTeXMLMath on LaTeXMLMath .
The LaTeXMLMath -ball with its standard symplectic form is a symplectic cobordism from LaTeXMLMath to LaTeXMLMath .
As a generalization , if LaTeXMLMath is any compact surface with LaTeXMLMath and LaTeXMLMath , and if we let LaTeXMLMath be LaTeXMLMath with “ rounded corners ” and LaTeXMLMath be the sum of a volume form on LaTeXMLMath and a volume form on LaTeXMLMath , then LaTeXMLMath is a symplectic cobordism form LaTeXMLMath to LaTeXMLMath .
We will call this the standard convex filling of LaTeXMLMath .
The strategy of this paper is to translate cobordism questions for contact LaTeXMLMath -manifolds into questions about the relationship between mapping class group elements of various surfaces .
We will rely on the following fundamental fact : The mapping class group of a surface is generated by Dehn twists .
It is important to distinguish between right-handed and left-handed Dehn twists .
A right-handed Dehn twist about a simple closed curve LaTeXMLMath is the twist LaTeXMLMath such that , if LaTeXMLMath is an arc transverse to LaTeXMLMath and we travel along LaTeXMLMath towards LaTeXMLMath , LaTeXMLMath diverges from LaTeXMLMath by forking off to the right , going around LaTeXMLMath and then rejoining LaTeXMLMath on the other side of LaTeXMLMath .
This distinction depends on the orientation of LaTeXMLMath but not of LaTeXMLMath .
We will shorten the phrase “ right-handed Dehn twist ” to “ right twist ” .
If we are given a single curve LaTeXMLMath , we will use the notation LaTeXMLMath to refer to the right twist about LaTeXMLMath .
If we are given a sequence of curves LaTeXMLMath , we will use the notation LaTeXMLMath to refer to the right twist about LaTeXMLMath .
It will also be important to distinguish between twists about homologically trivial and nontrivial curves .
We will call a twist about a homologically nontrivial curve a homologically nontrivial twist .
We now present the fundamental building blocks for our constructions .
Throughout , suppose we are given a pair LaTeXMLMath , where LaTeXMLMath is a compact surface with LaTeXMLMath and LaTeXMLMath .
The first two results below are more or less immediate consequences of results of Eliashberg LaTeXMLCite and Weinstein LaTeXMLCite on symplectic handlebodies , reinterpreted in the context of supporting open book decompositions .
As stated here the results are probably well known to experts , but for completeness we will provide proofs in section LaTeXMLRef .
Let LaTeXMLMath be LaTeXMLMath with a LaTeXMLMath -dimensional LaTeXMLMath -handle attached ( at points LaTeXMLMath ) .
Then LaTeXMLMath , and the cobordism is diffeomorphic to LaTeXMLMath with a LaTeXMLMath -dimensional LaTeXMLMath -handle attached to LaTeXMLMath ( at the corresponding points LaTeXMLMath and LaTeXMLMath in the binding ) .
Note that this result is straightforward if we ignore the contact and symplectic structures ; simply observe that the pages intersect the boundary LaTeXMLMath of each foot of the LaTeXMLMath -handle in longitudes running from the north to the south pole , and that these extend across the surgery as LaTeXMLMath -dimensional LaTeXMLMath -handles attached to each page To discuss LaTeXMLMath -handles , notice that a knot LaTeXMLMath lying in a page of an open book decomposition is given a framing by a vector tangent to the page and transverse to the knot ; call this the page-framing of the knot and abbreviate it LaTeXMLMath .
Let LaTeXMLMath , where LaTeXMLMath is a right twist along a homologically nontrivial curve LaTeXMLMath .
Then LaTeXMLMath , and the cobordism is diffeomorphic to LaTeXMLMath with a LaTeXMLMath -dimensional LaTeXMLMath -handle attached along LaTeXMLMath with framing LaTeXMLMath .
This proposition without the contact structures is simply the familiar observation due to Lickorish LaTeXMLCite that surgery on a knot LaTeXMLMath with framing LaTeXMLMath relative to a surface LaTeXMLMath is equivalent to splitting LaTeXMLMath along LaTeXMLMath and reglueing with a right twist along LaTeXMLMath .
The requirement that LaTeXMLMath be homologically nontrivial does not arise if we ignore the contact structures .
One should think of the above two propositions as giving two allowable moves on the set of pairs LaTeXMLMath .
We will refer to the first move as “ attaching a LaTeXMLMath -handle ” and to the second move as “ appending a homologically nontrivial right twist ” .
A standard fact about Dehn twists is that , if LaTeXMLMath is any orientation-preserving homeomorphism and LaTeXMLMath is a simple closed curve , then LaTeXMLMath .
Now suppose we are given a pair LaTeXMLMath , where LaTeXMLMath for LaTeXMLMath , and let LaTeXMLMath .
Then the operation of replacing LaTeXMLMath with LaTeXMLMath is the same as replacing LaTeXMLMath with LaTeXMLMath .
Thus we can immediately generalize the second move to allow “ inserting a homologically nontrivial right twist ” .
In other words , if LaTeXMLMath , then LaTeXMLMath .
The next result follows from earlier work by this author LaTeXMLCite on attaching symplectic LaTeXMLMath -handles along transverse knots ; here we will attach LaTeXMLMath -handles along the binding of an open book decomposition .
Notice that each component LaTeXMLMath of the binding is given a framing as a boundary component of a page ; call this the page-framing for the binding and again abbreviate it LaTeXMLMath .
By a right twist about a component LaTeXMLMath of LaTeXMLMath , we really mean a right twist about a curve parallel to LaTeXMLMath .
Let LaTeXMLMath , where LaTeXMLMath is the product of one right twist around each component of LaTeXMLMath .
Then LaTeXMLMath .
The cobordism is diffeomorphic to LaTeXMLMath with a LaTeXMLMath -dimensional LaTeXMLMath -handle attached to LaTeXMLMath along each component LaTeXMLMath of the binding with framing LaTeXMLMath .
In section LaTeXMLRef , we will show how this follows from the results in LaTeXMLCite .
A key point in our construction will depend on a relation among right twists called the “ chain relation ” by Wajnryb LaTeXMLCite .
A sequence of simple closed curves LaTeXMLMath in a surface LaTeXMLMath is called a chain if each LaTeXMLMath intersects each LaTeXMLMath transversely in exactly one point .
Let LaTeXMLMath be a chain in a surface LaTeXMLMath and let LaTeXMLMath be a regular neighborhood of LaTeXMLMath .
If LaTeXMLMath is even then LaTeXMLMath has one component .
Let LaTeXMLMath be a right twist about LaTeXMLMath and let LaTeXMLMath be a right twist about LaTeXMLMath .
Then the following relation holds : LaTeXMLEquation .
There is a similar relation when LaTeXMLMath is odd and LaTeXMLMath has two components , which could be used here as well , but for simplicity we only work with the case where LaTeXMLMath is even .
Notice that the regular neighborhood LaTeXMLMath is actually a surface of genus LaTeXMLMath with one disk removed .
Any surface LaTeXMLMath of genus LaTeXMLMath with one boundary component contains a chain LaTeXMLMath such that LaTeXMLMath is a regular neighborhood of LaTeXMLMath .
Given such a LaTeXMLMath and a fixed homologically nontrivial simple closed curve LaTeXMLMath , we may choose the chain so that LaTeXMLMath .
Furthermore all the curves in such a chain are necessarily homologically nontrivial .
Given a pair LaTeXMLMath , where LaTeXMLMath is a compact surface with LaTeXMLMath and LaTeXMLMath , there exists a pair LaTeXMLMath obtained from LaTeXMLMath by adding LaTeXMLMath -handles and inserting homologically nontrivial right twists ( see remark LaTeXMLRef ) such that : LaTeXMLMath has one boundary component and , LaTeXMLMath , where LaTeXMLMath is a right twist about LaTeXMLMath and LaTeXMLMath is a composition of homologically nontrivial right twists .
By attaching LaTeXMLMath -handles , we can get from LaTeXMLMath to LaTeXMLMath where LaTeXMLMath has one boundary component .
By theorem LaTeXMLRef , LaTeXMLMath is equal to a product of right-handed and left-handed Dehn twists .
First note that each homologically trivial right or left twist is equal to a product of homologically nontrivial right or left ( respectively ) twists , due to the chain relation ( proposition LaTeXMLRef ) .
To see this , observe that any homologically trivial curve LaTeXMLMath is the boundary of a compact subsurface of LaTeXMLMath .
Thus we may assume that all the twists are homologically nontrivial .
Let LaTeXMLMath be the genus of LaTeXMLMath and let LaTeXMLMath be a right twist about LaTeXMLMath .
Now suppose we express LaTeXMLMath as LaTeXMLMath where LaTeXMLMath is a right twist about the homologically nontrival curve LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are arbitrary elements of LaTeXMLMath .
Then we may find a chain LaTeXMLMath in LaTeXMLMath , with LaTeXMLMath , such that : LaTeXMLEquation .
Thus by inserting homologically nontrivial right twists we can change LaTeXMLMath to LaTeXMLMath ( since boundary twists commute with interior twists ) .
Repeating this process for every right twist in LaTeXMLMath , we can change LaTeXMLMath to LaTeXMLMath where LaTeXMLMath is a product of homologically nontrivial left twists and LaTeXMLMath is some ( possibly large ) positive integer .
Our task is to reduce LaTeXMLMath to LaTeXMLMath .
Note that , by inserting more right twists if necessary , we may assume that LaTeXMLMath is odd .
We write LaTeXMLMath where LaTeXMLMath is the corresponding product of right twists .
Add more LaTeXMLMath -handles to LaTeXMLMath to get LaTeXMLMath , with LaTeXMLMath connected and with the genus LaTeXMLMath of LaTeXMLMath equal to LaTeXMLMath , so that LaTeXMLMath .
Fix a particular chain LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath is a regular neighborhood of LaTeXMLMath and LaTeXMLMath is a regular neighborhood of LaTeXMLMath .
Then we have : LaTeXMLEquation .
Thus , by inserting homologically nontrivial right twists we can change this expression to : LaTeXMLEquation .
Finally this shows that LaTeXMLMath may be changed to LaTeXMLMath by inserting right twists .
∎ Given a contact LaTeXMLMath -manifold LaTeXMLMath , theorem LaTeXMLRef tells us that LaTeXMLMath is supported by an open book decomposition , with page LaTeXMLMath and monodromy LaTeXMLMath .
Apply lemma LaTeXMLRef to get a new pair LaTeXMLMath with LaTeXMLMath as in the lemma .
Since we only added LaTeXMLMath -handles and inserted right twists , propositions LaTeXMLRef and LaTeXMLRef tell us that LaTeXMLMath .
Consider the pair LaTeXMLMath as in proposition LaTeXMLRef and notice that LaTeXMLMath , a composition of homologically nontrivial right twists .
Let LaTeXMLMath be a disk and LaTeXMLMath the identity in LaTeXMLMath .
Then we can get from the pair LaTeXMLMath to LaTeXMLMath by attaching LaTeXMLMath -handles and inserting homologically nontrivial right twists , so that LaTeXMLMath ( see remark LaTeXMLRef ) .
Thus we have the following cobordism : LaTeXMLEquation .
Finally , apply proposition LaTeXMLRef to conclude that LaTeXMLEquation and put the two cobordisms together to get a concave filling of LaTeXMLMath .
∎ The reader familiar with Lefschetz pencils ( see LaTeXMLCite ) may notice that these building blocks can also be put together to give a symplectic structure on any topological Lefschetz pencil with homologically nontrivial vanishing cycles .
A Lefschetz pencil is the result of blowing down a Lefschetz fibration over LaTeXMLMath along LaTeXMLMath disjoint sections , which must therefore each have self-intersection LaTeXMLMath .
A Lefschetz fibration over LaTeXMLMath is completely determined by LaTeXMLMath vanishing cycles LaTeXMLMath in the ( closed ) fiber surface LaTeXMLMath , with the property that LaTeXMLMath , where LaTeXMLMath is a right twist about LaTeXMLMath .
The sections correspond to LaTeXMLMath points LaTeXMLMath , disjoint from the vanishing cycles , such that in fact LaTeXMLMath is isotopic to the identity via an isotopy fixing LaTeXMLMath pointwise .
The fact that the sections have self-intersection LaTeXMLMath means that , if we require the isotopy to fix a disk LaTeXMLMath around each point LaTeXMLMath , then LaTeXMLMath is isotopic to the product of one right twist around each LaTeXMLMath .
Thus the Lefschetz pencil may be built as follows .
Remove the interior of each LaTeXMLMath to get a compact surface LaTeXMLMath and let LaTeXMLMath ( where LaTeXMLMath is the product of one right twist about each component of LaTeXMLMath ) .
First build two copies of a symplectic cobordism from LaTeXMLMath to LaTeXMLMath using remark LaTeXMLRef .
Then attach to one of them a cobordism from LaTeXMLMath to LaTeXMLMath built using proposition LaTeXMLRef to get a cobordism from LaTeXMLMath to LaTeXMLMath .
Since LaTeXMLMath , proposition LaTeXMLRef gives us a cobordism from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be the union of the LaTeXMLMath -handles used in this last cobordism .
These three cobordisms piece together to make a closed symplectic LaTeXMLMath -manifold LaTeXMLMath .
Since we attached one LaTeXMLMath -handle for each vanishing cycle to LaTeXMLMath , with the appropriate framings , LaTeXMLMath is diffeomorphic to the Lefschetz pencil built with this data with a neighborhood of each section removed .
Analysis of the framings involved in LaTeXMLMath shows that removing a neighborhood of each section and replacing them with LaTeXMLMath is equivalent to blowing down these sections .
For this section it will be useful to have an explicit model of the unique contact structure LaTeXMLMath supported by an open book decomposition LaTeXMLMath .
The following construction is essentially the construction of Thurston and Winkelnkemper in LaTeXMLCite , done with a little more care to keep track of a Reeb vector field .
Recall that we chose to measure the monodromy of an open book decomposition using a flow transverse to the pages which had closed meridinal orbits near the binding .
If instead we required closed longitudinal orbits , realizing the framing LaTeXMLMath relative to the page framing , we would change the monodromy by a single left-handed Dehn twist along each component of LaTeXMLMath .
Given LaTeXMLMath , let LaTeXMLMath , where LaTeXMLMath is a right twist around each component of LaTeXMLMath .
We will construct LaTeXMLMath using LaTeXMLMath as our return map , but arrange that the return flow orbits are LaTeXMLMath longitudes near the binding .
Given a contact form LaTeXMLMath , let LaTeXMLMath denote the Reeb vector field for LaTeXMLMath .
Let LaTeXMLMath be coordinates on ( each component of ) a collar neighborhood LaTeXMLMath of LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Assume that LaTeXMLMath is the identity on LaTeXMLMath .
Now choose a LaTeXMLMath -form LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath on LaTeXMLMath ( this is where we need LaTeXMLMath ) and choose a smooth nonincreasing function LaTeXMLMath which equals LaTeXMLMath on LaTeXMLMath and LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is the LaTeXMLMath coordinate and LaTeXMLMath is some positive constant .
Notice that LaTeXMLMath descends to a smooth LaTeXMLMath -form on LaTeXMLMath , where LaTeXMLMath .
For LaTeXMLMath large enough , LaTeXMLMath and LaTeXMLMath is transverse to the fibers of the fibration LaTeXMLMath .
Also notice that , after choosing LaTeXMLMath , we may enlarge LaTeXMLMath to arrange that LaTeXMLMath with LaTeXMLMath and still have LaTeXMLMath on LaTeXMLMath .
Near LaTeXMLMath , LaTeXMLMath .
Now we construct LaTeXMLMath by Dehn fillings on LaTeXMLMath .
For each component of LaTeXMLMath , attach a solid torus LaTeXMLMath to LaTeXMLMath with a map LaTeXMLMath defined as follows : Use polar coordinates LaTeXMLMath on LaTeXMLMath and coordinates LaTeXMLMath near LaTeXMLMath ; let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
This is exactly the right filling so that the LaTeXMLMath circles become LaTeXMLMath longitudes , so that we are producing the correct monodromy .
Then LaTeXMLMath , which extends across LaTeXMLMath , and LaTeXMLMath .
This is tangent to the binding and satisfies the orientation condition in definition LaTeXMLRef .
Since any two contact structures supported by the same open book decomposition are isotopic ( theorem LaTeXMLRef ) , we may always assume that our contact structures are of the form described above .
We will call this model of the contact structure supported by an open book decomposition “ the standard model ” .
A Legendrian knot is a knot LaTeXMLMath in a contact LaTeXMLMath -manifold LaTeXMLMath which is everywhere tangent to LaTeXMLMath .
A Legendrian knot LaTeXMLMath has a canonical framing LaTeXMLMath , the “ Thurston-Bennequin ” framing , given by any vector field in LaTeXMLMath transverse to LaTeXMLMath .
If we are given a homologically nontrivial curve LaTeXMLMath , the standard model may be refined to arrange that LaTeXMLMath in a neighborhood of LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath is Legendrian for LaTeXMLMath , and furthermore LaTeXMLMath .
The construction of the LaTeXMLMath -dimensional symplectic handles of proposition LaTeXMLRef and proposition LaTeXMLRef is due to Weinstein LaTeXMLCite and Eliashberg LaTeXMLCite ; our task is to show that the contact surgeries associated with these handles behave well with respect to supporting open book decompositions .
This can be shown by explicit calculations using Weinstein ’ s description of the handles , but here we give less computational proofs .
Here we essentially ignore Weinstein ’ s construction of a symplectic LaTeXMLMath -handle and instead build the LaTeXMLMath -handle from scratch .
Let LaTeXMLMath be the standard convex filling of LaTeXMLMath and let LaTeXMLMath be the standard convex filling of LaTeXMLMath , as in remark LaTeXMLRef .
We can clearly construct LaTeXMLMath and LaTeXMLMath so that LaTeXMLMath and so that LaTeXMLMath is a LaTeXMLMath -dimensional symplectic LaTeXMLMath -handle .
From the standard model it is clear that there exists a contactomorphism from a neighborhood LaTeXMLMath of the binding LaTeXMLMath to a neighborhood LaTeXMLMath of the binding LaTeXMLMath .
Furthermore we can arrange that the handle LaTeXMLMath is attached inside LaTeXMLMath .
Thus LaTeXMLMath can just as well be attached to a cobordism from LaTeXMLMath to LaTeXMLMath to get a cobordism from LaTeXMLMath to LaTeXMLMath .
∎ To prove proposition LaTeXMLRef , we will use Torisu ’ s characterization in LaTeXMLCite of the unique contact structure supported by an open book decomposition .
Suppose that an open book decomposition of a LaTeXMLMath -manifold LaTeXMLMath has binding LaTeXMLMath and fibration LaTeXMLMath .
Notice that the sets LaTeXMLMath and LaTeXMLMath give a Heegard splitting of LaTeXMLMath .
A surface LaTeXMLMath is said to be convex if there exists a contact vector field LaTeXMLMath for LaTeXMLMath which is transverse to LaTeXMLMath .
The set LaTeXMLMath where LaTeXMLMath is tangent to LaTeXMLMath is called the dividing set of LaTeXMLMath .
( See LaTeXMLCite for background on convexity . )
We have the following result , given a Heegard splitting LaTeXMLMath of LaTeXMLMath coming from an open book decomposition as above : There exists a unique ( up to isotopy ) contact structure LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath is tight on both LaTeXMLMath and LaTeXMLMath and such that LaTeXMLMath is convex with dividing set equal to the binding LaTeXMLMath .
Using the standard model of LaTeXMLMath , one can show that the splitting surface LaTeXMLMath is convex with dividing set LaTeXMLMath .
To see that LaTeXMLMath is tight on both LaTeXMLMath and LaTeXMLMath , we note that LaTeXMLMath and LaTeXMLMath can both be contactomorphically embedded into LaTeXMLMath , again using the standard model .
LaTeXMLMath is tight because of the existence of the standard convex filling of LaTeXMLMath ( see remark LaTeXMLRef ) , and Eliashberg LaTeXMLCite shows that any contact LaTeXMLMath -manifold with a convex filling is tight .
Putting together the uniqueness in Torisu ’ s theorem with the uniqueness in theorem LaTeXMLRef , we get the following corollary , which in particular shows that Torisu ’ s theorem is actually equivalent to theorem LaTeXMLRef .
Given an open book decomposition on LaTeXMLMath , let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be as in the preceding paragraphs .
A contact structure LaTeXMLMath on LaTeXMLMath is supported by this open book decomposition if and only if LaTeXMLMath is tight on both LaTeXMLMath and LaTeXMLMath and LaTeXMLMath is convex with dividing set LaTeXMLMath .
Let LaTeXMLMath .
Use the standard model as described in remark LaTeXMLRef , so that LaTeXMLMath is Legendrian , with LaTeXMLMath .
Weinstein LaTeXMLCite shows that we can attach a symplectic handle along an arbitrarily small neighborhood of LaTeXMLMath with framing LaTeXMLMath , so that we get a cobordism from LaTeXMLMath to a new contact LaTeXMLMath -manifold LaTeXMLMath .
As mentioned earlier , we know that LaTeXMLMath , and we need to show that LaTeXMLMath is supported by this open book decomposition .
Split LaTeXMLMath into the sets LaTeXMLMath and LaTeXMLMath as above ; since the surgery that produces LaTeXMLMath from LaTeXMLMath is localized near LaTeXMLMath , we get a corresponding Heegard splitting of LaTeXMLMath into two handlebodies LaTeXMLMath and LaTeXMLMath , with LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath also agree on a neighborhood of LaTeXMLMath , we still have that the splitting surface LaTeXMLMath is convex with dividing set equal to the binding LaTeXMLMath .
Thus , by corollary LaTeXMLRef , we need only show that LaTeXMLMath is tight to complete the proof .
Let LaTeXMLMath be the contactomorphic embedding of LaTeXMLMath into LaTeXMLMath .
We can also attach a symplectic handle along LaTeXMLMath to the standard convex filling of LaTeXMLMath to get a convex filling of a new contact LaTeXMLMath -manifold LaTeXMLMath , which is therefore tight .
Since the contact surgery along LaTeXMLMath must be the same as the surgery along LaTeXMLMath , we see that LaTeXMLMath embeds contactomorphically in LaTeXMLMath .
Therefore LaTeXMLMath is tight .
∎ Before proving proposition LaTeXMLRef , we provide a summary of the relevant definitions and results from LaTeXMLCite .
In that paper we made a definition very similar to definition LaTeXMLRef , except that we worked with structures more general than open book decompositions .
Suppose LaTeXMLMath is a contact LaTeXMLMath -manifold , LaTeXMLMath is a link and LaTeXMLMath is a fibration .
The pair LaTeXMLMath is a nicely fibered link supporting LaTeXMLMath if there exists a Reeb field LaTeXMLMath for LaTeXMLMath with LaTeXMLMath and polar coordinates LaTeXMLMath near each component of LaTeXMLMath such that the following conditions are satisfied on each coordinate neighborhood of LaTeXMLMath : LaTeXMLMath for some functions LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath are constant near each component of LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are positive .
Note that this definition implies that LaTeXMLMath is tangent to LaTeXMLMath .
We may think of LaTeXMLMath as a “ fake open book decomposition ” , since the boundary of a “ page ” may multiply cover the “ binding ” .
There appears to be much more stringent control of LaTeXMLMath near LaTeXMLMath than in definition LaTeXMLRef and the orientation condition ( that LaTeXMLMath and LaTeXMLMath are positive ) looks different .
However the following lemma can be proved by direct computation using the standard model .
If the fibers of LaTeXMLMath meet LaTeXMLMath as longitudes so that we have an honest open book decomposition of LaTeXMLMath , then the condition that LaTeXMLMath is supported by the open book decomposition is equivalent , after an isotopy , to the condition that LaTeXMLMath is a nicely fibered link supporting LaTeXMLMath .
Notice that we can compare framings of LaTeXMLMath to the fibration LaTeXMLMath .
In terms of polar coordinates LaTeXMLMath near LaTeXMLMath , such a fibration LaTeXMLMath determines a “ slope ” LaTeXMLMath , while a framing LaTeXMLMath determines a family of parallel longitudes with “ slope ” LaTeXMLMath .
We say that the framing LaTeXMLMath is positive with respect to LaTeXMLMath if , on each component of LaTeXMLMath , LaTeXMLMath .
When LaTeXMLMath gives an honest open book decomposition , LaTeXMLMath is positive with respect to LaTeXMLMath if , on each component LaTeXMLMath of LaTeXMLMath , LaTeXMLMath for some positive integer LaTeXMLMath .
Following is the main result we need from LaTeXMLCite .
This is essentially theorem 1.2 and addendum 5.1 in LaTeXMLCite , but we also include some points that are made in their proofs .
Suppose that LaTeXMLMath is a nicely fibered link in LaTeXMLMath supporting a contact structure LaTeXMLMath and that LaTeXMLMath is a framing of LaTeXMLMath which is positive with respect to LaTeXMLMath .
Let LaTeXMLMath be LaTeXMLMath with a LaTeXMLMath -handle attached along each component of LaTeXMLMath in LaTeXMLMath with framing LaTeXMLMath .
Let LaTeXMLMath and let LaTeXMLMath be the other component of LaTeXMLMath , both oriented opposite to the boundary orientation .
Let LaTeXMLMath be the union of the ascending spheres of the LaTeXMLMath -handles .
Then LaTeXMLMath supports a symplectic structure LaTeXMLMath and a Liouville vector field LaTeXMLMath defined near LaTeXMLMath pointing in along both LaTeXMLMath and LaTeXMLMath and inducing the given contact structure LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the restrictions of LaTeXMLMath to LaTeXMLMath and LaTeXMLMath , respectively .
Then we have the following relation between LaTeXMLMath and LaTeXMLMath : There exist polar coordinates LaTeXMLMath on a neighborhood LaTeXMLMath of LaTeXMLMath as in definition LaTeXMLRef , a closed tubular neighborhood LaTeXMLMath of LaTeXMLMath , a positive constant LaTeXMLMath , a function LaTeXMLMath which is LaTeXMLMath outside LaTeXMLMath and a function of LaTeXMLMath inside LaTeXMLMath , and an orientation-reversing diffeomorphism LaTeXMLMath such that : LaTeXMLMath The pair LaTeXMLMath is a nicely fibered link supporting LaTeXMLMath .
Here is the idea of the relationship between LaTeXMLMath and LaTeXMLMath : First we perform the topological surgery by removing the interior of LaTeXMLMath from LaTeXMLMath and then collapsing each component of LaTeXMLMath to a circle , to get the new link LaTeXMLMath .
The theorem makes a judicious choice of LaTeXMLMath and LaTeXMLMath , so as to arrange that LaTeXMLMath is a negative contact form on LaTeXMLMath and furthermore extends across LaTeXMLMath to a contact form LaTeXMLMath on LaTeXMLMath .
Then we reverse the orientation to treat LaTeXMLMath as a positive contact structure on LaTeXMLMath .
Let LaTeXMLMath , with binding LaTeXMLMath and fibration LaTeXMLMath .
Proposition LaTeXMLRef asks us to attach a handle along each component LaTeXMLMath of LaTeXMLMath with framing LaTeXMLMath .
Theorem LaTeXMLRef tells us that we get a symplectic cobordism from LaTeXMLMath to LaTeXMLMath .
We must show that LaTeXMLMath is supported by the open book decomposition LaTeXMLMath .
Because the framing of the surgery is LaTeXMLMath for each component LaTeXMLMath , the fibers of LaTeXMLMath still meet LaTeXMLMath as longitudes , so that we do have an honest open book decomposition of LaTeXMLMath which supports LaTeXMLMath and with pages diffeomorphic to LaTeXMLMath .
To compute the monodromy , note that a meridian before the surgery becomes a longitude with framing LaTeXMLMath , so that our original monodromy LaTeXMLMath is now measuring the monodromy via a flow which is longitudinal near LaTeXMLMath .
Correcting this flow to be meridinal changes the monodromy by a left twist on each boundary component , so that , properly measured , the monodromy for LaTeXMLMath is LaTeXMLMath .
We should reverse the flow to get the monodromy for LaTeXMLMath , which is thus LaTeXMLMath .
∎
Stochastic antiderivational equations on Banach spaces over local non-Archimedean fields are investigated .
Theorems about existence and uniqueness of the solutions are proved under definite conditions .
In particular Wiener processes are considered in relation with the non-Archimedean analog of the Gaussian measure .
address : Laboratoire de Mathèmatiques Pures , Complexe Scientifique des Cèzeaux , 63177 AUBIÈRE Cedex , France .
permanent address : Theoretical Department , Institute of General Physics , Str .
Vavilov 38 , Moscow , 119991 GSP-1 , Russia .
This article continues investigations of stochastic processes on non-Archimedean spaces ( LaTeXMLCite ) .
In the first part stochastic processes were defined on Banach spaces over non-Archimedean local fields and the analogs of It LaTeXMLMath formula were proved .
This part is devoted to stochastic antiderivational equations .
In the non-Archimedean case antiderivational equations are used instead of stochastic integral or differential equations in the classical case .
Stochastic differential equations on real Banach spaces and manifolds are widely used for solutions of mathematical and physical problems and for construction and investigation of measures on them LaTeXMLCite .
Wide classes of quasi-invariant measures including analogous to Gaussian type on non-Archimedean Banach spaces , loops and diffeomorphisms groups were investigated in LaTeXMLCite .
Quasi-invariant measures on topological groups and their configuration spaces can be used for the investigations of their unitary representations ( see LaTeXMLCite and references therein ) .
In view of this developments non-Archimedean analogs of stochastic equations and diffusion processes need to be investigated .
Some steps in this direction were made in LaTeXMLCite .
There are different variants for such activity , for example , LaTeXMLMath -adic parameters analogous to time , but spaces of complex-valued functions .
At the same time measures may be real , complex or with values in a non-Archimedean field .
In the classical stochastic analysis indefinite integrals are widely used , but in the non-Archimedean case they have quite another meaning , because the field of LaTeXMLMath -adic numbers LaTeXMLMath has not any linear order structure compatible with its normed field structure ( see Part I ) .
This work treats the case which was not considered by another authors and that is suitable and helpful for the investigation of stochastic processes and quasi-invariant measures on non-Archimedean topological groups .
In §2 suitable analogs of Gaussian measures are considered .
Certainly they have not any complete analogy with the classical one , some of their properties are similar and some are different .
They are used for the definiton of the standard ( Wiener ) stochastic process .
Integration by parts formula for the non-Archimedean stochastic processes is studied .
Some particular cases of the general It LaTeXMLMath formula from Part I are dicussed here more concretely .
In §3 with the help of them stochastic antiderivational equations are defined and investigated .
Analogs of theorems about existence and uniquiness of solutions of stochastic antiderivational equations are proved .
Generating operators of solutions of stochastic equations are investigated .
All results of this paper are obtained for the first time .
In this part the notations of Part I also are used .
2.1 .
Let LaTeXMLMath be a Banach space over a local field LaTeXMLMath with an ordinal LaTeXMLMath and the standard orthonormal base LaTeXMLMath , LaTeXMLMath with LaTeXMLMath on the LaTeXMLMath -th place .
Let LaTeXMLMath be a cylindrical algebra generated by projections on finite-dimensional over LaTeXMLMath subspaces LaTeXMLMath in LaTeXMLMath and Borel LaTeXMLMath -algebras LaTeXMLMath .
Denote by LaTeXMLMath the minimal LaTeXMLMath -algebra LaTeXMLMath generated by LaTeXMLMath .
When LaTeXMLMath , then LaTeXMLMath .
Each vector LaTeXMLMath is considered as continuous linear functional on LaTeXMLMath by the formula LaTeXMLMath for each LaTeXMLMath , so there is the natural embedding LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath .
The field LaTeXMLMath is the finite algebraic extension of of the field LaTeXMLMath of LaTeXMLMath -adic numbers and as the Banach space over LaTeXMLMath it is isomorphic with LaTeXMLMath , that is , each LaTeXMLMath has the form LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , in particular for values LaTeXMLMath for LaTeXMLMath .
All continuous characters LaTeXMLMath of LaTeXMLMath as the additive group have the form LaTeXMLEquation for each LaTeXMLMath , LaTeXMLMath for LaTeXMLMath where LaTeXMLMath is a root of unity , LaTeXMLMath LaTeXMLMath , LaTeXMLMath ( see §25 LaTeXMLCite and §I.3.6 , about the spaces LaTeXMLMath of operators see §I.2 ) .
Each LaTeXMLMath is locally constant , hence LaTeXMLMath is also continuous , where LaTeXMLMath denotes the discrete group of all roots of LaTeXMLMath ( by multiplication ) .
Let us consider functions , whose Fourier transform has the form : LaTeXMLEquation where the Fourier transform was defined in §7 LaTeXMLCite and LaTeXMLCite , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Definition .
A cylindrical measure LaTeXMLMath on LaTeXMLMath is called LaTeXMLMath -Gaussian , if each its one-dimensional projection is LaTeXMLMath -Gaussian , that is , LaTeXMLEquation where LaTeXMLMath is the Haar measure on LaTeXMLMath with values in LaTeXMLMath , where LaTeXMLMath is a continuous LaTeXMLMath -linear functional on LaTeXMLMath giving projection on one-dimensional subspace in LaTeXMLMath , LaTeXMLMath are constants such that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath may depend on LaTeXMLMath , LaTeXMLMath is independent of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is the first countable ordinal .
If LaTeXMLMath is a measure on LaTeXMLMath , then LaTeXMLMath denotes its characteristic functional , that is , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath is the character of LaTeXMLMath as the additive group ( see §I.3.6 ) .
2.2 .
Theorem .
A non-negative LaTeXMLMath -Gaussian measure LaTeXMLMath on LaTeXMLMath is LaTeXMLMath -additive on LaTeXMLMath if and only if there exists an injective compact operator LaTeXMLMath for a chosen LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation are measures on LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Moreover , each one-dimensional projection LaTeXMLMath has the following characteristic functional : LaTeXMLEquation where LaTeXMLMath .
Proof .
Let LaTeXMLMath be a characteristic functional of LaTeXMLMath .
By the non-Archimedean analog of the Minlos-Sazonov Theorem ( see §2.31 in LaTeXMLCite , LaTeXMLCite ) a measure LaTeXMLMath is LaTeXMLMath -additive if and only if for each LaTeXMLMath there exists a compact operator LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath with LaTeXMLMath , where LaTeXMLMath .
From the definition of LaTeXMLMath to be LaTeXMLMath -Gaussian it follows , that each its projection LaTeXMLMath on LaTeXMLMath has the form given by Equation LaTeXMLMath .
It remains to establish that LaTeXMLMath is LaTeXMLMath -additive if and only if LaTeXMLMath and LaTeXMLMath .
In view of Lemma 2.3 LaTeXMLCite LaTeXMLMath is LaTeXMLMath -additive if and only if each sequence of finite-dimensional ( over LaTeXMLMath distributions ) satisfies two conditions : LaTeXMLMath for each LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath , LaTeXMLMath LaTeXMLMath .
Take in particular LaTeXMLMath for each LaTeXMLMath .
We have LaTeXMLMath LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are constants independent from LaTeXMLMath for LaTeXMLMath and each LaTeXMLMath , LaTeXMLMath is fixed ( see also the proof of Lemma 2.8 LaTeXMLCite and Theorem II.2.1 LaTeXMLCite ) .
Evidently , LaTeXMLMath is correctly defined for each LaTeXMLMath if and only if LaTeXMLMath .
In this case the character LaTeXMLMath is defined and LaTeXMLMath .
Therefore , if LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath is LaTeXMLMath -additive .
Let LaTeXMLMath .
Since LaTeXMLMath is the local field there exists LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath Put LaTeXMLMath .
Then LaTeXMLMath , since LaTeXMLMath , where LaTeXMLMath with LaTeXMLMath .
Consequently , LaTeXMLMath .
We denumerate the standard orthonormal basis LaTeXMLMath such that LaTeXMLMath .
There exists an operator LaTeXMLMath on LaTeXMLMath with matrix elements LaTeXMLMath for each LaTeXMLMath , LaTeXMLMath for each LaTeXMLMath .
Then LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath are the standard projectors on LaTeXMLMath .
When LaTeXMLMath , then evidently , LaTeXMLMath has the form given by Equation LaTeXMLMath , since LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath for each LaTeXMLMath .
Suppose now that LaTeXMLMath .
For this we consider LaTeXMLMath , where LaTeXMLMath .
On the other hand , there exists a constant LaTeXMLMath such that for LaTeXMLMath and each LaTeXMLMath there is the following inequality : LaTeXMLMath LaTeXMLMath .
From the estimates of Lemma II.1.1 LaTeXMLCite and using the substitution LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath we get that LaTeXMLMath is not LaTeXMLMath -additive , consequently , LaTeXMLMath is not LaTeXMLMath -additive , since LaTeXMLMath are cylindrical Borel subsets for each LaTeXMLMath , where LaTeXMLMath is the induced projection on LaTeXMLMath for each LaTeXMLMath .
For the verification of Formula LaTeXMLMath it is sufficient at first to consider the measure LaTeXMLMath on the algebra LaTeXMLMath of cylindrical subsets in LaTeXMLMath .
Then for each projection LaTeXMLMath , where LaTeXMLMath , we have : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , consequently , LaTeXMLMath , since LaTeXMLMath for each LaTeXMLMath Since LaTeXMLMath , then LaTeXMLMath is the Radon measure , consequently , the continuation of LaTeXMLMath from LaTeXMLMath produces LaTeXMLMath on the Borel LaTeXMLMath -algebra of LaTeXMLMath , hence LaTeXMLMath where LaTeXMLMath is the natural projection on LaTeXMLMath for each LaTeXMLMath such that LaTeXMLMath .
Using expressions of LaTeXMLMath we get Formula LaTeXMLMath .
From this follows , that if LaTeXMLMath , then LaTeXMLMath exists for each LaTeXMLMath if and only if LaTeXMLMath , since LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath .
2.3 .
Corollary .
LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath .
Proof .
In view of the ultrametric inequality LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath for each LaTeXMLMath , then from Formula 2.2 .
( iv ) the statement of this Corollary follows .
2.4 .
Remark .
Let LaTeXMLMath be a compact subset without isolated points in a local field LaTeXMLMath , for example , LaTeXMLMath .
Then the Banach space LaTeXMLMath has the Amice polynomial orthonormal base LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath LaTeXMLCite .
Suppose LaTeXMLMath are antiderivations from §80 LaTeXMLCite , where LaTeXMLMath .
Each LaTeXMLMath has a decomposition LaTeXMLMath , where LaTeXMLMath .
These decompositions establish the isometric isomorphism LaTeXMLMath such that LaTeXMLMath .
Since LaTeXMLMath is homeomorphic with LaTeXMLMath , then LaTeXMLMath is a linear injective compact operator such that LaTeXMLMath , where LaTeXMLMath here corresponds to LaTeXMLMath antiderivation operator by Schikhof ( see also §§54 , 80 LaTeXMLCite and §I.2.1 ) .
The Banach space LaTeXMLMath is dense in LaTeXMLMath .
Using Theorem 2.2 and Note I.2.3 for LaTeXMLMath we get a LaTeXMLMath -Gaussian measure on LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath , we put LaTeXMLMath for each LaTeXMLMath , LaTeXMLMath are projectors , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is the generator of the valuation group of LaTeXMLMath .
If LaTeXMLMath then the Banach space LaTeXMLMath is isomorphic with the tensor product LaTeXMLMath ( see §4.R LaTeXMLCite ) .
Therefore , the antiderivation LaTeXMLMath on LaTeXMLMath induces the antiderivation LaTeXMLMath on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath ( see also Theorem 4.33 LaTeXMLCite ) .
Put LaTeXMLMath and LaTeXMLMath , then each LaTeXMLMath induces the LaTeXMLMath -Gaussian measure LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath are LaTeXMLMath -Gaussian measures on LaTeXMLMath induced by LaTeXMLMath as above .
In particular for LaTeXMLMath we also can take LaTeXMLMath .
The LaTeXMLMath -Gaussian measure on LaTeXMLMath induced by LaTeXMLMath with LaTeXMLMath we call standard .
Analogously considering the following Banach subspace LaTeXMLMath LaTeXMLMath and operators LaTeXMLMath we get the LaTeXMLMath -Gaussian measures LaTeXMLMath on it also , where LaTeXMLMath is a marked point .
Certainly , we can take others operators LaTeXMLMath not related with the antiderivation as above .
3.1 .
A measurable space LaTeXMLMath with a normalised non-negative measure LaTeXMLMath is called a probability space and is denoted by LaTeXMLMath , where LaTeXMLMath is a LaTeXMLMath -algebra of LaTeXMLMath .
Points LaTeXMLMath are called elementary events and values LaTeXMLMath are called probabilities of events LaTeXMLMath .
A measurable map LaTeXMLMath is called a random variable with values in LaTeXMLMath , where LaTeXMLMath is a LaTeXMLMath -algebra of LaTeXMLMath ( see §I.4.1 ) .
3.2 .
We define a ( non-Archimedean ) Wiener process LaTeXMLMath with values in LaTeXMLMath as a stochastic process such that : LaTeXMLMath the differences LaTeXMLMath and LaTeXMLMath are independent for each chosen LaTeXMLMath , LaTeXMLMath and LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , either LaTeXMLMath or LaTeXMLMath is not in the two-element set LaTeXMLMath where LaTeXMLMath LaTeXMLMath the random variable LaTeXMLMath has a distribution LaTeXMLMath where LaTeXMLMath is a probability Gaussian measure on LaTeXMLMath described in §§2.1 , 2.4 , LaTeXMLMath for LaTeXMLMath and each LaTeXMLMath a continuous linear functional LaTeXMLMath is given by the formula LaTeXMLMath for each LaTeXMLMath where LaTeXMLMath LaTeXMLMath we also put LaTeXMLMath that is , we consider a Banach subspace LaTeXMLMath of LaTeXMLMath , where LaTeXMLMath .
If LaTeXMLMath is not a Gaussian measure on LaTeXMLMath and a stochastic process LaTeXMLMath satisfies conditions LaTeXMLMath , then it is called the ( non-Archimedean ) stochastic process ( see §I.4.2 ) .
If LaTeXMLMath is the standard Gaussian measure on LaTeXMLMath , then the Wiener process is called standard ( see also Theorem 3.23 , Lemmas 2.3 , 2.5 , 2.8 and §3.30 in LaTeXMLCite ) .
3.3 .
Remark .
In Part I the non-Archimedean analogs of the It LaTeXMLMath formula were proved .
In the particular case LaTeXMLMath we have LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are functions ( see §§4.2 , 4.6 LaTeXMLCite and §3.2 ) , so that LaTeXMLEquation .
LaTeXMLEquation for each LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath LaTeXMLMath and LaTeXMLMath , that is LaTeXMLMath commute .
In particular LaTeXMLMath , that is LaTeXMLMath where LaTeXMLMath is the Schikhof linear continuous antiderivation operator ( see for comparison §80 LaTeXMLCite ) .
In the classical case measures are real-valued and functions LaTeXMLMath are with values in Banach spaces over LaTeXMLMath or LaTeXMLMath .
But in the considered here case measures are real-valued and functions are with values in Banach spaces over non-Archimedean fields LaTeXMLMath , so the mean value LaTeXMLMath is real and not with values in LaTeXMLMath .
This leads to differences with the classical case , in particular formula LaTeXMLMath ( see Lemma 3.5 LaTeXMLCite ) is not valid , but there exists its another analog .
Let LaTeXMLMath be a locally compact Hausdorff space and LaTeXMLMath denotes a subspace of LaTeXMLMath consisting of bounded continuous functions LaTeXMLMath such that for each LaTeXMLMath there exists a compact subset LaTeXMLMath for which LaTeXMLMath for each LaTeXMLMath .
In particular for LaTeXMLMath , LaTeXMLMath and a fixed LaTeXMLMath in accordance with Theorem 7.22 LaTeXMLCite there exists a LaTeXMLMath -valued tight measure LaTeXMLMath on the LaTeXMLMath -algebra LaTeXMLMath of clopen subsets in LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is a topologically conjugate space , LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath is a continuous character of LaTeXMLMath as the additive group , then LaTeXMLMath due to Condition LaTeXMLMath For LaTeXMLMath independent from LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ( so that LaTeXMLMath ) it takes a simpler form , which can be considered as another analog of the classical formula .
For the evaluation of appearing integrals tables from §1.5.5 LaTeXMLCite can be used .
Another important result is the following theorem .
Theorem .
Let LaTeXMLMath LaTeXMLMath be the stochastic process on the Banach space LaTeXMLMath over LaTeXMLMath .
Then there exists a function LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath and each LaTeXMLMath and for each LaTeXMLMath .
Proof .
Let LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is the approximation of the identity in LaTeXMLMath , LaTeXMLMath for LaTeXMLMath ( see §I.2.1 LaTeXMLCite and §3.2 ) .
In view of Conditions LaTeXMLMath and the Hahn-Banach theorem ( see LaTeXMLCite ) there exists a projection operator LaTeXMLMath such that LaTeXMLMath , since LaTeXMLMath for each LaTeXMLMath and for each LaTeXMLMath , where LaTeXMLMath is the characteristic functional of the measure LaTeXMLMath corresponding to LaTeXMLMath , that is , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath is the character of LaTeXMLMath as the additive group , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is the Borel measure on LaTeXMLMath ( see also §I.3.6 ) .
The random variable LaTeXMLMath has the distribution LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath .
On the other hand the projection operator LaTeXMLMath commutes with the antiderivation operator LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath is defined pointwise for each LaTeXMLMath .
In LaTeXMLMath the family of step functions LaTeXMLMath is dense , where LaTeXMLMath , LaTeXMLMath is the characteristic function of LaTeXMLMath , LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath is nonnegative .
For each LaTeXMLMath there exists LaTeXMLMath in LaTeXMLMath ( see Theorem I.2.14 ) .
If LaTeXMLMath , then LaTeXMLMath LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , LaTeXMLMath LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , LaTeXMLMath LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath On the other hand LaTeXMLMath is completely defined by the family LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is the standard orthonormal base of LaTeXMLMath .
Hence the family LaTeXMLMath completely characterize LaTeXMLMath due to Equations LaTeXMLMath , when LaTeXMLMath .
For each LaTeXMLMath and each LaTeXMLMath the function LaTeXMLMath is continuous by LaTeXMLMath , consequently , there exists a continuous function LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , since characters LaTeXMLMath are continuous from LaTeXMLMath to LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath and the LaTeXMLMath -linear span of the family LaTeXMLMath of characters is dense in LaTeXMLMath by the Stone-Weierstrass theorem LaTeXMLCite .
On the other hand , LaTeXMLMath , when LaTeXMLMath for a sequence LaTeXMLMath in LaTeXMLMath .
Therefore , LaTeXMLEquation .
LaTeXMLEquation From the equality LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath the statement of this theorem follows for each LaTeXMLMath .
3.4 .
Theorem .
Let LaTeXMLMath and LaTeXMLMath LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath and LaTeXMLMath and LaTeXMLMath where LaTeXMLMath and LaTeXMLMath satisfy the local Lipschitz condition : LaTeXMLMath for each LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Then the stochastic process of the following type : LaTeXMLMath LaTeXMLMath has the unique solution .
Proof .
We have LaTeXMLMath , hence LaTeXMLMath for each LaTeXMLMath and for each LaTeXMLMath and each LaTeXMLMath and each LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are positive constants , LaTeXMLMath and LaTeXMLMath are short notations of LaTeXMLMath and LaTeXMLMath for LaTeXMLMath respectively .
For solving equation LaTeXMLMath we use iterations : LaTeXMLMath , … , LaTeXMLMath , consequently , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are short notations of LaTeXMLMath and LaTeXMLMath respectively .
Let LaTeXMLMath be a mean value of a real-valued distribution LaTeXMLMath by LaTeXMLMath , where LaTeXMLMath is the probabilty space , then LaTeXMLMath , where LaTeXMLMath for each LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
While LaTeXMLMath we put LaTeXMLMath , for LaTeXMLMath we take LaTeXMLMath .
Also LaTeXMLEquation .
LaTeXMLEquation in particular for LaTeXMLMath .
On the other hand , LaTeXMLEquation consequently , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
For each LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Therefore , there exists the unique solution on each LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , hence there exists LaTeXMLMath , where LaTeXMLMath here LaTeXMLMath is an arbitrary ball of radius LaTeXMLMath in LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are two solutions , then LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , since LaTeXMLMath has a disjoint covering by balls LaTeXMLMath , on each such ball there exists the unique solution with a given initial condition on it ( that is , in a chosen point LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are independent from LaTeXMLMath ) .
Therefore , LaTeXMLMath , hence LaTeXMLMath for each LaTeXMLMath , LaTeXMLMath due to Condition LaTeXMLMath , where LaTeXMLMath is the short notation of LaTeXMLMath .
The term LaTeXMLMath has the infinite-dimensional over LaTeXMLMath range in LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
If LaTeXMLMath for each LaTeXMLMath and almost all LaTeXMLMath , then LaTeXMLMath which is possible only for LaTeXMLMath .
If LaTeXMLMath and the function LaTeXMLMath is locally constant by LaTeXMLMath and independent from LaTeXMLMath , then LaTeXMLMath is locally constant by LaTeXMLMath and independent from LaTeXMLMath only for LaTeXMLMath and LaTeXMLMath due to definitions of LaTeXMLMath and LaTeXMLMath , hence LaTeXMLMath , since it is evident for LaTeXMLMath and LaTeXMLMath depending on LaTeXMLMath locally polynomially or polyhomogeneously for each LaTeXMLMath , but such locally polynomial or polyhomogeneous functions by LaTeXMLMath are dense in LaTeXMLEquation .
LaTeXMLEquation 3.5 .
Theorem .
Let LaTeXMLMath and LaTeXMLMath LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath and LaTeXMLMath where LaTeXMLMath and LaTeXMLMath satisfy the local Lipschitz condition ( see 3.4 . ( LLC ) ) .
A stochastic process of the type LaTeXMLMath LaTeXMLMath LaTeXMLMath such that LaTeXMLMath ( continuous and bounded on its domain ) for each LaTeXMLMath LaTeXMLMath and LaTeXMLMath LaTeXMLMath for each LaTeXMLMath when LaTeXMLMath , or each LaTeXMLMath when LaTeXMLMath , for each LaTeXMLMath Then LaTeXMLMath has the unique solution in LaTeXMLMath .
Proof .
Let LaTeXMLMath , … , LaTeXMLEquation consequently , LaTeXMLEquation .
LaTeXMLMath where in general LaTeXMLMath , LaTeXMLMath for each LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath for each LaTeXMLMath , LaTeXMLMath is the same constant as in §3.4 , LaTeXMLMath .
On the other hand , LaTeXMLEquation consequently , LaTeXMLEquation .
Due to Condition LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLEquation .
Therefore , there exists the unique solution on each LaTeXMLMath , since LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath , hence there exists LaTeXMLMath , where LaTeXMLMath here LaTeXMLMath is an arbitrary ball of radius LaTeXMLMath in LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are two solutions , then LaTeXMLMath as in §3.4 .
If LaTeXMLMath is a polyhomogeneous function , then there exists LaTeXMLMath such that differentials LaTeXMLMath for each LaTeXMLMath , but its antiderivative LaTeXMLMath has LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath , which we can apply to a convergent series considering terms LaTeXMLMath for each LaTeXMLMath .
Therefore , LaTeXMLMath , where the function LaTeXMLMath is locally constant by LaTeXMLMath and independent from LaTeXMLMath , hence LaTeXMLMath , since it is evident for LaTeXMLMath and LaTeXMLMath and LaTeXMLMath depending on LaTeXMLMath locally polynomially or polyhomogeneously for each LaTeXMLMath , but such locally polynomial or polyhomogeneous functions by LaTeXMLMath are dense in LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation respectively .
3.6 .
Proposition .
Let LaTeXMLMath be the Wiener process given by Equation LaTeXMLMath with the LaTeXMLMath -Gaussian measure associated with the operator LaTeXMLMath as in §2.4 and let also LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath LaTeXMLMath -almost everywhere by LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are non-negative constants .
Then LaTeXMLMath with probability LaTeXMLMath has a LaTeXMLMath -modification and LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
Proof .
For the following function LaTeXMLMath in accordance with Theorem I.4.6 LaTeXMLCite we have LaTeXMLMath LaTeXMLEquation hence LaTeXMLMath , since LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath , since LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation hence LaTeXMLMath , since LaTeXMLMath as a function by LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath and due to the definition of LaTeXMLMath .
Considering in particular polyhomogeneous LaTeXMLMath on which LaTeXMLMath takes its maximum value we get LaTeXMLMath .
Since LaTeXMLMath for the Markov measure LaTeXMLMath induced by the transition measures LaTeXMLMath for LaTeXMLMath of the non-Archimedean Wiener process ( see §2.2 ) , then LaTeXMLMath has with the probability LaTeXMLMath a LaTeXMLMath -modification .
Note .
If to consider a general stochastic process as in §I.4.3 , then from the proof of Proposition 3.6 it follows , that LaTeXMLMath with the probability LaTeXMLMath has a modification in the space LaTeXMLMath , where LaTeXMLMath is a nondegenerate correlation operator of the product measure LaTeXMLMath on LaTeXMLMath .
3.7 .
Proposition .
Let LaTeXMLMath be a stochastic process given by Equation LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath LaTeXMLMath -almost everywhere by LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are non-negative constants .
Then two solutions LaTeXMLMath and LaTeXMLMath with initial conditions LaTeXMLMath and LaTeXMLMath satisfy the following inequality : LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
Proof .
From §3.6 it follows , that LaTeXMLMath , since LaTeXMLMath .
3.8 .
Remark .
Let LaTeXMLMath and LaTeXMLMath be two stochastic processes corresponding to LaTeXMLMath and a Banach algebra LaTeXMLMath over LaTeXMLMath in §I.4.6 LaTeXMLCite .
Then LaTeXMLMath , where LaTeXMLMath .
Hence LaTeXMLMath .
Therefore , LaTeXMLEquation which is the non-Archimedean analog of the integration by parts formula , where in all terms LaTeXMLMath is displayed on the left from LaTeXMLMath .
For two LaTeXMLMath functions LaTeXMLMath and LaTeXMLMath we have LaTeXMLMath or LaTeXMLMath , that is terms with LaTeXMLMath are absent , consequently , LaTeXMLMath .
In a particular case LaTeXMLMath this leads two LaTeXMLMath , where the last term corresponds two LaTeXMLMath .
This means that LaTeXMLEquation .
For LaTeXMLMath and LaTeXMLMath the integration by parts formula gives LaTeXMLMath .
Such that LaTeXMLMath , for example , for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath this gives LaTeXMLMath .
Therefore , LaTeXMLMath , that is the important difference of the non-Archimedean and classical cases ( see for comparison Exer .
4.3 and Theorem 4.5 LaTeXMLCite ) .
If LaTeXMLMath is a Banach space over the local field LaTeXMLMath and LaTeXMLMath is a LaTeXMLMath -bilinear functional on it , where LaTeXMLMath is an image of LaTeXMLMath under an embedding LaTeXMLMath associated with the standard orthonormal base LaTeXMLMath in LaTeXMLMath , then LaTeXMLEquation hence LaTeXMLMath and LaTeXMLMath 3.9 .
Definition .
If LaTeXMLMath is a stochastic process and LaTeXMLMath is a family of bounded linear operators satisfying the following Conditions LaTeXMLMath LaTeXMLMath LaTeXMLMath , where LaTeXMLMath LaTeXMLMath LaTeXMLMath , LaTeXMLMath LaTeXMLMath for each LaTeXMLMath , LaTeXMLMath LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath is a positive nonrandom constant , LaTeXMLMath , then LaTeXMLMath is called a multiplicative operator functional of the stochastic process LaTeXMLMath .
If LaTeXMLMath is a system of random variables on LaTeXMLMath with values in LaTeXMLMath , satisfying almost surely Conditions LaTeXMLMath and uniformly by LaTeXMLMath Condition LaTeXMLMath such that LaTeXMLMath LaTeXMLMath , then such multiplicative operator functional is called homogeneous .
An operator LaTeXMLMath LaTeXMLMath is called the generating operator of the evolution family LaTeXMLMath .
If LaTeXMLMath depends on LaTeXMLMath , then LaTeXMLMath is also considered as the random variable on LaTeXMLMath ( depending on the parameter LaTeXMLMath ) with values in LaTeXMLMath .
3.10 .
Remark .
Let LaTeXMLMath be a linear continuous operator on a Banach space LaTeXMLMath over LaTeXMLMath such that it depends strongly continuously on LaTeXMLMath , that is LaTeXMLMath is continuous by LaTeXMLMath for each chosen LaTeXMLMath and LaTeXMLMath .
Then the solution of the differential equation LaTeXMLMath LaTeXMLMath LaTeXMLMath though LaTeXMLMath may be non-unique , where LaTeXMLMath is an initial condition , LaTeXMLMath .
The solution of Equation LaTeXMLMath exists using the method of iterations ( see §3.4 ) .
Indeed , in view of Lemma I.2.3 LaTeXMLCite LaTeXMLMath and LaTeXMLMath LaTeXMLMath .
If to consider a solution of the antiderivational equation LaTeXMLMath , then it is a solution of the Cauchy problem LaTeXMLMath LaTeXMLMath , LaTeXMLMath .
Therefore , LaTeXMLMath hence LaTeXMLMath is not dependent from LaTeXMLMath , consequently , there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath LaTeXMLMath for each LaTeXMLMath .
From this it follows , that LaTeXMLMath LaTeXMLMath for each LaTeXMLMath .
In particular , if LaTeXMLMath is a constant operator , then there exists a solution LaTeXMLMath ( see about LaTeXMLMath in Proposition 45.6 LaTeXMLCite ) .
Equation LaTeXMLMath has a solution under milder conditions , for example , LaTeXMLMath is weakly continuous , that is LaTeXMLMath is continuous for each LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath is differentiable by LaTeXMLMath and LaTeXMLMath satisfies Equation LaTeXMLMath in the weak sense and there exists a weak solution of LaTeXMLMath coinciding with LaTeXMLMath .
If to substitute LaTeXMLMath on another operator LaTeXMLMath , then for the corresponding evolution operator LaTeXMLMath there is the following inequality : LaTeXMLMath LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath is for LaTeXMLMath .
Proposition .
Let LaTeXMLMath and two sequences LaTeXMLMath and LaTeXMLMath be given of strongly continuous on LaTeXMLMath bounded linear operators and LaTeXMLMath be evolution operators corresponding to LaTeXMLMath , where LaTeXMLMath .
If LaTeXMLMath , then there exists a sequence LaTeXMLMath which is also uniformly bounded .
If there exists LaTeXMLMath strongly and uniformly converging to LaTeXMLMath in LaTeXMLMath , then LaTeXMLMath also can be chosen strongly and uniformly convergent .
Proof .
From the use of Equations LaTeXMLMath iteratively for LaTeXMLMath and LaTeXMLMath and also for LaTeXMLMath and taking LaTeXMLMath it follows , that LaTeXMLMath LaTeXMLMath for each LaTeXMLMath .
Therefore , LaTeXMLMath , hence LaTeXMLMath , since LaTeXMLMath .
If LaTeXMLMath in LaTeXMLMath and LaTeXMLMath is uniformly convergent to LaTeXMLMath , then for each LaTeXMLMath there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath for each LaTeXMLMath and LaTeXMLMath due to Equality LaTeXMLMath .
3.11 .
Proposition .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be the same as in §3.5 .
Then Equation LaTeXMLMath has the unique solution LaTeXMLMath in LaTeXMLMath for each initial value LaTeXMLMath and it can be represented in the following form : LaTeXMLMath LaTeXMLMath , where LaTeXMLMath is the multiplicative operator functional .
Proof .
In view of Theorem 3.5 , Definition 3.9 , Remark and Proposition 3.10 with the use of a parameter LaTeXMLMath the statement of Proposition 3.11 follows .
3.12 .
Let now consider the case LaTeXMLMath ( see §3.6 ) , for example , the standard Wiener process .
Corollary .
Let a function LaTeXMLMath satisfies conditions of §I.4.8 LaTeXMLCite , then a generating operator of an evolution family LaTeXMLMath of a stochastic process LaTeXMLMath is given by the following equation : LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Proof .
In view of Theorem I.4.8 LaTeXMLCite and Proposition 3.11 there exists a generating operator of an evolution family .
From Lemma I.2.3 and Formula LaTeXMLMath LaTeXMLCite it follows the statement of this Corollary .
Remark .
If LaTeXMLMath satisfies conditions either of §I.4.6 or of §I.4.7 , then Formula LaTeXMLMath takes simpler forms , since the corresponding terms vanish .
The author is sincerely grateful to I.V .
Volovich for his interest to this work and fruitful discussions .
H ilbert functions and resolutions for ideals of LaTeXMLMath fat points in P LaTeXMLMath Brian Harbourne Department of Mathematics and Statistics University of Nebraska-Lincoln Lincoln , NE 68588-0323 email : bharbour @ math.unl.edu WEB : http : //www.math.unl.edu/ LaTeXMLMath bharbour/ University of Nebraska-Lincoln Lincoln , NE 68588-0323 email : bharbour @ math.unl.edu WEB : http : //www.math.unl.edu/ LaTeXMLMath bharbour/ Joaquim Roé Departament d ’ Àlgebra i Geometria Universitat de Barcelona Barcelona 08007 , Spain email : jroevell @ mat.ub.es April 24 , 2001 Universitat de Barcelona Barcelona 08007 , Spain email : jroevell @ mat.ub.es April 24 , 2001 Abstract : Conjectures for the Hilbert function of the LaTeXMLMath th symbolic power of the ideal of LaTeXMLMath general points of P LaTeXMLMath are verified for infinitely many LaTeXMLMath for each square LaTeXMLMath , using an approach developed by the authors in a previous paper .
In those cases that LaTeXMLMath is even , conjectures for the resolution are also verified .
Previously , by work of Evain ( for the Hilbert function ) and of Harbourne , Holay and Fitchett ( for the resolution ) , these conjectures were known for infinitely many LaTeXMLMath for a given LaTeXMLMath only for LaTeXMLMath being a power of 4 .
I .
Introduction Consider the ideal LaTeXMLMath generated by all forms having multiplicity at least LaTeXMLMath at LaTeXMLMath given general points of P LaTeXMLMath .
This is a graded ideal , and thus we can consider the Hilbert function LaTeXMLMath whose value at each nonnegative integer LaTeXMLMath is the dimension LaTeXMLMath of the homogeneous component LaTeXMLMath of LaTeXMLMath of degree LaTeXMLMath .
In spite of a good deal of work , until recently LaTeXMLMath was known only when either LaTeXMLMath or LaTeXMLMath was small ( see Figure 1 , a color postscript graphic , which can be viewed , in color , also at http : //www.math.unl.edu/ LaTeXMLMath bharbour/Hilb1.jpg ) .
Alexander and Hirschowitz [ AH ] determined LaTeXMLMath for each LaTeXMLMath for all LaTeXMLMath sufficiently large compared with LaTeXMLMath ( in all dimensions , not just for P LaTeXMLMath ) ; however , it is unclear how large is large enough .
The first explicit determination when both LaTeXMLMath and LaTeXMLMath can be large was given by Evain [ Ev ] , whose result applies for all LaTeXMLMath whenever LaTeXMLMath is a power of 4 .
( Evain ’ s method also seems to work as long as LaTeXMLMath is a square divisible only by 2 , 3 or 5 . )
We [ HR ] recently found a different method and applied it to determine LaTeXMLMath in a great many additional cases in which both LaTeXMLMath and LaTeXMLMath can be large , but , unlike Evain ’ s result , for these cases the larger LaTeXMLMath is , the larger LaTeXMLMath must be also .
Here we show that our method also applies , like Evain ’ s , to determine LaTeXMLMath for infinitely many LaTeXMLMath for various LaTeXMLMath ( in fact , whenever LaTeXMLMath is a square ) .
In addition , applying the results of [ HHF ] , we also obtain the resolution of the ideal LaTeXMLMath for infinitely many LaTeXMLMath whenever LaTeXMLMath is an even square .
Thus with this paper and with our recent results , we can update the data of Figure 1 ; the updated data is shown in Figure 2 ( on the web this is located at http : //www.math.unl.edu/ LaTeXMLMath bharbour/Hilb2.jpg ) .
See Figures 3 and 4 for the corresponding data for resolutions ( on the web at http : //www.math.unl.edu/ LaTeXMLMath bharbour/Res1.jpg and http : //www.math.unl.edu/ LaTeXMLMath bharbour/Res2.jpg ) .
[ Note that these figures do not show all cases for which the algorithm of [ HR ] determines the Hilbert function or resolution , just certain cases for which the algorithm is especially easy to analyze ( i.e. , Corollaries V.2 and V.4 of [ HR ] ) and those additional cases we analyze here ( i.e. , Theorem II.3 ) .
There are additional cases which a computer search shows that the algorithm handles , but for which a simple statement seems hard to give . ]
II .
Conjectures and Main Result As shown in Figure 1 , LaTeXMLMath was determined for all LaTeXMLMath by Nagata , when LaTeXMLMath .
The result in these cases turns out to be a bit complicated .
Conjectures have been given ( [ H1 ] , [ Hi1 ] ) which imply that these complications disappear for LaTeXMLMath : Conjecture II.1 : For LaTeXMLMath general points of P LaTeXMLMath , LaTeXMLMath for each integer LaTeXMLMath .
Similarly , as shown in Figure 3 , the resolution of LaTeXMLMath is known ( [ Cat ] , [ H2 ] ) when LaTeXMLMath , but it also is somewhat complicated .
These complications are conjectured to disappear for LaTeXMLMath ( [ H2 ] , [ HHF ] ) : Conjecture II.2 : For LaTeXMLMath , the minimal free resolution of LaTeXMLMath is LaTeXMLEquation where LaTeXMLMath is the least LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is the direct sum of LaTeXMLMath copies of the ring LaTeXMLMath , regarded as an LaTeXMLMath -module with the grading LaTeXMLMath .
We now show that Conjecture II.1 holds for infinitely many LaTeXMLMath whenever LaTeXMLMath is a square , and ( as a consequence of [ HHF ] ) that Conjecture II.2 holds for infinitely many LaTeXMLMath whenever LaTeXMLMath is an even square .
For the purpose of stating the theorem , given any positive integer LaTeXMLMath , let LaTeXMLMath be the largest integer LaTeXMLMath such that LaTeXMLMath .
Theorem II.3 : Consider LaTeXMLMath general points of P LaTeXMLMath .
Let LaTeXMLMath be any nonnegative integer , and let LaTeXMLMath , where LaTeXMLMath is an integer satisfying LaTeXMLMath if LaTeXMLMath is even , or LaTeXMLMath if LaTeXMLMath is odd .
Then Conjecture II.1 holds for LaTeXMLMath if LaTeXMLMath is odd , and Conjecture II.1 and Conjecture II.2 both hold if LaTeXMLMath is even .
To prove Theorem II.3 we will , in the particular case of uniform multiplicities , use an algorithm developed in [ HR ] .
The algorithm gives bounds on LaTeXMLMath , defined to be the least LaTeXMLMath such that LaTeXMLMath .
We begin by briefly recalling the algorithm .
The parenthetical remarks below can be ignored ; they are meant only as a reminder of the geometry underlying the algorithm as developed in [ HR ] , and are not necessary for applying the algorithm .
Let LaTeXMLMath ( thus LaTeXMLMath is the genus of a plane curve of degree LaTeXMLMath ) .
Let LaTeXMLMath denote the LaTeXMLMath -tuple LaTeXMLMath , where the entry 1 occurs LaTeXMLMath times .
Given any LaTeXMLMath -tuple LaTeXMLMath of integers , with respect to the obvious bilinear form we have LaTeXMLMath .
( The LaTeXMLMath -tuple LaTeXMLMath corresponds to a divisor LaTeXMLMath on the blow up LaTeXMLMath of P LaTeXMLMath at the LaTeXMLMath general points , LaTeXMLMath corresponds to the proper transform LaTeXMLMath of a curve of degree LaTeXMLMath passing through LaTeXMLMath of the points and LaTeXMLMath is just the usual intersection product LaTeXMLMath , and hence equals the degree of LaTeXMLMath . )
Assume that LaTeXMLMath satisfies LaTeXMLMath .
Define LaTeXMLMath by LaTeXMLMath , and then define LaTeXMLMath to be what is obtained from LaTeXMLMath by first converting any of the 2nd through LaTeXMLMath st entries to 0 if it is negative and then permuting the 2nd through LaTeXMLMath st entries to be in descending order .
For convenience , define LaTeXMLMath to be the first entry of LaTeXMLMath ( thus LaTeXMLMath where LaTeXMLMath is the total transform to LaTeXMLMath of a line in P LaTeXMLMath ) .
We thus get a sequence of tuples , LaTeXMLMath , LaTeXMLMath , etc .
; let LaTeXMLMath be the least LaTeXMLMath such that LaTeXMLMath .
By [ HR ] , we then have LaTeXMLMath if the following two conditions hold : ( 1 ) LaTeXMLMath for LaTeXMLMath , and ( 2 ) LaTeXMLMath , where LaTeXMLMath is the sum of the second to LaTeXMLMath th entries of LaTeXMLMath .
The algorithm is simply to determine the biggest LaTeXMLMath such that the given conditions are satisfied .
( Geometrically , LaTeXMLMath and hence LaTeXMLMath is not linearly equivalent to an effective divisor .
According to the analysis in [ HR ] , LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath mean that LaTeXMLMath has no global sections , which by induction implies the divisor corresponding to LaTeXMLMath is not linearly equivalent to an effective divisor . )
In order to analyze the algorithm , we will use the following lemma , for which we define LaTeXMLMath to be the least LaTeXMLMath such that all entries of LaTeXMLMath except possibly the first are 0 .
Lemma II.4 : Let LaTeXMLMath .
Then for all LaTeXMLMath LaTeXMLEquation .
Proof : Let LaTeXMLMath be the LaTeXMLMath -tuple LaTeXMLMath , and for LaTeXMLMath let LaTeXMLMath , where 1 ’ s occupy positions LaTeXMLMath through LaTeXMLMath .
For LaTeXMLMath , it is not hard to check that LaTeXMLMath , where LaTeXMLMath with LaTeXMLMath , and therefore LaTeXMLEquation .
On the other hand , LaTeXMLMath , and it is easy to see that LaTeXMLEquation from which the claim follows .
LaTeXMLMath Our next result improves ( for uniform multiplicities ) on Theorem I.1 ( a ) ( i ) of [ HR ] , which assumed that LaTeXMLMath , whereas here we only require that LaTeXMLMath .
Proposition II.5 : Consider LaTeXMLMath general points of P LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be nonnegative integers such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , and let LaTeXMLMath .
Then LaTeXMLMath .
Proof : It is easy to check that LaTeXMLMath , so if LaTeXMLMath , it follows that LaTeXMLMath , and thus LaTeXMLMath .
Since LaTeXMLMath , it follows from Lemma II.4 that LaTeXMLMath for all LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
To conclude that LaTeXMLMath , it is now enough to check that LaTeXMLMath for LaTeXMLMath , where LaTeXMLMath is the sum of the second to LaTeXMLMath th entries of LaTeXMLMath .
If LaTeXMLMath ( i.e. , LaTeXMLMath ) we have LaTeXMLMath by hypothesis and hence LaTeXMLMath by definition of LaTeXMLMath .
If LaTeXMLMath ( so LaTeXMLMath ) , by definition of LaTeXMLMath we at least have LaTeXMLMath , so LaTeXMLMath .
But LaTeXMLMath implies LaTeXMLMath , and by hypothesis LaTeXMLMath ( so LaTeXMLMath ) ; therefore LaTeXMLMath as we wanted .
LaTeXMLMath We now give the proof of our main result : Proof of Theorem II.3 : We apply Proposition II.5 with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
We claim that LaTeXMLMath , where LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd .
But LaTeXMLMath because LaTeXMLMath .
To see LaTeXMLMath , note that LaTeXMLMath simplifies to LaTeXMLMath if LaTeXMLMath is even and to LaTeXMLMath if LaTeXMLMath is odd .
Therefore ( by definition of LaTeXMLMath ) we have to check that LaTeXMLMath and LaTeXMLMath if LaTeXMLMath is even , and that LaTeXMLMath and LaTeXMLMath if LaTeXMLMath is odd .
The first inequality follows from LaTeXMLMath and LaTeXMLMath respectively .
For the second , substituting LaTeXMLMath for LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath for LaTeXMLMath if LaTeXMLMath is odd , LaTeXMLMath and LaTeXMLMath resp .
become LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd .
Thus LaTeXMLMath and LaTeXMLMath resp .
hold if LaTeXMLMath is an integer satisfying LaTeXMLMath if LaTeXMLMath is even , and LaTeXMLMath if LaTeXMLMath is odd .
This shows by Proposition II.5 that LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd .
But since LaTeXMLMath points of multiplicity LaTeXMLMath impose at most LaTeXMLMath conditions on forms of degree LaTeXMLMath , it follows that LaTeXMLMath , and it is easy to check that LaTeXMLMath whenever LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd .
Thus in fact we have LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd , whenever LaTeXMLMath is of the form LaTeXMLMath , with LaTeXMLMath as given in the statement of Theorem II.3 .
Of course , LaTeXMLMath for all LaTeXMLMath , and by [ HHF ] , we know that LaTeXMLMath for all LaTeXMLMath ( apply Lemma 5.3 of [ HHF ] , keeping in mind our explicit expression for LaTeXMLMath ) .
This proves our claims regarding verification of Conjecture II.1 .
As for Conjecture II.2 , when LaTeXMLMath is an even square , apply Theorem 5.1 ( a ) of [ HHF ] .
This concludes the proof .
LaTeXMLMath III .
Figures Here we show graphs of what was known until recently regarding Conjecture II.1 and Conjecture II.2 , and what is known now .
For Figure 1 , the references are : [ N ] for Nagata , [ Hi2 ] for Hirschowitz , [ CM1 ] and [ CM2 ] for Ciliberto and Miranda , and [ Ev ] for Evain .
The additional data shown in Figure 2 is simply a graphical representation of Corollary V.2 of [ HR ] , and Theorem II.3 .
For Figure 3 , the references are : [ GGR ] for Geramita , Gregory and Roberts , [ Cat ] for Catalisano , [ H2 ] for Harbourne , [ Id ] for Idà , and [ HHF ] for Harbourne , Holay , and Fitchett .
As before , the additional data shown in Figure 4 is simply a graphical representation of Corollary V.4 of [ HR ] , and Theorem II.3 .
Figure 1 Figure 2 Figure 3 Figure 4 References [ AH ] J. Alexander and A. Hirschowitz .
An asymptotic vanishing theorem for generic unions of multiple points , Invent .
Math .
140 ( 2000 ) , no .
2 , 303–325 .
[ CM1 ] C. Ciliberto and R. Miranda .
Degenerations of planar linear systems , J. Reine Angew .
Math .
501 ( 1998 ) , 191-220 .
[ CM2 ] C. Ciliberto and R. Miranda .
Linear systems of plane curves with base points of equal multiplicity , Trans .
Amer .
Math .
Soc .
352 ( 2000 ) , 4037–4050 .
[ Cat ] M. V. Catalisano .
‘ ‘ Fat ’ ’ points on a conic , Comm .
Alg .
19 ( 8 ) ( 1991 ) , 2153–2168 .
[ Ev ] L. Evain .
La fonction de Hilbert de la réunion de LaTeXMLMath gros points génériques de P LaTeXMLMath de même multiplicité , J. Alg .
Geom .
8 ( 1999 ) , 787–796 .
[ GGR ] A. V. Geramita , D. Gregory and L. Roberts .
Monomial ideals and points in projective space , J .
Pure and Appl .
Alg .
40 ( 1986 ) , 33–62 .
[ H1 ] B. Harbourne .
The geometry of rational surfaces and Hilbert functions of points in the plane , Can .
Math .
Soc .
Conf .
Proc .
6 ( 1986 ) , 95–111 .
[ H2 ] .
The Ideal Generation Problem for Fat Points , J .
Pure and Applied Alg .
145 ( 2000 ) , 165–182 .
[ HHF ] B. Harbourne , S. Holay , and S. Fitchett .
Resolutions of Ideals of Quasiuniform Fat Point Subschemes of P LaTeXMLMath , preprint ( 2000 ) .
[ HR ] B. Harbourne and J. Roé .
Linear systems with multiple base points in P LaTeXMLMath , preprint ( 2000 ) .
[ Hi1 ] A. Hirschowitz .
Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques , Journ .
Reine Angew .
Math .
397 ( 1989 ) , 208–213 .
[ Hi2 ] A. Hirschowitz .
La méthode d ’ Horace pour l ’ interpolation à plusieurs variables , Manus .
Math .
50 ( 1985 ) , 337–388 .
[ Id ] M. Idà .
The minimal free resolution for the first infinitesimal neighborhoods of LaTeXMLMath general points in the plane , J. Alg .
216 ( 1999 ) , 741–753 .
[ N ] M. Nagata .
On rational surfaces , II , Mem .
Coll .
Sci .
Univ .
Kyoto , Ser .
A Math .
33 ( 1960 ) , 271–293 .
Chern classes for representations of reductive groups Arnaud B EAUVILLE Introduction Let LaTeXMLMath be a complex connected reductive group , and let LaTeXMLMath be its representation ring .
As an abelian group LaTeXMLMath is spanned by the finite-dimensional representations LaTeXMLMath of LaTeXMLMath , with the relations LaTeXMLMath ; the ring structure is defined by the tensor product .
Moreover the exterior product of representations give rise to a sequence of operations LaTeXMLMath which make LaTeXMLMath into a LaTeXMLMath -ring – see ( 1.1 ) below for the definition .
In the course on his work on the Riemann-Roch theorem , Grothendieck had the remarkable insight that this purely algebraic structure is enough to define Chern classes , without any reference to a cohomology or Chow ring .
He associated to any LaTeXMLMath -ring LaTeXMLMath a filtration of LaTeXMLMath , the LaTeXMLMath - filtration LaTeXMLMath ; the Chern classes take value in the associated graded ring LaTeXMLMath , and the Chern character is a ring homomorphism LaTeXMLMath , where LaTeXMLMath .
When applied to the Grothendieck ring LaTeXMLMath of vector bundles on a smooth algebraic variety LaTeXMLMath these definitions give back the classical ones , at least modulo torsion : the graded ring LaTeXMLMath coincides with the Chow ring LaTeXMLMath , and the Chern classes with the usual ones .
The aim of this note is to compute the Chern classes for the representation ring LaTeXMLMath – a simple exercise which I have been unable to find in the literature .
Let LaTeXMLMath be the Lie algebra of LaTeXMLMath ; we will assume that LaTeXMLMath , and therefore LaTeXMLMath , are defined over LaTeXMLMath ( alternatively , we could without any loss take our Chern classes in LaTeXMLMath ) .
We denote by LaTeXMLMath the ring of polynomial functions on LaTeXMLMath which are invariant under the action of the adjoint group .
Theorem LaTeXMLMath a ) The graded ring LaTeXMLMath is canonically isomorphic to the ring LaTeXMLMath of invariant polynomial functions on LaTeXMLMath .
b ) Let LaTeXMLMath be a representation of LaTeXMLMath , and LaTeXMLMath the corresponding representation of LaTeXMLMath .
The total Chern class LaTeXMLMath is equal to the invariant function LaTeXMLMath , and the Chern character LaTeXMLMath to LaTeXMLMath .
We have of course an explicit description of the ring LaTeXMLMath .
Let LaTeXMLMath be a maximal torus of LaTeXMLMath ; the Weyl group LaTeXMLMath acts on the ring LaTeXMLMath , and the restriction map LaTeXMLMath identifies LaTeXMLMath with the invariant sub-ring LaTeXMLMath .
As an abelian group LaTeXMLMath is spanned by one-dimensional elements ( we will say that it is a split LaTeXMLMath -ring ) , which makes easy to compute its LaTeXMLMath -filtration and Chern classes .
The theorem follows easily once we know that the LaTeXMLMath -filtration of LaTeXMLMath induces that of LaTeXMLMath .
This turns out to be a general fact for invariant sub-rings of split LaTeXMLMath -rings ; we will deduce it from the behaviour of the LaTeXMLMath -filtration under the Adams operations ( Proposition 1.6 ) .
In section 3 we discuss an application .
Let LaTeXMLMath be a principal LaTeXMLMath -bundle over a base LaTeXMLMath ( which may be a variety , or an arbitrary topos ) ; the Theorem provides a simple definition of the characteristic classes of LaTeXMLMath in the graded ring LaTeXMLMath , and a simple way of computing the Chern classes of the associated vector bundles .
1 .
Generalities on ˘ -rings In this section we recall the definition and basic properties of LaTeXMLMath -rings .
Standard references are [ SGA6 ] or [ F-L ] ; we follow the terminology of [ SGA6 ] , Exposé V. ( 1.1 ) A LaTeXMLMath - ring LaTeXMLMath is a commutative ring with two more pieces of structure : – An augmentation , that is a ring homomorphism LaTeXMLMath ; – A LaTeXMLMath -structure , that is a sequence of maps LaTeXMLMath such that , for any LaTeXMLMath in LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
If we put LaTeXMLMath , the last condition is equivalent to LaTeXMLEquation .
Moreover we want formulas giving LaTeXMLMath and LaTeXMLMath as polynomials in LaTeXMLMath and LaTeXMLMath respectively .
A convenient way of expressing these is to introduce a sequence LaTeXMLMath of additive endomorphisms of LaTeXMLMath , the Adams operations , defined by LaTeXMLEquation .
Then the condition on the LaTeXMLMath -structure means that the LaTeXMLMath are ring endomorphisms , and satisfy LaTeXMLMath for LaTeXMLMath .
( 1.2 ) Grothendieck associates to this situation a second LaTeXMLMath -structure , defined by LaTeXMLMath , and a decreasing filtration of LaTeXMLMath , the LaTeXMLMath - fitration LaTeXMLMath : we put LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is spanned by the elements LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the associated graded ring .
The Chern classes LaTeXMLMath of an element LaTeXMLMath are defined by LaTeXMLEquation ( 1.3 ) A crucial point in what follows will be the behaviour of the LaTeXMLMath -filtration with respect to the Adams operations .
Let us say that an element LaTeXMLMath of LaTeXMLMath has LaTeXMLMath - dimension LaTeXMLMath if it satisfies LaTeXMLMath and LaTeXMLMath for LaTeXMLMath ; we say that LaTeXMLMath is Q - finite LaTeXMLMath -dimensional if the LaTeXMLMath -vector space LaTeXMLMath is spanned by elements of finite LaTeXMLMath -dimension .
For such a LaTeXMLMath -ring we have , for each LaTeXMLMath and LaTeXMLMath : LaTeXMLEquation ( [ F-L ] , III , Proposition 3.1 ) .
( 1.5 ) Let us say that the LaTeXMLMath -ring LaTeXMLMath is split if is generated as an abelian group by elements of LaTeXMLMath -dimension 1 .
In this case we have LaTeXMLMath , where LaTeXMLMath is the augmentation ideal of LaTeXMLMath : indeed we have LaTeXMLMath because LaTeXMLMath is the identity , and LaTeXMLMath for all LaTeXMLMath because this holds for LaTeXMLMath if it holds for LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath if LaTeXMLMath has LaTeXMLMath -dimension 1 .
Proposition 1.6 LaTeXMLMath Let LaTeXMLMath be a split LaTeXMLMath -ring , and LaTeXMLMath a finite group of automorphisms of the LaTeXMLMath -ring LaTeXMLMath .
Then the LaTeXMLMath -invariant elements form a sub- LaTeXMLMath -ring LaTeXMLMath of LaTeXMLMath .
The filtrations LaTeXMLMath and LaTeXMLMath coincide in LaTeXMLMath .
Proof : Since the action of LaTeXMLMath commutes with the LaTeXMLMath the LaTeXMLMath -structure of LaTeXMLMath induces a LaTeXMLMath -structure on LaTeXMLMath .
Moreover the LaTeXMLMath -vector space LaTeXMLMath is spanned by the elements LaTeXMLMath , where LaTeXMLMath is an element of LaTeXMLMath of LaTeXMLMath -dimension 1 ; these elements have LaTeXMLMath -dimension LaTeXMLMath , thus LaTeXMLMath is LaTeXMLMath -finite LaTeXMLMath -dimensional .
To alleviate the notation we write LaTeXMLMath instead of LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath instead of LaTeXMLMath .
We have LaTeXMLMath for each LaTeXMLMath .
Let us first prove that the two filtrations define the same topology .
Let LaTeXMLMath be the augmentation ideal of LaTeXMLMath , and LaTeXMLMath .
Since LaTeXMLMath is a finite LaTeXMLMath -module , the ring LaTeXMLMath is artinian ; thus there exists an integer LaTeXMLMath such that LaTeXMLMath , and therefore LaTeXMLMath for all LaTeXMLMath .
As LaTeXMLMath -modules LaTeXMLMath is a direct summand of LaTeXMLMath ( consider the projector LaTeXMLMath LaTeXMLMath ) .
Thus for every ideal LaTeXMLMath of LaTeXMLMath , the induced homomorphism LaTeXMLMath is injective , which means LaTeXMLMath .
Using ( 1.5 ) we obtain LaTeXMLEquation .
Let us prove now the inclusion LaTeXMLMath by induction on LaTeXMLMath , the case LaTeXMLMath being obvious .
We have just seen that there exists an integer LaTeXMLMath such that LaTeXMLMath ; let LaTeXMLMath be the smallest integer with that property .
Assume LaTeXMLMath .
Let LaTeXMLMath , and let LaTeXMLMath be an integer ; by ( 1.4 ) we have LaTeXMLEquation .
On the other hand we have LaTeXMLMath by the induction hypothesis ; since LaTeXMLMath is LaTeXMLMath -finite LaTeXMLMath -dimensional ( 1.4 ) gives LaTeXMLEquation .
Since LaTeXMLMath we get LaTeXMLMath for all LaTeXMLMath , contradicting the choice of LaTeXMLMath .
Therefore we have LaTeXMLMath , hence LaTeXMLMath .
2 .
The ˘ -ring R ( G ) ( 2.1 ) Let LaTeXMLMath be a connected multiplicative group , and LaTeXMLMath its character group ; this is a free finitely generated abelian group .
The representation ring LaTeXMLMath is isomorphic to the group algebra LaTeXMLMath ; we denote by LaTeXMLMath its canonical basis .
The augmentation and the LaTeXMLMath -structure are characterized by LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the augmentation ideal in LaTeXMLMath .
We have a canonical isomorphism LaTeXMLMath which maps a character LaTeXMLMath to the class of LaTeXMLMath ( observe that LaTeXMLMath LaTeXMLMath ( mod .
LaTeXMLMath ) .
Since the rings LaTeXMLMath and LaTeXMLMath are regular , the canonical map LaTeXMLMath LaTeXMLMath is bijective ; using ( 1.5 ) we get a canonical isomorphism LaTeXMLEquation .
Under this isomorphism the element LaTeXMLMath of LaTeXMLMath corresponds to LaTeXMLMath .
Thus the total Chern class LaTeXMLMath of an element LaTeXMLMath is given by LaTeXMLMath .
Let LaTeXMLMath be the Lie algebra of LaTeXMLMath , viewed as a vector space over LaTeXMLMath .
We have a canonical isomorphism LaTeXMLMath , which associates to a character LaTeXMLMath its derivative LaTeXMLMath .
Thus we can identify LaTeXMLMath to the algebra LaTeXMLMath of polynomial maps LaTeXMLMath ; the above formula becomes LaTeXMLMath in LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath denote the one-dimensional representation with character LaTeXMLMath .
The element LaTeXMLMath of LaTeXMLMath is the class of the representation LaTeXMLMath .
In the corresponding representation LaTeXMLMath of LaTeXMLMath , an element LaTeXMLMath of LaTeXMLMath acts through the diagonal matrix LaTeXMLMath .
Thus LaTeXMLMath is equal to the function LaTeXMLMath in LaTeXMLMath .
The Chern character gives an homomorphism LaTeXMLMath of LaTeXMLMath into the ring LaTeXMLMath of formal series on LaTeXMLMath .
We have LaTeXMLMath , hence LaTeXMLEquation .
Remark 2.2 LaTeXMLMath The exponential morphism of formal groups LaTeXMLMath induces an injective homomorphism LaTeXMLMath , which maps LaTeXMLMath to LaTeXMLMath .
The Chern character is the composition of this map with the injection LaTeXMLMath .
( 2.3 ) Let LaTeXMLMath be a complex connected reductive group , LaTeXMLMath its Lie algebra , LaTeXMLMath a maximal torus of LaTeXMLMath and LaTeXMLMath its Lie algebra ; we can assume that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are defined over LaTeXMLMath .
The Weyl group LaTeXMLMath acts on LaTeXMLMath , hence on the ring LaTeXMLMath ; restriction to LaTeXMLMath induces a homomorphism of LaTeXMLMath -rings LaTeXMLMath , whose image is the invariant sub-ring LaTeXMLMath ( [ SGA6 ] , Exposé 0 App. , Th .
1.1 ) .
By Proposition 1.6 , the graded ring LaTeXMLMath is isomorphic to LaTeXMLMath , that is to the ring LaTeXMLMath of invariant polynomials on LaTeXMLMath ( 2.1 ) .
Let LaTeXMLMath be a representation of LaTeXMLMath .
From the commutative diagrams LaTeXMLEquation we obtain LaTeXMLEquation .
The restriction map LaTeXMLMath induces an isomorphism LaTeXMLMath ( [ B ] , §8 , Th .
1 ) .
Since the functions LaTeXMLMath and LaTeXMLMath are invariant , the above equalities hold as well in LaTeXMLMath and LaTeXMLMath respectively .
This proves the theorem stated in the introduction .
3 .
Application : Chern classes of associated bundles ( 3.1 ) Let LaTeXMLMath be an algebraic variety , and LaTeXMLMath a principal LaTeXMLMath -bundle over LaTeXMLMath ( for what follows LaTeXMLMath could be as well a scheme , or even a topos ) .
To any representation LaTeXMLMath is associated a vector bundle LaTeXMLMath on LaTeXMLMath ; we define in this way a homomorphism of LaTeXMLMath -rings LaTeXMLMath 1 In fancy terms , the principal bundle LaTeXMLMath corresponds to a morphism LaTeXMLMath of LaTeXMLMath into the classifying topos ( or stack ) LaTeXMLMath , and LaTeXMLMath is just the pull-back map .
.
In view of the isomorphism described above , it induces a homomorphism of graded rings In fancy terms , the principal bundle LaTeXMLMath corresponds to a morphism LaTeXMLMath of LaTeXMLMath into the classifying topos ( or stack ) LaTeXMLMath , and LaTeXMLMath is just the pull-back map .
LaTeXMLEquation called the characteristic homomorphism .
Let LaTeXMLMath .
Recall that there are homogeneous functions LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath ( [ B ] , §8 , Théorème 1 ) .
The elements LaTeXMLMath may be called the characteristic classes of LaTeXMLMath ( but note that they depend on the choice of the generating sequence LaTeXMLMath ) .
If LaTeXMLMath is a smooth variety , the graded ring LaTeXMLMath is canonically isomorphic to the rational Chow ring LaTeXMLMath ( [ SGA6 ] , Exposé XIV , n o 4 ) ; our characteristic classes correspond under this isomorphism to those defined in [ V ] and [ E-G ] .
Proposition 3.2 LaTeXMLMath Let LaTeXMLMath be a representation of LaTeXMLMath ; write LaTeXMLMath LaTeXMLMath , where LaTeXMLMath is a polynomial in LaTeXMLMath indeterminates .
Let LaTeXMLMath be a principal LaTeXMLMath -bundle on LaTeXMLMath , with characteristic classes LaTeXMLMath in LaTeXMLMath .
Then the total Chern class in LaTeXMLMath of the associated bundle LaTeXMLMath is LaTeXMLEquation .
Similarly , if LaTeXMLMath , with LaTeXMLMath , we have LaTeXMLMath .
Proof : This follows from the Theorem , the commutative diagram LaTeXMLEquation and the corresponding diagram for the Chern character .
Examples 3.3 LaTeXMLMath The Proposition provides a way of computing the characteristic classes in terms of Chern classes , at least for classical groups .
We use the standard generating system LaTeXMLMath given for instance in [ B ] , §13 .
We denote by LaTeXMLMath the vector bundle on LaTeXMLMath associated to LaTeXMLMath through the standard representation of LaTeXMLMath in LaTeXMLMath .
a ) For LaTeXMLMath , we have LaTeXMLMath for LaTeXMLMath and LaTeXMLMath ; this gives LaTeXMLMath for LaTeXMLMath .
b ) For LaTeXMLMath or LaTeXMLMath , we have LaTeXMLMath ; this gives LaTeXMLMath for LaTeXMLMath ( the vector bundle LaTeXMLMath is isomorphic to its dual , thus its odd Chern classes with rational coefficients vanish ) .
c ) For LaTeXMLMath , we realize LaTeXMLMath as the space of skew-symmetric matrices in LaTeXMLMath ; we have LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath .
We obtain LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath is a class in LaTeXMLMath with square LaTeXMLMath .
REFERENCES [ B ] N. B OURBAKI : Groupes et algèbres de Lie , chapitre VIII .
Hermann , Paris ( 1975 ) .
[ E-G ] D. E DIDIN , W. G RAHAM : Characteristic classes in the Chow ring .
J. Algebraic Geom .
6 ( 1997 ) , 431–443 .
[ F-L ] W. F ULTON , S. L ANG : Riemann-Roch algebra .
Grundlehren der Math .
277 .
Springer-Verlag , New York ( 1985 ) .
[ SGA6 ] Théorie des intersections et théorème de Riemann-Roch .
Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 ( SGA 6 ) .
Dirigé par P. Berthelot , A. Grothendieck et L. Illusie .
Lecture Notes in Math .
225 , Springer-Verlag , Berlin-New York ( 1971 ) .
[ V ] A. V ISTOLI : Characteristic classes of principal bundles in algebraic intersection theory .
Duke Math .
J .
58 ( 1989 ) , 299–315 .
Arnaud B EAUVILLE Laboratoire J.-A .
Dieudonné UMR 6621 du CNRS U NIVERSITÉ DE N ICE Parc Valrose F-06108 N ICE Cedex 2
We discuss the mathematical aspects of wave field measurements used in traveltime inversion from seismograms .
The primary information about the medium is assumed to be carried by the wave front set and its perturbation with repsect to a hypothetical background medium is to be estimated .
By a convincing heuristics a detection procedure for this perturbation was proposed based on optimization of wave field correlations .
We investigate its theoretical foundation in simple mathematical case studies using the distribution theoretic definition of oscillatory integrals .
In this paper , we investigate how to carry out tomography directly in terms of wavefield measurements .
Tomography , in its original form , uses a ‘ measured ’ wavefront set as input in an inversion procedure which is solely ( symplectic ) geometric in nature , viz .
based upon finding bicharacteristics that result through a canonical relation in matching the measurement .
In ‘ wave-equation ’ tomography , one aims at replacing the geometric procedure by a wave-solution procedure , but keeping the wavefront set of the measurements as the primary source of information about the medium .
Following an embedding procedure to formulate the inverse problem , i.e .
introducing a background medium and incident field and a medium contrast and scattered field , we then face the problem of detecting perturbations in the wavefront set associated with the scattered ( perturbed LaTeXMLMath incident ) field .
An intuitive choice is based upon correlating the perturbed field with the incident field .
We will show , by example , that such procedure should be carried out delicately .
In fact , we conclude that the perturbation of the wavefront set can be derived from the singular support ( of the derivative ) of the proposed time correlation .
The outline of the paper is as follows .
We briefly review the microlocal representation of solutions to the scalar wave equation ( Section 2 ) .
In Section 3 we introduce the measuring process and its mathematical implementation ; we describe how the wavefront set of the wavefield propagates through this measuring process .
When we perturb the coefficient function in the wave equation ( the wave speed ) the solution representation will be perturbed .
In particular , its wavefront set will shift in the measurement-variables cotangent bundle .
We formulate the process of correlating , within the measuring process , the perturbed representation with the original representation , and identify how such shift appears in the result .
It is conjectured that the derivative of the ( time ) correlation at any given measurement position has its singular support precisely at the time shift associated with the perturbation of the wavefront set .
In Section 4 we give examples to illustrate the conjecture .
Special attention is paid how to define the product of distribution solutions within the correlation process .
Finally , in Section 5 , we discuss a method of detecting the singular support of the correlation in time at any measuring position by means of ‘ localized ’ Fourier transforms .
The procedure defines a criterion to develop wave-equation tomography .
The scalar wave equation for acoustic waves in a constant density medium is given by LaTeXMLEquation with LaTeXMLEquation where LaTeXMLMath .
The equation is considered on an open domain LaTeXMLMath and in a time interval LaTeXMLMath .
We decouple the wave equation into its forward and backward components .
To this end , we introduce the elliptic operator LaTeXMLMath and its square root LaTeXMLMath .
Decomposing the field according to LaTeXMLEquation in combination with the source decomposition LaTeXMLEquation then results in the equivalent system of equations LaTeXMLEquation .
Throughout , we assume that LaTeXMLMath .
We will construct operators LaTeXMLMath with distribution kernels LaTeXMLMath that solve the initial value problem equivalent to ( LaTeXMLRef ) with LaTeXMLMath .
Let LaTeXMLMath denote the Hamiltonian either for the forward or backward wave propagation .
The Hamilton system of equations that generates the Hamiltonian flow or bicharacteristics is given by LaTeXMLEquation .
Observe that LaTeXMLMath implies LaTeXMLMath .
Equation ( LaTeXMLRef ) can be solved , microlocally , in the form of a Fourier integral representation .
The phase of the associated Fourier integral operator follows from the canonical relations LaTeXMLEquation .
Let LaTeXMLEquation denote coordinates on LaTeXMLMath .
A function LaTeXMLMath will locally describe LaTeXMLMath according to LaTeXMLEquation and generates the non-degenerate phase function LaTeXMLEquation .
In our notation , we will suppress the dependence on LaTeXMLMath and collect LaTeXMLMath in the phase variables LaTeXMLMath .
The canonical relation can then be written as LaTeXMLEquation .
We synthesize the canonical relation LaTeXMLMath with associated ( non-degenerate ) phase function LaTeXMLMath if LaTeXMLMath , LaTeXMLMath if LaTeXMLMath .
In accordance with ( LaTeXMLRef ) we obtain LaTeXMLEquation .
With this fundamental solution , the solution of ( LaTeXMLRef ) and its dependence on the initial conditions can then be written in the form of a Fourier integral operator ( FIO ) with amplitude LaTeXMLMath .
In fact , LaTeXMLMath is a section of the tensor product LaTeXMLMath of the Keller-Maslov line bundle and the half-densities on LaTeXMLMath .
The kernel of the FIO admits an oscillatory integral ( OI ) representation .
In the remainder of this paper we consider such OIs to represent ‘ the wavefield ’ .
Perturbation of this wavefield are induced by perturbation of the coefficient function LaTeXMLMath .
As described above , each component of the wave field as well as the perturbed wave field can be represented by an OI , LaTeXMLEquation where LaTeXMLMath is a non-degenerate phase function and LaTeXMLMath a symbol ( LaTeXMLCite , Sect .
7.8 ) ; note that the wave front set satisfies the inclusion ( LaTeXMLCite , Thm .
8.1.9 ) LaTeXMLEquation .
Measurements are recordings of the wave field LaTeXMLMath in stations at certain points LaTeXMLMath in the acquisition manifold over some time interval LaTeXMLMath ; mathematically , this corresponds to the restriction of the distribution LaTeXMLMath to the one-dimensional submanifolds LaTeXMLMath followed by further restriction of the resulting one-dimensional distribution LaTeXMLMath of time to the open interval LaTeXMLMath .
While the second of those restrictions is always possible and straightforward , the first can be carried out as continuous map only on distributions satisfying the following condition ( LaTeXMLCite , Thm .
8.2.4 and Cor .
8.2.7 ) LaTeXMLEquation .
Note that by ( LaTeXMLRef ) this condition is satisfied if and only if LaTeXMLMath whenever LaTeXMLMath .
If it holds , the restriction LaTeXMLMath can be defined as the pullback LaTeXMLMath of LaTeXMLMath under the embedding map LaTeXMLMath and by ( LaTeXMLRef ) we have the wave front set relation LaTeXMLEquation .
Let LaTeXMLMath be a phase function , LaTeXMLMath a symbol , both with the same domains and supports as LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath be the oscillatory integral defined by them ; assume that LaTeXMLMath also satisfies ( LaTeXMLRef ) and set LaTeXMLMath .
In case we are interested only in a certain time window of measurement we may use further cut-offs and achieve that LaTeXMLMath and LaTeXMLMath are compactly supported .
For LaTeXMLMath denote by LaTeXMLMath the translation by LaTeXMLMath on LaTeXMLMath .
If the distributional product LaTeXMLMath can be defined and yields an integrable distribution ( LaTeXMLCite , Sect .
4.5 ) we define the value of the correlation function at LaTeXMLMath by LaTeXMLEquation .
The correlation is bilinear in LaTeXMLMath .
Whenever there is no ambiguity about the distributions LaTeXMLMath and LaTeXMLMath and the point LaTeXMLMath under consideration we will denote the correlation briefly by LaTeXMLMath .
Whenever LaTeXMLMath and LaTeXMLMath represent the unperturbed and perturbed solution , then typically LaTeXMLMath ( the frequencies coincide ) and therefore for certain values of LaTeXMLMath we expect the cotangent components of the wave front sets of LaTeXMLMath and LaTeXMLMath to be identical on the overlap of singular supports .
That means that , unless both cotangent parts are only half rays on the same side of LaTeXMLMath , Hörmander ’ s condition ( LaTeXMLCite , Thm .
8.2.1 ) for defining the product does not apply .
But within the hierarchy of distributional products described by Oberguggenberger ( LaTeXMLCite , Ch .
II ) this condition , ‘ WF favorable ’ , appears only as one out of a variety of consistent possibilities to give a distributional meaning to the product under consideration .
We apply some of these to the analysis of the correlation function in some examples below to explore and illustrate whether and how the correlation , after restriction , can provide information about shifts in wave front set from LaTeXMLMath to LaTeXMLMath .
It will become clear that the customary criterion of searching for the ‘ stationary point ’ of the correlation ( Dahlen , Hung and Nolet LaTeXMLCite , Zhao , Jordan and Chapman LaTeXMLCite and Luo and Schuster LaTeXMLCite ) for detecting the shift in wave front sets is generally incorrect .
Here , we would like to point out that the appropriate mathematical framework to deal with the multiplication ( and also the restrictability ) in a uniform and systematic manner is Colombeau ’ s theory of generalized functions ( cf .
LaTeXMLCite ) .
Such framework will enable us to cope with the integrability question ( forming the correlation ) at the same time ( LaTeXMLCite ) .
Practically , we will have to consider regularizations or approximations to the formal expression LaTeXMLMath of the correlation either to give a meaning to the product or to make the integration ( i.e. , distributional action on LaTeXMLMath ) well-defined .
This amounts to the attempt of defining LaTeXMLMath as the pointwise ( in LaTeXMLMath ) limit of sequences LaTeXMLEquation as LaTeXMLMath where LaTeXMLMath is a suitable regularization or approximation of LaTeXMLMath .
We compare the singular supports , or rather the wave front sets , of LaTeXMLMath and LaTeXMLMath .
Their offset expresses the amount of time shift of the wave fronts ( or rather singularities ) at location LaTeXMLMath by the perturbation .
First we observe that under a natural time evolution condition on the phase function a restrictable OI is representable as an OI in one dimension .
If LaTeXMLMath satisfies condition ( LaTeXMLRef ) at LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath then the restriction LaTeXMLMath to LaTeXMLMath is the OI on LaTeXMLMath ( i.e. , in the time variable ) where LaTeXMLMath is considered as a parameter in the phase and amplitude .
Therefore LaTeXMLEquation .
By assumption LaTeXMLMath defines a phase function on LaTeXMLMath .
We have LaTeXMLMath and LaTeXMLMath is continuous on the subspace of restrictable distributions .
Therefore we may use any standard OI regularization LaTeXMLMath and obtain LaTeXMLMath .
Since the latter is an OI regularization in one dimension with phase function LaTeXMLMath and symbol LaTeXMLMath the assertion is proved .
∎ Note that the usual stationary phase argument applied to this one-dimensional OI gives the same upper bound for the wave front set as established above in ( LaTeXMLRef ) .
Assuming that the perturbed solution LaTeXMLMath is given as an OI with phase function LaTeXMLMath and amplitude LaTeXMLMath we can compare the wave front sets of their measurements at LaTeXMLMath ( restrictions to LaTeXMLMath ) .
As pointed out above , the perturbation will affect the phase function only in its LaTeXMLMath - and LaTeXMLMath -gradient , i.e. , we may assume that LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath for some LaTeXMLMath with LaTeXMLMath ; similarly if LaTeXMLMath then LaTeXMLMath for some LaTeXMLMath with LaTeXMLMath .
In any microlocal representation of the solution to the wave equation , in the absence of attenuation , the phase contains LaTeXMLMath only linearly , say , in the form LaTeXMLMath for some conjugate ( frequency ) variable LaTeXMLMath .
As was shown in Section 1 , typical phase functions are of the special form LaTeXMLMath and LaTeXMLMath .
In this case the stationary phase conditions ( see eq .
( LaTeXMLRef ) ) in the wave front sets read LaTeXMLEquation .
LaTeXMLEquation and the respective LaTeXMLMath -derivatives of the phases yield cotangent components LaTeXMLMath and LaTeXMLMath .
By the ( positive ) homogeneity of LaTeXMLMath and LaTeXMLMath w.r.t .
LaTeXMLMath their first order derivatives w.r.t .
those variables are ( positively ) homogeneous of degree LaTeXMLMath .
Hence , if we are detecting time-like forward ( resp .
backward ) cotangent directions , i.e. , LaTeXMLMath ( resp .
LaTeXMLMath ) , we may rescale the arguments in the phase and obtain the time shifts LaTeXMLEquation for the corresponding ( slowness co-vector ) projections LaTeXMLMath , LaTeXMLMath satisfying the conditions LaTeXMLEquation .
In LaTeXMLCite a traveltime inversion method is described that uses optimal fitting of traveltimes from synthetic seismograms according to wave equation solutions of velocity model perturbations .
The fitting criterion is based upon a crosscorrelation function of the observed ( LaTeXMLMath ) and the synthetic ( LaTeXMLMath ) seismic data .
This crosscorrelation of LaTeXMLCite corresponds to the correlation function defined in ( LaTeXMLRef ) above .
We give a brief schematic description of this interesting fitting strategy and test its theoretical validity in three simple examples below .
Assume that LaTeXMLMath represents the observed ( or perturbed ) wave field and LaTeXMLMath is the solution of a velocity model which is parametrized by the variable velocity LaTeXMLMath .
We assume that LaTeXMLMath is a real-valued smooth function .
Therefore the correlation function is actually dependent on time LaTeXMLMath and the velocity LaTeXMLMath which we indicate in the notation LaTeXMLEquation where LaTeXMLMath denotes the scalar and LaTeXMLMath the functional argument of LaTeXMLMath .
An intuitive expectation would then be that at the exact traveltime shift induced by the perturbation , we find optimum match ( overlap ) of the corresponding seismograms and therefore the crosscorrelation should be maximal .
Leaving possible maxima at time interval boundaries aside , we search for a LaTeXMLMath relation that gives stationarity of the crosscorrelation , i.e. , LaTeXMLEquation .
Naively speaking we can consider this to be an implicit definition of a functional relationship between LaTeXMLMath and LaTeXMLMath .
( Observe that LaTeXMLMath is an infinite-dimensional variable and therefore more attention is to be paid to the exact meaning of applying an ‘ implicit function theorem ’ below . )
Under the condition that LaTeXMLMath we would therefore try to solve equation ( LaTeXMLRef ) locally for LaTeXMLMath as a function of LaTeXMLMath and find a quasi-explicit representation by LaTeXMLEquation .
Consider LaTeXMLMath and LaTeXMLMath , two Dirac deltas travelling along the lines LaTeXMLMath and LaTeXMLMath respectively .
( These are distributional pullbacks of LaTeXMLMath , the Dirac measure located at LaTeXMLMath , via the maps LaTeXMLMath and LaTeXMLMath . )
Assume that LaTeXMLMath ; the opposite sign case is completely symmetric .
We clearly have LaTeXMLMath and LaTeXMLMath , and therefore LaTeXMLEquation yielding a singularity shift of LaTeXMLMath .
Observe that LaTeXMLMath and LaTeXMLMath have disjoint singular supports unless LaTeXMLMath in which case their product would require to multiply LaTeXMLMath with itself .
This can not be done consistently within the hierarchy of distributional products ( cf .
LaTeXMLCite ) and calls for a systematic treatment in the framework of algebras of generalized functions .
However , here we touch upon those aspects only in terms of regularizations .
Choose a rapidly decaying smooth function LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath , in other words LaTeXMLMath is a mollifier , and set LaTeXMLMath .
Denote by LaTeXMLMath and LaTeXMLMath the convolutions of LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
Then we have LaTeXMLEquation and upon integration of LaTeXMLMath w.r.t .
LaTeXMLMath with a change of the variable LaTeXMLMath we obtain for the regularized correlation function LaTeXMLEquation .
If we let LaTeXMLMath we observe that LaTeXMLMath pointwise for LaTeXMLMath and LaTeXMLMath .
Hence , in an approximative sense , the singular support of the correlation LaTeXMLMath contains the time shift information .
To be more precise , it is not difficult to show that in the sense of distributions LaTeXMLEquation .
For this , we just note that for arbitrary LaTeXMLMath one may change the variable in LaTeXMLMath to LaTeXMLMath and use the fact that LaTeXMLMath for rapidly decreasing functions LaTeXMLMath and LaTeXMLMath .
In particular , this shows that here the correlation is stable under changes within the chosen class of regularizations since the limit does not depend on LaTeXMLMath .
Curiously enough , the regularization approach also gives the correct answer when using the procedure of LaTeXMLCite .
Define the short-hand notation LaTeXMLMath and consider LaTeXMLEquation and set LaTeXMLMath .
We see that LaTeXMLEquation which is proportional to LaTeXMLMath at LaTeXMLMath and stays nonzero for all LaTeXMLMath and LaTeXMLMath close enough .
This in particular true at LaTeXMLMath in which case LaTeXMLMath .
Therefore , in a ( possibly smaller ) neighborhood of these values for LaTeXMLMath and LaTeXMLMath we can solve the implicit equation LaTeXMLMath for LaTeXMLMath and find locally LaTeXMLEquation .
We find from this by integration over LaTeXMLMath ( close to LaTeXMLMath ) that LaTeXMLEquation which is the correct shift of the singular support .
We set LaTeXMLMath and LaTeXMLMath ( where LaTeXMLMath is the Heaviside function ) yielding exactly the same configuration of wave front sets as in the previous case .
In this case , restricting our attention again to LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
The only critical product appears if LaTeXMLMath : At this point we have to deal with LaTeXMLMath which exists as a so-called ‘ strict product ( 7.4 ) ’ in the notion of LaTeXMLCite , Ch .
II , assigning the value LaTeXMLMath to it .
For LaTeXMLMath we obtain LaTeXMLMath and for LaTeXMLMath we have LaTeXMLMath because the Heaviside contribution is constant LaTeXMLMath or LaTeXMLMath in those regions .
In summary LaTeXMLEquation .
If we interpret LaTeXMLMath via the Fourier transform of LaTeXMLMath as LaTeXMLMath then we obtain LaTeXMLMath as the measurable function LaTeXMLEquation .
As in the previous case , we observe that it is exactly the singular support of LaTeXMLMath – here , the point LaTeXMLMath – that reveals the information of the correct shift .
Observe , however , that the travel time LaTeXMLMath is in fact the only point where the ( distributional ) derivative LaTeXMLMath does not vanish .
The previous evaluation based upon the implicit function theorem hence does not apply .
We now return to the wave equation ( Section 2 ) , assume constant coefficients , and invoke an exact solution representation rather than an asymptotic one .
We consider propagation in one spatial dimension .
Let LaTeXMLMath be real-valued , LaTeXMLMath , LaTeXMLMath in a neighborhood of LaTeXMLMath and LaTeXMLMath for LaTeXMLMath ; LaTeXMLMath a constant LaTeXMLMath .
In the sense of OIs LaTeXMLEquation .
LaTeXMLEquation ( LaTeXMLMath respectively LaTeXMLMath are the complex conjugates of LaTeXMLMath times the subtrahends in OI representation of the fundamental solutions for the d ’ Alembert operator with wavespeed equal to LaTeXMLMath and LaTeXMLMath , respectively . )
The general LaTeXMLMath -bounds according to ( LaTeXMLRef ) give LaTeXMLEquation .
LaTeXMLEquation Observe that half rays in cotangent components are minimal closed cones in LaTeXMLMath .
We will show that , in fact , the inclusion should be replaced by equality .
For symmetry reasons , we give detailed arguments in quadrant LaTeXMLMath , LaTeXMLMath only .
Since LaTeXMLMath and LaTeXMLMath , the theorem on the propagation of singularities LaTeXMLCite , Thm .
8.3.3 , applies ; in particular , if LaTeXMLMath ( resp .
LaTeXMLMath ) then the whole line through this point with directional vector LaTeXMLMath ( resp .
LaTeXMLMath ) is in the singular support ( with the same perpendicular cotangent component in the wave front set attached to it ) .
Therefore , to prove equality in the above LaTeXMLMath inclusion relations , it suffices to show that LaTeXMLMath ( resp .
LaTeXMLMath ) is not smooth near LaTeXMLMath Assuming the contrary , would imply that the function LaTeXMLMath is smooth ; but it is also equal to the Fourier transform of LaTeXMLMath , which can not be smooth since LaTeXMLMath where the first term is a smooth function ( of rapid decay ) since LaTeXMLMath is smooth and of compact support .
The argument for LaTeXMLMath is the same .
We conclude that LaTeXMLEquation .
LaTeXMLEquation It follows immediately that both LaTeXMLMath and LaTeXMLMath are restrictable to LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are represented as the one-dimensional OIs where LaTeXMLMath appears as parameter in the phase only .
Assuming LaTeXMLMath , we clearly have LaTeXMLEquation .
LaTeXMLEquation But then the time shift is given by LaTeXMLEquation as expected from physical intuition .
In the remainder of this section we analyze the correlation in detail , and investigate how the time shift appears .
In the correlation we have to multiply the distributions LaTeXMLEquation and LaTeXMLEquation which have wave front sets LaTeXMLEquation .
LaTeXMLEquation Hence , whenever LaTeXMLMath , the distributions have disjoint singular supports and in case LaTeXMLMath the cotangent vectors in their wave front set can not add up to LaTeXMLMath .
We conclude that for all LaTeXMLMath the wave front sets are in favorable position and the product LaTeXMLMath can be defined in the sense of LaTeXMLCite , Thm .
8.2.10 .
The following lemma states that we are even allowed to use the naive product of the OI expressions .
LaTeXMLMath is ( essentially ) an OI given by LaTeXMLEquation and LaTeXMLMath is weakly continuous LaTeXMLMath .
We introduce the following notation : LaTeXMLEquation .
LaTeXMLEquation for the phase function and the amplitude .
For the justification of ( LaTeXMLRef ) we use the construction of the distributional product in LaTeXMLCite , Thm .
8.2.10 via the pullback of the tensor product on LaTeXMLMath under the map LaTeXMLMath which embeds LaTeXMLMath as the diagonal into LaTeXMLMath .
In doing so the original OIs may be approximated by smooth regularizations ( e.g. , amplitude cut-offs in the integrands ) the tensor products thereof being pulled back simply as smooth functions ( meaning restriction to LaTeXMLMath in this case ) .
It is easily seen then that the smooth functions obtained thereby converge weakly ( as OI regularizations ) to the OI given in ( LaTeXMLRef ) .
By continuity of the pullback ( under the given wave front set conditions ) this limit equals the pullback of the tensor product of the corresponding limits and therefore , in turn , is the distributional product LaTeXMLMath .
Note that LaTeXMLMath is smooth in LaTeXMLMath ( due to the cut-off LaTeXMLMath ) and homogeneous of degree LaTeXMLMath outside the set LaTeXMLMath and is therefore a symbol of order LaTeXMLMath .
The function LaTeXMLMath is smooth on LaTeXMLMath and homogeneous of degree LaTeXMLMath in LaTeXMLMath .
If LaTeXMLMath then the gradient LaTeXMLMath for all LaTeXMLMath and hence LaTeXMLMath is a phase function .
In case LaTeXMLMath the gradient vanishes exactly along one half-ray component of the set LaTeXMLMath ( e.g. , along LaTeXMLMath if LaTeXMLMath ) .
Although it is no longer a phase function in the strict sense , the distribution LaTeXMLMath is then defined as the sum of a classical integral , an OI , and a Fourier transform of an LaTeXMLMath -function .
We discuss this for the case LaTeXMLMath in detail , the other cases are completely analogous .
Let LaTeXMLMath be a smooth function that is equal to LaTeXMLMath near LaTeXMLMath , has support in LaTeXMLMath , and satisfies LaTeXMLMath .
Let LaTeXMLMath be smooth with compact support and LaTeXMLMath when LaTeXMLMath .
We can split the integral defining LaTeXMLMath into three terms according to LaTeXMLMath .
The first integral , then , is a classical one defining a smooth function , the second is an OI since the gradient of LaTeXMLMath does not vanish on the support of the integrand .
In the third integral , we have LaTeXMLEquation ( insert LaTeXMLMath and use the fact that LaTeXMLMath and LaTeXMLMath on the support of the integrand ) and hence the last term is equal to LaTeXMLEquation which we interpret via the Fourier transform of the LaTeXMLMath -function LaTeXMLMath on LaTeXMLMath as LaTeXMLMath in the sense of locally integrable functions — hence it is distribution on LaTeXMLMath .
The weak continuity w.r.t .
LaTeXMLMath follows from the smooth dependence of the phase function in the OI representation ( cf .
LaTeXMLCite , before Thm .
2.2.2 ) and the continuity of the Fourier transform on LaTeXMLMath .
∎ From the last part of the proof it follows that LaTeXMLMath is weakly smooth on LaTeXMLMath .
In order to define the correlation function , we need to check whether the action of LaTeXMLMath on LaTeXMLMath is well defined .
We will do so by showing that LaTeXMLMath is tempered with Fourier transform LaTeXMLMath being in fact a continuous function .
This function can be evaluated at LaTeXMLMath yielding the interpretation LaTeXMLMath .
We use an OI regularization of LaTeXMLMath via the symmetric cut-off function LaTeXMLMath where LaTeXMLMath with LaTeXMLMath when LaTeXMLMath , LaTeXMLMath when LaTeXMLMath and LaTeXMLMath .
Writing LaTeXMLMath ( LaTeXMLMath ) we obtain LaTeXMLMath and LaTeXMLMath uniformly over compact subsets of LaTeXMLMath as LaTeXMLMath .
Hence LaTeXMLEquation where LaTeXMLEquation .
Since LaTeXMLMath is compact LaTeXMLMath is smooth and by differentiating inside the integral we see that for all LaTeXMLMath LaTeXMLMath is bounded by some constant ( depending on LaTeXMLMath and LaTeXMLMath ) .
Hence LaTeXMLMath is a sequence in the space LaTeXMLMath of tempered distributions .
Therefore , to prove that LaTeXMLMath is in LaTeXMLMath , it suffices to show that LaTeXMLMath converges weakly in LaTeXMLMath , i.e. , for all rapidly decaying smooth functions LaTeXMLMath the sequence LaTeXMLMath is convergent .
We have LaTeXMLEquation .
Here , the integrand tends pointwise to LaTeXMLMath as LaTeXMLMath and is dominated by LaTeXMLMath .
It remains to show that LaTeXMLMath is in LaTeXMLMath ; then an application of Lebesgue ’ s dominated convergence theorem will provide us with existence of an explicit integral expression for the limit LaTeXMLMath .
Since LaTeXMLMath , using the explicit structure of LaTeXMLMath , we have for any LaTeXMLMath a bound of the form LaTeXMLEquation .
While integrating the right-hand side of this inequality over LaTeXMLMath , we split the integration into four parts according to the sign combinations of LaTeXMLMath and LaTeXMLMath .
By symmetry , this boils down to estimating only the two kinds of integrals LaTeXMLEquation .
In LaTeXMLMath we only have to note that LaTeXMLMath which together with the remaining factors gives a finite integral as soon as LaTeXMLMath .
In LaTeXMLMath we change variables to LaTeXMLMath , LaTeXMLMath to obtain LaTeXMLEquation .
In the inner integral we use LaTeXMLMath yielding an upper bound LaTeXMLMath and hence LaTeXMLEquation which is finite if LaTeXMLMath .
This proves the assertion that LaTeXMLMath is indeed in LaTeXMLMath and establishes the following result .
LaTeXMLMath and for any LaTeXMLMath LaTeXMLEquation .
We are now in a position to determine the Fourier transform of LaTeXMLMath explicitly .
LaTeXMLMath is the continuous function on LaTeXMLMath given by ( the classical integral ) LaTeXMLEquation .
Let LaTeXMLMath then LaTeXMLMath and from ( LaTeXMLRef ) we obtain LaTeXMLEquation where in the last line we have made use of the symmetry properties of LaTeXMLMath .
Changing coordinates in the inner integrals to LaTeXMLMath and again by the symmetry of LaTeXMLMath this reads LaTeXMLEquation .
Finally , since LaTeXMLMath and LaTeXMLMath for some polynomial in LaTeXMLMath , we may interchange the order of integration and arrive at LaTeXMLEquation .
Since LaTeXMLMath was arbitrary and the above upper bound for LaTeXMLMath shows that the inner integrand is in LaTeXMLMath w.r.t .
LaTeXMLMath , the proposition is proved .
∎ From ( LaTeXMLRef ) we immediately obtain the correlation by setting LaTeXMLMath , in the form LaTeXMLEquation .
This shows that LaTeXMLMath is continuous and can be represented as the difference of two ( classically convergent ) OIs with symbols of order LaTeXMLMath , and hence LaTeXMLMath .
Note that the ( distributional ) derivative LaTeXMLMath can be obtained by differentiating w.r.t .
LaTeXMLMath inside the OI raising the order of the symbol by one .
Therefore LaTeXMLMath will not be continuous on the whole line .
Finally , we observe that again the information about the singularity shift is revealed by the singular support of LaTeXMLMath .
By the stationarity condition on the phase functions , we find LaTeXMLEquation where LaTeXMLMath represent the true shifts from LaTeXMLMath to LaTeXMLMath whereas LaTeXMLMath are the distances from LaTeXMLMath to LaTeXMLMath .
It is easily seen that LaTeXMLMath can not be smooth at the points LaTeXMLMath , e.g. , by noting that each time derivative brings down a new factor of LaTeXMLMath in each integrand , and at the LaTeXMLMath values in question one of the phase functions vanishes identically along a half-line in LaTeXMLMath .
Hence , we have in fact the exact information LaTeXMLEquation which also fits nicely with remark LaTeXMLRef on the weak smoothness of LaTeXMLMath .
From the case studies , we conjecture that the singular support of the correlation of two wave fields reveals the relative shift in wave front sets between them .
As we pointed out , in general , the critical point set of the correlation need not be compatible with this shift .
Here , we propose an alternative approach to extract the shift from the correlation , viz. , by detecting its singular support .
We design a pseudodifferential operator that enables this detection .
Our approach can be applied invariably to any derivative of the correlation also .
In the generic case , the correlation LaTeXMLMath with Fourier transform LaTeXMLMath .
Let LaTeXMLMath be the Gaussian in one dimension , define LaTeXMLEquation .
Introduce LaTeXMLEquation a continuous wavelet transform that can be written as the action of a pseudodifferential operator LaTeXMLMath ( in LaTeXMLMath ) on LaTeXMLMath .
The growth properties reveal the wave front set at LaTeXMLMath in the direction LaTeXMLMath .
In fact , LaTeXMLMath if for any LaTeXMLMath , LaTeXMLEquation ( see LaTeXMLCite ) .
Effectively , this leads to a scanning procedure over LaTeXMLMath : whenever the condition is not satisfied , LaTeXMLMath .
In particular , this applies if LaTeXMLMath for some fixed LaTeXMLMath .
If LaTeXMLMath would allow an OI representation , as is the case in the examples of Section 4 , we could apply a stationary phase argument instead , as in ( LaTeXMLRef ) .
Starting from the microlocal representation , we analyzed the measurement process of wave fields .
Such process can be described by a restriction operator .
We then adressed the issue of how the detection of wave front sets propagates through the measurement process .
Then we focused on the detection of ( base ) shifts in wave front sets due to perturbation of the wave field within the measurement .
We introduced the distributional cross-correlation as a tool for this purpose , and analyzed its properties .
In a series of case studies , we investigated in what way the cross-correlation reveals the shifts .
In the first case the correlation was a measure , in the second case it was a bounded measurable function , and in the third case it was a continuous function .
It was conjectured that the time shift coincides with the singular support of the correlation .
We proposed a procedure ( a pseudodifferential operator ) to detect the shift based on microlocalization .
Such procedure would comprise the foundation for wave-equation tomography .
We thank Jérôme Le Rousseau for valuable mathematical remarks and improvements of the text .
1 notationSection1 eg1Example3.3 linear part 1Corollary3.6 ecoh-thm1Theorem2.6 reciprocityTheorem3.7 sheaf cohomologyTheorem4.1 May 31 , 2001 Sheaf Cohomology and Free Resolutions over Exterior Algebras David Eisenbud , Gunnar Fløystad and Frank-Olaf Schreyer LaTeXMLMath The first and third authors are grateful to the NSF for partial support during the preparation of this paper .
The third author wishes to thank MSRI for its hospitality .
AMS Classification .
Primary : 14F05 , 14Q20 , 16E05 Abstract : In this paper we derive an explicit version of the Bernstein-Gel ’ fand-Gel ’ fand ( BGG ) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its “ Koszul dual ” exterior algebra .
Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the “ linear part ” of a resolution over the exterior algebra .
We explore the structure of free resolutions over an exterior algebra .
For example , we show that such resolutions are eventually dominated by their “ linear parts ” in the sense that erasing all terms of degree LaTeXMLMath in the complex yields a new complex which is eventually exact .
As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms .
The explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology of sheaves .
We also construct all the monads for a sheaf that can be built from sums of line bundles , and show that they are often characterized by numerical data .
Let V be a finite dimensional vector space over a field LaTeXMLMath , and let LaTeXMLMath be the dual space .
In this paper we will study complexes and resolutions over the exterior algebra LaTeXMLMath and their relation to modules over LaTeXMLMath and sheaves on projective space LaTeXMLMath .
In this paper we study the Bernstein-Gel ’ fand-Gel ’ fand ( BGG ) correspondence [ 1978 ] , usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on LaTeXMLMath and the stable category of finitely generated graded modules over LaTeXMLMath .
Its essential content is a functor LaTeXMLMath from complexes of graded LaTeXMLMath -modules to complexes of graded LaTeXMLMath -modules , and its adjoint LaTeXMLMath .
For example , if LaTeXMLMath is a graded LaTeXMLMath -module ( regarded as a complex with just one term ) then as a bigraded LaTeXMLMath -module LaTeXMLMath , with differential LaTeXMLMath defined from the multiplication map on LaTeXMLMath .
Similarly , for a graded LaTeXMLMath -module LaTeXMLMath , we have LaTeXMLMath .
In fact ( LaTeXMLMath is an equivalence from the category of graded LaTeXMLMath -modules to the category of linear complexes of free LaTeXMLMath -modules ; here linear means essentially that the maps are represented by matrices of linear forms .
A similar statement holds for LaTeXMLMath .
We show that finitely generated modules LaTeXMLMath go to left-bounded complexes that are exact far to the right , and characterize the point at which exactness begins as the Castelnuovo-Mumford regularity of LaTeXMLMath .
A strong form of this is If LaTeXMLMath is a graded LaTeXMLMath -module and LaTeXMLMath is a graded LaTeXMLMath -module , then LaTeXMLMath is an injective resolution of LaTeXMLMath if and only if LaTeXMLMath is a free resolution of LaTeXMLMath .
Let LaTeXMLMath be a coherent sheaf on projective space and take LaTeXMLMath .
The results above show that the complex LaTeXMLMath associated to the truncation of LaTeXMLMath is acyclic for LaTeXMLMath .
If we take a minimal free resolution of the kernel of the first term in this complex , we obtain a doubly infinite exact free complex , independent of LaTeXMLMath , which we call the Tate resolution LaTeXMLMath : LaTeXMLEquation .
It was first studied in Gel ’ fand [ 1984 ] .
Our first main theorem ( LaTeXMLMath term of the Tate resolution is LaTeXMLMath ; that is it is made from the cohomology of the twists of LaTeXMLMath .
This leads to a new algorithm for computing sheaf cohomology .
We have programmed this method in the computer algebra system Macaulay2 of Grayson and Stillman [ http : //www.math.uiuc.edu/Macaulay2/ ] .
In some cases it gives the fastest known computation of the cohomology .
We apply the Tate resolution to study a result of Beilinson [ 1978 ] , which gives , for each sheaf LaTeXMLMath on projective space , a complex LaTeXMLEquation called the Beilinson Monad whose homology is precisely LaTeXMLMath and whose terms depend only on the cohomology of a few twists of LaTeXMLMath .
Our second main result is a constructive version of Beilinson ’ s Theorem [ 1978 ] , which clarifies its connection of the BGG-correspondence ( Beilinson ’ s original paper sketches a proof that leads easily to a weak form of the result , the “ Beilinson spectral sequence ” , which determines the sheaf LaTeXMLMath only up to filtration .
That version is explained in the book of Okonek , Schneider , and Spindler [ 1980 ] .
Kapranov [ 1988 ] and Ancona and Ottaviani [ 1989 ] have given full proofs .
However their use of the derived category makes it difficult to compute the Beilinson monad effectively , and also makes it hard to obtain information about the maps in the monad .
Our construction of the Beilinson Monad leads to new results about its structure .
There are natural candidates for the linear components of the maps in the monad for a sheaf LaTeXMLMath ; and given such a monad , there are natural candidates for most of the maps in the monad of LaTeXMLMath .
Our techniques allow us to prove that these natural candidates really do occur ( A remarkable feature of the theory of resolutions over the exterior algebra , not visible for the corresponding theory over a polynomial ring , is that the linear terms of any resolution eventually predominate .
To state this precisely , we introduce the linear part of a free complex LaTeXMLMath over LaTeXMLMath or LaTeXMLMath .
The linear part is the complex obtained from LaTeXMLMath by taking a minimal free complex LaTeXMLMath homotopic to LaTeXMLMath , and then erasing all terms of absolute degree LaTeXMLMath from the matrices representing the differentials of LaTeXMLMath .
In fact taking the linear part is functorial in a suitable sense : under the BGG correspondence it corresponds to the homology functor ( complex itself is mysterious .
Of course free resolutions may have maps with no linear terms at all , that is , with linear part equal to zero .
And they can have infinitely many maps with nonlinear terms unavoidably present ( this is even the case for periodic resolutions ) .
But the linear terms eventually predominate in the following sense : If LaTeXMLMath is the free resolution of a finitely generated module over the exterior algebra LaTeXMLMath then the linear part of LaTeXMLMath is eventually exact .
This predominance can take arbitrarily long to assert itself : the resolution of the millionth syzygy of the residue field of LaTeXMLMath has a million linear maps follows by a map with linear part 0 , and linear dominance happens only at the million and first term .
In the case of a resolution of a monomial ideal , however , Herzog and Römer [ 1999 ] have shown that the linear part becomes exact after at most LaTeXMLMath steps .
It would be interesting to know more results of this sort .
Beilinson [ 1978 ] also proved the existence of a different monad for a sheaf LaTeXMLMath , using the sheaves LaTeXMLMath for LaTeXMLMath in place of the LaTeXMLMath .
Bernstein-Gel ’ fand-Gel ’ fand also introduced a “ linear ” monad using sums of line bundles and only having maps given by matrices of linear forms .
In the last section we show that such a monad “ partitions ” the cohomology of the sheaf into a “ positive ” part that appears as the homology of the corresponding complex of free LaTeXMLMath -modules and a “ negative ” part that appears as the cohomology of the dual complex .
We explain how these and other free monads of a sheaf LaTeXMLMath arise from the Tate resolution LaTeXMLMath .
We show that many such monads are characterized by simple numerical data .
Basic references for the BGG correspondence are Gel ’ fand [ 1984 ] , and Gel ’ fand-Manin [ 1996 ] ) .
Much of the elementary material of this paper could be done for an arbitrary pair of homogeneous Koszul algebras ( in the sense of Priddy [ 1970 ] ) in place of the pair of algebras LaTeXMLMath .
We use a tiny bit of this for the pair LaTeXMLMath .
See Buchweitz [ 1987 ] for a sketch of the general case and a statement of general conditions under which the BGG correspondence holds .
Buchweitz has also written a general treatment of the BGG correspondence over Gorenstein rings [ 1985 ] .
Versions of Beilinson ’ s theorem have been established for some other varieties through work of Swan [ 1985 ] , Kapranov [ 1988,1989 ] , and Orlov [ 1992 ] .
Yet other derived category equivalences have been pursued under the rubric of “ tilting ” ( see Happel [ 1988 ] ) .
Fløystad [ 2001a ] gives a general theory for Koszul pairs , and also studies how far the equivalences can be extended to unbounded complexes .
The material of our paper grew from two independent preprints of Eisenbud and Schreyer [ 2000 ] and Fløystad [ 2000b ] .
Since there was considerable overlap we wrote a more complete joint paper , which also includes new joint results .
The original preprint by the second author has also been altered so that the notation and terminology are more aligned with the present paper .
The material in this paper has been applied to study the cohomology of hyperplane arrangements ( Eisenbud , Popescu , and Yuzvinsky [ 2001 ] ) and to constructing counterexamples to the Minimal Free Resolution conjecture for points in projective space ( Eisenbud , Popescu , Schreyer , and Walter [ 2001 ] ) .
The technique developed here for the Beilinson monad has been used by Eisenbud and Schreyer to construct complexes on various Grassmanians that can be used to compute and study Chow forms [ 2001 ] .
In a direction related to Green ’ s proof of the Linear Syzygy Conjecture [ 1999 ] , Eisenbud and Weyman have found a general analogue for the Fitting lemma over Z/2 -graded algebras , including the exterior algebra .
This paper owes much to the experiments we were able to make using the computer algebra system Macaulay2 of Grayson and Stillman , and we would like to thank them for their help and patience with this project .
We are also grateful to Luchezar Avramov for getting us interested in resolutions over exterior algebras .
notation Notation and Background Throughout this paper we write LaTeXMLMath for a fixed field , and LaTeXMLMath for dual vector spaces of finite dimension LaTeXMLMath over LaTeXMLMath .
We give the elements of LaTeXMLMath degree 1 , so that the elements of LaTeXMLMath have degree LaTeXMLMath .
We write LaTeXMLMath and LaTeXMLMath for the exterior and symmetric algebras ; these algebras are graded by their internal degrees whereby LaTeXMLMath has degree LaTeXMLMath and LaTeXMLMath has degree LaTeXMLMath .
We think of LaTeXMLMath as LaTeXMLMath and LaTeXMLMath as LaTeXMLMath .
We will always write the index indicating the degree of a homogeneous component of a graded module as subscripts .
Thus if LaTeXMLMath is a graded module over LaTeXMLMath or LaTeXMLMath , then LaTeXMLMath denotes the component of degree LaTeXMLMath .
We let LaTeXMLMath be the shifted module , so that LaTeXMLMath .
We write complexes cohomologically , with upper indices and differentials of degree LaTeXMLMath .
Thus if LaTeXMLEquation is a complex , then LaTeXMLMath denotes the term of cohomological degree LaTeXMLMath .
We write LaTeXMLMath for the complex whose term of cohomological degree LaTeXMLMath is LaTeXMLMath .
We will write LaTeXMLMath for the module associated to the canonical bundle of LaTeXMLMath ; note that LaTeXMLMath is a vector space concentrated in degree LaTeXMLMath , so that LaTeXMLMath is noncanonically isomorphic to LaTeXMLMath .
Similarly , we set LaTeXMLMath , which is noncanonically isomorphic to LaTeXMLMath .
It is easy to check that for any graded vector space LaTeXMLMath we have LaTeXMLMath as left LaTeXMLMath -modules .
For any LaTeXMLMath -module LaTeXMLMath , we set LaTeXMLMath .
We often use the fact that the exterior algebra is Gorenstein and finite dimensional over LaTeXMLMath , which follows from the fact that LaTeXMLMath as above .
As a consequence , the dual of any exact sequence is exact and the notions free module , injective module , and projective module coincide .
We also use the notion of Castelnuovo-Mumford regularity .
The most convenient definition for our purposes is that the Castelnuovo-Mumford regularity of a graded LaTeXMLMath -module LaTeXMLMath is the smallest integer LaTeXMLMath such that the truncation LaTeXMLMath is generated by LaTeXMLMath and has a linear free resolution —that is , all the maps in its free resolution are represented by matrices of linear forms .
See for example Eisenbud-Goto [ 1984 ] or Eisenbud [ 1995 ] for a discussion .
The regularity of a sheaf LaTeXMLMath on projective space ( equal to the regularity of LaTeXMLMath if this module is finitely generated ) can also be expressed as the minimal LaTeXMLMath for which LaTeXMLMath for all LaTeXMLMath .
A free complex over LaTeXMLMath or a graded free complex over LaTeXMLMath is called minimal if all its maps can be represented by matrices with entries in the appropriate maximal ideal .
For example , any linear complex is minimal .
intro BGG The Bernstein-Gel ’ fand-Gel ’ fand Correspondence In this section we give a brief exposition of the main idea of Bernstein-Gel ’ fand-Gel ’ fand [ 1978 ] : a construction of a pair of adjoint functors between the categories of complexes over LaTeXMLMath and over LaTeXMLMath .
However , we avoid a peculiar convention , used in the original , according to which the differentials of complexes over LaTeXMLMath were not homomorphisms of LaTeXMLMath -modules .
Let LaTeXMLMath and LaTeXMLMath be dual bases of LaTeXMLMath and LaTeXMLMath respectively , so that LaTeXMLMath corresponds to the identity element under the isomorphism LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be vector spaces .
Giving a map LaTeXMLMath is the same as giving a map LaTeXMLMath ( where the tensor products are taken over LaTeXMLMath ) .
For example , given LaTeXMLMath we set LaTeXMLMath .
We begin with a special case that will play a central role .
We regard a graded LaTeXMLMath -module LaTeXMLMath as a complex with only one term , in cohomological degree 0 , and define LaTeXMLMath to be the complex LaTeXMLEquation .
Here the term LaTeXMLMath has cohomological index LaTeXMLMath , and a map LaTeXMLMath has degree LaTeXMLMath if it factors through the projection from LaTeXMLMath onto LaTeXMLMath .
Note that the complex LaTeXMLMath is linear in a strong sense : the LaTeXMLMath free module LaTeXMLMath has socle in degree LaTeXMLMath ; in particular all the maps are represented by matrices of linear forms .
basic correspondence The functor LaTeXMLMath is an equivalence between the category of graded left LaTeXMLMath -modules and the category of linear free complexes over LaTeXMLMath ( those for which the LaTeXMLMath free module has socle in degree LaTeXMLMath . )
A collection of maps LaTeXMLMath defines a module structure on the graded vector space LaTeXMLMath if and only if it satisfies a commutativity and associativity condition expressed by saying that , for each LaTeXMLMath , the composition of the multiplication maps LaTeXMLEquation factors through LaTeXMLMath .
Since LaTeXMLMath is the kernel of LaTeXMLMath , this is the same as saying that the induced map LaTeXMLMath is 0 , or again that the map LaTeXMLMath , is zero .
This last is equivalent to LaTeXMLMath being a complex .
As the whole construction is reversible , we are done .
As a first step in extending LaTeXMLMath to all complexes , we consider the case of a module regarded as a complex with a single term , but in arbitrary cohomological degree .
Let LaTeXMLMath be an LaTeXMLMath -module , regarded as a complex concentrated in cohomological degree 0 .
Then LaTeXMLMath is a complex concentrated in cohomological degree LaTeXMLMath , and we set LaTeXMLEquation .
Now consider the general case of a complex of graded LaTeXMLMath -modules LaTeXMLEquation .
Applying LaTeXMLMath to each LaTeXMLMath , regarded as a complex concentrated in cohomological degree LaTeXMLMath , we get a double complex , and we define LaTeXMLMath to be the total complex of this double complex .
Thus LaTeXMLMath is the total complex of LaTeXMLEquation where the vertical maps are induced by the differential of LaTeXMLMath and the horizontal complexes are the complexes LaTeXMLMath defined above .
As LaTeXMLMath -modules we have LaTeXMLEquation where LaTeXMLMath is regarded as a vector space concentrated in degree LaTeXMLMath .
Thus as a bigraded LaTeXMLMath -module , LaTeXMLMath , and the formula for the graded components is LaTeXMLEquation .
The functor LaTeXMLMath has a left adjoint LaTeXMLMath defined in an analogous way by tensoring with LaTeXMLMath : on a graded LaTeXMLMath -module LaTeXMLMath the functor LaTeXMLMath takes the value LaTeXMLEquation where the map takes LaTeXMLMath to LaTeXMLMath and the term LaTeXMLMath has cohomological degree LaTeXMLMath .
If LaTeXMLMath is a complex of graded LaTeXMLMath -modules , then we can apply LaTeXMLMath to each term to get a double complex , and we define LaTeXMLMath to be the total complex of this double complex , so that LaTeXMLEquation .
To see that LaTeXMLMath is the left adjoint of LaTeXMLMath we proceed as follows .
First , if LaTeXMLMath and LaTeXMLMath are left modules over LaTeXMLMath and LaTeXMLMath respectively , then LaTeXMLEquation .
If now LaTeXMLMath and LaTeXMLMath are complexes of graded modules over LaTeXMLMath and LaTeXMLMath , we must prove that LaTeXMLMath , where on each side we take the maps of modules that preserve the internal and cohomological degrees and commute with the differentials .
As a bigraded module , LaTeXMLMath , and similarly for LaTeXMLMath .
Direct computation shows that these maps of complexes correspond to the maps of bigraded LaTeXMLMath -modules LaTeXMLEquation such that LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath takes an element LaTeXMLMath to LaTeXMLMath .
We have proved : BGG theorem ( Bernstein-Gel ’ fand-Gel ’ fand [ 1978 ] ) The functor LaTeXMLMath , from the category of complexes of graded LaTeXMLMath -modules to the category of complexes of graded LaTeXMLMath -modules , is a left adjoint to the functor LaTeXMLMath .
It is not hard to compute the homology of the complexes produced by LaTeXMLMath and LaTeXMLMath : koszul homology If LaTeXMLMath is a graded LaTeXMLMath -module and LaTeXMLMath is a graded LaTeXMLMath -module then LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath The LaTeXMLMath free module in the free resolution of LaTeXMLMath over LaTeXMLMath is LaTeXMLMath , which is generated by the vector space LaTeXMLMath of degree LaTeXMLMath .
We can use this to compute the right hand side of the equality in LaTeXMLMath : the LaTeXMLMath graded component of the module of homomorphisms of this into LaTeXMLMath may be identified with LaTeXMLMath .
The differential is the same as that of LaTeXMLMath , and part LaTeXMLMath follows .
Part LaTeXMLMath is similar ( and even more familiar , from Koszul cohomology . )
It follows that the exactness of LaTeXMLMath or LaTeXMLMath are familiar conditions .
First the case a module over the symmetric algebra : exactness criterion LaTeXMLMath If LaTeXMLMath is a finitely generated graded LaTeXMLMath -module , then the truncated complex LaTeXMLEquation is acyclic ( that is , has homology only at LaTeXMLMath ) if and only if LaTeXMLMath is LaTeXMLMath -regular .
By LaTeXMLMath the given sequence is acyclic if and only if LaTeXMLMath has linear free resolution .
Since any linear complex is of the form LaTeXMLMath for a unique graded LaTeXMLMath -module LaTeXMLMath it is perhaps most interesting to interpret part LaTeXMLMath of LaTeXMLMath .
The result below is implicitly used in Green ’ s [ 1999 ] proof of the Linear Syzygy Conjecture .
We call a right bounded linear complex LaTeXMLEquation irredundant if it is a subcomplex of the minimal free resolution of LaTeXMLMath ( or equivalently of any module whose presentation has linear part equal to LaTeXMLMath . )
( Eisenbud-Popescu [ 1999 ] called this property linear exactness , but to follow this usage would risk overusing the adjective “ linear ” . )
linear exactness Let LaTeXMLMath be a minimal linear complex of free LaTeXMLMath -modules ending on the right with LaTeXMLMath as above , and let LaTeXMLMath be the LaTeXMLMath -module such that LaTeXMLMath .
The complex LaTeXMLMath is irredundant if and only the module LaTeXMLMath is generated by LaTeXMLMath .
The complex LaTeXMLMath is the linear part of a minimal free resolution if and only if the module LaTeXMLMath is linearly presented .
Let LaTeXMLMath be the differential of LaTeXMLMath , let LaTeXMLEquation be the minimal free resolution of LaTeXMLMath , and let LaTeXMLMath be a comparison map lifting the identity on LaTeXMLMath .
( This comparison map is unique because LaTeXMLMath is minimal and LaTeXMLMath is linear . )
By induction one sees that the comparison map is an injection if and only if LaTeXMLMath for all LaTeXMLMath , and it is an isomorphism onto the linear part of LaTeXMLMath if and only if in addition LaTeXMLMath for all LaTeXMLMath .
LaTeXMLMath injects into a direct sum of copies of LaTeXMLMath , while both conditions are true if and only if the minimal injective resolution begins with LaTeXMLEquation for some numbers LaTeXMLMath .
Dualizing , we get the desired linear presentation LaTeXMLEquation of LaTeXMLMath .
We now return to the BGG-correspondence .
Both the functors LaTeXMLMath and LaTeXMLMath preserve mapping cones and homotopies of maps of complexes .
For mapping cone this is immediate .
For the second note that two maps LaTeXMLMath of complexes are homotopic if and only if the induced map from LaTeXMLMath to the mapping cone of LaTeXMLMath is split .
This condition is preserved by any additive functor that preserves mapping cones .
Recall that a free resolution of a right bounded complex LaTeXMLEquation of graded LaTeXMLMath -modules is a graded free complex LaTeXMLMath with a morphism LaTeXMLMath , homogeneous of degree 0 , which induces an isomorphism on homology .
We say that LaTeXMLMath is minimal if LaTeXMLMath has trivial differential .
Every right bounded complex LaTeXMLMath of finitely generated modules has a minimal free resolution , unique up to isomorphism .
It is the minimal part of any free resolution .
The functors LaTeXMLMath and LaTeXMLMath give a general construction of resolutions .
ecoh-thm1 For any complex of graded LaTeXMLMath -modules LaTeXMLMath , the complex LaTeXMLMath is a free resolution of LaTeXMLMath which surjects onto LaTeXMLMath ; and for any complex of graded LaTeXMLMath -modules LaTeXMLMath , the complex LaTeXMLMath is an injective resolution of LaTeXMLMath into which LaTeXMLMath injects .
In fact we shall see that every free complex whose homology is LaTeXMLMath up to finite length comes as LaTeXMLMath of a complex that agrees with LaTeXMLMath in high degrees .
An immediate consequence is : BGG-result The functors LaTeXMLMath and LaTeXMLMath define an equivalence LaTeXMLMath .
of Corollary LaTeXMLMath of bounded complexes of finitely generated LaTeXMLMath -modules is equivalent to the derived category of complexes of finitely generated LaTeXMLMath -modules with bounded cohomology ( that is , having just finitely many cohomology modules ) , see for example Hartshorne [ 1977 ] , III Lemma 12.3 , and similarly for LaTeXMLMath .
The functors LaTeXMLMath and LaTeXMLMath carry bounded complexes into complexes with bounded cohomology .
This is clear for LaTeXMLMath .
For LaTeXMLMath this follows from LaTeXMLMath and LaTeXMLMath are well defined and by LaTeXMLMath are both equivalent to the identity .
Proof of The proofs of the two statements are similar , so we treat only the first .
( A slight simplification is possible in the second case since finitely generated modules over LaTeXMLMath have finite composition series . )
Because LaTeXMLMath is the left adjoint functor of LaTeXMLMath there is a natural map LaTeXMLMath adjoint to the identity map LaTeXMLMath .
We claim that this is a surjective quasi-isomorphism .
To see that it is a surjection , consider a map LaTeXMLMath such that the composite LaTeXMLMath is zero .
It follows that the adjoint composition LaTeXMLMath is also zero , and since the first map is the identity , we get LaTeXMLMath .
Since LaTeXMLMath is a faithful functor , LaTeXMLMath , proving surjectivity .
The functor LaTeXMLMath preserves direct limits because it is a left adjoint , while the functor LaTeXMLMath preserves direct limits because LaTeXMLMath is a finite dimensional vector space .
Thus it suffices to prove our claim in the case where LaTeXMLMath is a bounded complex of finitely generated LaTeXMLMath -modules .
If LaTeXMLMath has the form LaTeXMLEquation then LaTeXMLMath admits LaTeXMLMath ( that is , the module LaTeXMLMath considered as a complex concentrated in cohomological degree LaTeXMLMath ) as a subcomplex , and the quotient is a complex of smaller length .
Using the “ five lemma ” the naturality of the map LaTeXMLMath , and the exactness of the functor LaTeXMLMath , the claims will follow , by induction on the length of the complex , from the case where LaTeXMLMath has the form LaTeXMLMath for some finitely generated graded LaTeXMLMath -module LaTeXMLMath and integer LaTeXMLMath .
This reduces immediately to the case LaTeXMLMath .
It thus suffices to to see that LaTeXMLMath is a quasi-isomorphism when LaTeXMLMath is a finitely generated graded LaTeXMLMath -module .
Now LaTeXMLMath is the linear complex LaTeXMLMath , so LaTeXMLMath is the total complex of the following double complex : LaTeXMLEquation .
In this picture the terms below what is shown are all zero .
The terms of cohomological degree 0 in the total complex are those along the diagonal going northwest from LaTeXMLMath .
The generators of LaTeXMLMath have internal degree 0 , while those of LaTeXMLMath have internal degree 1 , etc .
The LaTeXMLMath row of this double complex is LaTeXMLMath , which is equal to the complex obtained by tensoring the Koszul complex LaTeXMLEquation with LaTeXMLMath .
It is thus acyclic , its one cohomology module being LaTeXMLMath , in cohomological degree LaTeXMLMath .
The spectral sequence starting with the horizontal cohomology of the double complex thus degenerates , and we see that the cohomology of the total complex LaTeXMLMath is a graded module with component of internal degree equal to LaTeXMLMath , concentrated in cohomological degree 0 .
Thus LaTeXMLMath is acyclic and the Hilbert function of LaTeXMLMath is the same as that of LaTeXMLMath .
As LaTeXMLMath has no terms in positive cohomological degree , and LaTeXMLMath is in cohomological degree 0 , the surjection LaTeXMLMath induces a surjection LaTeXMLMath , and we are done .
( One can show that LaTeXMLMath is the tensor product , over LaTeXMLMath , of the Koszul complex and LaTeXMLMath , the action of LaTeXMLMath being the diagonal action , but the isomorphism is complicated to write down . )
Though the statement of LaTeXMLMath are nearly always infinite ) .
linear part section The Linear Part of a Complex If LaTeXMLMath is a matrix over LaTeXMLMath then we define the linear part , written LaTeXMLMath , to be the matrix obtained by erasing all the terms of entries of LaTeXMLMath that are of degree LaTeXMLMath .
For example , if LaTeXMLMath are linear forms of LaTeXMLMath , then the linear part of LaTeXMLEquation .
Taking the linear part is a functorial operation on maps ( see LaTeXMLMath are linear forms , LaTeXMLEquation then LaTeXMLMath Suppose that LaTeXMLMath is a second map of free modules and that the composition LaTeXMLMath .
It need not be the case that that LaTeXMLMath ; but if we assume in addition that LaTeXMLMath is in the maximal ideal times LaTeXMLMath and LaTeXMLMath is in the maximal ideal times LaTeXMLMath , then LaTeXMLMath does follow .
Thus if LaTeXMLMath is a minimal free complex over LaTeXMLMath we can define a new complex LaTeXMLMath by replacing each differential LaTeXMLMath of LaTeXMLMath by its linear part , LaTeXMLMath .
Note that LaTeXMLMath is the direct sum of complexes LaTeXMLMath whose LaTeXMLMath term is a direct sum of copies of LaTeXMLMath and whose maps are of degree 1 .
In general , we define the linear part of any free complex LaTeXMLMath to be the linear part of a minimal complex homotopic to LaTeXMLMath .
linear dominance Let LaTeXMLMath be a free or injective resolution of a finitely generated module over the exterior algebra LaTeXMLMath .
The linear part of LaTeXMLMath is eventually exact .
We treat only the case where LaTeXMLMath is an injective resolution ; by duality , the statement for a free resolution is equivalent .
By LaTeXMLMath is the value of LaTeXMLMath on the LaTeXMLMath -module LaTeXMLMath .
Since any finitely generated LaTeXMLMath module has finite regularity ( see Eisenbud-Goto [ 1984 ] ) , it suffices by LaTeXMLMath is a finitely generated LaTeXMLMath -module .
This was done by Aramova , Avramov , and Herzog [ 2000 ] .
For the reader ’ s convenience we repeat the argument : we prove that LaTeXMLMath is a finitely generated LaTeXMLMath -module by induction on the length of LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is free of rank 1 over LaTeXMLMath .
If LaTeXMLMath is a proper submodule of LaTeXMLMath then from the exact sequence LaTeXMLEquation we get an exact triangle of LaTeXMLMath -modules LaTeXMLEquation .
The two LaTeXMLMath -modules in the top row are finitely generated by induction , and thus LaTeXMLMath is finitely generated too .
If LaTeXMLMath is an LaTeXMLMath -module , then we write LaTeXMLMath for the cokernel of LaTeXMLMath , where LaTeXMLMath is the map in a minimal free presentation of LaTeXMLMath .
We can further define a family of modules connecting LaTeXMLMath and LaTeXMLMath as follows : Let LaTeXMLMath be a minimal free presentation of LaTeXMLMath , choose a representation of LaTeXMLMath as a matrix , and let LaTeXMLMath be a basis of LaTeXMLMath .
Let LaTeXMLMath be the result of substituting LaTeXMLMath for LaTeXMLMath in the entries of LaTeXMLMath , and then dividing each entry by LaTeXMLMath .
The entries of LaTeXMLMath have no constant terms because LaTeXMLMath is minimal , and it follows that LaTeXMLMath is a matrix over LaTeXMLMath .
Let LaTeXMLMath be the cokernel of LaTeXMLMath .
It has fibers LaTeXMLMath at LaTeXMLMath and LaTeXMLMath at 0 .
The module LaTeXMLMath may not be flat over LaTeXMLMath , but the module LaTeXMLMath is flat over LaTeXMLMath : in fact , it is isomorphic to the module obtained from the trivial family LaTeXMLMath by pulling back along the automorphism LaTeXMLMath of LaTeXMLMath .
deformations If LaTeXMLMath is a finitely generated LaTeXMLMath module , then any sufficiently high syzygy LaTeXMLMath of LaTeXMLMath is a flat deformation of its linear part LaTeXMLMath .
If LaTeXMLMath is a sufficiently high syzygy , then by LaTeXMLMath is the resolution of LaTeXMLMath , so that ( with the notation of the preceding paragraph ) this free resolution of LaTeXMLMath lifts to a free resolution of LaTeXMLMath over LaTeXMLMath .
Thus LaTeXMLMath is flat , and the result follows .
eg1 It is sometimes not so obvious what the linear part of the minimal version of a complex will be , and in particular it may be hard to read from the linear terms in a nonminimal version .
For example , suppose that LaTeXMLMath has dimension 2 and that LaTeXMLMath is a dual basis to LaTeXMLMath .
Consider the complex LaTeXMLEquation where the notation means that the class of 1 goes to the class of LaTeXMLMath .
Applying LaTeXMLMath to LaTeXMLMath , we get the double complex LaTeXMLEquation whose total complex is LaTeXMLEquation .
Despite the presence of the linear terms in the differential of LaTeXMLMath , the minimal complex LaTeXMLMath homotopic to LaTeXMLMath is LaTeXMLEquation so the differential of LaTeXMLMath is 0 .
Fortunately , we can construct the linear part of a complex directly and conceptually , without passing to a minimal complex or to matrices .
First note that if LaTeXMLMath is a minimal free complex over LaTeXMLMath , then giving its linear part is equivalent , by LaTeXMLMath corresponding to the linear terms in the differential of LaTeXMLMath .
If LaTeXMLMath is a ( possibly nonminimal ) free complex homotopic to LaTeXMLMath , then LaTeXMLMath .
We will construct natural maps LaTeXMLMath , and prove that LaTeXMLMath .
We identify LaTeXMLMath with LaTeXMLMath and use the well-known LaTeXMLMath -module structure on LaTeXMLMath .
To formulate this explicitly , we make use of the exact sequence LaTeXMLEquation .
The extension class LaTeXMLEquation corresponds to the inclusion LaTeXMLMath .
Since LaTeXMLMath is a free complex , the sequence LaTeXMLMath is an exact sequence of complexes , and we obtain the homomorphism LaTeXMLMath from the connecting homomorphism LaTeXMLEquation linear part and tor If LaTeXMLMath is a complex of free modules over LaTeXMLMath , then LaTeXMLEquation where the LaTeXMLMath -module structure on LaTeXMLMath is given by the action of LaTeXMLMath .
We use the notation LaTeXMLMath introduced just before the theorem .
¿From the definition of LaTeXMLMath we see that it depends only on the homotopy class of LaTeXMLMath , so we may assume that LaTeXMLMath is minimal .
By LaTeXMLEquation .
Let LaTeXMLMath be a generator which is mapped by LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be a generator of the socle of LaTeXMLMath .
To prove LaTeXMLMath we have to show that an element of the form LaTeXMLMath is mapped to LaTeXMLMath under the connecting homomorphism LaTeXMLMath .
The element LaTeXMLMath corresponds to LaTeXMLMath in LaTeXMLMath .
We must lift LaTeXMLMath to an element of LaTeXMLMath .
The image of LaTeXMLMath will be an element LaTeXMLMath which satisfies LaTeXMLMath for all LaTeXMLMath and any such element defines a lifting .
We can take LaTeXMLMath .
The image of LaTeXMLMath under the connecting homomorphism is the map LaTeXMLMath , where LaTeXMLMath is the differential of LaTeXMLMath .
With LaTeXMLEquation we arrive at the desired formula .
To understand the linear parts of complexes obtained from the functor LaTeXMLMath , we will employ a general result : if the vertical differential of suitable double complex splits , then the associated total complex is homotopic to one built from the homology of the vertical differential in a simple way .
degenerate double complex Let LaTeXMLMath be a double complex LaTeXMLEquation in some abelian category such that LaTeXMLMath for LaTeXMLMath .
Suppose that the vertical differential of LaTeXMLMath splits , so that for each LaTeXMLMath there is a decomposition LaTeXMLMath such that the kernel of LaTeXMLMath in LaTeXMLMath is LaTeXMLMath , and such that LaTeXMLMath maps LaTeXMLMath isomorphically to LaTeXMLMath .
If we write LaTeXMLMath for the projection corresponding to this decomposition , and LaTeXMLMath for the composition of the projection with the inverse of LaTeXMLMath restricted to LaTeXMLMath , then the total complex of LaTeXMLMath is homotopic to the complex LaTeXMLEquation with differential LaTeXMLEquation .
We write LaTeXMLMath for the differential of the total complex .
Note first that LaTeXMLMath takes LaTeXMLMath to LaTeXMLMath .
Since LaTeXMLMath for LaTeXMLMath , the sum defining LaTeXMLMath is finite .
Let LaTeXMLMath denote LaTeXMLMath without the differential , that is , as a bigraded module .
We will first show that LaTeXMLMath is the direct sum of the three components LaTeXMLEquation and that LaTeXMLMath is a monomorphism on LaTeXMLMath .
The same statements , with LaTeXMLMath replaced by LaTeXMLMath , are true by hypothesis .
In particular , any element of LaTeXMLMath is a sum of elements of the form LaTeXMLMath with LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath .
Modulo LaTeXMLMath this element can be written as LaTeXMLMath .
As LaTeXMLMath for LaTeXMLMath , we may do induction on LaTeXMLMath , and assume that LaTeXMLMath , so we see that LaTeXMLMath .
Suppose LaTeXMLEquation and LaTeXMLMath ; we must show that LaTeXMLMath .
Write LaTeXMLMath with LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath and the desired result is a special case of the hypothesis .
In any case , there is no component of LaTeXMLMath in LaTeXMLMath so the component of LaTeXMLMath in LaTeXMLMath is equal to LaTeXMLMath .
From the hypothesis we see that LaTeXMLMath , so LaTeXMLMath , and we are done by induction on LaTeXMLMath .
This shows that LaTeXMLMath and that LaTeXMLMath is an isomorphism from LaTeXMLMath to LaTeXMLMath .
The modules LaTeXMLMath form a double complex contained in LaTeXMLMath that we will call LaTeXMLMath .
Since LaTeXMLMath is an isomorphism , the total complex LaTeXMLMath is split exact .
It follows that the total complex LaTeXMLMath is homotopic to LaTeXMLMath , and the modules of this last complex are isomorphic to LaTeXMLMath .
We will complete the proof by showing that the induced differential on LaTeXMLMath is the differential LaTeXMLMath defined above .
Choose LaTeXMLMath .
The image of LaTeXMLMath under the induced differential is the unique element LaTeXMLMath such that LaTeXMLMath .
Now LaTeXMLEquation .
However , LaTeXMLEquation .
Continuing this way , and using again the fact that LaTeXMLMath for LaTeXMLMath we obtain LaTeXMLEquation as required .
We apply LaTeXMLMath : linear part 1 If LaTeXMLMath is a left-bounded complex of graded LaTeXMLMath -modules , then LaTeXMLEquation where LaTeXMLMath is regarded as a complex of one term , concentrated in cohomological degree LaTeXMLMath .
A similar statement holds for the linear part of LaTeXMLMath when LaTeXMLMath is a left-bounded complex of graded LaTeXMLMath -modules .
As LaTeXMLMath is a left-bounded complex of finitely generated modules , the double complex whose total complex is LaTeXMLMath satisfies the conditions of bigraded module underlying LaTeXMLMath is precisely the module LaTeXMLMath of LaTeXMLMath restricted to LaTeXMLMath .
This is a linear map .
But the other terms in the sum LaTeXMLMath all involve two or more iterations of LaTeXMLMath , and are thus represented by matrices whose entries have degree at least 2. eg1 Continued : Note that the homology of LaTeXMLMath is LaTeXMLMath .
We may write LaTeXMLMath as required by Here is the promised information about the minimal resolution of a module : reciprocity LaTeXMLMath Reciprocity : If LaTeXMLMath is a finitely generated graded LaTeXMLMath -module and LaTeXMLMath is a finitely generated graded LaTeXMLMath -module , then LaTeXMLMath is a free resolution of LaTeXMLMath if and only if LaTeXMLMath is an injective resolution of LaTeXMLMath .
LaTeXMLMath More generally , for any minimal bounded complex of finitely generated graded LaTeXMLMath -modules LaTeXMLMath , the linear part of the minimal free resolution of LaTeXMLMath is LaTeXMLMath and for any minimal bounded complex of finitely generated graded LaTeXMLMath -modules LaTeXMLMath , the linear part of the minimal injective resolution of LaTeXMLMath is LaTeXMLMath The two parts of LaTeXMLMath being similar , we prove only the first statement .
By LaTeXMLMath is a free resolution .
The complex LaTeXMLMath is left-bounded because LaTeXMLMath is bounded and contains only finitely generated modules .
Thus we may apply For the reciprocity statement LaTeXMLMath , suppose that LaTeXMLMath is a minimal free resolution of LaTeXMLMath .
By part b ) the linear part of the minimal injective resolution of LaTeXMLMath is LaTeXMLMath .
Since LaTeXMLMath is a resolution of LaTeXMLMath , this is LaTeXMLMath .
All the terms of cohomological degree LaTeXMLMath of this complex have degree LaTeXMLMath , so there is no room for nonlinear differentials , and the linear part of the resolution is the resolution .
tate Sheaf cohomology and exterior syzygies In this section we establish a formula for the free modules that appear in resolutions over LaTeXMLMath .
Because LaTeXMLMath is Gorenstein , it is natural to work with doubly infinite resolutions : A Tate resolution over LaTeXMLMath is a doubly infinite free complex LaTeXMLEquation that is everywhere exact .
There is a Tate resolution naturally associated to a coherent sheaf LaTeXMLMath on LaTeXMLMath , defined as follows .
Let LaTeXMLMath be a finitely generated graded LaTeXMLMath -module representing LaTeXMLMath , for example LaTeXMLMath .
If LaTeXMLMath , then by LaTeXMLMath is acyclic .
Thus if LaTeXMLMath then , since LaTeXMLMath is minimal , LaTeXMLMath minimally covers the kernel of the map LaTeXMLMath Fixing LaTeXMLMath , we may complete LaTeXMLMath to a minimal Tate resolution LaTeXMLMath by adjoining a free resolution of LaTeXMLEquation .
Since any two modules representing LaTeXMLMath are equal in large degree , the Tate resolution is independent of which LaTeXMLMath and which large LaTeXMLMath is chosen , and depends only on the coherent sheaf LaTeXMLMath .
It has the form LaTeXMLEquation where the LaTeXMLMath are graded free LaTeXMLMath -modules .
The main theorem of this section expresses the linear part of this Tate resolution in terms of the LaTeXMLMath -modules LaTeXMLMath given by the ( Zariski ) cohomology of LaTeXMLMath .
We regard LaTeXMLMath as a complex of LaTeXMLMath -modules concentrated in cohomological degree LaTeXMLMath .
sheaf cohomology If LaTeXMLMath is a coherent sheaf on LaTeXMLMath , then the linear part of the Tate resolution LaTeXMLMath is LaTeXMLMath .
In particular , LaTeXMLEquation where LaTeXMLMath is regarded as a vector space of internal degree LaTeXMLMath .
A special case of the theorem appears without proof as Remark 3 after Theorem 2 in Bernstein-Gel ’ fand-Gel ’ fand [ 1978 ] .
The proof below could be extended to cover the case of a bounded complex of coherent sheaves , replacing the cohomology in the formula with hypercohomology .
We will postpone the proof of Rewriting the indices in LaTeXMLMath from the Tate resolution .
comp of shf coho For all LaTeXMLMath , LaTeXMLMath comp of shf coho provides the basis for an algorithm computing the cohomology of LaTeXMLMath with any computer program that can provide free resolutions of modules over the symmetric and exterior algebras , such as the program Macaulay2 of Grayson and Stillman [ http : //www.math.uiuc.edu/Macaulay2/ ] .
For an explanation of the algorithm in practical terms , see Decker and Eisenbud [ 2001 ] .
To prove LaTeXMLMath for the homogeneous maximal ideal LaTeXMLMath of LaTeXMLMath , and for any graded LaTeXMLMath -module LaTeXMLMath we write LaTeXMLMath for the LaTeXMLMath local cohomology module of LaTeXMLMath , regarded as a graded LaTeXMLMath -module .
local cohomology Let LaTeXMLMath be a graded LaTeXMLMath -module generated in degree LaTeXMLMath , and having linear free resolution LaTeXMLMath .
Let LaTeXMLMath be the minimal free resolution of LaTeXMLMath .
The linear part of LaTeXMLMath is LaTeXMLEquation where LaTeXMLMath is regarded as a complex with one term , concentrated in cohomological degree LaTeXMLMath .
In particular , we have LaTeXMLEquation .
Proof of We compute the linear part of the free resolution of LaTeXMLMath by taking the dual ( into LaTeXMLMath ) of the linear part of the injective resolution of LaTeXMLMath .
By LaTeXMLMath is LaTeXMLMath .
It follows at once from the definitions that LaTeXMLMath .
By once more LaTeXMLMath is the minimal free resolution of LaTeXMLMath , so LaTeXMLMath .
Thus the linear part of the free resolution of LaTeXMLMath is LaTeXMLMath , where LaTeXMLMath is thought of as a module concentrated in cohomological degree LaTeXMLMath .
Because LaTeXMLMath we have , for any graded vector space LaTeXMLMath , natural identifications LaTeXMLEquation ( Here all the duals of LaTeXMLMath -modules are Hom into LaTeXMLMath . )
If LaTeXMLMath has the structure of a graded LaTeXMLMath -module then LaTeXMLMath is again a graded LaTeXMLMath -module , and this becomes an isomorphism of graded LaTeXMLMath -modules .
If we think of LaTeXMLMath as a complex with just one term , in cohomological degree LaTeXMLMath , then LaTeXMLMath where , to make all the indices come out right , we must think of LaTeXMLMath as a complex of one term concentrated in cohomological degree LaTeXMLMath .
If we take LaTeXMLMath then by local duality LaTeXMLEquation .
Thus LaTeXMLEquation where LaTeXMLMath is regarded as a complex with just one term , of cohomological degree LaTeXMLMath , as required .
Proof of For each LaTeXMLMath we write LaTeXMLMath for the cohomology module LaTeXMLMath .
If we choose LaTeXMLMath as in the construction of LaTeXMLMath , the module LaTeXMLMath has a linear free resolution , so we may apply LaTeXMLMath is LaTeXMLMath If we insist that LaTeXMLMath then LaTeXMLMath .
From the exactness of the sequence LaTeXMLEquation it follows that the local cohomology module LaTeXMLMath agrees with the global cohomology module LaTeXMLMath in all degrees strictly less than d , and of course we have LaTeXMLMath .
This concludes the proof .
powers example Powers of the maximal ideal of LaTeXMLMath In this section we provide a basic example of the action of the functors LaTeXMLMath and LaTeXMLMath .
Among the most interesting graded LaTeXMLMath -modules are the syzygy modules that occur in the Koszul complex .
We write LaTeXMLEquation where as usual the elements of LaTeXMLMath have internal degree 1 , so that the generators of LaTeXMLMath have internal degree LaTeXMLMath .
For example LaTeXMLMath while LaTeXMLMath and LaTeXMLMath , a free module of rank one generated in degree LaTeXMLMath .
The sheafifications of these modules are the exterior powers of the cotangent bundle on projective space ( see Eisenbud [ 1995 ] Section 17.5 for more details . )
In this section we shall show that under the functors LaTeXMLMath and LaTeXMLMath introduced in LaTeXMLMath correspond to powers of the maximal ideal LaTeXMLMath .
To make the correspondence completely functorial , we make use of the LaTeXMLMath -modules LaTeXMLMath , where LaTeXMLMath .
Recall that LaTeXMLMath is a rank one free LaTeXMLMath -module generated in degree LaTeXMLMath ; its generators may be identified with the nonzero elements of LaTeXMLMath .
powers theorem The minimal LaTeXMLMath -free resolution of LaTeXMLMath is LaTeXMLMath ; the minimal LaTeXMLMath -injective resolution of LaTeXMLMath is LaTeXMLMath .
Since LaTeXMLMath is generated in degree LaTeXMLMath , the complex LaTeXMLMath begins in cohomological degree LaTeXMLMath , and we regard LaTeXMLMath as concentrated in cohomological degree LaTeXMLMath .
The complex LaTeXMLMath is the Koszul complex over LaTeXMLMath , so LaTeXMLMath is the truncation LaTeXMLEquation which is the resolution of LaTeXMLMath , proving the first statement .
The second statement follows from Since the LaTeXMLMath -dual of a minimal LaTeXMLMath -injective resolution is a minimal LaTeXMLMath -free resolution , we may immediately derive the free resolution of LaTeXMLEquation free res of powers The minimal LaTeXMLMath -free resolution of LaTeXMLMath is LaTeXMLEquation .
These resolutions can be made explicit using the Schur functors LaTeXMLMath associated to “ hook ” diagrams ( see for example Buchsbaum and Eisenbud [ 1975 ] or Akin , Buchsbaum , Weyman [ 1985 ] ) .
We may define LaTeXMLMath ( called LaTeXMLMath by Buchsbaum and Eisenbud ) by the formula LaTeXMLEquation .
Note that LaTeXMLEquation .
Buchsbaum and Eisenbud use these functors to give ( among other things ) a LaTeXMLMath -equivariant resolution LaTeXMLEquation of the LaTeXMLMath power LaTeXMLMath of the maximal ideal of LaTeXMLMath .
The LaTeXMLMath also provide the terms in the resolutions above : explicit exterior powers For LaTeXMLMath the minimal free resolution of LaTeXMLMath has the form LaTeXMLEquation .
For LaTeXMLMath the minimal injective resolution of LaTeXMLMath has the form LaTeXMLEquation .
From the exactness of the Koszul complex we see that LaTeXMLMath so the second statement follows from Using the exact sequence LaTeXMLEquation we may paste together the injective and free resolutions considered above into the Tate resolution LaTeXMLMath .
explicit tate There is an exact sequence LaTeXMLMath LaTeXMLEquation where LaTeXMLMath is the term in cohomological degree LaTeXMLMath .
The following well-known result now follows from homology of omegas In the range LaTeXMLMath or LaTeXMLMath LaTeXMLEquation .
Writing the ranks of the free modules in the Tate resolution for LaTeXMLMath in Macaulay notation we find LaTeXMLEquation with the rank LaTeXMLMath module sitting in homological degree LaTeXMLMath and the in-going and out-going map from it given by bases of the forms in LaTeXMLMath and LaTeXMLMath respectively .
If we shift the rank 1 module into homological degree O then we have the Tate resolution of LaTeXMLMath .
Following Beilinson ( [ 1978 ] Lemma 2 ) we can also compute LaTeXMLMath for any LaTeXMLMath , which will play major role in Section hom of omega example If LaTeXMLMath are the LaTeXMLMath -modules defined in section LaTeXMLMath then LaTeXMLEquation where in each case LaTeXMLMath denotes the ( degree 0 ) homomorphisms ; for other values of LaTeXMLMath the left hand side is 0 .
The product of homomorphisms corresponds to the product in LaTeXMLMath .
The modules LaTeXMLMath are 0 for LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath they have linear resolution , so we may apply LaTeXMLMath and generated in degree 1 , we have LaTeXMLMath if LaTeXMLMath by LaTeXMLMath maps LaTeXMLMath are in one-to-one correspondence with maps LaTeXMLMath .
Since LaTeXMLMath is a rank one free LaTeXMLMath -module , these may be identified with elements of LaTeXMLMath .
beilinson Beilinson ’ s Monad Beilinson ’ s paper [ 1978 ] contains two main results .
The first says that given a sheaf LaTeXMLMath on a projective space LaTeXMLMath there is a complex LaTeXMLEquation with LaTeXMLEquation such that LaTeXMLMath is exact except at LaTeXMLMath and the homology at LaTeXMLMath is LaTeXMLMath .
We show that the complex LaTeXMLMath may be obtained by applying a certain functor to the Tate resolution LaTeXMLMath over LaTeXMLMath .
Beilinson ’ s second main result gives another monad , which we will treat in Given any graded free complex LaTeXMLMath over LaTeXMLMath we may write each module of LaTeXMLMath as a direct sum of copies of LaTeXMLMath with varying LaTeXMLMath .
We define LaTeXMLMath to be the complex of sheaves on LaTeXMLMath obtained by replacing each summand LaTeXMLMath by the sheaf LaTeXMLMath and using the isomorphism of Hom in beilinson-thm If LaTeXMLMath is a coherent sheaf on LaTeXMLMath with associated Tate resolution LaTeXMLMath , then the only homology of LaTeXMLMath is in cohomological degree 0 , and is isomorphic to LaTeXMLMath , To simplify the notation we set LaTeXMLMath , and we let LaTeXMLMath be LaTeXMLMath modulo the elements of internal degree LaTeXMLMath .
Let LaTeXMLMath be the double complex of sheaves that arises by sheafifying the double complex of LaTeXMLMath -modules used to construct the complex LaTeXMLMath ; that is , if LaTeXMLMath is the component of LaTeXMLMath of cohomological degree LaTeXMLMath , and LaTeXMLMath is its component of internal degree LaTeXMLMath , then the double complex LaTeXMLMath has the form LaTeXMLEquation .
Since LaTeXMLMath is exact , the rows are exact ; since the columns are direct sums of sheafified Koszul complexes over LaTeXMLMath , they are exact as well .
Choose an integer LaTeXMLMath ( greater than the regularity of LaTeXMLMath will be sufficient ) and let LaTeXMLMath be the double complex obtained from LaTeXMLMath by taking only those terms LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath is generated in negative degrees , so the double complex LaTeXMLMath is finite , and is exact except at the right ( LaTeXMLMath ) and at j=1 .
An easy spectral sequence argument shows that the complex obtained as the vertical homology of LaTeXMLMath has the same homology as the complex obtained as the horizontal homology of LaTeXMLMath .
If we write LaTeXMLMath as a sum of copies of LaTeXMLMath for various LaTeXMLMath , then the LaTeXMLMath column of LaTeXMLMath is correspondingly a sum of copies of the sheafification of LaTeXMLMath .
As in LaTeXMLMath ; that is , it is LaTeXMLMath .
Thus the complex obtained as the vertical homology of LaTeXMLMath is LaTeXMLMath .
As LaTeXMLMath goes to infinity the degrees of the generators of LaTeXMLMath become more and more positive ; thus for LaTeXMLMath large the LaTeXMLMath column of LaTeXMLMath is the same as that of LaTeXMLMath , that is , it is LaTeXMLMath .
Since LaTeXMLMath the horizontal homology of LaTeXMLMath is the sheafification of LaTeXMLMath , where LaTeXMLMath is the homology of LaTeXMLMath .
As LaTeXMLMath is exact , LaTeXMLMath may also be written as the homology of LaTeXMLMath .
Taking LaTeXMLMath and using LaTeXMLMath is a free resolution of the module LaTeXMLMath , whose sheafification is LaTeXMLMath , as required .
beilinson cor 2 The map in the complex LaTeXMLMath corresponding to LaTeXMLEquation corresponds to the multiplication map LaTeXMLMath .
This follows from beilinson cor 1 The maps in the complex LaTeXMLMath correspond to the maps in the complex LaTeXMLMath under the natural correspondence LaTeXMLEquation whenever LaTeXMLMath .
The Tate resolution LaTeXMLMath is obtained by simply shifting LaTeXMLMath .
Examples Examples elliptic quartic Let LaTeXMLMath be an elliptic quartic curve in LaTeXMLMath , and consider LaTeXMLMath as a sheaf on LaTeXMLMath .
Write LaTeXMLMath as usual .
Computing cohomology one sees that LaTeXMLMath has the form LaTeXMLEquation .
If LaTeXMLMath is taken to be Heisenberg invariant , say LaTeXMLMath for some LaTeXMLMath , then LaTeXMLMath can be represented by the matrix LaTeXMLEquation rat normal curve The rational normal curve Let LaTeXMLMath be the curve parametrized by LaTeXMLMath .
We consider the line bundles LaTeXMLMath on LaTeXMLMath associated to LaTeXMLMath for LaTeXMLMath .
The Tate resolution LaTeXMLMath has betti numbers LaTeXMLEquation .
The LaTeXMLMath matrix near the middle and the matrices surrounding it , have in case LaTeXMLMath and LaTeXMLMath the following shapes : LaTeXMLEquation and LaTeXMLEquation .
All other matrices look similar to the last one .
In case LaTeXMLMath we obtain a LaTeXMLMath symmetric matrix of 2-forms : LaTeXMLEquation .
If we interpret 2-forms as coordinate functions LaTeXMLEquation on the Grassmanian of codimension 2 linear subspaces in LaTeXMLMath , then the determinant of the matrix above defines the Chow divisor of LaTeXMLMath , which is by definition the hypersurface LaTeXMLMath .
Eisenbud and Schreyer [ 2001 ] give a general computation of Chow forms along these lines .
Horrocks-Mumford The Horrocks-Mumford bundle in LaTeXMLMath .
A famous Beilinson monad was discovered by Horrocks and Mumford [ 1973 ] : Consider for LaTeXMLMath the Tate resolution LaTeXMLMath of the matrix LaTeXMLEquation .
By direct computation we find the betti numbers LaTeXMLEquation .
To deduce that this Tate resolution comes from a sheaf we use : row bound Let LaTeXMLMath be a Tate resolution over LaTeXMLMath .
Suppose that LaTeXMLMath for all LaTeXMLMath .
Then LaTeXMLMath if LaTeXMLMath and LaTeXMLMath , or if LaTeXMLMath and LaTeXMLMath .
Pictorially the statement is , that vanishing in a single LaTeXMLMath implies vanishing in the indicated range : LaTeXMLEquation .
The first vanishing follows , because LaTeXMLMath is also a minimal complex .
For the second we note for LaTeXMLMath that LaTeXMLMath holds for all LaTeXMLMath by our assumption .
By LaTeXMLMath which is a subquotient of LaTeXMLMath .
Thus this group vanishes in all degrees LaTeXMLMath .
Horrocks-Mumford Continued : By applying LaTeXMLMath and LaTeXMLMath we see that the LaTeXMLMath has terms only in the indicated range of rows , inparticular the rows with the question marks contain only zeroes .
So LaTeXMLMath is the Tate resolution of some sheaf LaTeXMLMath .
Moreover LaTeXMLMath is a bundle , since the middle cohomology has only finitely many terms .
The LaTeXMLMath difference function of LaTeXMLMath has constant value LaTeXMLMath .
So LaTeXMLMath has rank LaTeXMLMath .
It is the famous bundle on LaTeXMLMath discovered by Horrocks and Mumford [ 1973 ] .
In Decker and Schreyer [ 1986 ] it is proved that any stable rank LaTeXMLMath vector bundle on LaTeXMLMath with the same Chern classes equals LaTeXMLMath up to projectivities .
free monad section Free monads A free monad LaTeXMLMath for a coherent sheaf LaTeXMLMath is a finite complex LaTeXMLEquation on LaTeXMLMath , whose components LaTeXMLMath are direct sums of line bundles and whose homology is LaTeXMLMath : LaTeXMLEquation .
The complex of twisted global sections of LaTeXMLMath is a complex LaTeXMLMath of free LaTeXMLMath -modules .
If LaTeXMLMath is a minimal complex , then we speak of a minimal free monad .
The most familiar free monads are the sheafifications of the minimal free resolutions of the modules LaTeXMLMath for various LaTeXMLMath .
Free monads were constructed by Horrocks [ 1964 ] , Barth [ 1977 ] , Bernstein , Gel ’ fand and Gel ’ fand [ 1978 ] and Beilinson [ 1978 ] , mainly for the study of vector bundles on projective spaces .
Rao [ 1981 ] , Martin-Deschamps and Perrin [ 1990 ] used free monads in their studies of space curves .
Fløystad [ 2000c ] gives a complete classification of a certain class of linear monads on projective spaces .
The general construction of free monads is the following : free monads Let LaTeXMLMath be a coherent sheaf on LaTeXMLMath and let LaTeXMLMath be a left bounded complex of finite free LaTeXMLMath -modules with LaTeXMLMath for some LaTeXMLMath .
Let LaTeXMLMath be the minmalized complex of the BGG transform LaTeXMLMath .
Its sheafication LaTeXMLMath is a free monad for LaTeXMLMath .
Every minimal free monad LaTeXMLMath of LaTeXMLMath arises as LaTeXMLMath in this way with LaTeXMLMath .
Suppose LaTeXMLMath satisfies the assumption .
Since LaTeXMLMath is left bounded and acyclic for large degrees , LaTeXMLMath is a finite complex by the second statement in LaTeXMLMath can be computed by taking linear parts : LaTeXMLMath by LaTeXMLMath is of finite length for LaTeXMLMath and sheafifying gives LaTeXMLEquation .
Conversely if LaTeXMLMath is a free monad for LaTeXMLMath and LaTeXMLMath then LaTeXMLMath has finite length for LaTeXMLMath .
Thus LaTeXMLMath is a left bounded complex with LaTeXMLMath by LaTeXMLMath .
bad monad Consider LaTeXMLMath the structure sheaf of a point in LaTeXMLMath .
Its Tate resolution is periodic : LaTeXMLEquation .
If we take LaTeXMLMath to be the truncation LaTeXMLEquation then the monad LaTeXMLMath is the sheafified free resolution LaTeXMLEquation .
If instead we take LaTeXMLMath to be the complex LaTeXMLEquation then LaTeXMLMath is the free resolution of LaTeXMLMath which has sheafification LaTeXMLMath of the form LaTeXMLEquation .
For the rest of this section we will study a class of free monads we call partition monads ( because they partition the cohomology of LaTeXMLMath into two simple pieces , which occur as LaTeXMLMath and LaTeXMLMath .
This class includes the sheafified free resolutions and most of the other free monads found in the literature .
partition monads Partition monads .
Given a weakly increasing sequence of integers LaTeXMLEquation we define LaTeXMLMath to be the subcomplex of LaTeXMLMath given by LaTeXMLEquation .
We shall also make use of the complementary complex LaTeXMLMath defined by the exact sequence LaTeXMLEquation .
We set LaTeXMLMath and write LaTeXMLMath for the monad which is its sheafification .
free resolutions Free resolutions .
Let LaTeXMLMath be any integer , and choose LaTeXMLMath greater than the Castelnuovo-Mumford regularity of LaTeXMLMath .
The complex LaTeXMLMath is LaTeXMLMath .
Thus by LaTeXMLMath is the minimal free resolution of LaTeXMLMath .
linear monads Linear monads .
Consider the case LaTeXMLMath .
In this case LaTeXMLMath is an injective resolution of LaTeXMLMath and LaTeXMLMath has only linear maps .
Like free resolutions , partition monads enjoy a strong homotopy functoriality : funct of partition The partition monad LaTeXMLMath is functorial in LaTeXMLMath up to homotopy of complexes in such a way that if LaTeXMLMath is a map , then LaTeXMLMath .
Moreover , any map of complexes LaTeXMLMath is determined up to homotopy by the induced map LaTeXMLMath .
The first statement follows from the homotopy functoriality of the LaTeXMLMath and LaTeXMLMath .
For the second statement , it suffices to show that every map LaTeXMLMath is homotopic to a map of the form LaTeXMLMath .
But every map LaTeXMLMath is homotopic to a map LaTeXMLMath , and since LaTeXMLMath is an injective resolution , it is homotopic to LaTeXMLMath .
Using the adjointness of LaTeXMLMath and LaTeXMLMath we see that up to homotopy , indeed every map is in the image of the the composite homomorphism LaTeXMLEquation coh of partition The cohomology of the complexes LaTeXMLMath and LaTeXMLMath are given by LaTeXMLEquation where LaTeXMLMath occurs in degree LaTeXMLMath and LaTeXMLMath occurs in degree LaTeXMLMath .
In particular , for LaTeXMLMath we have LaTeXMLMath .
Let LaTeXMLMath be the complex LaTeXMLEquation so that the complex LaTeXMLMath is an injective resolution of LaTeXMLMath .
By part LaTeXMLMath of LaTeXMLMath is the sum of the linear complexes LaTeXMLMath .
Thus LaTeXMLMath by the definition of LaTeXMLMath .
For the proof of the second formula we first observe that LaTeXMLMath .
Since LaTeXMLMath is exact , the induced map LaTeXMLMath is a quasi-isomorphism .
Moreover , this map factors through LaTeXMLMath .
Thus LaTeXMLMath is a projective resolution of LaTeXMLMath , and LaTeXMLMath is an injective resolution of LaTeXMLMath .
The terms with LaTeXMLMath on the right hand side of the desired equality correspond to the LaTeXMLMath linear strand of LaTeXMLMath .
Again by boundedness of partition monads Any partition monad LaTeXMLMath satisfies LaTeXMLMath for LaTeXMLMath .
If LaTeXMLMath but LaTeXMLMath then Nakayama ’ s Lemma implies that LaTeXMLMath and similarly for the dual .
It is easy to give bounds on the line bundles that can occur in a partition monad .
Given the sequence LaTeXMLMath it will be convenient to extend the definition of LaTeXMLMath to all LaTeXMLMath by the formulas LaTeXMLEquation range of partition monads If LaTeXMLMath is a summand of the LaTeXMLMath term of the partition monad LaTeXMLMath then LaTeXMLEquation where the definition of LaTeXMLMath is is extended to all LaTeXMLMath as above .
By LaTeXMLEquation so the LaTeXMLMath term of LaTeXMLMath is LaTeXMLMath For the first inequality we have to show that if LaTeXMLMath then LaTeXMLMath .
Since LaTeXMLEquation and LaTeXMLMath is zero in negative degrees , the condition LaTeXMLMath implies LaTeXMLMath for some LaTeXMLMath with LaTeXMLMath .
Thus LaTeXMLMath and LaTeXMLMath as desired .
For the second inequality we argue similarly using LaTeXMLMath Note that if LaTeXMLMath is a monad for a sheaf LaTeXMLMath then so is LaTeXMLMath where LaTeXMLMath is an acyclic complex—for example the sheafification of the free resolution of a module of finite length .
The main result of this section is that partition monads are characterized by the conditions in LaTeXMLMath .
In most cases , these summands can not occur .
uniqueness of partition monads Let LaTeXMLMath be a monad for a coherent sheaf LaTeXMLMath on LaTeXMLMath , and let LaTeXMLMath .
If LaTeXMLMath for LaTeXMLMath and the terms LaTeXMLMath satisfy LaTeXMLMath for all LaTeXMLMath , then LaTeXMLMath is isomorphic to the direct sum of LaTeXMLMath and a sum LaTeXMLMath of twisted Koszul complexes .
Moreover , LaTeXMLMath can be nonzero only if LaTeXMLMath .
In particular , if the LaTeXMLMath are strictly increasing , or if we assume that no direct summand of LaTeXMLMath is a monad for LaTeXMLMath , then LaTeXMLMath .
Set LaTeXMLMath , and let LaTeXMLMath and LaTeXMLMath be the kernel and the image of the differential LaTeXMLMath respectively .
We begin by identifying the homology of LaTeXMLMath .
Note that LaTeXMLMath for LaTeXMLMath , so LaTeXMLMath for LaTeXMLMath too .
Since LaTeXMLMath is exact at LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath , we can use the sequences LaTeXMLMath to show that LaTeXMLMath for LaTeXMLMath and all LaTeXMLMath .
Thus LaTeXMLMath surjects onto LaTeXMLMath for LaTeXMLMath and all LaTeXMLMath .
It follows that LaTeXMLMath for LaTeXMLMath , while LaTeXMLMath is the cokernel of LaTeXMLMath .
For LaTeXMLMath the space LaTeXMLMath injects into LaTeXMLMath .
But LaTeXMLMath is the image of LaTeXMLMath ; the hypothesis on the LaTeXMLMath implies that this cohomology vanishes for LaTeXMLMath .
In particular , LaTeXMLMath for LaTeXMLMath , and it follows that LaTeXMLMath .
We next prove that for each LaTeXMLMath there is a short exact sequence LaTeXMLEquation where LaTeXMLMath unless LaTeXMLMath .
In fact we shall identify this sequence with the direct sum , over LaTeXMLMath , of the sequences LaTeXMLEquation which come from the sequence LaTeXMLMath expressing the fact that LaTeXMLMath is a monad for LaTeXMLMath .
We have already seen that LaTeXMLMath for LaTeXMLMath .
It follows that the right hand term of LaTeXMLMath is 0 for LaTeXMLMath , and the left hand term is 0 unless LaTeXMLMath , in which case it is LaTeXMLMath where LaTeXMLMath .
¿From the long exact sequences in cohomology—and in case LaTeXMLMath the hypothesis on the LaTeXMLMath —we see that LaTeXMLMath .
For all LaTeXMLMath we have LaTeXMLEquation .
These identifications and vanishing identify the two exact sequences as required .
Set LaTeXMLMath so that LaTeXMLMath by LaTeXMLMath is a complex of free LaTeXMLMath -modules which coincides with the exact complex LaTeXMLMath in large cohomological degrees we can construct a map of complexes LaTeXMLMath By LaTeXMLMath By the first part of the proof , the terms of LaTeXMLMath can be nonzero only in the range of ( internal and cohomological ) degrees where LaTeXMLMath is equal to LaTeXMLMath .
Hence LaTeXMLMath is mapped to LaTeXMLMath , and we obtain a morphism of monads from the composition LaTeXMLMath .
The morphism of monads induces an isomorphism in homology LaTeXMLMath , because by LaTeXMLMath and LaTeXMLMath coincide in large cohomological degrees .
The induced map LaTeXMLMath is the surjection of the first part of the proof .
Hence the map LaTeXMLMath is onto .
Its kernel has terms LaTeXMLMath and degree considerations show that it is a trivial complex .
Because these terms occur in degrees where LaTeXMLMath coincides with the acyclic complex LaTeXMLMath , the differential of LaTeXMLMath carries the generators of these modules into boundaries of LaTeXMLMath .
Thus after a change of generators in LaTeXMLMath we see that LaTeXMLMath is the direct sum of LaTeXMLMath and the trivial complex LaTeXMLMath .
Beilinson ’ s free monad Beilinson ’ s free monad .
Beilinson ’ s free monad LaTeXMLMath for LaTeXMLMath with terms LaTeXMLEquation is the partition monad for LaTeXMLMath .
This follows from Walter ’ s monads Walter ’ s monads .
Let LaTeXMLMath be an integer and let LaTeXMLMath be a sheaf such that LaTeXMLMath is finitedly generated for LaTeXMLMath .
Choose LaTeXMLEquation such that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
The monad LaTeXMLMath does not depend on the precise values of the LaTeXMLMath ’ s and hence has only terms , LaTeXMLEquation by LaTeXMLMath is the unique minimal free monad of LaTeXMLMath with nonzero components only from LaTeXMLMath up to LaTeXMLMath .
Thus LaTeXMLMath is the monad constructed by Walter [ 1990 ] with cohomology LaTeXMLMath and zero otherwise .
Consider a smooth rational surface LaTeXMLMath of degree LaTeXMLMath and sectional genus 10 .
The existence of three families of rational surfaces with these invariants is known , see [ Schreyer , 1996 ] or [ Decker , Schreyer , 2000 ] .
The Tate resolution of the ideal sheaf of these surfaces has shape LaTeXMLEquation with LaTeXMLMath as a reference point .
We display four monads for LaTeXMLMath .
LaTeXMLEquation are monads with only 4 terms .
The first monad is linear because LaTeXMLMath .
The two following monads are somewhat more complicated and hence are less convenient .
LaTeXMLEquation .
These monads up to twist , are Beilinson monads for LaTeXMLMath for LaTeXMLMath and LaTeXMLMath respectively .
The construction of such surfaces in [ Schreyer , 1996 ] was done by a Computer search for monads of shape LaTeXMLMath .
The degree of smooth rational surfaces in LaTeXMLMath is bounded according to Ellingsrud and Peskine [ 1989 ] .
Smooth rational surfaces with sectional genus LaTeXMLMath ( this excludes the cubic scroll and the projected Veronese surface ) have a linear Walter monad .
Indeed by Severi ’ s Theorem LaTeXMLMath and hence LaTeXMLMath .
The numerical type of these monads is LaTeXMLEquation with LaTeXMLMath .
The conjectured bound is LaTeXMLMath .
Perhaps even LaTeXMLMath is true .
However at present the best known bound is LaTeXMLMath , see Decker and Schreyer [ 2000 ] .
K. Akin , D. A. Buchsbaum , and J. Weyman : Schur functors and Schur complexes .
Adv .
in Math .
44 ( 1982 ) 207–278 .
V. Ancona and G. Ottaviani : An introduction to the derived categories and the theorem of Beilinson .
Atti Accademia Peloritana dei Pericolanti , Classe I de Scienze Fis .
Mat .
et Nat .
LXVII ( 1989 ) 99–110 .
A. Aramova , L.A. Avramov , J. Herzog : Resolutions of monomial ideals and cohomology over exterior algebras .
Trans .
Amer .
Math .
Soc .
352 ( 2000 ) 579–594 .
W. Barth : Moduli of vector bundles on the projective plane .
Invent .
Math 42 ( 1977 ) 63-91 .
A. Beilinson : Coherent sheaves on LaTeXMLMath and problems of linear algebra .
Funct .
Anal .
and its Appl .
12 ( 1978 ) 214–216 .
( Trans .
from Funkz .
Anal .
i. Ego Priloz 12 ( 1978 ) 68–69 . )
I. N. Bernstein , I. M. Gel ’ fand and S. I. Gel ’ fand : Algebraic bundles on LaTeXMLMath and problems of linear algebra .
Funct .
Anal .
and its Appl .
12 ( 1978 ) 212–214 .
( Trans .
from Funkz .
Anal .
i. Ego Priloz 12 ( 1978 ) 66–67 . )
D. A. Buchsbaum and D. Eisenbud : Generic Free resolutions and a family of generically perfect ideals .
Adv .
Math .
18 ( 1975 ) 245–301 .
R.-O .
Buchweitz : Appendix to Cohen-Macaulay modules on quadrics , by R.-O .
Buchweitz , D. Eisenbud , and J. Herzog .
In Singularities , representation of algebras , and vector bundles ( Lambrecht , 1985 ) , Springer-Verlag Lecture Notes in Math .
1273 ( 1987 ) 96–116 .
R.-O .
Buchweitz : Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein ring Preprint ( 1985 ) .
W. Decker and D. Eisenbud : Sheaf algorithms using the exterior algebra , in Computations in Algebraic Geometry with Macaulay2 , ed .
D. Eisenbud , D. Grayson , M. Stillman , and B. Sturmfels .
Springer-Verlag ( 2001 ) .
W. Decker and F.-O .
Schreyer : On the uniqueness of the Horrocks-Mumford bundle .
Math .
Ann .
273 ( 1986 ) 415–443 .
W. Decker and F.-O .
Schreyer : Non-general type surfaces in LaTeXMLMath : Some remarks on bounds and constructions .
J .
Symbolic Comp .
29 ( 2000 ) , 545–582 .
W. Decker : Stable rank 2 bundles with Chern-classes LaTeXMLMath .
Math .
Ann .
275 ( 1986 ) 481–500 .
D. Eisenbud : Commutative Algebra with a View Toward Algebraic Geometry .
Springer Verlag , 1995 .
D. Eisenbud and S. Goto : Linear free resolutions and minimal multiplicity .
J. Alg .
88 ( 1984 ) 89–133 .
D. Eisenbud and S. Popescu : Gale Duality and Free Resolutions of Ideals of Points .
Invent .
Math .
136 ( 1999 ) 419–449 .
D. Eisenbud , S. Popescu , F.-O .
Schreyer and C. Walter : Exterior algebra methods for the Minimal Resolution Conjecture .
Preprint ( 2000 ) .
D. Eisenbud , S. Popescu , and S. Yuzvinsky : Hyperplane arrangements and resolutions of monomial ideals over an exterior algebra .
Preprint ( 2001 ) .
D. Eisenbud and F.-O .
Schreyer : Sheaf cohomology and free resolutions over the exterior algebras , http : //arXiv.org/abs/math.AG/0005055 Preprint ( 2000 ) .
D. Eisenbud and F.-O .
Schreyer : Chow forms via exterior syzygies .
Preprint ( 2001 ) .
G. Ellingsrud and C. Peskine : Sur le surfaces lisse de LaTeXMLMath .
Invent .
Math .
95 ( 1989 ) 1–12 .
D. Eisenbud and J. Weyman : A Fitting Lemma for LaTeXMLMath -graded modules .
Preprint ( 2001 ) .
G. Fløystad : Koszul duality and equivalences of categories .
http : //arXiv.org/ abs/math.RA/0012264 Preprint ( 2000a ) .
G. Fløystad : Describing coherent sheaves on projective spaces via Koszul duality .
http : //arXiv.org/abs/math.RA/0012263 Preprint ( 2000b ) .
G. Fløystad : Monads on projective spaces .
Communications in Algebra .
28 ( 2000c ) 5503-5516 .
S. I. Gel ’ fand and Yu .
I. Manin : Methods of Homological Algebra .
Springer-Verlag , New York 1996 .
S. I. Gel ’ fand : Sheaves on LaTeXMLMath and problems in linear algebra .
Appendix to the Russian edition of C. Okonek , M. Schneider , and H. Spindler .
Mir , Moscow , 1984 .
M. Green : The Eisenbud-Koh-Stillman Conjecture on Linear Syzygies .
Invent .
Math .
136 ( 1999 ) 411–418 .
D. Grayson and M. Stillman : Macaulay2 .
http : //www.math.uiuc.edu/Macaulay2/ .
A. Grothendieck and J. Dieudonné : Éléments de la Géometrie Algébrique IV : Étude locale des schémas et de morphismes de schémas ( deuxième partie ) .
Publ .
Math .
de l ’ I.H.E.S .
24 ( 1965 ) .
D. Happel : Triangulated Categories In The Representation Theory Of Finite-Dimensional Algebras .
Lect .
Notes of the London Math .
Soc .
119 , 1988 .
R. Hartshorne : Algebraic geometry , Springer Verlag , 1977 .
J. Herzog , T. Römer : Resolutions of modules over the exterior algebra , working notes , 1999 .
G. Horrocks : Vector bundles on the punctured spectrum of a local ring , Proc .
Lond .
Math .
Soc. , III , Ser .
14 ( 1964 ) , 714–718 .
G. Horrocks , D. Mumford : A rank LaTeXMLMath vector bundle on LaTeXMLMath with LaTeXMLMath symmetries .
Topology 12 ( 1973 ) 63–81 .
M. M. Kapranov : On the derived categories of coherent sheaves on some homogeneous spaces .
Invent .
Math .
92 ( 1988 ) 479–508 .
M. M. Kapranov : On the derived category and LaTeXMLMath -functor of coherent sheaves on intersections of quadrics .
( Russian ) Izv .
Akad .
Nauk SSSR Ser .
Mat .
52 M. Martin-Deschamps and D. Perrin : Sur la classification des courbes gauche .
Asterisque 184-185 ( 1990 ) .
C. Okonek , M. Schneider , and H. Spindler : Vector Bundles on Complex Projective Spaces .
Birkhäuser , Boston 1980 .
D. O. Orlov : Projective bundles , monoidal transformations , and derived categories of coherent sheaves .
( Russian ) Izv .
Ross .
Akad .
Nauk Ser .
Mat .
56 ( 1992 ) 852–862 ; translation in Russian Acad .
Sci .
Izv .
Math .
41 ( 1993 ) 133–141 .
S. B. Priddy : Koszul resolutions .
Trans .
Amer .
Math .
Soc .
152 ( 1970 ) 39–60 .
P. Rao : Liaison equivalence classes , Math .
Ann .
258 ( 1981 ) 169–173 .
F.-O .
Schreyer : Small fields in constructive algebraic geometry .
In S. Maruyama ed .
: Moduli of of Vector Bundles , Sanda 1994 .
New York , Dekker 1996 , 221–228 .
R. G. Swan : LaTeXMLMath -theory of quadric hypersurfaces .
Ann .
of Math .
122 ( 1985 ) 113–153 .
C. Walter : Algebraic cohomology methods for the normal bundle of algebraic sapce curves .
Preprint ( 1990 ) .
Author Addresses : David Eisenbud Department of Mathematics , University of California , Berkeley , Berkeley CA 94720 eisenbud @ math.berkeley.edu Gunnar Fløystad Mathematisk Institutt , Johs .
Brunsgt .
12 , N-5008 Bergen , Norway gunnar @ mi.uib.no Frank-Olaf Schreyer FB Mathematik , Universität Bayreuth D-95440 Bayreuth , Germany schreyer @ btm8x5.mat.uni-bayreuth.de
In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension .
A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent .
In Section 2 the problem in the title is formulated in terms of separable bimodules .
In Section 3 we specialize the problem to ring extensions , noting that a biseparable extension is a two-sided finitely generated projective , split , separable extension .
Some reductions of the problem are discussed and solutions in special cases are provided .
In Section 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title .
We show in Section 5 that characterizations of the separable extensions among the Frobenius extensions in LaTeXMLCite are special cases of a result for adjoint functors .
An old problem is the extent to which separable algebras are Frobenius algebras .
By a Frobenius algebra we mean a finite dimensional algebra LaTeXMLMath with a non-degenerate linear functional , which induces an LaTeXMLMath -module isomorphism LaTeXMLMath ; symmetric algebra if this isomorphism is an LaTeXMLMath -bimodule map .
Eilenberg and Nakayama observed in LaTeXMLCite that the ( reduced ) trace of a central simple algebra over a field is non-degenerate , which implies that a finite dimensional semisimple algebra is symmetric .
Passing to a commutative ground ring LaTeXMLMath , Hattori LaTeXMLCite and DeMeyer LaTeXMLCite showed that a LaTeXMLMath -projective separable LaTeXMLMath -algebra LaTeXMLMath is symmetric as well if the Hattori-Stallings rank of LaTeXMLMath over its center LaTeXMLMath is an invertible element in LaTeXMLMath .
Endo and Watanabe extended this result to LaTeXMLMath -projective separable faithful LaTeXMLMath -algebras essentially by using the Auslander-Goldman Galois theory for commutative rings to define a more general notion of reduced trace LaTeXMLCite .
The main theorem in LaTeXMLCite led to several general results by Sugano LaTeXMLCite for when separable extensions LaTeXMLCite are Frobenius extensions LaTeXMLCite .
These are noncommutative ring extensions and are natural objects for study from the point of view of induced representations LaTeXMLCite .
Sugano shows that a centrally projective separable extension LaTeXMLMath is Frobenius since it satisfies LaTeXMLMath where the centralizer LaTeXMLMath is faithfully projective and separable over the center LaTeXMLMath , whence Frobenius .
Somewhat similarly , it is shown that a split one-sided finite projective H-separable extension LaTeXMLMath is Frobenius , since in this case the endomorphism ring LaTeXMLMath with LaTeXMLMath again separable , the result following from the endomorphism ring theorem as developed in LaTeXMLCite : if LaTeXMLMath is generator module , LaTeXMLMath is Frobenius iff LaTeXMLMath is Frobenius .
However , it is implicit in the literature that there are several cautionary examples showing separable extensions are not always Frobenius extensions in the ordinary untwisted sense LaTeXMLCite : in Section 4 we show that a non-finite ring leads to an example of split , separable , two-sided projective extension which is not finitely generated , whence not Frobenius .
As an independent line of inquiry , algebraic axioms for finite Jones index subfactors have been investigated in LaTeXMLCite .
If we simplify the discussion somewhat , we may start with an irreducible subfactor LaTeXMLMath of finite index : from the Pimsner-Popa orthonormal base , the natural modules LaTeXMLMath and LaTeXMLMath are finite projective LaTeXMLCite , and the algebra extension LaTeXMLMath is ( 1 ) split , ( 2 ) separable , and ( 3 ) Frobenius .
A ring extension LaTeXMLMath is said to be split if there is a bimodule projection LaTeXMLMath .
At the same time , Axiom ( 2 ) yields a Casimir element LaTeXMLMath such that LaTeXMLMath .
A problem in the independence of the axioms above becomes whether Axioms ( 1 ) and ( 2 ) imply Axiom ( 3 ) in the presence of the assumption of two-sided finite projectivity of the ring extension : i.e. , whether a bimodule map LaTeXMLMath and Casimir element LaTeXMLMath may be chosen such that LaTeXMLMath .
Equivalently , can a bimodule map LaTeXMLMath be found such that the LaTeXMLMath -multiplication on LaTeXMLMath LaTeXMLCite is unital ?
That Axiom ( 3 ) in combination with ( 1 ) or ( 2 ) does not imply the other are easy examples disposed of in Section 3 .
Many other algebraic examples of split separable Frobenius can be found LaTeXMLCite but none thus far that are finite projective split separable and not Frobenius .
In this paper , we will formulate the problem of independence of axioms for subfactors in several algebraic ways .
In Section 2 we first formulate the problem using separable bimodules LaTeXMLCite , a theory in which separable extension and split extension become dual notions LaTeXMLCite .
In analogy with “ bialgebra , ” we will baptise finite projective split separable extensions as biseparable extensions .
Posed in the negative , our question then becomes if biseparable extensions are Frobenius .
This question will be formulated in several other ways in Section 3 , with one special case being answered in the affirmative .
We point out here that the problem has many interesting sub-problems if restrictions are placed on the rings ( e.g. , “ finite dimensional algebras , ” “ Hopf algebras , ” etc . ) .
In Section 5 we discuss a type of converse to the considerations above .
We find a common feature of the theorems in LaTeXMLCite on when a Frobenius extension or bimodule is separable : in each case , it is a specific example of a known theorem on adjoint functors , which we expose in this last section .
A ring LaTeXMLMath will mean a unital associative ring .
A ring homomorphism sends 1 into 1 .
A right module LaTeXMLMath or left module LaTeXMLMath is always unitary .
Bimodules are associative with respect to the left and right actions .
An LaTeXMLMath - LaTeXMLMath -bimodule LaTeXMLMath is denoted by LaTeXMLMath .
Its right dual is defined by LaTeXMLMath , an LaTeXMLMath - LaTeXMLMath -bimodule where LaTeXMLMath .
The left dual of LaTeXMLMath is LaTeXMLMath is also LaTeXMLMath - LaTeXMLMath -bimodule where LaTeXMLMath .
Both LaTeXMLMath and LaTeXMLMath are contravariant functors of bimodule categories , sending LaTeXMLMath .
If LaTeXMLMath in the last paragraph , denote LaTeXMLMath .
Define the group of LaTeXMLMath -central or Casimir elements by LaTeXMLMath .
Note that LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are bimodules , then LaTeXMLMath receives the natural LaTeXMLMath - LaTeXMLMath -bimodule structure indicated by LaTeXMLMath , and the group of right module homomorphisms LaTeXMLMath receives the natural LaTeXMLMath - LaTeXMLMath -bimodule structure indicated by LaTeXMLMath .
The group of left module homomorphisms LaTeXMLMath receives the natural LaTeXMLMath - LaTeXMLMath -bimodule structure indicated by LaTeXMLMath .
All bimodules arising from Hom and tensor in this paper are the natural ones unless otherwise indicated .
A ring extension LaTeXMLMath is a ring homomorphism LaTeXMLMath .
A ring extension is an algebra if LaTeXMLMath is commutative and LaTeXMLMath factors into LaTeXMLMath where LaTeXMLMath is the center of LaTeXMLMath .
A ring extension is proper if LaTeXMLMath is 1-1 , in which case identification is made .
The natural bimodule LaTeXMLMath is given by LaTeXMLMath .
In particular , we consider the natural modules LaTeXMLMath and LaTeXMLMath .
An adjective , such as right projective or projective , for the ring extension LaTeXMLMath refers to the same adjective for one or both of these natural modules .
The structure map LaTeXMLMath is usually suppressed .
Separable extensions are studied in LaTeXMLCite among others .
A ring extension LaTeXMLMath is separable if the natural ( multiplication ) map LaTeXMLMath is a split epimorphism of LaTeXMLMath -bimodules .
Examples are abundant among finite dimensional algebras since a separable algebra is a separable extension of any of its subalgebras .
The next proposition , whose proof follows Sugano LaTeXMLCite , is important to keep in mind when finding examples of separable extensions from the class of finite dimensional algebras .
If LaTeXMLMath is a split exact sequence of algebras and LaTeXMLMath is a separable extension , then LaTeXMLMath is an idempotent ideal ( i.e. , LaTeXMLMath ) .
We assume with no loss of generality that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a separability element in LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLMath satisfies LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath , it follows that LaTeXMLMath is idempotent .
Similarly , LaTeXMLMath satisfies LaTeXMLMath , LaTeXMLMath and is idempotent .
Then LaTeXMLEquation .
Then LaTeXMLMath and LaTeXMLMath is central .
Then LaTeXMLMath , whence LaTeXMLMath is idempotent .
∎ An example of a ring epi splitting in the next corollary would be the one implicit in the Wedderburn Principal Theorem for finite dimensional algebras .
If LaTeXMLMath is a split , separable extension with splitting map LaTeXMLMath a ring epimorphism with nilpotent kernel LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
Indeed a separable finitely generated ( f.g. ) extension of a separable algebra is itself separable LaTeXMLCite .
The next proposition builds new separable extensions from old using multiplicative bimodules LaTeXMLCite .
Suppose LaTeXMLMath is a ring extension and LaTeXMLMath is a multiplicative LaTeXMLMath -bimodule .
Then LaTeXMLMath is a separable extension if and only if LaTeXMLMath is a separable extension of LaTeXMLMath .
( LaTeXMLMath ) Let LaTeXMLMath be a separability element for LaTeXMLMath .
Let LaTeXMLMath be its image in LaTeXMLMath induced by LaTeXMLMath .
If LaTeXMLMath , then : LaTeXMLEquation .
We easily conclude that LaTeXMLMath is a separability element for LaTeXMLMath .
( LaTeXMLMath ) Since LaTeXMLMath is an ideal in both LaTeXMLMath and LaTeXMLMath , the canonical epimorphism LaTeXMLMath sends the separable extension LaTeXMLMath onto a separable extension LaTeXMLMath LaTeXMLCite .
∎ As our final preliminary topic we recall Frobenius and QF extensions .
LaTeXMLCite A ring extension LaTeXMLMath is Frobenius if LaTeXMLMath is f.g. projective and LaTeXMLMath as LaTeXMLMath - LaTeXMLMath -bimodules : note that this extends the notion of Frobenius algebra .
We recall also the Morita characterization of Frobenius extensions LaTeXMLCite : an extension LaTeXMLMath is Frobenius iff induction and co-induction ( of LaTeXMLMath -modules to LaTeXMLMath -modules ) are naturally isomorphic ( cf .
LaTeXMLCite ) .
A ring extension LaTeXMLMath is a left Quasi-Frobenius ( QF ) extension if LaTeXMLMath is finitely generated projective and LaTeXMLMath is isomorphic to a direct summand of a finite direct product of LaTeXMLMath with itself .
Equivalently , LaTeXMLMath and LaTeXMLMath are finitely generated projectives and LaTeXMLMath is a direct summand of a finite direct sum of copies of LaTeXMLMath .
We similarly define right QF extensions LaTeXMLCite .
There is no published example of a right QF extension that is not left QF .
In this section we pose our question in the more general terms of bimodules rather than ring extensions .
There are two reasons for this .
First , the problem has a more attractive symmetrical formulation in terms of bimodules .
Second , Morita has shown in LaTeXMLCite how to generate interesting examples of ring extensions from bimodules via the endomorphism ring .
Let LaTeXMLMath and LaTeXMLMath be rings .
Given a bimodule LaTeXMLMath , there is a natural LaTeXMLMath -bimodule homomorphism , LaTeXMLEquation .
We next recall the definition of a separable bimodule LaTeXMLCite .
LaTeXMLMath is separable , or LaTeXMLMath is LaTeXMLMath -separable over LaTeXMLMath , if LaTeXMLMath is a split LaTeXMLMath -epimorphism .
It follows trivially that LaTeXMLMath is a generator module LaTeXMLCite .
By applying a splitting map to LaTeXMLMath , we note that LaTeXMLMath is separable iff there is an element LaTeXMLEquation called an LaTeXMLMath -separability element , which satisfies LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
As is the case with separability elements and idempotents LaTeXMLCite , LaTeXMLMath -separability elements are usually not unique .
Retaining this notation , we recall a useful proposition and its proof LaTeXMLCite .
But first a lemma which does not require LaTeXMLMath to be separable .
If LaTeXMLMath is finitely generated projective , then LaTeXMLMath given by LaTeXMLEquation is a LaTeXMLMath -bimodule isomorphism .
Similarly , LaTeXMLMath f.g. projective implies that LaTeXMLEquation is an LaTeXMLMath -bimodule isomorphism .
The proof of this and the next two propositions are left to the reader .
Now define LaTeXMLMath by LaTeXMLEquation .
We have LaTeXMLMath ; i.e. , the diagram below is commutative .
Whence LaTeXMLMath is separable iff there is an LaTeXMLMath - LaTeXMLMath -bimodule homomorphism LaTeXMLMath such that LaTeXMLMath .
LaTeXMLEquation ¿From LaTeXMLMath we note that LaTeXMLMath corresponds under LaTeXMLMath to an LaTeXMLMath -separability element .
A bimodule LaTeXMLMath is said to be biseparable if LaTeXMLMath and LaTeXMLMath are separable and LaTeXMLMath , LaTeXMLMath are finite projective modules .
We derive some consequences of assuming LaTeXMLMath f.g. projective and LaTeXMLMath separable .
First , since LaTeXMLMath is reflexive , it follows that LaTeXMLMath via the “ evaluation mapping ” from LaTeXMLMath to LaTeXMLMath .
It follows that LaTeXMLMath under this identification is the evaluation map given by LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath be finite dual bases for the f.g. projective module LaTeXMLMath .
We have the following easy analog of Proposition LaTeXMLRef .
The triangle below is commutative .
Whence if LaTeXMLMath is separable there is a LaTeXMLMath - LaTeXMLMath -bimodule homomorphism LaTeXMLMath such that LaTeXMLMath .
LaTeXMLEquation .
The downward map to the right is given by LaTeXMLMath .
For example , a bimodule LaTeXMLMath yielding a Morita equivalence of LaTeXMLMath and LaTeXMLMath is biseparable , since LaTeXMLMath as LaTeXMLMath - LaTeXMLMath -bimodules LaTeXMLCite while LaTeXMLMath and LaTeXMLMath are isomorphisms .
There have been various studies of properties shared by rings LaTeXMLMath and LaTeXMLMath related by a bimodule LaTeXMLMath in a Morita context and generalizations of this LaTeXMLCite .
A precursor of these studies is the theorem of D.G .
Higman LaTeXMLCite that a finite group has finite representation type ( f.r.t . )
in characteristic LaTeXMLMath iff its Sylow LaTeXMLMath -subgroup is cyclic , which later became a corollary of the theorem of J.P. Jans LaTeXMLCite that for Artinian algebras LaTeXMLMath in a split separable extension , LaTeXMLMath has f.r.t .
iff LaTeXMLMath has f.r.t .
( cf .
LaTeXMLCite ) .
It is in this spirit that the next theorem offers a sample of shared properties of LaTeXMLMath and LaTeXMLMath linked by a biseparable bimodule LaTeXMLMath .
If LaTeXMLMath is biseparable , then ( Cf .
LaTeXMLCite . )
LaTeXMLMath is a QF ring if and only if LaTeXMLMath is a QF ring ; ( Cf .
Ibid . )
LaTeXMLMath is semisimple if and only if LaTeXMLMath is semisimple ; weak global dimension LaTeXMLMath .
( QF ) .
Assume LaTeXMLMath is QF and LaTeXMLMath is injective .
By the Faith-Walker theorem , it will suffice to show that LaTeXMLMath is projective .
Since LaTeXMLMath is projective , then flat , we note that LaTeXMLMath is injective , then projective .
Since LaTeXMLMath is projective , we note that LaTeXMLMath is projective .
But the evaluation mapping LaTeXMLEquation is a split epi , for if LaTeXMLMath is an LaTeXMLMath -separability element , then LaTeXMLMath defines a splitting LaTeXMLMath -monic , where LaTeXMLMath .
Hence LaTeXMLMath is isomorphic to a direct summand in LaTeXMLMath and is projective .
Assuming that LaTeXMLMath is QF , and LaTeXMLMath is injective , we argue similarly that LaTeXMLMath is injective-projective and that LaTeXMLMath is isomorphic to direct summand in the projective LaTeXMLMath -module LaTeXMLMath .
( SEMISIMPLICITY ) .
Suppose LaTeXMLMath is semisimple and LaTeXMLMath is a module .
It suffices to note that LaTeXMLMath is projective .
Since LaTeXMLMath is projective and the map LaTeXMLMath defined as above is a split LaTeXMLMath -epimorphism , it follows that LaTeXMLMath is isomorphic to a direct summand of the projective module LaTeXMLMath .
Similarly , we argue that given LaTeXMLMath semisimple and module LaTeXMLMath , LaTeXMLMath is projective .
( WEAK GLOBAL DIMENSION . )
If LaTeXMLMath is a projective resolution of LaTeXMLMath , then LaTeXMLMath is a projective resolution as well since LaTeXMLMath is flat and LaTeXMLMath is projective .
Recall that LaTeXMLMath is the LaTeXMLMath ’ th homology group of the chain complex LaTeXMLMath for each non-negative integer LaTeXMLMath , so LaTeXMLMath is the LaTeXMLMath ’ th homology group of LaTeXMLMath .
At the level of chain complex , there is a split epi LaTeXMLEquation which implies that LaTeXMLMath is isomorphic to a direct summand in LaTeXMLMath for each LaTeXMLMath .
This shows that LaTeXMLMath .
A similar argument with left modules shows that LaTeXMLMath .
∎ We remark that in trying to prove other shared homological properties of biseparable LaTeXMLMath , particularly one-sided notions , one may run into the following complications : although the modules LaTeXMLMath and LaTeXMLMath are ( quite easily seen to be ) f.g. projective , one should avoid assuming the same of LaTeXMLMath and LaTeXMLMath .
We next recall the definition of Frobenius bimodule LaTeXMLCite .
A bimodule LaTeXMLMath is Frobenius if LaTeXMLMath , LaTeXMLMath are f.g. projective and LaTeXMLMath as LaTeXMLMath - LaTeXMLMath -bimodules .
Based on the many examples in LaTeXMLCite and elsewhere , we propose the following problem , which turns out to be almost equivalent to the ring extension formulation in the title : Is a biseparable bimodule LaTeXMLMath a Frobenius bimodule ?
For example , can we choose LaTeXMLMath and LaTeXMLMath such that they are inverses to one another ?
The problem above subsumes many interesting questions in various restricted cases .
For example , what can be said for the problem above if LaTeXMLMath and LaTeXMLMath are finite dimensional algebras ?
There is an affirmative answer in the next section if one algebra is separable .
A generalization of Frobenius bimodule is a twisted Frobenius bimodule LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are ring automorphisms , and the bimodule structure is now given by LaTeXMLMath ( for the definition see LaTeXMLCite ) .
We might ask more widely Is a biseparable bimodule a twisted Frobenius bimodule ?
However , this problem is the same as the previous one if the following question has an affirmative answer : If a twisted Frobenius bimodule LaTeXMLMath is biseparable , does this imply that LaTeXMLMath ?
We say that a twisted Frobenius bimodule is nontrivial if it not isomorphic to an untwisted Frobenius bimodule ; for a LaTeXMLMath -Frobenius extension LaTeXMLMath nontriviality means that LaTeXMLMath is not an extended inner automorphism in the sense that there is a unit LaTeXMLMath such that LaTeXMLMath is conjugation by LaTeXMLMath LaTeXMLCite .
We pose the last question since we have never observed a nontrivial LaTeXMLMath -Frobenius extension ( e.g .
in LaTeXMLCite ) which was simultaneously split and separable ( cf .
next section ) .
In this more limited setting , which covers Hopf subalgebras of finite dimensional Hopf algebras , the question becomes : If a LaTeXMLMath -Frobenius extension is split and separable , is LaTeXMLMath an extended inner automorphism ?
We will return to a discussion of this problem in the next section .
Suppose LaTeXMLMath is a ring extension .
Letting LaTeXMLMath in the definition of separable bimodule , we observe the following lemma LaTeXMLCite .
LaTeXMLMath is a separable extension iff LaTeXMLMath is a separable bimodule .
Dually , we let LaTeXMLMath and observe the following lemma LaTeXMLCite .
LaTeXMLMath is a split extension iff LaTeXMLMath is a separable bimodule iff LaTeXMLMath is a separable bimodule .
The LaTeXMLMath -separability element in this case is a bimodule projection LaTeXMLMath , which implies LaTeXMLMath is a proper extension .
LaTeXMLMath is also called a conditional expectation if it satisfies additional properties in subfactor theory .
¿From the last two lemmas , it follows that : LaTeXMLMath is a split , separable , two-sided finite projective extension iff LaTeXMLMath and LaTeXMLMath are biseparable bimodules iff LaTeXMLMath is biseparable and LaTeXMLMath is f.g. projective iff LaTeXMLMath is biseparable and LaTeXMLMath is f.g. projective .
We call LaTeXMLMath a biseparable extension if any of the equivalent conditions in Lemma LaTeXMLRef are satisfied .
Additionally , we have the following lemma LaTeXMLCite .
LaTeXMLMath is a Frobenius extension iff LaTeXMLMath or LaTeXMLMath is a Frobenius bimodule .
The last two lemmas lead to the seemingly restricted formulation of Problem LaTeXMLRef , also the title of this article .
Are biseparable extensions Frobenius ?
Surprisingly , this problem is almost equivalent to Problem LaTeXMLRef because of the endomorphism ring theorems for Frobenius bimodules LaTeXMLCite .
Suppose we knew an affirmative answer to the somewhat weaker problem where biseparable extension includes left f.g. projective , split separable extensions .
Given a biseparable bimodule LaTeXMLMath , we know from Sugano that LaTeXMLMath is a left f.g. projective , split , separable extension of LaTeXMLMath ( whose elements are identified in LaTeXMLMath with left multiplication operators ) LaTeXMLCite ( cf .
LaTeXMLCite ) .
Then LaTeXMLMath is a Frobenius extension by our affirmative answer to the weak Problem LaTeXMLRef .
Since LaTeXMLMath is faithfully balanced by Morita ’ s Lemma , it follows from LaTeXMLCite that LaTeXMLMath and then from the endomorphism ring theorem-converse LaTeXMLCite that LaTeXMLMath is a Frobenius bimodule .
What evidence do we have then for proposing Problem LaTeXMLRef ?
First , if LaTeXMLMath is an LaTeXMLMath -algebra , we are in the situation of a faithfully projective separable algebra , which is Frobenius by the Endo-Watanabe Theorem LaTeXMLCite discussed in the introduction .
This implies by elementary considerations that LaTeXMLMath -algebra extensions of the form LaTeXMLMath over LaTeXMLMath are Frobenius if LaTeXMLMath is faithfully projective LaTeXMLMath -separable .
Second , there are the many examples of split , separable , Frobenius extensions LaTeXMLCite and apparently none that contradict in the literature for noncommutative rings and ring extensions LaTeXMLCite .
We recall from LaTeXMLCite that some of the examples of LaTeXMLMath split , separable Frobenius are the following : Let LaTeXMLMath a field of characteristic LaTeXMLMath ( zero or prime ) and LaTeXMLMath a subfield such that LaTeXMLMath is a finite separable extension where LaTeXMLMath does not divide LaTeXMLMath .
The classical trace map from LaTeXMLMath into LaTeXMLMath is a Frobenius homomorphism in this example .
Let LaTeXMLMath be the group algebra LaTeXMLMath for a discrete group LaTeXMLMath , LaTeXMLMath a field , and LaTeXMLMath , where LaTeXMLMath is a subgroup of LaTeXMLMath with finite index not divisible by the characteristic of LaTeXMLMath ( e.g. , char LaTeXMLMath and LaTeXMLMath a Sylow LaTeXMLMath -subgroup ) .
Note that if the characteristic of LaTeXMLMath divides LaTeXMLMath we have an example of a split Frobenius extension which is not separable .
By LaTeXMLCite the endomorphism ring extension LaTeXMLMath is a separable Frobenius extension which is not split .
Let LaTeXMLMath and LaTeXMLMath be algebras over a commutative ring LaTeXMLMath such that LaTeXMLMath is an Hopf-Galois extension over the f.g. projective Hopf LaTeXMLMath -algebra LaTeXMLMath which is separable and coseparable over LaTeXMLMath .
Let LaTeXMLMath be a type LaTeXMLMath factor , LaTeXMLMath a subfactor of LaTeXMLMath of finite Jones index , as discussed in the introduction .
Third , Sugano ’ s result LaTeXMLCite for when H-separable extensions are Frobenius is evidence for biseparable implies Frobenius .
This is because an H-separable extension is a strong type of separable extension LaTeXMLCite : see Section 4 for a separable extension which is not H-separable .
Thus the result that a ( one-sided ) f.g. projective split H-separable extension is a ( symmetric ) Frobenius extension is a particular case of a biseparable extension which is Frobenius ( cf .
LaTeXMLCite LaTeXMLCite should read “ Suppose LaTeXMLMath is a right progenerator split H-separable… ” as the proof clearly shows .
) .
The next proposition shows that a biseparable extension is almost a two-sided QF extension in a certain sense .
If a module LaTeXMLMath is isomorphic to a direct summand in another module LaTeXMLMath , we denote this by LaTeXMLMath .
Suppose LaTeXMLMath is a biseparable extension .
Then all LaTeXMLMath -modules are LaTeXMLMath -relative injective and LaTeXMLMath -relative projective ; moreover , LaTeXMLMath and LaTeXMLMath are generator modules .
The first statement follows from the fact that a separable extension is both right and left semisimple extension and properties of these LaTeXMLCite .
For the second statement , we first establish an interesting isomorphism below involving LaTeXMLMath and its dual LaTeXMLMath .
On the one hand , since LaTeXMLMath is a separable extension , LaTeXMLMath is a split extension of LaTeXMLMath , for if LaTeXMLMath is a separability element we define a bimodule projection by LaTeXMLMath by LaTeXMLMath ( cf .
LaTeXMLCite ) .
Then as LaTeXMLMath -bimodules , LaTeXMLMath for some LaTeXMLMath : moreover , by restriction this is true as LaTeXMLMath - LaTeXMLMath -bimodules .
On the other hand , since LaTeXMLMath is split , it follows that for some LaTeXMLMath -bimodule LaTeXMLMath , which is left and right projective LaTeXMLMath -module , LaTeXMLMath as LaTeXMLMath -bimodules ; whence LaTeXMLEquation as LaTeXMLMath - LaTeXMLMath -bimodules .
Putting together the two isomorphisms for LaTeXMLMath , we obtain LaTeXMLEquation .
Since LaTeXMLMath is f.g. projective , there is a module LaTeXMLMath such that LaTeXMLMath .
Then applying LaTeXMLMath to this : LaTeXMLEquation .
Combining this with Eq .
( LaTeXMLRef ) , we obtain LaTeXMLEquation .
This establishes that LaTeXMLMath .
We similarly conclude LaTeXMLMath by combining the split extension LaTeXMLMath with the LaTeXMLMath -bimodule isomorphism LaTeXMLMath and the existence of LaTeXMLMath such that LaTeXMLMath .
∎ ¿From the proof just completed , we obtain a corollary worth noting for its relatively easy proof .
LaTeXMLMath is said to be centrally projective bimodule centrally projective if LaTeXMLMath for some positive integer LaTeXMLMath .
( Cf .
LaTeXMLCite ) If LaTeXMLMath is centrally projective biseparable extension , then it is a QF extension .
Since there is a bimodule LaTeXMLMath such that LaTeXMLMath , we combine this with LaTeXMLMath to see that Eq .
( LaTeXMLRef ) is an LaTeXMLMath - LaTeXMLMath -isomorphism , whence LaTeXMLMath is a right QF extension LaTeXMLCite .
Similarly , we show LaTeXMLMath to be a left QF extension .
∎ The proposition and corollary LaTeXMLCite and ring extensions LaTeXMLCite , the corollary states that a depth one biseparable extension is QF .
A depth two biseparable extension is QF as well LaTeXMLCite , where depth two is the condition that LaTeXMLMath as LaTeXMLMath - LaTeXMLMath and LaTeXMLMath - LaTeXMLMath -bimodules .
above lead naturally to the problem below , a weakening of Problem LaTeXMLRef .
Are biseparable extensions QF ?
The next theorem provides a solution of Problem LaTeXMLRef in case LaTeXMLMath or LaTeXMLMath is a separable algebra .
We assume our algebras to be faithful .
Suppose LaTeXMLMath is a biseparable extension of LaTeXMLMath -algebras with LaTeXMLMath a commutative ring .
If either LaTeXMLMath is a LaTeXMLMath -projective separable LaTeXMLMath -algebra , or LaTeXMLMath is a separable LaTeXMLMath -algebra with LaTeXMLMath a field , then LaTeXMLMath is a Frobenius extension .
The proof does not make use of LaTeXMLMath being f.g. projective .
Suppose LaTeXMLMath is a field and LaTeXMLMath is LaTeXMLMath -separable .
Let LaTeXMLMath denote the right global dimension of a ring LaTeXMLMath and LaTeXMLMath denote the projective dimension of a module LaTeXMLMath .
Then LaTeXMLMath since LaTeXMLMath is finite dimensional semisimple .
By Cohen ’ s Theorem for split extensions LaTeXMLCite , LaTeXMLEquation whence LaTeXMLMath and LaTeXMLMath is semisimple LaTeXMLCite .
Then LaTeXMLMath and LaTeXMLMath are finite dimensional semisimple algebras .
It follows from LaTeXMLCite that LaTeXMLMath and LaTeXMLMath are symmetric algebras .
Similarly , we arrive at symmetric algebras LaTeXMLMath and LaTeXMLMath via LaTeXMLCite under the assumption that LaTeXMLMath is LaTeXMLMath -projectively LaTeXMLMath -separable with no restriction on LaTeXMLMath .
For then LaTeXMLMath is LaTeXMLMath -projective and LaTeXMLMath -separable by transitivity for projectivity and separability .
Now we compute using the bimodule isomorphisms LaTeXMLMath and LaTeXMLMath and the hom-tensor adjunction : LaTeXMLEquation .
Then , since LaTeXMLMath is f.g. projective , LaTeXMLMath is a Frobenius extension .
∎ Part of the theorem is true without the hypothesis of biseparable extension for a finite-dimensional Hopf subalgebra pair LaTeXMLMath : if LaTeXMLMath is semisimple , then LaTeXMLMath is semisimple LaTeXMLCite , and LaTeXMLMath is a Frobenius extension ( cf .
LaTeXMLCite ) .
Finally , Problem ( LaTeXMLRef ) can be widened to twisted extensions , as Problem ( LaTeXMLRef ) was widened to twisted Frobenius bimodules in Problem ( LaTeXMLRef ) .
Are biseparable extensions LaTeXMLMath - LaTeXMLMath -Frobenius ?
We refer the reader to LaTeXMLCite for the definition of these twisted extensions , which are more general than the usual LaTeXMLMath -Frobenius extensions .
Also , Problem ( LaTeXMLRef ) has a twisted enlargement .
In this section , we consider weakening the definition of biseparability in various ways , and find examples of non-Frobenius extension for each such case .
We will see an example of non-finitely generated projective separable extension , which is an obstruction to extending Villamayor ’ s theorem LaTeXMLCite and Tominaga ’ s theorem LaTeXMLCite .
Suppose LaTeXMLMath is a commutative ring and LaTeXMLMath is a LaTeXMLMath -algebra with LaTeXMLMath but LaTeXMLMath .
Then LaTeXMLMath is a separable extension over LaTeXMLMath .
We note that LaTeXMLMath is a nontrivial idempotent in LaTeXMLMath .
Consider LaTeXMLMath .
Of course , LaTeXMLMath .
We compute with LaTeXMLMath : LaTeXMLEquation .
Now if LaTeXMLMath is a finitely generated , projective LaTeXMLMath -algebra , it is well-known that LaTeXMLMath implies LaTeXMLMath .
So we let LaTeXMLMath be a countably infinite rank free LaTeXMLMath -module and LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are column-finite and LaTeXMLMath denotes transpose .
LaTeXMLMath is a ring satisfying the hypothesis in the lemma with elements LaTeXMLMath given in terms of the matrix units LaTeXMLMath by LaTeXMLEquation .
Clearly , LaTeXMLMath but LaTeXMLEquation and the LaTeXMLMath -subalgebra , LaTeXMLEquation .
LaTeXMLMath is a split , separable , projective and non-finitely generated extension .
We have seen in the lemma that LaTeXMLMath is separable .
It is split since we easily check that LaTeXMLMath below is a bimodule projection : LaTeXMLEquation .
LaTeXMLMath is countably generated projective , since LaTeXMLMath defined by LaTeXMLEquation and LaTeXMLMath , where LaTeXMLMath , which satisfies the dual bases equation , LaTeXMLEquation .
By tranposing the elements and maps above , we similarly find a countable projective base for LaTeXMLMath .
The rest of the proof is now clear .
∎ LaTeXMLMath is not Frobenius and not H-separable .
For LaTeXMLMath to be a Frobenius extension , we must have LaTeXMLMath finitely generated from the very start .
A right projective H-separable extension is right f.g. by LaTeXMLCite .
∎ Finally , we consider various weakenings of Problem LaTeXMLRef and note that they all have known counterexamples .
There are easy examples of separable extensions which are not Frobenius , such as the rationals LaTeXMLMath extending the integers LaTeXMLMath .
There are even f.g. free separable extensions that are not Frobenius in the ordinary sense , but are LaTeXMLMath -Frobenius LaTeXMLCite .
As for a f.g. split , separable extension that is not Frobenius , here is one that is not flat , whence not projective : consider LaTeXMLMath as a LaTeXMLMath -algebra , which has projection onto its first factor as splitting bimodule projection , is a direct sum of separable algebras – whence separable – and its second factor is of course not flat over LaTeXMLMath .
The following example of a f.g. projective H-separable ( therefore separable LaTeXMLCite ) extension LaTeXMLMath which is not even twisted QF is certainly worth a mention : let LaTeXMLMath be the LaTeXMLMath by LaTeXMLMath matrices over a field with LaTeXMLMath the upper triangular matrices ( including diagonal matrices ) .
Although this example is not Frobenius in any sense that we have mentioned in this paper ( left as an exercise to the reader ) , we note that LaTeXMLMath .
However , there is no published example of a one-sided progenerator H-separable extension which is not Frobenius .
There are clearly many examples of split extensions that are not Frobenius , let alone f.g .
Asking for a split , f.g. projective extension that fails to be Frobenius is not hard : for example , let LaTeXMLMath be the upper triangular LaTeXMLMath matrix algebra with splitting LaTeXMLMath where LaTeXMLMath is the subalgebra of diagonal matrices and LaTeXMLMath .
It is well-known that LaTeXMLMath is not a QF-algebra , and certainly not Frobenius LaTeXMLCite , but LaTeXMLMath is semisimple , so LaTeXMLMath and LaTeXMLMath are f.g. projective ; moreover , LaTeXMLMath can not be Frobenius by the transitivity property of Frobenius extension ( cf .
LaTeXMLCite ) .
This is an example too of LaTeXMLMath not being a projective right LaTeXMLMath -module .
As a last cautionary example , we consider LaTeXMLMath and LaTeXMLMath .
It is easy to check that LaTeXMLMath is split , separable and Frobenius , even f.g. free .
But there are only two bimodule projections LaTeXMLMath , neither of which is a Frobenius homomorphism , i.e .
in possession of dual bases LaTeXMLCite .
The Frobenius homomorphism in this example is unique , since the group of units in LaTeXMLMath consists only of the identity LaTeXMLCite .
Let LaTeXMLMath be a contravariant functor .
LaTeXMLMath induces a natural transformation LaTeXMLMath .
LaTeXMLMath is called a separable functor LaTeXMLCite if LaTeXMLMath splits , i.e .
there exists a natural transformation LaTeXMLMath such that LaTeXMLMath is the identity natural transformation on LaTeXMLMath .
LaTeXMLCite Assume that LaTeXMLMath has a right adjoint LaTeXMLMath , and let LaTeXMLMath and LaTeXMLMath be the unit and counit of the adjunction .
LaTeXMLMath is separable if and only if there exists a natural transformation LaTeXMLMath such that LaTeXMLMath is the identity natural transformation on LaTeXMLMath .
LaTeXMLMath is separable if and only if there exists a natural transformation LaTeXMLMath such that LaTeXMLMath is the identity natural transformation on LaTeXMLMath .
The terminology stems from the fact that , for a ring homomorphism LaTeXMLMath , the restriction of scalars functor is separable if and only if LaTeXMLMath is separable ( see LaTeXMLCite , LaTeXMLCite ) ; Separable functors satisfy the following version of Maschke ’ s Theorem : if a morphism LaTeXMLMath in LaTeXMLMath is such that LaTeXMLMath has a left or right inverse in LaTeXMLMath , then LaTeXMLMath has a left or right inverse in LaTeXMLMath .
Separable functors have been studied in several particular cases recently , see e.g .
LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
The functor LaTeXMLMath is called Frobenius if LaTeXMLMath has a right adjoint LaTeXMLMath that is at the same time a right adjoint .
We will then say that LaTeXMLMath is a Frobenius pair .
Now the terminology is inspired by the property that a ring homomorphism LaTeXMLMath is Frobenius if and only if the restriction of scalars functor is Frobenius .
Frobenius pairs were introduced in LaTeXMLCite , and studied more recently in LaTeXMLCite and LaTeXMLCite .
For more details and examples of separable functors and Frobenius functors , we refer the reader to LaTeXMLCite .
Suppose we know that LaTeXMLMath is a Frobenius pair .
Then Rafael ’ s Theorem can be simplified : we can give an easier criterion for LaTeXMLMath or LaTeXMLMath to be separable .
First we need a result on adjoint functors .
Let LaTeXMLMath and LaTeXMLMath be the counit and unit of an adjunction LaTeXMLMath , and recall that LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be an adjoint pair functors , then we have isomorphisms LaTeXMLEquation .
In fact this is an easy consequence of LaTeXMLCite .
We restrict to giving a description of the isomorphism LaTeXMLMath , the other isomorphisms are similar .
For a natural transformation LaTeXMLMath , we define LaTeXMLMath by LaTeXMLEquation .
Conversely , for LaTeXMLMath , LaTeXMLMath is defined by LaTeXMLEquation .
The fact that LaTeXMLMath and LaTeXMLMath are each others inverses is proved using ( LaTeXMLRef ) .∎ Let LaTeXMLMath be a Frobenius pair of functors , then we have isomorphisms LaTeXMLEquation .
For a Frobenius pair LaTeXMLMath , we will write LaTeXMLMath and LaTeXMLMath for the counit and unit of the adjunction LaTeXMLMath .
For all LaTeXMLMath and LaTeXMLMath , we then have LaTeXMLEquation .
Let LaTeXMLMath be a Frobenius pair , and let LaTeXMLMath and LaTeXMLMath be as above .
The following statements are equivalent : LaTeXMLMath is separable ; LaTeXMLMath for all LaTeXMLMath ; LaTeXMLMath for all LaTeXMLMath .
We have a similar characterization for the separability of LaTeXMLMath : the statements LaTeXMLMath is separable ; LaTeXMLMath for all LaTeXMLMath ; LaTeXMLMath for all LaTeXMLMath .
are equivalent .
Assume that LaTeXMLMath is separable .
By Rafael ’ s Theorem , there exists LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLMath be the corresponding natural transformation of Corollary LaTeXMLRef , i.e .
LaTeXMLMath , and LaTeXMLMath , and the first implication of the Proposition follows .
The converse follows trivially from Rafael ’ s Theorem .
All the other equivalences can be proved in a similar way .
∎ We use the notation of Section LaTeXMLRef : let LaTeXMLMath and LaTeXMLMath be rings , and LaTeXMLMath a LaTeXMLMath -bimodule .
We have already seen that LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -bimodules .
As given in the preliminaries , LaTeXMLMath and LaTeXMLMath are respectively the natural LaTeXMLMath -bimodule and LaTeXMLMath -bimodule .
Furthermore , the induction functor LaTeXMLEquation has a right adjoint LaTeXMLEquation .
For LaTeXMLMath , LaTeXMLMath via LaTeXMLMath .
For LaTeXMLMath in LaTeXMLMath , we put LaTeXMLMath .
We also have a functor LaTeXMLEquation and a natural transformation LaTeXMLMath given by LaTeXMLEquation .
We will also write LaTeXMLMath .
LaTeXMLMath is well-defined on the tensor product , and left LaTeXMLMath -linear , so LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is a LaTeXMLMath -bimodule ( for example , LaTeXMLMath ) , then LaTeXMLMath is also right LaTeXMLMath -linear , and we have LaTeXMLEquation .
Now assume that LaTeXMLMath is finitely generated and projective , and consider a dual basis LaTeXMLMath , i.e .
LaTeXMLEquation for all LaTeXMLMath , or LaTeXMLMath .
Then LaTeXMLMath is a natural isomorphism , and for all left LaTeXMLMath -linear LaTeXMLMath , we have LaTeXMLEquation .
In order to decide whether LaTeXMLMath or LaTeXMLMath is separable , or whether LaTeXMLMath is a Frobenius pair , we have to investigate natural transformations LaTeXMLMath and LaTeXMLMath .
This is done in the next two Propositions .
We thank the referee for pointing out to us that Proposition LaTeXMLRef and LaTeXMLRef can be derived from LaTeXMLCite .
Let LaTeXMLMath and LaTeXMLMath be as above , and consider LaTeXMLEquation .
LaTeXMLEquation Then we have maps LaTeXMLEquation which are isomorphisms if LaTeXMLMath is finitely generated and projective as a left LaTeXMLMath -module .
For LaTeXMLMath , we put LaTeXMLMath .
By definition , LaTeXMLMath is left LaTeXMLMath -linear .
For any LaTeXMLMath , we consider the left LaTeXMLMath -linear map LaTeXMLMath , LaTeXMLMath .
The naturality of LaTeXMLMath implies that LaTeXMLMath , so LaTeXMLMath is also right LaTeXMLMath -linear .
LaTeXMLMath is given by LaTeXMLMath , with LaTeXMLEquation .
Using ( LaTeXMLRef ) and ( LaTeXMLRef ) , we easily deduce that LaTeXMLMath is right LaTeXMLMath -linear , and that LaTeXMLMath is left LaTeXMLMath -linear and right LaTeXMLMath -linear .
Assume that LaTeXMLMath is finitely generated projective , and , as above , assume that LaTeXMLMath is a dual basis .
We can then define the inverse LaTeXMLMath of LaTeXMLMath as follows .
We view LaTeXMLMath as a map LaTeXMLMath , and we identify LaTeXMLMath and LaTeXMLMath .
We then define LaTeXMLMath by LaTeXMLEquation .
It is clear that LaTeXMLMath is natural and that LaTeXMLMath and LaTeXMLMath are each others inverses .
For LaTeXMLMath , we define LaTeXMLMath by LaTeXMLEquation for all LaTeXMLMath in LaTeXMLMath .
Straightforward computations yield that LaTeXMLMath is well-defined , and is indeed an inverse of LaTeXMLMath .
∎ In a similar fashion , we have : Let LaTeXMLMath and LaTeXMLMath be as above , and consider LaTeXMLEquation .
LaTeXMLEquation We have maps LaTeXMLEquation .
LaTeXMLMath is an isomorphism , and LaTeXMLMath is an isomorphism if LaTeXMLMath is finitely generated as a right LaTeXMLMath -module .
The proof is similar to the previous one , so we restrict to giving the connecting maps .
For a natural transformation LaTeXMLMath , we define LaTeXMLMath .
Conversely , take LaTeXMLMath .
LaTeXMLMath is defined as follows : for all LaTeXMLMath , we put LaTeXMLEquation .
Now we define LaTeXMLMath .
For LaTeXMLMath , we let LaTeXMLEquation be given by LaTeXMLEquation .
Straightforward computations show that LaTeXMLMath , and that LaTeXMLMath is LaTeXMLMath -bilinear .
∎ The two previous results can be used to decide when the induction functor LaTeXMLMath and its adjoint LaTeXMLMath are separable or Frobenius .
Let us first look at separability .
Let LaTeXMLMath be a LaTeXMLMath -bimodule , and assume that LaTeXMLMath is finitely generated projective as a left LaTeXMLMath -module , with finite dual basis LaTeXMLMath .
Then the following assertions are equivalent : LaTeXMLMath is a separable functor ; there exists LaTeXMLMath such that LaTeXMLMath ; there exists LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -bimodule .
The functor LaTeXMLMath is separable if and only if LaTeXMLMath is LaTeXMLMath -separable over LaTeXMLMath , in the sense of Definition LaTeXMLRef .
If LaTeXMLMath is finitely generated and projective as a left LaTeXMLMath -module , with finite dual basis LaTeXMLMath , then this is also equivalent to the existence of LaTeXMLMath in LaTeXMLMath such that LaTeXMLEquation .
The Frobenius analog of Corollary LaTeXMLRef is that LaTeXMLMath is a Frobenius bimodule if and only if LaTeXMLMath is a Frobenius functor .
This was stated explicitely in LaTeXMLCite , where it is also proved that any additive Frobenius functor between module categories is of this type .
Let us show how this result can be deduced easily from Propositions Proposition LaTeXMLRef and Proposition LaTeXMLRef .
Let LaTeXMLMath be a LaTeXMLMath -bimodule , and consider the functors LaTeXMLMath and LaTeXMLMath .
The following assertions are equivalent .
LaTeXMLMath is a Frobenius pair ; LaTeXMLMath is finitely generated and projective , and there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath ; LaTeXMLMath is Frobenius in the sense of Definition LaTeXMLRef .
LaTeXMLMath .
If LaTeXMLMath is Frobenius , then LaTeXMLMath is a left adjoint , and therefore right exact and preserving direct limits , so LaTeXMLMath is necessarily finitely generated and projective .
Let LaTeXMLMath and LaTeXMLMath be the counit and unit of the adjunction LaTeXMLMath , i.e .
LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
Putting LaTeXMLMath in ( LaTeXMLRef ) , we find ( LaTeXMLRef ) .
Then take LaTeXMLMath in ( LaTeXMLRef ) .
Making the identification LaTeXMLMath ( LaTeXMLMath is finitely generated projective ) , we find LaTeXMLEquation and LaTeXMLMath , so ( LaTeXMLRef ) follows .
LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath satisfy ( LaTeXMLRef ) , so LaTeXMLMath is Frobenius .
LaTeXMLMath .
( LaTeXMLRef ) implies that LaTeXMLMath is finitely generated projective .
Let LaTeXMLMath and LaTeXMLMath .
We easily compute that LaTeXMLMath and LaTeXMLMath are each others inverses .
LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are isomorphic as LaTeXMLMath -bimodules , then there exist LaTeXMLMath and LaTeXMLMath that are each others inverses .
Put LaTeXMLMath and LaTeXMLMath .
Straightforward computations show that LaTeXMLMath and LaTeXMLMath satisfy ( LaTeXMLRef - LaTeXMLRef ) .
∎ Obviously our results also hold for functors between categories of right modules .
As before , let LaTeXMLMath be a LaTeXMLMath -bimodule , and consider the functors LaTeXMLEquation .
Then LaTeXMLMath is an adjoint pair .
We have a natural transformation LaTeXMLEquation .
For all LaTeXMLMath , LaTeXMLMath is given by LaTeXMLEquation .
We denote LaTeXMLMath .
The analogs of Propositions LaTeXMLRef and LaTeXMLRef are the following : With notation as above , we have maps LaTeXMLEquation and LaTeXMLEquation where LaTeXMLMath .
LaTeXMLMath is always an isomorphism , LaTeXMLMath and LaTeXMLMath are isomorphisms if LaTeXMLMath is finitely generated projective , and LaTeXMLMath is an isomorphism if LaTeXMLMath is finitely generated projective .
Completely similar to the proof of Propositions LaTeXMLRef and LaTeXMLRef .
Let us mention that LaTeXMLEquation .
LaTeXMLEquation As a consequence , we obtain relations between the separability and Frobenius properties of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a LaTeXMLMath -bimodule , and assume that LaTeXMLMath is finitely generated projective .
Then the following assertions are equivalent : LaTeXMLMath is separable ; there exists LaTeXMLMath : LaTeXMLMath ; there exists LaTeXMLMath : LaTeXMLMath ; LaTeXMLMath is separable .
Here LaTeXMLMath is a finite dual basis of LaTeXMLMath as a right LaTeXMLMath -module .
The equivalence of the first three statements is obtained in exactly the same way as Corollary LaTeXMLRef .
The equivalence of the third and the fourth statement is one of the equivalences in Corollary LaTeXMLRef .
∎ Let LaTeXMLMath be a LaTeXMLMath -bimodule .
The following statements are equivalent : LaTeXMLMath is separable ; there exists LaTeXMLMath such that LaTeXMLMath .
If LaTeXMLMath is finitely generated projective , then they are also equivalent to There exists LaTeXMLMath such that LaTeXMLMath ; LaTeXMLMath is separable .
Here LaTeXMLMath is a finite dual basis of LaTeXMLMath as a left LaTeXMLMath -module .
Let LaTeXMLMath be a LaTeXMLMath -bimodule .
The following statements are equivalent : LaTeXMLMath is a Frobenius pair and LaTeXMLMath is finitely generated projective ; LaTeXMLMath is finitely generated projective and there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath ; LaTeXMLMath is a Frobenius bimodule .
We now address the following problem : assume that we know that LaTeXMLMath is a Frobenius bimodule .
When is LaTeXMLMath LaTeXMLMath -separable over LaTeXMLMath ?
In view of the above considerations , it suffices to apply Proposition LaTeXMLRef .
We first need a Lemma .
Let LaTeXMLMath be a LaTeXMLMath -bimodule , and LaTeXMLMath .
Then LaTeXMLEquation .
Let LaTeXMLMath be a natural transformation .
Then LaTeXMLMath is left LaTeXMLMath -linear , and right LaTeXMLMath -linear since LaTeXMLMath is natural .
Given a LaTeXMLMath -linear map LaTeXMLMath , we define a natural transformation LaTeXMLMath as follows : LaTeXMLMath .
∎ Assume that LaTeXMLMath is a Frobenius LaTeXMLMath -bimodule , and let LaTeXMLMath and LaTeXMLMath be as in the second statement of Proposition LaTeXMLRef .
Then LaTeXMLMath is LaTeXMLMath -separable over LaTeXMLMath ( i.e .
LaTeXMLMath is separable ) if and only if there exists a LaTeXMLMath -linear map LaTeXMLMath such that LaTeXMLEquation .
This proposition recovers LaTeXMLCite .
One can similarly recover LaTeXMLCite .
This work reports on the construction of a nonlinear distributional geometry ( in the sense of Colombeau ’ s special setting ) and its applications to general relativity with a special focus on the distributional description of impulsive gravitational waves .
Key words .
Algebras of generalized functions , Colombeau algebras , generalized tensor fields , generalized pseudo-Riemannian geometry , general relativity , impulsive gravitational waves .
Mathematics Subject Classification ( 2000 ) .
Primary 46F30 ; Secondary 46T30 , 46F10 , 83C05 , 83C35 .
Idealizations play an overall role in modeling physical situations ; a particularly useful one is to replace smooth extended densities by “ concentrated sources ” whenever the density is confined to a “ small region ” in space and its internal structure is negligible ( e.g .
point charges in electrodynamics ) .
On trying to describe this idealization mathematically one is led to distributions in a natural way .
In the case of , e.g. , electrodynamics distribution theory in fact furnishes a consistent framework , i.e. , provides the following two features : first since Maxwell equations are linear with respect to sources and fields they make sense within distributions and second it is guaranteed that ( say smooth ) charge densities close—in the sense of LaTeXMLMath -convergence—to , e.g. , a point charge produce fields that are close to the Coulomb field .
While the first property allows for a mathematically sound formulation it is precisely the latter one which renders the idealization physically sensible .
One would wish for a similar mathematical description of concentrated sources in the theory of general relativity .
However , its field equations , i.e. , Einstein ’ s equations form a ( complicated ) system of nonlinear PDEs .
More precisely , since the spacetime metric and its first derivatives enter nonlinearly , the field equations simply can not be formulated for distributional metrics .
For a more detailed discussion of the geometrical aspects and , in particular , weak singularities in general relativity see LaTeXMLCite .
Despite this conceptual obstacle spacetimes involving an energy-momentum tensor supported on a hypersurface of spacetime ( so-called thin shells ) have long since been used in general relativity ( see LaTeXMLCite for the final formulation of this widely applied approach ) .
The description of gravitational sources supported lower dimensional submanifolds of spacetime ( e.g .
cosmic strings and point particles ) , however , is more delicate .
In fact by a result of Geroch and Traschen LaTeXMLCite a mathematically sound and at the same time physically resonable description ( in the sense of a “ limit consistency ” as discussed in the context of Maxwell fields above ) explicitly excludes the treatment of sources of the gravitational field concentrated on a submanifold of codimension greater than one .
Recently nonlinear generalized function methods have been used to overcome this conceptual obstacle in the context of such different topics in general relativity as cosmic strings ( e.g .
LaTeXMLCite ) , ( ultrarelativistic ) black holes ( LaTeXMLCite ) , impulsive gravitational waves ( e.g .
LaTeXMLCite ) and signature change ( e.g .
LaTeXMLCite ) .
For an overview see LaTeXMLCite .
In this work we are going to discuss the recently developed global approach to nonlinear distributional ( in the sense of the special version of Colombeau ’ s construction ) geometry ( LaTeXMLCite ) and its applications to general relativity .
While the following section is devoted to a review of the former and , in particular , to generalized pseudo-Riemannian geometry , applications to the distributional description of impulsive gravitational pp-waves will be presented in Section LaTeXMLRef .
We shall see that despite the absence of a canonical embedding of distributions Colombeau ’ s special setting , due to the fact that the basic building blocks automatically are diffeomorphism invariant provides a particularly flexible tool to model singular metrics in the nonlinear context of general relativity .
For an introduction into the diffeomorphism invariant full algebras of generalized functions of LaTeXMLCite and in particular its global formulation ( LaTeXMLCite ) , however , we refer to LaTeXMLCite in this volume ; its applications to general relativity are discussed in LaTeXMLCite .
In the following we use the notational conventions of LaTeXMLCite .
The ( special ) algebra of generalized functions on the ( separable , smooth Hausdorff ) manifold LaTeXMLMath is defined as the quotient LaTeXMLMath of the space of moderately growing nets of smooth functions LaTeXMLMath modulo negligible nets , where the respective notions of moderateness and negligibility are defined ( denoting by LaTeXMLMath the space of linear differential operators on LaTeXMLMath ) by LaTeXMLEquation .
LaTeXMLEquation Elements of LaTeXMLMath are denoted by capital letters , i.e. , LaTeXMLMath .
LaTeXMLMath is a fine sheaf of differential algebras with respect to the Lie derivative ( w.r.t .
smooth vector fields ) defined by LaTeXMLMath .
The spaces of moderate resp .
negligible sequences and hence the algebra itself may be characterized locally , i.e. , LaTeXMLMath iff LaTeXMLMath for all charts LaTeXMLMath .
Smooth functions are embedded into LaTeXMLMath simply by the “ constant ” embedding LaTeXMLMath , i.e. , LaTeXMLMath , hence LaTeXMLMath is a faithful subalgebra of LaTeXMLMath .
In the absence of a canonical embedding compatibility with respect to the distributional setting is established via the notion of association , defined as follows : a generalized function LaTeXMLMath is called associated to LaTeXMLMath , LaTeXMLMath , if LaTeXMLMath ( LaTeXMLMath ) for all compactly supported one-densities and one ( hence every ) representative LaTeXMLMath of LaTeXMLMath .
The equivalence relation induced by this notion gives rise to a linear quotient space of LaTeXMLMath .
If LaTeXMLMath for some LaTeXMLMath then LaTeXMLMath is called the distributional shadow ( or macroscopic aspect ) of LaTeXMLMath and we write LaTeXMLMath .
Similarly we call a generalized function LaTeXMLMath LaTeXMLMath -associated to LaTeXMLMath ( LaTeXMLMath ) , LaTeXMLMath , if for all LaTeXMLMath , all LaTeXMLMath and one ( hence any ) representative LaTeXMLMath uniformly on compact sets .
Also we say that LaTeXMLMath admits LaTeXMLMath as LaTeXMLMath -associated function , LaTeXMLMath , if for all LaTeXMLMath , all LaTeXMLMath and one ( hence any ) representative LaTeXMLMath uniformly on compact sets .
Finally , inserting LaTeXMLMath into LaTeXMLMath yields a well defined element of the ring of constants LaTeXMLMath ( corresponding to LaTeXMLMath resp .
LaTeXMLMath ) , defined as the set of moderate nets of numbers ( LaTeXMLMath with LaTeXMLMath for some LaTeXMLMath ) modulo negligible nets ( LaTeXMLMath for each LaTeXMLMath ) .
The LaTeXMLMath -module of generalized sections LaTeXMLMath of a vector bundle LaTeXMLMath —and in particular the space of generalized tensor fields LaTeXMLMath —is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers .
We denote generalized sections by LaTeXMLMath .
Alternatively we may describe a section LaTeXMLMath by a family LaTeXMLMath , where LaTeXMLMath is called the local expression of LaTeXMLMath with its components LaTeXMLMath ( LaTeXMLMath a vector bundle atlas and LaTeXMLMath , with LaTeXMLMath denoting the dimension of the fibers ) satisfying LaTeXMLMath LaTeXMLMath for all LaTeXMLMath , where LaTeXMLMath denotes the transition functions of the bundle .
Smooth sections of LaTeXMLMath again may be embedded as constant nets , i.e. , LaTeXMLMath .
Since LaTeXMLMath is a subring of LaTeXMLMath , LaTeXMLMath also may be viewed as LaTeXMLMath -module and the two respective module structures are compatible with respect to the embeddings .
Moreover we have the following algebraic characterization of the space of generalized sections LaTeXMLEquation where LaTeXMLMath denotes the space of smooth sections and the tensor product is taken over the module LaTeXMLMath .
Compatibility with respect to the classical resp .
distributional setting again is accomplished using the concept of ( LaTeXMLMath - ) association which carries over from the scalar case by ( LaTeXMLRef ) .
Generalized tensor fields may be viewed likewise as LaTeXMLMath - resp .
LaTeXMLMath -multilinear mappings , i.e. , as LaTeXMLMath - resp .
LaTeXMLMath -modules we have LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath resp .
LaTeXMLMath denotes the space of smooth vector resp .
covector fields on LaTeXMLMath .
In LaTeXMLCite many concepts of classical tensor analysis like e.g .
Lie derivatives ( with respect to both smooth and generalized vector fields ) , Lie brackets , tensor products and contraction have been generalized to the new setting and we shall use them freely in the sequel .
Moreover several consistency results with respect to smooth resp .
distributional geometry ( cf .
LaTeXMLCite ) have been etsablished .
We now begin to develop the basics of a generalized pseudo-Riemannian geometry .
Definition .
A generalized LaTeXMLMath tensor field LaTeXMLMath is called a generalized Pseudo-Riemannian metric if it has a representative LaTeXMLMath satisfying LaTeXMLMath is a smooth Pseudo-Riemannian metric for all LaTeXMLMath , and LaTeXMLMath is strictly nonzero on compact sets , i.e. , LaTeXMLMath LaTeXMLMath : LaTeXMLMath .
We call a separable , smooth Hausdorff manifold LaTeXMLMath furnished with a generalized pseudo-Riemannian metric LaTeXMLMath generalized pseudo-Riemannian manifold or generalized spacetime and denote it by LaTeXMLMath or merely by LaTeXMLMath .
The action of the metric on a pair of generalized vector fields will be denoted by LaTeXMLMath and LaTeXMLMath , equivalently .
Note that condition ( b ) above is precisely equivalent to invertibility of LaTeXMLMath in the generalized sense .
The inverse metric LaTeXMLMath is a well defined element of LaTeXMLMath , depending exclusively on LaTeXMLMath i.e. , independent of the particular representative LaTeXMLMath LaTeXMLMath .
Moreover if LaTeXMLMath , where LaTeXMLMath is a classical LaTeXMLMath -pseudo-Riemannian metric then LaTeXMLMath .
¿From now on we denote the inverse metric ( using abstract index notation cf .
LaTeXMLCite , Ch .
2 ) by LaTeXMLMath , its components by LaTeXMLMath and the components of a representative by LaTeXMLMath .
Also we shall denote the generalized metric LaTeXMLMath by LaTeXMLMath and use summation convention .
Examples .
A sequence LaTeXMLMath of classical ( smooth ) metrics constitutes a representative of a generalized metric if it is moderate and zero-associated to a classical ( then necessarily continuous ) metric LaTeXMLMath .
The metric of a two-dimensional cone was modeled in LaTeXMLCite by a generalized metric ( in the full setting ) obtained by using the embedding via convolution .
The metric of an impulsive pp-wave will be modeled by a generalized one in Section LaTeXMLRef .
Further examples may be found e.g .
in LaTeXMLCite .
A generalized metric LaTeXMLMath is non-degenerate in the following sense : LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath .
Moreover LaTeXMLMath induces a LaTeXMLMath -linear isomorphism LaTeXMLMath by LaTeXMLEquation which—as in the classical context—extends naturally to generalized tensor fields of all types .
Hence from now on we shall use the common conventions on upper and lower indices also in the context of generalized tensor fields .
In particular , identifying a vector field LaTeXMLMath with its metrically equivalent one-form LaTeXMLMath we denote its contravariant respectively covariant components by LaTeXMLMath and LaTeXMLMath .
A similar convention will apply to representatives .
Definition .
A generalized connection LaTeXMLMath on a manifold LaTeXMLMath is a map LaTeXMLMath satisfying LaTeXMLMath is LaTeXMLMath -linear in LaTeXMLMath .
LaTeXMLMath is LaTeXMLMath -linear in LaTeXMLMath .
LaTeXMLMath for all LaTeXMLMath .
Let LaTeXMLMath be a chart on LaTeXMLMath with coordinates LaTeXMLMath .
The generalized Christoffel symbols for this chart are given by the LaTeXMLMath functions LaTeXMLMath defined by LaTeXMLEquation .
We are already in the position to state the “ Fundamental Lemma of ( pseudo ) -Riemannian Geometry ” in our setting .
Theorem .
Let LaTeXMLMath be a generalized pseudo-Riemannian manifold .
Then there exists a unique generalized connection LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath hold for all LaTeXMLMath in LaTeXMLMath .
LaTeXMLMath is called generalized Levi-Civita connection of LaTeXMLMath and characterized by the so-called Koszul formula LaTeXMLEquation .
As in the classical case from the torsion-free condition ( i.e. , ( D4 ) ) we immediately infer the symmetry of the Christoffel symbols of the Levi-Civita connection in its lower pair of indices .
Moreover , from ( D3 ) and the Koszul formula ( LaTeXMLRef ) we derive ( analogously to the classical case ) the following Proposition .
On every chart LaTeXMLMath we have for the generalized Levi-Civita connection LaTeXMLMath of LaTeXMLMath and any vector field LaTeXMLMath LaTeXMLEquation .
Moreover , the generalized Christoffel symbols are given by LaTeXMLEquation .
To be able to state the appropriate consistency results with respect to classical resp .
distributional geometry we need to define the action of a classical ( smooth ) connection LaTeXMLMath on generalized vector fields LaTeXMLMath , LaTeXMLMath by LaTeXMLMath .
Now we have Proposition .
Let LaTeXMLMath be a generalized pseudo-Riemannian manifold .
If LaTeXMLMath where LaTeXMLMath is a classical smooth metric then we have , in any chart , LaTeXMLMath ( with LaTeXMLMath denoting the Christoffel Symbols of LaTeXMLMath ) .
Hence for all LaTeXMLMath LaTeXMLEquation .
If LaTeXMLMath , LaTeXMLMath a classical smooth metric , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath or vice versa , i.e. , LaTeXMLMath , LaTeXMLMath then LaTeXMLEquation .
Let LaTeXMLMath , LaTeXMLMath a classical LaTeXMLMath -metric , then , in any chart , LaTeXMLMath .
If in addition LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath then LaTeXMLEquation .
Next we define the generalized Riemann , Ricci , scalar and Einstein curvature from an invariant point of view .
However , all the classical formulae will hold on the level of representatives , i.e. , all the symmetry properties of the respective classical tensor fields carry over to our setting .
Moreover , the Bianchi identities hold in the generalized sense .
Definition .
Let LaTeXMLMath be a generalized pseudo-Riemannian manifold with Levi-Civita connection LaTeXMLMath .
The generalized Riemannian curvature tensor LaTeXMLMath is defined by LaTeXMLEquation .
The generalized Ricci curvature tensor is defined by LaTeXMLMath .
The generalized curvature ( or Ricci ) scalar is defined by LaTeXMLMath .
Finally we define the generalized Einstein tensor by LaTeXMLMath .
The framework developed above opens a gate to a wide range of applications in general relativity .
Definition LaTeXMLRef is capable of modeling a large class of singular spacetimes while at the same time its ( generalized ) curvature quantities simply may be calculated by the usual coordinate formulae .
Hence we are in a position to mathematically rigorously formulate Einsteins equations for generalized metrics .
Moreover we have at our disposal several theorems ( which essentially are rooted in LaTeXMLCite , Prop .
3 ) guaranteeing consistency with respect to linear distributional geometry resp .
the smooth setting .
Theorem .
Let LaTeXMLMath a generalized pseudo-Riemannian manifold with LaTeXMLMath .
Then all the generalized curvature quantities defined above are LaTeXMLMath -associated to their classical counterparts .
In particular , if a generalized metric LaTeXMLMath is LaTeXMLMath -associated to a vacuum solution of Einstein equations then we have for the generalized Ricci tensor LaTeXMLEquation .
Hence LaTeXMLMath satisfies the vacuum Einstein equations in the sense of LaTeXMLMath -association ( cf .
the remarks in LaTeXMLCite ) .
Generally speaking when dealing with singular spacetime metrics in general relativity we may apply the steps of the following scheme : first we transfer the classically singular metric to the generalized setting .
This may be done by some “ canonical ” smoothing or by some other physically motivated regularization .
Of course diffeomorphism invariance of the procedure employed has to be carefully investigated .
Once the generalized setting has been entered , the relevant curvature quantities may be calculated componentwise according to the classical formulae .
All classical concepts carry over to the new framework and one may treat e.g .
the Ricci tensor , geodesics , geodesic deviation , etc .
within this nonlinear distributional geometry .
Finally one may use the concept of ( LaTeXMLMath ) -association to return to the distributional or LaTeXMLMath -level for the purpose of interpretation .
This program has been carried out for a conical metric ( representing a cosmic string ) by Clarke , Vickers and Wilson ( LaTeXMLCite , however , in the full setting of Colombeau ’ s construction ) rigorously assigning to it a distributional curvature and ( via the field equations ) the heuristically expected energy-momentum tensor .
In Section LaTeXMLRef we are going to review the distributional description of impulsive pp-wave spacetimes of LaTeXMLCite .
Further applications following the proceddure described above may be found e.g .
in LaTeXMLCite .
Plane fronted gravitational waves with parallel rays ( pp-waves ) are spacetimes characterized by the existence of a covariantly constant null vector field , which can be used to write the metric tensor in the form LaTeXMLEquation where LaTeXMLMath is a pair of null coordinates ( LaTeXMLMath , LaTeXMLMath ) and LaTeXMLMath are transverse ( Cartesian ) coordinates .
We are especially interested in impulsive pp-waves as introduced by R. Penrose ( see e.g .
LaTeXMLCite ) where the profile function LaTeXMLMath is proportional to a LaTeXMLMath -distribution , i.e. , takes the form LaTeXMLMath , where LaTeXMLMath is a smooth function of the transverse coordinates .
This metric is flat everywhere except on the null hypersurface LaTeXMLMath , where it has a LaTeXMLMath -shaped “ shock ” and—due to the appearance of a distribution in one component—clearly lies beyond the scope of linear distributional geometry .
Physically this form of the metric arises as the impulsive limit of a sequence of sandwich waves , i.e. , LaTeXMLMath taking the form LaTeXMLMath with LaTeXMLMath weakly .
This is our motivation to model the impulsive pp-wave metric by a generalized metric of the form LaTeXMLEquation where LaTeXMLMath denotes a generalized delta function which allows for a strict delta net LaTeXMLMath as a representative , i.e. , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In LaTeXMLCite it has been shown that the geodesic as well as the geodesic deviation equation for the metric ( LaTeXMLRef ) may be solved uniquely within our present setting .
Moreover these unique generalized solutions possess physically reasonable distributional shadows which shows that we have achieved a physically sensible distributional description of impulsive pp-waves .
Diffeomorphism invariance of these results is assured by diffeomorphism invariance of the class of strict delta nets .
Here , however , we shall be interested in modeling the heuristically motivated singular transformation of the distributional pp-wave metric first given by R. Penrose ( LaTeXMLCite ) within our framework ( cf .
LaTeXMLCite ) .
For later use we introduce the notation LaTeXMLMath , LaTeXMLMath ( LaTeXMLMath ) for the unique generalized geodesics of ( LaTeXMLRef ) with vanishing initial speeds .
Here the fourth coordinate LaTeXMLMath —due to the special geometry—may be used as an affine parameter along the geodesics and the real constants LaTeXMLMath denote the initial positions , i.e. , LaTeXMLMath and analogously for LaTeXMLMath .
In the literature impulsive pp-waves have frequently been described in different coordinates where the metric tensor is actually continuous , i.e. , ( in the special case of a plane wave ( LaTeXMLMath and LaTeXMLMath denoting the kink function ) , LaTeXMLEquation .
Clearly a transformation relating these two metrics can not even be continuous , hence in addition to involving ill-defined products of distributions it changes the topological structure of the manifold .
In the special case envisaged above this discontinuous change of variables was given in LaTeXMLCite ( denoting by LaTeXMLMath the Heaviside function ) LaTeXMLEquation .
LaTeXMLEquation However , the two mathematically distinct spacetimes are equivalent from a physical point of view , i.e. , the geodesics and the particle motion agree on a heuristic level ( see LaTeXMLCite ) .
We are now going to model this transformation by a generalized coordinate transformation , that is Definition .
Let LaTeXMLMath be an open subset of LaTeXMLMath .
We call LaTeXMLMath a generalized diffeomorphism if there exists LaTeXMLMath such that There exists a representative LaTeXMLMath such that LaTeXMLMath is a diffeomorphism for all LaTeXMLMath and there exists LaTeXMLMath open , LaTeXMLMath .
LaTeXMLMath and there exists LaTeXMLMath open such that LaTeXMLMath LaTeXMLMath LaTeXMLMath .
Writing LaTeXMLMath , LaTeXMLMath as well as LaTeXMLMath are well-defined elements of LaTeXMLMath resp .
LaTeXMLMath .
It is then clear that LaTeXMLMath resp .
LaTeXMLMath .
Let us now consider the transformation LaTeXMLMath depending on the regularization parameter LaTeXMLMath according to LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are representatives of the generalized geodesics LaTeXMLMath resp .
LaTeXMLMath and are given by LaTeXMLEquation .
Now one may prove the following Theorem .
The generalized function LaTeXMLMath defined above is a generalized coordinate transformation on a suitable open subset LaTeXMLMath of LaTeXMLMath containing the shock hyperplane at LaTeXMLMath .
The new coordinates are constant along the geodesics given by ( LaTeXMLRef ) .
Moreover the macroscopic apsect of LaTeXMLMath coincides with the discontinous transformation used in the literature ( hence in the special case of a plane wave is given by ( LaTeXMLRef ) ) .
Applying this generalized transformation to the metric ( LaTeXMLRef ) we find that in the new coordinates it is given by the class of LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath denote derivatives with respect to LaTeXMLMath and LaTeXMLMath , respectively .
Moreover we find the following distributional shadow of the metric ( LaTeXMLRef ) LaTeXMLEquation which is precisely the continuous ( or so-called Rosen- ) form of the metric of an impulsive pp-wave ( cf .
LaTeXMLCite ) .
Summing up we have shown the following : after modeling the ( distributional form of the ) impulsive pp-wave metric in a diffeomorphism invariant way by the generalized metric ( LaTeXMLRef ) we have subjected the latter to the generalized change of coordinates LaTeXMLMath .
In either coordinates the distributional shadow is computed giving the distributional resp .
the continuous form of the pp-wave metric .
( Note that although the action of a smooth diffeomorphism is compatible with the notion of association , generalized coordinate transformations clearly are not . )
Physically speaking the two forms of the impulsive metric arise as the ( distributional ) limits of a sandwich wave in different coordinate systems .
Hence impulsive pp-waves indeed are sensibly modeled by the generalized spacetime metric ( LaTeXMLRef ) : in different coordinate systems related by generalized coordinate transformations , different distributional pictures arise .
Questions of Connectedness of the Hilbert Scheme of Curves in LaTeXMLMath Robin Hartshorne Department of Mathematics University of California Berkeley , California 94720–3840 Dedicated to S. Abhyankar on the occasion of his 70th birthday .
We review the present state of the problem , for each degree LaTeXMLMath and genus LaTeXMLMath , is the Hilbert scheme of locally Cohen–Macaulay curves in LaTeXMLMath connected ?
In studying algebraic curves in projective spaces , our forefathers in the 19th century noted that curves naturally move in algebraic families .
In the projective plane , this is a simple matter .
A curve of degree LaTeXMLMath is defined by a single homogeneous polynomial in the homogeneous coordinates LaTeXMLMath .
The coefficients of this polynomial give a point in another projective space , and in this way curves of degree LaTeXMLMath in the plane are parametrized by the points of a LaTeXMLMath with LaTeXMLMath .
For an open set of LaTeXMLMath , the corresponding curve is irreducible and nonsingular .
The remaining points of LaTeXMLMath correspond to curves that are singular , or reducible , or have multiple components .
In particular , the nonsingular curves of degree LaTeXMLMath in LaTeXMLMath form a single irreducible family .
In LaTeXMLMath , the situation is more complicated .
For a given degree LaTeXMLMath , there may be curves with several different values of the genus LaTeXMLMath .
Even for fixed LaTeXMLMath , the family of curves with given LaTeXMLMath may not be irreducible .
An early example , noted by Halphen and Weyr in 1874 is the case LaTeXMLMath and LaTeXMLMath .
One type consists of curves LaTeXMLMath of bidegree ( 3,6 ) on a nonsingular quadric surface LaTeXMLMath .
The other type consists of curves LaTeXMLMath that are the complete intersection of two cubic surfaces LaTeXMLMath and LaTeXMLMath .
These curves form two irreducible components of the Hilbert scheme LaTeXMLMath of smooth curves of degree LaTeXMLMath and genus LaTeXMLMath in LaTeXMLMath .
Furthermore , it is not hard to see that every curve in LaTeXMLMath belongs to one of these two types , and that there is no flat family of curves whose general member belongs to one type and whose special member belongs to the other type LaTeXMLCite .
Thus the Hilbert scheme of smooth curves of given degree and genus in LaTeXMLMath need not be connected .
On the other hand , if one phrases the question more generally , by letting a “ curve ” mean an arbitrary closed subscheme of dimension 1 in LaTeXMLMath , then the Hilbert scheme for each degree LaTeXMLMath and arithmetic genus LaTeXMLMath is connected .
In fact , in my thesis LaTeXMLCite , I showed that the Hilbert scheme of closed subschemes of LaTeXMLMath with Hilbert polynomial LaTeXMLMath is connected ( provided it is nonempty ) for any LaTeXMLMath and any LaTeXMLMath .
In the proof , non-reduced schemes play an essential role .
Here is the main idea of the proof for the case of curves in LaTeXMLMath .
Suppose , for example , that we start with a nonsingular curve LaTeXMLMath .
Its general projection to LaTeXMLMath will be a plane curve LaTeXMLMath with nodes .
Using the projection we can construct a flat family whose general member is LaTeXMLMath and whose special member LaTeXMLMath is a curve with support LaTeXMLMath , having embedded points at the nodes ( see LaTeXMLCite ) for an example showing how these embedded points arise ) .
Then we can make another flat family , pulling the embedded points off LaTeXMLMath , to get LaTeXMLMath union a number of points in LaTeXMLMath .
Finally , we move LaTeXMLMath in a flat family of plane curves to a union of lines in LaTeXMLMath meeting at a single point .
If the original curve LaTeXMLMath had degree LaTeXMLMath and genus LaTeXMLMath , we obtain in this way a “ fan ” of LaTeXMLMath lines in the plane together with LaTeXMLMath isolated points in the plane .
Any other curve LaTeXMLMath with the same LaTeXMLMath can be connected by a sequence of flat specializations and generalizations to the same configuration , so LaTeXMLMath and LaTeXMLMath are connected within the Hilbert scheme LaTeXMLMath , of all closed subschemes of LaTeXMLMath of dimension 1 , degree LaTeXMLMath , and arithmetic genus LaTeXMLMath .
For nonreduced curves a slightly more complicated , but similar method applies .
Thus we have a connectedness theorem for the Hilbert scheme of curves in LaTeXMLMath , but it is unsatisfactory in that , even if we want to connect one smooth curve to another , we must pass by way of schemes with embedded points and isolated points , which one can argue should not really count as “ curves ” .
With the development of liaison theory in recent years LaTeXMLCite , LaTeXMLCite , an intermediate class of curves has received much attention , the locally Cohen-Macaulay curves .
We say a curve is locally Cohen-Macaulay if it is a scheme of equidimension 1 , and all its local rings are Cohen-Macaulay rings .
Equivalently it is a 1-dimensional scheme with no embedded points or isolated points .
It is clear that this class of curves is the natural class in which to do liaison : even if one is primarily interested in nonsingular curves , the minimal curves in a biliaison class may be reducible and non-reduced .
So we pose the question : Is the Hilbert scheme LaTeXMLMath of locally Cohen-Macaulay curves of degree LaTeXMLMath and arithmetic genus LaTeXMLMath in LaTeXMLMath connected ?
The answer is unknown at present , so we devote this paper to a survey of the current state of this question .
Before discussing whether a given Hilbert scheme is connected , one should at least know when it is nonempty .
For smooth curves in LaTeXMLMath , the result was stated by Halphen LaTeXMLCite with an incorrect proof , and proved one hundred years later by Gruson and Peskine LaTeXMLCite , LaTeXMLCite .
There exists an irreducible smooth curve LaTeXMLMath of degree LaTeXMLMath and genus LaTeXMLMath in LaTeXMLMath if and only if either a ) LaTeXMLMath and LaTeXMLMath ( these are the plane curves ) , or b ) there exist LaTeXMLMath with LaTeXMLMath and LaTeXMLMath ( these are curves on quadric surfaces ) , or c ) LaTeXMLMath and LaTeXMLMath .
The hardest part of the proof is the existence of curves for all LaTeXMLMath in the range c ) , which they construct on suitable cubic and quartic surfaces in LaTeXMLMath .
If one considers all one-dimensional closed subschemes of LaTeXMLMath , the answer was known to Macaulay LaTeXMLCite , and rediscovered in LaTeXMLCite .
Then LaTeXMLMath is nonempty for all LaTeXMLMath and all arithmetic genus LaTeXMLMath .
The existence is simple .
Just take a plane curve of degree LaTeXMLMath and add lots of isolated points .
Note that the arithmetic genus LaTeXMLMath can become arbitrarily negative .
For locally Cohen-Macaulay curves , the answer is slightly more complicated , but not too difficult LaTeXMLCite .
A locally Cohen-Macaulay curve with given LaTeXMLMath exists if and only if either a ) LaTeXMLMath , LaTeXMLMath ( a plane curves ) , or b ) LaTeXMLMath , LaTeXMLMath For LaTeXMLMath one can exhibit a multiplicity two structure on a line with any given arithmetic genus LaTeXMLMath .
For example , the scheme in LaTeXMLMath defined by the homogeneous ideal LaTeXMLMath , for any LaTeXMLMath , has LaTeXMLMath .
Then one can construct curves for all LaTeXMLMath in case b ) above by taking a plane curve of degree LaTeXMLMath containing a line , and putting a suitable multiplicity two structure on the line .
From now on , we will consider only locally Cohen-Macaulay curves , and denote LaTeXMLMath by LaTeXMLMath .
There are some values of LaTeXMLMath for which LaTeXMLMath is irreducible , and hence trivially connected LaTeXMLCite .
These are a ) LaTeXMLMath , LaTeXMLMath , the plane curves b ) LaTeXMLMath , LaTeXMLMath .
For LaTeXMLMath we have a plane curve ; for LaTeXMLMath , two disjoint lines or a double line on a quadric ; and for LaTeXMLMath , double structures on a line .
c ) Some special values of LaTeXMLMath for higher degree , namely LaTeXMLMath , and LaTeXMLEquation .
For all other LaTeXMLMath , namely LaTeXMLMath , LaTeXMLMath ; LaTeXMLMath , LaTeXMLMath ; and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath has two or more irreducible components .
For any curve LaTeXMLMath , an important invariant is the Rao module LaTeXMLMath .
The dimensions of the graded components of the Rao module are the Rao function LaTeXMLMath .
A curve is arithmetically Cohen-Macaulay ( ACM ) if and only if its Rao module is 0 .
For non-plane curves , there are explicit bounds on the Rao function in terms of LaTeXMLMath and LaTeXMLMath LaTeXMLCite .
In particular , for all LaTeXMLMath we have LaTeXMLEquation .
Thus , if LaTeXMLMath , the Rao function is 0 , so the curve is necessarily ACM , and one knows in this case that the Hilbert scheme is irreducible LaTeXMLCite .
If LaTeXMLMath , then one has the more precise result that LaTeXMLMath is bounded by a function that is equal to LaTeXMLMath for LaTeXMLMath , and decreases with slope 1 ( resp .
–1 ) to zero on both ends of this range .
In their paper LaTeXMLCite , Martin-Deschamps and Perrin define an extremal curve to be a non-ACM curve whose Rao function is equal to this bound for all LaTeXMLMath .
For any LaTeXMLMath , they show the existence of extremal curves , and show that they form an irreducible component of the Hilbert scheme .
For curves that are not extremal , Nollet has established a stronger bound on the Rao function LaTeXMLCite .
If LaTeXMLMath and the curve is neither ACM nor extremal , then LaTeXMLMath .
In particular this implies LaTeXMLMath .
Thus any curve of LaTeXMLMath and LaTeXMLMath must be extremal , and we conclude that the Hilbert scheme is irreducible in that range .
Curves satisfying Nollet ’ s stronger bounds are called subextremal .
If the Hilbert scheme has two or more irreducible components , which happens for LaTeXMLMath , LaTeXMLMath ; LaTeXMLMath , LaTeXMLMath ; and LaTeXMLMath , LaTeXMLMath we can ask if it is connected .
Here are some cases in which it is known to be connected .
If LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath is connected LaTeXMLCite , and has approximately LaTeXMLMath irreducible components .
If LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath is connected LaTeXMLCite .
If LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath has two irreducible components , consisting of the extremal curves in one and the ACM curves in the other , and is connected LaTeXMLCite .
If LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath has 2 , 3 , or 4 irreducible components , and is connected LaTeXMLCite .
If LaTeXMLMath , LaTeXMLMath , LaTeXMLMath has four irreducible components and is connected LaTeXMLCite .
If LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath has four or five irreducible components , and is connected LaTeXMLCite .
These cases , together with the cases of LaTeXMLMath irreducible listed above , are the only cases in which it is known that LaTeXMLMath is connected at present .
The problem falls into two halves .
The first is to list the irreducible components of LaTeXMLMath , and the second is to show the existence of flat families of curves connecting the different components .
It is the first of these that is blocking further progress at the moment , because it requires a classification of all curves of the given LaTeXMLMath .
The most difficult part is to understand all the possible nonreduced structures on a curve of lesser degree .
Thus already the case of multiplicity four structures on a line is extremely complicated .
To avoid the first problem , we formulate the question differently .
For a given LaTeXMLMath , there is always one irreducible component of LaTeXMLMath consisting of the extremal curves .
So we ask , which classes of curves can be connected by flat families in LaTeXMLMath to an extremal curve ?
If every curve with the given LaTeXMLMath is connected to an extremal curve , then LaTeXMLMath is connected .
The advantage of this question is that we do not have to classify all curves of type LaTeXMLMath .
Here are some cases that are known , namely curves that can be connected within LaTeXMLMath to an extremal curve of the same degree and genus .
Any disjoint union of lines LaTeXMLCite .
Any smooth curve with LaTeXMLMath LaTeXMLCite .
Any ACM curve LaTeXMLCite .
Any curve in the biliaison equivalence class of an extremal curve LaTeXMLCite .
Any curve whose Rao module is a complete intersection ( also called a Koszul module ) .
LaTeXMLCite .
The classification part of the problem uses standard methods .
What is new in studying connectedness questions is to prove the existence of flat families of curves , whose general member lies in one irreducible component , and whose special member lies in another irreducible component .
We discuss here the different methods used to construct such families .
If one knows the equations of the two types of curves , one can attempt to make a flat family by writing equations depending on a parameter LaTeXMLMath .
A simple example of this is the family of twisted cubic curves having as limit a plane nodal curve with an embedded point LaTeXMLCite .
Explicit equations are used in the papers LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
This technique is obviously limited to situations where one has only to deal with very explicit examples of curves .
This method , used in LaTeXMLCite , is an extension of the first .
Some families of multiple structures on lines are proved by explicit equations .
These are then used as lemmas in drawings of much more complicated curves , supported on unions of lines .
Combined with the complete description of curves contained in a double plane [ 15 ] , this allows one to show existence of families for many types of smooth curves that specialize to stick figures , such as the nonspecial curves with LaTeXMLMath .
This method is the most sophisticated , and potentially the most powerful , but also the most technically difficult .
This method is developed in the three papers LaTeXMLCite , LaTeXMLCite , LaTeXMLCite and applied in the papers LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
The idea is to develop an algebraic theory of flat families , biliaison , and Rao modules similar to the well known theory of biliaison and Rao module for individual curves LaTeXMLCite .
In a flat family LaTeXMLMath of curves in LaTeXMLMath parametrized by a parameter scheme LaTeXMLMath , the Rao module LaTeXMLMath is not in general constant in the family .
Also the sheaf analogue LaTeXMLMath as a sheaf of graded LaTeXMLMath -modules over LaTeXMLMath , does not commute with base extension .
So , for example , if LaTeXMLMath is affine , one is led to consider the functor on LaTeXMLMath -modules LaTeXMLEquation .
This is a coherent functor in the sense of Auslander LaTeXMLCite , but it is still not a fine enough invariant to play the role of the Rao module for a family .
So instead we consider the triad associated to the family LaTeXMLMath : it is a 3-term complex LaTeXMLMath of graded LaTeXMLMath -modules , where LaTeXMLMath is the homogeneous coordinate ring of LaTeXMLMath , whose middle cohomology retrieves the cohomology LaTeXMLMath of the family , and which satisfies certain other technical conditions ( see LaTeXMLCite for the precise definition ) .
There is a notion of pseudoisomorphism for triades LaTeXMLCite , and then one obtains the analogue of Rao ’ s theorem , that two families of curves are in the same biliaison equivalence class if and only if their triades are pseudoisomorphic up to shift in degrees LaTeXMLCite .
There is also an algorithmic method of constructing the universal family of curves associated to a triad LaTeXMLCite , and this becomes the basic method of constructing flat families of curves .
The difficulty is that the triad is not determined simply by knowing the Rao modules of the general curve and the special curve : there are other choices to be made to determine the triad .
Thus to show the existence of a family connecting curves of particular types , one has to choose carefully a suitable triad to give the family .
This means also that while the method of triads is good for making families , it is more difficult to prove the non-existence of families between given types of curves .
See also LaTeXMLCite for a slightly less brief introduction to the theory of triades .
Here we describe an example for which it is not yet known whether LaTeXMLMath is connected or not .
We consider smooth curves LaTeXMLMath of bidegree ( 3,7 ) on a smooth quadric surface LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath .
We do not know if these curves can be connected to extremal curves .
Because of semicontinuity , these curves can not be specializations of a family of curves not contained in quadric surfaces .
So these curves form an open subset of an irreducible component of LaTeXMLMath .
The only possibility for connecting them to other curves requires specializing the quadric surface LaTeXMLMath to a quadric cone , the union of two planes , or a double plane .
One can show that if LaTeXMLMath specializes to a cone or to a union of two planes , the curves must necessarily acquire embedded points LaTeXMLCite .
So the only case remaining is when LaTeXMLMath specializes to a double plane .
Since one knows all about curves in the double plane LaTeXMLCite , it would be sufficient to show the existence of a flat family going from the curves LaTeXMLMath to a locally Cohen-Macaulay curve in the double plane , but this question has so far resisted analysis .
Another approach is to use biliaison .
If one has a flat family going from a curve LaTeXMLMath to an extremal curve LaTeXMLMath , then by biliaison of the family one obtains a flat family from LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are in the biliaison classes of LaTeXMLMath and LaTeXMLMath , respectively .
Schlesinger ’ s result LaTeXMLCite shows that LaTeXMLMath can be connected to an extremal curve with the same degree and genus .
Now our curve LaTeXMLMath of bidegree ( 3,7 ) on LaTeXMLMath is in the biliaison class of a curve LaTeXMLMath consisting of four skew lines , and one knows that four skew lines can be connected to an extremal curve ( cf .
LaTeXMLMath 2E above ) .
The catch is that in order to perform a biliaison of the family on the quadric surface , the entire family must be contained in quadric surfaces .
In the case of four skew lines on LaTeXMLMath , we do not know if they can be specialized on a quadric surface to an extremal curve .
The way we know they are connected to an extremal curve is to pull them off the quadric surface , giving more room to move around and then specialize .
So it seems that this example is a test case for the connectedness question , and might possibly lead to a counterexample .
We show that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable ( i.e. , is tangent to a smooth variation through harmonic maps ) .
This provides one of the few known answers to this problem of integrability , which was raised in different contexts of geometry and analysis .
It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from LaTeXMLMath to LaTeXMLMath thus giving the nullity of any such harmonic map ; it also has bearing on the behaviour of weakly harmonic LaTeXMLMath -minimizing maps from a 3-manifold to LaTeXMLMath near a singularity and the structure of the singular set of such maps from any manifold to LaTeXMLMath .
2000 Mathematics Subject Classification : 58E20 , 53C43 , 53A10 .
Throughout this paper , LaTeXMLMath and LaTeXMLMath will denote smooth compact Riemannian manifolds without boundary , and LaTeXMLMath a smooth map .
Then LaTeXMLMath is said to be harmonic if it is an extremal of the energy functional LaTeXMLEquation i.e. , if for every smooth variation LaTeXMLMath with LaTeXMLMath we have LaTeXMLMath .
Equivalently , LaTeXMLMath satisfies the Euler-Lagrange equation for the problem : LaTeXMLMath , where LaTeXMLMath is the tension field given by LaTeXMLEquation .
Indeed , LaTeXMLEquation .
Here and in the sequel , LaTeXMLMath and LaTeXMLMath denote the inner product and connection induced on the relevant bundle by the metrics and Levi-Civita connections on LaTeXMLMath and LaTeXMLMath .
In particular , in ( LaTeXMLRef ) , LaTeXMLMath is the inner product on LaTeXMLMath and , in ( LaTeXMLRef ) , LaTeXMLMath is the connection induced on the bundle LaTeXMLMath of which LaTeXMLMath is a section ( see , for example , LaTeXMLCite for this formalism ) .
The second variation of the energy is described as follows .
For a smooth two-parameter variation LaTeXMLMath of LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , the Hessian of LaTeXMLMath is defined by LaTeXMLEquation .
We have LaTeXMLEquation where LaTeXMLEquation is called the Jacobi operator , a self-adjoint linear elliptic differential operator .
Here LaTeXMLMath denotes the Laplacian induced on LaTeXMLMath and the sign conventions on LaTeXMLMath and the curvature LaTeXMLMath are those of LaTeXMLCite .
Let LaTeXMLMath be a vector field along LaTeXMLMath , i.e .
a smooth section of LaTeXMLMath .
Then LaTeXMLMath is called a Jacobi field ( for the energy ) if LaTeXMLMath .
The space of Jacobi fields , LaTeXMLMath , is finite dimensional , its dimension is called the ( LaTeXMLMath ) -nullity of LaTeXMLMath .
The following proposition expresses the standard fact that the Jacobi operator is the linearisation of the tension field ( up to a sign coming from the choice of conventions ) .
Let LaTeXMLMath be a vector field along a smooth map LaTeXMLMath .
Then for any one-parameter family LaTeXMLMath of smooth map with LaTeXMLMath and LaTeXMLMath we have LaTeXMLEquation .
Proof .
Choose a two-parameter variation LaTeXMLMath with LaTeXMLMath .
We compare LaTeXMLEquation with LaTeXMLEquation and obtain the statement .
In particular when LaTeXMLMath is a smooth family of harmonic maps , the vector field LaTeXMLMath is a Jacobi field along LaTeXMLMath .
Indeed , LaTeXMLMath for all LaTeXMLMath , so this follows from ( LaTeXMLRef ) .
This suggests the following definition .
A Jacobi field LaTeXMLMath along a harmonic map LaTeXMLMath is said to be integrable if there is a smooth family LaTeXMLMath of harmonic maps such that LaTeXMLMath and LaTeXMLMath .
It is natural to ask whether all Jacobi fields along harmonic maps between given Riemannian manifolds are integrable , or to determine those which are .
As we shall see shortly , only very few cases are known , and the main result of the present paper is the following , where we take LaTeXMLMath to be the 2-sphere LaTeXMLMath with its unique conformal structure and LaTeXMLMath the complex projective space LaTeXMLMath with the standard Fubini-Study metric of holomorphic sectional curvature 1 : Any Jacobi field along a harmonic map from LaTeXMLMath to LaTeXMLMath is integrable .
This tells us the nullity of any harmonic map from LaTeXMLMath to LaTeXMLMath , see Corollary LaTeXMLRef .
In the remainder of this section we shall describe ( i ) another interpretation of integrability , ( ii ) the present state of knowledge on this question , ( iii ) motivation for the study of the problem ( in particular for these special manifolds ) .
Consider a ( fixed ) harmonic map LaTeXMLMath .
In LaTeXMLCite , T. Sunada showed that the set of harmonic maps LaTeXMLMath -close to LaTeXMLMath is a subset of a ball in a finite dimensional manifold of maps whose tangent space at LaTeXMLMath is precisely the kernel of LaTeXMLMath .
In other words , all harmonic maps close to LaTeXMLMath form a subset of the image LaTeXMLMath of a projection of LaTeXMLMath to the space of maps ( see LaTeXMLCite ) .
In general , this subset need not be a manifold and can be strictly contained in LaTeXMLMath .
In LaTeXMLCite , D. Adams and L. Simon proved the following : Let LaTeXMLMath be a harmonic map between real-analytic manifolds .
Then all Jacobi fields along LaTeXMLMath are integrable if and only if the space of harmonic maps ( LaTeXMLMath - ) close to LaTeXMLMath is a smooth manifold , whose tangent space at LaTeXMLMath is LaTeXMLMath .
It follows that for two real-analytic manifolds , all Jacobi fields along all harmonic maps are integrable if and only if the space of harmonic maps is a manifold whose tangent bundle is given by the Jacobi fields .
It is easy to construct an example where these conditions are not satisfied , as follows : Recall that a map LaTeXMLMath from a circle LaTeXMLMath to a manifold LaTeXMLMath is harmonic if and only if it is a closed geodesic .
Start from such a geodesic in a flat 2-torus .
The space of Jacobi fields is two dimensional and is tangent to the two-dimensional space of geodesics obtained by rotations and translations of the given one .
Say that the torus is a quotient of the cylinder in LaTeXMLMath obtained by revolution of the curve LaTeXMLMath , and the geodesic is the circle generated by revolution of the point LaTeXMLMath , LaTeXMLMath .
Now modify the cylinder by replacing LaTeXMLMath by LaTeXMLMath .
The circle remains a geodesic and the Jacobi fields are unchanged , because the curvature of the surface is zero along the circle .
However , translations of the curve are not geodesics anymore , so the space of closed geodesics has become one-dimensional .
Note that the modified torus has real-analytic metric , and that the curvature is negative near the geodesic and zero along it .
( A similar example was mentioned by F.J. Almgren in the seventies . )
In LaTeXMLCite , M. Mukai obtains a remarkable description of the integrability question for a one-parameter family of Clifford tori , which are harmonic maps from the flat 2-torus to the Euclidean 3-sphere .
Explicitly , she considers the family of maps LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Her results are as follows : For LaTeXMLMath , LaTeXMLMath , the space of harmonic maps around LaTeXMLMath is a LaTeXMLMath -dimensional manifold , whose tangent space is LaTeXMLMath .
For LaTeXMLMath , the space is still a 7-dimensional manifold but LaTeXMLMath .
For LaTeXMLMath , the space of harmonic maps is not a manifold around LaTeXMLMath .
So even in small dimensions and for very simple manifolds , all possible cases occur .
Concerning the pairs of manifolds such that all Jacobi fields are integrable , so far only the following cases are known : LaTeXMLMath is the circle and LaTeXMLMath is a globally symmetric space LaTeXMLCite .
LaTeXMLMath is the circle and LaTeXMLMath a manifold all of whose geodesics are closed ( i.e .
periodic ) and of the same length LaTeXMLCite .
LaTeXMLMath is a locally symmetric space of non-positive sectional curvature LaTeXMLCite .
LaTeXMLMath , the LaTeXMLMath -sphere LaTeXMLCite .
In this case , all harmonic maps are holomorphic or antiholomorphic .
Theorem ( LaTeXMLRef ) above concludes this short list at the present time .
Note that the case of maps from LaTeXMLMath to LaTeXMLMath LaTeXMLMath ) seems much more difficult to handle and does not appear to follow from the present methods .
The case of maps from LaTeXMLMath to LaTeXMLMath for LaTeXMLMath is also unsolved ; some results on the space of harmonic maps have been obtained in a series of papers by J. Bolton and L.M .
Woodward ( see , for example , LaTeXMLCite and LaTeXMLCite ) .
To motivate the search for situations where all Jacobi fields are integrable , we refer to their appearance in two aspects of the study of singularities of weakly harmonic maps .
Recall ( LaTeXMLCite , and LaTeXMLCite , LaTeXMLCite for a special case ) that an energy minimizing map LaTeXMLMath from an LaTeXMLMath -dimensional manifold can have a singular set of Hausdorff dimension at most LaTeXMLMath , and that a singular point is characterized by the appearance of a minimizing tangent map from LaTeXMLMath to LaTeXMLMath , which is the composition of the projection of LaTeXMLMath to LaTeXMLMath ( given by LaTeXMLMath ) and a harmonic map from LaTeXMLMath to LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are real-analytic , the minimizing tangent map is unique ( L. Simon LaTeXMLCite and LaTeXMLCite ) .
As shown by A. Adams and L. Simon LaTeXMLCite , and R. Gulliver and B .
White LaTeXMLCite , the convergence of a sequence of blow-ups of a harmonic map at a singular point to its minimizing tangent map is fast ( in a precisely defined sense ) if and only if all Jacobi fields for harmonic maps from LaTeXMLMath to LaTeXMLMath are integrable .
Theorem ( LaTeXMLRef ) implies that this is the case for maps from a LaTeXMLMath -dimensional manifold LaTeXMLMath to LaTeXMLMath .
In a second direction , L. Simon has analysed the structure of the LaTeXMLMath -dimensional part LaTeXMLMath of the singular set for a minimizing map from an LaTeXMLMath -dimensional manifold LaTeXMLMath to LaTeXMLMath .
When LaTeXMLMath and LaTeXMLMath are real analytic , he showed that LaTeXMLMath is contained in a countable union of LaTeXMLMath -submanifolds of LaTeXMLMath ( see LaTeXMLCite and LaTeXMLCite ) .
Whether LaTeXMLMath and LaTeXMLMath are analytic or not , but with the supplementary hypothesis that Jacobi fields along harmonic maps from LaTeXMLMath to LaTeXMLMath form the tangent space to the manifold of harmonic maps , he gets the stronger assertion that LaTeXMLMath is LaTeXMLMath .
As he explained to us in private communication , the proof of this result parallels step by step the proof of the analogous result for minimal submanifolds , which can be found in LaTeXMLCite .
We stress the fact that this result , valid for manifolds LaTeXMLMath of any dimension LaTeXMLMath , always involves maps of LaTeXMLMath to LaTeXMLMath .
This is because the method involves a blow up in the transversal direction to LaTeXMLMath , i.e .
in a three dimensional vector space .
Thus , Theorem ( LaTeXMLRef ) implies : Let LaTeXMLMath be a Riemannian manifold of dimension LaTeXMLMath and LaTeXMLMath a weakly harmonic energy minimizing map from LaTeXMLMath to LaTeXMLMath .
Then the singular set of LaTeXMLMath is contained in a countable union of LaTeXMLMath -dimensional LaTeXMLMath -manifolds together with a locally compact set of Hausdorff dimension LaTeXMLMath .
In fact , the singular set also has locally finite LaTeXMLMath -dimensional Hausdorff measure in the neighbourhood of each of its points .
Notice that this is exactly the statement of LaTeXMLCite Theorem 1 and LaTeXMLCite Chapter 4 , Theorem 1 , without change in the proof since our integrability theorem guarantees the required Łojasiewicz inequality .
Another motivation for this work is to extend the intense research done in the topological framework on the space of harmonic maps from LaTeXMLMath to LaTeXMLMath and LaTeXMLMath ( see , for example , M. Guest and Y. Ohnita LaTeXMLCite ) .
The remainder of this paper is organized as follows .
In Section 2 , we describe the construction of all harmonic maps from LaTeXMLMath to LaTeXMLMath , gathering together the aspects of the construction that we shall use .
Section 3 gives an overview of the proof , the case of holomorphic maps being treated in Section 4 , and the detailed computations required for various steps postponed to Section 5 .
In Section 6 , we relate energy- and area-integrability , and in Section 7 note that every Jacobi field is locally integrable .
We would like to thank L. Simon for posing questions which led to this work during the first MSJ International Research Institute ( Sendai , 1993 ) , F.E .
Burstall and M.A .
Guest for numerous discussions over the years , and R.L .
Bryant and the referee for some useful comments .
The first author ’ s research is supported by the Ministère de la Communauté Française de Belgique through an Action de Recherche Concertée and the second author benefited from several invitations to the U.L.B .
on a scientific mission from the Belgian F.N.R.S .
Following work of others LaTeXMLCite LaTeXMLCite , see also LaTeXMLCite , J. Eells and the second author LaTeXMLCite classified all harmonic maps from LaTeXMLMath to LaTeXMLMath .
Later , F.E .
Burstall and the second author gave another interpretation of this construction LaTeXMLCite .
We now describe the aspects of this construction which will be of use in the proof , for the case of LaTeXMLMath .
We view LaTeXMLMath as usual as the quotient of LaTeXMLMath by the equivalence relation LaTeXMLMath LaTeXMLMath ) .
Each point LaTeXMLMath is thus identified with a complex line in LaTeXMLMath .
The tautological bundle LaTeXMLMath over LaTeXMLMath is the subbundle of the trivial bundle LaTeXMLMath whose fibre at LaTeXMLMath is the line LaTeXMLMath in LaTeXMLMath .
Using the complex structure on LaTeXMLMath , we can decompose its complexified tangent bundle LaTeXMLMath into LaTeXMLMath and LaTeXMLMath parts : LaTeXMLEquation there is a well-known connection-preserving isomorphism LaTeXMLEquation where LaTeXMLMath denotes the orthogonal complement of LaTeXMLMath with respect to the standard Hermitian inner product on the trivial bundle LaTeXMLMath , given by LaTeXMLEquation where LaTeXMLMath is a local section of LaTeXMLMath ; see LaTeXMLCite , and LaTeXMLCite for an alternative description .
Consider now a smooth map LaTeXMLMath .
The complex extension of its differential LaTeXMLMath induces by inclusion and projection the maps LaTeXMLEquation and LaTeXMLEquation .
It will frequently be convenient to work in a complex chart LaTeXMLMath for LaTeXMLMath ; our constructions will be independent of choice of chart .
Given such a complex chart we introduce the notations LaTeXMLEquation and , for any connection LaTeXMLMath , LaTeXMLEquation .
Then , with LaTeXMLMath the ( pull-back ) connection on LaTeXMLMath , the map LaTeXMLMath is harmonic if and only if LaTeXMLEquation .
Indeed these expressions essentially give the LaTeXMLMath part LaTeXMLMath of the tension field of LaTeXMLMath .
Now to each map LaTeXMLMath , we associate the bundle LaTeXMLMath , the pull-back of the tautological bundle .
Thus LaTeXMLMath is the complex line subbundle of the trivial bundle LaTeXMLMath over LaTeXMLMath whose fibre at LaTeXMLMath is the line LaTeXMLMath .
Conversely , to each complex line subbundle of LaTeXMLMath is associated a map from LaTeXMLMath to LaTeXMLMath .
In the following description , all the bundles are subbundles of LaTeXMLMath over LaTeXMLMath or over a complex chart LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath be a subbundle of LaTeXMLMath .
The standard derivation on LaTeXMLMath induces a connection LaTeXMLMath on LaTeXMLMath by composing with orthogonal projection LaTeXMLMath on LaTeXMLMath : for a section LaTeXMLMath in LaTeXMLMath , LaTeXMLEquation where we write LaTeXMLMath for LaTeXMLMath .
On the other hand , given two mutually orthogonal subbundles LaTeXMLMath and LaTeXMLMath , we can define the LaTeXMLMath -second fundamental form LaTeXMLMath of LaTeXMLMath in LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath is a section of LaTeXMLMath .
One can check that LaTeXMLMath is tensorial ( it is similar to the second fundamental form of a submanifold ) .
We make similar definitions for LaTeXMLMath , using LaTeXMLMath in place of LaTeXMLMath .
As remarked on page 260 of LaTeXMLCite , it is easy to check that the adjoint of LaTeXMLMath is LaTeXMLMath .
As a special case , we set LaTeXMLEquation explicitly , for any smooth nowhere-zero section LaTeXMLMath of LaTeXMLMath , LaTeXMLEquation .
Now the pull-back over LaTeXMLMath of the isomorphism ( LaTeXMLRef ) yields a connection-preserving isomorphism of bundles over LaTeXMLMath : LaTeXMLEquation which we use to identify LaTeXMLEquation .
LaTeXMLEquation We have then LaTeXMLCite : LaTeXMLMath is holomorphic ( resp .
antiholomorphic ) if and only if LaTeXMLMath ( resp .
LaTeXMLMath ) ; LaTeXMLMath is harmonic if and only if LaTeXMLMath is holomorphic , i.e .
LaTeXMLEquation and this holds if and only if LaTeXMLMath is antiholomorphic , i.e .
LaTeXMLMath .
Indeed , a ) is immediate from the above isomorphism and b ) follows from ( LaTeXMLRef ) .
Now let LaTeXMLMath be a harmonic map .
By a theorem of Koszul-Malgrange LaTeXMLCite , we can interpret ( LaTeXMLRef ) as saying that LaTeXMLMath is a holomorphic section of the holomorphic bundle LaTeXMLMath , see LaTeXMLCite .
In particular , provided LaTeXMLMath is not antiholomorphic ( in which case LaTeXMLMath ) , the zeros of LaTeXMLMath form a finite set LaTeXMLMath , and , in any local complex coordinate LaTeXMLMath centred on a point LaTeXMLMath , we can write LaTeXMLEquation where LaTeXMLMath is a smooth section of LaTeXMLMath , non-zero at LaTeXMLMath , and LaTeXMLMath is a positive integer called the LaTeXMLMath -ramification index of LaTeXMLMath at LaTeXMLMath .
For any LaTeXMLMath , set LaTeXMLMath .
This defines a smooth subbundle of LaTeXMLMath over LaTeXMLMath .
By ( LaTeXMLRef ) , it can be extended to a smooth subbundle LaTeXMLMath called the LaTeXMLMath -Gauss bundle of LaTeXMLMath ; this corresponds to a smooth map LaTeXMLMath called the LaTeXMLMath -Gauss transform of LaTeXMLMath .
( If LaTeXMLMath is antiholomorphic , LaTeXMLMath and does not define a map ) .
Similarly , if LaTeXMLMath is harmonic but not holomorphic , the set LaTeXMLMath of zeros of LaTeXMLMath is finite and , in any local complex coordinate LaTeXMLMath centred on a point LaTeXMLMath of LaTeXMLMath , we can write LaTeXMLMath with LaTeXMLMath and LaTeXMLMath as above , LaTeXMLMath now being called the LaTeXMLMath -ramification index of LaTeXMLMath at LaTeXMLMath .
This allows us to define a smooth subbundle LaTeXMLMath called the LaTeXMLMath -Gauss bundle and the corresponding smooth map LaTeXMLMath called the LaTeXMLMath -Gauss transform of LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath transform harmonic maps to harmonic maps and are inverse ; precisely , from LaTeXMLCite we have Let LaTeXMLMath be harmonic .
If LaTeXMLMath is not antiholomorphic then LaTeXMLMath is harmonic and LaTeXMLMath .
If LaTeXMLMath is not holomorphic then LaTeXMLMath is harmonic and LaTeXMLMath .
We now describe all harmonic maps from LaTeXMLMath to LaTeXMLMath , following LaTeXMLCite and LaTeXMLCite .
We first recall that all holomorphic or antiholomorphic maps are harmonic ( we refer to those as LaTeXMLMath -holomorphic maps ) .
A map is called full if its image is not contained in a projective line LaTeXMLMath of LaTeXMLMath .
Note that a non-full harmonic map reduces to a harmonic map from LaTeXMLMath to itself , which is necessarily LaTeXMLMath -holomorphic .
Thus , a non- LaTeXMLMath -holomorphic harmonic map is necessarily full .
Following LaTeXMLCite , a map LaTeXMLMath is called ( complex ) isotropic if LaTeXMLEquation for all LaTeXMLMath .
Here , LaTeXMLMath denotes the extension of the inner product on LaTeXMLMath to a Hermitian inner poduct on LaTeXMLMath ( here restricted to LaTeXMLMath ) .
The case LaTeXMLMath in ( LaTeXMLRef ) is the condition of weak conformality .
In LaTeXMLCite it is shown that any harmonic map from LaTeXMLMath to LaTeXMLMath is isotropic .
So in their description of harmonic isotropic maps from a surface to LaTeXMLMath , the word ‘ isotropic ’ can be omitted when the surface is the sphere .
Now , following LaTeXMLCite and LaTeXMLCite , all non LaTeXMLMath -holomorphic harmonic maps LaTeXMLMath from LaTeXMLMath to LaTeXMLMath occur in a triple LaTeXMLMath where LaTeXMLMath is full and holomorphic and LaTeXMLMath is full and antiholomorphic , the associated bundles form an orthogonal decomposition LaTeXMLEquation and each map LaTeXMLMath determines the other two by the formulae LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
For a more explicit description of the construction LaTeXMLMath , see LaTeXMLCite .
In particular , the maps LaTeXMLMath and LaTeXMLMath which define LaTeXMLMath and LaTeXMLMath can be included in the following diagram of LaTeXMLMath -second fundamental forms : LaTeXMLEquation .
We note that by definition of LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Furthermore , since LaTeXMLMath is antiholomorphic , LaTeXMLMath , i.e .
LaTeXMLMath .
This can be expressed by saying that the LaTeXMLMath -second fundamental forms not shown in the diagram ( LaTeXMLRef ) are zero .
Likewise , we have a diagram showing the only non-zero LaTeXMLMath -second fundamental forms : LaTeXMLEquation .
In LaTeXMLCite , it is shown that the assignments LaTeXMLMath , LaTeXMLMath with inverses LaTeXMLMath , LaTeXMLMath define smooth bijections between : LaTeXMLMath , the space of full holomorphic maps LaTeXMLMath of degree LaTeXMLMath and total ramification index LaTeXMLMath ; LaTeXMLMath , the space of harmonic non- LaTeXMLMath -holomorphic maps of degree LaTeXMLMath and energy LaTeXMLMath ; LaTeXMLMath , the space of antiholomorphic maps of degree LaTeXMLMath and total ramification index LaTeXMLMath ; where LaTeXMLEquation with LaTeXMLMath , LaTeXMLMath .
The components of the space of harmonic maps from LaTeXMLMath to LaTeXMLMath consist of ( i ) the spaces of LaTeXMLMath -holomorphic maps ( not necessarily full ) of degree LaTeXMLMath , these have energy LaTeXMLMath where LaTeXMLMath ; ( ii ) the spaces LaTeXMLMath for integers LaTeXMLMath and LaTeXMLMath where LaTeXMLMath for some non-negative integer LaTeXMLMath .
In LaTeXMLCite , we showed that these components are smooth submanifolds of the space of all LaTeXMLMath ( or LaTeXMLMath ) maps from LaTeXMLMath to LaTeXMLMath , of dimension LaTeXMLMath in case ( i ) and LaTeXMLMath in case ( ii ) .
Theorem LaTeXMLRef and Proposition LaTeXMLRef show that the tangent bundles are given precisely by the Jacobi fields .
Hence The nullity of the LaTeXMLMath -Jacobi operator of a harmonic map from LaTeXMLMath to LaTeXMLMath of energy LaTeXMLMath is LaTeXMLMath if the map is holomorphic or antiholomorphic , and LaTeXMLMath otherwise .
We shall now present the outline of the proof of Theorem LaTeXMLRef , deferring to Sections 4 and 5 the detailed proofs of the various steps .
First , we consider the case of holomorphic maps .
Extending in a straightforward manner the results of LaTeXMLCite , we show that any Jacobi field along a holomorphic map between Kähler manifolds is a holomorphic vector field ( Proposition LaTeXMLRef ) and that every holomorphic vector field along a holomorphic map from LaTeXMLMath to LaTeXMLMath is integrable by means of holomorphic maps ( Proposition LaTeXMLRef ) .
Of course the corresponding statement applies to antiholomorphic maps .
There remains to consider the case of non- LaTeXMLMath -holomorphic harmonic maps LaTeXMLMath , recall that these are necessarily full .
The idea of the proof is as follows .
We use the map LaTeXMLEquation and its inverse LaTeXMLEquation .
Given a Jacobi field LaTeXMLMath along LaTeXMLMath , we would like to assert that LaTeXMLMath is a Jacobi field along the holomorphic map LaTeXMLMath , therefore integrable through a family LaTeXMLMath of holomorphic maps .
Setting LaTeXMLMath would then provide a family of harmonic maps with LaTeXMLMath .
However , the map LaTeXMLMath is only defined on harmonic and isotropic maps , because its definition requires extension of sections through branch points .
So , since we don ’ t know that LaTeXMLMath is tangent to LaTeXMLMath ( that is what we want to prove !
) , LaTeXMLMath is not defined on LaTeXMLMath .
An additional problem is that LaTeXMLMath is not continuous on the connected components LaTeXMLMath of the space of holomorphic maps , so that the family LaTeXMLMath may be discontinuous at LaTeXMLMath .
To get around these problems , we work away from the branch points , so that we get explicit formulae for the constructions , and verify that we can extend these over the branch points .
So let LaTeXMLMath be a fixed harmonic non- LaTeXMLMath -holomorphic map , LaTeXMLMath a Jacobi field along LaTeXMLMath and set LaTeXMLMath , LaTeXMLMath .
As in §2 , write LaTeXMLEquation .
To verify that these two sets are indeed equal , note first that , as mentioned in Section 2 , LaTeXMLMath is minus the adjoint of LaTeXMLMath , but taking into account the remarks following ( LaTeXMLRef ) this means that LaTeXMLMath is minus the adjoint of LaTeXMLMath .
So the zero sets of these operators coincide , and are precisely the set LaTeXMLMath .
Likewise , write LaTeXMLEquation and set LaTeXMLMath .
Note that LaTeXMLMath is a finite set in LaTeXMLMath .
Starting from a given Jacobi field LaTeXMLMath , choose a smooth one-parameter variation LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath .
Note that the maps LaTeXMLMath are not in general harmonic or isotropic for LaTeXMLMath .
Let LaTeXMLMath .
Since LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath , so that LaTeXMLMath is well-defined , non-zero and smooth in LaTeXMLMath for LaTeXMLMath , LaTeXMLMath .
Set LaTeXMLEquation so that LaTeXMLMath is a smooth section of LaTeXMLMath .
Let LaTeXMLMath denote the LaTeXMLMath -part of LaTeXMLMath .
We shall prove successively : Step 1 : LaTeXMLMath is a smooth section of LaTeXMLMath over LaTeXMLMath , which depends only on LaTeXMLMath and not on the choice of LaTeXMLMath .
Step 2 : LaTeXMLMath is a holomorphic section of LaTeXMLMath over LaTeXMLMath .
Step 3 : LaTeXMLMath extends to a holomorphic section of LaTeXMLMath on the whole of LaTeXMLMath .
Step 4 : For any smooth one-parameter family of maps LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath is well-defined and smooth for LaTeXMLMath , LaTeXMLMath .
Furthermore , LaTeXMLMath on LaTeXMLMath .
Step 5 : If LaTeXMLMath , then LaTeXMLMath is tangent to LaTeXMLMath .
Step 6 : There is a smooth one-parameter family LaTeXMLMath in LaTeXMLMath LaTeXMLMath ) with LaTeXMLMath and LaTeXMLMath .
Therefore LaTeXMLMath is a smooth variation of LaTeXMLMath through harmonic maps with LaTeXMLMath , showing that LaTeXMLMath is integrable .
Let LaTeXMLMath be a holomorphic map between Kähler manifolds , with LaTeXMLMath compact , and let LaTeXMLMath be a Jacobi field along LaTeXMLMath .
Then LaTeXMLMath is holomorphic , i.e .
LaTeXMLMath LaTeXMLMath , where LaTeXMLMath denotes the LaTeXMLMath -components of LaTeXMLMath , LaTeXMLMath the connection on LaTeXMLMath , and LaTeXMLMath are local complex coordinates on LaTeXMLMath .
Proof .
Straightforward computations show that for a holomorphic map LaTeXMLMath and a variation LaTeXMLMath such that LaTeXMLMath , we have LaTeXMLEquation ( see , for example , LaTeXMLCite for properties of LaTeXMLMath ) .
Therefore LaTeXMLMath LaTeXMLMath , and LaTeXMLMath is holomorphic .
Let LaTeXMLMath be a holomorphic map and let LaTeXMLMath be a holomorphic vector field along LaTeXMLMath .
Then LaTeXMLMath is integrable by a one-parameter family of holomorphic maps , i.e. , there is a smooth one-parameter family of holomorphic maps LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Proof .
Consider a standard chart LaTeXMLMath given by LaTeXMLEquation .
Then the holomorphic map LaTeXMLMath has the form LaTeXMLMath where LaTeXMLMath for coprime polynomials LaTeXMLMath and LaTeXMLMath .
In the same chart , LaTeXMLMath , for some meromorphic functions LaTeXMLMath .
We now show that each LaTeXMLMath can be written in the form LaTeXMLMath for some polynomial LaTeXMLMath .
To see that , consider a second chart LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , in other words , LaTeXMLMath .
Then , LaTeXMLEquation .
Now suppose that LaTeXMLMath has a zero at LaTeXMLMath so that LaTeXMLMath has a pole there .
In the LaTeXMLMath coordinates , LaTeXMLMath is therefore smooth at LaTeXMLMath and so is LaTeXMLMath .
Thus LaTeXMLMath must be smooth .
If LaTeXMLMath has no pole at LaTeXMLMath , then LaTeXMLMath must be smooth there , and again LaTeXMLMath is smooth .
The same reasoning applies at LaTeXMLMath , and we see that LaTeXMLMath can be at most as singular as LaTeXMLMath .
Thus LaTeXMLMath for some polynomial LaTeXMLMath .
Finally , we shall construct the variation LaTeXMLMath by choosing LaTeXMLMath of the form LaTeXMLEquation where LaTeXMLEquation with LaTeXMLMath two polynomials .
Then LaTeXMLEquation and this equals LaTeXMLMath if and only if LaTeXMLMath LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are coprime , these equations have solutions LaTeXMLMath by the Euclidean algorithm in the ring of polynomials .
Note As already mentioned , this is a direct extension of a result of LaTeXMLCite , who considered only maps from LaTeXMLMath to LaTeXMLMath .
We do not require LaTeXMLMath to be full in this result .
Step 1 LaTeXMLMath is a smooth section of LaTeXMLMath over LaTeXMLMath , which depends only on LaTeXMLMath and not on the choice of LaTeXMLMath .
Proof .
This follows from the fact that over LaTeXMLMath , LaTeXMLMath is obtained by an explicit formula involving only LaTeXMLMath and LaTeXMLMath .
Indeed , from ( LaTeXMLRef ) , it follows that , for any smooth nowhere-zero family of sections LaTeXMLMath of LaTeXMLMath , the subbundle LaTeXMLMath is spanned by LaTeXMLMath .
Taking a derivative with respect to LaTeXMLMath and setting LaTeXMLMath gives an explicit formula for LaTeXMLMath involving only LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Step 2 LaTeXMLMath is a holomorphic section of LaTeXMLMath over LaTeXMLMath .
Proof .
As we have already mentioned , LaTeXMLMath is in general neither harmonic nor isotropic for LaTeXMLMath .
However , by Remark 1.1 , it is harmonic ‘ up to first order in LaTeXMLMath ’ in the sense that LaTeXMLEquation .
LaTeXMLEquation We shall express this by LaTeXMLMath .
We now show that LaTeXMLMath is also isotropic up to first order in LaTeXMLMath .
Let LaTeXMLMath be a one-parameter family of smooth maps from a Riemann surface with LaTeXMLMath harmonic and LaTeXMLMath a Jacobi field along LaTeXMLMath .
Then , on any complex chart LaTeXMLMath , LaTeXMLMath if LaTeXMLMath , then LaTeXMLMath for arbitrary LaTeXMLMath , if ( ii ) holds , then LaTeXMLEquation if LaTeXMLMath , then LaTeXMLEquation .
It is further shown in LaTeXMLCite that , when LaTeXMLMath , LaTeXMLEquation in fact , this is shown for harmonic maps from LaTeXMLMath to LaTeXMLMath for any LaTeXMLMath .
Proof [ of lemma ( LaTeXMLRef ) ] .
( i ) Since LaTeXMLMath is essentially the LaTeXMLMath part LaTeXMLMath of the tension field of LaTeXMLMath , we have LaTeXMLEquation which is LaTeXMLMath since LaTeXMLMath .
Hence LaTeXMLMath and LaTeXMLMath are well-defined holomorphic quadratic differentials on LaTeXMLMath , so they must vanish identically .
We have LaTeXMLEquation where LaTeXMLMath denotes the curvature of LaTeXMLMath and LaTeXMLEquation .
Using the explicit formula for the curvature of LaTeXMLMath ( see LaTeXMLCite ) , we get LaTeXMLEquation .
Each of these terms contains the factor LaTeXMLMath and is therefore LaTeXMLMath .
All terms in the expression for LaTeXMLMath are therefore LaTeXMLMath .
That LaTeXMLMath is LaTeXMLMath can be proven similarly .
LaTeXMLMath and LaTeXMLMath are well-defined holomorphic cubic differentials on LaTeXMLMath , and are therefore zero .
Hence LaTeXMLMath is LaTeXMLMath .
Similarly , LaTeXMLMath is LaTeXMLMath .
Proof [ of the end of step 2 ] .
Given LaTeXMLMath and LaTeXMLMath along LaTeXMLMath , we associate to any family of sections LaTeXMLMath of the bundles LaTeXMLMath the vector field LaTeXMLEquation .
Using the isomorphism ( LaTeXMLRef ) , we can see LaTeXMLMath ( the LaTeXMLMath -component of LaTeXMLMath ) as a section of LaTeXMLMath ; by applying the formula ( LaTeXMLRef ) to the pull-back of LaTeXMLMath to LaTeXMLMath by the map LaTeXMLMath we see that LaTeXMLEquation .
Let LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , there exists LaTeXMLMath such that those functions are non-zero for LaTeXMLMath , so that LaTeXMLEquation are well-defined , non-zero and smooth in LaTeXMLMath for LaTeXMLMath , LaTeXMLMath .
We set LaTeXMLMath , so that LaTeXMLMath is a smooth section of LaTeXMLMath .
We denote by LaTeXMLMath its LaTeXMLMath -part .
We shall show that LaTeXMLMath is holomorphic in a neighbourhood LaTeXMLMath of any point LaTeXMLMath .
To do this , choose on LaTeXMLMath a nowhere zero section LaTeXMLMath of LaTeXMLMath , with LaTeXMLMath holomorphic .
We set LaTeXMLEquation so that LaTeXMLMath .
Likewise , we denote by LaTeXMLMath a nowhere zero section of LaTeXMLMath over LaTeXMLMath and by LaTeXMLMath a nowhere zero section of LaTeXMLMath over LaTeXMLMath .
Then LaTeXMLEquation so that LaTeXMLMath is holomorphic if and only if LaTeXMLMath is .
Now LaTeXMLMath is a section of LaTeXMLMath , and we shall show that it is holomorphic by proving that LaTeXMLMath is perpendicular to LaTeXMLMath and LaTeXMLMath .
By construction of LaTeXMLMath , we have LaTeXMLMath , taking the derivative with respect to LaTeXMLMath gives LaTeXMLEquation .
Now LaTeXMLMath is a non-zero multiple of LaTeXMLMath which itself equals LaTeXMLMath since , by definition of LaTeXMLMath , we have LaTeXMLMath + a multiple of LaTeXMLMath .
This last expression is a multiple of the conjugate of LaTeXMLEquation which is LaTeXMLMath by Lemma LaTeXMLRef .
Hence LaTeXMLMath .
Taking the derivative with respect to LaTeXMLMath and setting LaTeXMLMath yields LaTeXMLEquation so that LaTeXMLMath , since LaTeXMLMath is holomorphic .
Next , note that LaTeXMLMath is a multiple of LaTeXMLMath .
Since LaTeXMLMath is a multiple of LaTeXMLMath and LaTeXMLMath a multiple of LaTeXMLMath ( modulo a section of LaTeXMLMath ) we see that LaTeXMLMath is a multiple of LaTeXMLMath .
Up to complex conjugation , this is a multiple of LaTeXMLEquation which is LaTeXMLMath by Lemma LaTeXMLRef .
So LaTeXMLMath .
Differentiating as before yields LaTeXMLMath .
We conclude that LaTeXMLMath is holomorphic on LaTeXMLMath , and since it is smooth on LaTeXMLMath , it is holomorphic on that larger set .
Step 3 LaTeXMLMath extends to a holomorphic section of LaTeXMLMath on the whole of LaTeXMLMath .
Proof .
Consider a point LaTeXMLMath , and a small neighbourhood LaTeXMLMath of LaTeXMLMath with LaTeXMLMath .
On LaTeXMLMath , consider LaTeXMLMath and LaTeXMLMath as above .
Since LaTeXMLMath and LaTeXMLMath are holomorphic sections of LaTeXMLMath and LaTeXMLMath respectively , LaTeXMLMath is a holomorphic section of LaTeXMLMath on LaTeXMLMath .
Decompose LaTeXMLMath .
We shall prove that both LaTeXMLMath and LaTeXMLMath extend to smooth sections across LaTeXMLMath .
Since by construction LaTeXMLMath , we have LaTeXMLMath , and taking the derivative , LaTeXMLEquation .
Putting LaTeXMLMath and using the fact that LaTeXMLMath and LaTeXMLMath , we get LaTeXMLEquation i.e .
LaTeXMLEquation .
It follows that LaTeXMLEquation which extends smoothly across LaTeXMLMath .
Since LaTeXMLMath is holomorphic on LaTeXMLMath , we have , on that domain , LaTeXMLEquation .
However , as noted in Section 2 , LaTeXMLMath , so that , on taking components in LaTeXMLMath and LaTeXMLMath , the above equation yields the pair of equations LaTeXMLEquation .
LaTeXMLEquation Equation ( LaTeXMLRef ) tells us that LaTeXMLMath is a holomorphic section of LaTeXMLMath on LaTeXMLMath .
Next , since LaTeXMLMath is harmonic , Lemma LaTeXMLRef implies that LaTeXMLMath is antiholomorphic .
Therefore , with respect to a complex coordinate LaTeXMLMath centred on LaTeXMLMath , LaTeXMLMath can be written locally in the form LaTeXMLMath times an antiholomorphic section which is nonzero at LaTeXMLMath , and LaTeXMLMath extends smoothly across LaTeXMLMath since the right hand side of ( LaTeXMLRef ) is smooth on LaTeXMLMath .
Now , LaTeXMLMath could at worst have a pole or an isolated essential singularity at LaTeXMLMath .
But this is not possible with LaTeXMLMath smooth , hence LaTeXMLMath has a removable singularity at LaTeXMLMath and so extends to a smooth section on LaTeXMLMath .
Putting a ) and b ) together , LaTeXMLMath and therefore LaTeXMLMath extend smoothly over LaTeXMLMath , yielding holomorphic sections on the whole of LaTeXMLMath .
Step 4 For any smooth one-parameter family of maps LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath is well-defined and smooth for LaTeXMLMath , LaTeXMLMath .
Furthermore , LaTeXMLMath on LaTeXMLMath .
Proof .
The construction of LaTeXMLMath from LaTeXMLMath is similar to that of LaTeXMLMath from LaTeXMLMath , and LaTeXMLMath is likewise well-defined and smooth .
To compute LaTeXMLMath , start again with LaTeXMLMath .
We have LaTeXMLMath on LaTeXMLMath .
Since LaTeXMLMath is harmonic and LaTeXMLMath is a Jacobi field , we have by Lemma LaTeXMLRef ( b ) LaTeXMLEquation .
Therefore , LaTeXMLEquation since LaTeXMLMath has image in LaTeXMLMath by construction .
Thus , LaTeXMLEquation hence LaTeXMLMath .
Taking the derivative with respect to LaTeXMLMath , we see that LaTeXMLMath — tangent to LaTeXMLMath at LaTeXMLMath — is mapped to LaTeXMLMath by the directional derivative of LaTeXMLMath , i.e .
LaTeXMLMath on LaTeXMLMath .
Step 5 If LaTeXMLMath , then LaTeXMLMath is tangent to LaTeXMLMath .
Proof .
LaTeXMLMath is a holomorphic vector field along the holomorphic map LaTeXMLMath .
By Proposition ( LaTeXMLRef ) , there exists a family of holomorphic maps LaTeXMLMath such that LaTeXMLMath .
When LaTeXMLMath is full and LaTeXMLMath , for LaTeXMLMath small enough LaTeXMLMath remains full and in the component LaTeXMLMath ( but not in LaTeXMLMath in general ) .
Set LaTeXMLMath .
Then LaTeXMLMath is harmonic for each LaTeXMLMath and along LaTeXMLMath it is smooth on LaTeXMLMath , in the LaTeXMLMath variables .
By step 4 , LaTeXMLMath on LaTeXMLMath .
Since LaTeXMLMath is a finite set , LaTeXMLMath so that LaTeXMLEquation .
Since the integrand is smooth this integral can be taken over LaTeXMLMath and integration by parts yields LaTeXMLEquation .
Similarly , LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath , we deduce that LaTeXMLEquation and LaTeXMLEquation where LaTeXMLMath denotes the degree of the map LaTeXMLMath .
Using formulae ( LaTeXMLRef ) , we see that LaTeXMLMath and LaTeXMLMath , so that LaTeXMLMath is tangent to LaTeXMLMath .
Step 6 There is a smooth one-parameter family LaTeXMLMath in LaTeXMLMath LaTeXMLMath ) with LaTeXMLMath , LaTeXMLMath , and therefore LaTeXMLMath is a smooth variation of LaTeXMLMath with LaTeXMLMath .
Proof .
Since LaTeXMLMath is a smooth submanifold of LaTeXMLMath LaTeXMLCite , we can project the family LaTeXMLMath to a family LaTeXMLMath in LaTeXMLMath which is also tangent to LaTeXMLMath .
Then , by LaTeXMLCite , LaTeXMLMath is a smooth family of harmonic maps , and by step 4 , LaTeXMLMath on LaTeXMLMath .
By continuity of both sides , LaTeXMLMath on LaTeXMLMath , and we are done .
Any harmonic map LaTeXMLMath is weakly conformal , and hence is precisely the same as a minimal branched immersion of LaTeXMLMath in LaTeXMLMath in the sense of LaTeXMLCite .
The question of integrability of Jacobi fields can therefore be asked in the setting of the first and second variation of the area , rather than energy .
As proven by N. Ejiri and M. Micallef ( private communication ) , for any harmonic map LaTeXMLMath , the map LaTeXMLMath the normal component of LaTeXMLMath is a surjective linear map from the space of LaTeXMLMath -Jacobi fields ( i.e .
Jacobi fields for the energy ) to the space of LaTeXMLMath -Jacobi fields , with kernel the tangential conformal fields .
Our result translates immediately to show that each LaTeXMLMath -Jacobi field along a minimal branched immersion LaTeXMLMath is integrable .
In the case of immersions , the space of tangential conformal fields is LaTeXMLMath -dimensional , so that Corollary LaTeXMLRef implies that the LaTeXMLMath -nullity of LaTeXMLMath , i.e .
the dimension of the space of LaTeXMLMath -Jacobi fields , for a non- LaTeXMLMath -holomorphic harmonic immersion of LaTeXMLMath in LaTeXMLMath with energy LaTeXMLMath is LaTeXMLMath .
This was proved in the framework of minimal surfaces by S. Montiel and F. Urbano LaTeXMLCite , using a different method that does not seem to extend easily to branched immersions .
The question treated above is global : to find a variation of a harmonic map generating a Jacobi field on the whole manifold .
The analogous local question always has the following positive answer : Let LaTeXMLMath be a harmonic map between Riemannian manifolds .
For each LaTeXMLMath , there is a neighbourhood LaTeXMLMath of LaTeXMLMath such that the restriction of any Jacobi field LaTeXMLMath to LaTeXMLMath is integrable .
Proof .
This can be deduced from the general results of LaTeXMLCite , but we show here how it follows from familiar results in the theory of harmonic maps .
We restrict LaTeXMLMath and LaTeXMLMath to a closed ball LaTeXMLMath such that LaTeXMLMath is contained in a ball whose radius is half that of a ‘ geodesically small ball ’ of LaTeXMLMath as defined in LaTeXMLCite .
Recall that a geodesically small ball is disjoint from the cut locus of its centre and has radius LaTeXMLMath , where the sectional curvature of LaTeXMLMath with LaTeXMLMath .
Consider then a smooth variation LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath .
For LaTeXMLMath small enough , LaTeXMLMath remains in the geodesically small ball .
Consider for each LaTeXMLMath the Dirichlet problem of finding LaTeXMLMath with LaTeXMLMath harmonic and LaTeXMLMath .
By LaTeXMLCite , it has a solution , which is unique , and has non-degenerate Hessian by LaTeXMLCite .
By LaTeXMLCite , LaTeXMLMath depends smoothly on LaTeXMLMath , so that LaTeXMLMath defines a Jacobi field along LaTeXMLMath equal to LaTeXMLMath along LaTeXMLMath .
By LaTeXMLCite , this Jacobi field must coincide with LaTeXMLMath , and so LaTeXMLMath is integrable .
Département de Mathématique Université Libre de Bruxelles CP 218 Campus Plaine Bd du Triomphe B-1050-Bruxelles , Belgium e-mail : llemaire @ ulb.ac.be School of Mathematics University of Leeds Leeds LS2 9JT G.B .
e-mail : j.c.wood @ leeds.ac.uk
We address the problem of identifying the ( nonstationary ) quantum systems that admit supersymmetric dynamical invariants .
In particular , we give a general expression for the bosonic and fermionic partner Hamiltonians .
Due to the supersymmetric nature of the dynamical invariant the solutions of the time-dependent Schrödinger equation for the partner Hamiltonians can be easily mapped to one another .
We use this observation to obtain a class of exactly solvable time-dependent Schrödinger equations .
As applications of our method , we construct classes of exactly solvable time-dependent generalized harmonic oscillators and spin Hamiltonians .
The problem of the solution of the time-dependent Schrödinger equation , LaTeXMLEquation is as old as quantum mechanics .
It is well-known that this equation may be reduced to the time-independent Schrödinger equation , i.e. , the eigenvalue equation for the Hamiltonian , provided that the eigenstates of the Hamiltonian are time-independent .
LaTeXMLCite .
The search for exact solutions of the eigenvalue equation for the Hamiltonian has been an ongoing effort for the past seven decades .
A rather recent development in this direction is the application of the ideas of supersymmetric quantum mechanics LaTeXMLCite .
The main ingredient provided by supersymmetry is that the eigenvectors of the bosonic and fermionic partner Hamiltonians are related by a supersymmetry transformation LaTeXMLCite .
Therefore , one can construct the solutions of the eigenvalue problem for one of the partner Hamiltonians , if the other is exactly solvable .
In general , this method can not be used to relate the solutions of the time-dependent Schrödinger equation unless the partner Hamiltonians have time-independent eigenvectors .
The aim of this article is to explore the utility of supersymmetry in solving time-dependent Schrödinger equation for a general class of time-dependent Hamiltonians .
This problem has been considered by Bagrov and Samsonov LaTeXMLCite and Cannata et al .
LaTeXMLCite for the standard Hamiltonians of the form LaTeXMLMath in one dimension .
Our method differs from those of these authors in the following way .
First , we approach the problem from the point of view of the theory of dynamical invariants LaTeXMLCite .
Dynamical invariants are certain ( time-dependent ) operators with a complete set of eigenvectors that are exact solutions of the time-dependent Schrödinger equation .
We can easily use the ideas of supersymmetric quantum mechanics to relate the solutions of the time-dependent Schrödinger equation for two different Hamiltonians , if we can identify them with the bosonic and fermionic Hamiltonians of a ( not necessarily supersymmetric ) LaTeXMLMath -graded quantum system admitting a supersymmetric dynamical invariant .
Unlike Refs .
LaTeXMLCite and LaTeXMLCite , we consider general even supersymmetric dynamical invariants and use our recent results on the geometrically equivalent quantum systems LaTeXMLCite to give a complete characterization of the time-dependent Hamiltonians that admit supersymmetric dynamical invariants .
The organization of the article is as follows .
In Sections 2 , we present a brief review of the dynamical invariants and survey our recent results on identifying the Hamiltonians that admit a given dynamical invariant .
In Section 3 , we discuss the supersymmetric dynamical invariants .
In section 4 , we give a characterization of the quantum systems that admit a Hermitian supersymmetric dynamical invariant .
In sections 5 and 6 , we apply our general results to obtain classes of exactly solvable time-dependent generalized harmonic oscillators and spin systems , respectively .
In section 6 , we compare our method with that of Refs .
LaTeXMLCite and LaTeXMLCite and present our concluding remarks .
By definition LaTeXMLCite , a dynamical invariant is a nontrivial solution LaTeXMLMath of the Liouville-von-Neumann equation LaTeXMLEquation where LaTeXMLMath denotes the Hamiltonian .
Consider a Hermitian Hamiltonian LaTeXMLMath admitting a Hermitian dynamical invariant LaTeXMLMath , and suppose that LaTeXMLMath has a discrete spectrum .
Then , Eq .
( LaTeXMLRef ) may be used to show that the eigenvalues LaTeXMLMath of LaTeXMLMath are constant and the eigenvectors LaTeXMLMath yield the evolution operator LaTeXMLMath for the Hamiltonian LaTeXMLMath according to LaTeXMLEquation .
Here LaTeXMLMath is a spectral label , LaTeXMLMath is a degeneracy label , LaTeXMLMath is the degree of degeneracy of LaTeXMLMath , the eigenvectors LaTeXMLMath are assumed to form a complete orthonormal basis of the Hilbert space , LaTeXMLMath are the entries of the solution of the matrix Schrödinger equation : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLMath and LaTeXMLMath are matrices with entries LaTeXMLEquation respectively , LaTeXMLCite .
Note that LaTeXMLMath are Hermitian matrices and LaTeXMLMath is unitary .
In view of Eq .
( LaTeXMLRef ) , LaTeXMLEquation are solutions of the Schrödinger equation ( LaTeXMLRef ) .
These solutions actually form a complete orthonormal set of eigenvectors of LaTeXMLMath .
We may use this observation or alternatively Eq .
( LaTeXMLRef ) to show LaTeXMLEquation .
Now , suppose that LaTeXMLMath is obtained from a parameter-dependent operator LaTeXMLMath as LaTeXMLMath where LaTeXMLMath , LaTeXMLMath are real parameters denoting the coordinates of points of a parameter manifold LaTeXMLMath ; LaTeXMLMath determines a smooth curve in LaTeXMLMath ; LaTeXMLMath is a Hermitian operator with a discrete spectrum ; ( in local coordinate patches of LaTeXMLMath ) the eigenvectors LaTeXMLMath of LaTeXMLMath , i.e. , the solutions of LaTeXMLEquation are smooth ( single-valued ) functions of LaTeXMLMath ; LaTeXMLMath and LaTeXMLMath are independent of LaTeXMLMath ; LaTeXMLMath form a complete orthonormal basis .
In the following , we shall identify LaTeXMLMath with LaTeXMLMath and express LaTeXMLMath in the form LaTeXMLEquation where LaTeXMLMath is a unitary operator and LaTeXMLMath defines a single-valued function of LaTeXMLMath .
Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) suggest LaTeXMLEquation .
For a closed curve LaTeXMLMath , there exists LaTeXMLMath such that LaTeXMLMath , and the quantity LaTeXMLEquation yields the non-Abelian cyclic geometric phase LaTeXMLCite associated with the solution LaTeXMLMath .
In Eq .
( LaTeXMLRef ) , LaTeXMLMath and LaTeXMLMath respectively denote the time-ordering and path-ordering operators , the loop integral is over the closed path LaTeXMLMath , and LaTeXMLMath is the nondegenerate non-Abelian generalization of the Berry connection one-form LaTeXMLCite .
The latter is defined in terms of its matrix elements : LaTeXMLEquation where LaTeXMLMath is the exterior derivative operator on LaTeXMLMath .
If LaTeXMLMath is nondegenerate , LaTeXMLMath is just a phase factor .
It coincides with the ( nonadiabatic ) geometric phase of Aharonov and Anandan LaTeXMLCite .
Next , we introduce LaTeXMLMath .
Then as discussed in LaTeXMLCite , Hermitian Hamiltonians that admit the invariant LaTeXMLEquation have the form LaTeXMLEquation where LaTeXMLMath is any Hermitian operator commuting with LaTeXMLMath .
Note that according to Eq .
( LaTeXMLRef ) , LaTeXMLMath is related to LaTeXMLMath by a time-dependent ( canonical ) unitary transformation of the Hilbert space LaTeXMLCite , namely LaTeXMLMath .
This observation may be used to express the evolution operator LaTeXMLMath of LaTeXMLMath in the form LaTeXMLEquation where LaTeXMLMath is the evolution operator for LaTeXMLMath .
Note that LaTeXMLMath commutes with LaTeXMLMath , therefore if LaTeXMLMath has a nondegenerate spectrum , LaTeXMLMath has a constant eigenbasis .
In this case , LaTeXMLMath with different LaTeXMLMath commute and LaTeXMLEquation .
Having expressed LaTeXMLMath in terms of LaTeXMLMath and LaTeXMLMath , we can write the solutions ( LaTeXMLRef ) of the Schrödinger equation in the form LaTeXMLEquation where LaTeXMLMath .
If LaTeXMLMath with different values of LaTeXMLMath commute , this equation takes the form LaTeXMLEquation where LaTeXMLMath .
We conclude this section by emphasizing that Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) are valid for any time-dependent unitary operator LaTeXMLMath satisfying Eq .
( LaTeXMLRef ) .
For example , one may identify LaTeXMLMath with the evolution operator of another Hamiltonian that admits the same invariant LaTeXMLMath .
Note that in general such a choice of LaTeXMLMath can not be expressed as the image of a curve LaTeXMLMath under a single-valued function LaTeXMLMath .
In particular , LaTeXMLMath can not be written as LaTeXMLMath for parameter-dependent vectors LaTeXMLMath that are single-valued functions of LaTeXMLMath .
This in turn implies that LaTeXMLMath can not be used in the calculation of the geometric phases .
A LaTeXMLMath -graded quantum system LaTeXMLCite is a system whose Hilbert space LaTeXMLMath is the direct sum of two of its nontrivial subspaces LaTeXMLMath , i.e. , LaTeXMLMath , and whose Hamiltonian maps LaTeXMLMath to LaTeXMLMath .
The elements of LaTeXMLMath and LaTeXMLMath are respectively called bosonic and fermionic state vectors , or graded state vectors with definite grading ( or chirality ) 0 and 1 .
Operators preserving the grading of the graded state vectors are called even operators .
Those that change the grading of these state vectors are called odd operators .
In the two-component representation of the Hilbert space , where the first component LaTeXMLMath denotes the bosonic and the second component LaTeXMLMath denotes the fermionic part of a state vector LaTeXMLMath , the Hamiltonian has the form LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath are Hermitian operators .
They are respectively called the bosonic and fermionic Hamiltonians .
Now , suppose that LaTeXMLMath and consider a parameter-dependent odd operator LaTeXMLMath and an even Hermitian operator LaTeXMLMath that satisfy the algebra of LaTeXMLMath supersymmetric quantum mechanics LaTeXMLCite : LaTeXMLEquation .
In particular , suppose that in the two-component representation of the Hilbert space , LaTeXMLEquation where LaTeXMLMath is a linear operator .
This choice of LaTeXMLMath satisfies the superalgebra ( LaTeXMLRef ) provided that LaTeXMLEquation with LaTeXMLEquation .
As is well-known from the study of the spectral properties of supersymmetric systems , one can use Eqs .
( LaTeXMLRef ) to derive the following properties of LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath have nonnegative spectra with the same set of positive eigenvalues LaTeXMLMath ; The degree of degeneracy LaTeXMLMath of LaTeXMLMath as an eigenvalue of LaTeXMLMath is the same as its degree of degeneracy as an eigenvalue of LaTeXMLMath ; Orthonormal eigenvectors LaTeXMLMath of LaTeXMLMath associated with LaTeXMLMath are related according to LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath are the entries of a unitary LaTeXMLMath matrix LaTeXMLMath .
In particular , for a given orthonormal set LaTeXMLMath of the eigenvectors of LaTeXMLMath , LaTeXMLEquation form a complete orthonormal eigenbasis of LaTeXMLMath for LaTeXMLMath .
Here ‘ Ker ’ denotes the kernel or the eigenspace with zero eigenvalue .
Next , we introduce LaTeXMLMath and LaTeXMLMath for some curve LaTeXMLMath in the parameter space LaTeXMLMath and demand that LaTeXMLMath is a dynamical invariant for the Hamiltonian LaTeXMLMath .
In view of Eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) , and ( LaTeXMLRef ) , LaTeXMLMath is a dynamical invariant for LaTeXMLMath .
We can write LaTeXMLMath in the form ( LaTeXMLRef ) by requiring LaTeXMLMath to satisfy LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath fulfil LaTeXMLEquation .
In view of Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , LaTeXMLEquation .
Moreover , employing Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , we can express the Hamiltonians LaTeXMLMath and their evolution operators LaTeXMLMath in the form LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath are Hermitian operators satisfying LaTeXMLEquation .
For example , we can choose LaTeXMLMath where LaTeXMLMath is a polynomial with time-dependent coefficients .
In view of Eq .
( LaTeXMLRef ) , we have the following set of orthonormal solutions of the Schrödinger equation for the Hamiltonians LaTeXMLMath .
LaTeXMLEquation where LaTeXMLMath .
As we discussed above , given an eigenbasis LaTeXMLMath for LaTeXMLMath we can set LaTeXMLEquation .
This identification may be used to relate the solutions ( LaTeXMLRef ) according to LaTeXMLEquation .
For the case where LaTeXMLMath with different LaTeXMLMath commute , Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) take the form LaTeXMLEquation .
LaTeXMLEquation respectively .
Here LaTeXMLMath .
The above construction is valid for any choice of time-dependent unitary operators LaTeXMLMath satisfying LaTeXMLEquation .
These observations together with Eq .
( LaTeXMLRef ) suggests a method of generating a class of exactly solvable time-dependent Schrödinger equations .
This is done according to the following prescription .
Choose a Hamiltonian LaTeXMLMath whose time-dependent Schrödinger equation is exactly solvable , i.e. , its evolution operator LaTeXMLMath is known ; Choose an arbitrary constant operator LaTeXMLMath and a unitary operator LaTeXMLMath satisfying LaTeXMLMath ; Set LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then , by construction LaTeXMLMath is a dynamical invariant for LaTeXMLMath .
It also satisfies LaTeXMLMath for LaTeXMLMath .
Note that , in view of Eq .
( LaTeXMLRef ) and the Schrödinger equation LaTeXMLEquation this choice of LaTeXMLMath corresponds to taking LaTeXMLMath .
Let LaTeXMLMath be a Hermitian operator commuting with LaTeXMLMath .
Then according to Eq .
( LaTeXMLRef ) , LaTeXMLMath and LaTeXMLEquation are partner Hamiltonians , and LaTeXMLMath admits the invariant LaTeXMLMath .
The choice LaTeXMLMath also implies that LaTeXMLMath are solutions of the Schrödinger equation ( LaTeXMLRef ) for LaTeXMLMath .
Furthermore , for all LaTeXMLMath , LaTeXMLEquation are solutions of the Schrödinger equation for the Hamiltonian LaTeXMLMath .
These solutions span LaTeXMLMath .
Again if LaTeXMLMath with different values of LaTeXMLMath commute , we have LaTeXMLEquation .
One can also employ an alternative construction for the Hamiltonian LaTeXMLMath in which one still defines LaTeXMLMath according to LaTeXMLMath but uses another unitary operator LaTeXMLMath to express it as LaTeXMLMath .
In this way , one may choose LaTeXMLMath to be the image of a curve LaTeXMLMath in a parameter space LaTeXMLMath under a single-valued function LaTeXMLMath .
This is especially convenient for addressing the geometric phase problem for the Hamiltonians LaTeXMLMath .
Following this approach , one must determine LaTeXMLMath according to Eq .
( LaTeXMLRef ) , i.e. , LaTeXMLEquation .
One then obtains the Hamiltonian LaTeXMLMath by substituting ( LaTeXMLRef ) in ( LaTeXMLRef ) .
In this section we explore the partner Hamiltonians for the Hamiltonian of the unit simple harmonic oscillator : LaTeXMLEquation .
Here LaTeXMLMath and LaTeXMLMath are respectively the momentum and position operators and LaTeXMLMath .
Let LaTeXMLMath be a unitary operator satisfying LaTeXMLMath and LaTeXMLEquation .
Then LaTeXMLMath is a dynamical invariant for LaTeXMLMath .
This invariant together with LaTeXMLEquation form a supersymmetric dynamical invariant .
The associated ‘ fermionic ’ partner Hamiltonian is given by Eq .
( LaTeXMLRef ) where LaTeXMLMath is a Hermitian operator commuting with LaTeXMLMath .
For example , let LaTeXMLEquation where LaTeXMLMath is a real-valued function , and LaTeXMLMath where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath , and LaTeXMLMath .
Note that LaTeXMLMath with different values of LaTeXMLMath commute and the operators LaTeXMLMath are generators of the group LaTeXMLMath in its oscillator representation .
The parameter space of the operator LaTeXMLMath is the unit hyperboloid : LaTeXMLEquation .
We have made the choices ( LaTeXMLRef ) and ( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath in view of the following considerations .
Up to a trivial addition of a multiple of identity , ( LaTeXMLRef ) is the most general expression for a second order differential operator commuting with LaTeXMLMath .
Every element of ( the oscillator representation of the ) Lie algebra of LaTeXMLMath may be expressed as LaTeXMLMath with LaTeXMLMath and LaTeXMLMath given by Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , respectively .
In particular , as we show in the following , these choices lead to the most general expression for an invariant LaTeXMLMath and a Hamiltonian LaTeXMLMath belonging to ( the oscillator representation of the ) Lie algebra of LaTeXMLMath .
In order to compute the Hamiltonian LaTeXMLMath , we substitute Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) in Eq .
( LaTeXMLRef ) and use the LaTeXMLMath algebra , LaTeXMLEquation and the Backer-Campbell-Hausdorff formula to compute the right-hand side of the resulting equation .
We then find , after a rather lengthy calculation , LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and a dot denotes a time derivative .
As seen from Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , LaTeXMLMath is the Hamiltonian of a time-dependent generalized harmonic oscillator LaTeXMLCite with three free functions LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
According to our general analysis , the corresponding Schrödinger equation is exactly solvable .
The evolution operator is given by LaTeXMLEquation where LaTeXMLMath .
Furthermore , we can use the stationary solutions of the Schrödinger equation for the unit simple harmonic oscillator ( LaTeXMLRef ) to construct solutions of the Schrödinger equation for LaTeXMLMath .
The stationary solutions for the Hamiltonian ( LaTeXMLRef ) are LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is the ground state vector for the unit simple harmonic oscillator ( LaTeXMLRef ) given by LaTeXMLMath .
In view of Eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) , and ( LaTeXMLRef ) , we have the following orthonormal solutions of the Schrödinger equation for LaTeXMLMath .
LaTeXMLEquation where LaTeXMLMath .
Next , we use the identity LaTeXMLCite LaTeXMLEquation and the expression for the propagator of the unit simple harmonic oscillator LaTeXMLCite , namely LaTeXMLEquation to compute the solutions ( LaTeXMLRef ) in the position representation .
This yields LaTeXMLEquation where LaTeXMLMath are the eigenfunctions of the unit simple harmonic oscillator Hamiltonian ( LaTeXMLRef ) .
Consider the dipole interaction Hamiltonian of a spinning particle in a constant magnetic field : LaTeXMLEquation where LaTeXMLMath is constant ( -Larmor frequency ) , the magnetic field is assumed to be directed along the LaTeXMLMath -direction , and LaTeXMLMath denotes the LaTeXMLMath -component of the angular momentum operator LaTeXMLMath of the particle .
Let LaTeXMLMath be a unitary operator satisfying LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath .
Then , in view of the identity LaTeXMLEquation the operator LaTeXMLEquation is a dynamical invariant for LaTeXMLMath .
Eq .
( LaTeXMLRef ) follows from the LaTeXMLMath algebra , LaTeXMLEquation satisfied by LaTeXMLMath , the fact that LaTeXMLMath is a Casimir operator , i.e. , LaTeXMLMath , and the relation LaTeXMLEquation .
The invariant LaTeXMLMath together with LaTeXMLEquation form a supersymmetric dynamical invariant .
The associated ‘ fermionic ’ partner Hamiltonian is given by Eq .
( LaTeXMLRef ) where LaTeXMLMath is a Hermitian operator commuting with LaTeXMLMath .
Next , we note that LaTeXMLMath .
This suggests that we may choose LaTeXMLMath as a polynomial in LaTeXMLMath with time-dependent coefficients .
For example , we may set LaTeXMLEquation where LaTeXMLMath is a real-valued function .
With this choice of LaTeXMLMath , we can construct a class of partner Hamiltonians LaTeXMLMath for LaTeXMLMath representing the dipole interaction of a spinning particle in a time-dependent magnetic field , provided that we choose LaTeXMLMath according to LaTeXMLCite LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
Note that again LaTeXMLMath with different values of LaTeXMLMath commute , the parameter space of the operator LaTeXMLMath is the unit sphere , and LaTeXMLMath and LaTeXMLMath are respectively the polar and azimuthal angles .
LaTeXMLCite , it turns out that LaTeXMLMath as given by Eq .
( LaTeXMLRef ) fails to be single-valued at the south pole ( LaTeXMLMath ) .
One can alternatively change the sign of LaTeXMLMath on the right-hand side of ( LaTeXMLRef ) , in which case LaTeXMLMath becomes single-valued for all values of LaTeXMLMath and LaTeXMLMath except for LaTeXMLMath , i.e. , the north pole .
The calculation of Hamiltonian LaTeXMLMath for these choices of LaTeXMLMath and LaTeXMLMath is similar to that of section 4 .
Substituting Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) in Eq .
( LaTeXMLRef ) and using the LaTeXMLMath algebra ( LaTeXMLRef ) and the Backer-Campbell-Hausdorff formula , we find LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
As seen from these equations , the fermionic partner Hamiltonians ( LaTeXMLRef ) to the bosonic Hamiltonian ( LaTeXMLRef ) also belong to the Lie algebra LaTeXMLMath ; they form a three-parameter family of dipole Hamiltonians describing spinning particles in time-dependent magnetic fields .
The solution of the Schrödinger equation for this type of Hamiltonians has been extensively studied in the literature .
A rather comprehensive list of references may be found in LaTeXMLCite .
In view of Eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) , and ( LaTeXMLRef ) , the evolution operator for the Hamiltonian ( LaTeXMLRef ) , for arbitrary choices of functions LaTeXMLMath and LaTeXMLMath , is given by LaTeXMLEquation where LaTeXMLMath .
Moreover , using the supersymmetric nature of our construction , we may construct a set of orthonormal solutions LaTeXMLMath of the Schrödinger equation for this Hamiltonian from those of the constant Hamiltonian ( LaTeXMLRef ) .
In order to compute these solutions , we first note that LaTeXMLMath .
Therefore , in view of Eq .
( LaTeXMLRef ) , LaTeXMLMath .
Furthermore , because LaTeXMLMath commutes with LaTeXMLMath , we may set LaTeXMLEquation where LaTeXMLMath are the well-known orthonormal angular basis vectors satisfying LaTeXMLEquation .
LaTeXMLMath labels the total angular momentum ( spin ) of the particle , and LaTeXMLMath is the magnetic quantum number .
Now , in view of Eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) , the eigenvalues of LaTeXMLMath are given by LaTeXMLEquation .
The solutions of the Schrödinger equation for the Hamiltonian ( LaTeXMLRef ) that are associated with this choice of LaTeXMLMath are the stationary solutions LaTeXMLEquation .
Under the supersymmetry transformation , LaTeXMLMath , with LaTeXMLMath , are mapped to the following solutions of the Schrödinger equation for the Hamiltonian ( LaTeXMLRef ) .
LaTeXMLEquation .
Note that here LaTeXMLMath and we have made use Eq .
( LaTeXMLRef ) and the relations LaTeXMLEquation .
LaTeXMLEquation Next , consider the special case of the Hamiltonians ( LaTeXMLRef ) obtained by choosing LaTeXMLMath =constant and LaTeXMLMath for some LaTeXMLMath , namely LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
These correspond to the dipole Hamiltonians for which the direction of the magnetic field precesses about the LaTeXMLMath -axis and its magnitude is an arbitrary function of time .
The case of the constant magnitude is obtained by setting LaTeXMLMath =constant .
This is the well-known case of a spin in a precessing magnetic field originally studied in Ref .
LaTeXMLCite .
For a more recent treatment see LaTeXMLCite .
In Section 4 , we restricted ourselves to the study of the quadratic invariants ( the invariants that are second order differential operators ) .
This restriction determined the expression for the operator LaTeXMLMath .
A choice of LaTeXMLMath which includes cubic or higher powers of LaTeXMLMath would lead to the fermionic partner Hamiltonians that can not be expressed as a second order differential operator .
An analogue of this restriction for the systems considered in this section is the condition that LaTeXMLMath should belong to the Lie algebra LaTeXMLMath .
Unlike the case of the harmonic oscillators , a violation of this condition does not lead to any serious problem for the spin systems .
For example , if we take LaTeXMLEquation but keep the same choice for LaTeXMLMath , i.e. , ( LaTeXMLRef ) , we are led to a class of exactly solvable fermionic partner Hamiltonians of the form LaTeXMLEquation where LaTeXMLMath is given by Eq .
( LaTeXMLRef ) and LaTeXMLMath is a general quadratic Stark Hamiltonian describing the quadrupole interaction of a spinning particle with the magnetic field LaTeXMLCite .
A straightforward calculation yields LaTeXMLEquation .
LaTeXMLEquation The Hamiltonian ( LaTeXMLRef ) belongs to the class of quadrupole Hamiltonians LaTeXMLEquation whose algebraic and geometric structure has been studied in Refs .
LaTeXMLCite .
In particular , up to trivial addition of a multiple of identity , any quadrupole Hamiltonian may be written in the form LaTeXMLMath where LaTeXMLMath are real parameters and LaTeXMLEquation .
LaTeXMLEquation Furthermore , the commutators LaTeXMLMath generate the group LaTeXMLMath that acts on the set of all quadrupole Hamiltonians , LaTeXMLCite .
LaTeXMLRef ) also arise in the study of the adiabatic evolution of a complex scalar field in a Bianchi type IX background spacetime LaTeXMLCite .
These observations suggest that one may construct supersymmetric dynamical invariants whose bosonic and fermionic components are linear combinations of the generators LaTeXMLMath , i.e. , they belong to the Lie algebra of LaTeXMLMath , i.e. , LaTeXMLMath .
This in turn implies that they may be obtained from constant elements of LaTeXMLMath by LaTeXMLMath rotations , LaTeXMLCite .
Next , observe that both LaTeXMLMath and LaTeXMLMath commute with LaTeXMLMath .
Therefore , in our construction of the partner Hamiltonians for the constant dipole Hamiltonian ( LaTeXMLRef ) , we may take LaTeXMLMath , where LaTeXMLMath is a real-valued function of time .
Now , if we take LaTeXMLEquation for arbitrary functions LaTeXMLMath , we obtain the most general invariant LaTeXMLMath belonging to LaTeXMLMath .
By construction , the corresponding fermionic partner Hamiltonians LaTeXMLMath will constitute a large class of time-dependent exactly solvable Hamiltonians belonging to the Lie algebra LaTeXMLMath .
The explicit calculation of LaTeXMLMath requires an appropriate parameterization of the operator LaTeXMLMath in terms of the coordinates of its parameter space .
In this article we studied supersymmetric dynamical invariants .
For a given time-dependent Hamiltonian LaTeXMLMath , we have constructed a supersymmetric dynamical invariant LaTeXMLMath and an associated partner Hamiltonian LaTeXMLMath such that the bosonic part of LaTeXMLMath is a dynamical invariant for LaTeXMLMath and the fermionic part of LaTeXMLMath is a dynamical invariant for LaTeXMLMath .
We have shown how the solutions of the Schrödinger equation for LaTeXMLMath may be used to obtain solutions of the Schrödinger equation for LaTeXMLMath .
In order to compare our approach with those of Refs .
LaTeXMLCite , we note that we could construct an even supersymmetric invariant of the form ( LaTeXMLRef ) by requiring the supersymmetric charge LaTeXMLMath to be a dynamical invariant .
It is not difficult to show that substituting LaTeXMLMath in the Liouville-von-Neumann equation yields the intertwining relation LaTeXMLEquation for the operator LaTeXMLMath .
Note that this relation is only a sufficient condition for LaTeXMLMath to be a dynamical invariant .
This in turn implies that our method is more general than that of Refs .
LaTeXMLCite .
One way to see this is to substitute Eq .
( LaTeXMLRef ) in Eq .
( LaTeXMLRef ) .
Using Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , one can then reduce Eq .
( LaTeXMLRef ) to LaTeXMLEquation .
It is not difficult to construct operators LaTeXMLMath that commute with LaTeXMLMath but do not satisfy this equation .
1 .
Introduction In the present paper , we consider the eigenvalue problems which concern a differential equation LaTeXMLEquation .
Here LaTeXMLMath is a function on the Siegel domain of type II LaTeXMLEquation and LaTeXMLMath is a differential operator given by LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is an integer .
Let LaTeXMLMath be an arithmetic subgroup or a convex cocompact subgroup , where LaTeXMLEquation and assume that LaTeXMLMath is a LaTeXMLMath -automorphic form of weight LaTeXMLMath in the sense that it is invariant under LaTeXMLEquation for every LaTeXMLMath .
In fact , LaTeXMLMath commutes with ( 1.5 ) .
Now , let us recall some basic facts about complex hyperbolic geometry ( see LaTeXMLCite and LaTeXMLCite ) .
Geometry of complex hyperbolic space LaTeXMLMath is the geometry of the unit ball LaTeXMLMath in LaTeXMLMath with the Kähler structure given by the Bergman metric whose automorphisms are biholomorphic automorphisms of the ball , i.e. , elements of LaTeXMLMath .
Any complex hyperbolic manifold can be represented as the quotient LaTeXMLMath by a discrete torsion free isometric action of the fundamental group of LaTeXMLMath , LaTeXMLMath , its boundary at infinity LaTeXMLMath is naturally identified as the quotient LaTeXMLMath of the discontinuity set of LaTeXMLMath at infinity .
Here the discontinuity set LaTeXMLMath is the maximal subset of LaTeXMLMath where LaTeXMLMath acts discretely , its complement LaTeXMLMath is the limit set of LaTeXMLMath , LaTeXMLMath for any LaTeXMLMath .
In general , let LaTeXMLMath be a connected , linear , real simple Lie group of rank one , LaTeXMLMath be an Iwasawa decomposition of LaTeXMLMath , LaTeXMLMath be the corresponding Iwasawa decomposition of the Lie algebra LaTeXMLMath , and LaTeXMLMath be a minimal parabolic subgroup .
The group LaTeXMLMath acts isometrically on the rank-one symmetric space LaTeXMLMath .
Let LaTeXMLMath be its geodesic boundary .
We regard LaTeXMLMath as a compact manifold with boundary .
By the classification of symmetric spaces with strictly negative sectional curvature , we know that LaTeXMLMath is one of the following spaces : a real hyperbolic space LaTeXMLMath ( LaTeXMLMath ) , a complex hyperbolic space LaTeXMLMath ( LaTeXMLMath ) , a quaternionic hyperbolic space LaTeXMLMath ( LaTeXMLMath ) or the Cayley hyperbolic plane LaTeXMLMath , and LaTeXMLMath is a linear group finitely covering the orientation-preserving isometric group of LaTeXMLMath .
Following LaTeXMLCite , we consider a torsion-free discrete subgroup LaTeXMLMath such that LaTeXMLMath admits a LaTeXMLMath -invariant partition LaTeXMLMath , where LaTeXMLMath is open and LaTeXMLMath acts freely and cocompactly on LaTeXMLMath .
The closed subset LaTeXMLMath is called the limit set of LaTeXMLMath .
The locally symmetric space LaTeXMLMath is a complete Riemannian manifold of infinite volume without cusps .
It can be compactified by adjoining the geodesic boundary LaTeXMLMath .
A subgroup LaTeXMLMath satisfying this assumption is called convex cocompact or geometrically cocompact since it acts cocompactly on the convex hull of the limit set .
The quotient LaTeXMLMath is called a Kleinian manifold .
For LaTeXMLMath , we define LaTeXMLMath by LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath .
Now , we need the following definition ( see LaTeXMLCite ) : Definition .
For any discrete group LaTeXMLMath , the critical exponent LaTeXMLMath is the infimum of all LaTeXMLMath such that LaTeXMLEquation converges for some ( or any ) LaTeXMLMath , where LaTeXMLMath is the Riemannian distance from LaTeXMLMath to LaTeXMLMath .
The critical exponent LaTeXMLMath has been extensively studied ( see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite and LaTeXMLCite ) .
It is known that LaTeXMLMath if LaTeXMLMath is nontrivial .
In fact , if LaTeXMLMath , then LaTeXMLMath is equal to the Hausdorff dimension of the limit set with respect to the natural class of sub-Riemannian metrics on LaTeXMLMath .
In the case of LaTeXMLMath , LaTeXMLMath for LaTeXMLMath .
The theory of Eisenstein series and the spectral theory for Kleinian groups on real hyperbolic spaces LaTeXMLMath has been extensively studied , in particular , by Mandouvalos LaTeXMLCite , LaTeXMLCite , Patterson LaTeXMLCite , and Perry LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
A Kleinian group LaTeXMLMath is a discrete subgroup acting on LaTeXMLMath , which acts discontinuously on LaTeXMLMath and has a fundamental domain in LaTeXMLMath of infinite hyperbolic volume .
This in particular implies that it is a non-arithmetic subgroup .
Their theory extended previous work of Roelcke LaTeXMLCite , Elstrodt LaTeXMLCite , Patterson LaTeXMLCite and Fay LaTeXMLCite on the spectral theory and Eisenstein series for Fuchsian groups of the second kind .
Now , we give the corresponding results on the complex hyperbolic space LaTeXMLMath .
We give the Eisenstein series for the Picard modular group , which is an arithmetic subgroup of LaTeXMLMath .
We also give the Eisenstein series for any discrete subgroup ( either arithmetic or non-arithmetic ) .
In particular , for a convex cocompact subgroup LaTeXMLMath which satisfies that LaTeXMLMath , we give the product formulas .
Now , we state the results of this paper .
The Poisson kernel is given by LaTeXMLEquation where LaTeXMLEquation .
Let LaTeXMLEquation .
It is a point-pair invariant under the action of LaTeXMLMath .
The critical exponent is given by LaTeXMLEquation .
For a discrete subgroup LaTeXMLMath , the Eisenstein series is defined by LaTeXMLEquation .
The automorphic Green function is given by LaTeXMLEquation with LaTeXMLEquation .
Here LaTeXMLEquation .
Now , the first main result of this paper is stated as follows : Theorem 1.1 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following product formula holds : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath , set LaTeXMLEquation .
The S-matrix is given by LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
By the S-matrix , we give the functional equation of Eisenstein series : Theorem 1.2 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following functional equation for Eisenstein series holds : LaTeXMLEquation for LaTeXMLMath .
The Poisson kernel of weight LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
The Eisenstein series of weight LaTeXMLMath is defined by LaTeXMLEquation .
Set LaTeXMLEquation .
LaTeXMLEquation The automorphic Green function of weight LaTeXMLMath is given by LaTeXMLEquation .
Now , we state the main theorem of this paper : Theorem 1.3 ( Main Theorem ) .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following product formula holds : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath or LaTeXMLMath .
The S-matrix of weight LaTeXMLMath is defined by LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
By the S-matrix of weight LaTeXMLMath , we obtain the functional equation of Eisenstein series of weight LaTeXMLMath .
Theorem 1.4 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following functional equation for Eisenstein series of weight LaTeXMLMath holds : LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
In fact , when LaTeXMLMath , Theorem 1.3 and Theorem 1.4 reduces to Theorem 1.1 and Theorem 1.2 .
In the appendix , we give the product formulas on LaTeXMLMath .
Theorem 1.5 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following product formula on LaTeXMLMath holds : LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
2 .
Epstein zeta function and Eisenstein series on Picard modular groups Let LaTeXMLMath be the Siegel domain LaTeXMLEquation .
The imaginary quadratic field LaTeXMLMath is called the field of Eisenstein numbers .
Its ring of integers LaTeXMLEquation with LaTeXMLMath is called the ring of Eisenstein integers .
In their paper LaTeXMLCite , Korányi and Wolf studied the general Cayley transform which carries the bounded domain of the Harish-Chandra realization into a generalized half-plane .
In LaTeXMLCite , Korányi and Reimann gave the Cayley transform on LaTeXMLMath .
Now , we define the Cayley transform LaTeXMLMath , where LaTeXMLEquation .
It is known that LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath matrix with a LaTeXMLMath at the LaTeXMLMath -th component , zeros elsewhere .
Using Kronecker ’ s delta , we have LaTeXMLEquation .
A basis of LaTeXMLMath is as follows : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation The action of LaTeXMLMath on LaTeXMLMath is as follows : LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation So , the corresponding operator is LaTeXMLEquation .
Similarly , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We denote LaTeXMLMath as the differential operator corresponding to LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Now , if LaTeXMLMath is any positive integer , we claim that LaTeXMLEquation lies in the center of LaTeXMLMath .
To prove this , without loss of generality , let us assume LaTeXMLMath .
Then LaTeXMLEquation .
By LaTeXMLMath , this equals LaTeXMLEquation .
LaTeXMLEquation Thus the above sum commutes with the generators LaTeXMLMath of LaTeXMLMath , and lies in the center of LaTeXMLMath .
The proof is similar if LaTeXMLMath .
Set LaTeXMLEquation .
LaTeXMLEquation In the category of differential operators , LaTeXMLEquation .
Therefore , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation is the Laplace-Beltrami operator of LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath is an arithmetic subgroup of LaTeXMLMath and LaTeXMLMath is called a Picard modular surface ( see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite and LaTeXMLCite for the details ) .
The complex hyperbolic Gauss-Bonnet formula states that ( see LaTeXMLCite and LaTeXMLCite ) LaTeXMLEquation where LaTeXMLMath is the Euler number of LaTeXMLMath .
Holzapfel showed that LaTeXMLEquation .
As LaTeXMLMath has index LaTeXMLMath in LaTeXMLMath , we see that LaTeXMLMath , and so LaTeXMLEquation .
Denote LaTeXMLMath , then LaTeXMLMath .
It follows that LaTeXMLEquation .
This is the other realization of LaTeXMLMath .
For simplicity , from now on , we use the same symbol LaTeXMLMath to denote as LaTeXMLMath .
In fact , LaTeXMLMath .
Let LaTeXMLMath be the parabolic subgroup of LaTeXMLMath which consists of upper triangular matrices .
This group LaTeXMLMath has the Langlands decomposition LaTeXMLMath , where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Then LaTeXMLMath is the Iwasawa decomposition .
Set LaTeXMLMath , then LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
Write explicitly , LaTeXMLEquation .
We have LaTeXMLMath .
Note that LaTeXMLMath .
Therefore , LaTeXMLMath if and only if LaTeXMLMath .
Define LaTeXMLMath by LaTeXMLMath .
It induces a bijection LaTeXMLMath .
For LaTeXMLMath , LaTeXMLEquation .
Thus , if we set LaTeXMLMath , then LaTeXMLMath .
Following LaTeXMLCite , let LaTeXMLMath and LaTeXMLMath be the positive integer such that LaTeXMLMath generates LaTeXMLMath .
For LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath for LaTeXMLMath , LaTeXMLMath .
Hence , LaTeXMLMath .
By Cayley transform , we have Theorem 2.1 .
The Laplace-Beltrami operator of LaTeXMLMath on LaTeXMLMath is LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath .
Denote LaTeXMLMath , set LaTeXMLMath , we have LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath .
By a straightforward calculation , we have LaTeXMLEquation .
For the Picard modular group LaTeXMLMath , the Eisenstein series is defined as LaTeXMLEquation .
The unit group of LaTeXMLMath is LaTeXMLMath , its order is LaTeXMLMath .
LaTeXMLEquation where LaTeXMLMath denotes the ideal generated by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Theorem 2.2 .
The Eisenstein series satisfies the following equation : LaTeXMLEquation .
Proof .
For LaTeXMLMath , LaTeXMLEquation .
LaTeXMLMath Let LaTeXMLEquation .
LaTeXMLMath and denote LaTeXMLMath , for LaTeXMLMath .
For LaTeXMLMath as above , LaTeXMLEquation .
We can identify LaTeXMLMath with LaTeXMLMath by LaTeXMLMath , where LaTeXMLEquation .
Lemma 2.3 .
If LaTeXMLMath , where LaTeXMLMath , then LaTeXMLEquation .
Proof .
In fact , LaTeXMLEquation .
LaTeXMLMath Definition 2.4 .
For the Picard modular group LaTeXMLMath the Epstein zeta function is given by LaTeXMLEquation where LaTeXMLMath Lemma 2.5 .
LaTeXMLMath , where LaTeXMLMath is the Dedekind zeta function of the number field LaTeXMLMath .
Because LaTeXMLMath , any zero of LaTeXMLMath will give a pole of LaTeXMLMath unless LaTeXMLMath vanishes at that value of LaTeXMLMath .
The trivial zeros of LaTeXMLMath and LaTeXMLMath are both LaTeXMLMath , and both have order LaTeXMLMath .
So the trivial zeros cancel out in LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath as above , then LaTeXMLMath , thus LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath , thus LaTeXMLMath .
Suppose LaTeXMLMath , we have the following lemma : Lemma 2.6 .
For LaTeXMLMath , LaTeXMLMath .
Proof .
In fact , LaTeXMLEquation .
LaTeXMLMath By Lemma 2.6 , it follows that Theorem 2.7 .
The functional equation of Epstein zeta functions : LaTeXMLEquation if LaTeXMLMath .
Following LaTeXMLCite , we define the Picard modular forms on LaTeXMLMath as follows : Definition 2.8 .
A holomorphic function LaTeXMLMath is a Picard modular form of LaTeXMLMath and of weight LaTeXMLMath , if it satisfies : LaTeXMLEquation where LaTeXMLMath is the Jacobi determinant of LaTeXMLMath at LaTeXMLMath .
Now , for the Picard modular group LaTeXMLMath , we define the Eisenstein series of weight LaTeXMLMath : LaTeXMLEquation .
In fact , in LaTeXMLCite and LaTeXMLCite , the authors studied the Poincaré-Eisenstein series .
The following proposition is important .
Proposition 2.9 .
LaTeXMLMath satisfies the following identity : LaTeXMLEquation .
Proof .
For LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
Thus , LaTeXMLEquation .
On the other hand .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
So , LaTeXMLEquation .
LaTeXMLMath 3 .
Product formulas on LaTeXMLMath In his paper LaTeXMLCite , the Poisson kernel LaTeXMLMath of LaTeXMLMath is defined by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is the distinguished boundary of LaTeXMLMath : LaTeXMLEquation .
The Szegö kernel function LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLEquation for LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
Now , we define the Poisson kernel as follows : LaTeXMLEquation where LaTeXMLEquation .
In fact , for LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLMath if and only if LaTeXMLMath .
In their papers LaTeXMLCite , LaTeXMLCite , Korányi and Wolf studied the realization of Siegel domain of type II and the Poisson integral .
Our definition has a little difference from theirs .
We have LaTeXMLEquation .
In fact , by Helgason ’ s conjecture , which was proved by Kashiwara et al .
in LaTeXMLCite , that the eigenfunctions on Riemannian symmetric spaces can be represented as Poisson integrals of their hyperfunction boundary values .
The boundary of LaTeXMLMath : LaTeXMLEquation .
In the homogeneous coordinates , one has LaTeXMLEquation .
In particular , LaTeXMLEquation .
The Green function associated to LaTeXMLMath on LaTeXMLMath is defined as LaTeXMLEquation .
It satisfies that LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath , i.e. , LaTeXMLEquation .
By a straightforward calculation , we have LaTeXMLEquation .
Hence , LaTeXMLEquation .
Therefore , LaTeXMLEquation .
By Proposition 2.9 , we have LaTeXMLEquation .
The point-pair invariant on LaTeXMLMath is defined by LaTeXMLEquation which is invariant under the action of LaTeXMLMath .
The critical exponent is given by LaTeXMLEquation .
The number LaTeXMLMath does not depend on the choice of LaTeXMLMath , LaTeXMLMath .
It is an invariant of the group LaTeXMLMath .
The Eisenstein series is given by LaTeXMLEquation where LaTeXMLMath is a discrete subgroup of LaTeXMLMath .
We have the following lemma : Lemma 3.1 .
The properties of Eisenstein series : We have the following lemma : Lemma 3.2 .
Let LaTeXMLMath with LaTeXMLMath , then LaTeXMLEquation .
By LaTeXMLMath , we have LaTeXMLEquation .
A solution is LaTeXMLEquation .
Here LaTeXMLMath is the Gauss hypergeometric function .
Let LaTeXMLEquation where LaTeXMLMath .
By the integral representation of hypergeometric functions , we can only consider the series LaTeXMLEquation which is convergent for LaTeXMLMath .
Thus , the right hand side of ( 3.14 ) is convergent when LaTeXMLMath .
We have LaTeXMLEquation where LaTeXMLMath as LaTeXMLMath .
Consequently , LaTeXMLEquation .
In the real hyperbolic geometry , the double transitivity of the action of the isometric groups on the real hyperbolic space has been studied extensively ( see LaTeXMLCite for three dimensional hyperbolic space ) .
Now , we give the corresponding results on the complex hyperbolic space .
Proposition 3.3 .
The group LaTeXMLMath acts in the following sense doubly transitively on LaTeXMLMath : For all LaTeXMLMath , such that LaTeXMLMath , there exists an element LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath .
Proof .
Set LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath .
LaTeXMLEquation .
Moreover , LaTeXMLMath , i.e .
LaTeXMLMath maps LaTeXMLMath onto LaTeXMLMath .
Set LaTeXMLMath , then LaTeXMLEquation i.e. , LaTeXMLMath has the form of LaTeXMLMath .
If LaTeXMLMath , where LaTeXMLMath , then we obtain two complex equations about LaTeXMLMath and LaTeXMLMath .
In fact , the number of real linear equations is four .
While , the real dimension of LaTeXMLMath is five .
Hence , the existence of LaTeXMLMath is proved .
Applying a suitable element LaTeXMLMath of the stabilizer LaTeXMLMath of LaTeXMLMath in LaTeXMLMath to LaTeXMLMath we get a transform LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath with LaTeXMLMath .
Since the isometric group of LaTeXMLMath is LaTeXMLMath , we have LaTeXMLEquation .
So LaTeXMLMath .
Similarly , there exists an element LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath with the same LaTeXMLMath , since LaTeXMLMath .
Thus , LaTeXMLMath has the desired properties .
LaTeXMLMath In LaTeXMLCite , the concept of real hyperbolic distance was introduced and the basic properties was studied .
Now , we give the complex hyperbolic distance and obtain the triangle inequality for our function LaTeXMLMath .
Proposition 3.4 .
The complex hyperbolic distance LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath is defined by LaTeXMLEquation and LaTeXMLMath , LaTeXMLMath .
Proof .
If LaTeXMLMath and LaTeXMLMath LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Thus the proposition is true in the special case LaTeXMLMath , LaTeXMLMath .
Note that LaTeXMLMath is a point-pair invariant .
Since LaTeXMLMath acts doubly transitively on LaTeXMLMath , there exists for all LaTeXMLMath an element LaTeXMLMath , such that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Therefore , by point-pair invariance , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath Proposition 3.5 .
For LaTeXMLMath , the following formulas hold : Proof .
It is obtained by a straightforward calculation .
LaTeXMLMath Proposition 3.6 .
The triangle inequality for the metric function LaTeXMLMath on the complex hyperbolic space : For all LaTeXMLMath , the following inequality holds : LaTeXMLEquation .
Proof .
Since LaTeXMLMath is a point-pair invariant , it is sufficient to prove ( 3.19 ) only in the special case LaTeXMLMath .
Set LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation and LaTeXMLEquation .
Note that LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
By LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLMath .
Interchanging LaTeXMLMath and LaTeXMLMath , we obtain the other part of the inequality .
LaTeXMLMath Theorem 3.7 .
Let LaTeXMLEquation .
Then one has the following formula : LaTeXMLEquation where LaTeXMLMath is the Lebesque measure on LaTeXMLMath , LaTeXMLMath is the point-pair invariant , and LaTeXMLMath is equal to the first term on the right hand side with LaTeXMLMath replaced by LaTeXMLMath .
Proof .
LaTeXMLMath is a point-pair invariant .
Thus , it suffices to calculate it for LaTeXMLMath and LaTeXMLMath , i.e. , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath stand for LaTeXMLMath and LaTeXMLMath , respectively .
Now , LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath .
Hence , LaTeXMLEquation .
By transform LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
Set LaTeXMLMath , then LaTeXMLEquation where LaTeXMLMath .
By ( see LaTeXMLCite , p.115 , ( 5 ) ) LaTeXMLEquation for LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
Now , we need the following two formulas : ( 1 ) LaTeXMLMath , LaTeXMLMath .
By ( see LaTeXMLCite , p.105 , 2.9 .
( 3 ) ) LaTeXMLEquation one has LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLMath .
By ( see LaTeXMLCite , p.108 , 2.10 .
( 1 ) ) LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , one has LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLMath , where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath implies that LaTeXMLMath .
Thus , LaTeXMLMath and LaTeXMLMath .
Without loss of generality , we can assume that LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Here we use the following formula : LaTeXMLEquation .
Similarly , LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath .
Set LaTeXMLMath , then LaTeXMLMath .
By ( see LaTeXMLCite , p.111 , 2.11 .
( 5 ) ) LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLEquation .
Moreover , we need the following lemma .
Lemma 3.8 .
The following formula about hypergeometric functions holds : LaTeXMLEquation where LaTeXMLMath denotes LaTeXMLMath , LaTeXMLMath stands for LaTeXMLMath and LaTeXMLMath stands for LaTeXMLMath .
Proof .
It is obtained by the integral representation of hypergeometric functions .
LaTeXMLMath By Lemma 3.8 and the following formula ( see LaTeXMLCite , p.103 , ( 35 ) ) : LaTeXMLEquation we get LaTeXMLEquation .
LaTeXMLEquation Thus , we have LaTeXMLEquation .
Substituting this identity to the expression of LaTeXMLMath , We get the desired results .
LaTeXMLMath Consequently , we have Corollary 3.9 .
The following formula holds : LaTeXMLEquation where LaTeXMLEquation .
Corollary 3.10 .
The functional equation for the trivial group on LaTeXMLMath : If LaTeXMLMath , then the following identity holds : LaTeXMLEquation where LaTeXMLMath .
Proof .
The integral on the left-hand side is absolutely convergent .
We set LaTeXMLEquation in the formula of Theorem 3.7 .
Next , we multiply both sides of the formula by LaTeXMLMath and take the limit as LaTeXMLMath .
Thus we get the desired results .
LaTeXMLMath Proposition 3.11 .
The integral formula for the Poisson kernel : LaTeXMLEquation .
Proof .
Denote the left hand side of ( 3.24 ) as LaTeXMLMath , then LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By the well known formula ( see LaTeXMLCite , p.15 ) : LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLMath We define the S-matrix as follows : LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
Theorem 3.12 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following functional equation for Eisenstein series holds : LaTeXMLEquation for LaTeXMLMath .
Proof .
By Corollary 3.10 , LaTeXMLEquation .
On the other hand , LaTeXMLEquation .
LaTeXMLEquation Hence , the left hand side of ( 3.27 ) : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLMath Proposition 3.13 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following product formula holds : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
For LaTeXMLMath , similar as in LaTeXMLCite and LaTeXMLCite , we set LaTeXMLEquation .
If LaTeXMLMath , we define the Eisenstein integral : LaTeXMLEquation .
LaTeXMLEquation and the scattering operator : LaTeXMLEquation where LaTeXMLMath .
Now , the functional equation for Eisenstein series can be written in terms of Eisenstein integrals as LaTeXMLEquation and in terms involving the scattering operator as : LaTeXMLEquation 4 .
Product formulas of weight LaTeXMLMath on LaTeXMLMath In the case of LaTeXMLMath , many people , especially Maass LaTeXMLCite , Selberg LaTeXMLCite , Roelcke LaTeXMLCite , Fay LaTeXMLCite , Elstrodt LaTeXMLCite , Hejhal LaTeXMLCite , and Shimura LaTeXMLCite studied the eigenvalue problems which concern a homogeneous equation LaTeXMLEquation .
Here LaTeXMLMath is a function on the complex upper plane LaTeXMLEquation and LaTeXMLMath is a differential operator given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is an integer .
Let LaTeXMLMath be a congruence subgroup of LaTeXMLMath and assume that LaTeXMLMath is a LaTeXMLMath -automorphic form of weight LaTeXMLMath in the sense that it is invariant under LaTeXMLEquation for any LaTeXMLMath .
In fact , LaTeXMLMath commutes with the above map .
One of the most remarkable facts about such LaTeXMLMath and LaTeXMLMath is the existence of Selberg zeta function ( see LaTeXMLCite ) , whose set of zeros coincides essentially with the set LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath occurs as an eigenvalue of LaTeXMLMath as in the above equation with a cusp form LaTeXMLMath ( see LaTeXMLCite ) .
Now , we study the corresponding problems in the case of LaTeXMLMath .
Proposition 4.1 .
Let LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation where LaTeXMLMath is the Picard modular group .
Then the following formula holds : LaTeXMLEquation .
Proof .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Hence , LaTeXMLMath .
LaTeXMLMath Set LaTeXMLEquation with LaTeXMLMath .
In particular , when LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Proposition 4.2 .
The operator LaTeXMLMath has the following automorphic property : LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is a real analytic function .
In other words , LaTeXMLMath commutes with the following map LaTeXMLEquation .
Proof .
For LaTeXMLMath , one has LaTeXMLMath , i.e. , LaTeXMLEquation .
Denote LaTeXMLEquation .
Set LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We have LaTeXMLEquation .
LaTeXMLEquation while LaTeXMLMath .
On the other hand , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLEquation .
Now , we will prove that LaTeXMLMath .
Note that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
On the other hand , LaTeXMLEquation .
Now , we need the following six identities : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By these identities , we get the desired formula .
Similarly , we have LaTeXMLEquation and LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath For the Picard modular group LaTeXMLMath , the Eisenstein series of weight LaTeXMLMath is defined by LaTeXMLEquation .
Theorem 4.3 .
The Eisenstein series satisfies the following equation : LaTeXMLEquation .
Proof .
First , we note that LaTeXMLEquation .
By Proposition 4.2 , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath Set LaTeXMLEquation .
Then LaTeXMLMath .
A generalized automorphic form on the Picard modular group LaTeXMLMath of weight LaTeXMLMath is given by LaTeXMLEquation .
Put LaTeXMLEquation .
Definition 4.4 .
For any function LaTeXMLMath , the associated real point-pair invariant is given by LaTeXMLEquation where LaTeXMLMath .
Proposition 4.5 .
The following formulas hold : Definition 4.6 .
Let LaTeXMLMath be a real-valued function .
The complex point-pair invariant is given by LaTeXMLEquation where LaTeXMLMath .
Proposition 4.7 .
The following formulas hold : Proof .
Since LaTeXMLMath for LaTeXMLMath .
We have LaTeXMLEquation .
The others are trivial .
LaTeXMLMath Lemma 4.8 .
The following formula holds : LaTeXMLEquation where LaTeXMLMath .
Proof .
It is obtained by a straightforward calculation .
LaTeXMLMath By LaTeXMLMath , one has LaTeXMLEquation .
It is known that the Fuchsian equation with three regular singularities LaTeXMLMath is given by LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath are exponents belonging to LaTeXMLMath , respectively , and , they satisfy that LaTeXMLMath .
The solution of this equation is given in Riemann P-notation by LaTeXMLEquation .
When LaTeXMLMath , it reduces to LaTeXMLEquation .
LaTeXMLEquation Now , in our case , LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Now , we set LaTeXMLEquation .
LaTeXMLEquation It has the following property : LaTeXMLEquation .
Moreover , note that LaTeXMLMath , we have LaTeXMLEquation .
The automorphic Green function of weight LaTeXMLMath is given by LaTeXMLEquation .
By the same method , we can prove that the right hand side of ( 4.18 ) is convergent if LaTeXMLMath .
The automorphic Green function of weight LaTeXMLMath has the following properties : LaTeXMLEquation .
Furthermore , LaTeXMLEquation .
LaTeXMLEquation The Poisson kernel of weight LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
By LaTeXMLEquation we have LaTeXMLEquation .
The Eisenstein series with respect to LaTeXMLMath of weight LaTeXMLMath is defined by LaTeXMLEquation .
Here LaTeXMLMath is a discrete subgroup of LaTeXMLMath .
Now , we have the following formula : LaTeXMLEquation where LaTeXMLMath as LaTeXMLMath .
Consequently , LaTeXMLEquation .
Lemma 4.9 .
The Eisenstein series of weight LaTeXMLMath satisfies the following three equations : Proof .
In fact , LaTeXMLEquation .
Since LaTeXMLMath , LaTeXMLMath .
By Proposition 4.2 , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation To prove the third identity , we note that LaTeXMLEquation .
By Proposition 2.9 , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath Theorem 4.10 .
Set LaTeXMLEquation .
Then the following formula holds : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath or LaTeXMLMath .
Proof .
In fact , if LaTeXMLMath , this is Theorem 3.7 .
Now , we only consider the nontrivial case LaTeXMLMath .
For LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLEquation .
Thus , LaTeXMLMath is a point-pair invariant covariant with respect to the weight LaTeXMLMath .
By the double transitivity of LaTeXMLMath on LaTeXMLMath , there exists a LaTeXMLMath , such that LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath stand for LaTeXMLMath and LaTeXMLMath , respectively .
Set LaTeXMLMath , then LaTeXMLEquation .
Denote LaTeXMLEquation then LaTeXMLMath .
Since LaTeXMLEquation .
LaTeXMLEquation By ( 4.24 ) , we have LaTeXMLEquation .
Hence , LaTeXMLEquation .
LaTeXMLEquation For LaTeXMLMath , i.e. , LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , LaTeXMLMath .
LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Assume that LaTeXMLMath , set LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath , LaTeXMLMath .
We have LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath , then LaTeXMLEquation .
We have LaTeXMLEquation .
It is known that the Tchebichef polynomial LaTeXMLEquation .
Hence LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation We know that LaTeXMLEquation .
Thus LaTeXMLEquation .
LaTeXMLEquation By the following formula LaTeXMLEquation for LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLEquation .
LaTeXMLEquation Note that LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation ( 1 ) LaTeXMLMath , LaTeXMLMath .
By LaTeXMLMath , we have LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLMath .
By LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation It is known that the incomplete beta function is defined by ( see LaTeXMLCite , p.87 ) LaTeXMLEquation .
Set LaTeXMLMath .
Note that LaTeXMLMath .
Hence , LaTeXMLEquation .
LaTeXMLEquation Similarly , LaTeXMLEquation .
LaTeXMLEquation We have LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLEquation .
LaTeXMLEquation ( 4.26 ) is equal to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Similarly , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Now , we need the following lemma : Lemma 4.11 .
The explicit expression of LaTeXMLMath : For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath with LaTeXMLMath , the following formula holds : LaTeXMLEquation .
Proof .
LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLMath By Lemma 4.11 , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation If LaTeXMLMath , then LaTeXMLMath .
( 1 ) LaTeXMLMath , LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLEquation .
On the other hand , when LaTeXMLMath , ( 4.27 ) is equal to LaTeXMLEquation .
Hence , LaTeXMLEquation .
LaTeXMLEquation If LaTeXMLMath , then LaTeXMLMath .
( 1 ) LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLEquation ( 3 ) LaTeXMLMath , LaTeXMLEquation ( 4 ) LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
On the other hand , when LaTeXMLMath , ( 4.27 ) is equal to LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Now , we need the following relation of Gauss between contiguous functions ( see LaTeXMLCite , p.103 , ( 36 ) ) : LaTeXMLEquation where LaTeXMLMath denotes LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath stands for LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath respectively .
We have LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLCite , p.104 , ( 46 ) , LaTeXMLEquation we have LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLCite , p.111 , ( 5 ) , LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath .
Hence , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Similarly , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By Lemma 4.11 , we have LaTeXMLEquation .
Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath .
( 1 ) LaTeXMLMath , LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLEquation .
On the other hand , when LaTeXMLMath , ( 4.28 ) is equal to LaTeXMLEquation .
Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath .
( 1 ) LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation ( 3 ) LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation ( 3 ) LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
On the other hand , when LaTeXMLMath , ( 4.28 ) is equal to LaTeXMLEquation .
Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Now , we need the following relation of Gauss between contiguous functions ( see LaTeXMLCite , p.103 , ( 33 ) ) : LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLCite , p.104 , ( 46 ) , LaTeXMLEquation we have LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By LaTeXMLCite , p.111 , ( 5 ) , LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Now , we need the following formulas : LaTeXMLEquation .
LaTeXMLEquation We have LaTeXMLEquation and LaTeXMLEquation .
Thus , LaTeXMLEquation .
Hence , we have LaTeXMLEquation .
LaTeXMLEquation Similarly , LaTeXMLEquation .
LaTeXMLEquation Therefore , LaTeXMLEquation .
LaTeXMLEquation For LaTeXMLMath , note that LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Similarly , LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
For LaTeXMLMath , if LaTeXMLMath , LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath .
Let us add the terms of LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath , we have LaTeXMLEquation .
Hence , LaTeXMLEquation .
LaTeXMLEquation Thus , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Set LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation At first , we calculate LaTeXMLMath .
It is known that ( see LaTeXMLCite , p.103 , ( 29 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation i.e. , LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation i.e. , LaTeXMLEquation .
In the formula ( see LaTeXMLCite , p.103 , ( 34 ) ) LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Now , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation It is known that LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Thus , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation It is known that ( see LaTeXMLCite , p.103 , ( 37 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
By LaTeXMLCite , p.103 , ( 30 ) , LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation It is known that ( see LaTeXMLCite , p.103 , ( 35 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
Hence , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We know that ( see LaTeXMLCite , p.103 , ( 35 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation For LaTeXMLMath , set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Therefore , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Since ( see LaTeXMLCite , p.103 , ( 38 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It is known that ( see LaTeXMLCite , p.103 , ( 35 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation In the following formula ( see LaTeXMLCite , p.103 , ( 36 ) ) LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Now , the sum of the latter two terms in ( 4.30 ) is equal to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLEquation and LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Therefore , we have LaTeXMLEquation .
Now , we compute LaTeXMLMath .
It is known that ( see LaTeXMLCite , p.103 , ( 28 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
By LaTeXMLCite , p.103 , ( 39 ) , LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We know that ( see LaTeXMLCite , p.103 , ( 30 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation It is known that ( see LaTeXMLCite , p.103 , ( 35 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
Hence , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By LaTeXMLCite , p.103 , ( 39 ) LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It is known that ( see LaTeXMLCite , p.103 , ( 28 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
By LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation So , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We know that ( see LaTeXMLCite , p.103 , ( 38 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation It is known that LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It is known that ( see LaTeXMLCite , p.103 , ( 28 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Note that LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It is known that ( see LaTeXMLCite , p.103 , ( 38 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Note that LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We know that ( see LaTeXMLCite , p.103 , ( 35 ) ) LaTeXMLEquation .
Set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Note that LaTeXMLEquation .
LaTeXMLEquation Therefore , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Note that LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation In the formula ( see LaTeXMLCite , p.103 , ( 36 ) ) LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Now , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Therefore , we have LaTeXMLEquation .
Consequently , LaTeXMLEquation where LaTeXMLMath .
LaTeXMLMath Corollary 4.12 .
The functional equation of Eisenstein series of weight LaTeXMLMath for the trivial group on LaTeXMLMath : If LaTeXMLMath , then the following identity holds : LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath , where LaTeXMLMath .
Proof .
The integral on the left-hand side is absolutely convergent .
We set LaTeXMLEquation in Theorem 4.10 .
Next , we multiply both sides of the formula by LaTeXMLMath and take the limit as LaTeXMLMath .
This completes the proof .
LaTeXMLMath We define the S-matrix of weight LaTeXMLMath as follows : LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
Theorem 4.13 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following functional equation for Eisenstein series of weight LaTeXMLMath holds : LaTeXMLEquation .
LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
Proof .
By Corollary 4.12 , LaTeXMLEquation .
LaTeXMLEquation On the other hand , LaTeXMLEquation .
LaTeXMLEquation By Lemma 4.9 , ( 3 ) , the left hand side of ( 4.34 ) is equal to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath By Theorem 4.10 , we get the main theorem of this paper .
Theorem 4.14 .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following product formula holds : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath or LaTeXMLMath .
Appendix .
Product formulas on LaTeXMLMath In this appendix , by the same method as in the above argument we give the product formulas on LaTeXMLMath .
For simplicity , we omit some details .
Let LaTeXMLEquation .
Set LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
The Poisson kernel of weight LaTeXMLMath is defined as follows : LaTeXMLEquation .
LaTeXMLEquation Here LaTeXMLMath is a discrete subgroup of LaTeXMLMath .
The point-pair invariant LaTeXMLEquation .
In fact , LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLEquation .
By LaTeXMLMath , we have LaTeXMLEquation .
A solution is LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
The automorphic Green function of weight LaTeXMLMath is given by LaTeXMLEquation .
Let LaTeXMLEquation .
Then LaTeXMLMath is a point-pair invariant covariant with respect to the weight LaTeXMLMath .
Put LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By LaTeXMLMath , LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation we have LaTeXMLEquation .
LaTeXMLEquation Set LaTeXMLMath , then LaTeXMLMath .
By the same method as the above argument for LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation ( 1 ) LaTeXMLMath , LaTeXMLEquation ( 2 ) LaTeXMLMath , LaTeXMLEquation .
By LaTeXMLCite , p.103 , ( 28 ) , LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
By LaTeXMLCite , p.103 , ( 36 ) , LaTeXMLEquation set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation .
Consequently , LaTeXMLEquation .
Thus , we have the following theorem : Theorem .
Assume that LaTeXMLMath is convex cocompact and LaTeXMLMath .
Then the following product formula on LaTeXMLMath holds : LaTeXMLEquation for LaTeXMLMath and LaTeXMLMath .
Department of Mathematics , Peking University Beijing 100871 , P. R. China E-mail address : yangleisxx0.math.pku.edu.cn
A rank- LaTeXMLMath tensor on a Lorentzian manifold whose contraction with LaTeXMLMath arbitrary causal future directed vectors is non-negative is said to have the dominant property .
These tensors , up to sign , are called causal tensors , and we determine their general mathematical properties in arbitrary dimension LaTeXMLMath .
Then , we prove that rank-2 tensors which map the null cone on itself are causal tensors .
Previously it has been shown that , to any tensor field LaTeXMLMath on a Lorentzian manifold there is a corresponding “ superenergy ” tensor field LaTeXMLMath ( defined as a quadratic sum over all Hodge duals of LaTeXMLMath ) which always has the dominant property .
Here we prove that , conversely , any symmetric rank-2 tensor with the dominant property can be written in a canonical way as a sum of LaTeXMLMath superenergy tensors of simple forms .
We show that the square of any rank-2 superenergy tensor is proportional to the metric in dimension LaTeXMLMath , and that the square of the superenergy tensor of any simple form is proportional to the metric in arbitrary dimension .
Conversely , we prove in arbitrary dimension that any symmetric rank-2 tensor LaTeXMLMath whose square is proportional to the metric must be a causal tensor and , up to sign , the superenergy of a simple LaTeXMLMath -form , and that the trace of LaTeXMLMath determines the rank LaTeXMLMath of the form .
This generalises , both with respect to the dimension LaTeXMLMath and the rank LaTeXMLMath , the classical algebraic Rainich conditions , which are necessary and sufficient conditions for a metric to originate algebraically in some physical field .
Furthermore , it has the important geometric interpretation that the set of superenergy tensors of simple forms is precisely the set of tensors which leave the null cone invariant and preserve its time orientation .
It also means that all involutory Lorentz transformations can be represented as superenergy tensors of simple forms , and that any rank-2 superenergy tensor is the sum of at most LaTeXMLMath conformally involutory Lorentz transformations .
Non-symmetric null cone preserving maps are shown to have a symmetric part with the dominant property and are classified according to the null eigenvectors of the skew-symmetric part .
We therefore obtain a complete classification of all conformal Lorentz transformations and singular null cone preserving maps on any Lorentzian manifold of any dimension .
The Bel-Robinson tensor LaTeXMLCite , a rank-4 tensor constructed from the Weyl curvature tensor and its dual ( it has only one dual in four dimensions ) , was until some ten years ago not a widely known tensor outside part of the general relativity community .
That it has the dominant property —the contraction with any four causal future directed vectors is non-negative— was certainly known LaTeXMLCite , and many relations to gravitational energy were found ( see e.g .
LaTeXMLCite and references therein ) .
Its precise physical meaning was , and still is , however not clear , and it is possible that no fundamental physical interpretation can be given .
Thus , interest in the Bel-Robinson tensor was limited .
This all changed with the work of Christodoulou and Klainerman on the global non-linear stability of Minkowski spacetime LaTeXMLCite , ( Bel-Robinson estimates were in fact previously considered in the works by Friedrich on hyperbolic formulations of the field equations , see LaTeXMLCite . )
It became clear that the Bel-Robinson tensor is mathematically a very useful quantity , its positivity ( the dominant property ) and divergence properties being the main reasons .
Today , the tensor is established as a key ingredient in many mathematical studies of Einstein ’ s vacuum equations , see e.g .
LaTeXMLCite and references therein .
Considering this rise of interest in the Bel-Robinson tensor , it is remarkable that the Bel tensor seems virtually unknown .
This is the full Riemann curvature tensor analogue of the Bel-Robinson tensor , so it is constucted from the Riemann tensor and its duals LaTeXMLCite .
A fundamental fact is that also the Bel tensor has the dominant property LaTeXMLCite , and it is essentially the only tensor with this property one can construct from the Riemann tensor .
Its divergence can often be controlled if some suitable field equations for the matter are given , and it should therefore be the natural candidate to replace the Bel-Robinson tensor if the full Einstein ’ s equations are studied .
More recently , it was discovered that this way of constructing a tensor with the dominant property from a given tensor and its duals is universal LaTeXMLCite .
Given any tensor field LaTeXMLMath on a Lorentzian manifold of arbitrary dimension , one can always in an essentially unique way construct from LaTeXMLMath a corresponding tensor LaTeXMLMath with the dominant property LaTeXMLCite .
It is perhaps unfortunate that , by historical reasons , LaTeXMLMath has become to be known as the superenergy tensor of LaTeXMLMath , as this terminology may have prevented attention from those studying differential equations on curved manifolds .
Superenergy tensors provide a very natural and geometric way to define norms ( including Sobolev norms ) and inner products ( corresponding to the positive norms ) on Lorentzian manifolds .
Like the Bel-Robinson tensor , there is no need of a physical interpretation of LaTeXMLMath for it to be mathematically useful .
A first example of how the general superenergy tensors can be used in this sense was given in LaTeXMLCite , where causal propagation of fields on Lorenztian manifolds was studied generalising techniques from LaTeXMLCite .
Note that for energy-momentum tensors ( symmetric rank-2 tensors ) the dominant property , first introduced in LaTeXMLCite , is usually called the dominant energy condition LaTeXMLCite .
Such tensors map the future cone on itself , something we refer to as a causal map or causal tensor .
Superenergy tensors have also been used to construct new conserved quantities LaTeXMLCite , and to study the propagation of shock-waves LaTeXMLCite .
In this paper we develop the mathematical structure of tensors having the dominant property and prove some new basic results about superenergy tensors .
We prove that the product LaTeXMLMath of the superenergy tensor LaTeXMLMath of a simple form LaTeXMLMath is always proportional to the metric .
This is further shown to be true for any arbitrary rank-2 superenergy tensor in dimensions LaTeXMLMath .
While any superenergy tensor has the dominant property , we prove that any symmetric rank-2 tensor with the dominant property can be written as a sum of LaTeXMLMath superenergy tensors of simple forms in a canonical way .
We also present some geometrical interpretation of these forms and emphasize that , in this sense , superenergy tensors of simple forms are the basic building blocks of positive or causal quantities .
The classical Rainich conditions LaTeXMLCite , sometimes referred to as RMW ( Rainich-Misner-Wheeler ) theory or already unified theory , are necessary and sufficient conditions in 4 dimensions for an energy-momentum tensor to originate in a Maxwell field .
One may also express this as saying that they are conditions on a metric , which then via the Ricci tensor and Einstein ’ s equations give the energy-momentum tensor .
The algebraic Rainich conditions as stated in LaTeXMLCite are that the energy-momentum tensor is trace-free , satisfies the dominant energy condition , and has a square that is proportional to the metric .
The Rainich conditions have also been generalised to cover some other physical situations ( e.g .
LaTeXMLCite ) .
Here , we prove a much more general result , namely that in LaTeXMLMath dimensions any symmetric rank-2 tensor with a square proportional to the metric must be the superenergy tensor of a simple LaTeXMLMath -form .
We prove that the trace can only have certain discrete values related to the rank LaTeXMLMath of this form .
This result , being an equivalence , has an important geometrical interpretation .
It says that on any Lorentzian manifold of any dimension , the set of superenergy tensors of simple forms is precisely the set of tensors which leave the null cone invariant and preserve its time orientation .
This also leads to an extended algebraic Rainich theory which includes the previously known results as special cases .
It also has the interesting implication that all symmetric ( i.e .
involutory ) Lorentz transformations can be represented as superenergy tensors of simple forms .
Furthermore , the combination with the previous results proves that any superenergy tensor is the sum of at most LaTeXMLMath conformally involutory Lorentz transformations .
We also study non-symmetric null cone preserving maps , which are proven to have a symmetric part with the dominant property , and classify them according to the null eigenvectors of its skew-symmetric part .
All this provides a complete classification of all conformal Lorentz transformations as well as the singular null cone preserving maps in any Lorentzian manifold of arbitrary dimension .
In our notation we sometimes use indices on the tensors .
These indices may be considered as abstract indices in the sense of Penrose and Rindler LaTeXMLCite , and it is clear that all results are geometric and independent of any basis or coordinate system .
We will also use the standard arrows for vectors and boldface characters for 1-forms .
The tensor and exterior products are denoted by LaTeXMLMath and LaTeXMLMath respectively .
As usual , ( square ) round brackets enclosing any set of indices indicate ( anti ) symmetrization .
Equalities by definition are denoted by LaTeXMLMath .
The symbol is used to mark the end of proofs .
We shall use the signature LaTeXMLMath of the metric .
Note that this is the opposite of LaTeXMLCite .
The outline of the paper is as follows .
In section 2 we develop some general mathematical properties of tensors with the dominant property and in section 3 we recall the definition of superenergy tensors and prove certain new results for superenergy tensors of simple forms .
We also extend the previous definition to LaTeXMLMath -forms in LaTeXMLMath dimensions and motivate why their superenergy is essentially the metric .
Various properties of null cone preserving maps , their relation to superenergy tensors , and how these are used to construct any tensor satisfying the dominant energy condition are described in section 4 .
The classification of the ( conformally ) non-involutory Lorentz transformation is then given in section 5 while that of the ( conformally ) involutory ones and the generalised algebraic Rainich conditions with their important geometrical consequences are presented in section 6 .
We assume that we work on an LaTeXMLMath -dimensional manifold LaTeXMLMath endowed with a Lorentzian metric LaTeXMLMath and that a time-orientation has been chosen .
Most of our considerations are algebraic and are implicitly assumed to hold in a point LaTeXMLMath ; of course , they can be straightforwardly translated to tensor fields .
We begin by giving the basic definition .
A tensor LaTeXMLMath is said to have the dominant property if LaTeXMLEquation for any set LaTeXMLMath , … , LaTeXMLMath of causal future-pointing vectors .
The set of tensors with the dominant property will be denoted by LaTeXMLMath .
By LaTeXMLMath we mean the set of tensors LaTeXMLMath such that LaTeXMLMath .
We will see below , in Properties LaTeXMLRef and LaTeXMLRef respectively , that the definition of LaTeXMLMath implies in fact that the strict inequality holds if the future-pointing LaTeXMLMath are all timelike , and that the use of only null vectors LaTeXMLMath is also enough .
By a natural extension , the non-negative real numbers are also considered to have the dominant property : LaTeXMLMath .
Rank-1 tensors with the dominant property are simply the future-pointing causal vectors ( while those in LaTeXMLMath are the past-directed ones ) .
For rank-2 tensors , the dominant property was introduced by Plebański LaTeXMLCite in General Relativity and is usually called the dominant energy condition LaTeXMLCite because it is a requirement for physically acceptable energy-momentum tensors .
The elements of LaTeXMLMath could thus be termed as “ future tensors ” , and those of LaTeXMLMath will be called “ causal tensors ” .
As in the case of past- and future-pointing vectors , any statement concerning LaTeXMLMath has its counterpart concerning LaTeXMLMath , and they will be taken as obvious unless otherwise stated .
The basic properties of tensors in the class LaTeXMLMath are given in what follows .
First of all , the class is closed under linear combinations with non-negative coefficients as well as under tensor products LaTeXMLCite .
If LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) then LaTeXMLMath .
Moreover , if LaTeXMLMath , LaTeXMLMath then LaTeXMLMath .
Proof : This is an immediate consequence of the definition of LaTeXMLMath .
Given any tensor LaTeXMLMath , one can immediately construct many other tensors in LaTeXMLMath by simply permuting the indices , as is obvious from Definition LaTeXMLRef .
Then , we also have ( see Section 5 in LaTeXMLCite ) If LaTeXMLMath , then for any set of non-negative constants LaTeXMLMath the family of tensors LaTeXMLMath belongs to LaTeXMLMath where the sum is over all permutations LaTeXMLMath of LaTeXMLMath .
In particular , any symmetric part of LaTeXMLMath is in LaTeXMLMath .
Proof : Given that LaTeXMLMath for any permutation LaTeXMLMath the first part follows from Property LaTeXMLRef .
Since any symmetric part is in fact a linear combination of such terms with particular positive coefficients LaTeXMLMath the Lemma is proven .
It must be remarked that , sometimes , linear combinations LaTeXMLMath with some negative coefficients LaTeXMLMath may also be in LaTeXMLMath .
On the other hand , we also have If LaTeXMLMath is antisymmetric in any pair of indices , then LaTeXMLMath can not be in LaTeXMLMath .
Proof : Assume , for instance , that LaTeXMLMath .
Then , for any future-pointing LaTeXMLMath , the scalars LaTeXMLMath and LaTeXMLMath have opposite signs .
This implies that a constant sign can not be maintained .
LaTeXMLMath for all future-pointing vectors LaTeXMLMath .
Proof : Again this is trivial from Definition LaTeXMLRef .
Of course , this can be equally proven for the contraction of LaTeXMLMath with any index of LaTeXMLMath .
The previous property can be generalized to show that the class LaTeXMLMath is also closed under tensor products with one contraction applied .
To that end , we introduce the following products for any two tensors LaTeXMLMath and LaTeXMLMath : LaTeXMLEquation where the contraction is taken with the LaTeXMLMath index of the first tensor and the LaTeXMLMath of the second .
There are of course many different products LaTeXMLMath depending on where the contraction is made .
For all LaTeXMLMath and all LaTeXMLMath , if LaTeXMLMath or if LaTeXMLMath , then LaTeXMLMath ; and if LaTeXMLMath and LaTeXMLMath then LaTeXMLMath and LaTeXMLMath .
Proof : If LaTeXMLMath , or if they are in LaTeXMLMath , then , by Property LaTeXMLRef , LaTeXMLEquation .
LaTeXMLEquation are causal vectors with the same time orientation for any set LaTeXMLMath of future-pointing vectors .
Hence LaTeXMLEquation and the first result follows .
The other is similar .
If for some LaTeXMLMath , LaTeXMLMath , then LaTeXMLMath can not be in LaTeXMLMath .
Proof : For if LaTeXMLMath were in either LaTeXMLMath or LaTeXMLMath , by Lemma LaTeXMLRef LaTeXMLMath should be in LaTeXMLMath .
Notice that LaTeXMLMath .
LaTeXMLMath for all LaTeXMLMath .
Proof : The implication from left to right is in Lemma LaTeXMLRef .
The converse can be proved by taking in particular LaTeXMLMath and using Property LaTeXMLRef .
Corollary LaTeXMLRef is the evident generalization of the well-known fact that a causal vector LaTeXMLMath is future-pointing if and only if LaTeXMLMath for all future-pointing vectors LaTeXMLMath .
The concept of “ positivity ” does not capture all that is behind the definition of LaTeXMLMath , and the terminology dominant property ( or dominant energy condition ) is preferable because the pure time component dominates any other component in orthogonal bases .
LaTeXMLMath for all LaTeXMLMath , where LaTeXMLMath are the components of LaTeXMLMath with respect to any orthonormal basis LaTeXMLMath with a future-pointing timelike LaTeXMLMath .
Proof : See Lemma 4.1 in LaTeXMLCite .
If LaTeXMLMath and LaTeXMLMath for a timelike vector LaTeXMLMath , then LaTeXMLMath .
Proof : By choosing the sign LaTeXMLMath we have that LaTeXMLMath is unit and future-pointing .
Thus , by Lemma LaTeXMLRef , all components of LaTeXMLMath vanish in any orthonormal basis including LaTeXMLMath , which means that LaTeXMLMath is the zero tensor .
If LaTeXMLMath and LaTeXMLMath for a timelike vector LaTeXMLMath , then LaTeXMLMath .
Definition LaTeXMLRef involves all causal future-pointing vectors , but in fact the class LaTeXMLMath can be equally characterized by using timelike vectors exclusively , or also only null vectors .
Concerning the timelike case we have : A tensor LaTeXMLMath is in LaTeXMLMath for any set LaTeXMLMath of timelike future-pointing vectors .
Proof : The implication from right to left follows by continuity .
Conversely , first for rank 1 , LaTeXMLMath for LaTeXMLMath would imply that LaTeXMLMath and LaTeXMLMath are parallel and null as this is the only way two causal vectors can be orthogonal .
Thus , if LaTeXMLMath is timelike and LaTeXMLMath then LaTeXMLMath .
Suppose now that the property has been proved for rank- LaTeXMLMath tensors and define LaTeXMLMath .
By Property LaTeXMLRef LaTeXMLMath and hence , if LaTeXMLMath then LaTeXMLMath which in turn , by Corollary LaTeXMLRef , would imply that LaTeXMLMath .
But then Corollary LaTeXMLRef would lead to LaTeXMLMath .
Thus the result follows by induction on LaTeXMLMath .
In order to give the characterization with null vectors we first need a basic result stating that future-pointing null vectors are the basic “ building blocks ” of all future-pointing vectors , i.e .
rank-1 tensors in LaTeXMLMath .
In section 4 we shall generalize this by identifying the analogous building blocks of rank-2 tensors in LaTeXMLMath .
Given a future-pointing timelike vector LaTeXMLMath and a future-pointing null vector LaTeXMLMath , there is another future-pointing null vector LaTeXMLMath such that LaTeXMLMath where LaTeXMLMath .
Proof : See , e.g. , LaTeXMLCite .
We can now show that in order to check that a tensor is in LaTeXMLMath it is sufficient to check it for null vectors .
This is very helpful because obviously it is easier to work with null vectors exclusively rather than with both null and timelike vectors .
LaTeXMLMath for any set LaTeXMLMath , … , LaTeXMLMath of future-pointing null vectors .
Proof : By Lemma LaTeXMLRef , LaTeXMLMath with LaTeXMLMath timelike vectors can be written as a sum with positive coefficients of LaTeXMLMath terms of the type LaTeXMLMath involving null vectors only and the result follows immediately .
Now we can prove a partial but important converse of the Lemma LaTeXMLRef .
LaTeXMLMath for some LaTeXMLMath .
Proof : For any set of timelike future pointing vectors LaTeXMLMath define LaTeXMLMath as in ( LaTeXMLRef ) of Lemma LaTeXMLRef .
Now , if LaTeXMLMath then LaTeXMLMath which implies that LaTeXMLMath is causal , either future- or past-pointing .
To see that the time orientation of these LaTeXMLMath is consistent take any other arbitrary set of timelike future-pointing vectors LaTeXMLMath and define LaTeXMLMath analogously to ( LaTeXMLRef ) .
As LaTeXMLMath and is not zero by assumption , using Property LaTeXMLRef we have that LaTeXMLMath , so that LaTeXMLMath and LaTeXMLMath have the same time orientation .
As all the vectors LaTeXMLMath and LaTeXMLMath are arbitrary and future pointing , this means that LaTeXMLMath or LaTeXMLMath is in LaTeXMLMath .
LaTeXMLMath for some LaTeXMLMath for all LaTeXMLMath .
Proof : If LaTeXMLMath for some LaTeXMLMath then by Proposition LaTeXMLRef LaTeXMLMath with LaTeXMLMath , but then by Lemma LaTeXMLRef LaTeXMLMath for all LaTeXMLMath .
In the last two results , Proposition LaTeXMLRef and Corollary LaTeXMLRef , the special case with LaTeXMLMath , which is in LaTeXMLMath , has been excluded .
Similar results apply for this extreme case , but they need some refinement .
LaTeXMLMath for some LaTeXMLMath there is a null vector LaTeXMLMath such that LaTeXMLMath for some tensor LaTeXMLMath .
Proof : Using the same notation as in Proposition LaTeXMLRef we have that all the LaTeXMLMath are causal and furthermore , as LaTeXMLMath , all of them are orthogonal to each other .
This means that all of them must be proportional to a null vector LaTeXMLMath , and the result follows .
The converse is immediate .
Therefore , it can happen that LaTeXMLMath and therefore is in LaTeXMLMath and yet neither LaTeXMLMath nor LaTeXMLMath is in LaTeXMLMath .
It is enough that LaTeXMLMath .
Nevertheless , we have the following result .
LaTeXMLMath for all LaTeXMLMath for a set of LaTeXMLMath null vectors LaTeXMLMath .
Therefore , LaTeXMLMath for all LaTeXMLMath .
Proof : The first part follows from repeated application of Proposition LaTeXMLRef .
Then , depending on how many of the null vectors LaTeXMLMath are future-pointing , either LaTeXMLMath or LaTeXMLMath is in LaTeXMLMath .
Assume that LaTeXMLMath is completely symmetric .
Then , LaTeXMLMath for some future-pointing null vector LaTeXMLMath .
Proof : If LaTeXMLMath is completely symmetric then all the products LaTeXMLMath are the same .
Thus , from Corollary LaTeXMLRef and the symmetry of LaTeXMLMath the result follows .
Of course , under the assumptions of Corollary LaTeXMLRef , LaTeXMLMath .
On the other hand , we have If LaTeXMLMath is completely antisymmetric and LaTeXMLMath then LaTeXMLMath .
Proof : If LaTeXMLMath is completely antisymmetric again all the products LaTeXMLMath are the same .
Thus , from Corollary LaTeXMLRef and the antisymmetry of LaTeXMLMath the only possibility is LaTeXMLMath .
In the previous section we have defined the set LaTeXMLMath and analyzed its general properties .
However , we must still face the question of how general is the class LaTeXMLMath and how we can build such causal tensors .
Actually this has been already solved and the result is that , given an arbitrary tensor LaTeXMLMath , there is a general procedure to construct its “ positive square ” : a tensor quadratic in LaTeXMLMath and with the dominant property .
This general procedure was introduced in LaTeXMLCite and extensively considered in LaTeXMLCite , and the positive tensors thus constructed receive the generic name of “ super-energy tensors ” ( due to historical reasons LaTeXMLCite ) .
In what follows , we recall here the definition of a general superenergy tensor ( see section 3 of LaTeXMLCite ) .
Let LaTeXMLMath be an arbitrary rank- LaTeXMLMath tensor .
Let LaTeXMLMath denote the set of indices containing LaTeXMLMath and all other indices LaTeXMLMath such that LaTeXMLMath is anti-symmetric in LaTeXMLMath .
The number LaTeXMLMath is the number of indices in LaTeXMLMath .
Then LaTeXMLMath is the next set formed from anti-symmetries with LaTeXMLMath ( or LaTeXMLMath if LaTeXMLMath is already in LaTeXMLMath and so on ) .
Note that LaTeXMLMath for each LaTeXMLMath .
In this way LaTeXMLMath are divided into LaTeXMLMath blocks LaTeXMLMath with LaTeXMLMath .
We can therefore consider LaTeXMLMath as an LaTeXMLMath -fold LaTeXMLMath -form and we write LaTeXMLMath .
There are LaTeXMLMath different ( multiple ) Hodge duals of LaTeXMLMath .
The dual with respect to the block LaTeXMLMath is denoted LaTeXMLMath , with respect to LaTeXMLMath by LaTeXMLMath , the dual with respect to LaTeXMLMath and LaTeXMLMath by LaTeXMLMath and so on .
Note that different duals may be tensors of different rank but all duals are LaTeXMLMath -folded forms .
Denote by LaTeXMLMath , LaTeXMLMath , all the possible duals , where LaTeXMLMath with LaTeXMLMath if there is no dual with respect to the block LaTeXMLMath and LaTeXMLMath if there is a dual with respect to this block ( so LaTeXMLMath and LaTeXMLMath is the LaTeXMLMath -fold form where the dual has been taken with respect to all blocks ) .
In order to define the superenergy tensor of LaTeXMLMath we need a product LaTeXMLMath of an LaTeXMLMath -fold form by itself resulting in a LaTeXMLMath -tensor .
Let LaTeXMLMath be the tensor obtained by permuting the indices in LaTeXMLMath so that the LaTeXMLMath first indices in LaTeXMLMath are precisely the indices in the block LaTeXMLMath , the following LaTeXMLMath indices are the ones in LaTeXMLMath and so on .
Now define the product LaTeXMLMath by LaTeXMLCite .
LaTeXMLEquation .
From each block in LaTeXMLMath two indices are obtained in LaTeXMLMath .
We can form LaTeXMLMath for any LaTeXMLMath but observe that LaTeXMLMath could contain LaTeXMLMath -blocks ( with dual 0-blocks ) for which the expression ( LaTeXMLRef ) has no meaning .
Therefore , assuming LaTeXMLMath for all LaTeXMLMath , we make the following definition .
The superenergy tensor of LaTeXMLMath is defined to be LaTeXMLEquation .
Observe that any dual LaTeXMLMath of the original tensor LaTeXMLMath generates the same superenergy tensor .
We note that LaTeXMLEquation and if LaTeXMLMath is symmetric with respect to two blocks LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath is symmetric with respect to the pairs LaTeXMLMath and LaTeXMLMath .
Important is also the property that LaTeXMLMath if and only if LaTeXMLMath LaTeXMLCite .
Another property of superenergy tensors is : If LaTeXMLMath for some LaTeXMLMath - and LaTeXMLMath -folded forms LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation .
Proof : This follows at once from Definition LaTeXMLRef because the LaTeXMLMath terms and the LaTeXMLMath terms of LaTeXMLMath and LaTeXMLMath , respectively , produce precisely the LaTeXMLMath terms needed .
An r-fold form LaTeXMLMath is said to be decomposable if there are LaTeXMLMath forms LaTeXMLMath ( LaTeXMLMath ) such that LaTeXMLMath .
If LaTeXMLMath is decomposable , then LaTeXMLEquation .
Proof : This is evident from Property LaTeXMLRef and Definition LaTeXMLRef .
The last result shows that rank-2 superenergy tensors may be used as basic set to build up more general superenergy tensors in many occasions .
Actually , we will later be interested ( for other important reasons ) in rank-2 tensors , specially those in LaTeXMLMath .
It is remarkable that , after expanding all duals in the Definition LaTeXMLRef , one obtains an explicit expression for the general superenergy tensor which is independent of the dimension LaTeXMLMath , see LaTeXMLCite .
In the case of a general LaTeXMLMath -form LaTeXMLMath , the rank-2 superenergy tensor becomes LaTeXMLCite LaTeXMLEquation .
Here we have used a notation which will be useful on many occasions : for any two tensors of the same rank LaTeXMLMath and LaTeXMLMath , we write LaTeXMLMath , i.e .
we have contracted over all indices in order .
In the Definition LaTeXMLRef we assumed that there were no LaTeXMLMath -blocks .
The expression ( LaTeXMLRef ) however is perfectly well defined for an LaTeXMLMath -form .
If LaTeXMLMath where LaTeXMLMath is the canonical volume form and LaTeXMLMath a scalar , then ( LaTeXMLRef ) gives LaTeXMLEquation .
If we combine ( LaTeXMLRef ) with Property LaTeXMLRef the Definition LaTeXMLRef is naturally extended to include LaTeXMLMath -blocks : The superenergy tensor of LaTeXMLMath , with LaTeXMLMath is defined to be LaTeXMLEquation .
This definition is to be understood recursively , if there are LaTeXMLMath LaTeXMLMath -blocks one continues until a tensor LaTeXMLMath given by Definition LaTeXMLRef is obtained .
We note that the tensor obtained by taking the dual of LaTeXMLMath with respect to the LaTeXMLMath -block to get a 0-block does not have the same superenergy tensor as LaTeXMLMath has , the difference being the LaTeXMLMath .
This is a special situation only for LaTeXMLMath -blocks and it is the price one has to pay to extend the definition .
The advantages however will be seen in a more consistent presentation of several definitions and results , the first being the following definition .
The set LaTeXMLMath is the set of all superenergy tensors according to Definitions LaTeXMLRef and LaTeXMLRef .
By LaTeXMLMath we denote the set of tensors such that LaTeXMLMath .
The sets LaTeXMLMath and LaTeXMLMath will denote the classes of rank-n tensors in LaTeXMLMath and LaTeXMLMath , respectively .
The metric is an essential element in this set .
In LaTeXMLCite it was shown that the metric LaTeXMLMath is not a superenergy tensor of any LaTeXMLMath -form LaTeXMLMath with LaTeXMLMath so without the extended definition elements of the form LaTeXMLMath would have to be added artificially to LaTeXMLMath .
A fundamental result is that superenergy tensors always have the dominant property .
LaTeXMLMath .
Proof : The first proof for 4 dimensions was given in LaTeXMLCite and used spinors .
In arbitrary dimension the first proof is in LaTeXMLCite while a proof that uses Clifford algebras and which is also valid in arbitrary dimension was presented in LaTeXMLCite .
Of course , LaTeXMLMath .
It is important to remark that the superenergy tensor LaTeXMLMath and its derived tensors by permutation of indices are the only ( up to linear combinations ) tensors quadratic in LaTeXMLMath and with the dominant property LaTeXMLCite .
Therefore , there is a unique ( up to a proportionality factor ) completely symmetric tensor in LaTeXMLMath which is quadratic in LaTeXMLMath , and this is simply LaTeXMLMath LaTeXMLCite .
In LaTeXMLMath , the superenergy tensor of a 2-form LaTeXMLMath is its Maxwell energy-momentum tensor , and the superenergy tensor of an exact 1-form LaTeXMLMath has the form of the energy-momentum tensor for a massless scalar field LaTeXMLMath .
If we compute the superenergy tensor of the Riemann tensor , which is a double symmetrical ( 2,2 ) -form , we get the so-called Bel tensor LaTeXMLCite .
The superenergy tensor of the Weyl curvature tensor is the well-known Bel-Robinson tensor LaTeXMLCite .
For these and other interesting physical examples see LaTeXMLCite .
The dominant property of the Bel-Robinson tensor was used by Christodoulou and Klainerman LaTeXMLCite in their study of the global stability of Minkowski spacetime , and in LaTeXMLCite to study the causal propagation of gravity in vacuum .
The dominant property of more general superenergy tensors was used in LaTeXMLCite to find criteria for the causal propagation of fields on Lorentzian spacetimes of LaTeXMLMath dimensions .
In order to study relations between LaTeXMLMath and LaTeXMLMath , and to see how LaTeXMLMath builds up LaTeXMLMath , we prove now some results for rank-2 tensors .
First , we need a very simple Lemma to fix the notation .
For any LaTeXMLMath we have that LaTeXMLMath and LaTeXMLMath are symmetric .
Furthermore , if LaTeXMLMath and LaTeXMLMath are symmetric , then LaTeXMLEquation if LaTeXMLMath and LaTeXMLMath are antisymmetric , then LaTeXMLEquation finally in the mixed case LaTeXMLEquation so that in the three cases the simple LaTeXMLMath -notation will be used .
Proof : First , LaTeXMLMath which is obviously symmetric in LaTeXMLMath , and analogously for LaTeXMLMath .
If LaTeXMLMath are symmetric then LaTeXMLMath and similarly for the other cases .
If LaTeXMLMath is a 2-form , then LaTeXMLMath .
Proof : If LaTeXMLMath were in LaTeXMLMath and non-zero , then from Proposition LaTeXMLRef LaTeXMLMath should be in LaTeXMLMath , which is impossible due to Lemma LaTeXMLRef .
If LaTeXMLMath , then from Corollary LaTeXMLRef it follows that LaTeXMLMath .
Notice that , still , LaTeXMLMath can certainly be in LaTeXMLMath .
In dimension LaTeXMLMath , LaTeXMLMath .
On the other hand , if LaTeXMLMath there exist tensors LaTeXMLMath such that LaTeXMLMath is not proportional to the metric .
Proof : If LaTeXMLMath then LaTeXMLMath so the property is trivial .
As the superenergy of a LaTeXMLMath -form is the same as the superenergy of its dual LaTeXMLMath -form , we just have to confirm the proposition for 1-forms for LaTeXMLMath , and for 1-forms and 2-forms for LaTeXMLMath .
By ( LaTeXMLRef ) , the superenergy tensor of a 1-form LaTeXMLMath in any dimension LaTeXMLMath is LaTeXMLEquation and this gives LaTeXMLEquation so LaTeXMLMath is proportional to the metric in any dimension .
For a 2-form LaTeXMLMath , the superenergy tensor reads LaTeXMLEquation which again , by ( LaTeXMLRef ) , holds in any LaTeXMLMath .
Now , if LaTeXMLMath , and only in this case , a very well-known result is ( see , e.g. , LaTeXMLCite ) LaTeXMLEquation where LaTeXMLMath is the 2-form dual to LaTeXMLMath in 4 dimensions .
This is the basis of the Rainich theory LaTeXMLCite and , as was pointed out by Lovelock LaTeXMLCite , formula ( LaTeXMLRef ) is an explicit example of a dimensionally-dependent identity , being valid only in LaTeXMLMath .
Not even by changing the proportionality factor on the righthand side the above expression ( LaTeXMLRef ) holds in LaTeXMLMath .
To check it , we can construct explicit counterexamples .
Let LaTeXMLMath be an orthonormal basis and let LaTeXMLMath .
Then the computation of ( LaTeXMLRef ) gives LaTeXMLMath from where one immediately obtains LaTeXMLMath which is ( proportional to ) the metric in 4 but not higher dimension .
Thus , for LaTeXMLMath , LaTeXMLMath is the set of tensors with the property that LaTeXMLMath is porportional to the metric , but this is not true for LaTeXMLMath .
The natural question arises of which super-energy tensors satisfy this property in arbitrary LaTeXMLMath .
This is going to be answered now , and in a more complete way in the next section .
The generalization of the algebraic Rainich condition ( LaTeXMLRef ) will be dealt with in the last section .
Recall that a LaTeXMLMath -form LaTeXMLMath is called simple LaTeXMLCite if it is a product of LaTeXMLMath linearly independent 1-forms LaTeXMLMath , i.e .
LaTeXMLMath .
By standard techniques , the set LaTeXMLMath can be chosen to be orthogonal by simply taking the appropriate linear combinations LaTeXMLMath with LaTeXMLMath , because LaTeXMLMath .
A LaTeXMLMath -form LaTeXMLMath is simple if and only if LaTeXMLMath is simple , and if and only if LaTeXMLMath , see e.g .
LaTeXMLCite .
We denote by LaTeXMLMath the set of superenergy tensors of simple LaTeXMLMath -forms .
Observe that LaTeXMLMath .
We define LaTeXMLMath as usual .
LaTeXMLMath .
Proof : From the proof of Proposition LaTeXMLRef the result is already proved for LaTeXMLMath and for the superenergy tensor LaTeXMLMath of any 1-form .
Using ( LaTeXMLRef ) for the superenergy tensor of a general LaTeXMLMath -form LaTeXMLMath , a straightforward calculation gives LaTeXMLEquation .
LaTeXMLEquation Now , in the first term LaTeXMLMath .
By LaTeXMLCite ( p.23 ) ( or LaTeXMLCite ( p.165 ) ) we have LaTeXMLMath if and only if LaTeXMLMath is simple .
Thus , in this case we are left with LaTeXMLEquation which proves the proposition .
If LaTeXMLMath is a simple LaTeXMLMath -form , then LaTeXMLMath are eigenvectors , all with the same eigenvalue LaTeXMLMath , of its superenergy tensor LaTeXMLMath .
Proof : The case LaTeXMLMath is trivial so assume LaTeXMLMath .
The dual LaTeXMLMath of LaTeXMLMath is obviously orthogonal to any of the LaTeXMLMath .
But the superenergy tensors LaTeXMLMath and LaTeXMLMath are identical , so that using the explicit expression ( LaTeXMLRef ) for LaTeXMLMath and contracting with LaTeXMLMath for LaTeXMLMath we get LaTeXMLEquation for any LaTeXMLMath .
Recall finally that a LaTeXMLMath -form LaTeXMLMath is called null if it is simple and LaTeXMLMath .
Then , LaTeXMLMath defines canonically a null direction LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Equivalently , LaTeXMLMath can be written in the form LaTeXMLMath where the LaTeXMLMath 1-forms LaTeXMLMath are mutually orthogonal and orthogonal to LaTeXMLMath .
We denote by LaTeXMLMath the set of superenergy tensors of null LaTeXMLMath -forms .
Obviously , LaTeXMLMath .
LaTeXMLMath is defined as usual .
LaTeXMLMath and LaTeXMLMath where LaTeXMLMath is null and LaTeXMLMath .
Proof : As LaTeXMLMath for a null LaTeXMLMath -form LaTeXMLMath , so that LaTeXMLMath , from ( LaTeXMLRef ) we get LaTeXMLMath .
Furthermore , as LaTeXMLMath where LaTeXMLMath is its canonical null direction , a simple calculation produces LaTeXMLMath where LaTeXMLMath which is positive .
LaTeXMLMath has N independent eigenvectors , p of them with eigenvalue LaTeXMLMath and ( N-p ) of them with the opposite eigenvalue .
LaTeXMLMath has N-1 independent eigenvectors , all with zero eigenvalues .
It has a unique null eigenvector .
Proof : Again LaTeXMLMath is trivial so assume LaTeXMLMath .
As LaTeXMLMath is simple if LaTeXMLMath is simple , and as LaTeXMLMath , the LaTeXMLMath 1-forms that generate LaTeXMLMath are also eigenvectors of LaTeXMLMath .
From ( LaTeXMLRef ) one immediately finds the eigenvalues LaTeXMLMath .
The case LaTeXMLMath is trivial from Corollary LaTeXMLRef .
In this section we are going to show two important properties of the set LaTeXMLMath : on one hand , its elements are the basic building blocks of all rank-2 tensors in LaTeXMLMath , and on the other they define maps which leave the null cone invariant .
The converse of this result also holds but is left for the last section .
We say that LaTeXMLMath defines a null-cone preserving map if LaTeXMLMath is null for any null vector LaTeXMLMath .
A map that preserves the null cone is said to be orthochronus ( respectively time reversal ) if it keeps ( resp .
reverses ) the cone ’ s time orientation , and is called proper , improper or singular if LaTeXMLMath is positive , negative , or zero , respectively .
If the map is proper and orthochronus then it is called restricted .
A null-cone preserving map is involutory if LaTeXMLMath , and bi-preserving if LaTeXMLMath is also null for any null 1-form LaTeXMLMath .
Most of the above terminology is taken from that of Lorentz transformations , see e.g .
LaTeXMLCite .
Notice that involutory null-cone preserving maps are necessarily non-singular .
In order to characterize all these maps and relate them to LaTeXMLMath we first recall a simple result .
LaTeXMLMath for any LaTeXMLMath that is null .
Proof : The implication from left to right is trivial .
Conversely , if LaTeXMLMath , take an orthonormal basis LaTeXMLMath with a timelike LaTeXMLMath .
Using first as null LaTeXMLMath the vectors LaTeXMLMath for LaTeXMLMath one immediately deduces LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath .
Using then as null LaTeXMLMath the vectors LaTeXMLMath for LaTeXMLMath one gets LaTeXMLMath for all LaTeXMLMath .
The following Lemma gives important geometrical interpretations to some results .
LaTeXMLMath is a null cone preserving map .
Proof : The basic formula is LaTeXMLMath .
If LaTeXMLMath then for any null LaTeXMLMath we have that LaTeXMLMath must be null .
Conversely , if LaTeXMLMath is null for any LaTeXMLMath that is null , and given that LaTeXMLMath is symmetric according to Lemma LaTeXMLRef , then by Lemma LaTeXMLRef LaTeXMLMath must be proportional to the metric .
If LaTeXMLMath then LaTeXMLMath ( for LaTeXMLMath ) .
Proof : From Lemma LaTeXMLRef we know that LaTeXMLMath is null for any LaTeXMLMath that is null .
If LaTeXMLMath were negative , then for any null and future-pointing vectors LaTeXMLMath and LaTeXMLMath we would have LaTeXMLMath , so that any two null vectors of type LaTeXMLMath and LaTeXMLMath would have opposite time orientations .
But this is evidently impossible for all the null vectors of type LaTeXMLMath unless there are only two , that is , LaTeXMLMath .
Similar results can be shown for the product LaTeXMLMath .
However , they are mainly redundant because of the following In LaTeXMLMath , LaTeXMLMath and LaTeXMLMath defines a non-singular null-cone preserving map .
A fortiori , all non-singular maps preserving the null cone are automatically bi-preserving , proportional to an LaTeXMLMath -dimensional Lorentz transformation , and in LaTeXMLMath .
Proof : If LaTeXMLMath , from Corollary LaTeXMLRef we get LaTeXMLMath so that we can define LaTeXMLMath and the condition becomes LaTeXMLMath .
This means that LaTeXMLMath is a Lorentz transformation ( ergo non-singular ) , which as is well-known also satisfies LaTeXMLMath , see e.g .
LaTeXMLCite .
This is exactly LaTeXMLMath .
Now , a reasoning identical to that in the proof of Lemma LaTeXMLRef implies that LaTeXMLMath is null for any LaTeXMLMath that is null , that is , LaTeXMLMath is bi-preserving .
Finally , as LaTeXMLMath , LaTeXMLMath so that from Proposition LaTeXMLRef LaTeXMLMath .
The singular case must be treated separately because of some minor subtleties .
( a ) LaTeXMLMath is a singular null cone preserving map LaTeXMLMath where LaTeXMLMath is null .
( b ) LaTeXMLMath is a singular null-cone bi-preserving map LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are null and LaTeXMLMath .
Proof : From Lemma LaTeXMLRef we know that LaTeXMLMath if and only if the map defined by LaTeXMLMath preserves the null cone , and by Lemma LaTeXMLRef this map must be singular .
Thus , from Proposition LaTeXMLRef there exists a null LaTeXMLMath such that LaTeXMLMath .
This proves ( a ) .
Then , ( b ) follows from Corollary LaTeXMLRef in a similar way .
The tensors in LaTeXMLMath ( respectively in LaTeXMLMath ) are proportional to involutory orthochronus ( resp .
time-reversal ) Lorentz transformations .
The tensors in LaTeXMLMath ( resp .
LaTeXMLMath ) define singular orthochronus ( resp .
time-reversal ) null-cone bi-preserving maps .
Proof : This follows at once from Proposition LaTeXMLRef , Corollary LaTeXMLRef , Lemmas LaTeXMLRef , LaTeXMLRef , and the fact that if LaTeXMLMath is involutory then by Lemma LaTeXMLRef it must coincide with an involutory Lorentz transformation LaTeXMLMath , which are symmetric LaTeXMLMath LaTeXMLCite .
If a null-cone preserving map is non-symmetric ( ergo not proportional to an involutory Lorentz transformation if non-singular ) , then it can be divided into its symmetric and anti-symmetric parts : LaTeXMLEquation .
Notice that , by definition , if LaTeXMLMath is proportional to an involutory Lorentz transformation then LaTeXMLMath and ( up to sign ) LaTeXMLMath ( later we shall prove that , in fact , LaTeXMLMath , see Theorem LaTeXMLRef ) .
The general characterization is ( see LaTeXMLCite for LaTeXMLMath ) : The symmetric and antisymmetric parts of LaTeXMLMath satisfy LaTeXMLEquation if and only if LaTeXMLMath defines a null cone bi-preserving map .
Furthermore , LaTeXMLMath .
Proof : If LaTeXMLMath then LaTeXMLEquation .
LaTeXMLEquation and by adding and substracting these two equations the expresions ( LaTeXMLRef ) are obtained .
Moreover , due to Lemmas LaTeXMLRef and LaTeXMLRef ( b ) we know that LaTeXMLMath .
Then , from Lemma LaTeXMLRef it follows that LaTeXMLMath .
Recall that , from elementary considerations , any eigenvector of a 2-form LaTeXMLMath with non-zero eigenvalue must be null .
If there is one such eigenvector , then there are exactly two of them with non-zero eigenvalues of opposite signs , and any other eigenvector must be spacelike .
Thus , if there is a timelike eigenvector then all eigenvectors have zero eigenvalue .
The possible number of null eigenvectors for a 2-form is : ( i ) if LaTeXMLMath , there are exactly two of them with nonzero eigenvalues of opposite sign ; ( ii ) if LaTeXMLMath , there can be LaTeXMLMath or LaTeXMLMath null eigenvectors ; if there are LaTeXMLMath then both of them have non-zero eigenvalues of opposite sign .
( iii ) for LaTeXMLMath and even , say LaTeXMLMath ( LaTeXMLMath ) , there can be either LaTeXMLMath null eigenvectors ( the only odd number in the list is LaTeXMLMath ) .
( iv ) for LaTeXMLMath and odd , say LaTeXMLMath , there can be either LaTeXMLMath null eigenvectors ( the only even numbers in the list are LaTeXMLMath ) .
In all cases , if there is only one null eigenvector its eigenvalue is zero .
This case includes the null 2-forms .
If LaTeXMLMath defines a null cone bi-preserving map then : ( a ) every null eigenvector of its symmetric part LaTeXMLMath is also a null eigenvector of its antisymmetric part LaTeXMLMath .
( b ) every eigenvector with non-zero eigenvalue of LaTeXMLMath is also a null eigenvector of LaTeXMLMath .
( c ) In the singular case , LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath ( which may coincide if LaTeXMLMath ) are the null eigenvectors of both LaTeXMLMath and LaTeXMLMath .
( d ) every null eigenvector LaTeXMLMath with zero eigenvalue of LaTeXMLMath is either a null eigenvector of LaTeXMLMath or there is another independent null eigenvector LaTeXMLMath with vanishing eigenvalue of LaTeXMLMath such that the timelike 2-plane generated by LaTeXMLMath contains two eigenvectors of both LaTeXMLMath and LaTeXMLMath , one of them spacelike the other timelike , with opposite eigenvalues .
( e ) every timelike eigenvector of its symmetric part LaTeXMLMath is either an eigenvector also of LaTeXMLMath or there are two null vectors which are simultaneously eigenvectors of both LaTeXMLMath and LaTeXMLMath with non-zero eigenvalues .
( f ) if LaTeXMLMath has a timelike eigenvector then there is a common timelike eigenvector for LaTeXMLMath and LaTeXMLMath .
( g ) Furthermore , if LaTeXMLMath then LaTeXMLMath ( for LaTeXMLMath ) .
Proof : Let us start with the null eigenvectors .
From equations ( LaTeXMLRef ) we get for any null vector LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation Thus , if LaTeXMLMath is an eigenvector of LaTeXMLMath then by ( LaTeXMLRef ) LaTeXMLMath is null and obviosuly orthogonal to LaTeXMLMath so that they must be proportional LaTeXMLMath .
This proves ( a ) .
If LaTeXMLMath is an eigenvector of LaTeXMLMath then by ( LaTeXMLRef ) LaTeXMLMath is null , and by ( LaTeXMLRef ) it is orthogonal to LaTeXMLMath .
Hence , if LaTeXMLMath then LaTeXMLMath , which proves ( b ) .
The statement ( c ) for the singular case follows immediately from Lemma LaTeXMLRef ( b ) .
It remains the case with LaTeXMLMath .
In this case from ( LaTeXMLRef ) we get LaTeXMLEquation so that LaTeXMLMath is also a null eigenvector of LaTeXMLMath with zero eigenvalue .
If LaTeXMLMath and LaTeXMLMath are not colinear , that is LaTeXMLMath , then LaTeXMLMath are eigenvectors of LaTeXMLMath with eigenvalues LaTeXMLMath , respectively .
From ( c ) we know that LaTeXMLMath , so obviously one of these vectors is timelike and the other spacelike , and both of them are eigenvectors with zero eigenvalue of LaTeXMLMath .
This proves ( d ) .
Concerning timelike eigenvectors , let LaTeXMLMath be unit and such that LaTeXMLMath .
Contracting relations ( LaTeXMLRef ) with LaTeXMLMath we get LaTeXMLEquation .
Thus , either LaTeXMLMath vanishes or it is spacelike ( for it is orthogonal to LaTeXMLMath ) .
In the latter case from ( LaTeXMLRef ) we have LaTeXMLMath and the two null vectors LaTeXMLMath are eigenvectors of LaTeXMLMath with eigenvalue LaTeXMLMath , and also eigenvectors of LaTeXMLMath with eigenvalues LaTeXMLMath respectively , proving ( e ) .
To prove ( f ) , let LaTeXMLMath be unit and such that LaTeXMLMath .
Then , from ( LaTeXMLRef ) it follows LaTeXMLEquation .
From Lemma LaTeXMLRef we know that LaTeXMLMath , and then LaTeXMLMath is causal .
In fact , contracting the first relation in ( LaTeXMLRef ) with LaTeXMLMath we deduce LaTeXMLMath , so that LaTeXMLMath must be timelike , as otherwise LaTeXMLMath would vanish which is impossible due to ( c ) above .
Then , using ( LaTeXMLRef ) is easy to check that the two vectors LaTeXMLMath are eigenvectors of both LaTeXMLMath and LaTeXMLMath , with eigenvalues LaTeXMLMath respectively , one of them timelike and the other spacelike .
Finally , to prove ( g ) , if LaTeXMLMath is in LaTeXMLMath for LaTeXMLMath or LaTeXMLMath , then by Proposition LaTeXMLRef LaTeXMLMath so that from ( LaTeXMLRef ) we have LaTeXMLMath .
But then Corollary LaTeXMLRef and Lemma LaTeXMLRef imply that LaTeXMLMath unless LaTeXMLMath .
Thus , the maps preserving the null cone have a symmetric part which is in LaTeXMLMath and either in LaTeXMLMath ( if LaTeXMLMath , see Theorem LaTeXMLRef ) or not ( if LaTeXMLMath ) , in the second case algebraically determined by the antisymmetric part of the map and its null eigenvalues .
Hence , in order to classify all these maps we only need to know the structure of tensors in LaTeXMLMath ( defined as the rank-2 tensors in LaTeXMLMath ) in relation with LaTeXMLMath and with the null eigenvectors .
Curiously enough , this result is the analogue to Lemma LaTeXMLRef but for rank-2 symmetric tensors ( LaTeXMLMath and LaTeXMLMath playing the role analogous to causal and null future-pointing vectors , respectively ) : we now show that all symmetric tensors in LaTeXMLMath can be written as sums of terms in LaTeXMLMath .
This means that the elements in LaTeXMLMath can be used to build up LaTeXMLMath , and a fortiori LaTeXMLMath .
Furthermore , each term of LaTeXMLMath in the sum is related in a precise way to the null eigenvectors of the tensor in LaTeXMLMath .
More precisely , we have : In LaTeXMLMath dimensions , any symmetric rank-2 tensor LaTeXMLMath can be written LaTeXMLEquation where LaTeXMLMath are the superenergy tensors of simple LaTeXMLMath -forms LaTeXMLMath , LaTeXMLMath such that for LaTeXMLMath they have the structure LaTeXMLMath where LaTeXMLMath are appropriate null 1-forms .
The number of tensors in the sum ( LaTeXMLRef ) and the structure of the LaTeXMLMath depend on the particular LaTeXMLMath as follows : if LaTeXMLMath has LaTeXMLMath null eigenvectors LaTeXMLMath then at least LaTeXMLMath , with LaTeXMLMath , must appear in the sum , and possibly terms with LaTeXMLMath .
If it has no null eigenvectors , then at least LaTeXMLMath appears in the sum , and possibly terms with LaTeXMLMath , and LaTeXMLMath is the timelike eigenvector of LaTeXMLMath .
Remark : As already stated , the superenergy tensor of the dual of a LaTeXMLMath -form ( LaTeXMLMath ) is identical with that of the LaTeXMLMath -form itself .
Thus , in the sum ( LaTeXMLRef ) there are two superenergy tensors of 1-forms , namely LaTeXMLMath and LaTeXMLMath , but the first one is the superenergy tensor of a causal 1-form and the second of a spacelike 1-form .
This is an essential difference .
Similar remarks apply to the 2-forms LaTeXMLMath and LaTeXMLMath , and so on .
The choice of simple LaTeXMLMath -forms taken in Theorem LaTeXMLRef is such that LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath is causal .
Proof : Recall that for a symmetric tensor LaTeXMLMath any two eigenvectors with different eigenvalues must be orthogonal .
Then , any two linearly independent null eigenvectors of a symmetric tensor must have the same eigenvalue .
We divide up in cases depending on the number of null eigenvectors of LaTeXMLMath .
Suppose that LaTeXMLMath has LaTeXMLMath linearly independent null eigenvectors .
All their eigenvalues must be equal to some constant , say LaTeXMLMath , and LaTeXMLMath as LaTeXMLMath .
The LaTeXMLMath null eigenvectors span all tangent vectors so we get LaTeXMLMath where LaTeXMLMath is the volume LaTeXMLMath -form .
Suppose now that the Theorem is proven for the case with LaTeXMLMath linearly independent null eigenvectors and assume that LaTeXMLMath has LaTeXMLMath linearly independent null eigenvectors , LaTeXMLMath say , all with eigenvalue LaTeXMLMath .
With LaTeXMLMath , all LaTeXMLMath must be spacelike as they are orthogonal to all LaTeXMLMath and LaTeXMLMath ( a vector which is orthogonal to two null vectors must be spacelike ) .
We have LaTeXMLMath so LaTeXMLMath is orthogonal to all LaTeXMLMath and hence LaTeXMLMath .
Therefore LaTeXMLMath is a symmetric map on LaTeXMLMath which is a Euclidean space .
There are then LaTeXMLMath orthonormal ( LaTeXMLMath ) eigenvectors LaTeXMLMath to LaTeXMLMath in LaTeXMLMath with corresponding eigenvalues LaTeXMLMath .
We can assume LaTeXMLMath for LaTeXMLMath .
Now , by Proposition LaTeXMLRef and Corollary LaTeXMLRef , LaTeXMLMath , where LaTeXMLMath , has also the null eigenvectors LaTeXMLMath with some eigenvalue LaTeXMLMath and it has the spacelike eigenvectors LaTeXMLMath with eigenvalues LaTeXMLMath .
Define LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath .
Take LaTeXMLMath with LaTeXMLMath and let LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Hence LaTeXMLMath has LaTeXMLMath linearly independent null eigenvectors LaTeXMLMath .
Note that LaTeXMLMath are future-pointing since LaTeXMLMath , and therefore LaTeXMLMath for all LaTeXMLMath .
To show that LaTeXMLMath , then use that an arbitrary future-pointing null vector LaTeXMLMath can be written LaTeXMLMath where LaTeXMLMath and LaTeXMLMath , and where LaTeXMLMath .
Then LaTeXMLMath which has the squared length LaTeXMLMath .
Thus , LaTeXMLMath has the form required by the induction hypothesis so the statement of the theorem holds for the cases with at least 2 linearly independent null eigenvectors .
Next consider the case with precisely one null eigenvector LaTeXMLMath , with eigenvalue LaTeXMLMath .
Take a set LaTeXMLMath of linearly independent spacelike vectors , all orthogonal to LaTeXMLMath .
Again we have LaTeXMLMath so LaTeXMLMath is orthogonal to LaTeXMLMath .
For those LaTeXMLMath that LaTeXMLMath for some real number LaTeXMLMath , define LaTeXMLMath so LaTeXMLMath , LaTeXMLMath is spacelike , and LaTeXMLMath .
For those LaTeXMLMath that LaTeXMLMath is already spacelike , let LaTeXMLMath .
Let LaTeXMLMath be the other future-pointing null vector orthogonal to LaTeXMLMath and normalised by LaTeXMLMath .
As LaTeXMLMath we have LaTeXMLMath which means that LaTeXMLMath is orthogonal to all LaTeXMLMath .
Thus , LaTeXMLMath with LaTeXMLMath .
Define LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath so LaTeXMLMath and LaTeXMLMath are two linearly independent null eigenvectors of LaTeXMLMath .
To show LaTeXMLMath we use as above that in LaTeXMLMath there is an orthonormal basis of eigenvectors LaTeXMLMath where LaTeXMLMath .
For any LaTeXMLMath , LaTeXMLMath is future-pointing and null , and therefore LaTeXMLMath .
As LaTeXMLMath can be taken arbitrary large we get LaTeXMLMath .
An arbitrary future-pointing null vector can be written LaTeXMLMath where LaTeXMLMath .
We find LaTeXMLMath and conclude that LaTeXMLMath .
Thus , LaTeXMLMath has the required form which proves the case with one null eigenvector .
Finally we consider the case with no null eigenvector .
If there exist null vectors LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath then , as LaTeXMLMath , LaTeXMLMath for some LaTeXMLMath .
Then LaTeXMLMath is a timelike eigenvector which normalised we denote by LaTeXMLMath .
Otherwise , if all null vectors are mapped on timelike vectors then again LaTeXMLCite LaTeXMLMath has a ( unit ) timelike eigenvector LaTeXMLMath .
Thus we have a unit timelike eigenvector LaTeXMLMath with eigenvalue LaTeXMLMath , and on LaTeXMLMath there is an ON-basis LaTeXMLMath of eigenvectors with eigenvalues LaTeXMLMath .
As LaTeXMLMath is future-pointing and null LaTeXMLMath must be future-pointing which implies LaTeXMLMath for all LaTeXMLMath .
Assume LaTeXMLMath for LaTeXMLMath and define LaTeXMLMath .
Then LaTeXMLMath so LaTeXMLMath has two null eigenvectors .
To show that LaTeXMLMath , let LaTeXMLMath ; then LaTeXMLMath is proportional to an arbitrary future-pointing null vector .
One finds LaTeXMLMath , so LaTeXMLMath has the right properties and this finishes the proof .
Remarks : Recall that by Lemma LaTeXMLRef a future-pointing causal vector can be written as a sum of two future-pointing null vectors in infinitely many ways .
In the same manner , a symmetric LaTeXMLMath can be expressed as a sum of LaTeXMLMath elements of LaTeXMLMath in many ways .
As an example , let LaTeXMLMath be an orthonormal basis .
Then , by ( LaTeXMLRef ) , one easily find relations such as LaTeXMLEquation .
LaTeXMLEquation In Theorem LaTeXMLRef however , we construct the representation of LaTeXMLMath in a canonical way in which the simple LaTeXMLMath -forms LaTeXMLMath are constructed from the null eigenvectors of LaTeXMLMath .
We are now prepared to present the classification of the general conformally non-involutory null-cone preserving maps , which follows directly from the Theorem LaTeXMLRef and the Lemma LaTeXMLRef .
Given that the results are elementary but the number of different cases is increasing with the dimension LaTeXMLMath , we will restrict ourselves to the low-dimension cases in full , but this will show the way one has to follow as well as the general ideas which serve for a general LaTeXMLMath .
As the singular case has been already solved , in this section we only deal with the non-singular conformally non-involutory maps , so that LaTeXMLMath .
The conformally involutory ones are left for the next section .
Case LaTeXMLMath .
The simplest case is a 2-dimensional Lorentzian manifold .
In this case there are only two independent null directions , say LaTeXMLMath and LaTeXMLMath , and we can always write LaTeXMLMath .
Both LaTeXMLMath and LaTeXMLMath are null eigenvectors of LaTeXMLMath with non-zero eigenvalue , and then due to Lemma LaTeXMLRef ( b ) , they are also null eigenvectors of LaTeXMLMath .
Using then Theorem LaTeXMLRef the only possibility is that LaTeXMLMath .
Thus , we have In LaTeXMLMath , the maps proportional to non-involutory Lorentz transformations are given by LaTeXMLMath with arbitrary LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
They are proper ( resp .
improper ) if LaTeXMLMath ( resp .
LaTeXMLMath ) , and orthochronus ( resp .
time-reversal ) if LaTeXMLMath ( resp .
LaTeXMLMath ) .
Notice that in this particular case , an arbitrary 2-form LaTeXMLMath defines an improper null cone bi-preserving map .
This is the only possibility in which a 2-form can preserve the null cone , and it appears as an exceptional case as follows from Corollary LaTeXMLRef and Lemmas LaTeXMLRef and LaTeXMLRef .
Before we proceed with the non-trivial cases LaTeXMLMath , we need some simple lemmas .
From Corollary LaTeXMLRef we know that if LaTeXMLMath then the tangent space can be decomposed as LaTeXMLMath where LaTeXMLMath is LaTeXMLMath -dimensional , LaTeXMLMath is LaTeXMLMath -dimensional , and both LaTeXMLMath are eigensubspaces of LaTeXMLMath with opposite eigenvalues .
If LaTeXMLMath is a simple 2-form and LaTeXMLMath , then LaTeXMLMath if and only if LaTeXMLMath lies entirely in either LaTeXMLMath or LaTeXMLMath .
Proof : LaTeXMLMath for some 1-forms LaTeXMLMath and LaTeXMLMath .
Obviously LaTeXMLMath , LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
A straightforward computation gives then LaTeXMLEquation where LaTeXMLMath is the eigenvalue for LaTeXMLMath .
Then , the condition LaTeXMLMath holds if and only if either LaTeXMLMath or LaTeXMLMath .
Similarly , from Corollary LaTeXMLRef if LaTeXMLMath , there is an LaTeXMLMath -dimensional subspace LaTeXMLMath of eigenvectors with zero eigenvalue generated by LaTeXMLMath , where LaTeXMLMath is the canonical null direction of the null LaTeXMLMath .
If LaTeXMLMath is a simple 2-form and LaTeXMLMath , then LaTeXMLMath lies entirely in LaTeXMLMath .
Proof : Set LaTeXMLMath as before and choose LaTeXMLMath null , independent of LaTeXMLMath , and orthogonal to all LaTeXMLMath .
Obviously LaTeXMLMath , LaTeXMLMath with LaTeXMLMath .
As LaTeXMLMath , we have LaTeXMLEquation and given that LaTeXMLMath and LaTeXMLMath are linearly independent , the vanishing of this ( or of its symmetric part ) gives LaTeXMLMath , and conversely .
The notation of Lemma LaTeXMLRef for LaTeXMLMath and LaTeXMLMath is used in the remaining of this section .
Case LaTeXMLMath .
There are three possibilities , as LaTeXMLMath can have 0,1 , or 2 null eigenvectors .
( a ) If LaTeXMLMath has no null eigenvector , then it is proportional to the dual of a unit timelike vector LaTeXMLMath , i.e .
LaTeXMLMath .
Due to Lemma LaTeXMLRef ( a ) , LaTeXMLMath has no null eigenvectors , and due to Lemma LaTeXMLRef ( f ) , LaTeXMLMath is timelike eigenvector also of LaTeXMLMath .
Thus , Theorem LaTeXMLRef allows us to write LaTeXMLEquation .
Using Lemma LaTeXMLRef one has LaTeXMLMath and here the term in brackets is non-vanishing due again to Lemma LaTeXMLRef .
Thus , the second condition ( LaTeXMLRef ) implies LaTeXMLMath .
With this , it is easily checked that the first condition in ( LaTeXMLRef ) leads to LaTeXMLMath .
Thus , we obtain LaTeXMLEquation .
These maps are proper and orthochronus if LaTeXMLMath , and improper and time-reversal if LaTeXMLMath .
( b ) If LaTeXMLMath has one null eigenvector LaTeXMLMath , then LaTeXMLMath is null and can be written LaTeXMLMath with LaTeXMLMath .
Lemma LaTeXMLRef ( d ) implies that LaTeXMLMath is also a null eigenvector of LaTeXMLMath , and this is unique for LaTeXMLMath due to Lemma LaTeXMLRef ( a ) .
So , again Theorem LaTeXMLRef tells us that LaTeXMLEquation .
Analogously to case ( a ) above , Lemmas LaTeXMLRef and LaTeXMLRef lead to LaTeXMLMath , and the first relation in ( LaTeXMLRef ) gives again LaTeXMLMath .
Hence LaTeXMLEquation .
Notice that this can be considered a limit case of ( LaTeXMLRef ) when LaTeXMLMath becomes null .
( c ) If LaTeXMLMath has two independent null eigenvectors LaTeXMLMath and LaTeXMLMath , then they necessarily have non-zero eigenvalues , and by Lemma LaTeXMLRef ( b ) they are also eigenvectors of LaTeXMLMath , which can not have more null eigenvectors due to Lemma LaTeXMLRef ( a ) .
Thus , by Theorem LaTeXMLRef LaTeXMLEquation .
The computation of ( LaTeXMLRef ) leads now simply to LaTeXMLMath .
In summary , LaTeXMLEquation .
Observe that this case can be rewritten as LaTeXMLEquation where LaTeXMLMath is spacelike and defined by LaTeXMLMath .
Hence , the combination of ( LaTeXMLRef - LaTeXMLRef ) proves the following In LaTeXMLMath , the maps proportional to non-involutory Lorentz transformations are given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are arbitrary with LaTeXMLMath and LaTeXMLMath is any 1-form .
These maps leave none , one or two null directions invariant if LaTeXMLMath is time- , light- , or space-like , respectively .
Case LaTeXMLMath .
Now there are just two possibilities : either LaTeXMLMath has one or two null eigenvectors .
( a ) If LaTeXMLMath has one null eigenvector LaTeXMLMath , then LaTeXMLMath is null , LaTeXMLMath with LaTeXMLMath .
Due to Lemma LaTeXMLRef ( d ) and ( a ) this is also the unique null eigenvector of LaTeXMLMath so that from Theorem LaTeXMLRef LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath having the form LaTeXMLMath and LaTeXMLMath , respectively , for null LaTeXMLMath and LaTeXMLMath .
Lemmas LaTeXMLRef and LaTeXMLRef imply that the second equation in ( LaTeXMLRef ) reads LaTeXMLEquation which , as LaTeXMLMath can not be linear combination of LaTeXMLMath and LaTeXMLMath , becomes LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are positive as they are proportional to LaTeXMLMath and LaTeXMLMath , respectively .
The above expression can only be satisfied with non-negative LaTeXMLMath if LaTeXMLMath and LaTeXMLMath .
This also implies that LaTeXMLMath and we can write LaTeXMLEquation .
The remaining condition in ( LaTeXMLRef ) implies in particular that LaTeXMLMath , so that two possibilities arise ( assuming that LaTeXMLMath is unit ) : LaTeXMLMath and then LaTeXMLMath ; or LaTeXMLMath and LaTeXMLMath .
In summary , by setting LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Observe that in both cases one can replace LaTeXMLMath by LaTeXMLMath , because LaTeXMLMath is null .
Furthermore , the above expressions ( LaTeXMLRef - LaTeXMLRef ) are valid for arbitrary LaTeXMLMath so they are proportional to Lorentz transformations in any LaTeXMLMath ( where LaTeXMLMath is just any spacelike vector orthogonal to both LaTeXMLMath and LaTeXMLMath ) .
( b ) If LaTeXMLMath has two null eigenvectors LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation with LaTeXMLMath .
If LaTeXMLMath , from Lemma LaTeXMLRef ( a ) , ( b ) and ( d ) , LaTeXMLMath and LaTeXMLMath are the two null eigenvectors of LaTeXMLMath and we can write in principle , from Theorem LaTeXMLRef , LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath having the form LaTeXMLMath and LaTeXMLMath , respectively , for null LaTeXMLMath .
Lemmas LaTeXMLRef and LaTeXMLRef imply that the second equation in ( LaTeXMLRef ) leads to LaTeXMLMath .
Solving these two possibilites we arrive at LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is arbitrary .
If LaTeXMLMath , there also arises the possibility given by Lemma LaTeXMLRef ( d ) , ( f ) that LaTeXMLMath has a timelike eigenvector LaTeXMLMath and a spacelike one LaTeXMLMath with LaTeXMLMath , such that from Theorem LaTeXMLRef one has in principle LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath .
However , LaTeXMLMath and LaTeXMLMath have opposite eigenvalues due to Lemma LaTeXMLRef ( d ) , from where we get LaTeXMLMath .
Then , from Lemma LaTeXMLRef and the second equation in ( LaTeXMLRef ) it follows that LaTeXMLMath too .
Finally , taking LaTeXMLMath and LaTeXMLMath unit , the first relation in ( LaTeXMLRef ) leads to LaTeXMLMath so that LaTeXMLEquation .
In LaTeXMLMath , the maps proportional to non-involutory Lorentz transformations are given by ( LaTeXMLRef - LaTeXMLRef ) .
These results were obtained for the restricted case in LaTeXMLCite , and in general in LaTeXMLCite using spinors .
The case given by ( LaTeXMLRef ) may seem not included in the solution presented in LaTeXMLCite , but this is apparent .
In fact , one can rewrite ( LaTeXMLRef ) by using the identity ( LaTeXMLRef ) as LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath , and this last form is certainly included in the cases given in LaTeXMLCite .
The number of possibilities and the complexity of the equations increase with LaTeXMLMath , but the reasonings and techniques are always simple and the same : application of Lemmas LaTeXMLRef , LaTeXMLRef and LaTeXMLRef and Theorem LaTeXMLRef to the equations ( LaTeXMLRef ) .
The details will be omitted here but , as an illustrative example , we present the general solution for arbitrary odd dimension LaTeXMLMath .
Case LaTeXMLMath , ( LaTeXMLMath ) .
Let LaTeXMLMath be an orthonormal basis .
Then , the maps proportional to non-involutory Lorentz transformations are in one of the following cases : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath are arbitrary , the LaTeXMLMath are given , for all LaTeXMLMath by LaTeXMLEquation and LaTeXMLMath is a causal 1-form equal to LaTeXMLMath if LaTeXMLMath leaves no null direction invariant , and to LaTeXMLMath if it leaves exactly one null direction ( LaTeXMLMath ) invariant .
LaTeXMLEquation .
LaTeXMLEquation where now LaTeXMLEquation and for all LaTeXMLMath LaTeXMLEquation .
Generically , this leaves 2 null directions invariant , 3 if LaTeXMLMath , and in general LaTeXMLMath null directions invariant if LaTeXMLMath .
( 3 ) Those cases which effectively reduce to low-dimensional cases , such as for instance LaTeXMLEquation which is the analogue of ( LaTeXMLRef ) and has two invariant null directions .
And similarly for the appropriate generalizations of ( LaTeXMLRef - LaTeXMLRef ) and ( LaTeXMLRef ) .
We are now going to prove an important result : the converses of Proposition LaTeXMLRef , Lemma LaTeXMLRef and Corollary LaTeXMLRef hold .
One can also intrinsically determine the rank LaTeXMLMath of the LaTeXMLMath -form generating the tensor in LaTeXMLMath .
More precisely In LaTeXMLMath dimensions , if LaTeXMLMath is symmetric and LaTeXMLMath then : ( a ) LaTeXMLMath and LaTeXMLMath for a null LaTeXMLMath .
( b ) LaTeXMLMath and , for some LaTeXMLMath , LaTeXMLMath with LaTeXMLMath .
Moreover , LaTeXMLMath where LaTeXMLMath is the superenergy tensor of a simple LaTeXMLMath -form LaTeXMLMath of the type used in Theorem LaTeXMLRef .
Proof : By Corollary LaTeXMLRef , LaTeXMLMath symmetric and LaTeXMLMath implies that LaTeXMLMath .
Thus LaTeXMLMath is well defined as the positive square root of LaTeXMLMath .
Lemmas LaTeXMLRef and LaTeXMLRef give then LaTeXMLMath .
Then by Theorem LaTeXMLRef and using LaTeXMLMath , LaTeXMLMath .
We have to verify that only one term or proportional terms can remain if LaTeXMLMath .
We have LaTeXMLMath .
This expression contains four types of terms : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
By Theorem LaTeXMLRef and Lemma LaTeXMLRef every term is in LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath for every null vector LaTeXMLMath .
Since each term is non-negative this means that every term must be zero when contracted with LaTeXMLMath .
By Proposition LaTeXMLRef this is satisfied by LaTeXMLMath and LaTeXMLMath since they are proportional to LaTeXMLMath .
Take then LaTeXMLMath .
This means that LaTeXMLMath which is impossible for every null vector LaTeXMLMath unless LaTeXMLMath or LaTeXMLMath .
Thus if LaTeXMLMath then LaTeXMLMath for all LaTeXMLMath so LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath we need to study the implications of LaTeXMLMath for every null vector LaTeXMLMath .
By Proposition LaTeXMLRef and Lemma LaTeXMLRef , LaTeXMLMath and LaTeXMLMath are null vectors and hence they must be parallel , LaTeXMLMath .
Contraction with another null vector LaTeXMLMath gives LaTeXMLMath because of symmetry .
Therefore , there is some constant LaTeXMLMath such that LaTeXMLMath for all null vectors , and extending by linear combinations to all vectors we must have LaTeXMLMath , or that one of them is zero .
In total , all non-zero LaTeXMLMath , LaTeXMLMath , should be proportional , but this is not possible because as seen in the proof of Theorem LaTeXMLRef each of them has a strict different number of null eigenvectors .
Then LaTeXMLMath for some LaTeXMLMath and by ( LaTeXMLRef ) LaTeXMLMath so LaTeXMLMath and LaTeXMLEquation which , as LaTeXMLMath , gives the values LaTeXMLMath .
Finally , the particular case ( a ) , with LaTeXMLMath , follows immediately from the above or directly from Lemma LaTeXMLRef ( b ) .
Combining Proposition LaTeXMLRef , Lemma LaTeXMLRef , Corollary LaTeXMLRef and Theorem LaTeXMLRef , we immediately obtain the following important result .
In LaTeXMLMath dimensions , if LaTeXMLMath is symmetric then LaTeXMLMath .
This means that the tensors in LaTeXMLMath ( respectively in LaTeXMLMath ) are precisely those proportional to involutory orthochronus ( resp .
time-reversal ) Lorentz transformations .
The symmetric tensors in LaTeXMLMath ( resp .
LaTeXMLMath ) are exactly the symmetric singular orthochronus ( resp .
time-reversal ) null-cone bi-preserving maps .
Theorems LaTeXMLRef and LaTeXMLRef provide a complete characterization of the conformally involutory null-cone preserving maps .
Its classification also follows from the proof of Proposition LaTeXMLRef and Corollary LaTeXMLRef , for we know that LaTeXMLMath eigenvalues of LaTeXMLMath are equal to LaTeXMLMath while LaTeXMLMath are equal to LaTeXMLMath .
If an odd number of these are negative , LaTeXMLMath is an improper null cone preserving map , otherwise a proper one .
If LaTeXMLMath is a spacelike 1-form , then one eigenvalue is negative so LaTeXMLMath is improper and can be interpreted as a reflection in the hyperplane orthogonal to LaTeXMLMath .
If LaTeXMLMath is a timelike 1-form , then LaTeXMLMath eigenvalues are negative and LaTeXMLMath is proper in odd dimensions and improper in even dimensions .
It can be interpreted as a reflection in the line parallel to LaTeXMLMath .
For other LaTeXMLMath -forms one can develop the corresponding geometrical interpretations .
Proposition LaTeXMLRef and Theorem LaTeXMLRef also imply : If LaTeXMLMath then LaTeXMLMath , i.e .
in 2 , 3 , and 4 dimensions , LaTeXMLMath is precisely the set of symmetric tensors leaving invariant the null cone with its time orientation .
Thus , if LaTeXMLMath , LaTeXMLMath is constituted by all tensors proportional to involutory orthochronus Lorentz transformations plus the symmetric singular orthochronus null-cone bi-preserving maps .
LaTeXMLMath gives the time-reversal case .
For LaTeXMLMath this is trivial .
For LaTeXMLMath this also means that the energy-momentum of any Maxwell field is proportional to an involutory orthochronus ( and proper ) Lorentz transformation , and coincides with the energy-momentum of some ( possibly another ) Maxwell field corresponding to a simple 2-form .
This is well known and related to the duality rotations LaTeXMLCite .
With LaTeXMLMath and LaTeXMLMath in Theorem LaTeXMLRef we can state this as In LaTeXMLMath dimensions , a tensor LaTeXMLMath is ( up to sign ) algebraically the energy-momentum tensor of a Maxwell field ( a 2-form ) if and only if LaTeXMLMath and LaTeXMLMath .
These are the classical algebraic Rainich conditions LaTeXMLCite ( see also LaTeXMLCite ) .
They are necessary and sufficient conditions for a spacetime metric to originate algebraically ( via Einstein ’ s equations ) in a Maxwell field , i.e .
a way of determining the physics from the geometry .
By Theorem LaTeXMLRef we can find generalisations to arbitrary dimension and to many different physical fields .
In order to show the possibilities of our results , we can derive the following algebraic Rainich conditions .
For a scalar field ( compare with the partial results in LaTeXMLCite for LaTeXMLMath ) we have In LaTeXMLMath dimensions , a tensor LaTeXMLMath is algebraically the energy-momentum tensor of a minimally coupled massless scalar field LaTeXMLMath if and only if LaTeXMLMath and LaTeXMLMath where LaTeXMLMath .
Moreover , LaTeXMLMath is spacelike if LaTeXMLMath and LaTeXMLMath , timelike if LaTeXMLMath and LaTeXMLMath , and null if LaTeXMLMath .
Proof : Recall that LaTeXMLMath which is exactly LaTeXMLMath so LaTeXMLMath .
We get LaTeXMLMath and LaTeXMLMath so LaTeXMLMath with a plus sign if LaTeXMLMath and a minus sign if LaTeXMLMath .
Conversely , if LaTeXMLMath then LaTeXMLMath .
If LaTeXMLMath , then by Theorem LaTeXMLRef LaTeXMLMath is the superenergy tensor of a null 1-form in case LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath then LaTeXMLMath is the superenergy tensor of a timelike 1-form .
If LaTeXMLMath and LaTeXMLMath then LaTeXMLMath is the superenergy tensor of an LaTeXMLMath -form of the type used in Theorem LaTeXMLRef which is the same as the superenergy tensor of its dual spacelike 1-form .
We can also generalize the algebraic Rainich conditions for a perfect fluid as given by Coll and Ferrando LaTeXMLCite to the case of general LaTeXMLMath .
In LaTeXMLMath dimensions , a tensor LaTeXMLMath is algebraically the energy-momentum tensor of a perfect fluid satisfying the dominant energy condition if and only if LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is timelike ( so that LaTeXMLMath is intrinsically characterized as a tensor in LaTeXMLMath according to its trace , see previous Corollary ) .
The velocity vector of the fluid , its energy density and pressure are given by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , respectively .
Proof : Recall that a perfect fluid has the Segre type LaTeXMLMath , so that LaTeXMLEquation where LaTeXMLMath .
Thus , if ( LaTeXMLRef ) holds it is obvious that LaTeXMLMath takes the form ( LaTeXMLRef ) .
Conversely , if ( LaTeXMLRef ) holds , then LaTeXMLMath has every null LaTeXMLMath as eigenvector , as can be trivially checked .
Therefore , LaTeXMLMath is proportional to the metric , and the proportionality factor is obtained from the LaTeXMLMath .
In fact , we can get the conditions as stated in LaTeXMLCite generalized for LaTeXMLMath dimensions as follows .
From ( LaTeXMLRef ) we get LaTeXMLEquation and also LaTeXMLMath , LaTeXMLMath and LaTeXMLMath for all timelike LaTeXMLMath .
As another example , let us consider the case of dust ( LaTeXMLMath perfect fluids ) .
Of course , this case can be deduced from the previous one by setting LaTeXMLMath .
However , in dimension LaTeXMLMath some stronger results can be derived .
To see it , recall that any 2-form LaTeXMLMath with no null eigenvector can only exist in odd dimension LaTeXMLMath , and must take the form LaTeXMLEquation where LaTeXMLMath is an orthonormal basis and LaTeXMLMath ( LaTeXMLMath ) are non-zero constants .
Thus , in the particular case that all the LaTeXMLMath ’ s are equal we get for the superenergy tensor ( LaTeXMLRef ) of such an LaTeXMLMath LaTeXMLEquation .
In 5 dimensions , LaTeXMLMath is algebraically the energy-momentum tensor of a dust , that is LaTeXMLMath where LaTeXMLMath and LaTeXMLMath , if and only if LaTeXMLMath is the s-e tensor of a 2-form LaTeXMLMath with no null eigenvector having LaTeXMLMath .
Proof : This is the case LaTeXMLMath ( LaTeXMLMath ) of the previous formula ( LaTeXMLRef ) , identifying LaTeXMLMath and LaTeXMLMath .
Notice that the timelike direction LaTeXMLMath is intrinsically defined by LaTeXMLMath .
The authors are grateful to the Ministerio de Educación , Cultura y Deporte in Spain for support for a longer stay at the Department of Theoretical Physics at the University of the Basque Country ( SAB1999-0135 for G.B . )
and for a 3-month stay at the Institut für Theoretische Physik of the University of Vienna ( PR2000-0222 for J.M.M.S . ) .
JMMS is thankful to this Institut , where the paper was partly written , for kind hospitality .
We thank Anders Höglund for useful comments and Alfonso García-Parrado for pointing out some typos .
We introduce an n-ary Lie algebroid canonically associated to a Nambu-Poisson manifold .
We also prove that every Nambu-Poisson bracket defined on functions is induced by some differential operator on the exterior algebra , and characterize such operators .
Some physical examples are presented .
In the seventies , trying to describe the simultaneous classical dynamics of three particles as a previous step towards a quantum statistics for the quark model , Y. Nambu introduced a generalization of Poisson brackets formalism that today bears his name ( see [ Nam 73 ] ) .
The field underwent a revitalization with the work of L. Takhtajan in the nineties ( [ Tak 94 ] ) , which showed the algebraic setting underlying Nambu ideas and introduced an analog of the Jacobi Identity , the Fundamental Identity , allowing the connection with the theory of n-ary Lie algebras developed by Filippov and others ( see [ Fil 85 ] and [ Han-Wac 95 ] ) .
In [ AIP 97 ] , further references about the development of this subject can be found , along with an alternative generalization of Poisson structures .
Recently , much work has been done in this area , showing an interesting algebraic structure ( see , for instance , [ Ale-Guh 96 ] , [ Gau 96 ] , [ Gau 98 ] , [ Mic-Vin 96 ] , [ MVV 98 ] , [ Nak 98 ] , [ Vin-Vin 98 ] and references therein ) .
This is very interesting both from a mathematical and a physical point of view : an algebraic formulation not only provides a more concise and elegant framework , but also can be the source for new insights .
In this direction , the notion of Leibniz algebroid was introduced ( [ Dal-Tak 97 ] , [ ILMP 99 ] ) as a kind of analogue of the Lie algebroid associated to a Poisson manifold .
That concept of a Lie algebroid , has proved to be very useful in the study of Poisson manifolds and dynamics ( for a sample , see [ Lib 96 ] , [ Cou 94 ] , [ Wei 98 ] , [ Gra-Urb 97 ] ) , and recently has been considered for the quantization of Poisson algebras ( [ Lan-Ram 00 ] ) .
It is to be expected the same in the Nambu-Poisson case ( n-ary Poisson algebras ) , with the appropiate generalizations .
Thus , the main motivation for this work comes from the question : to what extent the constructions and properties of the Lie algebroids associated to Poisson manifolds , including quantization , carry over to the n-ary case ?
This paper deals with a very first step towards the answer , namely , the study of the canonical relation between Lie algebroids and Poisson manifolds in the n-ary case .
For this purpose , one could consider Leibniz algebroids ; however , this concept ( although very interesting in itself ) is not a genuine generalization of that of a Lie algebroid : it relies upon an algebraic construction called Leibniz ( or Loday ) algebra ( see [ Lod 93 ] ) , which is a non-commutative version of a Lie algebra .
In [ Gra-Mar 00 ] the notion of an n-ary Lie algebroid ( called by the authors Filippov algebroid ) has been introduced , and it seems to fit better to the aforementioned question .
In this notes , we construct an n-ary Lie algebroid canonically associated to a Nambu-Poisson manifold à la Koszul ( see [ Kos 85 ] ) , that is to say , by using differential operators on the ( graded ) exterior algebra of differential forms .
An interesting advantage of this approach , is that it allows for the possibility of characterizing the operators which generate the brackets under consideration , this characterization being in terms of the commutator of the operator with the exterior differential .
For the case of Poisson brackets and its extension as a graded brackets to the whole exterior algebra , such study was made in [ BMS 97 ] , where these operators were called Jacobi operators ( adopting the terminology from [ Gra 92 ] ) , and essentially the same techniques will be used here to show that every Nambu-Poisson bracket coincides with the bracket induced on functions by a certain differential operator on the exterior algebra , giving its explicit form in terms of the Nambu-Poisson multivector .
In the last section , some physical examples are described .
Acknowledgments : The author wants to express his gratitude to J. V. Beltrán , J. Grabowski , J .
A. de Azcárraga and specially J. Monterde , for very useful discussions and comments .
Our main tool in the study of the n-ary generalizations of Poisson manifolds and Lie algebroids , will be the theory of differential operators on the exterior algebra of a manifold , so we collect here the basics of this theory .
Let LaTeXMLMath be the space of LaTeXMLMath homomorphisms of the exterior algebra LaTeXMLMath .
We say that LaTeXMLMath has LaTeXMLMath -degree LaTeXMLMath ( sometimes LaTeXMLMath as exponent ) if LaTeXMLMath maps LaTeXMLMath forms on LaTeXMLMath forms , then we write LaTeXMLMath ( or LaTeXMLMath ) .
On the space LaTeXMLMath we introduce a bracket LaTeXMLMath ( called commutator ) by means of LaTeXMLEquation and it is easy to prove that this bracket turns LaTeXMLMath LaTeXMLMath into a graded Lie algebra .
A differential operator on LaTeXMLMath of degree LaTeXMLMath and order LaTeXMLMath , is a homomorphism LaTeXMLMath such that LaTeXMLEquation for all LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath denotes the homomorphism multiplication by LaTeXMLMath , which has order 0 and degree LaTeXMLMath , and so will be often denoted simply LaTeXMLMath .
The space of such operators is denoted LaTeXMLMath ; examples are the insertion operator LaTeXMLMath where LaTeXMLMath is a LaTeXMLMath multivector , and the generalized Lie derivative LaTeXMLMath , LaTeXMLMath being the exterior differential .
A useful result states that LaTeXMLEquation .
Also , we have that any operator LaTeXMLMath has the form LaTeXMLMath , for some LaTeXMLMath .
This is a consequence of ( LaTeXMLRef ) and the fact that any operator is determined by its action on forms with degree equal or less than the order of the operator .
The order defines a filtration on the space of all differential operators .
Indeed , given LaTeXMLMath with LaTeXMLMath , if LaTeXMLMath is an operator of order LaTeXMLMath , then it is also of order LaTeXMLMath , that is to say : LaTeXMLEquation .
Given LaTeXMLMath , we will call symbol of order LaTeXMLMath of LaTeXMLMath the LaTeXMLMath -module LaTeXMLEquation and the space of symbols of the exterior algebra LaTeXMLMath is defined as the graded LaTeXMLMath -module LaTeXMLEquation .
Note that , if LaTeXMLMath is a differential operator of order LaTeXMLMath , LaTeXMLMath , then we can speak about the symbol of order LaTeXMLMath of the operator LaTeXMLMath , that is to say , LaTeXMLEquation .
We will need some terminology from the theory of Nambu-Poisson manifolds .
A Nambu-Poisson bracket on a manifold LaTeXMLMath , is an LaTeXMLMath -linear mapping LaTeXMLMath , satisfying : Skew-symmetry .
For all LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath is the symmetric group of order LaTeXMLMath ) LaTeXMLEquation .
Leibniz rule .
For all LaTeXMLMath LaTeXMLEquation .
Fundamental Identity .
For all LaTeXMLMath LaTeXMLEquation .
If LaTeXMLMath is a Nambu-Poisson bracket , it has associated an LaTeXMLMath -multivector LaTeXMLMath through LaTeXMLEquation which is called the Nambu-Poisson multivector .
A Nambu-Poisson manifold is a pair LaTeXMLMath where LaTeXMLMath is a Nambu-Poisson bracket ( also can be denoted LaTeXMLMath ) .
A Lie ( or Filippov ) LaTeXMLMath -algebra structure on a vector space LaTeXMLMath , is an LaTeXMLMath -linear skew-symmetric bracket LaTeXMLMath satisfying the generalized Jacobi identity , also called Fundamental Identity : LaTeXMLEquation for all LaTeXMLMath .
A Lie ( or Filippov ) LaTeXMLMath -algebroid is a vector bundle LaTeXMLMath equipped with an LaTeXMLMath -bracket LaTeXMLMath on sections of LaTeXMLMath ( LaTeXMLMath ) and a vector bundle morphism LaTeXMLMath over the identity on LaTeXMLMath , called the anchor map , such that : The induced morphism on sections LaTeXMLMath satisfies the following relation with respect to the bracket of vector fields LaTeXMLEquation and LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath .
We will call these structures n-Lie brackets and algebroids , respectively , for short .
In [ Kos 85 ] , Koszul introduces the following notation : for LaTeXMLMath ( of any degree ) LaTeXMLEquation .
In fact , he considers that expression for LaTeXMLMath such that LaTeXMLMath , and uses it in order to define a bracket on LaTeXMLMath as LaTeXMLEquation .
Koszul also studies under what conditions the bracket LaTeXMLMath is a graded Lie one , and obtains the necessary and sufficient condition LaTeXMLEquation when a priori LaTeXMLMath lies in LaTeXMLMath .
This idea can be followed on to construct new kinds of brackets from a differential operator .
In the following , we will consider only operators such that LaTeXMLMath .
By a direct calculation involving only Jacobi identity and the skew-symmetry for the commutator LaTeXMLMath , it can be shown that , for any LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation If we want to induce a n-ary bracket on 1-forms , we need an operator of type LaTeXMLMath ( so acting on LaTeXMLMath 1-forms gives a 1-form , recall ( LaTeXMLRef ) ) .
In this case , with LaTeXMLMath , ( LaTeXMLRef ) reduces to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation and , as the operator degree is LaTeXMLMath , to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Taking into account that ( for odd degree operators ) LaTeXMLMath , we see that this expression generalizes the one given by Koszul ( in lemma 1.5 of [ Kos 85 ] ) for the Jacobi Identity when only 1-forms are considered : LaTeXMLEquation .
LaTeXMLEquation Thus , we are led to consider the following definition for a n-ary bracket on 1-forms , induced by an operator LaTeXMLMath For LaTeXMLMath and LaTeXMLMath , LaTeXMLMath LaTeXMLEquation .
Let LaTeXMLMath the bracket on 1-forms induced by an operator LaTeXMLMath .
Then , it has the following properties : ( i ) LaTeXMLMath -linearity on each argument ( ii ) skew-symmetry .
Proof .
( i ) Is a direct consequence of the corresponding property for the bracket on LaTeXMLMath .
( ii ) Here we use that , for any LaTeXMLMath , and LaTeXMLMath LaTeXMLEquation and the statement follows from a straightforward computation .
In order to construct an n-Lie bracket on 1-forms , we need a Fundamental Identity .
A glance at ( LaTeXMLRef ) tells us we are almost done , it suffices to give a condition on LaTeXMLMath similar to that of Koszul for the binary case .
Let LaTeXMLMath be a differential operator .
Then , LaTeXMLMath induces an n-Lie algebra structure on LaTeXMLMath if and only if , for any LaTeXMLMath , LaTeXMLEquation .
Proof .
Just observe that the condition above kills the last term in ( LaTeXMLRef ) .
In this section , we will construct an n-Lie algebroid on the space of LaTeXMLMath forms .
Given an operator LaTeXMLMath , we consider an extension of the bracket LaTeXMLMath to another one defined in LaTeXMLMath , that is , where the last argument is a differential form of any degree , with the same formula : LaTeXMLEquation .
LaTeXMLMath .
Although this bracket loses the skew-symmetry property , under the condition on LaTeXMLMath given in Proposition 2 it retains the Fundamental Identity , in the sense that ( recall ( LaTeXMLRef ) ) LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation when LaTeXMLMath .
This will enable us to construct an n-Lie ( or Filippov ) algebroid associated to LaTeXMLMath .
Indeed , the Leibniz property guarantees that LaTeXMLMath acts as a derivation on LaTeXMLMath , that is , it belongs to LaTeXMLMath : for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation Consider now the cotangent bundle LaTeXMLMath .
On the sections LaTeXMLMath , we define the anchor map as LaTeXMLEquation .
LaTeXMLEquation and extend it by linearity .
Let us check that the definition makes sense : the observation above tells us that for LaTeXMLMath LaTeXMLEquation and so LaTeXMLMath takes values in the right space .
Note also the LaTeXMLMath linearity of the anchor map , obtained as a consequence of the Leibniz property for the bracket on LaTeXMLMath and the degree of LaTeXMLMath : if LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ( the term with LaTeXMLMath inside the brackets vanishes because LaTeXMLMath ) .
Let us now verify the conditions of the n-Lie algebroid definition .
One of these is nothing more but the property ( ii ) of Proposition 1 : LaTeXMLEquation .
LaTeXMLEquation and the other one is equivalent to the Fundamental Identity .
On the one hand we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and on the other LaTeXMLEquation .
LaTeXMLEquation The Fundamental Identity equates these two expressions , so LaTeXMLEquation .
LaTeXMLEquation We summarize all this in the following result .
Let LaTeXMLMath be such that LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath is an n-Lie algebroid .
Given an operator LaTeXMLMath as in the previous Proposition , any isomorphism LaTeXMLMath ( for example , the canonical ones associated to riemannian , Poisson or symplectic manifolds ) induces the corresponding n-Lie algebroid on LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath is the extension of LaTeXMLMath as a homomorphism of exterior algebras .
Our goal in this section is to construct a basic example of n-Lie algebroid , and we shall obtain a result similar to that for Poisson manifolds ( namely , that each Poisson manifold has a canonical Lie algebroid structure associated to it ) for the n-ary case , i.e : each n-Poisson ( Nambu-Poisson ) manifold has a canonically associated n-Lie algebroid .
Let LaTeXMLMath be a Nambu-Poisson manifold .
Leibniz property for LaTeXMLMath tells us that we can express it through a n-multivector LaTeXMLMath ( called the Nambu-Poisson multivector ) such that , for any LaTeXMLMath : LaTeXMLEquation .
Let us translate the Fundamental Identity for LaTeXMLMath in terms of LaTeXMLMath .
Consider a hamiltonian vector field , which in this case will have the form LaTeXMLEquation and let us compute the Lie derivative of LaTeXMLMath , LaTeXMLMath , which is a tensor of the same type of LaTeXMLMath and , accordingly , will act on LaTeXMLMath functions LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Thus , the Fundamental Identity for the n-bracket LaTeXMLMath , translates into LaTeXMLEquation .
Now , this Lie derivative is nothing but the Schouten-Nijenhuis bracket of LaTeXMLMath and LaTeXMLMath ( taken as multivectors ) , so we can write LaTeXMLEquation .
Knowing what the Fundamental Identity means in terms of LaTeXMLMath , we can construct the promised example .
In order to do this , consider the operator LaTeXMLEquation which is of the order and degree we have seen can generate Filippov brackets on 1-forms .
An easy computation ( using that LaTeXMLMath and LaTeXMLMath ) gives LaTeXMLEquation and similarly ( using here that LaTeXMLMath and LaTeXMLMath commute ) LaTeXMLEquation .
As a consequence we get the following result .
For each LaTeXMLMath satisfying LaTeXMLMath ( for all LaTeXMLMath , LaTeXMLMath , the operator LaTeXMLMath induces an n-Lie bracket on 1-forms , and so each Nambu-Poisson bracket on functions has an associated n-Lie bracket on LaTeXMLMath .
Proof .
By Leibniz property , we only need to check that LaTeXMLEquation when the LaTeXMLMath are of the form LaTeXMLMath ( in fact , we will see that this expression vanishes ) .
Now , we can distinguish three different cases .
1 LaTeXMLMath case : there are at least two functions among the LaTeXMLMath ( LaTeXMLMath ) .
Rearrange the factors to get LaTeXMLEquation and then compute , having in mind the previous remark , LaTeXMLEquation 2 LaTeXMLMath case : there is exactly one function among the LaTeXMLMath ( LaTeXMLMath ) .
This time , rearrange to LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation 3 LaTeXMLMath case : there is no function among the LaTeXMLMath ( LaTeXMLMath ) .
Here , all we have are exact LaTeXMLMath -forms , and then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be a Nambu-Poisson manifold and LaTeXMLMath the induced Nambu-Poisson tensor .
Then LaTeXMLMath constructed as above is an n-Lie algebroid .
We have just seen how to construct Filippov algebroids from Nambu-Poisson structures .
Next , we would like to obtain examples of the later ones also by using differential operators techniques .
Given an operator LaTeXMLMath and a function LaTeXMLMath , we define a bracket LaTeXMLEquation .
LaTeXMLEquation and from it , a n-bracket on functions LaTeXMLEquation .
LaTeXMLEquation This bracket is linear in each argument and that it has the Leibniz property .
The following result , specifies another features .
Let LaTeXMLMath and LaTeXMLMath as above .
Then : ( i ) LaTeXMLMath is skew-symmetric if and only if LaTeXMLMath ( ii ) LaTeXMLMath verifies the Fundamental Identity if and only if for all LaTeXMLMath then LaTeXMLMath Proof .
Condition ( ii ) is known from Section 1 ( see ( LaTeXMLRef ) ) .
For the condition ( i ) to be understood , we only need to check it .
First , note that if LaTeXMLMath , then clearly LaTeXMLMath .
Next , consider any differential operator LaTeXMLMath and LaTeXMLMath ; we have LaTeXMLEquation .
LaTeXMLEquation thus , if we take LaTeXMLMath it results LaTeXMLMath , and so LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
It is the first term on the last member what destroys skew-symmetry , but writing LaTeXMLEquation we see that LaTeXMLEquation thus , we have from ( LaTeXMLRef ) and ( LaTeXMLRef ) , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Let LaTeXMLMath be such that LaTeXMLMath , for all LaTeXMLMath , and Simb LaTeXMLMath .
Then , the induced bracket on functions LaTeXMLMath , is a Nambu-Poisson bracket .
Now , we prove that any Nambu-Poisson bracket comes from an operator of this kind .
Let LaTeXMLMath be a Nambu-Poisson bracket on LaTeXMLMath .
Then , it coincides with LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath is the Nambu-Poisson n-multivector associated to LaTeXMLMath .
Proof .
If LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Our last result shows that , in a sense , the operators of the form LaTeXMLMath are the unique ones for which LaTeXMLMath is a Nambu-Poisson structure .
We will need some technical results first .
Let LaTeXMLMath be a vector bundle with finite rank over a manifold LaTeXMLMath .
Let LaTeXMLMath be the space of its sections .
Then , given a linear map LaTeXMLMath , there exists a ( necessarily unique ) section LaTeXMLMath of the dual vector bundle LaTeXMLMath such that , for any point LaTeXMLMath and any element LaTeXMLMath in LaTeXMLMath , LaTeXMLEquation if and only if LaTeXMLMath is LaTeXMLMath -linear .
( This is a standard result in Differential Geometry , see for example [ War 71 ] , pgs 64-65 ) .
A differential operator LaTeXMLMath ( of any order and degree ) is said to be tensorial if , for all LaTeXMLMath , LaTeXMLEquation .
Let LaTeXMLMath be a tensorial operator .
Then , there exist uniques LaTeXMLMath such that LaTeXMLEquation .
Proof .
As a consequence of LaTeXMLMath , we have that for all LaTeXMLMath , LaTeXMLMath , so LaTeXMLMath is LaTeXMLMath linear and -by the localization lemma- it defines an LaTeXMLMath such that LaTeXMLEquation .
Next , we study what happens when LaTeXMLMath acts on LaTeXMLMath forms .
Just because LaTeXMLMath and LaTeXMLMath are so , LaTeXMLMath is a tensorial operator , and then LaTeXMLEquation .
LaTeXMLEquation is LaTeXMLMath linear in all its arguments .
Then , again by the localization lemma , LaTeXMLMath such that LaTeXMLEquation .
Now , acting on any LaTeXMLMath form , with LaTeXMLMath , any of the previous operators gives LaTeXMLMath ( note the degrees ) ; thus , we have LaTeXMLMath Given LaTeXMLMath , then LaTeXMLMath if and only if LaTeXMLMath for some LaTeXMLMath .
Proof .
We will follow the ideas presented in [ Bel-Mon 94 ] , where the authors consider the LaTeXMLMath case .
If LaTeXMLMath , where LaTeXMLMath , we can write LaTeXMLMath , with LaTeXMLMath , and then , as LaTeXMLMath commutes with LaTeXMLMath , LaTeXMLEquation so LaTeXMLMath For the converse , consider an arbitrary LaTeXMLMath , neither necessarily verifying LaTeXMLMath , nor being tensorial .
Let us determine its non tensorial part by taking brackets with a function LaTeXMLMath .
We have LaTeXMLMath , so LaTeXMLMath such that LaTeXMLMath .
Now , the mapping LaTeXMLEquation .
LaTeXMLEquation is a derivation : if LaTeXMLMath , by the Leibniz property for the bracket LaTeXMLMath , LaTeXMLEquation .
Now , to each derivation from LaTeXMLMath to LaTeXMLMath , we can assign a LaTeXMLMath in the following manner .
Let LaTeXMLEquation and note that LaTeXMLMath as a consequence of LaTeXMLMath being a derivation .
Also , we have LaTeXMLMath , so LaTeXMLMath is LaTeXMLMath - linear and there exists a LaTeXMLMath such that LaTeXMLEquation .
Let us study under what conditions this LaTeXMLMath is skew-symmetric .
It suffices to consider only the case when LaTeXMLMath , and to observe that LaTeXMLEquation .
Now , from the proof of Proposition 10 , we know that this is skew-symmetric if and only if LaTeXMLMath .
So , under this condition we can take LaTeXMLEquation .
In the last step , we check that , for any LaTeXMLMath , LaTeXMLMath is a tensorial operator .
As LaTeXMLMath and a differential operator is characterized by its action on forms of degree less or equal to its order , we only need to consider the case of a LaTeXMLMath form .
So , for LaTeXMLMath , we compute LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLMath is a tensorial operator , and applying the previous lemma , LaTeXMLMath such that LaTeXMLEquation and then LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then must be LaTeXMLMath , so LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath , this result appeared in [ Bel 95 ] .
Given LaTeXMLMath , then LaTeXMLMath if and only if LaTeXMLMath for some LaTeXMLMath .
Proof .
If LaTeXMLMath for some LaTeXMLMath , it is clear that LaTeXMLMath .
For the converse , we have in the proof of the preceding proposition LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
We present here two examples of Nambu-Poisson structures with their corresponding n-Lie algebroids .
As it is not very usual to find explicit examples in the literature , we shall be rather detailed .
Kepler Dynamics .
This example is based on [ MVV 98 ] .
It is well known that the Kepler dynamics has five first integrals , which are given by the components of the angular momentum and those of the Runge-Lenz vector .
In action-angle coordinates LaTeXMLMath , such integrals are LaTeXMLMath .
Call them LaTeXMLMath respectively .
In the space LaTeXMLMath with coordinates LaTeXMLMath , consider the Nambu-Poisson LaTeXMLMath -vector LaTeXMLEquation and the induced Nambu-Poisson bracket LaTeXMLEquation .
It is immediate to prove that the hamiltonian vector field corresponding to the multi-hamiltonian LaTeXMLMath is LaTeXMLEquation so that LaTeXMLMath for all LaTeXMLMath .
Now , we can describe the associated Filippov algebroid .
The vector bundle is LaTeXMLMath , the Filippov bracket on the space of sections LaTeXMLMath is given by LaTeXMLMath , where LaTeXMLMath is the Nambu-Poisson LaTeXMLMath -vector ( LaTeXMLRef ) ; explicitly , we would write for LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Finally , the anchor map LaTeXMLMath , acts as LaTeXMLEquation thus LaTeXMLEquation .
Bihamiltonian systems .
This example is adapted from [ Mag-Mag 91 ] , and it is a system of the Calogero-Moser type .
Consider two particles on a line , interacting through a potential proportional to the inverse square power of their distance LaTeXMLEquation .
The newtonian equations of motion are readily derived : LaTeXMLEquation .
These equations can also be obtained from a hamiltonian description .
It suffices to take the hamiltonian LaTeXMLEquation and the canonical symplectic form on the phase space LaTeXMLMath , with coordinates LaTeXMLMath : LaTeXMLEquation .
In fact , this system is a bihamiltonian one .
We could consider , along with LaTeXMLMath , another conserved quantity : the total momentum LaTeXMLEquation and the symplectic form LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and we would obtain the same evolution equations ( LaTeXMLRef ) .
What we are now going to do , is to construct a Nambu-Poisson system from this bihamiltonian one .
For this purpose , it is better to introduce new coordinates for the position of the particles : LaTeXMLEquation .
LaTeXMLEquation It is straightforward to prove that the hamiltonians now adopt the form LaTeXMLEquation .
LaTeXMLEquation ( the expressions for the symplectic forms are , of course , also changed ; for example now we have LaTeXMLEquation .
Consider the Nambu-Poisson multivector LaTeXMLEquation associated to which we have a Nambu-Poisson bracket LaTeXMLMath , so we can compute the hamiltonian vector field with respect to the LaTeXMLMath -hamiltonian LaTeXMLMath : this is the vector field LaTeXMLEquation as can be readily seen by evaluating LaTeXMLMath for an arbitrary function LaTeXMLMath .
Then , it is quite easy to show the Filippov algebroid corresponding to this structure .
It is given by the vector bundle LaTeXMLMath , and as in the previous example , the bracket on the space of sections is determined by LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and the anchor map LaTeXMLMath , acts as LaTeXMLEquation where LaTeXMLEquation
We define a Fourier-Mukai transform for Higgs bundles on smooth curves ( over LaTeXMLMath or another algebraically closed field ) and study its properties .
The transform of a stable degree- LaTeXMLMath Higgs bundle is an algebraic vector bundle on the cotangent bundle of the Jacobian of the curve .
We show that the transform admits a natural extension to an algebraic vector bundle over projective compactification of the base .
The main result is that the original Higgs bundle can be reconstructed from this extension .
Higgs bundles on a compact Riemann surface LaTeXMLMath are pairs LaTeXMLMath consisting of a holomorphic vector bundle LaTeXMLMath and a holomorphic one-form LaTeXMLMath with values in LaTeXMLMath on LaTeXMLMath .
They originated essentially simultaneously in Nigel Hitchin ’ s study LaTeXMLCite of dimensionally reduced self-duality equations of Yang-Mills gauge theory and , over general Kähler manifolds , in Carlos Simpson ’ s work LaTeXMLCite on Hodge theory .
To explain Hitchin ’ s point of view , we consider solutions of the self-dual Yang-Mills equations on LaTeXMLMath that are invariant under translations in one or more directions in LaTeXMLMath .
Invariance in one direction reduces the SDYM equations to the Bogomolnyi equations describing magnetic monopoles in LaTeXMLMath , while invariance in three directions leads to Nahm ’ s equation on LaTeXMLMath .
Invariance in two directions leads to the conformally invariant Hitchin ’ s equations ( or Higgs bundle equations ) , which the conformal invariance allows to be considered on compact Riemann surfaces .
A solution to Hitchin ’ s equations has an interpretation as a triple LaTeXMLMath with LaTeXMLMath a holomorphic vector bundle , LaTeXMLMath a holomorphic section of LaTeXMLMath and LaTeXMLMath a Hermitian metric on LaTeXMLMath , satisfying LaTeXMLMath , where LaTeXMLMath is the curvature of the connection determined by the metric .
It was shown in Hitchin LaTeXMLCite for rank- LaTeXMLMath bundles on Riemann surfaces , and in Simpson LaTeXMLCite in general , that a pair LaTeXMLMath admits a unique such metric precisely when LaTeXMLMath has vanishing Chern classes and the pair LaTeXMLMath is stable in a sense which generalises the usual stability for vector bundles .
For an excellent overview of Simpson ’ s viewpoint of non-Abelian cohomology , see Simpson LaTeXMLCite .
An important class of transforms in Yang-Mills theory , including the ADHM construction and the Fourier transform for instantons ( Donaldson-Kronheimer LaTeXMLCite ) and the Nahm transform for monopoles , is based on using the kernel of the Dirac operator coupled to the ( twisted ) connection .
A recent work of Marcos Jardim LaTeXMLCite uses a version of the Nahm transform to establish a link between singular Higgs bundles on a LaTeXMLMath -torus and doubly-periodic instantons .
Our goal is to generalise this work to Riemann surfaces of genus LaTeXMLMath .
In this paper we shall consider the purely holomorphic aspect of the transform ; we plan to return to the properly gauge-theoretic questions in a future paper .
The holomorphic side of the Nahm transform is captured by the ( generalised ) Fourier-Mukai transform , which originated in the work of Shigeru Mukai LaTeXMLCite on Abelian varieties .
Let LaTeXMLMath be a complex torus and LaTeXMLMath its dual , and let LaTeXMLMath and LaTeXMLMath denote the derived categories of the categories of coherent sheaves on LaTeXMLMath and LaTeXMLMath respectively .
Using the Poincaré sheaf LaTeXMLMath on LaTeXMLMath , Mukai defined a functor LaTeXMLMath by LaTeXMLEquation and showed that it is a category equivalence .
This construction can be generalised to any varieties LaTeXMLMath and LaTeXMLMath together with a sheaf LaTeXMLMath on LaTeXMLMath .
The properties of these generalisations have been studied by A. Bondal and D. Orlov LaTeXMLCite , A. Maciocia LaTeXMLCite , T. Bridgeland LaTeXMLCite , and others .
We interpret the endomorphism-valued one-form LaTeXMLMath as a bundle map , making a Higgs bundle LaTeXMLMath into a sheaf complex LaTeXMLMath , where LaTeXMLMath is the canonical sheaf of LaTeXMLMath .
Hence a Higgs bundle gives us an object of the derived category LaTeXMLMath , and we can use the general machinery of Fourier-Mukai transforms .
The key observation is that it is necessary to consider relative transforms of families of Higgs bundles twisted by adding a scalar term to the Higgs field LaTeXMLMath .
Our transform produces sheaves on LaTeXMLMath , where LaTeXMLMath is the Jacobian of LaTeXMLMath .
This base can be identified with the cotangent bundle of the Jacobian .
While the motivation for the present work comes from differential and complex analytic geometry , we are actually working within the framework of algebraic geometry , noting that on an algebraic curve the Higgs bundle data is purely algebraic .
Translation between these categories is provided by Serre ’ s GAGA LaTeXMLCite .
Our approach has the advantage that all constructions are automatically algebraic ( or holomorphic ) , while the fact that we are dealing with rather high-dimensional base spaces would make some of the analytic techniques of Jardim hard to use .
The first part of this paper develops the machinery of generalised Fourier-Mukai transforms .
While some of the material presented in section 2 can not be found in the literature , it is mostly well known .
The main new contributions are the definition of a relative Fourier transform for curves and the reduction of it to the original Mukai transform .
The transform for Higgs bundles is developed in section 3 .
The first interesting application is that our Fourier transform takes stable Higgs bundles of degree zero to vector bundles on LaTeXMLMath .
Our approach has the advantage of giving directly an algebraic ( holomorphic ) extension of this bundle over the projective completion LaTeXMLMath of the base space , without a need to separately compactify a bundle on LaTeXMLMath .
Denote this extension of the transform of a Higgs bundle LaTeXMLMath by LaTeXMLMath .
The main theorem of this paper is the following : Let LaTeXMLMath and LaTeXMLMath be two Higgs bundles on a curve LaTeXMLMath of genus LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath as Higgs bundles .
We in fact prove this theorem by exhibiting a procedure for recovering a Higgs bundle from its transform .
Furthermore , it follows easily from the theorem that the transform functor is in fact fully faithful .
Acknowledgements .
The original idea of developing a Fourier transform for Higgs bundles is due to Nigel Hitchin LaTeXMLCite .
I am deeply grateful to him for generous comments , support and encouragement .
Unless otherwise specified , all rings and algebras are commutative and unital .
We fix an algebraically closed field LaTeXMLMath .
All schemes are assumed to be of finite type over LaTeXMLMath .
All morphisms are LaTeXMLMath -morphisms and all products are products over LaTeXMLMath unless stated otherwise .
A curve always means a smooth irreducible complete ( i.e. , projective ) curve over LaTeXMLMath .
If LaTeXMLMath is an LaTeXMLMath -module , LaTeXMLMath denotes its dual LaTeXMLMath .
The canonical sheaf of a curve LaTeXMLMath is denoted by LaTeXMLMath .
LaTeXMLMath denotes the derived category of the ( Abelian ) category of LaTeXMLMath -modules .
LaTeXMLMath ( resp .
LaTeXMLMath , resp .
LaTeXMLMath ) is the full subcategory of objects cohomologically bounded above ( resp .
bounded below , resp .
bounded ) .
LaTeXMLMath and LaTeXMLMath are the full subcategories of objects with quasi-coherent and coherent cohomology objects , respectively .
These superscripts and subscripts can be combined in the obvious way .
The category of LaTeXMLMath -modules is denoted by LaTeXMLMath , and LaTeXMLMath is the thick subcategory of quasi-coherent sheaves .
A commutative square LaTeXMLEquation is called Cartesian if the mapping LaTeXMLMath is an isomorphism .
We denote canonical isomorphisms often by `` LaTeXMLMath `` .
We develop here the general machinery of Fourier-Mukai transforms that will be necessary for our application to Higgs bundles .
We will be using the theory of derived categories ; the main reference to derived categories in algebraic geometry remains Hartshorne ’ s seminar LaTeXMLCite on Grothen-dieck ’ s duality theory .
Further references include Gelfand-Manin LaTeXMLCite , Kashiwara-Shapira LaTeXMLCite and Weibel LaTeXMLCite .
For a nice informal introduction , see Illusie LaTeXMLCite or the introduction of Verdier ’ s thesis LaTeXMLCite .
Consider the following diagram of schemes ( here not necessarily of finite type over a field ) : LaTeXMLEquation with LaTeXMLMath .
Recall the external tensor product over LaTeXMLMath of an LaTeXMLMath -module LaTeXMLMath and an LaTeXMLMath -module LaTeXMLMath : LaTeXMLEquation .
We get the corresponding left-derived bifunctor LaTeXMLEquation .
The following theorem should be part of folklore ; we include a proof of it for the lack of a suitable reference .
It is essentially the derived-category version of a part of Grothendieck ’ s theory of `` global hypertor functors '' ( EGA III LaTeXMLCite , §6 ) .
For LaTeXMLMath let LaTeXMLMath be an object of LaTeXMLMath .
Assume that the schemes are Noetherian and of finite dimension , and that the LaTeXMLMath are separated .
Then LaTeXMLEquation if either LaTeXMLMath or LaTeXMLMath is quasi-isomorphic to a complex of LaTeXMLMath -flat sheaves .
This is true in particular if either LaTeXMLMath or LaTeXMLMath is flat over LaTeXMLMath .
The Noetherian and dimensional hypotheses guarantee that the derived direct images are defined for complexes not bounded below .
There are natural `` adjunction '' maps LaTeXMLMath giving LaTeXMLEquation .
Notice that LaTeXMLEquation .
LaTeXMLEquation Now the adjunctions LaTeXMLMath give a natural map LaTeXMLEquation .
LaTeXMLEquation Composing gives us a natural transformation LaTeXMLEquation .
Whether this is an isomorphism is a local question ; hence we may assume that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Suppose LaTeXMLMath is quasi-isomorphic to a complex of LaTeXMLMath -flat sheaves ; replace LaTeXMLMath with this flat resolution .
Then LaTeXMLMath .
For LaTeXMLMath let LaTeXMLMath be a finite affine open cover of LaTeXMLMath .
Let LaTeXMLMath denote the open affine cover LaTeXMLMath of LaTeXMLMath .
Notice that in all these covers arbitrary intersections of the covering sets are affine .
Let LaTeXMLMath denote the simple complex associated to the Čech double complex of LaTeXMLMath with respect to LaTeXMLMath .
Similarly , let LaTeXMLMath be the Čech complex with respect to LaTeXMLMath .
Now LaTeXMLMath is quasi-isomorphic to LaTeXMLMath , and hence LaTeXMLMath is quasi-isomorphic to LaTeXMLMath .
But the sheaves of these complexes are LaTeXMLMath -flat by construction , whence LaTeXMLEquation .
Similarly LaTeXMLEquation .
Hence we are reduced to showing that the complex LaTeXMLMath is quasi-isomorphic to LaTeXMLMath .
But this is showed in the proof of ( 6.7.6 ) of EGA III LaTeXMLCite .
∎ If one wants to avoid the Noetherian hypothesis in the theorem , one could work with objects LaTeXMLMath of LaTeXMLMath and require the LaTeXMLMath to be quasi-compact .
This is essentially the viewpoint of EGA III .
Let LaTeXMLMath and LaTeXMLMath be morphisms of finite-dimensional Noetherian schemes .
Let LaTeXMLMath and LaTeXMLMath be the projections , and let LaTeXMLMath belong to LaTeXMLMath .
If LaTeXMLMath is quasi-isomorphic to a complex of LaTeXMLMath -flat sheaves ( in particular , if LaTeXMLMath is flat ) , then LaTeXMLEquation .
If LaTeXMLMath is flat , then LaTeXMLEquation .
Apply the Künneth formula with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
∎ Let LaTeXMLMath be a separated LaTeXMLMath -scheme and let LaTeXMLMath and LaTeXMLMath be flat LaTeXMLMath -schemes .
If LaTeXMLMath is an object of LaTeXMLMath , the relative integral transform defined by LaTeXMLMath is the functor LaTeXMLMath given by LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the canonical projections of LaTeXMLMath .
When LaTeXMLMath we call the transform the absolute integral transform and denote it by LaTeXMLMath .
Let LaTeXMLMath be the morphism LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLEquation .
We have the commutative diagram LaTeXMLEquation .
Notice that because both LaTeXMLMath and LaTeXMLMath are flat morphisms , we have LaTeXMLEquation .
Using this and the projection formula , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
But LaTeXMLMath fits in a Cartesian square LaTeXMLEquation .
As LaTeXMLMath is separated , LaTeXMLMath is a closed immersion , and consequently so is LaTeXMLMath .
In particular , LaTeXMLMath is an exact functor and therefore equal to LaTeXMLMath .
Hence LaTeXMLEquation as claimed .
∎ We can not avoid using the derived tensor product in the above result , even if LaTeXMLMath is a locally free sheaf , because LaTeXMLMath is not flat in general .
However , as LaTeXMLMath is proper , LaTeXMLMath belongs always to LaTeXMLMath .
For flat LaTeXMLMath -schemes LaTeXMLMath and LaTeXMLMath and for LaTeXMLMath , let LaTeXMLMath denote the fibre LaTeXMLMath , where LaTeXMLMath is the canonical projection .
We have then a commutative diagram LaTeXMLEquation in which all squares are Cartesian .
Let LaTeXMLMath denote the composition of the top arrows .
For an object LaTeXMLMath of LaTeXMLMath ( resp .
LaTeXMLMath ) , we denote by LaTeXMLMath the `` restriction '' LaTeXMLMath ( resp .
LaTeXMLMath ) to LaTeXMLMath .
For complexes of locally free sheaves these are just ordinary restrictions to LaTeXMLMath .
If LaTeXMLMath is a locally free sheaf on LaTeXMLMath , then for each LaTeXMLMath LaTeXMLEquation where LaTeXMLMath is the skyscraper sheaf LaTeXMLMath at LaTeXMLMath .
Indeed , consider the commutative diagram above : the claim follows from flat base change around the left-hand square and the projection formula applied to LaTeXMLMath .
Notice that LaTeXMLMath is exact .
Let LaTeXMLMath be an Abelian variety , LaTeXMLMath its dual , and let LaTeXMLMath be a separated scheme .
Let LaTeXMLMath be the Poincaré sheaf on LaTeXMLMath , normalised so that both LaTeXMLMath and LaTeXMLMath are the trivial line bundles .
Denote by LaTeXMLMath the pull-back of this Poincaré sheaf to LaTeXMLMath .
The relative Mukai transform functor LaTeXMLMath is the relative integral transform functor LaTeXMLMath .
If LaTeXMLMath , we denote the transform by LaTeXMLMath .
The following theorem of Mukai plays a crucial role in the proof of our invertibility result ( LaTeXMLRef ) .
If LaTeXMLMath is a smooth projective variety , then the relative Mukai transform LaTeXMLMath is an equivalence of categories from LaTeXMLMath to LaTeXMLMath .
See Mukai LaTeXMLCite .
The proof is a generalisation of Mukai ’ s original proof of this result for the absolute transform LaTeXMLMath in LaTeXMLCite .
∎ Let LaTeXMLMath and LaTeXMLMath be flat LaTeXMLMath -schemes and LaTeXMLMath an object of LaTeXMLMath .
Let LaTeXMLMath be a morphism of schemes .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath be the canonical projections .
Then LaTeXMLEquation .
Moreover , if LaTeXMLMath is a flat morphism , then all derived pull-backs above can be replaced with normal pull-backs .
Consider the commutative diagram LaTeXMLEquation .
It is immediate that all squares are Cartesian .
If LaTeXMLMath is flat , then so are LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ; this proves the claim about replacing derived pull-backs .
Since in any case LaTeXMLMath is flat , LaTeXMLMath is also flat .
So by ( LaTeXMLRef ) we can do a base change around the leftmost square .
We get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ∎ Let LaTeXMLMath and LaTeXMLMath be flat LaTeXMLMath -schemes and LaTeXMLMath an object of LaTeXMLMath .
Then LaTeXMLEquation .
We simply use the composition property of derived functors and the projection formula : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ∎ Let LaTeXMLMath and LaTeXMLMath be proper flat LaTeXMLMath schemes .
We fix a locally free sheaf LaTeXMLMath on LaTeXMLMath , and denote by LaTeXMLMath the relative integral transform functor LaTeXMLMath .
We leave it to the reader to generalise the results of this subsection to a more general setting .
We say that an object LaTeXMLMath of LaTeXMLMath is a LaTeXMLMath -complex Following Mukai , `` WIT '' stands for `` weak index theorem '' .
if LaTeXMLMath for all LaTeXMLMath .
If LaTeXMLMath is clear from the context , we shall omit the explicit reference to it .
An object of LaTeXMLMath is a WIT- complex if it is a LaTeXMLMath -complex for some LaTeXMLMath .
If LaTeXMLMath is a LaTeXMLMath -complex on LaTeXMLMath , the ( coherent ) sheaf LaTeXMLMath on LaTeXMLMath is called the integral transform of LaTeXMLMath , and is denoted by LaTeXMLMath .
We say that an object LaTeXMLMath of LaTeXMLMath is an LaTeXMLMath -complex if for each ( closed ) point LaTeXMLMath and each LaTeXMLMath we have LaTeXMLEquation where we are using the notation of ( LaTeXMLRef ) for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a proper morphism of ( Noetherian ) schemes and let LaTeXMLMath be an object of LaTeXMLMath which has a LaTeXMLMath -flat resolution .
Let LaTeXMLMath .
Then : if the natural map LaTeXMLMath is surjective , then it is an isomorphism .
If LaTeXMLMath is an isomorphism , then LaTeXMLMath is also an isomorphism if and only if LaTeXMLMath is free in a open neighbourhood of LaTeXMLMath .
This follows from EGA III LaTeXMLCite §7 .
However , that part of EGA can be somewhat hard to read ; one could also follow the simpler proof of Hartshorne LaTeXMLCite Theorem III.12.11 , making the fairly minor and obvious adjustments for hypercohomology .
∎ Let LaTeXMLMath be an LaTeXMLMath complex .
Then LaTeXMLMath is a LaTeXMLMath -complex , and LaTeXMLMath is locally free on LaTeXMLMath .
Our schemes are Jacobson , and so it suffices to restrict our attention to closed points .
Since LaTeXMLMath is flat , LaTeXMLMath is quasi-isomorphic to a complex of sheaves flat over LaTeXMLMath .
Moreover , LaTeXMLMath is proper over LaTeXMLMath , and so LaTeXMLMath is a proper morphism .
We are then in position to use ( LaTeXMLRef ) .
Let LaTeXMLMath be a closed point .
Now LaTeXMLEquation on LaTeXMLMath .
Hence by hypothesis the natural map LaTeXMLEquation is trivially surjective , and by the base change theorem in fact isomorphic , for all LaTeXMLMath .
As the hyper direct images of a complex of coherent sheaves are coherent for a proper map , we have LaTeXMLEquation for LaTeXMLMath by Nakayama ’ s lemma .
This proves the first part of the proposition .
Now in particular LaTeXMLMath .
Thus , by the second part of the base change theorem , LaTeXMLMath is an isomorphism .
But as LaTeXMLMath is also surjective and thus isomorphic , LaTeXMLMath is free in a neighbourhood of LaTeXMLMath , again by the second part of ( LaTeXMLRef ) .
∎ Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be as in ( LaTeXMLRef ) , and let LaTeXMLMath be a morphism of schemes .
Suppose that LaTeXMLMath is an LaTeXMLMath -complex on LaTeXMLMath .
Then , in the notation of ( LaTeXMLRef ) , LaTeXMLMath is a LaTeXMLMath -complex with respect to the pull-back LaTeXMLMath of LaTeXMLMath to LaTeXMLMath .
Furthermore , if LaTeXMLMath denotes the corresponding Fourier transform , then LaTeXMLEquation .
By the assumptions and ( LaTeXMLRef ) , LaTeXMLMath is a locally free sheaf shifted LaTeXMLMath places to the right .
Hence ( LaTeXMLRef ) gives LaTeXMLEquation .
But this shows that LaTeXMLMath is also a locally free sheaf shifted LaTeXMLMath places to the right .
Both statements of the proposition are now immediate .
∎ To fix terminology and notation , we first recall some basic facts about Jacobians of curves ; for details , see Milne LaTeXMLCite .
Let LaTeXMLMath be a smooth projective curve of genus LaTeXMLMath .
We denote by LaTeXMLMath a Jacobian of LaTeXMLMath , i.e. , a scheme representing the functor LaTeXMLMath .
Let LaTeXMLMath be the corresponding universal sheaf on LaTeXMLMath .
Recall that LaTeXMLMath is an Abelian variety of dimension LaTeXMLMath ; let LaTeXMLMath denote its dual Abelian variety , and let LaTeXMLMath be the Poincaré sheaf on LaTeXMLMath , normalised as in ( LaTeXMLRef ) .
Choosing a base point LaTeXMLMath gives the Abel-Jacobi map LaTeXMLMath , taking the base point to LaTeXMLMath .
Notice that LaTeXMLMath is a closed immersion .
Furthermore , this choice gives LaTeXMLMath a principal polarisation and hence an isomorphism LaTeXMLMath , which we use henceforth to identify LaTeXMLMath with its dual .
Under this identification , the pull-back LaTeXMLMath is just the universal sheaf LaTeXMLMath on LaTeXMLMath .
Let LaTeXMLMath be a separated LaTeXMLMath -scheme , LaTeXMLMath , and let LaTeXMLMath be the relative Jacobian of the trivial family LaTeXMLMath .
We have a Cartesian square LaTeXMLEquation .
Let LaTeXMLMath be the pull-back of LaTeXMLMath to LaTeXMLMath .
The relative integral transform functor LaTeXMLMath is given by LaTeXMLEquation where we can use the ordinary tensor product since LaTeXMLMath is locally free .
The relative integral transform LaTeXMLMath is called the relative Fourier functor on LaTeXMLMath and is denoted by LaTeXMLMath .
If LaTeXMLMath is LaTeXMLMath with respect to LaTeXMLMath , the integral transform LaTeXMLMath is called the Fourier transform of LaTeXMLMath .
Let LaTeXMLMath denote the relative Mukai transform .
Then LaTeXMLEquation .
Consider the diagram LaTeXMLEquation where the right-hand square is the fibre-product diagram and LaTeXMLMath .
It is clear that the left-hand square is also commutative , and that the composition of the two top arrows is just the canonical projection LaTeXMLMath .
But this means that the big rectangle is Cartesian , and hence so is the left-hand square too .
By definition , LaTeXMLEquation where LaTeXMLMath is the pull-back of the Poincaré sheaf onto LaTeXMLMath .
Clearly LaTeXMLMath .
Now by the projection formula LaTeXMLEquation .
Because LaTeXMLMath is flat as a base extension of a flat morphism , we can do a base change ( LaTeXMLRef ) around the left-hand square to get LaTeXMLEquation .
But LaTeXMLMath is a closed immersion and thus LaTeXMLMath .
Putting these observations together , we get LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ∎ Let LaTeXMLMath be a curve of genus LaTeXMLMath and choose a base point LaTeXMLMath as in ( LaTeXMLRef ) ; we suppose made the identifications given loc .
cit .
Let LaTeXMLMath be a LaTeXMLMath -scheme , and denote by LaTeXMLMath the embedding LaTeXMLMath of the fibre over LaTeXMLMath .
Let LaTeXMLMath be a bounded complex of locally free sheaves on LaTeXMLMath .
Then LaTeXMLEquation .
By ( LaTeXMLRef ) we have natural isomorphisms LaTeXMLEquation for all LaTeXMLMath .
With the notation of the proposition , LaTeXMLMath is the zero-differential complex LaTeXMLMath where LaTeXMLMath is the direct sum of LaTeXMLMath copies of LaTeXMLMath for LaTeXMLMath , zero otherwise .
Consider the Cartesian square LaTeXMLEquation .
By flat base change around the square we get LaTeXMLEquation .
In order to compute LaTeXMLMath on LaTeXMLMath , we consider the Cartesian square LaTeXMLEquation .
Now by the general base-change ( LaTeXMLRef ) we have LaTeXMLEquation .
LaTeXMLEquation But LaTeXMLMath , the skyscraper sheaf at LaTeXMLMath shifted LaTeXMLMath places to the right ( see the proof of the theorem of §13 in Mumford LaTeXMLCite ) .
Notice that LaTeXMLMath is a regular embedding ; using Koszul resolutions it follows that LaTeXMLMath , the zero-differential exterior-algebra complex of the conormal sheaf of LaTeXMLMath in LaTeXMLMath , concentrated in degrees LaTeXMLMath to LaTeXMLMath .
Similarly LaTeXMLMath , the exterior algebra of the fibre at LaTeXMLMath , whence the lemma follows immediately taking into account the shift by LaTeXMLMath .
Using the projection formula we have LaTeXMLEquation .
The proposition now follows from the lemma because hypercohomology commutes with direct sums .
∎ We shall now apply the Fourier-transform machinery developed in the previous section to stable Higgs bundles on curves .
A Higgs bundle on a smooth projective curve is a pair LaTeXMLMath , where LaTeXMLMath is a locally free sheaf on LaTeXMLMath , and LaTeXMLMath is a morphism LaTeXMLMath .
The morphism LaTeXMLMath is often called the Higgs field .
The Higgs bundle LaTeXMLMath is called trivial .
The rank and degree ( i.e. , the first Chern class ) of a Higgs bundle LaTeXMLMath mean the rank and degree of the underlying sheaf LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are Higgs bundles , by a morphism LaTeXMLMath we understand a morphism of sheaves LaTeXMLMath making the square LaTeXMLEquation commutative .
Let LaTeXMLMath be a Higgs bundle on LaTeXMLMath .
Then we can consider it as a complex of sheaves concentrated in degrees LaTeXMLMath and LaTeXMLMath , and hence as an object in LaTeXMLMath .
When we write LaTeXMLMath or LaTeXMLMath etc. , we consider the Higgs bundle as a sheaf complex this way .
Notice that the image of LaTeXMLMath in LaTeXMLMath does not uniquely determine the isomorphism class of the Higgs bundle LaTeXMLMath .
In fact , multiplying LaTeXMLMath by a non-zero constant gives a quasi-isomorphic complex ; however , the resulting Higgs bundle is not in general isomorphic .
A Higgs bundle LaTeXMLMath is called stable if for any locally free subsheaf LaTeXMLMath of LaTeXMLMath satisfying LaTeXMLMath , we have LaTeXMLEquation .
Let LaTeXMLMath be a non-trivial stable Higgs bundle on LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLEquation for LaTeXMLMath .
Hausel LaTeXMLCite Corollary ( 5.1.4 . ) .
Notice that LaTeXMLMath automatically for LaTeXMLMath because LaTeXMLMath and the length of the complex LaTeXMLMath is LaTeXMLMath .
∎ If a Higgs bundle LaTeXMLMath is stable , then so is LaTeXMLMath , where LaTeXMLMath is an element of LaTeXMLMath .
Let LaTeXMLMath be a subbundle stable under LaTeXMLMath .
Then LaTeXMLMath is a subbundle of LaTeXMLMath stable under LaTeXMLMath .
But tensoring with LaTeXMLMath affects neither the ranks nor the degrees of LaTeXMLMath and LaTeXMLMath , and hence the lemma follows from the stability of the Higgs bundle LaTeXMLMath .
∎ Let LaTeXMLMath be a Higgs bundle and LaTeXMLMath a global LaTeXMLMath -form .
Then LaTeXMLMath is canonically identified with a morphism LaTeXMLMath .
We denote the Higgs bundle LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be a stable Higgs bundle .
Then LaTeXMLMath is also stable for any LaTeXMLMath .
Let LaTeXMLMath be a subbundle stable under LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLMath .
But LaTeXMLMath too , and hence LaTeXMLMath .
Thus LaTeXMLMath is stable under LaTeXMLMath , and the lemma follows from the stability of LaTeXMLMath .
∎ We shall now introduce an important construction of algebraic families of Higgs bundles .
For details about projective bundles see for example EGA II LaTeXMLCite §4 .
Let LaTeXMLMath be a Higgs bundle on a curve LaTeXMLMath of genus LaTeXMLMath , and let LaTeXMLMath be the structural morphism .
Then the LaTeXMLMath -rational points of the vector bundle ( or affine space ) LaTeXMLMath are canonically identified with the elements of LaTeXMLMath ; we use the notation LaTeXMLMath also for this scheme if no confusion seems likely .
Let LaTeXMLMath ; we have the canonical adjunction morphism LaTeXMLEquation .
Let LaTeXMLMath be the morphism LaTeXMLEquation .
On the other hand , let LaTeXMLMath be the map that takes LaTeXMLMath to LaTeXMLMath .
Putting these together we get a morphism LaTeXMLEquation .
Because LaTeXMLMath , we have a canonical isomorphism LaTeXMLEquation .
Let LaTeXMLMath be the projection .
There is the canonical surjection LaTeXMLMath , and so by dualising a canonical LaTeXMLMath .
Composing this morphism with LaTeXMLMath we get a morphism LaTeXMLEquation or in other words a global section of LaTeXMLMath .
We interpret this section as a morphism LaTeXMLEquation and denote this complex of sheaves ( in degrees LaTeXMLMath and LaTeXMLMath ) on LaTeXMLMath by LaTeXMLMath .
In more pedestrian terms , let LaTeXMLMath be a basis of LaTeXMLMath , and let LaTeXMLMath be the dual basis of LaTeXMLMath .
Let LaTeXMLMath be the canonical coordinate on LaTeXMLMath ; then LaTeXMLMath forms a basis of the global sections of LaTeXMLMath , and LaTeXMLMath corresponds to the open affine subscheme of LaTeXMLMath with LaTeXMLMath .
Now LaTeXMLEquation .
Notice that for LaTeXMLMath the restriction of LaTeXMLMath to LaTeXMLMath is just LaTeXMLMath of ( LaTeXMLRef ) .
Let LaTeXMLMath be a stable Higgs bundle of degree LaTeXMLMath and rank LaTeXMLMath on a curve LaTeXMLMath of genus LaTeXMLMath .
Then the complex LaTeXMLMath on LaTeXMLMath is LaTeXMLMath with respect to the relative Fourier functor LaTeXMLMath .
Moreover , the Fourier transform LaTeXMLMath is a locally free sheaf on LaTeXMLMath .
By ( LaTeXMLRef ) we are reduced to showing that LaTeXMLMath is LaTeXMLMath with respect to LaTeXMLMath .
We consider two cases .
Let LaTeXMLMath denote the open subset LaTeXMLMath in LaTeXMLMath .
A ) Let LaTeXMLMath .
Then ( using the notation of ( LaTeXMLRef ) ) LaTeXMLEquation and we need to show that LaTeXMLEquation for LaTeXMLMath .
But this follows from ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Notice that for a rank- LaTeXMLMath Higgs bundle LaTeXMLMath one of the bundles LaTeXMLMath would be trivial , and the vanishing theorem ( LaTeXMLRef ) would fail .
B ) Let LaTeXMLMath .
We consider the second hypercohomology spectral sequence : LaTeXMLEquation .
But LaTeXMLEquation for a LaTeXMLMath -form LaTeXMLMath , determined up to multiplication by a non-zero scalar .
Now LaTeXMLMath is clearly an injective map of sheaves ; let LaTeXMLMath be its cokernel .
Thus the LaTeXMLMath -terms of the spectral sequence are LaTeXMLEquation .
But LaTeXMLMath is a direct sum of skyscraper sheaves supported on the divisor of zeroes of the one-form , and since skyscraper sheaves are flasque , we have LaTeXMLMath .
Hence LaTeXMLMath .
∎ Let LaTeXMLMath be a stable Higgs bundle of degree LaTeXMLMath and rank LaTeXMLMath on a curve LaTeXMLMath of genus LaTeXMLMath .
Then the locally free sheaf LaTeXMLMath on LaTeXMLMath is called ( by abuse of language ) the total Fourier transform of LaTeXMLMath and is denoted by LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be as in ( LaTeXMLRef ) , and let LaTeXMLMath .
Then LaTeXMLEquation where the left-hand side denotes the absolute Fourier transform .
By the proof of ( LaTeXMLRef ) LaTeXMLMath is LaTeXMLMath .
Now the proposition follows from Remark ( LaTeXMLRef ) and Proposition ( LaTeXMLRef ) applied to the immersion LaTeXMLMath .
∎ Let LaTeXMLMath be a non-trivial stable Higgs bundle of degree LaTeXMLMath on a curve LaTeXMLMath of genus LaTeXMLMath .
Then the rank of the total Fourier transform LaTeXMLMath is LaTeXMLMath .
It follows from ( LaTeXMLRef ) and ( LaTeXMLRef ) that LaTeXMLMath .
Consider the first hypercohomology spectral sequence LaTeXMLEquation .
The LaTeXMLMath -terms of the sequence are : LaTeXMLEquation .
The sequence clearly degenerates at LaTeXMLMath , i.e. , LaTeXMLMath , and hence LaTeXMLEquation .
LaTeXMLEquation But these hypercohomologies vanish by ( LaTeXMLRef ) , and thus LaTeXMLMath is injective and LaTeXMLMath is surjective .
On the other hand , LaTeXMLEquation and hence LaTeXMLEquation .
But as LaTeXMLMath , the Riemann-Roch theorem gives LaTeXMLEquation .
LaTeXMLEquation whence the result follows immediately .
∎ Let LaTeXMLMath be a stable Higgs bundle of rank LaTeXMLMath and degree LaTeXMLMath on a curve LaTeXMLMath of genus LaTeXMLMath .
Then LaTeXMLEquation when LaTeXMLMath , and zero otherwise .
Let LaTeXMLMath be a base point giving an embedding LaTeXMLMath , and denote by LaTeXMLMath the embedding LaTeXMLMath of the fibre LaTeXMLMath .
Then by ( LaTeXMLRef ) LaTeXMLEquation .
We apply the first hypercohomology spectral sequence LaTeXMLEquation .
The LaTeXMLMath -terms are given by LaTeXMLEquation .
The standard results on the cohomology of a projective space ( Hartshorne LaTeXMLCite III.5.1 ) show that the LaTeXMLMath .
Furthermore , it is clear from the definition ( LaTeXMLRef ) of LaTeXMLMath that LaTeXMLMath is an injection .
Thus we see that LaTeXMLEquation .
Thus in the direct sum of ( LaTeXMLRef ) we have non-zero cohomology only when LaTeXMLMath , and the result follows immediately .
∎ Let LaTeXMLMath be a stable non-trivial Higgs bundle on a smooth projective curve LaTeXMLMath of genus LaTeXMLMath , with LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation where LaTeXMLMath is the class of the LaTeXMLMath -divisor on LaTeXMLMath .
This is an easy application of the Grothendieck-Riemann-Roch formula .
∎ Let LaTeXMLMath and LaTeXMLMath be two Higgs bundles on a curve LaTeXMLMath of genus LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath as Higgs bundles .
We show this by actually exhibiting a process of recovering a Higgs bundle LaTeXMLMath from its total Fourier transform LaTeXMLMath .
Step LaTeXMLMath .
Choose a base point LaTeXMLMath as in ( LaTeXMLRef ) , and let LaTeXMLMath be the corresponding embedding .
Denote by LaTeXMLMath the immersion LaTeXMLMath .
Then by ( LaTeXMLRef ) LaTeXMLMath .
By ( LaTeXMLRef ) LaTeXMLMath is a category equivalence ; let LaTeXMLMath be its inverse .
Now by definition LaTeXMLMath , and hence LaTeXMLEquation .
The differential LaTeXMLMath of the complex LaTeXMLMath is injective .
Let LaTeXMLMath be an open subset and LaTeXMLMath a non-zero section .
There is a point LaTeXMLMath for which LaTeXMLMath .
Because LaTeXMLMath is locally free , it follows ( using Nakayama ’ s lemma ) that there is an open neighbourhood LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
If LaTeXMLMath , it follows from the definition of LaTeXMLMath that there is a point LaTeXMLMath with LaTeXMLMath , and in particular LaTeXMLMath .
But this shows that LaTeXMLMath is injective as a morphism of presheaves and hence as a sheaf morphism too .
Thus the lemma is proved .
By the lemma there is an exact sequence LaTeXMLEquation and consequently LaTeXMLMath is quasi-isomorphic to LaTeXMLMath .
It follows from this that LaTeXMLMath in LaTeXMLMath .
Since LaTeXMLMath is an honest sheaf , LaTeXMLMath also in LaTeXMLMath .
This means that we can recover the cokernel LaTeXMLMath of LaTeXMLMath on LaTeXMLMath as LaTeXMLMath .
Step LaTeXMLMath .
Tensor ( LaTeXMLRef ) with LaTeXMLMath and obtain the exact sequence LaTeXMLEquation .
We shall use the long exact LaTeXMLMath -sequence associated to ( LaTeXMLRef ) .
By the projection formula LaTeXMLEquation .
LaTeXMLEquation Now it follows from base change and the standard formulas for the cohomology of projective spaces that LaTeXMLEquation .
LaTeXMLEquation It follows then from the long exact sequence that LaTeXMLMath , and that we may consequently recover the underlying sheaf LaTeXMLMath of LaTeXMLMath from LaTeXMLMath by twisting by LaTeXMLMath , projecting down to LaTeXMLMath , and twisting by LaTeXMLMath .
Step LaTeXMLMath .
It remains to recover the Higgs field LaTeXMLMath .
This will be done after discarding much of the information contained in LaTeXMLMath .
We choose a non-zero LaTeXMLMath , and we let LaTeXMLMath be an open affine subscheme of LaTeXMLMath over which LaTeXMLMath does not vanish ; then LaTeXMLMath gives a trivialisation of LaTeXMLMath on LaTeXMLMath .
Clearly it is enough to recover LaTeXMLMath over LaTeXMLMath .
Let LaTeXMLMath be the subvectorspace of LaTeXMLMath generated by LaTeXMLMath .
We can consider LaTeXMLMath as a closed subscheme of the open subscheme LaTeXMLMath of LaTeXMLMath .
Furthermore , we consider LaTeXMLMath as a subscheme of LaTeXMLMath , and let LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath ; it is just the cokernel of LaTeXMLMath restricted to LaTeXMLMath .
Notice that LaTeXMLMath .
On LaTeXMLMath the underlying sheaf LaTeXMLMath of LaTeXMLMath corresponds to an LaTeXMLMath -module LaTeXMLMath and LaTeXMLMath corresponds to an endomorphism LaTeXMLMath of LaTeXMLMath .
Furthermore , the pull-back of LaTeXMLMath to LaTeXMLMath corresponds to LaTeXMLMath .
By the definition of LaTeXMLMath ( LaTeXMLRef ) , LaTeXMLMath corresponds to the LaTeXMLMath -linear map LaTeXMLEquation .
But LaTeXMLMath fits into the exact sequence LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath -module with LaTeXMLMath acting on LaTeXMLMath as LaTeXMLMath ( cf .
Bourbaki LaTeXMLCite , Ch .
III §8 no .
10 ) .
Hence LaTeXMLMath .
But the structure of LaTeXMLMath -module of LaTeXMLMath determines LaTeXMLMath and hence LaTeXMLMath .
∎ Lemma 6.8 in Simpson LaTeXMLCite gives a description of Higgs bundles on LaTeXMLMath as coherent sheaves on the total space of the cotangent bundle of LaTeXMLMath .
The scheme LaTeXMLMath in Step 3 of the proof is the total space of the cotangent bundle of LaTeXMLMath , and the coherent sheaf LaTeXMLMath on LaTeXMLMath is the one that corresponds to LaTeXMLMath under Simpson ’ s correspondence .
The functor LaTeXMLMath from the category of stable non-trivial Higgs bundles on LaTeXMLMath with vanishing Chern classes to LaTeXMLMath is fully faithful .
Let LaTeXMLMath and LaTeXMLMath be Higgs bundles on LaTeXMLMath and let LaTeXMLMath and LaTeXMLMath be the cokernels of LaTeXMLMath and LaTeXMLMath respectively .
Because the relative Mukai transform is an equivalence of categories , we have LaTeXMLEquation .
Thus faithfulness is clear .
On the other hand , let LaTeXMLMath ; using the notation of the proof of the theorem , the previous remark shows that LaTeXMLMath gives a morphism of Higgs bundles LaTeXMLMath .
But as the genus of LaTeXMLMath is at least LaTeXMLMath , the canonical linear system LaTeXMLMath has no base points .
Hence we can cover LaTeXMLMath by open sets like LaTeXMLMath ; it is clear that the morphisms thus obtained glue to give a morphism LaTeXMLMath .
∎
An explicit coupling construction of random-cluster measures is presented .
As one of the applications of the construction , the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity .
Coupling and Bernoullicity In the ( ferromagnetic ) Potts model , spins ( or colors ) from the set LaTeXMLMath are assigned to the vertices of a graph LaTeXMLMath randomly , in a way that favors configurations where many pairs of neighboring vertices take the same spin value .
More precisely , a spin configuration LaTeXMLMath is assigned probability proportional to LaTeXMLEquation where LaTeXMLMath is referred to as the inverse temperature parameter .
The case LaTeXMLMath is known as the Ising model .
The Potts model has received a considerable amount of attention in the statistical mechanics and probability literature for several decades .
In the last decade , perhaps the most important tool for analyzing the Potts model has been the random-cluster model , which is a kind of edge representation of the Potts model .
It was introduced by Fortuin and Kasteleyn LaTeXMLCite , and has been heavily exploited in the study of Potts models since the seminal papers by Swendsen and Wang LaTeXMLCite , Edwards and Sokal LaTeXMLCite , and Aizenman , Chayes , Chayes and Newman LaTeXMLCite .
One of the main points of working with the random-cluster representation , rather than directly with the Potts model , is that questions about spin correlations in the latter turn into questions about connectivity probabilities in the former , thereby allowing powerful percolation techniques to come into play .
Another interesting aspect of the random-cluster representation is that it makes sense also for noninteger LaTeXMLMath .
This paper is a contribution to the study of random-cluster and Potts models on infinite lattices .
After recalling some necessary prerequisites in Section LaTeXMLRef , we come in Sections LaTeXMLRef and LaTeXMLRef to the two main purposes of this paper , which are the following : In Section LaTeXMLRef , we present a useful device for the analysis of random-cluster and Potts models , namely an explicit pointwise dynamical construction of random-cluster measures .
The construction provides natural couplings between random-cluster measures with different parameter values or different boundary conditions .
To some extent , this construction can be viewed as known and our presentation of it can to the same extent be viewed as expository ; it consists of putting together a few well-known ingredients from Grimmett LaTeXMLCite , Propp and Wilson LaTeXMLCite , and Häggström , Schonmann and Steif LaTeXMLCite .
In Section LaTeXMLRef , we apply the dynamical construction from the preceding section to show that the Potts model with fixed-spin boundary condition on LaTeXMLMath ( and more generally on amenable Cayley graphs ) exhibits a rather strong mixing condition known as Bernoullicity .
Our proof appears to be the simplest to date even in cases where the result was known previously .
Finally , some additional consequences of , and questions on , the dynamical construction are discussed in Section LaTeXMLRef .
The following subsections are devoted to recalling known material that will be used in later sections .
The random-cluster and Potts models are introduced in Sections LaTeXMLRef and LaTeXMLRef , respectively .
Before that , however , we recall some graph terminology in Section LaTeXMLRef and some basics on stochastic domination in Section LaTeXMLRef .
A general reference for this background material is Georgii , Häggström and Maes LaTeXMLCite .
Let LaTeXMLMath be a graph with vertex set LaTeXMLMath and edge set LaTeXMLMath .
We shall always assume either that the graph is finite , or that it is countably infinite and locally finite .
An edge LaTeXMLMath will often be denoted LaTeXMLMath .
The number of edges incident to a vertex LaTeXMLMath is called the degree of LaTeXMLMath .
For LaTeXMLMath , we define the ( inner ) boundary LaTeXMLMath of LaTeXMLMath as LaTeXMLEquation .
A graph automorphism of LaTeXMLMath is a bijective mapping LaTeXMLMath with the property that for all LaTeXMLMath , we have LaTeXMLMath if and only if LaTeXMLMath .
Write LaTeXMLMath for the group of all graph automorphisms of LaTeXMLMath .
To each LaTeXMLMath , there is a corresponding mapping LaTeXMLMath defined by LaTeXMLMath .
The graph LaTeXMLMath is said to be transitive if and only if for some ( any ) LaTeXMLMath , one has that for any LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath .
One says that LaTeXMLMath is quasi-transitive if and only if for some finite subset LaTeXMLMath of LaTeXMLMath , one has that for any LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath for some LaTeXMLMath .
A probability measure LaTeXMLMath on LaTeXMLMath is said to be automorphism invariant if for any LaTeXMLMath , any LaTeXMLMath , any LaTeXMLMath , and any LaTeXMLMath we have LaTeXMLEquation .
LaTeXMLEquation In the sequel , we shall simplify the notation and omit the “ LaTeXMLMath ” as used in the preceding equation .
A graph property that turns out to be important in many situations is amenability : An infinite graph LaTeXMLMath is said to be amenable if LaTeXMLEquation where the infimum ranges over all finite LaTeXMLMath , and LaTeXMLMath denotes cardinality .
There are various alternative definitions of amenability of a graph that coincide for transitive graphs ( and more generally for graphs of bounded degree ) , but not in general .
For any graph LaTeXMLMath and LaTeXMLMath , define the stabilizer LaTeXMLMath as the set of graph automorphisms that fix LaTeXMLMath , i.e. , LaTeXMLEquation .
For LaTeXMLMath , define LaTeXMLEquation .
When LaTeXMLMath is given the weak topology generated by its action on LaTeXMLMath , all stabilizers are compact subgroups of LaTeXMLMath because LaTeXMLMath is locally finite and connected .
A transitive graph LaTeXMLMath is said to be unimodular if for all LaTeXMLMath we have the symmetry LaTeXMLEquation .
Another important class of graphs is the class of Cayley graphs .
If LaTeXMLMath is a finitely generated group with generating set LaTeXMLMath , then the Cayley graph associated with LaTeXMLMath and that particular set of generators is the ( unoriented ) graph LaTeXMLMath with vertex set LaTeXMLMath , and edge set LaTeXMLEquation .
Obviously , a Cayley graph is transitive , and furthermore it is not hard to show that it is unimodular .
Most graphs that have been studied in percolation theory are Cayley graphs .
Examples include LaTeXMLMath ( which , with a slight abuse of notation , is short for the graph with vertex set LaTeXMLMath and edges connecting pairs of vertices at Euclidean distance LaTeXMLMath from each other ) , and the regular tree LaTeXMLMath in which every vertex has exactly LaTeXMLMath neighbors .
The graph LaTeXMLMath is amenable , while LaTeXMLMath is nonamenable for LaTeXMLMath .
Also studied are certain nonamenable tilings of the hyperbolic plane ( see , e.g. , LaTeXMLCite and LaTeXMLCite ) , and further examples can be obtained , e.g. , by taking Cartesian products of two or more Cayley graphs .
Let LaTeXMLMath be any finite or countably infinite set .
( In our applications , LaTeXMLMath will be an edge set ; hence the notation . )
For two configurations LaTeXMLMath , we write LaTeXMLMath if LaTeXMLMath for all LaTeXMLMath .
A function LaTeXMLMath is said to be increasing if LaTeXMLMath whenever LaTeXMLMath .
For two probability measures LaTeXMLMath and LaTeXMLMath on LaTeXMLMath , we say that LaTeXMLMath is stochastically dominated by LaTeXMLMath , writing LaTeXMLMath , if LaTeXMLEquation for all bounded increasing LaTeXMLMath .
By a coupling of LaTeXMLMath and LaTeXMLMath , or of two random objects LaTeXMLMath and LaTeXMLMath with distributions LaTeXMLMath and LaTeXMLMath , we simply mean a joint construction of two random objects with the prescribed distributions on a common probability space .
By Strassen ’ s Theorem ( see , e.g. , LaTeXMLCite ) , LaTeXMLMath is equivalent to the existence of a coupling LaTeXMLMath of two random objects LaTeXMLMath and LaTeXMLMath with distributions LaTeXMLMath and LaTeXMLMath , such that LaTeXMLMath .
We call such a coupling a witness to the stochastic domination ( LaTeXMLRef ) .
A useful tool for establishing stochastic domination is the well-known Holley ’ s Inequality .
For LaTeXMLMath and LaTeXMLMath , we let LaTeXMLMath denote the restriction of LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be finite , and let LaTeXMLMath and LaTeXMLMath be probability measures on LaTeXMLMath that assign positive probability to all elements of LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath satisfy LaTeXMLEquation for all LaTeXMLMath , and all LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLMath .
This is not the most general form of Holley ’ s Inequality , but one that is sufficient for our purposes .
For a proof , see , e.g. , LaTeXMLCite ( Theorem 4.8 ) .
We shall also need the notion of weak convergence of probability measures on LaTeXMLMath , when LaTeXMLMath is countably infinite .
For such probability measures LaTeXMLMath and LaTeXMLMath , we say that LaTeXMLMath is the ( weak ) limit of LaTeXMLMath as LaTeXMLMath if LaTeXMLMath for all cylinder events LaTeXMLMath .
Let LaTeXMLMath be a finite graph .
An element LaTeXMLMath of LaTeXMLMath will be identified with the subgraph of LaTeXMLMath that has vertex set LaTeXMLMath and edge set LaTeXMLMath .
An edge LaTeXMLMath with LaTeXMLMath ( resp .
LaTeXMLMath ) is said to be open ( resp .
closed ) .
A central quantity to the random-cluster model is the number of connected components of LaTeXMLMath , which will be denoted LaTeXMLMath .
We emphasize that in the definition of LaTeXMLMath , isolated vertices in LaTeXMLMath also count as connected components .
The random-cluster measure LaTeXMLMath ( sub- and superscripts will be dropped whenever possible ) with parameters LaTeXMLMath and LaTeXMLMath , is defined as the probability measure on LaTeXMLMath that to each LaTeXMLMath assigns probability LaTeXMLEquation where LaTeXMLMath is a normalizing constant making LaTeXMLMath a probability measure .
When LaTeXMLMath , we see that all edges are independently open and closed with respective probabilities LaTeXMLMath and LaTeXMLMath , so that we get the usual i.i.d .
bond percolation model on LaTeXMLMath .
All other choices of LaTeXMLMath yield dependence between the edges .
Throughout the paper , we shall assume ( as in most studies of the random-cluster model ) that LaTeXMLMath .
The main reason for doing so is that when LaTeXMLMath , the conditional probability in eq .
( LaTeXMLRef ) below becomes increasing not only in LaTeXMLMath but also in LaTeXMLMath , and this allows some very powerful stochastic domination arguments , based on Holley ’ s Inequality ( Lemma LaTeXMLRef ) , to come into play ; these are not available for LaTeXMLMath .
Furthermore , it is only random-cluster measures with LaTeXMLMath that have proved to be useful in the analysis of Potts models .
It is immediate from the definition that if LaTeXMLMath is a LaTeXMLMath -valued random object with distribution LaTeXMLMath , then we have , for each LaTeXMLMath and each LaTeXMLMath , that LaTeXMLEquation where LaTeXMLMath is the event that there is an open path ( i.e. , a path of open edges ) from LaTeXMLMath to LaTeXMLMath in LaTeXMLMath .
As a first application of Holley ’ s Inequality , we get from ( LaTeXMLRef ) that LaTeXMLEquation whenever LaTeXMLMath and LaTeXMLMath .
Our next task is to define the random-cluster model on infinite graphs .
Let LaTeXMLMath be infinite and locally finite .
The definition ( LaTeXMLRef ) of random-cluster measures does not work in this case , because there are uncountably many different configurations LaTeXMLMath .
Instead , there are two other approaches to defining random-cluster measures on infinite graphs : one via limiting procedures , and the other via local specifications , also known as the Dobrushin-Lanford-Ruelle ( DLR ) equations .
We shall sketch the first approach .
Let LaTeXMLMath be a sequence of finite vertex sets increasing to LaTeXMLMath in the sense that LaTeXMLMath and LaTeXMLMath .
For any finite LaTeXMLMath , define LaTeXMLEquation set LaTeXMLMath and note that LaTeXMLMath increases to LaTeXMLMath in the same sense that LaTeXMLMath increases to LaTeXMLMath .
Let LaTeXMLMath be the ( inner ) boundary of LaTeXMLMath ( defined as in ( LaTeXMLRef ) ) .
Also set LaTeXMLMath , and let LaTeXMLMath be the probability measure on LaTeXMLMath corresponding to picking LaTeXMLMath by letting LaTeXMLMath have distribution LaTeXMLMath and setting LaTeXMLMath for all LaTeXMLMath .
Since the projection of LaTeXMLMath on LaTeXMLMath is nonrandom , we can also view LaTeXMLMath as a measure on LaTeXMLMath , in which case it coincides with LaTeXMLMath .
Applying ( LaTeXMLRef ) to the graph LaTeXMLMath with LaTeXMLMath and LaTeXMLMath gives LaTeXMLEquation so that LaTeXMLEquation .
This implies the existence of a limiting ( as LaTeXMLMath ) probability measure LaTeXMLMath on LaTeXMLMath .
This limit is independent of the choice of LaTeXMLMath , and we call it the random-cluster measure on LaTeXMLMath with free boundary condition ( hence the LaTeXMLMath in LaTeXMLMath ) and parameters LaTeXMLMath and LaTeXMLMath .
Next , define LaTeXMLMath as the probability measure on LaTeXMLMath corresponding to first setting LaTeXMLMath , and then picking LaTeXMLMath in such a way that LaTeXMLEquation where LaTeXMLMath is the number of connected components of LaTeXMLMath that do not intersect LaTeXMLMath , and LaTeXMLMath is again a normalizing constant .
Similarly as in ( LaTeXMLRef ) , we get LaTeXMLEquation ( with the inequalities reversed compared to ( LaTeXMLRef ) ) , and thus also a limiting measure LaTeXMLMath that we call the random-cluster measure on LaTeXMLMath with wired boundary condition and parameters LaTeXMLMath and LaTeXMLMath .
Note that the free and wired random-cluster measures LaTeXMLMath and LaTeXMLMath are both automorphism invariant .
This follows from their construction , in particular from the independence of the choice of LaTeXMLMath .
Fix a finite graph LaTeXMLMath and the inverse temperature parameter LaTeXMLMath .
We define the Gibbs measure for the LaTeXMLMath -state Potts model on LaTeXMLMath at inverse temperature LaTeXMLMath , denoted LaTeXMLMath , as the probability measure that to each LaTeXMLMath assigns probability LaTeXMLEquation where LaTeXMLMath is yet another normalizing constant .
The main link between random-cluster and Potts models is the following well-known result .
( See , e.g. , LaTeXMLCite . )
Fix a finite graph LaTeXMLMath , an integer LaTeXMLMath and LaTeXMLMath .
Pick a random edge configuration LaTeXMLMath according to the random-cluster measure LaTeXMLMath .
Then , for each connected component LaTeXMLMath of LaTeXMLMath , pick a spin uniformly from LaTeXMLMath , and assign this spin to all vertices of LaTeXMLMath .
Do this independently for different connected components .
The LaTeXMLMath -valued random spin configuration arising from this procedure is then distributed according to the Gibbs measure LaTeXMLMath for the LaTeXMLMath -state Potts model on LaTeXMLMath at inverse temperature LaTeXMLMath .
This provides the way ( mentioned in the introduction ) to reformulate problems about pairwise dependencies in the Potts model into problems about connectivity probabilities in the random-cluster model .
Aizenman et al .
LaTeXMLCite were the first to exploit such ideas to obtain results about the phase transition behavior of the Potts model , and the technique has been of much use since then .
The case of infinite graphs is slightly more intricate .
Let LaTeXMLMath be infinite and locally finite , and let LaTeXMLMath be as in Section LaTeXMLRef .
For LaTeXMLMath and LaTeXMLMath , define probability measures LaTeXMLMath on LaTeXMLMath in such a way that the projection of LaTeXMLMath on LaTeXMLMath equals LaTeXMLMath , and the spins on LaTeXMLMath are i.i.d .
uniformly distributed on LaTeXMLMath and independent of the spins on LaTeXMLMath .
Using Proposition LaTeXMLRef , one can show that LaTeXMLMath has a limiting distribution LaTeXMLMath as LaTeXMLMath .
Furthermore , for a fixed spin LaTeXMLMath , define LaTeXMLMath to be the distribution corresponding to picking LaTeXMLMath by letting LaTeXMLMath , and letting LaTeXMLMath be distributed according to LaTeXMLMath conditioned on the event that LaTeXMLMath .
Again , it turns out that LaTeXMLMath has a limiting distribution as LaTeXMLMath , and we denote it by LaTeXMLMath .
The existence of the limiting distributions LaTeXMLMath and LaTeXMLMath are nontrivial results , and in fact the shortest route to proving them goes via random-cluster arguments : First carry out the stochastic monotonicity arguments for the random-cluster model outlined in Section LaTeXMLRef , and then use Propositions LaTeXMLRef and LaTeXMLRef below .
A probability measure LaTeXMLMath on LaTeXMLMath is said to be a Gibbs measure ( in the DLR sense ) for the LaTeXMLMath -state Potts model on LaTeXMLMath at inverse temperature LaTeXMLMath , if it admits conditional distributions such that for all LaTeXMLMath , all LaTeXMLMath , and all LaTeXMLMath , we have LaTeXMLEquation where the normalizing constant LaTeXMLMath may depend on LaTeXMLMath and LaTeXMLMath but not on LaTeXMLMath .
The limiting measures LaTeXMLMath and LaTeXMLMath are both Gibbs measures in this sense .
The following extensions of Proposition LaTeXMLRef provide the relations between LaTeXMLMath and LaTeXMLMath on one hand , and LaTeXMLMath and LaTeXMLMath on the other .
Let LaTeXMLMath be an infinite locally finite graph , and fix LaTeXMLMath and LaTeXMLMath .
Pick a random edge configuration LaTeXMLMath according to LaTeXMLMath .
Then , for each connected component LaTeXMLMath of LaTeXMLMath independently , pick a spin uniformly from LaTeXMLMath , and assign this spin to all vertices of LaTeXMLMath .
The LaTeXMLMath -valued random spin configuration arising from this procedure is then distributed according to the Gibbs measure LaTeXMLMath for the LaTeXMLMath -state Potts model on LaTeXMLMath at inverse temperature LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be as in Proposition LaTeXMLRef .
Pick a random edge configuration LaTeXMLMath according to the random-cluster measure LaTeXMLMath .
Then , for each finite connected component LaTeXMLMath of LaTeXMLMath independently , pick a spin uniformly from LaTeXMLMath , and assign this spin to all vertices of LaTeXMLMath .
Finally assign value LaTeXMLMath to all vertices of infinite connected components .
The LaTeXMLMath -valued random spin configuration arising from this procedure is then distributed according to the Gibbs measure LaTeXMLMath for the LaTeXMLMath -state Potts model on LaTeXMLMath at inverse temperature LaTeXMLMath .
Let LaTeXMLMath be infinite and locally finite , and let LaTeXMLMath be as in Section LaTeXMLRef .
We know from Section LaTeXMLRef that LaTeXMLEquation .
Other well-known stochastic inequalities are that for LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation and LaTeXMLEquation .
For all of the above stochastic inequalities , it is desirable to find some natural construction of couplings that witness them .
What we shall construct in this section is a coupling of all of the above probability measures ( for all LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ) simultaneously that provides witnesses to the stochastic inequalities ( LaTeXMLRef ) – ( LaTeXMLRef ) above .
Some additional useful aspects of the construction are the following .
Not only are LaTeXMLMath and LaTeXMLMath automorphism invariant separately , but also their joint behavior in our coupling is automorphism invariant .
This remains true also if we consider the realizations simultaneously for different parameter values .
See Section LaTeXMLRef , where we describe an application where this property is crucial .
If LaTeXMLMath is obtained as an automorphism-invariant percolation process on another graph LaTeXMLMath , then the construction is easily set up in such a way that the joint distribution of LaTeXMLMath and the random-cluster measures on LaTeXMLMath becomes an automorphism-invariant process on LaTeXMLMath .
( See LaTeXMLCite for an example where an analogous property turns out to be important in the context of Ising models with external field on percolation clusters . )
Nevertheless , there are still some desirable aspects of couplings of random-cluster processes for which we do not know whether or not they hold for our construction ; see Conjecture LaTeXMLRef and Question LaTeXMLRef in the final section .
The construction is based on time dynamics for the random-cluster model .
Such time dynamics have previously been considered , e.g. , by Bezuidenhout , Grimmett and Kesten LaTeXMLCite and by Grimmett LaTeXMLCite for the random-cluster model on LaTeXMLMath .
To some extent our construction will resemble Grimmett ’ s analysis .
However , one feature of our construction that differs from Grimmett ’ s is that the dynamics are run “ from the past ” rather than “ into the future ” , along the lines of the very fashionable CFTP ( coupling from the past ) algorithm of Propp and Wilson LaTeXMLCite ; see also LaTeXMLCite for an early treatment of dynamics from the past , and LaTeXMLCite for a survey putting the ideas in a more general mathematical context .
For the case of finite graphs , CFTP was applied to simulate the random-cluster model in LaTeXMLCite .
Simulation on infinite graphs would require additional arguments , but our purpose is not simulation ; rather , it is to gain some theoretical information .
For models other than the random-cluster model , CFTP ideas have been extended to the setting of infinite graphs in LaTeXMLCite , LaTeXMLCite and LaTeXMLCite , but in all those cases the interaction of the dynamics had a strictly local character , which is not the case in our context .
Another feature of our construction is the simultaneity in the parameter space .
Such simultaneity , which is related to the level-set representations of Higuchi LaTeXMLCite , appears in both LaTeXMLCite and LaTeXMLCite ; Propp and Wilson use the term “ omnithermal ” to denote this particular feature of the construction .
Let us start with a simple finite case : how do we construct a LaTeXMLMath -valued random element with distribution LaTeXMLMath ( equivalently , with distribution LaTeXMLMath ) ?
If we are content with getting something that has only approximately the right distribution , then the following dynamical approach works fine : Define some ergodic Markov chain whose unique equilibrium distribution is LaTeXMLMath , and run it for time LaTeXMLMath starting from an arbitrary initial state LaTeXMLMath .
If LaTeXMLMath is large enough , then the distribution of the final state is close to LaTeXMLMath , regardless of the choice of LaTeXMLMath .
In particular , we may proceed as follows .
To each edge LaTeXMLMath , we independently assign an i.i.d .
sequence LaTeXMLMath of exponential random variables with mean LaTeXMLMath , and an independent i.i.d .
sequence LaTeXMLMath of uniform LaTeXMLMath random variables .
For LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath , so that LaTeXMLMath are the jump times of a unit rate Poisson process .
Now define a LaTeXMLMath -valued continuous-time Markov chain LaTeXMLMath with starting state LaTeXMLMath and evolution as follows .
For LaTeXMLMath , the value of LaTeXMLMath does not change other than ( possibly ) at the times LaTeXMLMath , at which times it takes the value LaTeXMLEquation where LaTeXMLMath denotes negation .
( Note that a.s. , LaTeXMLMath for all LaTeXMLMath when LaTeXMLMath . )
It is easy to see that this Markov chain is irreducible and reversible with LaTeXMLMath as stationary distribution , so that indeed LaTeXMLMath converges in distribution to LaTeXMLMath as LaTeXMLMath .
Note also that since LaTeXMLMath , the chain preserves the partial order LaTeXMLMath on LaTeXMLMath ; in other words , for all LaTeXMLMath we have LaTeXMLEquation .
To get a LaTeXMLMath -valued random object whose distribution is precisely LaTeXMLMath , we need to consider some limit as LaTeXMLMath .
On the other hand , LaTeXMLMath does not converge in any a.s. sense , so this may appear not to be feasible .
The solution , which turns the convergence in distribution into a.s. convergence , is to run the dynamics from the past up to time LaTeXMLMath , rather than from time LaTeXMLMath into the future .
For LaTeXMLMath , define the LaTeXMLMath -valued continuous-time Markov chain LaTeXMLEquation with starting state LaTeXMLMath and the following evolution , similar to the one of LaTeXMLMath .
The value at an edge LaTeXMLMath changes only at times LaTeXMLMath , when it takes the value LaTeXMLEquation as in ( LaTeXMLRef ) .
We have , for LaTeXMLMath , that LaTeXMLEquation ( essentially because of ( LaTeXMLRef ) ) , so by monotonicity LaTeXMLMath has an a.s. limit LaTeXMLMath , defined by setting LaTeXMLMath for each LaTeXMLMath .
Clearly , LaTeXMLMath has the same distribution as LaTeXMLMath with LaTeXMLMath , so LaTeXMLMath converges in distribution to LaTeXMLMath as LaTeXMLMath .
Hence LaTeXMLMath has distribution LaTeXMLMath , and if we furthermore define LaTeXMLMath by setting LaTeXMLEquation for each LaTeXMLMath , then LaTeXMLMath has distribution LaTeXMLMath .
Now suppose that we have defined the random variables LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath ( and not just all LaTeXMLMath ) in the obvious way .
By another application of the order-preserving property ( LaTeXMLRef ) , we get that LaTeXMLEquation so that the limiting object LaTeXMLMath , defined by taking LaTeXMLMath , exists .
For any cylinder set LaTeXMLMath , we have LaTeXMLEquation so that LaTeXMLMath has distribution LaTeXMLMath .
Thus , to summarize the construction so far , what we have is a coupling of LaTeXMLMath -valued random objects LaTeXMLMath and LaTeXMLMath that witnesses the stochastic inequalities in the first half of ( LaTeXMLRef ) .
Next , we go on to construct , in analogous fashion , the corresponding objects for wired random-cluster measures .
For LaTeXMLMath , define the LaTeXMLMath -valued continuous-time Markov chain LaTeXMLEquation with starting configuration LaTeXMLMath .
Edges LaTeXMLMath remain in state LaTeXMLMath forever , while the value of an edge LaTeXMLMath is updated at times LaTeXMLMath , when it takes the value LaTeXMLEquation here , LaTeXMLMath is the event LaTeXMLMath , where , in turn , LaTeXMLMath denotes the event that either ( a ) there is an open path from LaTeXMLMath to LaTeXMLMath ( not using LaTeXMLMath ) , or ( b ) both LaTeXMLMath and LaTeXMLMath have open paths ( not using LaTeXMLMath ) to LaTeXMLMath .
It is immediate from the definition of LaTeXMLMath that the conditional LaTeXMLMath -probability that an edge LaTeXMLMath is open , given the status of all other edges , is LaTeXMLMath or LaTeXMLMath , depending on whether or not the event LaTeXMLMath happens .
It follows that the distribution of LaTeXMLMath tends to LaTeXMLMath as LaTeXMLMath .
Moreover , the dynamics in ( LaTeXMLRef ) preserves LaTeXMLMath similarly as in ( LaTeXMLRef ) , implying that LaTeXMLEquation whenever LaTeXMLMath .
This establishes the existence of a limiting LaTeXMLMath -valued random object LaTeXMLMath defined by LaTeXMLMath for each LaTeXMLMath .
Clearly , LaTeXMLMath has distribution LaTeXMLMath .
Another use of the LaTeXMLMath -preserving property of the dynamics ( LaTeXMLRef ) shows that LaTeXMLEquation so that we have a limiting object LaTeXMLMath defined by setting LaTeXMLMath for each LaTeXMLMath .
By arguing as in ( LaTeXMLRef ) , we get that LaTeXMLMath has distribution LaTeXMLMath .
The random objects LaTeXMLMath and LaTeXMLMath witness the stochastic inequalities in the second half of ( LaTeXMLRef ) .
In order to fully establish that we have a witness to ( LaTeXMLRef ) , it remains to show that LaTeXMLMath and LaTeXMLMath witness the middle inequality in ( LaTeXMLRef ) , i.e. , we need to show that LaTeXMLMath .
From the observations that the right-hand sides of ( LaTeXMLRef ) and ( LaTeXMLRef ) are increasing in the configurations on LaTeXMLMath , and that for each such configuration the right-hand side of ( LaTeXMLRef ) is greater than that of ( LaTeXMLRef ) , we get that LaTeXMLEquation for any LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
By taking LaTeXMLMath , letting LaTeXMLMath and then LaTeXMLMath , we get LaTeXMLEquation as desired .
Hence our coupling is a witness to all the inequalities in ( LaTeXMLRef ) .
It remains to be demonstrated that the coupling is also a witness to the inequalities ( LaTeXMLRef ) – ( LaTeXMLRef ) .
Note first that the right-hand sides of ( LaTeXMLRef ) and ( LaTeXMLRef ) are increasing not only in the configurations on LaTeXMLMath , but also in LaTeXMLMath .
It follows that for LaTeXMLMath we have LaTeXMLEquation and LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Taking LaTeXMLMath and letting LaTeXMLMath yields LaTeXMLEquation and LaTeXMLEquation witnessing ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Letting LaTeXMLMath , we get LaTeXMLEquation and LaTeXMLEquation finally witnessing ( LaTeXMLRef ) and ( LaTeXMLRef ) .
In fact , examination also shows that as long as LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation witnessing more general well-known stochastic inequalities LaTeXMLCite .
Property ( A1 ) of the coupling is obvious from the construction .
In order for ( A2 ) to be true , we need only define random variables LaTeXMLMath for all edges in LaTeXMLMath and to take them to be independent of the percolation process that yields LaTeXMLMath from LaTeXMLMath .
Let LaTeXMLMath be a closed subgroup of LaTeXMLMath with LaTeXMLMath being any connected graph .
We shall be most interested in two cases : ( 1 ) that LaTeXMLMath is the Cayley graph of LaTeXMLMath with respect to some finite generating set of LaTeXMLMath ; and ( 2 ) that LaTeXMLMath and LaTeXMLMath is quasi-transitive .
Let LaTeXMLMath and LaTeXMLMath be arbitrary state spaces .
For LaTeXMLMath , define the map LaTeXMLMath ( or LaTeXMLMath ) by setting LaTeXMLMath for each LaTeXMLMath .
A measurable mapping LaTeXMLMath is said to be LaTeXMLMath -equivariant if it commutes with these actions of LaTeXMLMath , i.e. , if LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath -a.e .
LaTeXMLMath ; it is called measure-preserving if LaTeXMLMath .
The action of LaTeXMLMath on LaTeXMLMath is called free if for LaTeXMLMath -a.e .
LaTeXMLMath , the only element in LaTeXMLMath that leaves LaTeXMLMath fixed is the identity .
We say that a probability measure LaTeXMLMath on LaTeXMLMath is a LaTeXMLMath -factor of an i.i.d .
process if there exists a LaTeXMLMath -valued random element LaTeXMLMath with distribution LaTeXMLMath , a state space LaTeXMLMath , an LaTeXMLMath -valued random element LaTeXMLMath with distribution LaTeXMLMath , and a LaTeXMLMath -equivariant measure-preserving mapping LaTeXMLMath such that LaTeXMLMath is an i.i.d .
process , and LaTeXMLMath .
In case LaTeXMLMath is the Cayley graph of LaTeXMLMath , if LaTeXMLMath can be taken to be finite and LaTeXMLMath can be taken to be an invertible mapping , then LaTeXMLMath is said to be Bernoulli , a mixing property of fundamental importance in ergodic theory .
In LaTeXMLCite , it is shown that the following definition is a proper extension of the preceding definition : An action LaTeXMLMath is said to be Bernoulli if it is a free LaTeXMLMath -factor of a Poisson process on LaTeXMLMath .
We shall prove , using the dynamical construction in Section LaTeXMLRef , that Bernoullicity holds for the wired Potts model on LaTeXMLMath , and more generally on many amenable quasi-transitive graphs .
We shall need the following condition .
Let LaTeXMLMath denote the set of points at distance LaTeXMLMath from a vertex LaTeXMLMath .
Consider the condition on LaTeXMLMath that LaTeXMLEquation .
Let LaTeXMLMath be a Cayley graph of any amenable group LaTeXMLMath or be any amenable graph with a closed automorphism group LaTeXMLMath acting quasi-transitively on LaTeXMLMath and satisfying ( LaTeXMLRef ) .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then the Gibbs measure LaTeXMLMath is Bernoulli with respect to the action of LaTeXMLMath .
For the LaTeXMLMath case , this was previously known only for the cases where either LaTeXMLMath ( the Ising model ) or LaTeXMLMath is sufficiently small ; see , e.g. , LaTeXMLCite , LaTeXMLCite and LaTeXMLCite .
For the Ising model result on amenable graphs , see LaTeXMLCite , while for a proof of a stronger property than Bernoullicity in the case of LaTeXMLMath small , using CFTP ideas , see LaTeXMLCite .
The paper LaTeXMLCite uses ideas similar to ours to prove that the Ising model is Bernoulli .
Actually , we shall prove a slightly stronger result , which is the best possible .
That is , we shall show that as long as i.i.d .
variables on the vertices of LaTeXMLMath yield a free action of LaTeXMLMath , then LaTeXMLMath is Bernoulli .
It is not clear when the full automorphism group LaTeXMLMath satisfies this freeness condition , so we have supplied the condition ( LaTeXMLRef ) .
We call an i.i.d .
process LaTeXMLMath standard if LaTeXMLMath is a standard Borel space and the marginal of LaTeXMLMath on LaTeXMLMath is Borel .
Ornstein and Weiss LaTeXMLCite show that when LaTeXMLMath is amenable and discrete , then LaTeXMLMath is Bernoulli iff it is a free LaTeXMLMath -factor of a standard i.i.d .
process .
More generally , we have the following result : Let LaTeXMLMath be a countable set and LaTeXMLMath be a closed subgroup of the symmetric group on LaTeXMLMath .
Suppose that all orbits of the LaTeXMLMath -action on LaTeXMLMath are infinite and that LaTeXMLMath is amenable , unimodular , and not the union of an increasing sequence of compact proper subgroups of LaTeXMLMath .
Further , suppose that for each LaTeXMLMath , the LaTeXMLMath -stabilizer of LaTeXMLMath is compact .
Then every free LaTeXMLMath -factor of a standard i.i.d .
process LaTeXMLMath is Bernoulli .
Proof .
Assume that there is some free LaTeXMLMath -factor LaTeXMLMath of a standard i.i.d .
process LaTeXMLMath , since otherwise there is nothing to prove .
Let LaTeXMLMath be i.i.d .
Poisson point processes on LaTeXMLMath with Haar measure as the underlying intensity measure .
By LaTeXMLCite , the product process LaTeXMLMath is Bernoulli .
We shall show that LaTeXMLMath is a LaTeXMLMath -factor of LaTeXMLMath , whence is a factor of a Poisson process , whence is Bernoulli .
Let LaTeXMLMath be a selection of one point from each orbit of the action of LaTeXMLMath on LaTeXMLMath .
Given LaTeXMLMath , let LaTeXMLMath be the number of points in LaTeXMLMath that take LaTeXMLMath to LaTeXMLMath for LaTeXMLMath , where LaTeXMLMath .
Since LaTeXMLMath is a countable union of translates of stabilizers , each stabilizer has positive finite Haar measure , so that LaTeXMLMath is a nontrivial Poisson random variable .
Also , the random variables LaTeXMLMath are mutually independent .
Since LaTeXMLMath is a LaTeXMLMath -factor of LaTeXMLMath , it follows that LaTeXMLMath is a LaTeXMLMath -factor of LaTeXMLMath .
Since every standard i.i.d .
process LaTeXMLMath is a LaTeXMLMath -factor of LaTeXMLMath and LaTeXMLMath is a factor of LaTeXMLMath , we obtain the result we want .
LaTeXMLMath We also need the following fact : If LaTeXMLMath is a quasi-transitive amenable graph , then LaTeXMLMath is amenable , unimodular , and not the union of an increasing sequence of compact proper subgroups .
Proof .
LaTeXMLMath is amenable and unimodular by results of Soardi and Woess LaTeXMLCite and Salvatori LaTeXMLCite ; see also LaTeXMLCite for another proof .
Furthermore , in this case LaTeXMLMath is generated by , say , the compact set LaTeXMLMath , where LaTeXMLMath is such that every vertex of LaTeXMLMath is within distance LaTeXMLMath of some vertex in LaTeXMLMath and LaTeXMLMath denotes distance in LaTeXMLMath .
Thus , if LaTeXMLMath are compact increasing subgroups of LaTeXMLMath whose union is LaTeXMLMath , we have LaTeXMLMath , whence for some LaTeXMLMath , we have LaTeXMLMath .
Since LaTeXMLMath generates LaTeXMLMath , it follows that LaTeXMLMath .
LaTeXMLMath Because of the above , Theorem LaTeXMLRef is established once the following lemma is proved : For any graph LaTeXMLMath , any subgroup LaTeXMLMath of LaTeXMLMath , any LaTeXMLMath and LaTeXMLMath , and any LaTeXMLMath , the Gibbs measure LaTeXMLMath is a LaTeXMLMath -factor of a standard i.i.d .
process .
If either ( i ) LaTeXMLMath is countable and every element of LaTeXMLMath other than the identity moves an infinite number of vertices or ( ii ) LaTeXMLMath satisfies condition ( LaTeXMLRef ) , then the action of LaTeXMLMath on LaTeXMLMath is free .
Proof .
Let the degree of LaTeXMLMath be LaTeXMLMath .
For each LaTeXMLMath , let LaTeXMLMath be the set of neighbors of LaTeXMLMath in any fixed order .
Take LaTeXMLEquation .
Let LaTeXMLEquation be independent random variables with LaTeXMLMath exponential of mean 1 , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath uniform LaTeXMLMath , and LaTeXMLMath uniform on LaTeXMLMath .
For each LaTeXMLMath , put LaTeXMLEquation .
Set LaTeXMLMath , and construct a LaTeXMLMath -valued edge configuration LaTeXMLMath with distribution LaTeXMLMath by the dynamical construction in Section LaTeXMLRef , where for each LaTeXMLMath we take LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are chosen in such a way that LaTeXMLMath , and , if we denote LaTeXMLMath and LaTeXMLMath is such that LaTeXMLMath , then LaTeXMLMath .
This choice of LaTeXMLMath and LaTeXMLMath is a.s. unique .
From LaTeXMLMath , we obtain the desired spin configuration LaTeXMLMath with distribution LaTeXMLMath by assigning spins to the connected components of LaTeXMLMath as in Proposition LaTeXMLRef : All vertices in infinite connected components in LaTeXMLMath are assigned value LaTeXMLMath , whereas the vertices of each finite connected component LaTeXMLMath are assigned value LaTeXMLMath , where LaTeXMLMath is the vertex in LaTeXMLMath that minimizes LaTeXMLMath .
It is obvious that this mapping LaTeXMLMath from LaTeXMLMath to LaTeXMLMath is LaTeXMLMath -equivariant , and that the resulting spin configuration has distribution LaTeXMLMath .
Hence LaTeXMLMath is a factor of a standard i.i.d .
process .
To see that the action of LaTeXMLMath on LaTeXMLMath is free under the additional hypotheses ( i ) stated in the lemma , it suffices to show that for any LaTeXMLMath other than the identity , LaTeXMLMath .
From the hypotheses , we may find an infinite set LaTeXMLMath of vertices such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath for distinct LaTeXMLMath .
Because of ( LaTeXMLRef ) , by repeated conditioning we see that there is some LaTeXMLMath such that for any LaTeXMLMath , we have LaTeXMLMath .
Therefore LaTeXMLMath .
Consider now the hypothesis ( ii ) .
Again because of ( LaTeXMLRef ) , there is some LaTeXMLMath such that if LaTeXMLMath and LaTeXMLMath are two finite sets of vertices that are not identical , then the chance is at most LaTeXMLMath that the number of spins in LaTeXMLMath equal to 1 is the same as the number of spins in LaTeXMLMath equal to 1 , even given all spins outside LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath and LaTeXMLMath are in the same orbit .
Let LaTeXMLMath be the set of spin configurations such that for some LaTeXMLMath , the number of spins in LaTeXMLMath equal to 1 differs from the number in LaTeXMLMath .
By our assumption and the fact just noted , it follows that LaTeXMLMath has probability 1 .
Hence so does LaTeXMLMath .
It is clear that LaTeXMLMath acts freely on LaTeXMLMath .
LaTeXMLMath Let us mention another application of the pointwise construction in Section LaTeXMLRef .
Consider the random-cluster model on an infinite quasi-transitive graph LaTeXMLMath at some fixed value of LaTeXMLMath .
We shall let LaTeXMLMath vary .
Clearly , by stochastic monotonicity , the LaTeXMLMath - and LaTeXMLMath -probabilities of having some infinite open cluster are increasing in LaTeXMLMath .
Furthermore , by ergodicity , these probabilities must be LaTeXMLMath or LaTeXMLMath for any given LaTeXMLMath ( although the LaTeXMLMath -probability does not necessarily equal the LaTeXMLMath -probability ) .
Hence , there exist critical values LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation .
A very natural question is whether or not there is an infinite cluster at criticality .
In LaTeXMLCite , we proved that when LaTeXMLMath is a unimodular nonamenable quasi-transitive graph , then the answer is no for LaTeXMLMath .
In other words , LaTeXMLEquation .
The proof in LaTeXMLCite of ( LaTeXMLRef ) uses , as a key ingredient , the existence of an automorphism-invariant coupling of the measures LaTeXMLMath for different LaTeXMLMath that witnesses the stochastic domination ( LaTeXMLRef ) .
Such a coupling was provided in Section LaTeXMLRef of the present paper .
It seems reasonable to expect that ( LaTeXMLRef ) extends to all quasi-transitive graphs ( except those for which the critical value is LaTeXMLMath ) .
For LaTeXMLMath , this was conjectured by Benjamini and Schramm LaTeXMLCite .
The situation for LaTeXMLMath seems to be more complicated .
For instance , as shown in LaTeXMLCite and LaTeXMLCite , when LaTeXMLMath is the regular tree LaTeXMLMath with LaTeXMLMath , we get that the LaTeXMLMath -probability of seeing an infinite cluster is LaTeXMLMath or LaTeXMLMath depending on whether LaTeXMLMath or LaTeXMLMath .
For quasi-transitive graphs , the famous finite-energy argument of Newman and Schulman LaTeXMLCite shows that the number of infinite clusters must ( under either LaTeXMLMath or LaTeXMLMath , and for fixed LaTeXMLMath and LaTeXMLMath ) be an almost sure constant , and either LaTeXMLMath , LaTeXMLMath or LaTeXMLMath .
For unimodular quasi-transitive graphs , Lyons LaTeXMLCite recently obtained the necessary uniqueness monotonicity statement for deducing that ( in addition to the critical values in ( LaTeXMLRef ) and ( LaTeXMLRef ) ) , there exist critical values LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation ( For LaTeXMLMath this goes back to LaTeXMLCite and LaTeXMLCite . )
See LaTeXMLCite for a detailed discussion of how the four critical values LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath relate to each other .
It is not obvious that , in the coupling of Section LaTeXMLRef , ( LaTeXMLRef ) and ( LaTeXMLRef ) hold simultaneously for all LaTeXMLMath and LaTeXMLMath .
This is in fact an open problem , and we conjecture the following strengthening , analogous to the simultaneous uniqueness results of LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , and LaTeXMLCite : Let LaTeXMLMath be connected and quasi-transitive .
For a configuration LaTeXMLMath , write LaTeXMLMath for the number of infinite clusters in LaTeXMLMath .
Let LaTeXMLMath be the set of quadruples LaTeXMLMath such that LaTeXMLEquation with at least one of these inequalities being strict .
In the notation of Section LaTeXMLRef , we have a.s. for all quadruples LaTeXMLMath simultaneously , each infinite cluster of LaTeXMLMath contains LaTeXMLMath infinite clusters of LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath may be any of the following three pairs of random variables : LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let us finally discuss another open problem concerning our coupling in Section LaTeXMLRef .
For LaTeXMLMath , define LaTeXMLEquation and note that LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , write LaTeXMLMath for the event that LaTeXMLMath .
From the fact that LaTeXMLMath is a DLR random-cluster measure , it follows that for any LaTeXMLMath and almost any LaTeXMLMath with respect to the law of LaTeXMLMath under our coupling ( which implies that LaTeXMLMath ) , we have LaTeXMLEquation ( and similarly for wired random-cluster measures ; everything we say in relation to Question LaTeXMLRef applies as well to the wired case as to the free ) .
From this , one is easily seduced into thinking that LaTeXMLEquation but to conclude this directly from ( LaTeXMLRef ) is unwarranted , because conditioning on LaTeXMLMath and LaTeXMLMath jointly is not the same as conditioning on them separately .
It is nevertheless natural to ask whether something like ( LaTeXMLRef ) is true .
In particular , the following question asks for a weaker property .
For LaTeXMLMath and LaTeXMLMath , does there exist an LaTeXMLMath ( depending on LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ) such that for any LaTeXMLMath and almost any LaTeXMLMath , we have LaTeXMLEquation .
A positive answer to this question ( for our coupling or for some other automorphism-invariant witness to the stochastic inequality LaTeXMLMath ) is precisely the missing ingredient that prevented the authors of LaTeXMLCite from extending their uniqueness monotonicity result for i.i.d .
percolation ( LaTeXMLMath ) for unimodular quasi-transitive graphs to the more general case LaTeXMLMath ( i.e. , from proving the relations ( LaTeXMLRef ) and ( LaTeXMLRef ) that were later obtained in LaTeXMLCite ) .
Such a positive answer might perhaps also be an ingredient in applying the reasoning of Schonmann LaTeXMLCite in order to remove the unimodularity assumption in these results .
Acknowledgement .
We are grateful to Benjy Weiss for useful discussions .
We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization .
The proof is based on a new , geometric approach to the problem .
Among the fundamental problems concerning analytic ( real or complex ) Hamiltonian systems near an equilibrium point , one may mention the following two : 1 ) Convergent Birkhoff .
In this paper , by “ convergent Birkhoff ” we mean the existence of a convergent Birkhoff normalization , i.e .
the existence of a local analytic symplectic system of coordinates in which the Hamiltonian function will Poisson commute with the semisimple part of its quadratic part .
2 ) Analytic integrability .
By “ analytic integrability ” we mean the existence of a complete set of local analytic functionally independent first integrals in involution .
These problems have been studied by many classical and modern mathematicians , including Poincaré , Birkhoff , Siegel , Moser , Bruno , etc .
In this paper , we will be concerned with the relations between the two problems .
The starting point is that , since both the Birkhoff normal form and the search for first integrals are a way to simplify and solve Hamiltonian systems , these two problems must be very closely related .
Indeed , it has been known to Birkhoff LaTeXMLCite that , for nonresonant Hamiltonian systems , convergent Birkhoff implies analytic integrability .
The inverse is also true , though much more difficult to prove LaTeXMLCite .
What has been known to date concerning “ convergent Birkhoff vs. analytic integrability ” may be summarized in the following list .
Denote by LaTeXMLMath ( LaTeXMLMath ) the degree of resonance ( see Section LaTeXMLRef for a definition ) of an analytic Hamiltonian system at an equilibrium point .
Then we have : a ) When LaTeXMLMath ( i.e .
for non-resonant systems ) , convergent Birkhoff is equivalent to analytic integrability .
The part “ convergent Birkhoff implies analytic integrability ” is straightforward .
The inverse has been a difficult problem .
Under an additional nondegeneracy condition involving the momentum map , it was first proved by Rüssmann LaTeXMLCite in 1964 for the case with two degrees of freedom , and then by Vey LaTeXMLCite in 1978 for any number of degrees of freedom .
Finally Ito LaTeXMLCite in 1989 solved the problem without any additional condition on the momentum map .
b ) When LaTeXMLMath ( i.e .
for systems with a simple resonance ) , then convergent Birkhoff is still equivalent to analytic integrability .
The part “ convergent Birkhoff implies analytic integrability ” is again obvious .
The inverse has been proved some years ago by Ito LaTeXMLCite and Kappeler , Kodama and Némethi LaTeXMLCite .
c ) When LaTeXMLMath then convergent Birkhoff does not imply analytic integrability .
The reason is that the Birkhoff normal form in this case will give us LaTeXMLMath first integrals in involution , where LaTeXMLMath is the number of degrees of freedom , but additional first integrals don ’ t exist in general , not even formal ones .
( A counterexample can be found in Duistermaat LaTeXMLCite , see also Verhulst LaTeXMLCite and references therein ) .
The question “ does analytic integrability imply convergent Birkhoff ” when LaTeXMLMath has remained open until now .
The powerful analytical techniques , which are based on the fast convergent method and used in LaTeXMLCite , could not have been made to work in the case with non-simple resonances .
The main purpose of this paper is to complete the above list , by giving a positive answer to the last question .
Any real ( resp. , complex ) analytically integrable Hamiltonian system in a neighborhood of an equilibrium point on a symplectic manifold admits a real ( resp. , complex ) convergent Birkhoff normalization at that point .
An important consequence of Theorem LaTeXMLRef is that we may classify degenerate singular points of analytic integrable Hamiltonian systems by their analytic Birkhoff normal forms ( see , e.g. , LaTeXMLCite and references therein ) .
The proof given in this paper of Theorem LaTeXMLRef works for any analytically integrable system , regardless of its degree of resonance .
Our proof is based on a geometrical method involving homological cycles , period integrals , and torus actions , and it is completely different from the analytical one used in LaTeXMLCite .
In a sense , our approach is close to that of Eliasson LaTeXMLCite , who used torus actions to prove the existence of a smooth Birkhoff normal form for smooth integrable systems with a nondegenerate elliptic singularity .
The role of torus actions is given by the following proposition ( see Proposition LaTeXMLRef for a more precise formulation ) : The existence of a convergent Birkhoff normalization is equivalent to the existence of a local Hamiltonian torus action which preserves the system .
We also have the following result , which implies that it is enough to prove Theorem LaTeXMLRef in the complex analytic case : A real analytic Hamiltonian system near an equilibrium point admits a real convergent Birkhoff normalization if and only if it admits a complex convergent Birkhoff normalization .
Both Proposition LaTeXMLRef and Proposition LaTeXMLRef are very simple and natural .
They are often used implicitly , but they have not been written explicitly anywhere in the literature , to our knowledge .
The rest of this paper is organized as follows : In Section LaTeXMLRef we introduce some necessary notions , and prove the above two propositions .
In Section LaTeXMLRef we show how to find the required torus action in the case of integrable Hamiltonian systems , by searching 1-cycles on the local level sets of the momentum map , using an approximation method based on the existence of a formal Birkhoff normalization and Łojasiewicz inequalities .
This section contains the proof of our main theorem , modulo a lemma about analytic extensions .
This lemma , which may be useful in other problems involving the existence of first integrals of singular foliations ( see LaTeXMLCite ) , is proved in Section LaTeXMLRef , the last section .
Let LaTeXMLMath , where LaTeXMLMath ( resp. , LaTeXMLMath ) be a real ( resp. , complex ) analytic function defined on an open neighborhood LaTeXMLMath of the origin in the symplectic space LaTeXMLMath .
When LaTeXMLMath is real , we will also consider it as a complex analytic function with real coefficients .
Denote by LaTeXMLMath the symplectic vector field of LaTeXMLMath : LaTeXMLEquation .
Here the sign convention is taken so that LaTeXMLMath for any function LaTeXMLMath , where LaTeXMLEquation denotes the standard Poisson bracket .
Assume that LaTeXMLMath is an equilibrium of LaTeXMLMath , i.e .
LaTeXMLMath .
We may also put LaTeXMLMath .
Denote by LaTeXMLEquation the Taylor expansion of LaTeXMLMath , where LaTeXMLMath is a homogeneous polynomial of degree LaTeXMLMath for each LaTeXMLMath .
The algebra of quadratic functions on LaTeXMLMath , under the standard Poisson bracket , is naturally isomorphic to the simple algebra LaTeXMLMath of infinitesimal linear symplectic transformations in LaTeXMLMath .
In particular , LaTeXMLEquation where LaTeXMLMath ( resp. , LaTeXMLMath ) denotes the semi-simple ( resp. , nilpotent ) part of LaTeXMLMath .
For each natural number LaTeXMLMath , the Lie algebra of quadratic functions on LaTeXMLMath acts linearly on the space of homogeneous polynomials of degree LaTeXMLMath on LaTeXMLMath via the Poisson bracket .
Under this action , LaTeXMLMath corresponds to a linear operator LaTeXMLMath , whose semisimple part is LaTeXMLMath .
In particular , LaTeXMLMath admits a decomposition LaTeXMLEquation where LaTeXMLMath is some element in the space of homogeneous polynomials of degree LaTeXMLMath , and LaTeXMLMath is in the kernel of the operator LaTeXMLMath , i.e .
LaTeXMLMath .
Denote by LaTeXMLMath the time-one map of the flow of the Hamiltonian vector field LaTeXMLMath .
Then LaTeXMLMath ( where LaTeXMLMath , or also LaTeXMLMath , is a shorthand for LaTeXMLMath ) is a symplectic transformation of LaTeXMLMath whose Taylor expansion is LaTeXMLEquation where LaTeXMLMath denotes terms of order greater or equal to LaTeXMLMath .
Under the new local symplectic coordinates LaTeXMLMath , we have LaTeXMLEquation .
In other words , the local symplectic coordinate transformation LaTeXMLMath of LaTeXMLMath changes the term LaTeXMLMath to the term LaTeXMLMath satisfying LaTeXMLMath in the Taylor expansion of LaTeXMLMath , and it leaves the terms of order smaller than LaTeXMLMath unchanged .
By induction , one finds a series of local analytic symplectic transformations LaTeXMLMath ( LaTeXMLMath ) of type LaTeXMLEquation such that for each LaTeXMLMath , the composition LaTeXMLEquation is a symplectic coordinate transformation which changes all the terms of order smaller or equal to LaTeXMLMath in the Taylor expansion of LaTeXMLMath to terms that commute with LaTeXMLMath .
By taking limit LaTeXMLMath , we get the following classical result due to Birkhoff et al .
( see , e.g. , LaTeXMLCite ) : For any real ( resp. , complex ) Hamiltonian system LaTeXMLMath near an equilibrium point with a local real ( resp. , complex ) symplectic system of coordinates LaTeXMLMath , there exists a formal real ( resp. , complex ) symplectic transformation LaTeXMLMath such that in the coordinates LaTeXMLMath we have LaTeXMLEquation where LaTeXMLMath denotes the semisimple part of the quadratic part of LaTeXMLMath .
LaTeXMLMath When Equation ( LaTeXMLRef ) is satisfied , one says that the Hamiltonian LaTeXMLMath is in Birkhoff normal form , and the symplectic transformation LaTeXMLMath in Theorem LaTeXMLRef is called a Birkhoff normalization .
Birkhoff normal form is one of the basic tools in Hamiltonian dynamics , and it has already been used in the 19th century by Delaunay LaTeXMLCite and Linstedt LaTeXMLCite for some problems of celestial mechanics .
When a Hamiltonian function LaTeXMLMath is in normal form , then its first integrals are also normalized simultaneously to some extent .
More precisely , one has the following folklore lemma , whose proof is straightforward ( see , e.g. , LaTeXMLCite ) : If LaTeXMLMath , i.e .
LaTeXMLMath is in Birkhoff normal form , and LaTeXMLMath , i.e .
LaTeXMLMath is a first integral of LaTeXMLMath , then we also have LaTeXMLMath LaTeXMLMath Recall that the simple Lie algebra LaTeXMLMath has only one Cartan subalgebra up to conjugacy .
In terms of quadratic functions , there is a complex linear canonical system of coordinates LaTeXMLMath of LaTeXMLMath in which LaTeXMLMath can be written as LaTeXMLEquation where LaTeXMLMath are complex coefficients , called frequencies .
( The quadratic functions LaTeXMLMath span a Cartan subalgebra ) .
The frequencies LaTeXMLMath are complex numbers uniquely determined by LaTeXMLMath up to a sign and a permutation .
The reason why we choose to write LaTeXMLMath instead of LaTeXMLMath in Equation ( LaTeXMLRef ) is that this way monomial functions will be eigenvectors of LaTeXMLMath under the Poisson bracket : LaTeXMLEquation .
In particular , LaTeXMLMath if and only if every monomial term LaTeXMLMath with a non-zero coefficient in the Taylor expansion of LaTeXMLMath satisfies the following relation , called a resonance relation : LaTeXMLEquation .
In the nonresonant case , when there are no resonance relations except the trivial ones , the Birkhoff normal condition LaTeXMLMath means that LaTeXMLMath is a function of LaTeXMLMath variables LaTeXMLMath , implying complete integrability .
Thus any nonresonant Hamiltonian system is formally integrable LaTeXMLCite .
More generally , denote by LaTeXMLMath the sublattice of LaTeXMLMath consisting of elements LaTeXMLMath such that LaTeXMLMath .
The dimension of LaTeXMLMath over LaTeXMLMath , denoted by LaTeXMLMath , is called the degree of resonance of the Hamiltonian LaTeXMLMath .
Let LaTeXMLMath be a basis of the resonance lattice LaTeXMLMath .
Let LaTeXMLMath be a basis of LaTeXMLMath such that LaTeXMLMath ( LaTeXMLMath if LaTeXMLMath and LaTeXMLMath if LaTeXMLMath ) , and set LaTeXMLEquation for LaTeXMLMath .
Then we have LaTeXMLMath with no resonance relation among LaTeXMLMath .
The equation LaTeXMLMath is now equivalent to LaTeXMLEquation .
What is so good about the quadratic functions LaTeXMLMath is that each LaTeXMLMath ( where LaTeXMLMath ) is a periodic Hamiltonian function , i.e .
its holomorphic Hamiltonian vector field LaTeXMLMath is periodic with a real positive period ( which is LaTeXMLMath or a divisor of this number ) .
In other words , if we write LaTeXMLMath , where LaTeXMLMath is a real vector field called the real part of LaTeXMLMath ( i.e .
LaTeXMLMath is a vector field of LaTeXMLMath considered as a real manifold ; LaTeXMLMath denotes the operator of the complex structure of LaTeXMLMath ) , then the flow of LaTeXMLMath in LaTeXMLMath is periodic .
Of course , if LaTeXMLMath is a holomorphic function on a complex symplectic manifold , then the real part of the holomorphic vector field LaTeXMLMath is a real vector field which preserves the complex symplectic form and the complex structure .
Since the periodic Hamiltonian functions LaTeXMLMath commute pairwise ( in this paper , when we say “ periodic ” , we always mean with a real positive period ) , the real parts of their Hamiltonian vector fields generate a Hamiltonian action of the real torus LaTeXMLMath on LaTeXMLMath .
( One may extend it to a complex torus LaTeXMLMath -action , LaTeXMLMath , but we will only use the compact real part of this complex torus ) .
If LaTeXMLMath is in ( analytic ) Birkhoff normal form , it will Poisson-commute with LaTeXMLMath , and hence it will be preserved by this torus action .
Conversely , if there is a Hamiltonian torus action of LaTeXMLMath in LaTeXMLMath which preserves LaTeXMLMath , then the equivariant Darboux theorem ( which may be proved by an equivariant version of the Moser path method , see , e.g. , LaTeXMLCite ) implies that there is a local holomorphic canonical transformation of coordinates under which the action becomes linear ( and is generated by LaTeXMLMath ) .
Since this action preserves LaTeXMLMath , it follows that LaTeXMLMath .
Thus we have proved the following With the above notations , the following two conditions are equivalent : i ) There exists a holomorphic Birkhoff canonical transformation of coordinates LaTeXMLMath for LaTeXMLMath in a neighborhood of LaTeXMLMath in LaTeXMLMath .
ii ) There exists an analytic Hamiltonian torus action of LaTeXMLMath , in a neighborhood of LaTeXMLMath in LaTeXMLMath , which preserves LaTeXMLMath , and whose linear part is generated by the Hamiltonian vector fields of the functions LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath Proof of Proposition LaTeXMLRef .
Let LaTeXMLMath be a real analytic Hamiltonian function which admits a local complex analytic Birkhoff normalization , we will have to show that LaTeXMLMath admits a local real analytic Birkhoff normalization .
Let LaTeXMLMath be a Hamiltonian torus action which preserves LaTeXMLMath and which has an appropriate linear part , as provided by Proposition LaTeXMLRef .
To prove Proposition LaTeXMLRef , it suffices to linearize this action by a local real analytic symplectic transformation .
Let LaTeXMLMath be a holomorphic periodic Hamiltonian function generating a LaTeXMLMath -subaction of LaTeXMLMath .
Denote by LaTeXMLMath the function LaTeXMLMath , where LaTeXMLMath is the complex conjugation in LaTeXMLMath .
Since LaTeXMLMath is real and LaTeXMLMath , we also have LaTeXMLMath .
It follows that , if LaTeXMLMath is in complex Birkhoff normal form , we will have LaTeXMLMath , and hence LaTeXMLMath is preserved by the torus LaTeXMLMath -action .
LaTeXMLMath is a periodic Hamiltonian function by itself ( because LaTeXMLMath is ) , and due to the fact that LaTeXMLMath is real , the quadratic part of LaTeXMLMath is a real linear combination of the quadratic parts of periodic Hamiltonian functions that generate the torus LaTeXMLMath -action .
It follows that LaTeXMLMath must in fact be also the generator of an LaTeXMLMath -subaction of the torus LaTeXMLMath -action .
( Otherwise , by combining the action of LaTeXMLMath with the LaTeXMLMath -action , we would have a torus action of higher dimension than possible ) .
The involution LaTeXMLMath gives rise to an involution LaTeXMLMath in LaTeXMLMath .
The torus action is reversible with respect to this involution and to the complex conjugation : LaTeXMLEquation .
The above equation implies that the local torus LaTeXMLMath -action may be linearized locally by a real transformation of variables .
Indeed , one may use the following averaging formula LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is the linear part of LaTeXMLMath ( so LaTeXMLMath is a linear torus action ) , and LaTeXMLMath is the standard constant measure on LaTeXMLMath .
The action LaTeXMLMath will be linear with respect to LaTeXMLMath : LaTeXMLMath .
Due to Equation ( LaTeXMLRef ) , we have that LaTeXMLMath , which means that the transformation LaTeXMLMath is real analytic .
After the above transformation LaTeXMLMath , the torus action becomes linear ; the symplectic structure LaTeXMLMath is no longer constant in general , but one can use the equivariant Moser path method to make it back to a constant form ( see , e.g. , LaTeXMLCite ) .
In order to do it , one writes LaTeXMLMath and considers the flow of the time-dependent vector field LaTeXMLMath defined by LaTeXMLMath , where LaTeXMLMath is the constant symplectic form which coincides with LaTeXMLMath at point LaTeXMLMath .
One needs LaTeXMLMath to be LaTeXMLMath -invariant and real .
The first property can be achieved , starting from an arbitrary real analytic LaTeXMLMath such that LaTeXMLMath , by averaging with respect to the torus action .
The second property then follows from Equation ( LaTeXMLRef ) .
Proposition LaTeXMLRef is proved .
LaTeXMLMath Proof of theorem LaTeXMLRef .
According to Proposition LaTeXMLRef , it is enough to prove Theorem LaTeXMLRef in the complex analytic case .
In this section , we will do it by finding local Hamiltonian LaTeXMLMath -actions which preserve the momentum map of an analytically completely integrable system .
The Hamiltonian function generating such an action will be a first integral of the system , called an action function ( as in “ action-angle coordinates ” ) .
If we find LaTeXMLMath such LaTeXMLMath -actions , then they will automatically commute and give rise to a Hamiltonian LaTeXMLMath -action .
To find an action function , we will use the following period integral formula , known as Mineur-Arnold formula : LaTeXMLEquation where LaTeXMLMath denotes an action function , LaTeXMLMath denotes a primitive 1-form ( i.e .
LaTeXMLMath is the symplectic form ) , and LaTeXMLMath denotes an 1-cycle ( closed curve ) lying on a level set of the momentum map .
To show the existence of such 1-cycles LaTeXMLMath , we will use an approximation method , based on the existence of a formal Birkhoff normalization .
Denote by LaTeXMLMath the holomorphic momentum map germ of a given complex analytic integrable Hamiltonian system .
Let LaTeXMLMath be a small positive number such that LaTeXMLMath is defined in the ball LaTeXMLMath .
We will restrict our attention to what happens inside this ball .
As in the previous section , we may assume that in the symplectic coordinate system LaTeXMLMath we have LaTeXMLEquation with LaTeXMLEquation with no resonance relations among LaTeXMLMath .
We will fix this coordinate system LaTeXMLMath , and all functions will be written in this coordinate system .
The real and imaginary parts of the Hamiltonian vector fields of LaTeXMLMath are in involution and define an associated singular foliation in the ball LaTeXMLMath .
Hereafter the norm in LaTeXMLMath is given by the standard Hermitian metric with respect to the coordinate system LaTeXMLMath .
Similarly to the real case , the leaves of this foliation are called local orbits of the associated Poisson action ; they are complex isotropic submanifolds , and generic leaves are Lagrangian and have complex dimension LaTeXMLMath .
For each LaTeXMLMath we will denote the leaf which contains LaTeXMLMath by LaTeXMLMath .
Recall that the momentum map is constant on the orbits of the associated Poisson action .
If LaTeXMLMath is a point such that LaTeXMLMath is a regular value for the momentum map , then LaTeXMLMath is a connected component of LaTeXMLMath .
Denote by LaTeXMLEquation the singular locus of the momentum map , which is also the set of singular points of the associated singular foliation .
What we need to know about LaTeXMLMath is that it is analytic and of codimension at least 1 , though for generic integrable systems LaTeXMLMath is in fact of codimension 2 .
In particular , we have the following Łojasiewicz inequality ( see LaTeXMLCite ) : there exist a positive number LaTeXMLMath and a positive constant LaTeXMLMath such that LaTeXMLEquation for any LaTeXMLMath with LaTeXMLMath , where the norm applied to LaTeXMLMath is some norm in the space of LaTeXMLMath -vectors , and LaTeXMLMath is the distance from LaTeXMLMath to LaTeXMLMath with respect to the Euclidean metric .
We will choose an infinite decreasing series of small numbers LaTeXMLMath ( LaTeXMLMath ) , as small as needed , with LaTeXMLMath , and define the following open subsets LaTeXMLMath of LaTeXMLMath : LaTeXMLEquation .
We will also choose two infinite increasing series of natural numbers LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath , as large as needed , with LaTeXMLMath .
It follows from Birkhoff ’ s Theorem LaTeXMLRef and Lemma LaTeXMLRef that there is a series of local holomorphic symplectic coordinate transformations LaTeXMLMath , LaTeXMLMath , such that the following two conditions are satisfied : a ) The differential of LaTeXMLMath at LaTeXMLMath is identity for each LaTeXMLMath , and for any two numbers LaTeXMLMath with LaTeXMLMath we have LaTeXMLEquation .
In particular , there is a formal limit LaTeXMLMath .
b ) The momentum map is normalized up to order LaTeXMLMath by LaTeXMLMath .
More precisely , the functions LaTeXMLMath can be written as LaTeXMLEquation with LaTeXMLMath such that LaTeXMLEquation .
Here the functions LaTeXMLMath are quadratic functions LaTeXMLEquation in local symplectic coordinates LaTeXMLEquation .
Notice that LaTeXMLMath is a quadratic function in the coordinate system LaTeXMLMath .
But from now on we will use only the original coordinate system LaTeXMLMath .
Then LaTeXMLMath is not a quadratic function in LaTeXMLMath in general , and the quadratic part of LaTeXMLMath is LaTeXMLMath .
The norm in LaTeXMLMath used the estimations in this section will be given by the standard Hermitian metric with respect to the original coordinate system LaTeXMLMath .
Denote by LaTeXMLMath the orbit of the real part of the periodic Hamiltonian vector field LaTeXMLMath which goes through LaTeXMLMath .
Then for any LaTeXMLMath we have LaTeXMLMath , and LaTeXMLMath , i.e .
LaTeXMLMath .
( The reason is that real part of the linear periodic Hamiltonian vector field LaTeXMLMath also preserves the Hermitian metric of LaTeXMLMath , and the linear part of LaTeXMLMath is LaTeXMLMath ) .
As a consequence , we have LaTeXMLEquation .
Note that , for each LaTeXMLMath , we can choose the numbers LaTeXMLMath and LaTeXMLMath first , then choose the radius LaTeXMLMath sufficiently small so that the equivalence LaTeXMLMath makes sense for LaTeXMLMath .
On the other hand , we have LaTeXMLEquation .
We can assume that LaTeXMLMath .
Then for LaTeXMLMath small enough , the above inequality may be combined with Łojasiewicz inequality ( LaTeXMLRef ) to yield LaTeXMLEquation where LaTeXMLMath is a positive constant ( which does not depend on LaTeXMLMath ) .
If LaTeXMLMath , and assuming that LaTeXMLMath is small enough , we have LaTeXMLMath , which may be combined with the last inequality to yield : LaTeXMLEquation .
Assuming that LaTeXMLMath is much larger than LaTeXMLMath , we can use the implicit function theorem to project the curve LaTeXMLMath on LaTeXMLMath as follows : For each point LaTeXMLMath , let LaTeXMLMath be the complex LaTeXMLMath -dimensional disk centered at LaTeXMLMath , which is orthogonal to the kernel of the differential of the momentum map LaTeXMLMath at LaTeXMLMath , and which has radius equal to LaTeXMLMath .
Since the second derivatives of LaTeXMLMath are locally bounded by a constant near LaTeXMLMath , it follows from the definition of LaTeXMLMath that we have , for LaTeXMLMath small enough : LaTeXMLEquation where LaTeXMLMath denotes the differential of the momentum map at LaTeXMLMath , considered as an element of the linear space of LaTeXMLMath matrices .
Inequality ( LaTeXMLRef ) together with Inequality ( LaTeXMLRef ) imply that the momentum map LaTeXMLMath , when restricted to LaTeXMLMath , is a diffeomorphism from LaTeXMLMath to its image , and the image of LaTeXMLMath in LaTeXMLMath under LaTeXMLMath contains a ball of radius LaTeXMLMath .
( Because LaTeXMLMath , where LaTeXMLMath is the order of the radius of LaTeXMLMath , and LaTeXMLMath is a majorant of the order of the norm of the differential of LaTeXMLMath .
The differential of LaTeXMLMath is “ nearly constant ” on LaTeXMLMath due to Inequality ( LaTeXMLRef ) ) .
Thus , if LaTeXMLMath for example , then Inequality ( LaTeXMLRef ) implies that there is a unique point LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath .
The map LaTeXMLMath is continuous , and it maps LaTeXMLMath to some close curve LaTeXMLMath , which must lie on LaTeXMLMath because the point LaTeXMLMath maps to itself under the projection .
When LaTeXMLMath is large enough and LaTeXMLMath is small enough , then LaTeXMLMath is a smooth curve with a natural parametrization inherited from the natural parametrization of LaTeXMLMath , it has bounded derivative ( we can say that its velocity vectors are uniformly bounded by 1 ) , and it depends smoothly on LaTeXMLMath .
Define the following action function LaTeXMLMath on LaTeXMLMath : LaTeXMLEquation where LaTeXMLMath ( so that LaTeXMLMath is the standard symplectic form ) .
This function has the following properties : i ) Because the 1-form LaTeXMLMath is closed on each leaf of the Lagrangian foliation of the integrable system in LaTeXMLMath , LaTeXMLMath is a holomorphic first integral of the foliation .
( This fact is well-known in complex geometry : period integrals of holomorphic LaTeXMLMath -forms , which are closed on the leaves of a given holomorphic foliation , over LaTeXMLMath -cycles of the leaves , give rise to ( local ) holomorphic first integrals of the foliation ) .
The functions LaTeXMLMath Poisson commute pairwise , because they commute with the momentum map .
ii ) LaTeXMLMath is uniformly bounded by 1 on LaTeXMLMath , because LaTeXMLMath is small together with its first derivative .
iii ) Provided that the numbers LaTeXMLMath are chosen large enough , for any LaTeXMLMath we have that LaTeXMLMath coincides with LaTeXMLMath in the intersection of LaTeXMLMath with LaTeXMLMath .
To see this important point , recall that we have LaTeXMLEquation by construction , which implies that the curve LaTeXMLMath is LaTeXMLMath -close to the curve LaTeXMLMath in LaTeXMLMath -norm .
If LaTeXMLMath is large enough with respect to LaTeXMLMath ( say LaTeXMLMath ) , then it follows that the complex LaTeXMLMath -dimensional cylinder LaTeXMLEquation lies inside ( and near the center of ) the complex LaTeXMLMath -dimensional cylinder LaTeXMLEquation .
On the other hand , one can check that LaTeXMLMath is a retract of LaTeXMLMath in LaTeXMLMath , and the same thing is true for the index LaTeXMLMath .
It follows easily that LaTeXMLMath must be homotopic to LaTeXMLMath in LaTeXMLMath , implying that LaTeXMLMath coincides with LaTeXMLMath .
iv ) Since LaTeXMLMath coincides with LaTeXMLMath in LaTeXMLMath , we may glue these functions together to obtain a holomorphic function , denoted by LaTeXMLMath , on the union LaTeXMLMath .
Lemma LaTeXMLRef in the following section shows that if we have a bounded holomorphic function in LaTeXMLMath then it can be extended to a holomorphic function in a neighborhood of LaTeXMLMath in LaTeXMLMath .
Thus our action functions LaTeXMLMath are holomorphic in a neighborhood of LaTeXMLMath in LaTeXMLMath .
v ) LaTeXMLMath is a local periodic Hamiltonian function whose quadratic part is LaTeXMLMath .
To see this , remark that LaTeXMLEquation for LaTeXMLMath .
Since the curve LaTeXMLMath is LaTeXMLMath -close to the curve LaTeXMLMath by construction ( provided that LaTeXMLMath ) , we have that LaTeXMLEquation for LaTeXMLMath .
Due to the nature of LaTeXMLMath ( almost every complex line in LaTeXMLMath which contains the origin LaTeXMLMath intersects with LaTeXMLMath in an open subset ( of the line ) which surrounds the point LaTeXMLMath ) , it follows from the last estimation that in fact the coefficients of all the monomial terms of order LaTeXMLMath of LaTeXMLMath coincide with that of LaTeXMLMath , i.e .
we have LaTeXMLEquation in a neighborhood of LaTeXMLMath in LaTeXMLMath .
In particular , we have LaTeXMLEquation where the limit on the right-and side of the above equation is understood as the formal limit of Taylor series , and the left-hand side is also considered as a Taylor series .
This is enough to imply that LaTeXMLMath has LaTeXMLMath as its quadratic part , and that LaTeXMLMath is a periodic Hamiltonian of period LaTeXMLMath because each LaTeXMLMath is so .
( If a local holomorphic Hamiltonian vector field which vanishes at LaTeXMLMath is formally periodic then it is periodic ) .
Now we can apply Proposition LaTeXMLRef and Proposition LaTeXMLRef to finish the proof of Theorem LaTeXMLRef .
LaTeXMLMath The following lemma shows that the action functions LaTeXMLMath constructed in the previous section can be extended holomorphically in a neighborhood of LaTeXMLMath .
Let LaTeXMLMath , with LaTeXMLMath , where LaTeXMLMath is an arbitrary series of positive numbers and LaTeXMLMath is a local proper complex analytic subset of LaTeXMLMath ( LaTeXMLMath ) .
Then any bounded holomorphic function on LaTeXMLMath has a holomorphic extension in a neighborhood of LaTeXMLMath in LaTeXMLMath .
Proof .
Though we suspect that this lemma should have been known to specialists in complex analysis , we could not find it in the literature , so we will provide a proof here .
When LaTeXMLMath the lemma is obvious , so we will assume that LaTeXMLMath .
Without loss of generality , we can assume that LaTeXMLMath is a singular hypersurface .
We divide the lemma into two steps : Step 1 .
The case when LaTeXMLMath is contained in the union of hyperplanes LaTeXMLMath where LaTeXMLMath is a local holomorphic system of coordinates .
Clearly , LaTeXMLMath contains a product of non-empty annuli LaTeXMLMath , hence LaTeXMLMath is defined by a Laurent series in LaTeXMLMath there .
We will study the domain of convergence of this Laurent series , using the well-known fact that the domain of convergence of a Laurent series is logarithmically convex .
More precisely , denote by LaTeXMLMath the map LaTeXMLMath from LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath , and set LaTeXMLEquation .
Denote by LaTeXMLMath the convex hull of LaTeXMLMath in LaTeXMLMath .
Then since the function LaTeXMLMath is analytic and bounded in LaTeXMLMath , it can be extended to abounded analytic function on LaTeXMLMath .
On the other hand , by definition of LaTeXMLMath , there is a series of positive numbers LaTeXMLMath ( tending to infinity ) such that LaTeXMLMath , where LaTeXMLEquation .
It is clear that the convex hull of LaTeXMLMath , with each LaTeXMLMath defined as above , contains a neighborhood of LaTeXMLMath , i.e .
a set of the type LaTeXMLEquation .
It implies that the function LaTeXMLMath can be extended to a bounded analytic function in LaTeXMLMath , where LaTeXMLMath is a neighborhood of LaTeXMLMath in LaTeXMLMath .
Since LaTeXMLMath is bounded in LaTeXMLMath , it can be extended analytically on the whole LaTeXMLMath .
Step 1 is finished .
Step 2 .
Consider now the case with an arbitrary LaTeXMLMath .
Then we can use Hironaka ’ s desingularization theorem LaTeXMLCite to make it smooth .
The general desingularization theorem is a very hard theorem , but in the case of a singular complex hypersurface a relatively simple constructive proof of it can be found in LaTeXMLCite .
In fact , since the exceptional divisor will also have to be taken into account , after the desingularization process we will have a variety which may have normal crossings .
More precisely , we have the following commutative diagram LaTeXMLEquation where LaTeXMLMath denotes the germ of LaTeXMLMath at LaTeXMLMath presented by a ball which is small enough ; LaTeXMLMath is a complex manifold ; the projection LaTeXMLMath is surjective , and injective outside the exceptional divisor ; LaTeXMLMath denotes the union of the exceptional divisor with the smooth proper submanifold of LaTeXMLMath which is desingularization of LaTeXMLMath – the only singularities in LaTeXMLMath are normal crossings ; LaTeXMLMath is compact .
LaTeXMLMath is obtained from LaTeXMLMath by a finite number of blowing-ups along submanifolds .
Denote by LaTeXMLMath the preimage of LaTeXMLMath under the projection LaTeXMLMath .
One can pull back LaTeXMLMath from LaTeXMLMath to LaTeXMLMath to get a bounded holomorphic function on LaTeXMLMath , denoted by LaTeXMLMath .
An important observation is that the type of LaTeXMLMath persists under blowing-ups along submanifolds .
( Or equivalently , the type of its complement , which may be called a sharp-horn-neighborhood of LaTeXMLMath because it is similar to horn-type neighborhoods of LaTeXMLMath used by singularists but it is sharp of arbitrary order , is persistent under blowing-ups ) .
More precisely , for each point LaTeXMLMath , the complement of LaTeXMLMath in a small neighborhood of LaTeXMLMath is a “ sharp-horn-neighborhood ” of LaTeXMLMath at LaTeXMLMath .
Since LaTeXMLMath only has normal crossings , the pair LaTeXMLMath satisfies the conditions of Step 1 , and therefore we can extend LaTeXMLMath holomorphically in a neighborhood of LaTeXMLMath in LaTeXMLMath .
Since LaTeXMLMath is compact , we can extend LaTeXMLMath holomorphically in a neighborhood of LaTeXMLMath in LaTeXMLMath .
One can now project this extension of LaTeXMLMath back to LaTeXMLMath to get a holomorphic extension of LaTeXMLMath in a neighborhood of LaTeXMLMath .
The lemma is proved .
LaTeXMLMath Remark .
The “ sharp-horn ” type of the complement of LaTeXMLMath in the above lemma is essential .
If we replace LaTeXMLMath by LaTeXMLMath ( for any given number LaTeXMLMath ) then the lemma is false .
Acknowledgements .
I would like to thank Jean-Paul Dufour for proofreading this paper , Jean-Claude Sikorav for supplying me with the above proof of Step 1 of Lemma LaTeXMLRef , and Alexandre Bruno for some critical remarks .
I ’ m also thankful to the referee for his pertinent remarks which helped improve the presentation of this paper .
Le Théorème 2 de [ BR ] montre qu ’ il existe un entier impair LaTeXMLMath tel que LaTeXMLMath et LaTeXMLMath , LaTeXMLMath et LaTeXMLMath sont linéairement indépendants sur LaTeXMLMath : ce résultat implique l ’ irrationalité de LaTeXMLMath mais est bien sûr plus fort .
Dans cet article , nous améliorons la majoration LaTeXMLMath en ne recherchant que l ’ irrationalité de LaTeXMLMath : Il existe un entier impair LaTeXMLMath tel que LaTeXMLMath et LaTeXMLMath .
La démonstration de ce théorème repose sur la série suivante LaTeXMLEquation où LaTeXMLMath est un nombre complexe de module LaTeXMLMath et LaTeXMLMath un entier LaTeXMLMath .
L ’ étude de LaTeXMLMath , que nous écrirons désormais LaTeXMLMath , est similaire à celle de la série considérée dans [ R ] et [ BR ] : LaTeXMLMath Le Lem me 1 montre que , si LaTeXMLMath est pair , la série LaTeXMLMath s ’ écrit comme une combinaison linéaire ( à coefficients rationnels ) de 1 et des LaTeXMLMath pour LaTeXMLMath impair , LaTeXMLMath .
LaTeXMLMath Le Lem me 2 détermine un dénominateur commun aux coefficients de cette combinaison linéaire .
LaTeXMLMath L ’ estimation du comportement de LaTeXMLMath est délicate puisqu ’ une expression intégrale de type Beukers [ Be ] n ’ est pas connue pour LaTeXMLMath .
Néanmoins , en suivant Nesterenko [ Ne ] , le Lem me 4 montre que LaTeXMLMath peut s ’ écrire comme la partie réelle d ’ une intégrale complexe : le comportement asymptotique de cette intégrale est alors déterminé au Lem me 5 par la méthode du col ( Lem me 3 ) .
LaTeXMLMath Enfin , il n ’ y a pas lieu ici de borner la hauteur des coefficients de la combinaison : cela n ’ est nécessaire que pour l ’ indépendance linéaire .
Remerciements L ’ auteur tient à remercier F. Amoroso et D. Essouabri pour leurs conseils qui ont permis d ’ améliorer une précédente version .
Posons LaTeXMLEquation .
LaTeXMLMath et LaTeXMLMath : on a alors la décomposition en éléments simples LaTeXMLEquation .
Définissons également les polynômes à coefficients rationnels LaTeXMLEquation où LaTeXMLMath Pour tout LaTeXMLMath , LaTeXMLMath , on a LaTeXMLEquation et LaTeXMLMath .
De plus , si LaTeXMLMath est pair , alors pour tout LaTeXMLMath et pour tout entier pair LaTeXMLMath , on a LaTeXMLMath et donc LaTeXMLEquation .
Démonstration De la décomposition ( 1 ) de LaTeXMLMath , on déduit que si LaTeXMLMath LaTeXMLEquation .
Comme le degré total de la fraction rationnelle LaTeXMLMath est LaTeXMLMath , on a LaTeXMLEquation .
On peut réécrire LaTeXMLMath où LaTeXMLEquation .
On a LaTeXMLEquation .
En appliquant l ’ identité LaTeXMLMath aux trois symboles de Pochhammer de ( 3 ) , on obtient LaTeXMLEquation .
Donc pour tout LaTeXMLMath , LaTeXMLEquation .
En particulier , avec LaTeXMLMath et LaTeXMLMath , on a LaTeXMLEquation ce qui implique la relation LaTeXMLEquation .
Si LaTeXMLMath est pair , on en déduit que LaTeXMLMath .
Pour tout LaTeXMLMath on a LaTeXMLEquation où LaTeXMLMath .
Démonstration On écrit LaTeXMLMath où LaTeXMLMath et LaTeXMLEquation .
Décomposons LaTeXMLMath , LaTeXMLMath et LaTeXMLMath en fractions partielles : LaTeXMLEquation où LaTeXMLEquation .
LaTeXMLEquation et LaTeXMLEquation .
On a alors pour tout entier LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation avec LaTeXMLMath si LaTeXMLMath , LaTeXMLMath si LaTeXMLMath .
On a donc montré que LaTeXMLEquation sont des entiers pour tout LaTeXMLMath .
De plus , LaTeXMLMath .
Grâce à la formule de Leibniz LaTeXMLEquation ( où la somme est sur les multi-indices LaTeXMLMath tels que LaTeXMLMath ) , on en déduit alors que LaTeXMLMath .
Les expressions ( 2 ) des polynômes LaTeXMLMath et LaTeXMLMath permettent de conclure .
Pour estimer LaTeXMLMath , nous suivons la démarche utilisée par [ Ne ] et [ HP ] qui consiste à exprimer LaTeXMLMath à l ’ aide d ’ une intégrale complexe à laquelle on peut appliquer la méthode du col , méthode dont nous rappelons tout d ’ abord le principe ( voir par exemple [ Co ] , pp .
91-94 ou [ Di ] , pp .
279-285 ] ) .
Soit LaTeXMLMath une fonction analytique au voisinage d ’ un point LaTeXMLMath .
On appelle chemin de descente de LaTeXMLMath en LaTeXMLMath tout chemin du plan issu de LaTeXMLMath et le long duquel LaTeXMLMath est strictement décroissante quand LaTeXMLMath s ’ éloigne de LaTeXMLMath .
Les chemins de plus grande descente de LaTeXMLMath en LaTeXMLMath sont les chemins tels que LaTeXMLMath a ( localement ) la décroissance la plus rapide parmi tous les chemins de descente : il est en fait équivalent de demander que LaTeXMLMath soit constante le long de ces chemins , c ’ est à dire que la phase de LaTeXMLMath soit stationaire .
Supposons LaTeXMLMath telle que LaTeXMLMath et LaTeXMLMath .
Notons LaTeXMLMath la direction d ’ une droite LaTeXMLMath passant par LaTeXMLMath , c ’ est-à-dire LaTeXMLMath où LaTeXMLMath .
Il existe exactement deux chemins de plus grande descente de LaTeXMLMath en LaTeXMLMath , dont les directions des tangentes en LaTeXMLMath sont LaTeXMLMath et LaTeXMLMath : ces directions critiques sont opposées .
Il peut s ’ avérer difficile de déterminer exactement les chemins de plus grande descente .
On peut s ’ affranchir de ce problème en considérant n ’ importe quelle direction LaTeXMLMath en LaTeXMLMath telle que LaTeXMLMath : au voisinage de LaTeXMLMath , LaTeXMLEquation et sur un chemin LaTeXMLMath dont les deux directions en LaTeXMLMath vérifient la condition ci-dessus , on a alors LaTeXMLMath et LaTeXMLMath admet un maximum local en LaTeXMLMath le long de LaTeXMLMath .
Convenons de dire qu ’ un chemin LaTeXMLMath est admissible en LaTeXMLMath si les deux directions LaTeXMLMath en LaTeXMLMath vérifient LaTeXMLMath et si LaTeXMLMath est le maximum global de LaTeXMLMath le long de LaTeXMLMath .
Soit LaTeXMLMath et LaTeXMLMath deux fonctions analytiques dans un ouvert simplement connexe LaTeXMLMath du plan .
Supposons qu ’ il existe LaTeXMLMath tel que LaTeXMLMath et LaTeXMLMath .
Si LaTeXMLMath est un chemin inclus dans LaTeXMLMath et admissible en LaTeXMLMath , alors LaTeXMLEquation où le choix de LaTeXMLMath dépend de l ’ orientation de LaTeXMLMath .
De plus , cette estimation est encore valable si LaTeXMLMath est un chemin que l ’ on peut déformer en un chemin admissible en LaTeXMLMath .
Nous appliquons maintenant cette méthode à l ’ estimation asymptotique de LaTeXMLMath .
Considérons l ’ intégrale complexe LaTeXMLEquation où LaTeXMLMath est un nombre complexe tel que LaTeXMLMath et LaTeXMLMath , LaTeXMLMath est une droite verticale orientée de LaTeXMLMath à LaTeXMLMath et contenue dans la bande LaTeXMLMath , ce qui assure que l ’ intégrale LaTeXMLMath converge .
Dans ces conditions , on a i ) LaTeXMLEquation ii ) LaTeXMLEquation .
Démonstration i ) Comme LaTeXMLMath et LaTeXMLMath , on a LaTeXMLEquation .
De plus , la formule des compléments LaTeXMLMath ( pour LaTeXMLMath ) implique que LaTeXMLEquation .
On a donc LaTeXMLEquation où LaTeXMLMath est une droite verticale quelconque contenue dans LaTeXMLMath .
Le changement de variable LaTeXMLMath et le théorème de Cauchy justifient que LaTeXMLEquation ii ) Soit LaTeXMLMath et soit LaTeXMLMath tel que LaTeXMLMath .
Considérons le contour rectangulaire LaTeXMLMath orienté dans le sens direct , de sommets LaTeXMLMath et LaTeXMLMath : la fonction LaTeXMLMath est méromorphe dans le demi-plan LaTeXMLMath et ses pôles sont les entiers LaTeXMLMath .
En appliquant le théorème des résidus , il découle que LaTeXMLEquation où LaTeXMLEquation .
Sur les trois côtés LaTeXMLMath , LaTeXMLMath et LaTeXMLMath , on a LaTeXMLMath .
Sur LaTeXMLMath , en posant LaTeXMLMath , on a LaTeXMLEquation et donc LaTeXMLMath .
Comme LaTeXMLMath , on en déduit que LaTeXMLEquation puisque LaTeXMLMath et LaTeXMLMath .
De façon similaire , sur les deux côtés LaTeXMLMath et LaTeXMLMath , en posant LaTeXMLMath avec LaTeXMLMath , on a LaTeXMLEquation et donc LaTeXMLMath .
Comme LaTeXMLMath , on en déduit que LaTeXMLEquation .
Donc LaTeXMLEquation .
En particulier , LaTeXMLEquation et donc LaTeXMLMath .
Nous utilisons maintenant la formule de Stirling sous la forme suivante LaTeXMLEquation où LaTeXMLMath , LaTeXMLMath et où les fonctions LaTeXMLMath et LaTeXMLMath sont définies avec la détermination principale du logarithme .
Sur la droite LaTeXMLMath , les quantités LaTeXMLMath , LaTeXMLMath , LaTeXMLMath et LaTeXMLMath sont équivalentes à des multiples constants de LaTeXMLMath , d ’ où LaTeXMLEquation avec LaTeXMLEquation et LaTeXMLEquation les différentes fonctions racines et logarithmes de LaTeXMLMath et LaTeXMLMath étant de nouveau définies à l ’ aide de la détermination principale du logarithme .
L ’ expression ( 5 ) de LaTeXMLMath se prête maintenant à une estimation par la méthode du col. Dorénavant , nous supposons LaTeXMLMath .
Alors LaTeXMLEquation et l ’ équation LaTeXMLMath possède une seule solution LaTeXMLMath vérifiant LaTeXMLMath : LaTeXMLEquation .
On a LaTeXMLEquation et LaTeXMLEquation .
On constate que LaTeXMLMath et LaTeXMLMath vérifient LaTeXMLMath .
Montrons que la droite LaTeXMLMath est admissible , c ’ est à dire que LaTeXMLMath admet un maximum global en LaTeXMLMath le long de LaTeXMLMath .
Posons LaTeXMLMath ; donc LaTeXMLEquation .
On a LaTeXMLEquation .
Par ailleurs , LaTeXMLMath pour LaTeXMLMath , d ’ où LaTeXMLEquation où l ’ on a noté LaTeXMLEquation .
On vérifie que LaTeXMLMath a une seule racine dans LaTeXMLMath .
Donc LaTeXMLMath ne s ’ annule que pour LaTeXMLMath .
La fonction LaTeXMLMath est donc strictement croissante sur LaTeXMLMath , puis strictement décroissante sur LaTeXMLMath .
En conséquence , la droite LaTeXMLMath est admissible en LaTeXMLMath pour LaTeXMLMath .
On a : LaTeXMLEquation où LaTeXMLMath .
De plus , il existe une suite d ’ entiers LaTeXMLMath telle que LaTeXMLEquation .
Démonstration L ’ estimation de LaTeXMLMath résulte de l ’ estimation générale ( 4 ) , appliquée à ( 5 ) et à la droite admissible LaTeXMLMath .
Pour montrer la dernière affirmation , notons LaTeXMLMath et LaTeXMLMath , de sorte que LaTeXMLEquation .
LaTeXMLEquation où LaTeXMLMath est une suite de nombres complexes qui converge vers LaTeXMLMath .
Remarquons que LaTeXMLMath n ’ est pas un multiple entier de LaTeXMLMath et donc il existe une suite d ’ entiers LaTeXMLMath telle que LaTeXMLMath converge vers une limite LaTeXMLMath .
On en déduit que LaTeXMLEquation et donc LaTeXMLEquation .
Démonstration du Théorème 1 Posons LaTeXMLMath et LaTeXMLMath pour LaTeXMLMath : le Lem me 2 implique que ce sont des entiers .
Définissons également LaTeXMLMath : le Lem me 1 montre que LaTeXMLEquation .
Enfin , d ’ après le Théorème des nombres premiers , LaTeXMLMath .
Le Lem me 5 montre que LaTeXMLEquation ce qui prouve le Théorème 1 .
Références [ BR ] K. Ball et T. Rivoal , Irrationalité d ’ une infinité de valeurs de la fonction zêta aux entiers impairs , soumis .
[ Be ] F. Beukers , A note on the irrationality of LaTeXMLMath and LaTeXMLMath , Bull .
Lond .
Math .
Soc .
11 , no .
33 , 268-272 ( 1978 ) .
[ Co ] E. T. Copson , Asymptotic expansions , Cambridge University Press ( 1967 ) .
[ Di ] J. Dieudonné , Calcul infinitésimal , Collection ” Méthodes ” , Hermann ( 1980 ) .
[ HP ] T. G. Hessami Pilerhood , Linear independence of vectors withpolylogarithmic coordinates , Mosc .
Univ .
Math .
Bull .
54 , no .
6 , 40-42 ( 1999 ) .
[ Ne ] Yu.V .
Nesterenko , A few remarks on LaTeXMLMath , Math .
Notes , 59 , no .
6 , 625-636 ( 1996 ) .
[ R ] T. Rivoal , La fonction Zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs , C. R. Acad .
Sci .
Paris 331 , 267-270 ( 2000 ) .
The tensorial form of the Lax pair equations are given in a compact and geometrically transparent form in the presence of Cartan ’ s torsion tensor .
Three dimensional spacetimes admitting Lax tensors are analyzed in detail .
Solutions to Lax tensor equations include interesting examples as separable coordinates and the Toda lattice .
Geometrization of the Lax Pair Tensors D. Băleanu Department of Mathematics and Computer Sciences , Faculty of Arts and Sciences , Çankaya University , 06531 Ankara , Turkey S. Başkal Physics Department , Middle East Technical University , Ankara , 06531 , Turkey In a series of papers , Rosquist et all .
LaTeXMLCite introduced the Lax tensors and presented some models for which Lax tensors exist .
The Lax representation for a given dynamical system is not unique LaTeXMLCite .
Consequently , the Lax tensor equations LaTeXMLEquation depend on an arbitrary third rank object LaTeXMLMath , whose specific form can be found through a suitable geometrization of the system .
The solutions of the Lax tensors were investigated on the ” dual ” manifold LaTeXMLCite in two dimensions LaTeXMLCite .
The connection between Lax tensors and Killing-Yano tensors of order three has also been well-established LaTeXMLCite .
Killing-Yano tensors of order three were introduced long time ago by Bocnher LaTeXMLCite and Yano LaTeXMLCite as a natural generalization of a Killing vector .
Gibbons et all .
LaTeXMLCite found that Killing-Yano tensors can be understood as an object generating a ” non-generic symmetry ” , i.e. , a supersymmetry appearing only in the specific space-time .
An alternative way to introduce Lax tensors is to consider the Killing and Killing-Yano tensors of order three .
Since the Killing-Yano tensors of order three are the generators of the ” non-generic ” suppersymmetry on a given manifold LaTeXMLCite , investigation of the manifolds admitting Lax tensors becomes interesting .
Untill now a Killing tensor could only be found by solving the Killing or the Killing-Yano equations LaTeXMLCite of order two , or by calculating the Nijenhuis tensor LaTeXMLCite .
The existence of the Lax tensors of order three open a new possibility for finding Killing tensors .
The three-particle open Toda lattice was geometrized by a suitable canonical transformation and it was found that the tensor LaTeXMLMath is antisymmetric with respect to its first two indices LaTeXMLCite .
It is also known that the geometric duality can be generalized to spinning spaces , at an expense of introducing a torsion on the manifold LaTeXMLCite .
All these results suggest that torsion can play an important role in the description of Lax tensors .
There are interesting questions yet to be answered as to the Lax equations , and as to the geometrical interpretations of its constituent tensors .
Specifically , the properties of the manifold admitting Lax tensors deserve further investigation .
Although , on a very general context answers to these questions are not easy to provide , problems concerning Lax tensor equations seem to be manageable when some symmetry properties are imposed to its constituent tensors , in some particular dimensions , which is what we intend to present here .
This letter is organized as follows : In Sec .
2 the Lax tensor equations are rederived in the presence of torsion to provide a geometric meaning to the arbitrary tensor LaTeXMLMath .
In Sec .
3 the Lax tensor equations are analyzed in three dimensions for various symmetry properties of this tensor .
The integrability conditions of these equations are discussed for specific cases such as flat and curved space-times as well as spacetimes having torsion .
We present the explicit expressions for the Lax tensors for a number of interesting examples including separable orthogonal systems and the three-particle open Toda lattice .
The last section is devoted to our comments and conclusions .
In this section we rederive the tensorial Lax pair equations by taking into account a suitable definition for the Dirac-Poisson brackets when torsion is introduced on an n-dimensional manifold .
The manifold is endowed with a metric LaTeXMLEquation and with an affine connection LaTeXMLMath , satisfying the metricity condition , and thereby related to the Christoffel symbols and Cartan ’ s torsion tensor as : LaTeXMLEquation .
The torsion tensor is assumed to be completely antisymmetric to fit the autoparallels with the geodesics of the manifold .
The Hamiltonian for a dynamical system is constructed as LaTeXMLEquation .
On the phase space , the expression for the covariant derivative with torsion tensor is given by LaTeXMLEquation where F is any function .
The Poisson-Dirac brackets are expressed as LaTeXMLEquation and the fundamental brackets are LaTeXMLEquation .
In the phase space , the geodesic equations read LaTeXMLEquation .
The complete integrability of these equations are secured with the existence of the Lax pair equations : LaTeXMLCite LaTeXMLEquation .
Referring to LaTeXMLCite , it will be assumed that LaTeXMLMath is a homogeneous first order polynomial in momenta LaTeXMLMath .
The second matrix LaTeXMLMath is also of the same form with respect to the momenta LaTeXMLMath .
These third rank objects are referred as the Lax tensor and the Lax connection , respectively .
After these preliminaries the brackets LaTeXMLMath are evaluated for the time derivative of the Lax matrix LaTeXMLEquation .
The right hand side of ( LaTeXMLRef ) can be written as LaTeXMLEquation .
Comparing the right hand side of ( LaTeXMLRef ) with the left hand side , it is now possible to make the following identification : LaTeXMLEquation .
In general , LaTeXMLMath in ( LaTeXMLRef ) is an arbitrary third rank tensor , unless some symmetry properties are imposed on it .
However , assuming it to be completely antisymmetric , it can be identified with a geometrical object as Cartan ’ s torsion tensor .
Furthermore , with such an identification the Lax tensor equations reduce to a compact form : LaTeXMLEquation where , the semicolon denotes the covariant differentiation with respect to the affine connection .
Particularly , if LaTeXMLMath , then the right hand side of ( LaTeXMLRef ) vanishes .
In n-dimensional Euclidean or Minkowskian spacetimes they admit a solution of the form : LaTeXMLEquation .
Here the tensor LaTeXMLMath satisfies LaTeXMLMath , and LaTeXMLMath is a constant tensor .
If LaTeXMLMath is completely antisymmetric , then with LaTeXMLMath , the equations ( LaTeXMLRef ) and ( LaTeXMLRef ) are equivalent .
Specifically , the Lax tensor can be split into completely symmetric and antisymmetric parts LaTeXMLMath .
For a more detailed analysis we are confined to three dimensions , for the following reasons : First , the antisymmetric part of the Lax tensor becomes proportional to LaTeXMLMath .
Then equation ( LaTeXMLRef ) reduces to a simple form LaTeXMLEquation for the symmetric part when LaTeXMLMath , while for LaTeXMLMath the introduction of a completely antisymmetric third rank tensor is not possible , to identify it with torsion .
In view of ( LaTeXMLRef ) , LaTeXMLMath is a covariantly constant tensor .
Spacetimes admitting such tensors are analyzed in the context of recurrent tensors when LaTeXMLMath LaTeXMLCite .
The integrability condition for ( LaTeXMLRef ) can be expressed as : LaTeXMLEquation .
For simplicity in the notation and when no confusion is possible , LaTeXMLMath either denotes antisymmetrization , or cyclic permutations .
After a detailed analysis of the above consistency condition for an arbitrary diagonal metric with LaTeXMLMath , we have found that all six surviving components of the Riemann tensor are equal and the surviving Lax tensor components are related as : LaTeXMLMath .
However , these restrictions turned out to be very severe on the manifold , yielding the metric components to be constants .
Therefore , we can state that the Lax equations are integrable if and only if the manifold is flat .
In the following we will present some examples for the solutions of Lax equations in three dimensions .
One of the many possible generalizations of a two dimensional Rindler system to three dimensions can be obtained by assuming the coordinates as LaTeXMLEquation with LaTeXMLMath .
The associated metric is LaTeXMLEquation .
The LaTeXMLMath hypersurface defines the well-known two dimensional Rindler system LaTeXMLCite .
The symmetric Lax tensor has ten components in three dimensions and without torsion the solutions of ( LaTeXMLRef ) are found to be LaTeXMLEquation .
There are several relations between the Lax tensors and second rank Killing tensors .
A second rank Killing tensor satisfies LaTeXMLEquation .
When the spacetime is flat a Lax tensor can be constructed as : LaTeXMLCite LaTeXMLEquation .
A solution to ( LaTeXMLRef ) , for this metric is found as LaTeXMLEquation .
When a second rank Killing tensor is non-degenerate , it can be considered as a metric itself , defining a ” dual ” spacetime LaTeXMLCite .
Although , three dimensional Rindler system defines a flat spacetime , its dual spacetime is curved , with a curvature scalar LaTeXMLMath .
In view of ( LaTeXMLRef ) and ( LaTeXMLRef ) we obtain LaTeXMLEquation as the two surviving components of the Lax tensor .
Further relations between the Killing tensor and the Lax tensor can be established as LaTeXMLMath .
Such a Killing tensor is trivial LaTeXMLCite .
Its surviving components are : LaTeXMLEquation .
The dual spacetime associated to this Killing tensor is flat .
Separable coordinate systems in three-dimensional Minkowski space label confocal surfaces of order two LaTeXMLCite .
In the following coordinates will be denoted by LaTeXMLMath and LaTeXMLMath , defined on the intervals LaTeXMLMath , where LaTeXMLMath .
The metric corresponding of the Ellipsoidal coordinates systems , falls into this class LaTeXMLEquation .
Non-degenerate Killing metrics corresponding to the Ellipsoidal coordinate system are immediately calculated as : LaTeXMLEquation .
For LaTeXMLMath solutions to ( LaTeXMLRef ) are found as : LaTeXMLEquation .
The dynamics of the three-particle open Toda Lattice can be formulated through a purely kinetic Hamiltonian LaTeXMLEquation .
In view of ( LaTeXMLRef ) the components of the diagonal metric are found to be LaTeXMLEquation .
At this point we refer to Sec .
2 and relax the totally antisymmetric condition on the torsion tensor , but consider it in its most general form , which is LaTeXMLMath , still keeping the metricity condition .
Even with this relaxation the autoparalles are retained on the manifold .
Now , the affine connection differs from the Christoffel symbols by the contorsion tensor LaTeXMLMath as : LaTeXMLCite LaTeXMLEquation whose relation to the torsion tensor is defined through LaTeXMLMath .
The surviving components of LaTeXMLMath are found LaTeXMLEquation .
We give the surviving components of the contorsion tensor as : LaTeXMLEquation .
The tensorial Lax equation ( LaTeXMLRef ) is satisfied when LaTeXMLMath is as in ( LaTeXMLRef ) and LaTeXMLMath is as above .
Therefore , for this particular example LaTeXMLMath can be interpreted as the contorsion tensor .
In this paper , we generalized the Lax tensor equations introduced by Rosquist , by appropriately defining the Poisson brackets in the presence of torsion .
This way , otherwise arbitrary tensors of these equations can be identified with concrete geometrical objects , such as the torsion or the contorsion tensor , when some relevant symmetry properties are imposed on them .
The form of the equations are considerably simplified , when LaTeXMLMath is completely antisymmetric .
We have also found the conditions when the Lax equation on a three dimensional manifold admit solutions .
We analyzed separable coordinates and the three-particle open Toda lattice , in detail .
As was pointed in LaTeXMLCite Killing tensors can be trivial or non-trivial .
A similar characterization arises when we investigate the solutions of the Lax tensor equations .
If the Lax tensors satisfy LaTeXMLMath , then they are non-trivial tensors .
Further intriguing problems are to investigate the existence of the Lax tensors , when the manifold admits Runge-Lenz symmetry , or to find Lax tensors for superintegrable systems .
These problems are currently under investigation LaTeXMLCite .
One of us ( D .
B . )
is grateful to Ashok Das for valuable discussions .
We would also like to thank to Y. Güler for encouragements .
OU-HET 382 hep-th/0104184 April 2001 Supersymmetric Nonlinear Sigma Models on Ricci-flat Kähler Manifolds with LaTeXMLMath Symmetry Kiyoshi Higashijima E-mail : higashij @ phys.sci.osaka-u.ac.jp , Tetsuji Kimura E-mail : t-kimura @ het.phys.sci.osaka-u.ac.jp and Muneto Nitta E-mail : nitta @ het.phys.sci.osaka-u.ac.jp Department of Physics , Graduate School of Science , Osaka University , Toyonaka , Osaka 560-0043 , Japan We propose a class of LaTeXMLMath supersymmetric nonlinear sigma models on the Ricci-flat Kähler manifolds with LaTeXMLMath symmetry .
String theory propagating in a curved spacetime is described by a conformally invariant nonlinear sigma model in two dimensions .
Spacetime supersymmetry of the string theory requires LaTeXMLMath worldsheet supersymmetry .
The conformal invariance is realized in finite field theories with vanishing LaTeXMLMath -functions .
In this letter , we propose a class of nonlinear sigma models on Ricci-flat Kähler manifolds with LaTeXMLMath symmetry .
Two-dimensional LaTeXMLMath supersymmetric nonlinear sigma models are described by ( anti- ) chiral superfields : LaTeXMLMath ( LaTeXMLMath ) where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are complex scalar , Dirac fermion , auxiliary scalar fields , respectively .
To define supersymmetric nonlinear sigma models on the Ricci-flat Kähler manifolds with LaTeXMLMath symmetry , we prepare dynamical chiral superfields LaTeXMLMath ( LaTeXMLMath ; LaTeXMLMath ) , belonging to the vector representation of LaTeXMLMath , and an auxiliary chiral superfield LaTeXMLMath , belonging to an LaTeXMLMath singlet .
The most general Lagrangian with LaTeXMLMath symmetry , composed of these chiral superfields , is given by LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath invariant defined by LaTeXMLEquation and LaTeXMLMath is an arbitrary function of LaTeXMLMath .
In the Lagrangian ( LaTeXMLRef ) , we can assume that LaTeXMLMath is a positive real constant , using the field redefinition .
By the integration over the auxiliary field LaTeXMLMath , we obtain the constraint among the superfields LaTeXMLMath , LaTeXMLMath , whose bosonic part is LaTeXMLEquation .
The manifold defined by this constraint with the Kähler potential LaTeXMLMath is a non-compact Kähler manifold with the complex dimension LaTeXMLMath , where the LaTeXMLMath symmetry acts as a holomorphic isometry .
One of the component , say LaTeXMLMath , can be expressed in terms of the independent fields LaTeXMLMath ( LaTeXMLMath ) LaTeXMLEquation .
To obtain the Ricci-flat Kähler manifold with LaTeXMLMath symmetry , we calculate the Ricci form and solve the Ricci-flat condition .
To do this , we first calculate the Kähler metric LaTeXMLMath of the manifold : LaTeXMLEquation where the prime denotes differentiation with respect to LaTeXMLMath , and LaTeXMLMath is given by ( LaTeXMLRef ) .
Here we have used the same letter LaTeXMLMath for its lowest component : LaTeXMLMath .
Using this metric the nonlinear sigma model Lagrangian is written as LaTeXMLEquation where LaTeXMLMath is the Riemann curvature tensor and LaTeXMLMath LaTeXMLCite .
Without loss of generality , an arbitrary point on the manifold can be transformed by the LaTeXMLMath symmetry to LaTeXMLEquation where LaTeXMLMath is complex .
The Kähler metric on this point is LaTeXMLEquation and its determinant is given by LaTeXMLEquation .
The Ricci form is given by LaTeXMLMath .
Hence , the Ricci-flat condition LaTeXMLMath is equivalent to the condition that the determinant is a constant up to products of holomorphic and anti-holomorphic functions : LaTeXMLMath .
From this condition , we obtain a differential equation LaTeXMLEquation where LaTeXMLMath is a constant .
Using this equation , the metric ( LaTeXMLRef ) can be rewritten as LaTeXMLEquation where LaTeXMLMath is given by ( LaTeXMLRef ) .
Therefore , we only need the solution of LaTeXMLMath but not LaTeXMLMath itself , to calculate the Ricci-flat Kähler metric .
To solve the nonlinear differential equation ( LaTeXMLRef ) , we transform it to a linear differential equation : LaTeXMLEquation .
The general solution of LaTeXMLMath is immediately obtained as LaTeXMLEquation where LaTeXMLMath is an integration constant and LaTeXMLMath .
In order to obtain the finite solution at LaTeXMLMath , we must set the parameter LaTeXMLMath the solution of LaTeXMLEquation .
Using this equation , we obtain an integral representation of LaTeXMLMath : LaTeXMLEquation .
By performing the integration we can express LaTeXMLMath using the hypergeometric function LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation We can also obtain the Ricci-flat metric by substituting LaTeXMLMath to ( LaTeXMLRef ) .
We thus have obtained LaTeXMLMath -dimensional Ricci-flat Kähler manifolds with LaTeXMLMath symmetry .
When LaTeXMLMath is odd ( LaTeXMLMath and LaTeXMLMath ) , the hypergeometric function reduces to a polynomial : LaTeXMLEquation .
If LaTeXMLMath is even ( LaTeXMLMath and LaTeXMLMath ) except for LaTeXMLMath [ see ( LaTeXMLRef ) , below , for the LaTeXMLMath solution ] , the solution can be written as LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Explicit expressions of LaTeXMLMath for LaTeXMLMath to LaTeXMLMath are LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation For definiteness , we give the explicit solution of LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath can be obtained as LaTeXMLEquation .
By the field redefinition LaTeXMLMath ( LaTeXMLMath LaTeXMLMath C ) , this Kähler potential becomes LaTeXMLEquation .
Here , LaTeXMLMath and LaTeXMLMath are holomorphic and anti-holomorphic functions , respectively , which can be eliminated by the Kähler transformation .
Thus we obtain the free field theory with the flat metric LaTeXMLMath as we expected , since the Ricci-flatness implies the vanishing Riemann curvature in real two-dimensional manifolds .
The solution of LaTeXMLMath for LaTeXMLMath can be calculated , to yield LaTeXMLEquation .
The Ricci-flat Kähler metric for LaTeXMLMath is LaTeXMLEquation where LaTeXMLMath is given in ( LaTeXMLRef ) .
This defines a ( real ) four-dimensional hyper-Kähler manifold with LaTeXMLMath symmetry , the Eguchi-Hanson space LaTeXMLCite .
The metric for LaTeXMLMath calculated from ( LaTeXMLRef ) coincides with the metric of the deformed conifold , obtained earlier in LaTeXMLCite .
We discuss the limit of vanishing LaTeXMLMath .
In this limit , the manifold becomes a conifold , in which the point represented by LaTeXMLMath in ( LaTeXMLRef ) is singular .
In this limit , we can obtain the explicit solution of LaTeXMLMath for any LaTeXMLMath , given by LaTeXMLEquation .
In the LaTeXMLMath limit , the Kähler potential ( LaTeXMLRef ) becomes the simplest form : LaTeXMLEquation .
Therefore , when we discuss non-perturbative effects of the sigma model on the conifold defined by ( LaTeXMLRef ) , using the LaTeXMLMath expansion method , it is sufficient to consider the simplest Kähler potential LaTeXMLMath instead of ( LaTeXMLRef ) .
If we prepare invariants of other groups in the superpotential of ( LaTeXMLRef ) , we would obtain Ricci-flat Kähler manifolds with other symmetry .
We have studied the nonlinear sigma models on Ricci-flat Kähler manifolds .
These models have vanishing LaTeXMLMath -function up to the fourth order in the perturbation theory LaTeXMLCite .
Despite the appearance of non-zero LaTeXMLMath -function at the four-loop order , we will be able to obtain the conformally invariant field theory for the background metric related to the Ricci-flat manifolds through non-local field redefinition LaTeXMLCite .
After the completion of this work we came to know that our metric was also discussed in other context LaTeXMLCite .
Let us give a comment on the relation to their work .
Defining the new variable LaTeXMLMath by the relation LaTeXMLMath in Eq .
( LaTeXMLRef ) , the integral can be rewritten as LaTeXMLMath , where LaTeXMLMath has been defined by LaTeXMLMath .
Here LaTeXMLMath and LaTeXMLMath are the same ones introduced in LaTeXMLCite .
We obtain the Kähler potential differentiated by LaTeXMLMath , given by LaTeXMLEquation .
We would like to thank Gary Gibbons for pointing out references LaTeXMLCite .
We are also grateful to Takashi Yokono for useful comments .
FSU-TPI 03/01 WZW–Poisson manifolds [ 50pt ] C. Klimčík Institute de mathématiques de Luminy , 163 , Avenue de Luminy , 13288 Marseille , France and T. Strobl Institut für Theoretische Physik Friedrich Schiller Universität , Max–Wien–Platz 1 , D–07743 Jena , Germany We observe that a term of the WZW-type can be added to the Lagrangian of the Poisson LaTeXMLMath -model in such a way that the algebra of the first class constraints remains closed .
This leads to a natural generalization of the concept of Poisson geometry .
The resulting ” WZW–Poisson ” manifold LaTeXMLMath is characterized by a bivector LaTeXMLMath and by a closed three-form LaTeXMLMath such that LaTeXMLMath .
1 .
It turns out to be a fruitful idea to associate dynamical systems—and in particular topological field theories—to geometric data on manifolds .
Here we shall study the following example : Given a bivector LaTeXMLMath and a two-form LaTeXMLMath on a manifold LaTeXMLMath , we can immediately write down the action functional LaTeXMLEquation .
In the story that follows , LaTeXMLMath will be a cylindrical world-sheet , LaTeXMLMath is a collection of coordinates on the target space LaTeXMLMath , and LaTeXMLMath is a set of LaTeXMLMath -forms on LaTeXMLMath .
Of course this action can be written also in a coordinate-independent way .
Introducing the standard world-sheet coordinates LaTeXMLMath and LaTeXMLMath ( the loop and the evolution parameters , respectively ) we set LaTeXMLEquation and rewrite ( 1 ) in the following form LaTeXMLEquation .
Here LaTeXMLEquation 2 .
Let LaTeXMLMath be some ( possibly infinite-dimensional ) manifold equipped with a symplectic form LaTeXMLMath .
Suppose there is a set of functions LaTeXMLMath fulfilling LaTeXMLEquation where the indices take values in some set LaTeXMLMath and the Poisson bracket corresponds to LaTeXMLMath .
To these data we associate the constrained dynamical system described by the action LaTeXMLEquation where LaTeXMLMath is a set of Lagrange multipliers and LaTeXMLMath are the corresponding first class constraints .
Now the question arises : for which pair LaTeXMLMath the action ( 3 ) defines a constrained dynamical system in the sense described above ( i.e .
the relations ( 5 ) should hold ) .
Of course , LaTeXMLMath , LaTeXMLMath play the role of the Lagrange multipliers LaTeXMLMath and LaTeXMLMath .
It is simple to answer this question .
The symplectic form LaTeXMLMath has the canonical Darboux form and the calculation of the Poisson brackets is straightforward .
We obtain LaTeXMLEquation provided LaTeXMLEquation holds true .
The symbol LaTeXMLMath denotes the Schouten bracket and the functions LaTeXMLMath can be read off from ( 7 ) .
The contraction on the right hand side is with respect to the first , third and fifth entry of LaTeXMLMath .
We remark that the condition ( LaTeXMLRef ) is necessary and sufficient for the system of constraints following from ( 1 ) to be of the first class ( cf .
LaTeXMLCite for further details ) .
3 .
Our discussion can be slightly generalized .
Consider the bivector LaTeXMLMath and a closed LaTeXMLMath -form on the manifold LaTeXMLMath .
To these data we associate the following action LaTeXMLEquation .
Here LaTeXMLMath is the interior of the cylinder LaTeXMLMath and by LaTeXMLMath we really mean the pullback of LaTeXMLMath to LaTeXMLMath by an extension to LaTeXMLMath of the map LaTeXMLMath .
Of course , there are the subtleties concerning the boundaries of the cylinder and the WZW term .
We do not give the detailed discussion in this letter .
It is a straightforward generalization of the treatment in LaTeXMLCite , where the WZW model on the cylinder is studied from the point of view of Hamiltonian mechanics .
Note that ( 9 ) reduces to ( 1 ) for LaTeXMLMath .
By repeating the previous discussion , we arrive at the conclusion that the model ( 9 ) corresponds to a maximally constrained dynamical system iff LaTeXMLEquation 4 .
For LaTeXMLMath , the action ( 9 ) defines the Poisson LaTeXMLMath -model LaTeXMLCite and the condition ( 10 ) says that the bivector LaTeXMLMath satisfies the Jacobi identity .
Therefore Poisson geometry could have been invented by asking the question when the model ( 9 ) ( with LaTeXMLMath ) is a maximally constrained dynamical system or a topological field theory .
If we do not set LaTeXMLMath , the same logic gives a natural generalization : the concept of what one might call WZW-Poisson manifolds .
We repeat that the latter is characterized by a bivector LaTeXMLMath and a closed LaTeXMLMath -form LaTeXMLMath such that the condition ( 10 ) holds .
It remains to understand the properties of the WZW-Poisson manifolds in more detail .
It may be that there is a non-trivial intersection of this notion with the other generalizations of Poisson geometry like quasi-Poisson manifolds LaTeXMLCite , Dirac manifolds LaTeXMLCite or the manifolds leading to the nonassociative generalization LaTeXMLCite of the Kontsevich expansion .
Note added : After completion of this work we became aware that the relation ( 10 ) was obtained also in LaTeXMLCite within a BV approach .
We intend to comment on some of those aspects of the theory of differential equations which we think are clarified ( for us , at least ) by means of the synthetic method .
By this , we understand that the objects under consideration are seen as objects in one sufficiently rich category ( model for SDG ) , allowing us , for instance , to work with nilpotent numbers , say LaTeXMLMath with LaTeXMLMath ; but the setting should also permit the formation of function spaces , so that some of the methods of functional analysis , become available , in particular , the theory of distributions .
The specific topics we treat are generalities on vector fields and the solutions of corresponding first- and second-order ordinary differential equations ; and also some partial differential equations , which can be seen in this light , the wave- and heat-equation on some simple spaces , like the line LaTeXMLMath .
For these equations , distribution theory is not just a tool , but is rather the essence of the matter , since what develops through time , is a distribution ( of heat , say ) , which , as stressed by Lawvere , is an extensive quantity , and as such behaves covariantly , unlike density functions ( which behave contravariantly ) ; and the distributions may have no density function , in particular in the setting of model for SDG where all functions are smooth .
When we consider these partial differential equations , we shall follow an old practice and sometimes denote derivative LaTeXMLMath with respect to “ time ” by a dot , LaTeXMLMath , whereas differential operators with respect to space variables are denoted LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , etc .
We want to thank Henrik Stetkær for useful conversations on the topic of distributions .
Recall that an action of a set ( object ) LaTeXMLMath on a set ( object ) LaTeXMLMath is a map LaTeXMLMath , and a homomorphism of actions LaTeXMLMath is a map LaTeXMLMath with LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
The category of actions by a set LaTeXMLMath form a topos ; we shall in particular be interested in the exponent formation in this topos , when the action in the exponent is invertible .
An action LaTeXMLMath is called invertible , if for each LaTeXMLMath , LaTeXMLMath is invertible .
In this case , the exponent LaTeXMLMath may be described as LaTeXMLMath equipped with the following action by LaTeXMLMath : an element LaTeXMLMath acts on LaTeXMLMath by “ conjugation ” : LaTeXMLEquation where LaTeXMLMath denotes LaTeXMLMath , and similarly for LaTeXMLMath .
In the applications below , LaTeXMLMath is the usual set of square zero elements in LaTeXMLMath .
It is a pointed object , pointed by LaTeXMLMath , and the actions LaTeXMLMath we consider , are pointed actions in the sense that LaTeXMLMath for all LaTeXMLMath , or equivalently , LaTeXMLMath is the identity map on LaTeXMLMath .
A pointed action , in this situation , is the same thing as a vector field on LaTeXMLMath , cf .
LaTeXMLCite .
In the above situation , if LaTeXMLMath and LaTeXMLMath are pointed actions , then so is the exponent described .
The pointed actions likewise form a topos , and the exponent described is then also the exponent in the category of pointed actions ; cf .
LaTeXMLCite .
For the case of vector fields seen as actions by LaTeXMLMath , we want to describe the “ streamlines ” generated by a vector field in abstract action-theoretic terms ; this is going to involve a certain “ universal ” action LaTeXMLMath : LaTeXMLMath is an “ infinitesimally open subset ” of LaTeXMLMath i.e. , whenever LaTeXMLMath then LaTeXMLMath for every LaTeXMLMath The main examples of such subsets are LaTeXMLMath itself , the non-negative numbers LaTeXMLMath open intervals , and the set LaTeXMLMath of all nilpotent elements of the number line .
The action LaTeXMLMath is the vector field LaTeXMLMath , meaning the map LaTeXMLMath given by LaTeXMLMath .
( So it is not to be confused with the Laplace operatot LaTeXMLMath , to be considered later . )
The main property to be assumed is that the individual LaTeXMLMath ’ s are homomorphisms of LaTeXMLMath -actions ( which is a commutativity requirement ) ; the structure of LaTeXMLMath could probably be derived from this , but we shall be content with assuming that LaTeXMLMath is an additively written monoid , and that LaTeXMLMath ( with the LaTeXMLMath of LaTeXMLMath also being the zero of the monoid ) .
First , if LaTeXMLMath is a set with an action , a homomorphism LaTeXMLMath is to be thought of as a particular solution of the differential equation given by LaTeXMLMath , with initial value LaTeXMLMath , or as a “ streamline ” for the vector field LaTeXMLMath , starting in LaTeXMLMath .
One wants , however , also to include dependence on initial value into the notion of solution , and so one is led to consider maps LaTeXMLEquation satisfying at least LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath ; we shall consider and compare the following further conditions ( universally quantified over all LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ) : LaTeXMLEquation this is the main one , the two following conditions are included for systematic reasons only : LaTeXMLEquation .
LaTeXMLEquation Finally , one may consider the following equation LaTeXMLEquation .
Writing LaTeXMLMath for the map LaTeXMLMath , and similarly for LaTeXMLMath , condition ( LaTeXMLRef ) may be rewritten as LaTeXMLEquation .
The others may be rewritten in a similar way .
For instance ( LaTeXMLRef ) may be rewritten as LaTeXMLEquation .
Equation ( LaTeXMLRef ) expresses that , for each fixed LaTeXMLMath , the map LaTeXMLMath is a homomorphism ( and thus , by virtue of LaTeXMLMath , a “ solution with initial value LaTeXMLMath ” ) .
Writing the action of LaTeXMLMath in terms of the symbol LaTeXMLMath , we may write it LaTeXMLMath Equation ( LaTeXMLRef ) expresses a certain bi-homogeneity condition of LaTeXMLMath , LaTeXMLMath ; ( LaTeXMLRef ) says that for fixed LaTeXMLMath , LaTeXMLMath is an endomorphism of LaTeXMLMath -actions , LaTeXMLMath .
Finally ( LaTeXMLRef ) is the usual condition for action af a monoid on a set LaTeXMLMath .
Clearly , it implies all the others .
Let LaTeXMLMath be a vector field on LaTeXMLMath , thought of as a first-order differential equation .
We say that the map LaTeXMLMath is a complete solution or simply a solution if LaTeXMLMath and LaTeXMLMath satisfies ( LaTeXMLRef ) .
A solution in this sense does not satisfy the other conditions ( LaTeXMLRef ) - ( LaTeXMLRef ) , but it does , provided that LaTeXMLMath satisfies a certain axiom ( reflecting , synthetically , validity of the uniqueness assertion for solutions of differential equations on LaTeXMLMath ) .
— The axiom in question is the following Uniqueness property for LaTeXMLMath : If LaTeXMLMath is a LaTeXMLMath -action on LaTeXMLMath , and LaTeXMLMath are homomorphisms of actions , with LaTeXMLMath , then LaTeXMLMath .
Note that the validity of the axiom , for a given LaTeXMLMath , depends on the choice of LaTeXMLMath .
For instance , we shall prove below that it holds for any microlinear LaTeXMLMath if LaTeXMLMath is taken to be LaTeXMLMath ( and LaTeXMLMath ) .
Let LaTeXMLMath be a vector field on LaTeXMLMath and assume that LaTeXMLMath satisfies the uniqueness axiom .
Then any solution LaTeXMLMath of the differential equation LaTeXMLMath satisfies properties ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Furthermore , if LaTeXMLMath is a monoid ( under + ) then LaTeXMLMath also satisfies ( LaTeXMLRef ) .
Proof .
Since the proofs are quite similar , we shall do only ( LaTeXMLRef ) .
Fix LaTeXMLMath and LaTeXMLMath and define the couple of functions LaTeXMLMath by the formulas LaTeXMLEquation .
We have to check that LaTeXMLMath and LaTeXMLMath are homomorphisms of LaTeXMLMath -actions , i.e. , they satisfy ( LaTeXMLRef ) .
Let us do this for the first LaTeXMLEquation .
The proof that LaTeXMLMath is a homomorphism is similar .
Thus , the equality of the two expressions follows from the uniqueness property assumed for LaTeXMLMath .
Recall that a vector field LaTeXMLMath on LaTeXMLMath is called integrable if there exists a solution LaTeXMLMath .
If we assume the uniqueness property , the equation ( LaTeXMLRef ) holds ; if further the commutative monoid structure LaTeXMLMath on LaTeXMLMath actually is a group structure , then ( LaTeXMLRef ) implies that the action is invertible , with LaTeXMLMath as LaTeXMLMath ( in fact LaTeXMLMath ) .
Of course , both the uniqueness property and the question whether or not the vector field LaTeXMLMath is integrable , depends on which LaTeXMLMath is considered .
In particular , we shall say that LaTeXMLMath is formally integrable or has a formal solution if LaTeXMLMath is integrable for LaTeXMLMath ( which is a group under addition ) .
For the case of LaTeXMLMath , this amounts to integration by formal power series , whence the terminology .
The uniqueness property holds for any microlinear object , ( for LaTeXMLMath ) .
Furthermore , every vector field on a microlinear object is formally integrable .
Thus , every vector field on a microlinear object has a unique formal solution .
Proof .
We need to recall some infinitesimal objects from the literature on SDG , cf .
e.g .
LaTeXMLCite .
Besides LaTeXMLMath , consisting of LaTeXMLMath with LaTeXMLMath , we have LaTeXMLMath , the LaTeXMLMath -fold product of LaTeXMLMath with itself .
It has the subobject LaTeXMLMath consisting of those LaTeXMLMath -tuples LaTeXMLMath where LaTeXMLMath for all LaTeXMLMath .
There is also the object LaTeXMLMath consisting of LaTeXMLMath with LaTeXMLMath ; LaTeXMLMath is the union of all the LaTeXMLMath ’ s .
If LaTeXMLMath , then LaTeXMLMath .
— Now , let LaTeXMLMath be a microlinear object , and LaTeXMLMath a vector field on it .
We first recall that if LaTeXMLMath have the property that LaTeXMLMath , then LaTeXMLMath .
( For microlinear objects perceive LaTeXMLMath to be a pushout over LaTeXMLMath of the two inclusions LaTeXMLMath , and clearly both expressions given agree if either LaTeXMLMath or LaTeXMLMath . )
In particular , LaTeXMLMath and LaTeXMLMath commute .
But more generally , If LaTeXMLMath is a vector field on a microlinear object and LaTeXMLMath , the maps LaTeXMLMath and LaTeXMLMath commute .
Proof .
This is a consequence of the theory of Lie brackets , cf .
e.g .
LaTeXMLCite 3.2.2 , namely LaTeXMLMath .
Likewise If LaTeXMLMath is a vector field on a microlinear object and LaTeXMLMath are such that LaTeXMLMath , then LaTeXMLEquation ( = the identity map on LaTeXMLMath ) .
In particular , LaTeXMLMath Proof .
We first prove that LaTeXMLMath , and hence any microlinear object , perceives LaTeXMLMath to be the orbit space of LaTeXMLMath under the action of the symmetric group LaTeXMLMath in LaTeXMLMath letters : Assume that LaTeXMLMath coequalizes the action , i.e .
is symmetric in the LaTeXMLMath arguments .
By the basic axiom of SDG , LaTeXMLMath may be written in the form LaTeXMLEquation for unique LaTeXMLMath ’ s in LaTeXMLMath ( where LaTeXMLMath denotes LaTeXMLMath ) .
We claim that LaTeXMLMath for every LaTeXMLMath .
Indeed , LaTeXMLEquation since LaTeXMLMath is symmetric .
But LaTeXMLEquation .
By comparing coefficients and using uniqueness of coefficients , we conclude LaTeXMLMath , and this shows that LaTeXMLMath is ( the restriction to LaTeXMLMath of ) a symmetric polynomial LaTeXMLMath .
By Newton ’ s theorem ( which holds internally ) , LaTeXMLMath is a polynomial in the elementary symmetric polynomials LaTeXMLMath .
Recall that LaTeXMLMath : and each LaTeXMLMath , when restricted to LaTeXMLMath , is a function of LaTeXMLMath , since LaTeXMLMath ; e.g .
LaTeXMLEquation .
Now consider , for fixed LaTeXMLMath , the map LaTeXMLMath given by LaTeXMLMath .
By Lemma LaTeXMLRef , this map is invariant under the symmetric group LaTeXMLMath ( recall that this group is generated by transpositions ) , so there is a unique LaTeXMLMath such that LaTeXMLEquation .
So if LaTeXMLMath , LaTeXMLMath .
This proves the Lemma .
We can now prove the Theorem .
We need to define LaTeXMLMath when LaTeXMLMath .
Assume for instance that LaTeXMLMath .
By microlinearity of LaTeXMLMath , LaTeXMLMath perceives LaTeXMLMath to be the orbit space of LaTeXMLMath under the action of LaTeXMLMath ( see the proof of Lemma LaTeXMLRef ) , via the map LaTeXMLMath , so we are forced to define LaTeXMLMath if LaTeXMLMath is to extend LaTeXMLMath and to satisfy ( LaTeXMLRef ) .
The fact that this is well defined independently of the choice of LaTeXMLMath and the choice of LaTeXMLMath that add up to LaTeXMLMath follows from Lemma LaTeXMLRef .
As a particular case of special importance , we consider a linear vector field on a microlinear and Euclidean LaTeXMLMath -module LaTeXMLMath .
To say that the vector field is linear is to say that its principal-part formation LaTeXMLMath is a linear map , LaTeXMLMath , say .
We have then the following version of a classical result : Let a linear vector field on a microlinear Euclidean LaTeXMLMath -module LaTeXMLMath be given by the linear map LaTeXMLMath .
Then the unique formal solution of the corresponding differential equation , i.e. , the equation LaTeXMLMath with initial position LaTeXMLMath , is the map LaTeXMLMath given by LaTeXMLEquation where the right hand side here means the sum of the following “ series ” ( which has only finitely many non-vanishing terms , since LaTeXMLMath is assumed nilpotent ) : LaTeXMLEquation .
Here of course LaTeXMLMath means LaTeXMLMath , etc .
Proof .
We have to prove that LaTeXMLMath .
We calculate the left hand side by differentiating the series term by term ( there are only finitely many non-zero terms ) : LaTeXMLEquation using linearity of LaTeXMLMath .
But this is just LaTeXMLMath applied to LaTeXMLMath .
There is an analogous result for second order differential equations of the form LaTeXMLMath ( with LaTeXMLMath linear ) ; the proof is similar and we omit it : The formal solution of this second order differential equation LaTeXMLMath , with initial position LaTeXMLMath and initial speed LaTeXMLMath , is given by LaTeXMLEquation .
In this section , we show that solutions of an exponent vector field may be obtained by conjugating solutions of the vector fields that make up the exponent .
Furthermore , this method of conjugation is equivalent ( under some conditions ) to the method of change of variables , widely used to solve differential equations .
Assume that LaTeXMLMath and LaTeXMLMath are vector fields having solutions LaTeXMLMath and LaTeXMLMath , respectively , and assume that all LaTeXMLMath are invertible .
Then a solution LaTeXMLMath of the exponent LaTeXMLMath is obtained as the map LaTeXMLEquation given by conjugation : LaTeXMLMath .
Proof .
This is purely formal .
For LaTeXMLMath , we have LaTeXMLEquation where in the third step we used the equation ( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath , in the form LaTeXMLEquation together with invertibility of LaTeXMLMath for all LaTeXMLMath and invertibility of LaTeXMLMath .
A similar argument gives that if each of ( LaTeXMLRef ) - ( LaTeXMLRef ) holds for both LaTeXMLMath and LaTeXMLMath , then the corresponding property holds for LaTeXMLMath .
In most applications , the invertibility of the LaTeXMLMath will be secured by subtraction on LaTeXMLMath , with LaTeXMLMath .
Recall that an LaTeXMLMath -module LaTeXMLMath is called Euclidean if the canonical map LaTeXMLMath given by LaTeXMLMath is invertible ; the composite of LaTeXMLMath with projection to the second factor , LaTeXMLMath is called principal part formation .
If LaTeXMLMath is a vector field on a Euclidean module LaTeXMLMath , we may compose it with principal part formation to get a ( not necessarily linear ) map LaTeXMLMath , called the principal part of the vector field LaTeXMLMath ; it is thus characterized by the formula LaTeXMLEquation .
Recall also that if LaTeXMLMath is any map into a Euclidean LaTeXMLMath -module , and LaTeXMLMath is a vector field on LaTeXMLMath , then the directional derivative LaTeXMLMath of LaTeXMLMath along LaTeXMLMath is the composite LaTeXMLEquation where the last map is principal part formation .
Using function theoretic notation , LaTeXMLMath is characterized by validity of the equation LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath .
When LaTeXMLMath itself is a Euclidean LaTeXMLMath module , and LaTeXMLMath has principal part LaTeXMLMath , we usually write LaTeXMLMath instead of LaTeXMLMath .
Assume that LaTeXMLMath , LaTeXMLMath are vector fields on LaTeXMLMath , LaTeXMLMath , respectively , and that LaTeXMLMath is a homomorphism ( i.e. , it preserves the LaTeXMLMath -action ) .
Let LaTeXMLMath be a Euclidean LaTeXMLMath -module .
Then for any LaTeXMLMath , LaTeXMLEquation .
Proof .
This is a straightforward computation : LaTeXMLEquation on the other hand LaTeXMLEquation .
By comparing these two expressions we obtain the conclusion of the Proposition .
For any object LaTeXMLMath , let us consider its “ zero vector field ” LaTeXMLMath , i.e. , LaTeXMLMath is the identity map on LaTeXMLMath , for all LaTeXMLMath .
For a vector field LaTeXMLMath on an object LaTeXMLMath , we then also have the “ vertical ” vector field LaTeXMLMath on LaTeXMLMath .
If we have a complete solution LaTeXMLMath of a vector field LaTeXMLMath on LaTeXMLMath , we may consider the map LaTeXMLMath given by LaTeXMLMath The map LaTeXMLMath thus described is an automorphism of the vector field LaTeXMLMath on LaTeXMLMath .
Proof .
By a straightforward diagram chase , one sees that this is a restatement of ( LaTeXMLRef ) .
We now consider solutions LaTeXMLMath for such vector fields , so equation ( LaTeXMLRef ) holds : LaTeXMLMath .
In terms of principal parts , this equation may be rewritten as LaTeXMLEquation .
Similarly , equation ( LaTeXMLRef ) may be written as LaTeXMLEquation .
Using directional derivatives , we can give a more familiar expression to the vector field ( 1ODE ) LaTeXMLMath considered above on the object LaTeXMLMath , when the base LaTeXMLMath is a microlinear Euclidean LaTeXMLMath -module LaTeXMLMath , and the exponent LaTeXMLMath is mocrolinear .
In fact , letting LaTeXMLMath be the principal part of the vector field LaTeXMLMath on LaTeXMLMath , we have , for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ( recall that LaTeXMLMath ) LaTeXMLEquation ( at the third equality sign , a cancellation of LaTeXMLMath took place in the last term ) In other words , the principal part of LaTeXMLMath is LaTeXMLMath given by LaTeXMLEquation .
Recalling that the 1ODE corresponding to a vector field LaTeXMLMath on a Euclidean LaTeXMLMath -module LaTeXMLMath may be written as LaTeXMLMath where LaTeXMLMath is the principal part of LaTeXMLMath .
In these terms , the above equation may be rewritten ( leaving out the LaTeXMLMath , and modulo some obvious abuse of notation ) as LaTeXMLEquation or still , recalling that LaTeXMLMath is “ derivative with respect to time ” , LaTeXMLEquation .
This is a PDE of first order “ in time ” .
The following may be seen as a generalization of ( LaTeXMLRef ) , and is a form of the chain rule .
We consider a vector field LaTeXMLMath on LaTeXMLMath , with solution LaTeXMLMath .
Let LaTeXMLMath be any function with values in a Euclidean LaTeXMLMath -module .
Under these circumstances , we have LaTeXMLEquation for all LaTeXMLMath , LaTeXMLMath .
Proof .
Since LaTeXMLMath is a solution of LaTeXMLMath , LaTeXMLMath , and so for any LaTeXMLMath LaTeXMLMath .
Therefore , by definition of directional derivative , LaTeXMLEquation .
Putting LaTeXMLMath , we thus have LaTeXMLEquation .
LaTeXMLEquation by a standard cancellation of two LaTeXMLMath ’ s , after Taylor expansion .
Expanding the first term , we may continue : LaTeXMLEquation .
On the other hand , LaTeXMLEquation comparing these two expressions gives the result .
The method of change of variables has been used extensively to solve differential equations .
We shall prove that our method for solving the exponential differential equation LaTeXMLMath , where LaTeXMLMath is an integrable vector field on LaTeXMLMath , LaTeXMLMath an integrable vector field on a Euclidean LaTeXMLMath -module , and where LaTeXMLMath is symmetric with respect to the origin ( if LaTeXMLMath , then LaTeXMLMath ) , may be seen as an application of the method of change of variables .
We let LaTeXMLMath denote the principal part of LaTeXMLMath , as before .
Let LaTeXMLMath be the assumed solution of LaTeXMLMath , and let LaTeXMLMath be the map LaTeXMLEquation .
Then LaTeXMLMath ( which represents the change of variables LaTeXMLMath , LaTeXMLMath ) is invertible .
( “ Change of variables ” ) .
If LaTeXMLMath is a particular solution of LaTeXMLMath , or , equivalently , of LaTeXMLEquation then the unique map LaTeXMLMath given as the composite LaTeXMLEquation is a particular solution of LaTeXMLMath , or , equivalently , of LaTeXMLEquation and vice versa .
Proof .
Since LaTeXMLMath , we have LaTeXMLEquation by the chain rule , Proposition LaTeXMLRef .
On the other hand , LaTeXMLMath is an automorphism of the vector field LaTeXMLMath , by Proposition LaTeXMLRef , and so , by construction of LaTeXMLMath and Proposition LaTeXMLRef , LaTeXMLEquation .
Therefore , LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath , i.e. , LaTeXMLMath is solution of LaTeXMLEquation proving the theorem ( the vice versa part follows because LaTeXMLMath is invertible ) .
Example .
Let LaTeXMLMath be the set of elements of square zero in LaTeXMLMath , as usual .
It carries a vector field , namely the map LaTeXMLMath given by LaTeXMLMath .
It is easy to see that this vector field is integrable , with complete solution LaTeXMLMath given by LaTeXMLMath .
Now consider the tangent vector bundle LaTeXMLMath on LaTeXMLMath .
The zero vector field LaTeXMLMath on LaTeXMLMath is certainly integrable , and so we have by the theorem a complete integral for the vector field LaTeXMLMath on the tangent bundle .
We describe the integral explicitly ( this then also describes the vector field , by restriction ) : it is the map LaTeXMLMath given by LaTeXMLMath .— The vector field on LaTeXMLMath obtained this way is , except for the sign , the Liouville vector field , cf .
LaTeXMLCite , IX.2 .
We want to apply parts of the general theory of ordinary differential equations to some of the basic equations of mathematical physics , the wave- and heat- equations .
This takes us by necessity to the realm of distributions .
Not primarily as a technique , but because of the nature of these equations : they model evolution through time of ( say ) a heat distribution .
A heat distribution is an extensive quantity , and does not necessarily have a density function , which is an intensive quantity ; the most important of all distributions , the point distributions ( or Dirac distributions ) , for instance , do not .
For the case of the heat equation , it is well known that the evolution through time of any distribution “ instantaneously ” ( i.e. , after any positive lapse of time , LaTeXMLMath ) leads to distributions that do have smooth density functions .
But in SDG , we are interested also in what happens after a nilpotent lapse of time .
In more computational terms , we are interested in the Taylor expansion of the solutions of evolution equations .
For this , it is necessary to stay within one vector space , that of distributions .
The vector space of “ distributions of compact support ” on any object LaTeXMLMath can be introduced purely synthetically ( see LaTeXMLCite p. 393 , or LaTeXMLCite p. 94 ) as the LaTeXMLMath -linear dual of the vector space LaTeXMLMath ( which internally represents the vector space of smooth functions on LaTeXMLMath ) .
What follows could , to a certain extent ( in particular for the wave equation ) , be treated purely synthetically .
Presently , we shall only be interested in distributions on LaTeXMLMath , and LaTeXMLMath , so for the presentation , we have chosen to assume that we are working in a sufficiently good “ well-adapted ” model LaTeXMLMath of SDG , containing the category of smooth manifolds as a full subcategory .
In such models , for any given manifold LaTeXMLMath , we could define the linear subspace LaTeXMLMath of LaTeXMLMath consisting of functions with compact support , ( the “ test functions ” ) .
Then the vector space of distributions on LaTeXMLMath , LaTeXMLMath , is taken to be the LaTeXMLMath -linear dual of LaTeXMLMath .
One could take an alternative , slightly more concrete , approach : namely , take a model LaTeXMLMath of SDG which contains the category of smooth manifolds as above , but which also contains the category of Convenient Vector Spaces LaTeXMLCite and the smooth maps between them as a full subcategory .
The embedding is to preserve the cartesian closed structure .
Such models do exist : we provided in LaTeXMLCite , LaTeXMLCite such an embedding of Convenient Vector Spaces into the “ Cahiers ” topos of Dubuc LaTeXMLCite .
Note that the usual topological ( Fréchet ) vector spaces of smooth functions , test functions , distributions , etc .
on a smooth manifold LaTeXMLMath have canonical structure of Convenient Vector Spaces .
In such a model , we can construct internal functions , say curves LaTeXMLMath , by constructing , externally , a function by an “ excluded middle ” recipe of the form LaTeXMLEquation and then proving smoothness of LaTeXMLMath by a usual limit argument .
We have to resort to this kind of “ external ” constructions only for the heat equation , and there our embedding from LaTeXMLCite , LaTeXMLCite is not quite good enough , since it does not take manifolds with boundary into account ; for the heat equation , one constructs externally an “ evolution ” map LaTeXMLEquation by an excluded middle recipe .
So , for the justification of our treatment of the heat equation , we need an extension ( hopefully forthcoming ) of our work LaTeXMLCite , LaTeXMLCite , i.e. , we need to construct a Cahiers-like topos that includes also manifolds with boundary , and then to construct an embedding of Convenient Vector Spaces into that “ extended ” Cahiers Topos .
( Maybe even the Cahiers Topos itself will be good enough . )
For what follows about wave equation , the Cahiers Topos , and the embedding of Convenient Vector Spaces into it , is sufficient ; in fact , for these equations , a purely synthetic treatment alluded to will be sufficient , since the distributions considered there are all of compact support .
As stressed by Lawvere in LaTeXMLCite , distributions should not be thought of as generalized functions : functions are intensive quantities , and transform contravariantly ; distributions are extensive quantities and transform covariantly .
For functions , this is the fact that the “ space ” of functions on LaTeXMLMath , LaTeXMLMath is contravariant in LaTeXMLMath , by elementary cartesian-closed category theory .
Similarly , the “ space ” of distributions of compact support on LaTeXMLMath is a subspace of LaTeXMLMath ( carved out by the LaTeXMLMath linearity condition ) , and so for similar elementary reasons is covariant in LaTeXMLMath .
We shall write LaTeXMLMath for this subspace .
The space of functions of compact support on LaTeXMLMath is only functorial with respect to proper smooth maps , ( counterimages of compact set required to be compact ) , and so similarly , the space LaTeXMLMath of all distributions on LaTeXMLMath is covariant functorial only w.r.to proper maps .
The formula for covariant functorality looks the same for LaTeXMLMath and LaTeXMLMath ; let us make it explicit for the LaTeXMLMath case .
Let LaTeXMLMath be a proper map .
The map LaTeXMLMath is described by declaring LaTeXMLEquation where LaTeXMLMath is a distribution on LaTeXMLMath , and LaTeXMLMath is a test function on LaTeXMLMath , ( so LaTeXMLMath is a test function on LaTeXMLMath , by properness of LaTeXMLMath ) .
The brackets denote evaluation of distributions on test functions .
We shall also write just LaTeXMLMath instead of LaTeXMLMath .
Recall that a distribution LaTeXMLMath on LaTeXMLMath may be mulitplied by any function LaTeXMLMath , by the recipe LaTeXMLEquation observing that LaTeXMLMath is a test function ( has compact support ) if LaTeXMLMath is .
If LaTeXMLMath is a vector field on LaTeXMLMath , one defines the directional derivative LaTeXMLMath of a distribution LaTeXMLMath on LaTeXMLMath by the formula LaTeXMLEquation .
This in particular applies to the vector field LaTeXMLMath on LaTeXMLMath , and reads here LaTeXMLMath ( LaTeXMLMath denoting the ordinary derivative of the function LaTeXMLMath ) .
One has the following Leibniz rule : LaTeXMLEquation for any distribution LaTeXMLMath and function LaTeXMLMath on LaTeXMLMath .
This is an elementary consequence of the Leibniz rule for directional derivatives LaTeXMLMath of functions on LaTeXMLMath .
Remark .
The equation ( LaTeXMLRef ) becomes a theorem , rather than a definition , if one takes the following line of reasoning : let LaTeXMLMath be a covariant functor from microlinear spaces ( and invertible maps between them ) to Euclidean vector spaces .
Then one may define the Lie derivative along LaTeXMLMath , LaTeXMLMath , as a map LaTeXMLMath .
For the functor LaTeXMLMath , LaTeXMLMath becomes the LaTeXMLMath described .
We shall not pursue this line further here .
Applying LaTeXMLMath twice leads to LaTeXMLEquation .
In particular , for LaTeXMLMath a distribution on LaTeXMLMath LaTeXMLEquation and therefore for the Laplace operator LaTeXMLMath , we put LaTeXMLEquation .
The following Proposition is an application of the covariant functorality of the functor LaTeXMLMath , which will be used in connection with the wave equation in dimension 2 .
We consider the ( orthogonal ) projection LaTeXMLMath onto the LaTeXMLMath -plane .
( It is not a proper map , so functorality only works for compactly supported distributions . )
For any distribution LaTeXMLMath ( of compact support ) on LaTeXMLMath , LaTeXMLEquation ( The same result holds for any orthogonal projection LaTeXMLMath of LaTeXMLMath onto any linear subspace ; the proof is virtually the same , if one uses invariance of LaTeXMLMath under orthogonal transformations . )
Proof .
Let LaTeXMLMath be any test function on LaTeXMLMath .
Then LaTeXMLEquation .
But , with LaTeXMLMath , LaTeXMLMath is just LaTeXMLMath , considered as a function of LaTeXMLMath which happens not to depend on LaTeXMLMath ; so LaTeXMLEquation the last term vanishes because LaTeXMLMath does not depend on LaTeXMLMath , so the equation continues LaTeXMLEquation .
So the right hand expression in ( LaTeXMLRef ) may be rewritten as LaTeXMLEquation from which the result follows .
For LaTeXMLMath , we let LaTeXMLMath denote the distribution LaTeXMLMath .
Such distributions on the line , we of course call intervals ; the length of an interval LaTeXMLMath is defined to be LaTeXMLMath .
Note that the interval LaTeXMLMath as a distribution is not quite the same as the order theoretic interval , i.e. , the subset of LaTeXMLMath consisting of LaTeXMLMath with LaTeXMLMath .
For instance , the order theoretic interval from LaTeXMLMath to LaTeXMLMath contains all nilpotent elements , whereas the distribution LaTeXMLMath is the zero distribution .
The distribution theoretic interval LaTeXMLMath contains more information about LaTeXMLMath and LaTeXMLMath than does the order theoretic one .
We consider the question to which extent LaTeXMLMath determines the endpoints .
The answer is contained in Let LaTeXMLMath and LaTeXMLMath be two intervals in the distribution theoretic sense .
They are equal as distributions if and only if they have same length , LaTeXMLMath ( LaTeXMLMath , say ) , and LaTeXMLMath ( this then also implies LaTeXMLMath .
Proof .
Assume LaTeXMLMath .
The statement about length follows immediately by applying each of these two distributions to the function LaTeXMLMath which is constant LaTeXMLMath .
Generally , we have for any function LaTeXMLMath that LaTeXMLEquation .
LaTeXMLEquation by making the change of variables LaTeXMLMath .
Subtracting , we get LaTeXMLEquation .
Apply this equation to the function LaTeXMLMath , we get LaTeXMLEquation .
Conversely , assume LaTeXMLMath ( LaTeXMLMath , say ) , and LaTeXMLMath .
For any function LaTeXMLMath , we calculate the values of the distribution LaTeXMLMath on LaTeXMLMath .
We have LaTeXMLEquation .
Similarly LaTeXMLEquation .
The difference is LaTeXMLEquation .
By Hadamard ’ s Lemma , LaTeXMLMath may be written as LaTeXMLMath for some function LaTeXMLMath , and so the integral ( LaTeXMLRef ) can be written as LaTeXMLEquation which vanishes if LaTeXMLMath .
The assertions about LaTeXMLMath is similar .
Note the following Corollaries : First , if the length LaTeXMLMath of an interval LaTeXMLMath is invertible ( positive , say ) , then the endpoints LaTeXMLMath , LaTeXMLMath are uniquely determined by the distribution LaTeXMLMath .
Secondly , for any LaTeXMLMath , we have LaTeXMLEquation .
In fact , by the Proposition , their lengths must be equal , i.e. , LaTeXMLMath .
The distribution LaTeXMLMath will appear below under the name LaTeXMLMath , “ the ball of radius LaTeXMLMath in dimension One ” .
We shall also consider such “ balls ” in dimension Two and Three , where , however , LaTeXMLMath can not in general be recovered from the distribution , unless LaTeXMLMath is strictly positive .
We fix a positive integer LaTeXMLMath .
We shall consider the sphere LaTeXMLMath of radius LaTeXMLMath , and the ball LaTeXMLMath of radius LaTeXMLMath , for any LaTeXMLMath , as distributions on LaTeXMLMath ( of compact support , in fact ) , in the following sense : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath refers to the surface element of the unit sphere LaTeXMLMath in the first equation and to the volume element of the unit ball LaTeXMLMath in the second .
The expressions involving LaTeXMLMath and LaTeXMLMath are to be understood symbolically , unless LaTeXMLMath ; if LaTeXMLMath , they make sense literally as integrals over sphere and ball , respectively , of radius LaTeXMLMath , with LaTeXMLMath denoting surface- , resp .
volume element .
But the expression on the right in both equations make sense for any LaTeXMLMath , and so the distributions LaTeXMLMath and LaTeXMLMath are defined for all LaTeXMLMath ; in particular , for nilpotent ones .
It is natural to consider also the following distributions LaTeXMLMath and LaTeXMLMath on LaTeXMLMath ( likewise of compact support ) : LaTeXMLEquation .
LaTeXMLEquation For LaTeXMLMath , they may , modulo factors of the type LaTeXMLMath , be considered as “ average over LaTeXMLMath ” and “ average over LaTeXMLMath ” , respectively , since LaTeXMLMath differs from LaTeXMLMath by a factor LaTeXMLMath , which is just the surface area of LaTeXMLMath ( modulo the factor of type LaTeXMLMath ) , and similarly for LaTeXMLMath .
Note that LaTeXMLMath and LaTeXMLMath .
And also note that the definition of LaTeXMLMath and LaTeXMLMath can be formulated as LaTeXMLEquation where LaTeXMLMath is the homothetic transformation LaTeXMLMath , and where we are using the covariant functorality of distributions of compact support .
For low dimensions , we shall describe the distributions LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath explicitly : Dimension 1 LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Dimension 2 LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Dimension 3 LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Notice that these formulas make sense for all LaTeXMLMath ( positive , negative , nilpotent , … ) , using the standard convention : LaTeXMLMath ) , whereas set-theoretically LaTeXMLMath and LaTeXMLMath ( as point sets ) only make good sense for LaTeXMLMath .
It is clear from the very definition that LaTeXMLMath and LaTeXMLMath ( in any dimension LaTeXMLMath ) ; but since we are interested also in LaTeXMLMath ’ s that are not invertible , LaTeXMLMath and LaTeXMLMath can not be defined in terms of each other .
Note also that LaTeXMLMath , whereas LaTeXMLMath and LaTeXMLMath are constants times the Dirac distribution at the origin LaTeXMLMath .
The constants are the “ area ” of the unit sphere , or the “ volume ” of the unit ball , in the appropriate dimension .
Explicitly , LaTeXMLEquation and LaTeXMLEquation in dimensions 1,2 , and 3 , respectively .
We shall also have occasion to consider the distribution ( of compact support ) LaTeXMLMath on LaTeXMLMath as well as its projection LaTeXMLMath on the LaTeXMLMath -plane ( using functorality of LaTeXMLMath with respect to the projection map LaTeXMLMath ) .
For LaTeXMLMath ( more generally , for LaTeXMLMath invertible ) , we can give an explicit integral expression for it , but note that since LaTeXMLMath and LaTeXMLMath are defined for all LaTeXMLMath , then so is LaTeXMLMath , whether or not we have such an integral expression .
The integral expression ( for LaTeXMLMath ) goes under the name of Poisson kernel for the wave equation in dimension 2 and may be obtained as follows : using the above expression for LaTeXMLMath in dimension 3 , we have for a test function LaTeXMLMath that only depends on LaTeXMLMath , but not on LaTeXMLMath that LaTeXMLEquation .
We then make the change of variables LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and then the integral becomes LaTeXMLEquation using the explicit form of the ball distribution LaTeXMLMath in dimension 2 , we may rewrite the right hand side here as LaTeXMLEquation so that we have , for LaTeXMLMath ( or even for LaTeXMLMath invertible ) , LaTeXMLEquation .
The Main Theorem of vector calculus is Stokes ’ Theorem : LaTeXMLMath , for LaTeXMLMath an LaTeXMLMath -form , LaTeXMLMath a suitable LaTeXMLMath -dimensional figure ( with appropriate measure on it ) and LaTeXMLMath its geometric boundary .
In the synthetic context , the theorem holds at least for any singular cubical chain LaTeXMLMath ( LaTeXMLMath the LaTeXMLMath -dimensional coordinate cube ) , because the theorem may then be reduced to the fundamental theorem of calculus , which is the only way integration enters in the elementary synthetic context ; measure theory not being available therein .
For an account of Stokes ’ Theorem in this context , see LaTeXMLCite p.139 .
Below , we shall apply the result not only for singular cubes , but also for singular boxes , like the usual LaTeXMLMath , parametrizing the unit disk by polar coordinates , LaTeXMLEquation .
We shall need from vector calculus the Gauss-Ostrogradsky “ Divergence Theorem ” LaTeXMLEquation with LaTeXMLMath a vector field , for the geometric “ figure ” LaTeXMLMath = the unit ball in LaTeXMLMath For the case of the unit ball in LaTeXMLMath , the reduction of the Divergence Theorem to Stokes ’ Theorem is a matter of the differential calculus of vector fields , differential forms , inner products etc .
( See e.g .
LaTeXMLCite p. 204 ) .
For the convenience of the reader , we recall the case LaTeXMLMath .
Given a vector field LaTeXMLMath in LaTeXMLMath , apply Stokes ’ Theorem to the differential form LaTeXMLEquation for the singular rectangle LaTeXMLMath given by ( LaTeXMLRef ) above .
Then LaTeXMLEquation .
Since LaTeXMLMath , then LaTeXMLEquation .
On the other hand , LaTeXMLEquation ( all LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath to be evaluated ar LaTeXMLMath ) .
Therefore LaTeXMLEquation this is LaTeXMLMath .
On the other hand by Stokes ’ Theorem LaTeXMLMath which is a curve integral of the 1-form ( LaTeXMLRef ) around the boundary of the rectangle LaTeXMLMath .
This curve integral is a sum of four terms corresponding to the four sides of the rectangle .
Two of these ( corresponding to the sides LaTeXMLMath and LaTeXMLMath ) cancel , and the term corresponding to the side where LaTeXMLMath vanishes because of the LaTeXMLMath in LaTeXMLMath , so only the side with LaTeXMLMath , LaTeXMLMath remains , and its contribution is , with the correct orientation , LaTeXMLEquation where LaTeXMLMath is the outward unit normal of the unit circle .
This expression is the flux of LaTeXMLMath over the unit circle , which thus equals the divergence integral calculated above .
We insert for reference two obvious “ change of variables ” equations .
Recall that LaTeXMLMath is the homothetic transformation “ multiplying by LaTeXMLMath ” .
We have , for any vector field LaTeXMLMath on LaTeXMLMath ( viewed , via principal part , as a map LaTeXMLMath ) : LaTeXMLEquation and LaTeXMLEquation .
We now combine vector calculus with the calculus of the basic ball- and sphere-distributions , as introduced in Section 3 , to prove In LaTeXMLMath ( for any LaTeXMLMath ) , we have , for any LaTeXMLMath , LaTeXMLEquation ( LaTeXMLMath the Laplace operator ) .
Proof .
We prove first that LaTeXMLEquation .
In fact , for any test function LaTeXMLMath , LaTeXMLEquation ( by differentiating under the integral sign and using the chain rule ) LaTeXMLEquation where LaTeXMLMath is the homothetic transformation “ multiplying by LaTeXMLMath ” .
This , by the Divergence Theorem , may be rewritten as LaTeXMLEquation ( using ( LaTeXMLRef ) ) LaTeXMLEquation ( by a standard change of variables , cf .
( LaTeXMLRef ) ) , so LaTeXMLEquation .
From LaTeXMLEquation we may of course conclude the desired equality , by cancelling LaTeXMLMath on both sides , if LaTeXMLMath is invertible ; but we want the equation for all LaTeXMLMath .
We can get this from “ Lavendhomme ’ s principle ” , which says that if LaTeXMLMath satisfies LaTeXMLMath for all LaTeXMLMath , then LaTeXMLMath is constantly LaTeXMLMath .
This principle was derived from the integration axiom purely synthetically by Lavendhomme in LaTeXMLCite p.25 .
So the claim of the Theorem is valid for all LaTeXMLMath .
We collect information about LaTeXMLMath -derivatives of the four basic distributions LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath in LaTeXMLMath .
The results are valid for any LaTeXMLMath and any LaTeXMLMath .
For invertible LaTeXMLMath ( say positive LaTeXMLMath ) , some of the statements may be simplified by multiplying by LaTeXMLMath , but we prefer having formulae which are universally valid .
We have in dimension LaTeXMLMath for all LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation In dimension LaTeXMLMath , we also have LaTeXMLEquation .
Proof .
Equation ( LaTeXMLRef ) is an immediate consequence of the fundamental theorem of calculus ; e.g .
for LaTeXMLMath , consider the explicit formula for LaTeXMLMath given above in Section 3 ( “ Spheres and balls as distributions ” ) .
With LaTeXMLMath as the outer integral , the LaTeXMLMath of it is just the inner integral , i.e. , exactly the exhibited formula ( idem ) for LaTeXMLMath .
For ( LaTeXMLRef ) , we LaTeXMLMath -differentiate the equation LaTeXMLMath by the Leibniz rule and get LaTeXMLMath ; so by Theorem LaTeXMLRef , LaTeXMLEquation .
If we multiply this equation by LaTeXMLMath , we get LaTeXMLEquation using LaTeXMLMath and LaTeXMLMath , the result follows ( note that LaTeXMLMath commutes with multiplication by LaTeXMLMath ) .
The proof of ( LaTeXMLRef ) is similar : LaTeXMLMath -differentiating LaTeXMLMath , we get LaTeXMLEquation ( using ( LaTeXMLRef ) ) , so using LaTeXMLMath , this equation may be rewritten as LaTeXMLEquation .
The result now follows by cancelletion of the factor LaTeXMLMath by Lavendhomme ’ s principle , and rearranging .
Next , ( LaTeXMLRef ) is identical to Theorem LaTeXMLRef , and is included again for completeness ’ sake .
Finally , ( LaTeXMLRef ) follows from ( LaTeXMLRef ) : the first term vanishes , since LaTeXMLMath , and in the remaining equation , we may cancel the factor LaTeXMLMath by Lavendhomme ’ s principle .
Alternatively , ( LaTeXMLRef ) can be proved directly , by a very simple calculation .
Let LaTeXMLMath denote the Laplace operator LaTeXMLMath on LaTeXMLMath .
We shall consider the wave equation ( WE ) in LaTeXMLMath , ( for LaTeXMLMath ) , LaTeXMLEquation as a second order ordinary differential equation on the Euclidean vector space LaTeXMLMath of distributions of compact support ; in other words , we are looking for functions LaTeXMLEquation so that for all LaTeXMLMath LaTeXMLMath ( viewing LaTeXMLMath as a map LaTeXMLMath .
We shall only be looking for particular solutions , in fact , so called fundamental solutions : solutions whose initial value and initial speed is either the Dirac distribution at LaTeXMLMath , or LaTeXMLMath .
Given any other initial value and speed — these being both assumed to be distributions of compact support — , the corresponding particular solution may , as is well known , be obtained from the fundamental solution just by convolution LaTeXMLMath with these fundamental solutions .
This follows purely formally from the rules for convolution of distributions LaTeXMLMath and LaTeXMLMath , such as LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath is any differential operator on LaTeXMLMath with constant coefficients ; and from linearity of the convolution , implying that LaTeXMLMath ; see e.g .
LaTeXMLCite , Ch .
3 .
Dimension 1 The function LaTeXMLMath given by LaTeXMLEquation is a solution of the WE in dimension 1 ; its initial value and speed are , respectively LaTeXMLMath and LaTeXMLMath .
The function LaTeXMLMath given by LaTeXMLEquation is a solution of the WE ; its initial value and speed are , respectively , LaTeXMLMath and LaTeXMLMath .
Proof .
The statements about the initial values are immediate from the explicit integral formulas for LaTeXMLMath and LaTeXMLMath ( putting LaTeXMLMath ) .
The statements about the initial speeds are equally immediate from the following formulas ( LaTeXMLRef ) and ( LaTeXMLRef ) for the LaTeXMLMath -derivatives , ( putting LaTeXMLMath ) .
We have by ( LaTeXMLRef ) LaTeXMLEquation and so by further LaTeXMLMath differentiation LaTeXMLEquation now , LaTeXMLMath and LaTeXMLMath commute , so we may continue LaTeXMLEquation using ( LaTeXMLRef ) with LaTeXMLMath .
Now by linearity of LaTeXMLMath , the terms involving LaTeXMLMath in the last expression cancel , and we are left with LaTeXMLEquation which establishes WE for LaTeXMLMath and hence also for LaTeXMLMath .
Also , by ( LaTeXMLRef ) , we have that LaTeXMLEquation and so by further LaTeXMLMath differentiation LaTeXMLEquation using ( LaTeXMLRef ) , which establishes WE for LaTeXMLMath and hence for LaTeXMLMath .
So the theorem is proved .
Dimension 3 The function LaTeXMLMath given by LaTeXMLEquation is a solution of the WE in dimension 3 ; its initial value and speed are , respectively , LaTeXMLMath and LaTeXMLMath .
The function LaTeXMLMath given by LaTeXMLEquation is a solution of the WE ; its initial value and speed are , respectively , LaTeXMLMath and LaTeXMLMath .
Proof .
We calculate first LaTeXMLMath of LaTeXMLMath , using ( LaTeXMLRef ) : LaTeXMLEquation and so by Theorem LaTeXMLRef ( = ( LaTeXMLRef ) ) , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation using ( LaTeXMLRef ) , and now by linearity of LaTeXMLMath , the terms involving LaTeXMLMath cancel , so we are left with the equation LaTeXMLEquation which establishes WE for LaTeXMLMath and hence for LaTeXMLMath .
The statements about initial value and speed are immediate ( using ( LaTeXMLRef ) for the speed ) .
Because LaTeXMLMath and LaTeXMLMath commute , it is clear that if LaTeXMLMath is a distributional solution of WE , then so is LaTeXMLMath .
So since LaTeXMLMath is a solution , then so is its LaTeXMLMath -derivative ( calculated in ( LaTeXMLRef ) above ) , i.e .
LaTeXMLMath is a solution .
Its initial value and its initial speed can be found by putting LaTeXMLMath in ( LaTeXMLRef ) ( note LaTeXMLMath commutes with multiplication by LaTeXMLMath ) .
Dimension 2 Recall that we considered the orthogonal projection LaTeXMLMath Applying covariant functorality , we get for any distribution LaTeXMLMath on LaTeXMLMath of compact support a distribution LaTeXMLMath on LaTeXMLMath , also of compact support .
The function LaTeXMLMath given by LaTeXMLEquation is a solution of the WE in dimension 2 ; its initial value and speed are , respectively , LaTeXMLMath and LaTeXMLMath .
The function LaTeXMLMath given by LaTeXMLMath is also a solution of the WE in dimension 2 ; its initial value and speed are , respectively , LaTeXMLMath and LaTeXMLMath .
Recall that an explicit integral formula for LaTeXMLMath , for LaTeXMLMath , was given above , in ( LaTeXMLRef ) ( “ Poisson kernel ” ) .
Proof .
The fact that the two distributions in question are solutions of the WE is immediate from the Proposition LaTeXMLRef ( “ LaTeXMLMath commutes with LaTeXMLMath ” ) and from the fact that LaTeXMLMath is linear , and hence commutes with formation of LaTeXMLMath ; also , LaTeXMLMath sends Dirac distribution at LaTeXMLMath to Dirac distribution at LaTeXMLMath , so the initial values and speeds are as claimed .
The Taylor Series at LaTeXMLMath for the solutions given can be calculated directly , but they can more easily be obtained from the formal solution given in Proposition LaTeXMLRef .
In this section we deal with distributions that do not have compact support and we only consider the one-dimensional case .
We are thus considering solutions for the vector field on the Euclidean vector space LaTeXMLMath , whose principal part is given by LaTeXMLMath .
We consider the particular solution LaTeXMLMath whose initial value is the distribution LaTeXMLMath Thus , referring to the general treatment of solutions for ( differential equations given by ) vector fields , we are considering LaTeXMLMath ; for the heat equation , one can not do better , as is well known .
Also , as mentioned above , we rely on external ( classical ) calculus ; namely , we consider the classical “ heat kernel ” function , i.e. , the function LaTeXMLMath given by LaTeXMLEquation .
Here , for the case LaTeXMLMath , we described a function rather than a distribution , so here we do make the identification of functions LaTeXMLMath with distributions LaTeXMLMath .
Differentiation of distributions reduces to differentiation of the representing functions .
For LaTeXMLMath , we thus have LaTeXMLMath , a smooth function in two variables , described by the above expression .
It satisfies the heat equation LaTeXMLEquation for LaTeXMLMath .
Also the following limit expression is classical : LaTeXMLEquation for any test function LaTeXMLMath .
More generally , For any integer LaTeXMLMath , and any test function LaTeXMLMath LaTeXMLEquation .
Proof .
The case LaTeXMLMath is just ( LaTeXMLRef ) ; the general case follows by iteration .
Let us do the case LaTeXMLMath .
Then LaTeXMLEquation ( by the heat equation for LaTeXMLMath ) LaTeXMLEquation ( by integration by parts . )
We then use ( LaTeXMLRef ) , for the test function LaTeXMLMath to conclude ( LaTeXMLRef ) for LaTeXMLMath .
The function LaTeXMLMath is smooth .
Here , smoothness is taken in the following sense ( appropriate for convenient vector spaces ) : for each test function LaTeXMLMath , the function LaTeXMLMath given by LaTeXMLMath is smooth .
Proof .
It suffices to prove that LaTeXMLMath is infinitely often differentiable at LaTeXMLMath , since smoothness for LaTeXMLMath is clear .
For fixed LaTeXMLMath , we let LaTeXMLMath denote the function in LaTeXMLMath described in ( the first clause in ) ( LaTeXMLRef ) above .
Thus , LaTeXMLMath is given by the integral LaTeXMLEquation .
We first notice that , by Hadamard ’ s Lemma , LaTeXMLMath .
By linearity , LaTeXMLMath But LaTeXMLMath and this implies that the derivative of LaTeXMLMath at LaTeXMLMath is LaTeXMLEquation .
To compute this limit , we use the formulas and notations in Lang ’ s book LaTeXMLCite , with the exception that we use LaTeXMLMath for the Fourier transform .
We also use the following well known formulae , where all the functions under considerations belong to the class LaTeXMLMath of fast decreasing functions and thus LaTeXMLMath works with no limitations .
First , for any pair of functions LaTeXMLMath , LaTeXMLMath in this class , one has the “ adjointness ” formula LaTeXMLEquation .
Furthermore LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
To show the existence of the limit , we compute , using ( LaTeXMLRef ) , adjointness , and ( LaTeXMLRef ) LaTeXMLEquation .
The last step uses integration by parts .
Using ( LaTeXMLRef ) , this may be rewritten as LaTeXMLEquation .
LaTeXMLEquation using adjointness and ( LaTeXMLRef ) in the last step .
Now we divide by LaTeXMLMath , as requested in ( LaTeXMLRef ) , and let LaTeXMLMath .
Using ( LaTeXMLRef ) , we thus get that the limit in ( LaTeXMLRef ) equals LaTeXMLEquation .
But since LaTeXMLMath , LaTeXMLMath , This proves that the limit in ( LaTeXMLRef ) exists and equals LaTeXMLMath ; we conclude that LaTeXMLEquation .
To better understand what has been done and to develop this matter further , let us define for every LaTeXMLMath LaTeXMLEquation .
We can summarize the results of this section as follows LaTeXMLEquation .
Recall from Proposition LaTeXMLRef that LaTeXMLEquation and thus , by going to the limit when LaTeXMLMath LaTeXMLEquation .
These results suffice to summarize the result in the present Section in the following way : The function LaTeXMLMath is smooth and , furthermore , LaTeXMLMath Proof .
Let us show , for instance , that LaTeXMLMath exists and equals LaTeXMLMath Using the previous results , LaTeXMLMath and this implies the corollary , by going to the limit when LaTeXMLMath and using ( LaTeXMLRef ) with the function LaTeXMLMath instead of LaTeXMLMath Now , iterate .
The idea to use Fourier transform to prove smoothness was pointed out to us by H. Stetkær and E. Skibsted .
Summarizing : we have a smooth function LaTeXMLMath , satisfying the heat equation LaTeXMLMath for all LaTeXMLMath ; for LaTeXMLMath , this follows from Proposition LaTeXMLRef .
By the assumed fullness of the embedding of smooth manifolds with boundary and convenient vector spaces into the model of SDG , we have the desired solution internally in the model .
We may then ask for the values of LaTeXMLMath for nilpotent LaTeXMLMath .
The answer can be deduced from the Taylor Series at LaTeXMLMath for the function LaTeXMLMath , and the coefficients can be read off from Proposition LaTeXMLRef ; alternatively , by the uniqueness of formal solutions ( Theorem LaTeXMLRef ) , they can be read off from the formal solution we know already from Proposition LaTeXMLRef .
In any case , we get for nilpotent LaTeXMLMath LaTeXMLEquation the series being a finite sum , since LaTeXMLMath is nilpotent .
In particular , for LaTeXMLMath with LaTeXMLMath , we have LaTeXMLMath , or since LaTeXMLMath , LaTeXMLEquation .
In some sense , the motivation for our study of the heat equation in particular was to see how LaTeXMLMath evolves in nilpotent lapse LaTeXMLMath of time and specially for LaTeXMLMath with LaTeXMLMath ; the answer is ( LaTeXMLRef ) ( or more generally ( LaTeXMLRef ) ) .
Being an extensive quantity , a distribution like ( LaTeXMLRef ) should be drawable .
In fact , it can be exhibited as a finite linear combination of Dirac distributions LaTeXMLMath ( = “ evaluate at LaTeXMLMath ” ) .
This hinges on : Let LaTeXMLMath .
Then LaTeXMLEquation .
Proof .
It suffices to prove , for an arbitrary test function LaTeXMLMath , that LaTeXMLMath ; now just Taylor expand the two outer terms in the sum on the right ; the terms of odd degree cancel , the terms of even degree ( 0 and two ) give the result .
( There is a similar result for higher derivatives of LaTeXMLMath : for LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath denotes the binomial coefficient LaTeXMLMath .
This hinges on some combinatorics with binomial coefficients , cf .
LaTeXMLCite p. 63 , Problem 16 ) .
To make a “ drawing ” of LaTeXMLMath where LaTeXMLMath , we assume that LaTeXMLMath for some LaTeXMLMath with LaTeXMLMath ( we shall not deal here with the question whether this can always be done ) .
Then LaTeXMLEquation using ( LaTeXMLRef ) and ( LaTeXMLRef ) .
The drawing one can make of LaTeXMLMath ( as for any discrete distribution ) , is a column diagram : erect a column of heigth 1 at LaTeXMLMath .
The distribution above then comes about by removing LaTeXMLMath units from the unit column at LaTeXMLMath , and placing the small columns of heigth LaTeXMLMath at LaTeXMLMath and LaTeXMLMath .
This is the beginning of the diffusion of the Dirac distribution .
Several other ways of exhibiting LaTeXMLMath as linear combination of Dirac distributions are also possible .
Since LaTeXMLMath is a microlinear and Euclidean LaTeXMLMath -module , and LaTeXMLMath is linear , we may apply the general results of Propositions LaTeXMLRef and LaTeXMLRef to conclude that the formal solution of the heat equation LaTeXMLMath with initial value ( the distribution ) LaTeXMLMath , is the series LaTeXMLEquation .
Similarly , the formal solution of the wave equation LaTeXMLMath with initial value ( the distribution ) LaTeXMLMath , and initial speed the distribution LaTeXMLMath is the series LaTeXMLEquation .
Applying ( in the one-variable case , say ) these formulas to a test function LaTeXMLMath in the variable LaTeXMLMath and to the distributions LaTeXMLMath and LaTeXMLMath we obtain the following Maclaurin series for the heat equation LaTeXMLEquation .
Here LaTeXMLMath refers to the time derivative , whereas LaTeXMLMath to the space derivative LaTeXMLMath The variable LaTeXMLMath has been left unexpressed .
There is a similar series for the wave equation : LaTeXMLEquation .
For the sake of completeness , we also consider the function LaTeXMLMath given by LaTeXMLMath , the Dirac distribution at LaTeXMLMath .
This is the “ fundamental solution ” for the equation for “ simple transport ” , cf .
e.g .
LaTeXMLCite .
The function LaTeXMLMath is the solution for the differential equation for “ simple transport ” , LaTeXMLEquation with initial value LaTeXMLMath .
Proof .
For any test function LaTeXMLMath , LaTeXMLEquation
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques .
It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor .
Examples show how it behaves compared to other known bounds .
Subj .
Class .
: Differential Geometry .
2000 MSC : 53C27 , 53C25 .
Keywords : Dirac operator , eigenvalues , harmonic Weyl tensor .
If LaTeXMLMath is a compact Riemannian spin manifold with positive scalar curvature LaTeXMLMath , then each eigenvalue LaTeXMLMath of the Dirac operator LaTeXMLMath satisfies the inequality LaTeXMLEquation where LaTeXMLMath is the minimum of LaTeXMLMath on LaTeXMLMath .
The estimate LaTeXMLMath is sharp in the sense that there exist manifolds for which LaTeXMLMath is an equality for the first eigenvalue LaTeXMLMath of LaTeXMLMath .
If this is the case , then each eigenspinor LaTeXMLMath corresponding to LaTeXMLMath is a Killing spinor with the Killing number LaTeXMLMath , i.e. , LaTeXMLMath is a solution of the field equation LaTeXMLEquation and LaTeXMLMath must be an Einstein space ( see LaTeXMLCite ) .
A generalization of this inequality was proved in the paper LaTeXMLCite , where a conformal lower bound for the spectrum of the Dirac operator occured .
Moreover , for special Riemannian manifolds better estimates for the eigenvalues of the Dirac operator are known , see LaTeXMLCite , LaTeXMLCite .
However , all these estimates of the spectrum of the Dirac operators depend only on the scalar curvature of the underlying manifold .
Therefore it is a natural question whether or not one may relate the spectrum of the Dirac operator to more refined curvature data .
In this paper we shall prove an estimate depending on the Ricci tensor for the eigenvalues of the Dirac operator on compact Riemannian manifolds with harmonic curvature tensor .
The main idea is the investigation of the differential operators LaTeXMLEquation depending on a real parameter LaTeXMLMath and defined by LaTeXMLEquation where Ric denotes the Ricci tensor .
Under the assumption that the curvature tensor is harmonic we prove a formula expressing the length LaTeXMLMath by the Dirac operator LaTeXMLMath , the covariant derivatives LaTeXMLMath and LaTeXMLMath as well as by some curvature terms ( Theorem 1.6 ) .
Integrating this formula we obtain , for any LaTeXMLMath , an inequality for the eigenvalues of the Dirac operator depending on the scalar curvature , the minimum of the eigenvalues of the Ricci tensor and its length .
An optimal choice of the parameter LaTeXMLMath bounds the spectrum of the Dirac operator from below .
For example , we prove the inequality LaTeXMLEquation for compact Riemannian spin manifolds with harmonic curvature tensor and vanishing scalar curvature , where LaTeXMLMath and LaTeXMLMath denote the minimum of the eigenvalues and the length of the Ricci tensor , respectively .
First of all let us fix some notations .
In the following LaTeXMLMath is always any local frame of vector fields , and LaTeXMLMath is the associated frame defined by LaTeXMLMath , where the LaTeXMLMath denote the components of the inverse of the Riemannian metric LaTeXMLMath .
Using the twistor operator ( see LaTeXMLCite , Section 1.4 ) LaTeXMLEquation locally given by LaTeXMLMath and LaTeXMLMath , we may rewrite the operator LaTeXMLMath as LaTeXMLEquation .
The image of the twistor operator LaTeXMLMath is contained in the kernel of the Clifford multiplication LaTeXMLMath , i.e. , LaTeXMLEquation .
As endomorphisms acting on the spinor bundle the following identities are well known : LaTeXMLEquation .
In particular , we see that the image of the operators LaTeXMLMath is contained in the kernel of the Clifford multiplication .
By definition , a spinor field LaTeXMLMath belongs to the kernel of the operator LaTeXMLMath if and only if it satisfies the equation LaTeXMLEquation for each vector field LaTeXMLMath .
In the following we shall use the Weitzenböck formula LaTeXMLEquation for the twistor operator LaTeXMLMath .
Lemma 1.1 : For any spinor field LaTeXMLMath , the following formula holds : LaTeXMLEquation .
Proof : Using the formulas LaTeXMLMath and LaTeXMLMath we have LaTeXMLEquation .
Equation LaTeXMLMath is a preliminary version of the Weitzenböck formula , which we will apply in the proof of our main result .
Our next aim is to express the uncontrollable last term on the right-hand side by terms that are controllable .
For this purpose we need a condition on the covariant derivative of the Ricci tensor .
For vector fields LaTeXMLMath , we use the notation LaTeXMLEquation for the corresponding tensorial derivatives of second order in LaTeXMLMath as well as in LaTeXMLMath .
By LaTeXMLMath we denote the Riemannian curvature tensor and by LaTeXMLMath the curvature tensor in the spinor bundle LaTeXMLMath .
Then , for all LaTeXMLMath and all LaTeXMLMath , we have LaTeXMLEquation as well as the well known relation between the two curvatures LaTeXMLEquation .
Considering LaTeXMLMath as a map from LaTeXMLMath to LaTeXMLMath locally defined by LaTeXMLEquation the length LaTeXMLMath is just the scalar product LaTeXMLEquation .
Moreover , for two spinor fields LaTeXMLMath we introduce a complex vector field LaTeXMLMath defined by the formula LaTeXMLEquation .
Lemma 1.2 : For any LaTeXMLMath , we have the equation LaTeXMLEquation .
Proof : Let LaTeXMLMath be any point and let LaTeXMLMath be any orthonormal frame in a neighbourhood of the point such that LaTeXMLMath holds for LaTeXMLMath .
Then we have at LaTeXMLMath that LaTeXMLEquation and we obtain the following formula for the left-hand side of the expression ( 10 ) LaTeXMLEquation .
On the other hand , from ( 9 ) we obtain LaTeXMLMath and LaTeXMLEquation .
The Bianchi identity LaTeXMLEquation implies the relation LaTeXMLEquation .
The latter two equations yield LaTeXMLEquation .
Inserting this formula we obtain LaTeXMLMath .
LaTeXMLMath In the following we use the Schrödinger-Lichnerowicz formula LaTeXMLEquation .
The local expression of the Bochner Laplacian LaTeXMLMath is LaTeXMLEquation where the Christoffel symbols LaTeXMLMath are defined by LaTeXMLMath .
In the proof of the following lemma we also use the well known general formulas LaTeXMLEquation .
LaTeXMLEquation Moreover , for LaTeXMLMath , let LaTeXMLMath and LaTeXMLMath be the complex vector fields on LaTeXMLMath locally given by LaTeXMLEquation .
The vector field satisfies the relation LaTeXMLEquation .
Lemma 1.3 : Let LaTeXMLMath be any spinor field .
Then there is the identity LaTeXMLEquation .
Proof : Let LaTeXMLMath be any point and let LaTeXMLMath be any orthonormal frame in a neighbourhood of LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
We use the notations LaTeXMLEquation .
Then , we have LaTeXMLMath at the point LaTeXMLMath and LaTeXMLMath LaTeXMLMath Using this we calculate LaTeXMLEquation .
LaTeXMLEquation Hence , it holds that LaTeXMLEquation .
Further , we have LaTeXMLEquation .
LaTeXMLEquation Inserting the latter equation into LaTeXMLMath we obtain LaTeXMLMath .
LaTeXMLMath Comparing the equations LaTeXMLMath and LaTeXMLMath we obtain immediately Lemma 1.4 : For any spinor field LaTeXMLMath , we have the identity LaTeXMLEquation .
The following purely algebraic condition on the covariant derivative of the Ricci tensor implies that the second term in formula LaTeXMLMath vanishes .
The proof is an easy computation using the relations in the Clifford algebra .
A thorough geometric discussion of this condition will be provided in Section LaTeXMLMath .
Lemma 1.5 : If the covariant derivative of the Ricci tensor satisfies LaTeXMLEquation then , for any spinor field LaTeXMLMath and any vector field LaTeXMLMath , the Clifford product LaTeXMLEquation vanishes .
We thus obtain the following Weitzenböck formula for the length LaTeXMLMath , which is fundamental for all our further considerations .
Theorem 1.6 : Let LaTeXMLMath be a Riemannian spin manifold and suppose that LaTeXMLEquation .
Then , for any spinor field LaTeXMLMath , there exists a vector field LaTeXMLMath such that LaTeXMLEquation .
Proof : The formula follows from LaTeXMLMath and LaTeXMLMath if one defines LaTeXMLMath locally by LaTeXMLEquation .
In this section we assume that LaTeXMLMath is compact , connected and that the Ricci tensor satisfies the condition LaTeXMLEquation .
It is an easy consequence of the Bianchi idendity that the scalar curvature of the manifold must be constant .
Then the tensor LaTeXMLEquation has the same properties as the Ricci tensor .
In dimension LaTeXMLMath the manifold is conformally flat .
If LaTeXMLMath , we obtain the identity LaTeXMLEquation by computing the divergence of the Weyl tensor LaTeXMLMath ( see LaTeXMLCite ) .
Therefore , the manifold satisfies the mentioned condition for the Ricci tensor if and only if it has constant scalar curvature and a harmonic Weyl tensor .
Moreover , these two properties are equivalent to the condition that the curvature tensor is harmonic ( see Chapter LaTeXMLMath in LaTeXMLCite ) .
The following examples are known : Local products of Einstein manifolds ; conformally flat manifolds with constant scalar curvature ; warped products LaTeXMLMath of an Einstein manifold with positive scalar curvature LaTeXMLMath by LaTeXMLMath ( see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) , where the function LaTeXMLMath is a positive , periodic solution of the differential equation LaTeXMLEquation warped products over Riemann surfaces .
We denote by LaTeXMLMath the eigenvalues of the Ricci tensor at the point LaTeXMLMath and by LaTeXMLMath the minimum of LaTeXMLMath .
If LaTeXMLMath is an eigenspinor , then LaTeXMLMath yields the inequality LaTeXMLEquation .
In case of an Einstein manifold we get back the inequality LaTeXMLMath .
In general , the Schrödinger-Lichnerowicz formula and the estimation LaTeXMLEquation imply the inequality LaTeXMLEquation and , finally , for any LaTeXMLMath we obtain the condition LaTeXMLEquation .
This is a min-max principle and can be used in order to estimate the eigenvalues of the Dirac operator from below .
Of course , only parameters between LaTeXMLMath are interesting .
A similar result involving only the scalar curvature was proved in LaTeXMLCite .
For LaTeXMLMath we immediately obtain the following result Theorem 2.1 : Let LaTeXMLMath be a compact Riemannian spin manifold with harmonic curvature tensor .
If LaTeXMLMath and LaTeXMLMath denote the minimum of the eigenvalues and the length of the Ricci tensor , respectively , and if LaTeXMLEquation holds , then there are no harmonic spinors .
If the scalar curvature is positive , we know that LaTeXMLMath and the mini-max principle yields a better estimate only in case that the left-hand side is negative for LaTeXMLMath and some LaTeXMLMath .
This condition is equivalent to LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the minimum of the eigenvalues and the length of the Ricci tensor , respectively Example 1 : The warped product LaTeXMLMath of an Einstein manifold LaTeXMLMath with positive scalar curvature LaTeXMLMath by LaTeXMLMath never satisfies the latter condition .
Solving the differential equation LaTeXMLMath for LaTeXMLMath and LaTeXMLMath we obtain , for example , LaTeXMLMath and LaTeXMLMath .
One series of examples which we can apply our inequality to consist of products LaTeXMLMath with a sufficiently large number of factors .
A second series are products LaTeXMLMath by an Einstein manifold LaTeXMLMath with sufficiently large scalar curvature .
We describe the case of a two-dimensional sphere LaTeXMLMath and a LaTeXMLMath -dimensional Einstein spin manifold LaTeXMLMath with scalar curvature LaTeXMLMath in greater delail .
Consider the positive , periodic solution LaTeXMLMath of the differential equation LaTeXMLMath with initial values LaTeXMLMath and LaTeXMLMath .
The Ricci tensor of the manifold LaTeXMLMath has two eigenvalues LaTeXMLEquation .
The multiplicity of LaTeXMLMath is one , the multiplicity of LaTeXMLMath is four , the scalar curvature of the warped product equals LaTeXMLMath .
Denote by LaTeXMLMath the scalar curvature of the sphere LaTeXMLMath and consider the manifold LaTeXMLMath .
Then we have LaTeXMLEquation .
For the optimal parameter LaTeXMLMath we obtain the estimate LaTeXMLEquation whereas the inequality LaTeXMLMath yields the estimate LaTeXMLMath .
LaTeXMLMath The discussion of the limiting case yields a spinor field LaTeXMLMath in the kernel of one of the operators LaTeXMLMath with LaTeXMLMath .
Moreover , at every point we have LaTeXMLEquation i.e. , the derivative of LaTeXMLMath vanish in all directions LaTeXMLMath that are orthogonal to the LaTeXMLMath -eigenspace of the Ricci tensor , LaTeXMLMath .
The equation LaTeXMLMath means that the eigenspinor LaTeXMLMath satisfies the equation LaTeXMLEquation for each vector field LaTeXMLMath .
In particular , the length of the spinor field is constant .
If LaTeXMLMath , then LaTeXMLMath is a Killing spinor .
In case LaTeXMLMath , we consider the largest eigenvalue LaTeXMLMath at a minimum LaTeXMLMath of LaTeXMLMath and insert an eigenvector .
Then we obtain LaTeXMLEquation .
But LaTeXMLMath is positive , a contradiction .
Thus the limiting case in the inequality can not occur except that LaTeXMLMath is an Einstein manifold with a Killing spinor .
First we consider the case that the scalar curvature LaTeXMLMath vanishes .
Then LaTeXMLMath is negative and for any positive LaTeXMLMath we have LaTeXMLEquation .
An elementary discussion yields the proof of the following theorem .
Theorem 2.2 : Let LaTeXMLMath be a compact , non-Ricci flat Riemannian spin manifold with harmonic curvature tensor and vanishing scalar curvature .
If LaTeXMLMath and LaTeXMLMath denote the minimum of the eigenvalues and the length of the Ricci tensor , respectively , then the eigenvalues of the Dirac operator are bounded by LaTeXMLEquation .
Remark : The Schrödinger-Lichnerowicz formula implies the well known fact that a compact , non Ricci-flat Riemannian spin manifold with LaTeXMLMath does not admit harmonic spinors .
The estimate in Theorem 2.2 is a quantitative improvement of this fact for manifolds with harmonic curvature tensor and vanishing scalar curvature .
Example 2 : Let LaTeXMLMath be a geometrically finite Kleinian group of compact type and denote by LaTeXMLMath its limit set .
Then LaTeXMLMath is a closed manifold equipped with a flat conformal structure .
If the Hausdorff dimension of the limit set equals LaTeXMLMath , then there exists a Riemannian metric in the conformal class with vanishing scalar curvature ( see LaTeXMLCite ) .
S. Nayatani constructed this metric explicitly and studied its Ricci tensor ( LaTeXMLCite ) .
Example 3 : Let us continue Example 1 .
If LaTeXMLMath is a compact surface with scalar curvature LaTeXMLMath , then LaTeXMLMath has a harmonic curvature tensor and vanishing scalar curvature .
Theorem 2.2 proves the estimate LaTeXMLEquation .
Example 4 : If LaTeXMLMath is a compact surface with scalar curvature LaTeXMLMath , then LaTeXMLMath has a harmonic curvature tensor and negative scalar curvature .
We apply the mini-max principle and obtain LaTeXMLEquation .
In particular , LaTeXMLMath has no harmonic spinors .
Let us introduce the short cuts LaTeXMLEquation as well as the new parameter LaTeXMLMath .
Then , by definition , LaTeXMLMath and LaTeXMLMath are non-negative and the condition LaTeXMLMath of Theorem 2.1 is equivalent to LaTeXMLMath .
The mini-max principle yields immediately LaTeXMLEquation and then LaTeXMLEquation for any LaTeXMLMath .
The function LaTeXMLMath attains its maximum at the point LaTeXMLEquation .
Hence , in case LaTeXMLMath LaTeXMLMath , the parameter LaTeXMLMath is automatically positive .
In case LaTeXMLMath LaTeXMLMath the parameter LaTeXMLMath is positive if and only if LaTeXMLMath or , equivalently , if LaTeXMLEquation holds .
We summarize the main result .
Theorem 3.1 : Let LaTeXMLMath be a compact Riemannian spin manifold with harmonic curvature tensor such that the condition LaTeXMLEquation is satisfied .
Then every eigenvalue LaTeXMLMath of the Dirac operator satisfies LaTeXMLEquation .
Proof : In case LaTeXMLMath , the condition LaTeXMLMath is equivalent to LaTeXMLMath and the case of a positive scalar curvature we already discussed .
Then we obtain LaTeXMLMath and equality can not occur for an eigenvalue LaTeXMLMath of LaTeXMLMath .
LaTeXMLMath We remark that the compact , conformally flat LaTeXMLMath -manifolds with constant scalar curvature and constant length of the Ricci tensor are the LaTeXMLMath -dimensional space forms and the product of a LaTeXMLMath -dimensional space form LaTeXMLMath by LaTeXMLMath ( see LaTeXMLCite ) .
These manifolds do not satisfy the condition LaTeXMLMath .
In order to express the lower bound of the eigenvalue estimate in a convenient way we introduce the new variables LaTeXMLMath by the formulas LaTeXMLEquation .
They are polynomials of degree three and six , depending on the eigenvalues of the Ricci tensor .
The inequality LaTeXMLMath can be reformulated in the following form .
Corollary 3.2 : Let LaTeXMLMath be a compact Riemannian spin manifold with harmonic curvature tensor and positive scalar curvature LaTeXMLMath .
Suppose , moreover , that LaTeXMLEquation holds .
Then each eigenvalue of the Dirac operator satisfies the estimate LaTeXMLEquation .
We remark that the condition in Corollary 3.2 is satisfied in case that the scalar curvature is positive and at least one eigenvalue of the Ricci tensor is negative .
Consequently , we obtain Corollary 3.3 : Let LaTeXMLMath be a compact Riemannian spin manifold with parallel Ricci tensor , positive scalar curvature and at least one negative eigenvalue of the Ricci tensor .
Then the estimate of Corollary LaTeXMLMath holds .
Example 1 : Let us consider the product manifold LaTeXMLMath equipped with the Riemannian metric induced by the metric of the flat torus LaTeXMLMath and the metric of the standard sphere LaTeXMLMath .
Then the Ricci tensor of LaTeXMLMath is parallel and the set of eigenvalues of Ric is given by LaTeXMLMath .
Hence , we have LaTeXMLMath .
The condition LaTeXMLMath is satisfied since LaTeXMLEquation .
Moreover , we find LaTeXMLMath .
Inserting this into LaTeXMLMath we obtain LaTeXMLMath LaTeXMLMath .
The Riemannian estimate LaTeXMLMath yields the inequality LaTeXMLMath and the Kähler estimate ( see LaTeXMLCite ) gives the lower bound LaTeXMLMath .
Since LaTeXMLMath , the estimation of the first eigenvalue of the Dirac operator on the product considered as a Kähler manifold is the best one .
We remark that LaTeXMLMath becomes an equality for the first eigenvalue ( see LaTeXMLCite ) .
Example 2 : Let LaTeXMLMath be any compact Riemannian surface of constant Gaussian curvature LaTeXMLMath and let LaTeXMLMath be the standard sphere of radius LaTeXMLMath .
Then the Ricci tensor of the Riemannian product LaTeXMLMath is parallel and LaTeXMLMath is the corresponding set of eigenvalues of Ric .
Thus , here we have LaTeXMLEquation and LaTeXMLMath This shows that the condition LaTeXMLMath is satisfied and we find LaTeXMLEquation .
LaTeXMLEquation Inserting this into the inequality LaTeXMLMath we obtain the estimation LaTeXMLMath LaTeXMLMath Hence , we see that , on the product LaTeXMLMath , the Dirac operator has a trivial kernel even in case the scalar curvature is negative .
In case the scalar curvature is positive , we can compare the new estimation LaTeXMLMath with the estimation LaTeXMLMath and with the estimation for Kähler manifolds ( see LaTeXMLCite ) , respectively .
For positive scalar curvature LaTeXMLMath , the lower bound LaTeXMLMath is obviously better than the Riemannian estimate LaTeXMLMath , LaTeXMLEquation .
If we compare the new lower bound LaTeXMLMath and the lower bound LaTeXMLMath in the Kähler case then , in the region LaTeXMLMath , the inequality LaTeXMLMath is the better one ( see the figure ) : LaTeXMLEquation .
This example shows that in certain cases with positive scalar curvature the estimate given by Theorem 3.1 is even better than the bound in LaTeXMLCite for Kähler manifolds .
LaTeXMLEquation .
The preceding two examples are special cases of a more general situation .
Consider compact Einstein manifolds with spin structures LaTeXMLMath of dimensions LaTeXMLMath ( in the case of LaTeXMLMath , we assume that LaTeXMLMath is a surface of constant Gaussian curvature ) .
Then the Riemannian product LaTeXMLMath is a compact Riemannian spin manifold with parallel Ricci tensor .
Let LaTeXMLMath be the scalar curvature of LaTeXMLMath .
Then the scalar curvature LaTeXMLMath as well as the length of the Ricci tensor of LaTeXMLMath are given by LaTeXMLEquation .
Moreover , let us assume that the smallest eigenvalue of the Ricci tensor is LaTeXMLMath .
Then the conditions under which we can apply our estimate are equivalent to LaTeXMLEquation respectively .
Remark that , by the theorem of de Rham-Wu LaTeXMLCite , any compact , simply connected Riemannian manifold with parallel Ricci tensor splits into a Riemannian product of Einstein manifolds , i.e. , the product situation is the general one for a parallel Ricci tensor .
Thomas Friedrich and Klaus-Dieter Kirchberg Humboldt-Universität zu Berlin Institut für Reine Mathematik Sitz : WBC Adlershof D-10099 Berlin
We present in this short note an idea about a possible extension of the standard noncommutative algebra of coordinates to the formal differential operators framework .
In this sense , we develop an analysis and derive an extended noncommutative structure given by LaTeXMLMath where LaTeXMLMath is the standard noncommutativity parameter and LaTeXMLMath is an antisymmetric non-constant vector-field shown to play the role of the extended deformation parameter .
This idea was motivated by the importance of noncommutative geometry framework , with nonconstant deformation parameter , in the current subject of string theory and D-brane physics .
Keywords : Star product , differential operators , Noncommutative algebra , string theory .
Recently there has been a revival interest in the noncommutativity of coordinates in string theory and D-brane physics [ 1-6 ] .
This interest is known to concern also noncommutative quantum mechanics and noncommutative field theories [ 7 , 8 ] .
Before going into presenting the aim of our work , we will try in what follows to expose some of the results actually known in literature .
The sharing property between all the above interesting areas of research is that the corresponding space exhibits the following structure LaTeXMLEquation where LaTeXMLMath are non-commuting coordinates which can describe also the space-time coordinates operators and LaTeXMLMath is a constant antisymmetric tensor .
Quantum field theories living on this space are necessarily noncommutative field theories .
Their formulation is simply obtained when the algebra ( 1 ) is realized in the space of fields ( functions ) by means of the Moyal bracket according to which the usual product of functions is replaced by the star-product as follows [ 9 ] LaTeXMLEquation .
The link with string theory consist on the correspondence between the LaTeXMLMath -constant parameter and the constant antisymmetric two-form potential LaTeXMLMath on the brane as follows [ 1 ] .
LaTeXMLEquation such that in the presence of this LaTeXMLMath -field , the end points of an open string become noncommutative on the D-brane .
In this letter , we try to go beyond the standard noncommutative algebra ( 1 ) by presenting some computations leading to consider among other a non-constant antisymmetric LaTeXMLMath -parameter satisfying an extended noncommutative Heisenberg-type algebra given by LaTeXMLEquation where LaTeXMLMath and where LaTeXMLMath describes a non-constant vector-field deformation parameter .
This is important since the obtained algebra ( 4 ) can be reduced to the standard noncommutative algebra ( 1 ) once one forget about the operatorial part of LaTeXMLMath namely LaTeXMLMath .
This construction is also interesting as it may help to build a correspondence between the noncommutative geometry framework , based on the algebra ( 4 ) , and the string theory with a non-constant B-field .
Consider the noncommutative space defined by the relation ( 1 ) originated from the star product definition of two functions LaTeXMLMath and LaTeXMLMath of an algebra LaTeXMLMath that is given by LaTeXMLEquation where LaTeXMLMath .
We denote the star-product in ( 5 ) by a prime for some reasons that we will explain later .
With this star product , one can define the Moyal bracket as follows LaTeXMLEquation .
For functions LaTeXMLMath and LaTeXMLMath coinciding with the coordinates LaTeXMLMath and LaTeXMLMath , we recover in a simple way ( 1 ) .
Actually , our idea starts from the observation that the derivatives LaTeXMLMath in the exponential ( 2 ) are differential operators which act in the following way : LaTeXMLEquation such that the prime derivative is given by LaTeXMLEquation .
Furthermore , for two given functions LaTeXMLMath and LaTeXMLMath of the algebra LaTeXMLMath , the term LaTeXMLMath remains an element of LaTeXMLMath .
So , the prime introduced in the definition of the LaTeXMLMath -product ( 5 ) is just to express the prime character of the derivative LaTeXMLMath as shown in ( 7-8 ) .
Looking for a possible generalization of the above analysis to the formal differential operators framework , we shall now introduce another kind of star-product , denoted by LaTeXMLMath and associated to an operatorial action of the derivative LaTeXMLMath .
Before going into describing how does it works , let us first introduce the set LaTeXMLMath , LaTeXMLMath [ 10 ] .
This is the algebra of local differential operators of arbitrary spins and positive degrees .
The upper indices LaTeXMLMath carried by LaTeXMLMath are the lowest and the highest degrees .
A particular example is given by LaTeXMLMath which is nothing but the algebra LaTeXMLMath , the structure usually used in the standard LaTeXMLMath -product computations .
Furthermore , in terms of the spin quantum number LaTeXMLMath , the space LaTeXMLMath is given by LaTeXMLEquation .
Typical elements of ( 9 ) are given for LaTeXMLMath by LaTeXMLEquation .
Next , we assume that the derivative LaTeXMLMath acts on the function LaTeXMLMath as an operator in the following way LaTeXMLEquation a fact which means that our derivative should be defined as LaTeXMLEquation .
This way to define the derivative is induced from the extended LaTeXMLMath -product operation defined as follows LaTeXMLEquation .
In section 3 we summarize some non trivial relation satisfied by the product LaTeXMLMath .
The major difference between the two star products LaTeXMLMath and LaTeXMLMath is that , for a given function LaTeXMLMath of the algebra LaTeXMLMath , the term LaTeXMLMath belongs on the first case to the algebra LaTeXMLMath while on the second case it is an element of the space LaTeXMLMath .
In general LaTeXMLMath is an element of LaTeXMLMath which is a particular set of the space of local differential operators denoted by LaTeXMLMath and which we can realize as LaTeXMLEquation .
Now , consider the LaTeXMLMath -product definition for the coordinates LaTeXMLMath and LaTeXMLMath , we obtain by using the above analysis LaTeXMLEquation and explicitly we have , LaTeXMLEquation .
This result is obtained by using the derived recurrence formula ( 30 ) .
Furthermore , using the antisymmetry property of LaTeXMLMath , we can easily check that ( 16 ) can be more simplified .
Concrete examples are given by the first term LaTeXMLMath which vanishes .
Also LaTeXMLMath LaTeXMLMath as well as LaTeXMLMath are terms which contribute only for the value LaTeXMLMath .
On the other hand , performing straightforward but lengthy computations , we find the following noncommutative extended LaTeXMLMath -algebra LaTeXMLEquation where the only non-vanishing term among a long mathematical series is given by LaTeXMLEquation .
Later on , we will denote the vector-field appearing on the rhs of ( 18 ) simply by LaTeXMLEquation .
In this way , LaTeXMLMath is interpreted as a deformation parameter term such that the algebra ( 17 ) becomes LaTeXMLEquation .
Note by the way that the long series we obtained for the LaTeXMLMath -parameter , before simplifying to ( 18 ) , is given by LaTeXMLEquation with LaTeXMLEquation 1 let LaTeXMLMath and LaTeXMLMath be constant numbers , we have LaTeXMLEquation 2 For each function LaTeXMLMath on the algebra LaTeXMLMath , we can show by using explicit computations that LaTeXMLEquation where for example LaTeXMLMath is the prime derivative with respect to LaTeXMLMath 3 We have also LaTeXMLEquation 4 Combining ( 24-25 ) we find for the particular case LaTeXMLMath LaTeXMLEquation 5 Obviously LaTeXMLEquation 6 Also we have LaTeXMLEquation 7 The general formula LaTeXMLEquation 8 Applying to the coordinates LaTeXMLMath LaTeXMLEquation or equivalently LaTeXMLEquation .
Following this construction , some remarks are in order : { 1 } A first important remark concerning the obtained algebra ( 17 ) , is that it does not closes as a standard algebra .
This property is easily observed since the extended Moyal bracket of LaTeXMLMath and LaTeXMLMath ; which are coordinates elements of LaTeXMLMath ; gives LaTeXMLMath which is an element of LaTeXMLMath .
However , if we forget about the vector field term LaTeXMLMath in LaTeXMLMath , we recover the standard noncommutative structure ( 1 ) which is a closed algebra .
We can conclude for this point that the fact to transit from prime star product LaTeXMLMath to the operatorial one LaTeXMLMath is equivalent to introduce local vector fields contributions at the level of the deformation parameter LaTeXMLMath which therefore becomes coordinates dependent .
{ 2 } Related to { 1 } , we can also check that the associativity with respect to the operatorial LaTeXMLMath -product operation is not satisfied .
As an example consider LaTeXMLEquation where the differential operator LaTeXMLMath is just the result of LaTeXMLMath .
Then we can easily check that LaTeXMLMath as shown in the formulas ( 24-25 ) .
This property of non associativity of the operatorial star product exhibits a particular interest .
In fact it makes us recall the non associative algebra based on results about open string correlation functions proposed in [ 4a ] and which deal with D-branes in a background with non-vanishing H. { 3 } Concerning the mentioned properties { 1-2 } , the problem of closure of the derived algebra ( 17 ) can be approached by using the analogy with the non-linear Zamolodchikov LaTeXMLMath -algebra which exhibits a similar property .
Namely the non-closure of the algebra due to the presence of the spin-4 term in the commutation relations of LaTeXMLMath currents .
For a review see [ 11 ] .
{ 4 } The noncommutative extended parameter LaTeXMLMath is not a constant object contrary to LaTeXMLMath and thus the associated algebra ( 17 ) is not a trivial structure as it corresponds to a noncommutative deformation of the standard algebra ( 1 ) by the vector fields LaTeXMLMath .
{ 5 } Using the derived relation ( 26 ) , we can easily show that the non constant deformation parameter LaTeXMLMath is given by LaTeXMLMath .
{ 6 } LaTeXMLMath as well as the antisymmetric tensor LaTeXMLMath are objects of conformal weights LaTeXMLMath , since LaTeXMLMath .
{ 7 } From the mathematical point of view , LaTeXMLMath given in ( 21-22 ) are general objects which belong to the subspaces LaTeXMLEquation and LaTeXMLMath given in ( 18 ) is nothing but the first contribution for LaTeXMLMath and consequently is an object of LaTeXMLMath .
{ 8 } We easily obtain the standard noncommutative algebra ( 1 ) from ( 17 ) just by considering the following quotient space LaTeXMLEquation which consist simply on forgetting about vector fields LaTeXMLMath .
References A. Connes , M.R .
Douglas , A. Schwarz , Noncommutative geometry and matrix theory : Compactification on tori , JHEP 02 ( 1998 ) 003 , [ hep-th/9711162 ] and references there in .
N. Seiberg , E. Witten , String Theory and Noncommutative geometry , JHEP 09 ( 1999 ) 032 , [ hep-th/9908142 ] and references there in .
C.S.Chu , P.M.Ho , Noncommutative open string and D-brane , Nucl .
Phys.B550 , 151 ( 1999 ) , [ hep-th/9812219 ] M.R .
Douglas , C. Hull , D-branes and the noncommutative Torus , JHEP 02 ( 1998 ) 008 , [ hep-th/9711165 ] B. Morariu , B. Zumino , in Relativity , Particle Physics and Cosmology , World Scientific , Singapore , 1998 , hep-th/9807198 ] W. Taylor , D-brane field theory on compact spaces , Phys .
Lett .
B394 , 283 ( 1997 ) , [ hep-th/9611042 ] .
Y.K.E .
Cheung and M. Krogh , Noncommutative geometry from 0-branes in a background B field , Nucl .
Phys .
B 528 ( 1998 ) 185 , [ hep-th/9803031 ] ; F. Ardalan , H. Arfaei and M.M.Seikh-Jabbari , Noncommutative geometry from strings and branes , JHEP 02 ( 1999 ) 016 , [ hep-th/9810072 ] ; M.M.Seikh-Jabbari , Open strings in a B field background as electric dipole , Phys .
Lett .
B 455 ( 1999 ) 129 , [ hep-th/9901080 ] ; M.M.Seikh-Jabbari , One Loop Renormalizability of Supersymmetric Yang-Mills Theories on Noncommutative Two-Torus , JHEP 06 ( 1999 ) 015 , [ hep-th/9903107 ] ; V. Schomerus , D-branes and deformation quantization , JHEP , 06 ( 1999 ) 030 , [ hep-th/9903205 ] ; D. Bigatti and L. Susskind , Magnetic fields , branes and noncommutative geometry , Phys.Rev .
D62 ( 2000 ) 066004 , [ hep-th/9908056 ] .
L. Cornalba and R. Schiappa , Non associative star product deformations for D-brane world volume in curved backgrounds , [ hep-th/0101219 ] P.M. Ho , Y-T. Yeh , Noncommutative D-brane in nonconstant NS-NS B field background , [ hep-th/0005159 ] , Phys .
Rev .
Lett .
85 , 5523 ( 2000 ) P.M. Ho , Making non associative algebra associative , [ hep-th/0103024 ] .
M. Aganagic , R. Gopakumar , S. Minwalla , A. Strominger , Unstable Solitons in Noncommutative Gauge Theory , JHEP 0104 ( 2001 ) 001 , [ hep-th/0103256 ] , R. Gopakumar , J. Maldacena , S. Minwalla , A. Strominger , S-Duality and Noncommutative Gauge Theory , JHEP 0006 ( 2000 ) 036 , [ hep-th/0005048 ] .
B. Jurco , P. Schupp , J. Wess , Nonabelian noncommutative gauge theory via noncommutative extra dimensions [ hep-th/0102129 ] , B. Jurco , P. Schupp , J. Wess , Nonabelian noncommutative gauge fields and Seiberg-Witten map , [ hep-th/0012225 ] , B. Jurco , P. Schupp , J. Wess , Noncommutative gauge theory for Poisson manifolds , [ hep-th/0005005 ] , A. Micu and M.M.Seikh-Jabbari , Noncommutative LaTeXMLMath Theory at Two Loops , JHEP 01 ( 2001 ) 025 , [ hep-th/0008057 ] and references there in L. Bonora , M. Schnabl and A. Tomasiello , A note on consistent anomalies in noncommutative YM theories , [ hep-th/0002210 ] .
L. Bonora , M. Schnabl , M.M.Seikh-Jabbari , A. Tomasiello , Noncommutative LaTeXMLMath and LaTeXMLMath Gauge theories , [ hep-th/0006091 ] .
M. Chaichian , M.M.Seikh-Jabbari , A. Tureanu , Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED , Phys .
Rev .
Lett .
86 ( 2001 ) 2716 , [ hep-th/0010175 ] ; M. Chaichian , A. Demichev , P. Presnajder , M.M.Seikh-Jabbari , A. Tureanu , Quantum Theories on Noncommutative Spaces with Nontrivial Topology : Aharonov-Bohm and Casimir Effects , [ hep-th/0101209 ] .
M. Kontsevitch , Deformation quantization of Poisson manifolds I , [ q-alg/9709040 ] D.B .
Fairlie , Moyal Brackets , Star Products and the Generalized Wigner Function , [ hep-th/9806198 ] D.B .
Fairlie , Moyal Brackets in M-Theory , Mod.Phys.Lett .
A13 ( 1998 ) 263-274 , [ hep-th/9707190 ] , C. Zachos , A Survey of Star Product Geometry , [ hep-th/0008010 ] , C. Zachos , Geometrical Evaluation of Star Products , J.Math.Phys .
41 ( 2000 ) 5129-5134 , [ hep-th/9912238 ] , C. Zachos , T. Curtright , Phase-space Quantization of Field Theory , Prog.Theor .
Phys .
Suppl .
135 ( 1999 ) 244-258 , [ hep-th/9903254 ] , M.Bennai , M. Hssaini , B. Maroufi and M.B.Sedra , On the Fairlie ’ s Moyal formulation of M ( atrix ) - theory , Int .
Journal of Modern Physics A V16 ( 2001 ) , [ hep-th/0007155 ] .
E.H.Saidi , M.B.Sedra , On the Gelfand-Dickey algebra LaTeXMLMath and the Wn-symmetry .
I .
The bosonic case , Jour .
Math .
Phys .
V35N6 ( 1994 ) 3190 M.B.Sedra , On the huge Lie superalgebra of pseudo-superdifferential operators and super KP-hierarchies , Jour .
Math .
Phys .
V37 ( 1996 ) 3483 .
A .
B. Zamolodchikov , Infinite additional symmetries in two dimensional conformal quantum field theory , Teor .
Math .
Fiz .
65 , 347 ( 1985 )
Introduction : Let LaTeXMLMath denote a submanifold of LaTeXMLMath of codimension LaTeXMLMath .
Let LaTeXMLMath denote a restriction operator LaTeXMLEquation .
We wish to find an optimal range of exponents LaTeXMLMath such that LaTeXMLEquation where LaTeXMLMath is a compactly supported measure on LaTeXMLMath .
When LaTeXMLMath is a codimension one surface in LaTeXMLMath with non-vanishing Gaussian curvature , the estimate LaTeXMLMath is well understood .
A celebrated result due to Stein and Tomas says that if LaTeXMLMath has non-vanishing Gaussian curvature , then the estimate LaTeXMLMath holds with LaTeXMLMath .
In higher dimensions the situation is more complicated .
When LaTeXMLMath is a codimension two surface in LaTeXMLMath satisfying a non-degeneracy assumption , the estimate LaTeXMLMath holds with LaTeXMLMath .
( See LaTeXMLCite and Theorem A in the Section 2 below ) .
However , a sharp necessary and sufficient condition is not currently available .
It should also be noted that even in codimension one , the more general LaTeXMLMath estimates for the restriction operator are not fully understood , except in the dimension two .
( See LaTeXMLCite for a detailed discussion ) .
We shall not address this issue here .
The purpose of this paper is to establish the estimate LaTeXMLMath in the case when LaTeXMLEquation where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Let LaTeXMLMath denote the Fourier transform of LaTeXMLMath .
By a theorem of Greenleaf ( see LaTeXMLCite ) , the inequality LaTeXMLMath holds for LaTeXMLMath if LaTeXMLEquation .
We shall see below that isotropic Fourier transform estimates do not yield the sharp restriction theorem in codimension two or higher ( see e.g LaTeXMLCite ) .
It should be noted that even in codimension one , it is not known whether the exponent given by Greenleaf ’ s theorem is sharp .
The best possible isotropic rate of decay of LaTeXMLMath for the homogeneous manifold LaTeXMLMath defined above is LaTeXMLMath .
An application of Greenleaf ’ s theorem yields the estimate LaTeXMLMath with LaTeXMLEquation where LaTeXMLMath .
However , the following homogeneity argument due to Knapp suggests that the optimal exponent for the estimate LaTeXMLMath is LaTeXMLEquation .
Indeed , let LaTeXMLMath denote the restriction operator defined above .
Let LaTeXMLMath , where LaTeXMLMath is the characteristic function of a rectangle in LaTeXMLMath with sides of lengths LaTeXMLMath , LaTeXMLMath large .
Then LaTeXMLEquation .
Hence LaTeXMLMath can only hold if LaTeXMLMath .
We will establish the estimate LaTeXMLMath for a homogeneous manifold LaTeXMLMath , with the exponent LaTeXMLMath given by Knapp ’ s homogeneity argument , under a variety of conditions on the level sets of the graphing functions LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath .
Our main results are the following .
The first result gives us a good description of LaTeXMLMath restriction theorems for two-dimensional submanifolds in codimension LaTeXMLMath given as graphs of homogeneous polynomials .
( Please see Definition 2.1 and 2.2 below for the precise description of finite type .
Please see Definition 2.8 below for the description of the order of vanishing along a line . )
( See Theorem 2.9 in Section 2 ) .
Let LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath are homogeneous polynomials of degree LaTeXMLMath and LaTeXMLMath respectively , LaTeXMLMath .
Suppose that there exists a non-zero constant LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , where LaTeXMLMath denotes the determinant of the hessian matrix of LaTeXMLMath .
Suppose that the curve LaTeXMLMath is of finite type LaTeXMLMath at each point of LaTeXMLMath and that LaTeXMLMath vanishes of order LaTeXMLMath along the lines contained in LaTeXMLMath .
Then LaTeXMLMath holds for every LaTeXMLMath , where LaTeXMLMath is the sharp exponent LaTeXMLEquation if LaTeXMLMath .
Furthermore LaTeXMLMath holds with The following result answers a question posed by Fulvio Ricci about the restriction theorems for manifolds given as graphs of quadratic monomials .
The proof relies on the precise asymptotics of the Fourier transforms of certain distributions obtained by Shintani ( see LaTeXMLCite , and Lemma 3.3 below ) .
It has been brought to our attention that this result is implied by a theorem announced in the Bulletin of the AMS by Gerd Mockenhaupt ( see LaTeXMLCite ) .
We enclose the proof for reader ’ s convenience , and to motivate the related results proved in this paper .
( See e.g .
Theorem III below ) .
( See Theorem 3.1 below ) .
Let LaTeXMLMath , where LaTeXMLMath , and the LaTeXMLMath denote the distinct monomials of degree LaTeXMLMath .
Then the estimate LaTeXMLMath holds with the sharp exponent LaTeXMLMath .
The following result generalizes Theorem II to manifolds given as joint graphs of smooth functions of higher order of homogeneity .
The proof relies on the non-isotropic decay estimates for the associated Fourier transform of the surface carried measure .
The observation that the non-isotropic decay estimates are useful to obtain sharp restriction theorems in codimension LaTeXMLMath is not new .
See for example the work of M.Christ ( LaTeXMLCite ) and E.Prestini ( LaTeXMLCite ) .
( See Theorem 3.2 below ) .
Let LaTeXMLMath denote a compact piece of the manifold LaTeXMLMath , where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that no linear combination of the LaTeXMLMath ’ s vanishes on a subset of postive measure of LaTeXMLMath .
Let LaTeXMLMath .
Suppose that LaTeXMLMath , LaTeXMLMath .
Then the estimate LaTeXMLMath holds with the sharp exponent LaTeXMLEquation .
Many of our results are based on the non-isotropic decay estimates for the associated Fourier transform of the surface carried measure .
A sample result is the following .
( See Theorem 1.1B below ) Let LaTeXMLMath , where each LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLEquation where LaTeXMLMath is a smooth cutoff function .
Then LaTeXMLEquation .
This paper is organized as follows .
In Section 1 we will prove some estimates related to the decay of the Fourier transform of the surface carried measure .
In particular , we will estimate the decay of the Fourier transform of the surface carried measure , in any codimension , in terms of the integrability of the multiplicative inverses of the graphing functions LaTeXMLMath restricted to the unit sphere .
Using this technique we shall also obtain an accurate non-isotropic estimate for the Fourier transform of the surface carried measure in the case when every graphing function has the same homogeneity .
In Section 2 we will apply the results of Section 1 along with the results of M. Christ ( LaTeXMLCite ) , E. Prestini ( LaTeXMLCite ) , and a variety of scaling arguments to obtain a sharp estimate LaTeXMLMath with the exponent LaTeXMLMath given by LaTeXMLMath .
In Section 3 we will use the non-isotropic estimates from Section 1 to study restriction theorems in the case when every graphing function is homogeneous of the same degree LaTeXMLMath .
We will need the following definitions .
( See e.g .
LaTeXMLCite , LaTeXMLCite ) .
Nonvanishing Gaussian curvature : Let LaTeXMLMath be a submanifold of LaTeXMLMath of codimension LaTeXMLMath equipped with a smooth compactly supported measure LaTeXMLMath .
Let LaTeXMLMath be the usual Gauss map taking each point on LaTeXMLMath to the outward unit normal at that point .
We say that LaTeXMLMath has everywhere nonvanishing Gaussian curvature if the differential of the Gauss map LaTeXMLMath is always nonsingular .
Strong curvature condition : Let LaTeXMLMath be a submanifold of LaTeXMLMath of codimension LaTeXMLMath equipped with a smooth compactly supported measure LaTeXMLMath .
Suppose that LaTeXMLMath is a joint graph of smooth functions LaTeXMLMath , LaTeXMLMath , … , LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath denote the LaTeXMLMath -dimensional space of normals to LaTeXMLMath at a point LaTeXMLMath .
We say that LaTeXMLMath satisfies the strong curvature condition ( SCC ) if for all LaTeXMLMath in some neighborhood of the support of LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath denotes the Hessian matrix .
N-curvature condition : Let LaTeXMLMath be defined as above .
We say that LaTeXMLMath satisfies the LaTeXMLMath -curvature condition if the rank of the Hessian matrix in LaTeXMLMath is greater than or equal to LaTeXMLMath everywhere .
Our main results are the following .
Let LaTeXMLMath , where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that no LaTeXMLMath is a constant multiple of any LaTeXMLMath for LaTeXMLMath .
Suppose that LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath denote the number of distinct LaTeXMLMath ’ s .
Suppose that LaTeXMLMath for each LaTeXMLMath .
Let LaTeXMLMath , and define LaTeXMLEquation where LaTeXMLMath is a smooth cutoff function .
Then LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLMath , where each LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLMath be defined as in LaTeXMLMath .
Then LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath be defined as above .
Let LaTeXMLEquation where LaTeXMLMath is a smooth cutoff function supported away from the origin , and each LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that LaTeXMLEquation .
Then LaTeXMLEquation if LaTeXMLMath .
Let LaTeXMLMath be defined as above .
Suppose that LaTeXMLMath , LaTeXMLMath , is an LaTeXMLMath -dimensional submanifold of codimension LaTeXMLMath , of the hyperplane LaTeXMLEquation satisfying the LaTeXMLMath -curvature condition .
Let LaTeXMLMath be defined as in LaTeXMLMath .
Then LaTeXMLEquation .
Let LaTeXMLMath be defined as above with LaTeXMLMath .
Suppose that there exists a constant LaTeXMLMath , such that LaTeXMLMath .
Suppose that LaTeXMLMath .
Suppose that LaTeXMLMath , the Hessian matrix of LaTeXMLMath , has rank LaTeXMLMath away from the origin .
Suppose that LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be defined as in LaTeXMLMath .
Then LaTeXMLEquation .
Let LaTeXMLMath be defined as in lemma LaTeXMLMath , with LaTeXMLMath .
Then the assumptions of Theorem B ( see Section 2 ) are satisfied at every LaTeXMLMath iff LaTeXMLMath , LaTeXMLMath , and the level set LaTeXMLMath has non-vanishing Gaussian curvature .
We shall prove the theorem under the assumption that all the LaTeXMLMath ’ s are distinct .
The general statement follows by combining the terms with the same homogeneity .
Recall that LaTeXMLEquation where without loss of generality LaTeXMLMath is radial , and each LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLEquation .
Taking absolute values inside the integral we see that LaTeXMLEquation which by Chebyshev ’ s inequality is bounded by LaTeXMLEquation for LaTeXMLMath large .
Let LaTeXMLMath be supported in the interval LaTeXMLMath , such that LaTeXMLMath inside LaTeXMLMath , and LaTeXMLEquation .
Let LaTeXMLEquation .
A change of variables sending LaTeXMLMath shows that LaTeXMLEquation .
Let LaTeXMLMath .
We must estimate LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation To estimate each LaTeXMLMath we shall need the fact that the curve LaTeXMLMath has non-vanishing curvature and torsion away from the origin , so long as all the LaTeXMLMath ’ s are distinct .
An elementary van der Corput type estimate shows that the Fourier transform of the measure carried by this curve decays of order LaTeXMLMath .
We shall also use the fact that LaTeXMLMath is bounded .
More precisely , LaTeXMLEquation .
LaTeXMLEquation provided that LaTeXMLMath , for each LaTeXMLMath .
Moreover , LaTeXMLEquation .
LaTeXMLEquation since LaTeXMLMath .
Hence , the expression LaTeXMLMath is bounded by LaTeXMLEquation for LaTeXMLMath large .
This completes the proof if LaTeXMLMath .
However , if this is not the case , the gradient of the phase function LaTeXMLMath is bounded away from zero , and an integration by parts argument ( see LaTeXMLCite , p.364 ) shows that LaTeXMLMath , for any LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLMath .
We rewrite LaTeXMLMath in the form LaTeXMLEquation where LaTeXMLMath is a smooth cutoff function .
Let LaTeXMLMath .
Let LaTeXMLEquation .
Taking the absolute values inside the integral , we see that LaTeXMLEquation which by Chebyshev ’ s inequality is bounded by LaTeXMLEquation .
Let LaTeXMLMath be supported in the interval LaTeXMLMath , such that LaTeXMLMath inside LaTeXMLMath , and LaTeXMLEquation .
Let LaTeXMLEquation .
A change of variables sending LaTeXMLMath shows that LaTeXMLMath We must estimate LaTeXMLEquation .
To estimate LaTeXMLMath we shall use the fact that away from zero the Fourier transform of the measure supported on the curve LaTeXMLMath decays of order LaTeXMLMath .
To estimate LaTeXMLMath we shall just use the fact that LaTeXMLMath is bounded .
More precisely , LaTeXMLEquation and LaTeXMLEquation as long as LaTeXMLMath .
This completes the proof if LaTeXMLMath .
However , if this is not the case , the gradient of the phase function LaTeXMLMath is bounded away from the origin , and an integration by parts argument ( see LaTeXMLCite , p.364 ) shows that LaTeXMLMath , for any LaTeXMLMath .
Let LaTeXMLEquation .
A change of variables sending LaTeXMLMath shows that LaTeXMLEquation .
Let LaTeXMLMath .
We must estimate LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation Using the assumed decay of LaTeXMLMath , and the fact that , in particular , LaTeXMLMath is bounded , we get LaTeXMLEquation .
LaTeXMLEquation as long as LaTeXMLMath .
This completes the proof if LaTeXMLMath .
However , if this is not the case , the gradient of the phase function LaTeXMLMath is bounded away from the origin .
An integration by parts argument , ( see LaTeXMLCite , p.364 ) , shows that LaTeXMLMath , for any LaTeXMLMath .
Assume that LaTeXMLMath , LaTeXMLMath .
In what follows we will denote LaTeXMLMath by LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath and by LaTeXMLMath ( resp .
LaTeXMLMath ) and LaTeXMLMath ( resp .
LaTeXMLMath ) the Jacobian of a function LaTeXMLMath and the Hessian matrix of a function LaTeXMLMath computed with respect to the LaTeXMLMath ( resp .
LaTeXMLMath ) variables .
Let LaTeXMLMath .
Take LaTeXMLMath on the support of LaTeXMLMath such that LaTeXMLMath Since LaTeXMLMath is by assumption a submanifold of codimension LaTeXMLMath of the hyperplane LaTeXMLMath , the Jacobian of the function LaTeXMLMath , LaTeXMLMath , has rank LaTeXMLMath at LaTeXMLMath .
There is no loss of generality if we assume that LaTeXMLMath is the identity in the space of LaTeXMLMath matrices , and that LaTeXMLMath By the implicit function theorem there exists a smooth function LaTeXMLMath such that LaTeXMLMath in a neighborhood of LaTeXMLMath .
An application of the chain rule yields : ¿From ( 2 ) we have that , for every LaTeXMLMath , the Hessian matrix of LaTeXMLMath at LaTeXMLMath , LaTeXMLMath , is LaTeXMLMath , and from ( 3 ) that LaTeXMLMath can be written as a linear combination of the Hessian matrices of the functions LaTeXMLMath and of the Hessian matrix of LaTeXMLMath at LaTeXMLMath .
Let LaTeXMLMath denote the phase function of LaTeXMLMath as in ( 1.2 ) .
By the above remark , LaTeXMLMath can be written as a linear combination of the Hessian matrices of the functions LaTeXMLMath and the function LaTeXMLMath at LaTeXMLMath .
Since LaTeXMLMath satisfies the LaTeXMLMath -curvature condition , the rank of every linear combination of the above matrices is LaTeXMLMath for every LaTeXMLMath .
This shows that the rank of the Hessian matrix of LaTeXMLMath is LaTeXMLMath and hence that ( 1.9 ) holds .
A theorem of Littman ( LaTeXMLCite ) says that if a surface in codimension one has at least LaTeXMLMath non-vanishing principal curvatures , then the Fourier transform of the surface carried measure decays of order LaTeXMLMath .
The proof of that theorem shows that LaTeXMLMath has the required decay if the rank of the Hessian matrix of the phase function LaTeXMLMath is LaTeXMLMath for every LaTeXMLMath on the support of LaTeXMLMath and for every LaTeXMLMath .
We observe now that LaTeXMLMath .
Indeed , if LaTeXMLMath , then LaTeXMLMath , and since LaTeXMLMath , then LaTeXMLMath .
Then LaTeXMLMath for every LaTeXMLMath such that LaTeXMLMath .
But the argument that we have just used shows that LaTeXMLMath iff LaTeXMLMath .
Hence LaTeXMLMath for every LaTeXMLMath .
By the chain rule , LaTeXMLEquation for LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be a point on the support of LaTeXMLMath .
If we set LaTeXMLMath , we observe that LaTeXMLMath equals a matrix whose rank is LaTeXMLMath if and only if LaTeXMLMath , LaTeXMLMath , and zero elsewhere .
Since we assumed that LaTeXMLMath doesn ’ t vanish away from the origin , Euler ’ s homogeneity relations guarantee that LaTeXMLMath doesn ’ t vanish .
Consequently , the rank of LaTeXMLMath is at most LaTeXMLMath .
To show that LaTeXMLMath can not be zero , we observe that if this were the case we would have LaTeXMLEquation .
The coefficient which multiplies LaTeXMLMath can not be zero because the matrix on the right-hand side has rank one .
On the other hand , the matrix on the left-hand side has rank LaTeXMLMath by assumption , hence the equality in LaTeXMLMath can never hold .
This concludes the proof of the theorem .
After perhaps rotating and dilating the coordinates , we can work in a neighborhood of the point LaTeXMLMath .
Since there exists a constant LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath .
( See proof of Lemma 1.4 ) .
By the chain rule LaTeXMLMath LaTeXMLEquation for LaTeXMLMath .
Observe that by Euler ’ s homogeneity relations In order to show that the sufficient condition of Theorem 1.1B is verified , we must show that the determinant of the matrix LaTeXMLMath whose rows are LaTeXMLMath LaTeXMLMath and LaTeXMLMath is not a square .
A direct computations shows that the discriminant of the determinant of LaTeXMLMath is LaTeXMLEquation which doesn ’ t vanish by the assumptions on LaTeXMLMath .
Hence , LaTeXMLMath is integrable for LaTeXMLMath .
This concludes the proof of the lemma .
We will need the following results .
Let LaTeXMLEquation where LaTeXMLMath .
Suppose that LaTeXMLMath holds with LaTeXMLMath .
Then for every LaTeXMLMath and every LaTeXMLMath , the expression LaTeXMLEquation does not vanish .
Let LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLMath denote the quadratic part of the Taylor expansion of LaTeXMLMath .
Suppose that the vectors LaTeXMLMath span LaTeXMLMath .
Let LaTeXMLMath denote the determinant of the matrix LaTeXMLMath .
Suppose that LaTeXMLMath , for any LaTeXMLMath .
Then LaTeXMLMath holds with LaTeXMLMath .
Note that when LaTeXMLMath , the assumptions of Theorem B are equivalent to the necessary condition LaTeXMLMath in Theorem A .
In particular , the conditions of Theorem B are necessary and sufficient in that case .
Before stating our main results , we need to introduce the following definitions .
Let LaTeXMLMath , where LaTeXMLMath is a compact interval in LaTeXMLMath , and LaTeXMLMath is smooth .
We say that LaTeXMLMath is finite type if LaTeXMLMath does not vanish of infinite order for any LaTeXMLMath , and any unit vector LaTeXMLMath .
We will also need a more precise definition to specify the order of vanishing at each point .
Let LaTeXMLMath denote a point in the compact interval LaTeXMLMath .
We can always find a smooth function LaTeXMLMath , such that in a small neighborhood of LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath .
Let LaTeXMLMath be defined as before .
Let LaTeXMLMath in a small neighborhood of LaTeXMLMath .
We say that LaTeXMLMath is finite type LaTeXMLMath at LaTeXMLMath if LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath .
Our main results are the following .
Let LaTeXMLMath , where each LaTeXMLMath , homogeneous of degree LaTeXMLMath .
Let LaTeXMLMath denote a compactly supported smooth measure on LaTeXMLMath , and let LaTeXMLMath , where LaTeXMLMath is a smooth cutoff function supported away from the origin .
Let LaTeXMLMath and let LaTeXMLMath .
Suppose that LaTeXMLMath is a bounded operator .
Then LaTeXMLMath is a bounded operator , where LaTeXMLEquation as long as LaTeXMLMath .
Let LaTeXMLEquation where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that LaTeXMLMath satisfies the assumptions of Theorem LaTeXMLMath A .
Then the estimate LaTeXMLMath holds with LaTeXMLEquation .
Let LaTeXMLEquation where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that LaTeXMLMath satisfies the assumptions of the Lemma 1.3 .
Then the estimate LaTeXMLMath holds with the sharp exponent LaTeXMLEquation provided that LaTeXMLMath .
Let LaTeXMLEquation where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that there exists a non-zero constant LaTeXMLMath , such that LaTeXMLEquation .
Suppose that LaTeXMLMath , LaTeXMLMath .
If the rank of the Hessian matrix of LaTeXMLMath is LaTeXMLMath , then LaTeXMLMath holds with LaTeXMLMath , provided that LaTeXMLMath .
In order to introduce the Theorem 2.9 below , we need the following result which was stated and proved by the second author in ( LaTeXMLCite ) .
Let LaTeXMLMath .
Let LaTeXMLMath .
Let LaTeXMLMath where LaTeXMLMath denotes the determinant of the Hessian matrix of LaTeXMLMath .
Then for each LaTeXMLMath , LaTeXMLMath , where each LaTeXMLMath is a line through the origin , and LaTeXMLMath .
Moreover , LaTeXMLMath .
Let LaTeXMLMath denote the partial derivative of LaTeXMLMath with respect to LaTeXMLMath .
Since LaTeXMLMath is homogeneous of degree LaTeXMLMath , LaTeXMLMath is homogeneous of degree LaTeXMLMath , and LaTeXMLMath is homogeneous of degree LaTeXMLMath .
By homogeneity , if LaTeXMLMath contains a point LaTeXMLMath , it also contains a line through the origin containing that point .
Since LaTeXMLMath is a polynomial , there can be at most a finite number of such lines .
This proves the first assertion of the lemma .
By the Euler homogeneity relations , LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation where the LaTeXMLMath denote the second partial derivatives .
Hence , LaTeXMLMath .
If we write the equations for LaTeXMLMath and LaTeXMLMath in matrix form we see that LaTeXMLMath is obtained by applying the Hessian matrix of LaTeXMLMath to the vector LaTeXMLMath .
Hence , LaTeXMLMath .
Putting these observations together we see that LaTeXMLMath .
Suppose that both LaTeXMLMath and LaTeXMLMath vanish along a line through the origin , which without loss of generality we take to be the LaTeXMLMath -axis .
Then LaTeXMLMath .
This implies that LaTeXMLMath .
Also , LaTeXMLMath .
This implies that LaTeXMLMath .
LaTeXMLMath .
By assumption , LaTeXMLEquation .
Since LaTeXMLMath , we must conclude that LaTeXMLMath which implies that LaTeXMLMath .
This proves that LaTeXMLMath and hence that LaTeXMLMath .
This completes the proof of the lemma .
We shall need the following definition : Definition 2.8 Let LaTeXMLMath .
We say that LaTeXMLMath vanishes of order LaTeXMLMath along the line LaTeXMLMath if LaTeXMLMath is the largest positive integer so that LaTeXMLMath , where LaTeXMLMath is allowed to vanish only at the origin .
Let LaTeXMLMath where LaTeXMLMath , LaTeXMLMath are homogeneous polynomials of degree LaTeXMLMath and LaTeXMLMath respectively , LaTeXMLMath .
Suppose that there exists a non-zero constant LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath be defined as in Lemma 2.7 with respect to LaTeXMLMath .
Suppose that the curve LaTeXMLMath is of finite type LaTeXMLMath at each point of LaTeXMLMath and that LaTeXMLMath vanishes of order LaTeXMLMath along the lines contained in LaTeXMLMath .
Then LaTeXMLMath holds for every LaTeXMLMath , where LaTeXMLMath is the sharp exponent LaTeXMLEquation if LaTeXMLMath .
Furthermore LaTeXMLMath holds with Let LaTeXMLEquation where LaTeXMLMath is a cutoff function .
Let LaTeXMLMath be a cutoff function supported in the interval LaTeXMLMath such that LaTeXMLMath for every LaTeXMLMath , and let LaTeXMLEquation .
If we make the change of variables sending LaTeXMLMath we can write : LaTeXMLEquation .
Let LaTeXMLMath denote the nonisotropic dilation LaTeXMLEquation .
Then LaTeXMLEquation .
A change of variables shows that LaTeXMLMath It follows that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The series LaTeXMLMath converges , provided that LaTeXMLEquation which yields LaTeXMLEquation .
This concludes the proof of the lemma .
The application of Greenleaf ’ s theorem ( see ( 0.3 ) above ) yields the result .
Let LaTeXMLMath be defined as in LaTeXMLMath .
By Lemma 1.3 LaTeXMLEquation .
By a theorem of Greenleaf , ( see ( 0.3 ) above ) , the inequality LaTeXMLMath holds with LaTeXMLMath .
An application of Lemma 2.3 completes the proof .
Let LaTeXMLMath be defined as in LaTeXMLMath .
By Lemma 1.4 LaTeXMLEquation .
Applying Greenleaf ’ s theorem as above we get LaTeXMLMath .
An application of Lemma 2.3 completes the proof .
Let LaTeXMLMath be the level set LaTeXMLMath , and let LaTeXMLEquation where LaTeXMLMath is a smooth cutoff function .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be defined as in Lemma LaTeXMLMath with respect to LaTeXMLMath .
Recall that LaTeXMLMath is the union of a finite number of lines through the origin .
Let LaTeXMLMath and LaTeXMLMath be two finite families of cones in LaTeXMLMath with the following properties : i ) Each LaTeXMLMath contains exactly one line of LaTeXMLMath , and each LaTeXMLMath contains exactly one line of LaTeXMLMath ii ) LaTeXMLMath if LaTeXMLMath , LaTeXMLMath if LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLMath be the characteristic function of LaTeXMLMath and let LaTeXMLMath be the characteristic function of LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We first consider LaTeXMLEquation where we have set LaTeXMLMath .
On the support of LaTeXMLMath the curvature of LaTeXMLMath never vanishes and LaTeXMLMath vanishes only at zero .
We recall that by the Stein-Tomas observation , ( 0.2 ) is equivalent to the inequality LaTeXMLEquation .
Let LaTeXMLMath be a smooth cutoff function supported in the interval LaTeXMLMath , such that LaTeXMLMath .
Let LaTeXMLEquation .
The assumptions of Lemma 1.5 are satisfied on the support of LaTeXMLMath , hence the inequality LaTeXMLMath holds for the measure LaTeXMLMath with LaTeXMLMath .
Since the sharp exponent LaTeXMLMath can not exceed LaTeXMLMath , the estimate LaTeXMLMath holds for LaTeXMLMath , provided that LaTeXMLMath .
If we make the change of variables sending LaTeXMLMath , and if we observe that LaTeXMLMath is invariant with respect to dilations , we see that LaTeXMLEquation .
LaTeXMLEquation Without loss of generality we can replace LaTeXMLMath by a function LaTeXMLMath , homogeneous of degree zero , whose support coincides with the support of LaTeXMLMath .
Let LaTeXMLMath denote the nonisotropic dilation LaTeXMLEquation .
Then LaTeXMLEquation .
A change of variables shows that LaTeXMLMath It follows that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The series LaTeXMLMath converges , provided that LaTeXMLEquation which yields LaTeXMLMath .
Hence the measure LaTeXMLMath satisfies the inequality LaTeXMLMath with LaTeXMLMath .
We consider now LaTeXMLMath .
Let LaTeXMLEquation where LaTeXMLMath is fixed .
In the proof of lemma 1.4 we observed that LaTeXMLMath implies that LaTeXMLMath for every LaTeXMLMath .
Observe that LaTeXMLMath is star-shaped with respect to the origin , because if LaTeXMLMath , then for every LaTeXMLMath , LaTeXMLMath .
In the polar coordinates associated to LaTeXMLMath , LaTeXMLEquation .
Without loss generality LaTeXMLMath is radial .
Consider the ( unique ) point of LaTeXMLMath , which can be taken to be LaTeXMLMath .
Suppose that LaTeXMLMath is supported in a sufficiently small neighborhood of LaTeXMLMath .
Since LaTeXMLMath is finite type LaTeXMLMath , it can be written as the graph of a smooth function LaTeXMLMath , where LaTeXMLMath , and LaTeXMLEquation .
Let LaTeXMLMath be supported in LaTeXMLMath , such that LaTeXMLMath in LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLEquation .
The integral with respect to LaTeXMLMath is supported over a dyadic piece of LaTeXMLMath where the Gaussian curvature does not vanish .
By Lemma 1.5 and the Stein-Thomas observation , the estimate LaTeXMLMath holds for the measure LaTeXMLMath , for LaTeXMLMath , with a constant LaTeXMLMath .
In order to estimate LaTeXMLMath we make a change of variables in the expression for LaTeXMLMath setting LaTeXMLMath .
We have LaTeXMLEquation .
Let LaTeXMLMath be the nonisotropic dilation LaTeXMLMath and let LaTeXMLEquation .
Then LaTeXMLEquation .
A change of variables shows that LaTeXMLMath = LaTeXMLMath , and that LaTeXMLMath for every LaTeXMLMath and LaTeXMLMath .
Then we can write the following string of inequalities : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We must prove that the constants LaTeXMLMath in the above expression are uniformly bounded .
In fact the sum LaTeXMLMath converges if LaTeXMLMath , hence if LaTeXMLMath .
Since LaTeXMLMath when LaTeXMLMath , the estimate LaTeXMLMath holds for the measure LaTeXMLMath , and consequently for the measure LaTeXMLMath , for LaTeXMLMath , provided that LaTeXMLMath , and with LaTeXMLMath if LaTeXMLMath .
¿From the proof of the theorem of Greenleaf it follows that the bounds for the constants LaTeXMLMath depend on a finite number of derivatives of the phase function of LaTeXMLMath , LaTeXMLMath .
Since LaTeXMLMath is a smooth function , then , for LaTeXMLMath large , LaTeXMLMath .
This shows that the constant LaTeXMLMath ’ s are uniformly bounded .
We now consider LaTeXMLMath .
Fix LaTeXMLMath .
After perhaps a rotation of coordinates we may assume that LaTeXMLMath vanishes along the LaTeXMLMath axis .
Then LaTeXMLMath can be written as LaTeXMLMath , where LaTeXMLMath does not vanish on the LaTeXMLMath axis , ( except perhaps at the origin ) , if LaTeXMLMath is small enough .
Let LaTeXMLEquation where we have set LaTeXMLMath .
Let LaTeXMLMath be a cutoff function supported in LaTeXMLMath , such that LaTeXMLMath in LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLEquation .
The above integral is defined over a cone of LaTeXMLMath where the curvature of LaTeXMLMath never vanishes , and LaTeXMLMath vanishes only at the origin .
Let LaTeXMLMath be supported in LaTeXMLMath , such that LaTeXMLMath in LaTeXMLMath , and LaTeXMLMath .
Let LaTeXMLEquation .
The assumptions of Lemma 1.5 are satisfied on the support of LaTeXMLMath , and hence the estimate LaTeXMLMath holds for the measure LaTeXMLMath with LaTeXMLMath .
If we make the change of variables sending LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath denote the nonisotropic dilation LaTeXMLEquation .
Then LaTeXMLEquation .
A change of variables shows that LaTeXMLMath It follows that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The series LaTeXMLMath converges , provided that LaTeXMLEquation which yields LaTeXMLMath .
The above argument shows that we can assume that the measure LaTeXMLMath is supported away from zero .
Hence , by Lemma 1.5 and the above observation , the inequality LaTeXMLMath holds for the measure LaTeXMLMath , for LaTeXMLMath , with a constant LaTeXMLMath .
In order to estimate LaTeXMLMath we perform the change of variables in LaTeXMLMath sending LaTeXMLMath , LaTeXMLMath .
We obtain LaTeXMLEquation where we have set LaTeXMLEquation .
Let LaTeXMLMath be the nonisotropic dilation LaTeXMLMath LaTeXMLMath .
Then LaTeXMLEquation .
A change of variables shows that LaTeXMLEquation and that LaTeXMLMath for every LaTeXMLMath .
Then we can write the following string of inequalities : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
If we show that the constants LaTeXMLMath in the above expression are uniformly bounded then we are done .
In fact the sum LaTeXMLMath converges if LaTeXMLMath hence if LaTeXMLEquation .
Since LaTeXMLMath when LaTeXMLMath , the estimate LaTeXMLMath holds for the measure LaTeXMLMath , and consequently for the measure LaTeXMLMath , with LaTeXMLMath , provided that LaTeXMLMath , and with LaTeXMLMath if LaTeXMLMath .
By a theorem of Greenleaf the bounds for the constants LaTeXMLMath in LaTeXMLMath depend only on a finite number of derivatives of the phase function of LaTeXMLMath , LaTeXMLMath Since the above function is smooth , then , for LaTeXMLMath large , LaTeXMLMath .
This shows that the constants LaTeXMLMath are uniformly bounded , thus concluding the proof of the theorem .
Let LaTeXMLMath , where LaTeXMLMath , and the LaTeXMLMath denote the distinct monomials of degree LaTeXMLMath .
Then the estimate LaTeXMLMath holds with the sharp exponent LaTeXMLMath .
Let LaTeXMLMath denote a compact piece of the manifold LaTeXMLMath , where LaTeXMLMath is homogeneous of degree LaTeXMLMath .
Suppose that no linear combination of the LaTeXMLMath ’ s vanishes on a subset of postive measure of LaTeXMLMath .
Let LaTeXMLMath .
Suppose that LaTeXMLMath , LaTeXMLMath .
Then the estimate LaTeXMLMath holds with the sharp exponent LaTeXMLMath given by LaTeXMLMath .
The restriction LaTeXMLMath in Theorem 3.2 is not necessary .
In fact , using the techniques in ( LaTeXMLCite ) , one can prove Theorem 3.2 under the weaker restriction LaTeXMLMath .
Theorem 3.2 implies the natural generalization of Theorem 3.1 to the case where LaTeXMLMath , LaTeXMLMath , and the LaTeXMLMath are the disitinct monomials of degree LaTeXMLMath .
Let LaTeXMLMath , and let LaTeXMLEquation .
Let LaTeXMLMath be the matrix associated to the quadratic form LaTeXMLMath , and LaTeXMLEquation .
Thus , LaTeXMLMath is the Fourier transform of LaTeXMLMath , and an easy generalization of the well-known formula for the Fourier transform of the Gaussian functions , ( see e.g .
LaTeXMLCite pg .
LaTeXMLMath ) , yields LaTeXMLEquation .
Let LaTeXMLEquation with LaTeXMLEquation where LaTeXMLMath is the standard Gamma function .
Let LaTeXMLMath .
We will prove that LaTeXMLMath is an continuous family of operators when Re LaTeXMLMath , is analytic when Re LaTeXMLMath , and that i ) LaTeXMLMath , when Re LaTeXMLMath , ii ) LaTeXMLMath , when Re LaTeXMLMath , iii ) LaTeXMLMath and LaTeXMLMath have at most exponential growth with respect to Im LaTeXMLMath .
Stein ’ s analytic interpolation theorem ( see e.g .
LaTeXMLCite ) will then imply that LaTeXMLMath is a bounded operator when LaTeXMLMath .
Since LaTeXMLMath , the Stein-Thomas observation ( see the proof of Theorem 2.9 ) implies the conclusion of our theorem .
To prove LaTeXMLMath we observe that when LaTeXMLMath , with LaTeXMLMath , then LaTeXMLMath is bounded by LaTeXMLMath .
By the Hausdorff -Young inequality we have that LaTeXMLMath .
LaTeXMLMath is then satisfied , and one can check , using Stirling ’ s formula , that LaTeXMLMath has at most exponential growth .
To prove LaTeXMLMath it is enough to show that LaTeXMLMath is a bounded function when Re LaTeXMLMath .
To compute the the Fourier transform of LaTeXMLMath with respect to LaTeXMLMath , LaTeXMLMath , we use again the formula LaTeXMLMath obtaining LaTeXMLEquation .
Hence LaTeXMLEquation where we have set LaTeXMLMath .
We recall that the above identities hold in distribution sense .
Since the phase of the above integral is a linear function of LaTeXMLMath , we reduce to computing the Fourier transform of det LaTeXMLMath .
We need the following lemma ( see LaTeXMLCite , pg .
LaTeXMLMath ) .
Let LaTeXMLMath be the space of the real and symmetric matrices and let LaTeXMLMath be the subset of the matrices with LaTeXMLMath positive and LaTeXMLMath negative eigenvalues .
Let LaTeXMLMath be the distribution LaTeXMLEquation where LaTeXMLMath is the standard Euclidean measure on LaTeXMLMath .
Then the distribution LaTeXMLMath , viewed as a function of LaTeXMLMath , has analytic continuation to a meromorphic function in the whole complex plane satisfying LaTeXMLEquation where LaTeXMLMath is as in LaTeXMLMath and the LaTeXMLMath are bounded coefficients .
¿From the above formula we deduce that , modulo bounded constants , LaTeXMLEquation where the above formula holds in distribution sense .
Since Re LaTeXMLMath , the above is a bounded function of LaTeXMLMath .
This shows that LaTeXMLMath is a bounded function of LaTeXMLMath , and completes the proof of the theorem .
Let LaTeXMLEquation where LaTeXMLMath is a smooth cutoff function .
By Theorem LaTeXMLMath we can write LaTeXMLEquation where LaTeXMLMath .
Let LaTeXMLEquation .
LaTeXMLEquation with LaTeXMLEquation where LaTeXMLMath is the standard Gamma function .
Let LaTeXMLMath .
We will prove that LaTeXMLMath is a continous family of operators when Re LaTeXMLMath , is analytic when Re LaTeXMLMath , and that i ) LaTeXMLMath , when Re LaTeXMLMath , ii ) LaTeXMLMath , when Re LaTeXMLMath , iii ) LaTeXMLMath and LaTeXMLMath have at most exponential growth with respect to Im LaTeXMLMath .
Stein ’ s analytic interpolation theorem ( see e.g .
LaTeXMLCite ) will then imply that LaTeXMLMath is a bounded operator when LaTeXMLMath .
Since LaTeXMLMath , Stein-Thomas observation ( see the proof of Theorem 2.9 ) implies the conclusion of our theorem .
To prove the estimate LaTeXMLMath we observe that when LaTeXMLMath , with LaTeXMLMath , then LaTeXMLMath is bounded by LaTeXMLMath .
By the Haussdorf -Young inequality we have that LaTeXMLMath .
The estimate LaTeXMLMath is then satisfied , and one can check using Stirling ’ s formula that LaTeXMLMath has at most exponential growth .
To prove the estimate LaTeXMLMath it is enough to show that LaTeXMLMath is a bounded function when Re LaTeXMLMath .
Since LaTeXMLMath is a finite measure it is enough to prove that LaTeXMLMath is bounded .
Let LaTeXMLMath be the vector defined by the equation LaTeXMLMath .
Then LaTeXMLEquation .
Since LaTeXMLMath is homogeneous of degree zero with respect to LaTeXMLMath , we can assume LaTeXMLMath .
In polar coordinates with respect to LaTeXMLMath , with LaTeXMLMath , we have LaTeXMLEquation .
Let LaTeXMLMath be such that LaTeXMLMath , when LaTeXMLMath , and LaTeXMLMath when LaTeXMLMath .
Let LaTeXMLEquation .
We will prove that LaTeXMLMath is bounded by a constant LaTeXMLMath independent of LaTeXMLMath and that LaTeXMLMath in distribution sense as LaTeXMLMath .
¿From the above it follows that LaTeXMLMath is bounded .
Indeed , since the balls are sequentially compact in the weak LaTeXMLMath topology in LaTeXMLMath , there exists a sequence LaTeXMLMath which converges to a bounded function in the weak LaTeXMLMath topology of LaTeXMLMath , and hence converges also in distribution sense .
Consequently , LaTeXMLMath , which is a bounded function .
Recalling that by assumption the vector LaTeXMLMath is never zero on LaTeXMLMath , we can construct an orthogonal matrix LaTeXMLMath with the property that LaTeXMLMath .
The first row of LaTeXMLMath is LaTeXMLMath and the other rows are a set of LaTeXMLMath vectors which , togeter with LaTeXMLMath , determine an orthonormal basis of LaTeXMLMath for every LaTeXMLMath .
We make the change of variables LaTeXMLMath in the expression for LaTeXMLMath .
Since LaTeXMLMath is orthogonal , the change of variables maps LaTeXMLMath into itself and the determinant of the Jacobian matrix of the transformation is LaTeXMLMath .
We obtain LaTeXMLEquation .
LaTeXMLEquation The integral with respect to LaTeXMLMath is a continous function of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , say LaTeXMLMath .
In particular LaTeXMLMath Since Re LaTeXMLMath by assumption , we can check that we can compute at least LaTeXMLMath derivatives of LaTeXMLMath with respect to LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation Here we set LaTeXMLMath , and we let LaTeXMLMath be the measure on the LaTeXMLMath -dimensional sphere LaTeXMLMath .
If we make a change of variables in the above integral letting LaTeXMLMath , we get LaTeXMLEquation .
LaTeXMLEquation We recall that the distribution LaTeXMLMath is an entire function of LaTeXMLMath which coincides with the Dirac distribution LaTeXMLMath when LaTeXMLMath , and that the distribution LaTeXMLMath is an entire function of LaTeXMLMath which coincides with the LaTeXMLMath th derivative of the Dirac distribution LaTeXMLMath when LaTeXMLMath and LaTeXMLMath .
With that in mind we consider a smooth function LaTeXMLMath which is LaTeXMLMath when LaTeXMLMath and is LaTeXMLMath when LaTeXMLMath , and we write LaTeXMLEquation .
Since LaTeXMLEquation is bounded and continous with respect to LaTeXMLMath and LaTeXMLMath , we can write LaTeXMLEquation .
Thus LaTeXMLMath is bounded by a constant with does not depend on LaTeXMLMath and has at most exponential growth with respect to Im LaTeXMLMath .
We shall now estimate LaTeXMLMath .
By our previous observations , the function LaTeXMLEquation can be differentiated LaTeXMLMath times with respect to LaTeXMLMath and its derivatives are continous functions of LaTeXMLMath .
Then , if LaTeXMLMath , LaTeXMLEquation .
If LaTeXMLMath , and if LaTeXMLMath we use the formula LaTeXMLMath in LaTeXMLCite , pg .
LaTeXMLMath , obtaining : LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLMath is bounded by a constant which does not depend on LaTeXMLMath .
This shows that LaTeXMLMath is bounded by a uniform constant .
An easy adaptation of the above argument shows that LaTeXMLMath converges to LaTeXMLMath in distribution sense as LaTeXMLMath .
This completes the proof of the theorem .
Acknowledgements : The authors wish to thank Eric Sawyer for financial support provided through his NSERC grant .
The second author wishes to thank Professor Sawyer for numerous helpful conversations over the last two years , which had a profound influence on his work .
The authors wish to thank Fulvio Ricci for suggesting Theorem 3.1 , and for related helpful suggestions .
References
We prove that the pure state space is homogeneous under the action of the group of asymptotically inner automorphisms for all the separable simple nuclear LaTeXMLMath -algebras .
If simplicity is not assumed for the LaTeXMLMath -algebras , the set of pure states whose GNS representations are faithful is homogeneous for the above action .
If LaTeXMLMath is a LaTeXMLMath -algebra , an automorphism LaTeXMLMath of LaTeXMLMath is asymptotically inner if there is a continuous family LaTeXMLMath in the group LaTeXMLMath of unitaries in LaTeXMLMath ( or LaTeXMLMath if LaTeXMLMath is non-unital ) such that LaTeXMLMath ; we denote by LaTeXMLMath the group of asymptotically inner automoprphisms of LaTeXMLMath , which is a normal subgroup of the group of approximately inner automorphisms .
Note that each LaTeXMLMath leaves each ( closed two-sided ) ideal of LaTeXMLMath invariant .
It is shown , in LaTeXMLCite , for a large class of separable nuclear LaTeXMLMath -algebras that if LaTeXMLMath and LaTeXMLMath are pure states of LaTeXMLMath such that the GNS representations associated with LaTeXMLMath and LaTeXMLMath have the same kernel , then there is an LaTeXMLMath such that LaTeXMLMath .
We shall show in this paper that this is the case for all separable nuclear LaTeXMLMath -algebras ; in particular the pure state space of a separable simple nuclear LaTeXMLMath -algebra LaTeXMLMath is homogeneous under the action of LaTeXMLMath .
We do not know of a single example of a separable LaTeXMLMath -algebra which does not have this property .
See LaTeXMLCite for some problems on this and see LaTeXMLRef and LaTeXMLRef for remarks on the non-separable case .
Choi and Effros LaTeXMLCite have shown that LaTeXMLMath is nuclear if and only if there is a net of pairs LaTeXMLMath of completely positive ( CP ) contractons such that LaTeXMLMath , where LaTeXMLEquation and LaTeXMLMath is a finite-dimensional LaTeXMLMath -algebra .
When LaTeXMLMath is a non-unital LaTeXMLMath -algebra , LaTeXMLMath is nuclear if and only if LaTeXMLMath is nuclear LaTeXMLCite .
If LaTeXMLMath is unital , we may assume that both LaTeXMLMath and LaTeXMLMath are unit-preserving .
We refer to LaTeXMLCite for some other facts on nuclear LaTeXMLMath -algebras .
We also quote LaTeXMLCite for a review on the subject .
Our proof of the homogeneity is a combination of the techniques leading up to the above result from LaTeXMLCite and the techniques from LaTeXMLCite .
In section 2 we shall show how the homogeneity follows from inductive use of Lemma LaTeXMLRef ( or LaTeXMLRef ) , whose conclusion is very similar to the properties already used in LaTeXMLCite ; this part follows closely LaTeXMLCite and so the proof will be sketchy .
In section 3 we shall prove Lemma LaTeXMLRef from another technical lemma , Lemma LaTeXMLRef , which shows some amenability of the nuclear LaTeXMLMath -algebras ; this is the arguments often used for individual examples treated in LaTeXMLCite and so the proof will be again sketchy .
Then we will give a proof of Lemma LaTeXMLRef , which constitutes the main body of this paper and uses the results and techniques from LaTeXMLCite .
We will conclude this paper , following LaTeXMLCite , by generalizing Lemma LaTeXMLRef and then extend the main result , Theorem LaTeXMLRef , to show that LaTeXMLMath acts on the pure state space of LaTeXMLMath strongly transitively .
See Theorem LaTeXMLRef for details .
We first give a main technical lemma , whose conclusion is a slightly weaker version of Property 2.6 in LaTeXMLCite .
We will give a proof in the next section .
Let LaTeXMLMath be a nuclear LaTeXMLMath -algebra .
Then for any finite subset LaTeXMLMath of LaTeXMLMath , any pure state LaTeXMLMath of LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath , there exist a finite subset LaTeXMLMath of LaTeXMLMath and LaTeXMLMath satisfying : If LaTeXMLMath is a pure state of LaTeXMLMath such that LaTeXMLMath , and LaTeXMLEquation then there is a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath , and LaTeXMLEquation .
In the above statement , LaTeXMLMath is the GNS representation of LaTeXMLMath associated with the state LaTeXMLMath ; LaTeXMLMath is the Hilbert space for this representation ; LaTeXMLMath is the LaTeXMLMath -algebra of compact operators on LaTeXMLMath ; LaTeXMLMath means that LaTeXMLMath is equivalent to LaTeXMLMath .
We could also impose the extra condition that the length of LaTeXMLMath is smaller than LaTeXMLMath for the choice of the path LaTeXMLMath ; see Property 8.1 in LaTeXMLCite .
The following is an easy consequence : Let LaTeXMLMath be a nuclear LaTeXMLMath -algebra .
Then for any finite subset LaTeXMLMath of LaTeXMLMath , any pure state LaTeXMLMath of LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath , there exist a finite subset LaTeXMLMath of LaTeXMLMath and LaTeXMLMath satisfying : If LaTeXMLMath is a pure state of LaTeXMLMath such that LaTeXMLMath , and LaTeXMLEquation then for any finite subset LaTeXMLMath of LaTeXMLMath and LaTeXMLMath there is a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath , and LaTeXMLEquation .
LaTeXMLEquation Proof .
Given LaTeXMLMath , choose LaTeXMLMath as in the previous lemma .
Let LaTeXMLMath be a pure state of LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
Let LaTeXMLMath be a finite subset of LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
We can mimic LaTeXMLMath as a vector state through LaTeXMLMath ; by Kadison ’ s transitivity there is a LaTeXMLMath such that LaTeXMLEquation ( see 2.3 of LaTeXMLCite ) .
Since LaTeXMLMath , we have , by applying Lemma LaTeXMLRef to the pair LaTeXMLMath and LaTeXMLMath , a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath , and LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath , this completes the proof .
LaTeXMLMath We shall now turn to the main result stated in the introduction .
We denote by LaTeXMLMath the set of LaTeXMLMath which has a continuous family LaTeXMLMath in LaTeXMLMath with LaTeXMLMath and LaTeXMLMath ; LaTeXMLMath can be smaller than LaTeXMLMath ( e.g. , LaTeXMLMath may not contain LaTeXMLMath ; see LaTeXMLCite ) .
Let LaTeXMLMath be a separable nuclear LaTeXMLMath -algebra .
If LaTeXMLMath and LaTeXMLMath are pure states of LaTeXMLMath such that LaTeXMLMath , then there is an LaTeXMLMath such that LaTeXMLMath .
Proof .
Once we have Lemma LaTeXMLRef , we can prove this in the same way as 2.5 of LaTeXMLCite .
We shall only give an outline here .
Let LaTeXMLMath and LaTeXMLMath be pure states of LaTeXMLMath such that LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath is equivalent to LaTeXMLMath .
Then by Kadison ’ s transitivity ( see , e.g. , 1.21.16 of LaTeXMLCite ) , there is a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
Suppose that LaTeXMLMath , which also implies that LaTeXMLMath .
Let LaTeXMLMath be a dense sequence in LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath ( or LaTeXMLMath ) .
Let LaTeXMLMath be the LaTeXMLMath for LaTeXMLMath as in Lemma LaTeXMLRef such that LaTeXMLMath .
For this LaTeXMLMath we choose a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
Let LaTeXMLMath and let LaTeXMLMath be the LaTeXMLMath for LaTeXMLMath as in Lemma LaTeXMLRef such that LaTeXMLMath and LaTeXMLMath .
By LaTeXMLRef there is a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath and let LaTeXMLMath be the LaTeXMLMath for LaTeXMLMath as in LaTeXMLRef such that LaTeXMLMath and LaTeXMLMath .
By LaTeXMLRef there is a continuous path LaTeXMLMath in LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation We shall repeat this process .
Assume that we have constructed LaTeXMLMath , and LaTeXMLMath inductively .
In particular if LaTeXMLMath is even , LaTeXMLEquation and LaTeXMLMath is the LaTeXMLMath for LaTeXMLMath as in LaTeXMLRef such that LaTeXMLMath and LaTeXMLMath .
And LaTeXMLMath is given by LaTeXMLRef for LaTeXMLMath and for LaTeXMLMath and LaTeXMLMath and it satisfies LaTeXMLEquation .
We define continuous paths LaTeXMLMath and LaTeXMLMath in LaTeXMLMath with LaTeXMLMath by : For LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation Then , since LaTeXMLMath , we can show that LaTeXMLMath ( resp .
LaTeXMLMath ) converges to an automorphism LaTeXMLMath ( resp .
LaTeXMLMath ) as LaTeXMLMath and that LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath is a group , this will complete the proof .
See the proofs of 2.5 and 2.8 of LaTeXMLCite for details .
LaTeXMLMath The notion of asymptotical innerness for automorphisms may be appropriate only for separable LaTeXMLMath -algebras .
Because any LaTeXMLMath can be obtained as the limit of a sequence in LaTeXMLMath , not just as the limit of a net there .
Hence the following remark will not be a surprise ; it may only suggest that we should take LaTeXMLMath or something bigger than LaTeXMLMath in place of LaTeXMLMath , in formulating LaTeXMLRef for non-separable LaTeXMLMath -algebras .
There is a unital simple non-separable nuclear LaTeXMLMath -algebra LaTeXMLMath such that the pure states space of LaTeXMLMath is not homogeneous under the action of LaTeXMLMath .
We can construct such an example as follows .
Let LaTeXMLMath be a unital simple separable nuclear LaTeXMLMath -algebra and LaTeXMLMath an uncountable set .
For each finite subset LaTeXMLMath of LaTeXMLMath we set LaTeXMLMath with LaTeXMLMath and take the natural inductive limit LaTeXMLMath of the net LaTeXMLMath .
Since LaTeXMLMath is nuclear , it follows that LaTeXMLMath is nuclear .
For each LaTeXMLMath we define LaTeXMLMath to be the LaTeXMLMath -subalgebra of LaTeXMLMath generated by LaTeXMLMath with finite LaTeXMLMath .
Note that for each LaTeXMLMath there is a countable LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath be a sequence in LaTeXMLMath such that LaTeXMLMath converges to LaTeXMLMath in the point-norm topology .
Since there is a countable subset LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath is non-trivial only on LaTeXMLMath , where LaTeXMLMath is countable .
Thus any LaTeXMLMath has the above property of countable support .
For each LaTeXMLMath let LaTeXMLMath and LaTeXMLMath be pure states of LaTeXMLMath such that LaTeXMLMath and let LaTeXMLMath and LaTeXMLMath .
Then it follows that LaTeXMLMath and LaTeXMLMath are pure states of LaTeXMLMath and that LaTeXMLMath for any LaTeXMLMath .
Hence LaTeXMLMath serves as an example for the above remark .
In this case , however , we have an LaTeXMLMath such that LaTeXMLMath ( since this is the case for each pair LaTeXMLMath from LaTeXMLRef ) and it may be the case that the pure state space of LaTeXMLMath is homogeneous under the action of LaTeXMLMath .
There is a unital simple non-separable non-nuclear LaTeXMLMath -algebra LaTeXMLMath such that the pure state space of LaTeXMLMath is not homogeneous under the action of LaTeXMLMath .
There are plenty of such LaTeXMLMath -algebras at hand .
Let LaTeXMLMath be a factor of type II LaTeXMLMath or type III with separable predual LaTeXMLMath .
Then LaTeXMLMath is a unital simple non-separable non-nuclear LaTeXMLMath -algebra ( see , e.g. , LaTeXMLCite for non-nuclearity ) .
Since LaTeXMLMath contains a LaTeXMLMath -subalgebra isomorphic to LaTeXMLMath and LaTeXMLMath has cardinality LaTeXMLMath , the pure state space of LaTeXMLMath has cardinality ( at least ) LaTeXMLMath , where LaTeXMLMath denotes the cardinality of the continuum .
( We owe this argument to J .
Anderson . )
On the other hand any LaTeXMLMath corresponds to an isometry on the predual LaTeXMLMath , a separable Banach space .
Thus , since the set of bounded operators on a separable Banach space has cardinality LaTeXMLMath , LaTeXMLMath has cardinality ( at most ) LaTeXMLMath .
Hence the pure state space of LaTeXMLMath can not be homogeneous under the action of LaTeXMLMath .
We note in passing that LaTeXMLMath for any factor LaTeXMLMath ( or any quotient of a factor ) , since any convergent sequence in LaTeXMLMath with the point-norm topology converges in norm LaTeXMLCite .
We also note that LaTeXMLMath for any full factor LaTeXMLCite , since then LaTeXMLMath is closed in LaTeXMLMath with the topology of point-norm convergence in LaTeXMLMath and so is closed in LaTeXMLMath with the topology of point-norm convergence in LaTeXMLMath .
If LaTeXMLMath is a non-unital LaTeXMLMath -algebra , LaTeXMLMath is nuclear if and only if the LaTeXMLMath -algebra LaTeXMLMath obtained by adjoining a unit is nuclear .
Hence to prove Lemma LaTeXMLRef we may suppose that LaTeXMLMath is unital .
In the following LaTeXMLMath denotes the connected component of LaTeXMLMath in the unitary group LaTeXMLMath of LaTeXMLMath .
Let LaTeXMLMath be a unital nuclear LaTeXMLMath -algebra .
Let LaTeXMLMath be a finite subset of LaTeXMLMath , LaTeXMLMath an irreducible representation of LaTeXMLMath on a Hilbert space LaTeXMLMath , LaTeXMLMath a finite-dimensional projection on LaTeXMLMath , and LaTeXMLMath .
Then there exist an LaTeXMLMath and a finite subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , and for any LaTeXMLMath there is a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath with LaTeXMLEquation .
In the above statement , LaTeXMLMath denotes the LaTeXMLMath by LaTeXMLMath matrices over LaTeXMLMath ; if LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation We shall first show that Lemma LaTeXMLRef implies Lemma LaTeXMLRef .
Let LaTeXMLMath be a finite subset of LaTeXMLMath , LaTeXMLMath a pure state of LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath .
Since LaTeXMLMath linearly spans LaTeXMLMath , we may suppose that LaTeXMLMath is a finite subset of LaTeXMLMath .
For LaTeXMLMath and the projection LaTeXMLMath onto the subspace LaTeXMLMath , we choose an LaTeXMLMath and a finite subset LaTeXMLMath of LaTeXMLMath as in Lemma LaTeXMLRef .
We take the finite subset LaTeXMLEquation for the subset LaTeXMLMath required in Lemma LaTeXMLRef .
We will choose LaTeXMLMath sufficiently small later .
Suppose that we are given a unit vector LaTeXMLMath satisfying LaTeXMLEquation for any LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath .
Note that LaTeXMLEquation which implies that LaTeXMLMath .
Thus the two finite sets of vectors LaTeXMLMath and LaTeXMLMath have similar geometric properties in LaTeXMLMath if LaTeXMLMath is sufficiently small .
Hence we are in a situation where we can apply 3.3 of LaTeXMLCite .
Let us describe how we proceed from here in a simplified case .
Suppose that the linear span LaTeXMLMath of LaTeXMLMath is orthogonal to the linear span LaTeXMLMath of LaTeXMLMath and that the map LaTeXMLMath and LaTeXMLMath extends to a unitary on LaTeXMLMath ; in particular we have assumed that LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath is a self-adjoint unitary , LaTeXMLMath is a projection and satisfies that LaTeXMLMath on the finite-dimensional subspace LaTeXMLMath .
By Kadison ’ s transitivity we choose an LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
We set LaTeXMLEquation where LaTeXMLEquation .
Since LaTeXMLEquation and LaTeXMLMath , it follows that LaTeXMLEquation .
Hence we have that LaTeXMLMath switches LaTeXMLMath and LaTeXMLMath .
On the other hand for LaTeXMLMath there is a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath such that LaTeXMLMath .
Since LaTeXMLEquation it follows that LaTeXMLMath .
Thus the path LaTeXMLMath almost commutes with LaTeXMLMath and is what is desired .
( Since what is required is LaTeXMLMath , we may take the path LaTeXMLMath , whose length is LaTeXMLMath . )
If LaTeXMLMath is not orthogonal to LaTeXMLMath , we still find a unit vector LaTeXMLMath such that LaTeXMLEquation and such that LaTeXMLMath is orthogonal to both LaTeXMLMath and LaTeXMLMath .
Here we use the assumption that LaTeXMLMath .
Then we combine the path of unitaries sending LaTeXMLMath to LaTeXMLMath and then the path sending LaTeXMLMath to LaTeXMLMath to obtain the desired path .
The above arguments can be made rigorous in the general case ; see LaTeXMLCite for details .
LaTeXMLMath We will now turn to the proof of Lemma LaTeXMLRef , by first giving a series of lemmas .
The following is an easy version of 3.4 of LaTeXMLCite .
Let LaTeXMLMath be a non-degenerate representation of a LaTeXMLMath -algebra LaTeXMLMath on a Hilbert space LaTeXMLMath , LaTeXMLMath a finite-dimensional projection on LaTeXMLMath , LaTeXMLMath a finite subset of LaTeXMLMath , and LaTeXMLMath .
Then there is a finite-rank self-adjoint operator LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
Proof .
We define finite-dimensional subspaces LaTeXMLMath of LaTeXMLMath as follows : LaTeXMLMath and if LaTeXMLMath is defined then LaTeXMLMath is the linear span of LaTeXMLMath and LaTeXMLMath , where we have omitted LaTeXMLMath .
Then LaTeXMLMath is increasing and LaTeXMLEquation with LaTeXMLMath .
Denoting by LaTeXMLMath the projection onto LaTeXMLMath we define LaTeXMLEquation .
Then LaTeXMLMath .
If LaTeXMLMath , we have , for LaTeXMLMath , that LaTeXMLEquation .
Hence for LaTeXMLMath , LaTeXMLEquation and thus , by splitting the above sum into three terms , each of which is the sum over LaTeXMLMath for LaTeXMLMath , and estimating each , we reach LaTeXMLEquation .
This implies that LaTeXMLMath for LaTeXMLMath .
LaTeXMLMath If LaTeXMLMath is a representation of LaTeXMLMath on a Hilbert space LaTeXMLMath , we denote by LaTeXMLMath the representation of LaTeXMLMath , the LaTeXMLMath by LaTeXMLMath matrix algebra over LaTeXMLMath , on the Hilbert space LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is naturally a diagonal element of LaTeXMLMath .
Let LaTeXMLMath be a non-degenerate representation of a unital LaTeXMLMath -algebra LaTeXMLMath on a Hilbert space LaTeXMLMath , LaTeXMLMath a finite-rank projection on LaTeXMLMath , LaTeXMLMath a finite subset of LaTeXMLMath , and LaTeXMLMath .
Then there exists an LaTeXMLMath such that each LaTeXMLMath has a diagonal element LaTeXMLMath in LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation .
Furthermore there exists a finite-rank projection LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
Proof .
Since LaTeXMLMath is path-wise connected , the first part is immediate .
Let LaTeXMLMath , which will be specified sufficiently small later .
By the previous lemma we choose a finite-rank self-adjoint operator LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation where we have omitted LaTeXMLMath .
Let LaTeXMLMath be the support projection of LaTeXMLMath and let LaTeXMLMath be a finite-rank self-adjoint operator on LaTeXMLMath such that LaTeXMLMath , and LaTeXMLEquation .
In this way we define LaTeXMLMath and set LaTeXMLMath , the support projection of LaTeXMLMath .
We define an operator LaTeXMLMath on LaTeXMLMath as a tri-diagonal matrix as follows : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath .
Noting that LaTeXMLMath and LaTeXMLMath , it is easy to check that LaTeXMLMath is a finite-rank projection and LaTeXMLMath dominates LaTeXMLMath .
For LaTeXMLMath , we have that LaTeXMLEquation .
LaTeXMLEquation Thus , since LaTeXMLMath , the norm of LaTeXMLMath is smaller than LaTeXMLEquation which can be made smaller than LaTeXMLMath for all LaTeXMLMath by choosing LaTeXMLMath small .
LaTeXMLMath When LaTeXMLMath is a projection on a Hilbert space LaTeXMLMath , we denote by LaTeXMLMath the bounded operators on the subspace LaTeXMLMath .
Let LaTeXMLMath be a unital nuclear LaTeXMLMath -algebra , LaTeXMLMath an irreducible representation of LaTeXMLMath on a Hilbert space LaTeXMLMath , and LaTeXMLMath a finite-rank projection on LaTeXMLMath .
Then the identity map on LaTeXMLMath can be approximated by a net of compositions of CP maps LaTeXMLEquation where LaTeXMLMath is a finite-dimensional LaTeXMLMath -algebra , LaTeXMLMath is an increasing net of finite-rank projections on LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are unital CP maps such that LaTeXMLMath , and LaTeXMLMath is a unital CP map such that LaTeXMLEquation .
LaTeXMLEquation Proof .
There is a non-degenerate representation LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath is disjoint from LaTeXMLMath and LaTeXMLMath is a universal representation , i.e. , LaTeXMLMath extends to a faithful representation of LaTeXMLMath .
Note that LaTeXMLMath .
If the nuclear LaTeXMLMath -algebra LaTeXMLMath is separable , LaTeXMLMath is semidiscrete LaTeXMLCite , which in turn implies that LaTeXMLMath is semidiscrete .
Hence the identity map on LaTeXMLMath can be approximated , in the point-weak LaTeXMLMath topology , by a net LaTeXMLMath of CP maps on LaTeXMLMath , where LaTeXMLMath ( resp .
LaTeXMLMath ) is a weak LaTeXMLMath -continuous unital CP map of LaTeXMLMath into a finite-dimensional LaTeXMLMath -algebra LaTeXMLMath ( resp .
of LaTeXMLMath into LaTeXMLMath ) .
By denoting LaTeXMLMath by LaTeXMLMath again , we obtain a net of diagrams LaTeXMLEquation such that LaTeXMLMath converges to LaTeXMLMath in the weak LaTeXMLMath topology for any LaTeXMLMath .
If LaTeXMLMath is separable or not , we have the characterization of nuclearity in terms of CP maps LaTeXMLCite ; there is a net of diagrams of unital CP maps : LaTeXMLEquation such that LaTeXMLMath is finite-dimensional and LaTeXMLMath converges to LaTeXMLMath in norm for any LaTeXMLMath .
By denoting LaTeXMLMath by LaTeXMLMath again , we obtain a net of diagrams : LaTeXMLEquation as above ; actually LaTeXMLMath converges to LaTeXMLMath in norm for any LaTeXMLMath .
Since LaTeXMLMath is semidiscrete , there is such a net of CP maps on LaTeXMLMath as for LaTeXMLMath as well .
But we shall construct one in a specific way .
Let LaTeXMLMath be an increasing net of finite-rank projections on LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
We define LaTeXMLMath by LaTeXMLMath and LaTeXMLMath by LaTeXMLMath , where LaTeXMLMath is a vector state , defined through a fixed unit vector in LaTeXMLMath .
Then it is immediate that LaTeXMLMath has the desired properties .
By denoting LaTeXMLMath by LaTeXMLMath again , we obtain a net of diagrams : LaTeXMLEquation such that LaTeXMLMath converges to LaTeXMLMath in the weak LaTeXMLMath topology for any LaTeXMLMath .
We may suppose that we use the same directed set LaTeXMLMath for both LaTeXMLMath and LaTeXMLMath .
We set LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
By identifying LaTeXMLMath with LaTeXMLMath , we have that LaTeXMLEquation approximate the identity map on LaTeXMLMath ( in the point-weak LaTeXMLMath topology ) , i.e. , LaTeXMLMath converges to LaTeXMLMath in the weak LaTeXMLMath topology for any LaTeXMLMath .
Following LaTeXMLCite we approximate LaTeXMLMath by unital CP maps of LaTeXMLMath into LaTeXMLMath .
This is done as follows .
If LaTeXMLMath denotes a family of matrix units of LaTeXMLMath , LaTeXMLMath is uniquely determined by the positive element LaTeXMLMath in LaTeXMLMath ( 2.1 of LaTeXMLCite ) .
Since LaTeXMLMath is dense in LaTeXMLMath in the weak LaTeXMLMath topology , we can , by general theory , approximate LaTeXMLMath by positive elements in LaTeXMLMath , in the weak LaTeXMLMath topology , which then determine CP maps of LaTeXMLMath into LaTeXMLMath ( see the proof of 3.1 of LaTeXMLCite ) .
In particular we approximate LaTeXMLMath by CP maps LaTeXMLMath satisfying LaTeXMLEquation and LaTeXMLMath by CP maps LaTeXMLMath satisfying LaTeXMLEquation .
This is indeed possible as shown by using Kadison ’ s transitivity .
Moreover , by taking convex combinations of LaTeXMLMath , we may assume that LaTeXMLMath is close to LaTeXMLMath in norm .
By replacing LaTeXMLMath by LaTeXMLMath etc .
we may suppose that LaTeXMLMath is a unital CP map .
Since LaTeXMLMath , this does not destroy the above properties imposed on LaTeXMLMath and LaTeXMLMath .
Restricting LaTeXMLMath to LaTeXMLMath and retaining the same symbol LaTeXMLMath for the CP maps into LaTeXMLMath ( instead of LaTeXMLMath ) , we now have a net of the compositions of unital CP maps : LaTeXMLEquation which approximates the identity map in the point-weak topology .
By taking convex combinations of the above CP maps , we will obtain such a net which now approximates the identity map in the point-norm topology .
For example , if LaTeXMLMath is such that LaTeXMLMath , LaTeXMLMath is finite , and LaTeXMLMath , then we define LaTeXMLEquation where LaTeXMLMath is such that LaTeXMLMath , and LaTeXMLEquation .
LaTeXMLEquation with LaTeXMLMath defined by the multiplication of LaTeXMLMath on both sides .
By doing so , the properties LaTeXMLMath and LaTeXMLMath are still retained , where LaTeXMLMath is the first component of LaTeXMLMath etc .
See LaTeXMLCite for technical details .
LaTeXMLMath Let LaTeXMLMath be as in LaTeXMLRef .
For any LaTeXMLMath there is a LaTeXMLMath such that if LaTeXMLMath satisfies that LaTeXMLMath , there is a LaTeXMLMath with LaTeXMLMath .
Proof .
Suppose that LaTeXMLMath is represented on a Hilbert space LaTeXMLMath .
Since LaTeXMLMath is a unital CP map , by Steinspring ’ s theorem there is a representation LaTeXMLMath of LaTeXMLMath on a Hilbert space LaTeXMLMath which contains LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath is the projection onto LaTeXMLMath .
If LaTeXMLMath satisfies that LaTeXMLMath , where LaTeXMLMath etc. , it follows that LaTeXMLEquation .
Let LaTeXMLMath denote LaTeXMLMath .
Since LaTeXMLMath , we have that LaTeXMLMath .
Since LaTeXMLMath , we also have that LaTeXMLMath .
For any LaTeXMLMath it follows that LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is the spectral projection of LaTeXMLMath corresponding to LaTeXMLMath for some LaTeXMLMath , then LaTeXMLMath and LaTeXMLEquation .
Let LaTeXMLMath .
Then the above inequality implies that LaTeXMLEquation or LaTeXMLMath .
Hence we have that LaTeXMLMath and LaTeXMLMath for a sufficiently small LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is also close to LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is also close to LaTeXMLMath ( up to the order of LaTeXMLMath in this rough estimate ) ; here LaTeXMLMath is the inverse of LaTeXMLMath in LaTeXMLMath .
We now define a unitary LaTeXMLMath in LaTeXMLMath by LaTeXMLMath , where LaTeXMLMath satisfies that LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath is indeed a unitary .
Since LaTeXMLMath , LaTeXMLMath is of the order of LaTeXMLMath .
Since LaTeXMLMath is close to LaTeXMLMath up to the order of LaTeXMLMath , we can conclude that LaTeXMLMath is close to zero up to the order of LaTeXMLMath .
LaTeXMLMath When LaTeXMLMath is a metric space , LaTeXMLMath , and LaTeXMLMath , we call LaTeXMLMath an LaTeXMLMath -net if LaTeXMLMath , where LaTeXMLMath .
When LaTeXMLMath has a finite LaTeXMLMath -net , we denote by LaTeXMLMath the minimum of orders over all the finite LaTeXMLMath -nets .
If LaTeXMLMath is compact , then LaTeXMLMath is well-defined for any LaTeXMLMath .
Let LaTeXMLMath be a compact metric space .
If LaTeXMLMath and LaTeXMLMath are LaTeXMLMath -nets consisting LaTeXMLMath points , then there is a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath such that LaTeXMLMath .
Proof .
Let LaTeXMLMath be a non-empty subset of LaTeXMLMath and set LaTeXMLEquation .
Since LaTeXMLMath , it follows that LaTeXMLMath is an LaTeXMLMath -net and that the order of LaTeXMLMath is greater than or equal to the order of LaTeXMLMath .
Then by the matching theorem we can find a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath such that LaTeXMLMath .
LaTeXMLMath Proof of Lemma LaTeXMLRef Let LaTeXMLMath be an irreducible representation of the unital nuclear LaTeXMLMath -algebra LaTeXMLMath on a Hilbert space LaTeXMLMath , LaTeXMLMath a finite-rank projection on LaTeXMLMath , LaTeXMLMath a finite subset of LaTeXMLMath , and LaTeXMLMath .
We apply Lemma LaTeXMLRef to this situation .
Thus there exist an LaTeXMLMath and a finite-rank projection LaTeXMLMath on LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath denotes the natural extension of LaTeXMLMath to a representation of LaTeXMLMath on LaTeXMLMath ; hereafter we shall simply denote LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be a finite-rank projection on LaTeXMLMath such that LaTeXMLMath .
By Lemma LaTeXMLRef we find a net of diagrams LaTeXMLEquation with LaTeXMLMath in place of LaTeXMLMath as described there ; in particular LaTeXMLMath .
We take tensor product of these diagrams with LaTeXMLMath ; denoting LaTeXMLMath by the same symbol LaTeXMLMath etc. , we obtain LaTeXMLEquation .
Noting that LaTeXMLMath , we denote LaTeXMLEquation which is a compact group .
Since LaTeXMLMath and LaTeXMLMath , we have that for each LaTeXMLMath LaTeXMLEquation .
Since the first term is LaTeXMLMath as LaTeXMLMath , the second term must be zero .
Hence it follows that LaTeXMLEquation which implies that LaTeXMLEquation .
By multiplying LaTeXMLMath from the right we have that LaTeXMLEquation .
Since LaTeXMLMath , we have that LaTeXMLMath for LaTeXMLMath .
Thus it follows that for LaTeXMLMath , LaTeXMLEquation .
By Lemma LaTeXMLRef ( applied to LaTeXMLMath instead of LaTeXMLMath ) we choose LaTeXMLMath such that each LaTeXMLMath has a unitary LaTeXMLMath such that LaTeXMLEquation as well as LaTeXMLEquation .
Since LaTeXMLEquation .
LaTeXMLEquation we have that LaTeXMLEquation .
By choosing LaTeXMLMath sufficiently large , we may assume that LaTeXMLEquation .
By taking the unitary part of the polar decomposition of LaTeXMLMath , we may assume that LaTeXMLEquation .
Since LaTeXMLMath , we can estimate that LaTeXMLEquation .
Since LaTeXMLMath , we have that for any LaTeXMLMath , LaTeXMLEquation ( See the proof of LaTeXMLRef . )
Hence for LaTeXMLMath LaTeXMLEquation .
We choose an LaTeXMLMath -net LaTeXMLMath of LaTeXMLMath consisting of LaTeXMLMath points and set LaTeXMLEquation .
Since LaTeXMLMath is also an LaTeXMLMath -net of LaTeXMLMath for LaTeXMLMath , Lemma LaTeXMLRef gives a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath such that LaTeXMLEquation .
Hence for each LaTeXMLMath there is a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath such that LaTeXMLEquation which implies that regarding LaTeXMLMath as a map of LaTeXMLMath onto LaTeXMLMath , LaTeXMLEquation .
This completes the proof .
LaTeXMLMath In Lemma LaTeXMLRef we could handle a mutually disjoint finite family of irreducible representations instead of just one .
By doing so we can derive : Let LaTeXMLMath be a unital nuclear LaTeXMLMath -algebra .
Let LaTeXMLMath be a finite subset of LaTeXMLMath , LaTeXMLMath a representation of LaTeXMLMath on a Hilbert space LaTeXMLMath such that LaTeXMLMath with LaTeXMLMath a mutually disjoint family of irreducible representations of LaTeXMLMath , LaTeXMLMath a finite-dimensional projection on LaTeXMLMath , and LaTeXMLMath .
Then there exist an LaTeXMLMath and a finite subset LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , and for any LaTeXMLMath there is a bijection LaTeXMLMath of LaTeXMLMath onto LaTeXMLMath with LaTeXMLEquation .
A straightforward generalization of LaTeXMLRef would require that LaTeXMLMath in the above statement .
But , since any finite-rank projection on LaTeXMLMath is dominated by such a one in LaTeXMLMath , we did not need it .
By having this at hand we can derive a stronger version of Lemma LaTeXMLRef and then strengthen Theorem LaTeXMLRef .
For example we will obtain : Let LaTeXMLMath be a separable nuclear LaTeXMLMath -algebra .
If LaTeXMLMath and LaTeXMLMath are finite sequences of pure states of LaTeXMLMath such that LaTeXMLMath ( resp .
LaTeXMLMath ) are mutually disjoint and LaTeXMLMath for all LaTeXMLMath , then there is an LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
We will have to use a general form of Kadison ’ s transitivity for the proofs of the above results as in LaTeXMLCite .
See Section 7 of LaTeXMLCite for details and for other consequences .
We do not know whether we could take an arbitrary non-degenerate representation of LaTeXMLMath for LaTeXMLMath in Lemma LaTeXMLRef ( perhaps by weakening the requirement LaTeXMLMath by LaTeXMLMath ) .
If this were the case , we would obtain a new characterization of nuclearity which manifests a close connection with amenability of LaTeXMLMath ( cf .
LaTeXMLCite ) .
Department of Mathematics , Hokkaido University , Sapporo , Japan 060-0810 5-1-6-205 , Odawara , Aoba-ku , Sendai , Japan 980-0003
Nobuhiro Asai Current address : International Institute for Advanced Studies , Kizu , Kyoto , 619-0225 , JAPAN .
Graduate School of Mathematics Nagoya University Nagoya , 464-8602 , JAPAN Izumi Kubo Department of Mathematics Graduate School of Science Hiroshima University Higashi-Hiroshima , 739-8526 , JAPAN and Hui-Hsiung Kuo Department of Mathematics Louisiana State University Baton Rouge , LA 70803 , USA In the recent paper LaTeXMLCite by Asai et al. , the growth order of holomorphic functions on a nuclear space has been considered .
For this purpose , certain classes of growth functions LaTeXMLMath are introduced and many properties of Legendre transform of such functions are investigated .
In LaTeXMLCite , applying Legendre transform of LaTeXMLMath under the conditions ( U0 ) , ( U2 ) and ( U3 ) ( see §2 ) , the Gel ’ fand triple LaTeXMLEquation associated with a growth function LaTeXMLMath is constructed .
The main purpose of this work is to prove Theorem 4.4 , so-called , the characterization theorem of Hida measures ( generalized measures ) .
As examples of such measures , we shall present the Poisson noise measure and the Grey noise measure in Example 4.5 and 4.6 , respectively .
The present paper is organized as follows .
In §2 , we give a quick review of some fundamental results in white noise analysis and introduce the notion of Legendre transform utilized by Asai et al .
in LaTeXMLCite , LaTeXMLCite .
In §3 , we simply cite some useful properties of the Legendre transform from LaTeXMLCite .
In §4 , we discuss the characterization of Hida measures ( generalized measures ) .
Acknowledgements .
N. Asai wants to thank the Daiko Foundation and the Kamiyama Foundation for research support .
In this section , we will summarize well-known results in white noise analysis LaTeXMLCite , LaTeXMLCite , LaTeXMLCite and notions from Asai et al .
LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Complete details and further developments will be appeared in LaTeXMLCite .
Some similar results have been obtained independently by Gannoun et al .
LaTeXMLCite .
Let LaTeXMLMath be a real separale Hilbert space with the norm LaTeXMLMath .
Suppose LaTeXMLMath is a sequence of densely defined inner product norms on LaTeXMLMath .
Let LaTeXMLMath be the completion of LaTeXMLMath with respect to the norm LaTeXMLMath .
In addition we assume There exists a constant LaTeXMLMath such that LaTeXMLMath .
For any LaTeXMLMath , there exists LaTeXMLMath such that the inclusion LaTeXMLMath is a Hilbert-Schmidt operator .
Let LaTeXMLMath and LaTeXMLMath denote the dual spaces of LaTeXMLMath and LaTeXMLMath , respectively .
We can use the Riesz representation theorem to identify LaTeXMLMath with its dual space LaTeXMLMath .
Let LaTeXMLMath be the projective limit of LaTeXMLMath .
Then we get the following continuous inclusions : LaTeXMLEquation .
The above condition ( b ) says that LaTeXMLMath is a nuclear space and so LaTeXMLMath is a Gel ’ fand triple .
Let LaTeXMLMath be the standard Gaussian measure on LaTeXMLMath with the characteristic function given by LaTeXMLEquation .
The probability space LaTeXMLMath is called a white noise space or Gaussian space .
For simplicity , we will use LaTeXMLMath to denote the Hilbert space of LaTeXMLMath -square integrable functions on LaTeXMLMath .
By the Wiener-Itô theorem , each LaTeXMLMath can be uniquely expressed as LaTeXMLEquation where LaTeXMLMath is the multiple Wiener integral of order LaTeXMLMath and LaTeXMLMath is the Wick tensor of LaTeXMLMath ( see LaTeXMLCite . )
Moreover , the LaTeXMLMath -norm of LaTeXMLMath is given by LaTeXMLEquation .
Let LaTeXMLMath be the set of all positive continuous functions on LaTeXMLMath satisfying LaTeXMLEquation .
In addition , we introduce conditions : LaTeXMLMath .
LaTeXMLMath is increasing and LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath is convex on LaTeXMLMath .
Obviously , ( U1 ) is a stronger condition than ( U0 ) .
Let LaTeXMLMath denote the set of all positive continuous functions LaTeXMLMath on LaTeXMLMath satisfying the condition : LaTeXMLEquation .
It is easy to see LaTeXMLMath .
The Legendre transform LaTeXMLMath of LaTeXMLMath is defined to be the function LaTeXMLEquation .
Some useful properties of the Legendre transform will be refered in section LaTeXMLRef .
From now on , we take a function LaTeXMLMath satisfying ( U0 ) ( U2 ) ( U3 ) .
We shall constract a Gel ’ fand triple associated with LaTeXMLMath .
For LaTeXMLMath being represented by Equation ( LaTeXMLRef ) and LaTeXMLMath , define LaTeXMLEquation .
Let LaTeXMLMath .
Define the space LaTeXMLMath of test functions on LaTeXMLMath to be the projective limit of LaTeXMLMath .
The dual space LaTeXMLMath of LaTeXMLMath is called the space of generalized functions on LaTeXMLMath .
Choose an appropriate LaTeXMLMath such that LaTeXMLMath for some LaTeXMLMath .
Then two conditions ( a ) and ( U2 ) imply that LaTeXMLMath for all LaTeXMLMath .
Hence LaTeXMLMath holds .
By identifying LaTeXMLMath with its dual space we get the following continuous inclusions : LaTeXMLEquation where LaTeXMLMath is the dual space of LaTeXMLMath .
Moreover , LaTeXMLMath is a nuclear space and so LaTeXMLMath is a Gel ’ fand triple .
Note that LaTeXMLMath and for LaTeXMLMath , LaTeXMLMath is the completion of LaTeXMLMath with respect to the norm LaTeXMLEquation .
For LaTeXMLMath belonging to the complexification LaTeXMLMath of LaTeXMLMath , the renormalized exponential function LaTeXMLMath is defined by LaTeXMLEquation .
Then we have the norm estimate , LaTeXMLEquation .
For later uses , let us define the notion of equivalent functions here .
Two positive functions LaTeXMLMath and LaTeXMLMath on LaTeXMLMath are called equivalent if there exist constants LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is given by LaTeXMLEquation .
Then the function LaTeXMLMath belongs to LaTeXMLMath and satisfies conditions ( U1 ) ( U2 ) ( U3 ) .
In the sense of Definition LaTeXMLRef , the function LaTeXMLMath is equivalent to the function given by LaTeXMLEquation where LaTeXMLMath is the k-th order Bell number .
Hence we get the Gel ’ fand triple , LaTeXMLEquation known as the CKS-space associated with LaTeXMLMath , which is the same as the one defined by the k-th order Bell number LaTeXMLMath .
See more details in LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
For LaTeXMLMath , let LaTeXMLMath be the function defined by LaTeXMLEquation .
It is easy to check that LaTeXMLMath belongs to LaTeXMLMath and satisfies conditions ( U1 ) ( U2 ) ( U3 ) .
Hence this Gel ’ fand triple , LaTeXMLEquation which is well-known as the Hida-Kubo-Takenaka space for LaTeXMLMath LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite and the Kondratiev-Streit space for a general LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
For LaTeXMLMath case , see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Remark .
We have the following chain of Gel ’ fand triples : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
First we mention the following notions of concave and convex functions which will be used frequently .
A positive function LaTeXMLMath on LaTeXMLMath is called log-concave if the function LaTeXMLMath is concave on LaTeXMLMath ; log-convex if the function LaTeXMLMath is convex on LaTeXMLMath ; ( log , exp ) -convex if the function LaTeXMLMath is convex on LaTeXMLMath ; ( log , LaTeXMLMath ) -convex if the function LaTeXMLMath is convex on LaTeXMLMath .
We will need the fact that if LaTeXMLMath is log-concave , then the sequence LaTeXMLMath is log-concave .
To check this fact , note that for any LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation .
In particular , take LaTeXMLMath , and LaTeXMLMath to get LaTeXMLEquation .
Hence the sequence LaTeXMLMath is log-concave .
The next theorem is from Lemma 3.4 in LaTeXMLCite .
Let LaTeXMLMath .
Then the Legendre transform LaTeXMLMath is log-concave .
( Hence LaTeXMLMath is continuous on LaTeXMLMath and the sequence LaTeXMLMath is log-concave . )
From Theorem 2 ( b ) in LaTeXMLCite we have the fact : If LaTeXMLMath is log-concave and LaTeXMLMath , then LaTeXMLEquation .
By Theorem LaTeXMLRef the sequence LaTeXMLMath is log-concave .
Hence we can apply the above fact to the sequence LaTeXMLMath to get the next theorem .
Let LaTeXMLMath .
Then for all integers LaTeXMLMath , we have LaTeXMLEquation .
In the next theorem we state some properties of the Legendre transform LaTeXMLMath of a ( log , exp ) -convex function LaTeXMLMath in LaTeXMLMath .
It is from Lemmas 3.6 and 3.7 in LaTeXMLCite .
Let LaTeXMLMath be ( log , exp ) -convex .
Then LaTeXMLMath is decreasing for large LaTeXMLMath , LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath .
On the other hand , for a ( log , LaTeXMLMath ) -convex function LaTeXMLMath in LaTeXMLMath , its Legendre transform LaTeXMLMath has the properties in the next theorem from Lemmas 3.9 and 3.10 in LaTeXMLCite .
If in addition LaTeXMLMath is increasing , then LaTeXMLMath is also ( log , exp ) -convex and hence LaTeXMLMath has the properties in the above Theorem LaTeXMLRef .
Let LaTeXMLMath .
We have the assertions : LaTeXMLMath is ( log , LaTeXMLMath ) -convex if and only if LaTeXMLMath is log-convex .
If LaTeXMLMath is ( log , LaTeXMLMath ) -convex , then for any integers LaTeXMLMath , LaTeXMLEquation .
Now , suppose LaTeXMLMath and assume that LaTeXMLMath .
We define the LaTeXMLMath - function LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation .
Note that LaTeXMLMath is an entire function .
By Theorem LaTeXMLRef ( 2 ) , LaTeXMLMath is defined for any ( log , exp ) -convex function LaTeXMLMath in LaTeXMLMath .
Moreover , we have the next theorem from Theorem 3.13 in LaTeXMLCite .
( 1 ) Let LaTeXMLMath be ( log , exp ) -convex .
Then its LaTeXMLMath -function LaTeXMLMath is also ( log , exp ) -convex and for any LaTeXMLMath , LaTeXMLEquation ( 2 ) Let LaTeXMLMath be increasing and ( log , LaTeXMLMath ) -convex .
Then there exists a constant LaTeXMLMath such that LaTeXMLEquation .
Recall from Proposition 2.3 ( 3 ) in LaTeXMLCite : If LaTeXMLMath is increasing and ( log , LaTeXMLMath ) -convex for some LaTeXMLMath , then LaTeXMLMath is ( log , exp ) -convex .
Hence the above Theorem LaTeXMLRef yields the next theorem .
Let LaTeXMLMath be increasing and ( log , LaTeXMLMath ) -convex .
Then the functions LaTeXMLMath and LaTeXMLMath are equivalent .
In the next section LaTeXMLRef , we will consider the characterization of Hida measures ( generalized measures ) .
We prepare two lemmas for this purpose .
The proof of Lemma LaTeXMLRef is simple application of Theorem LaTeXMLRef so that we just state it without proof .
Suppose LaTeXMLMath is ( log , LaTeXMLMath ) -convex .
Then LaTeXMLEquation .
Remark .
Note that LaTeXMLMath for all LaTeXMLMath .
Hence we have LaTeXMLEquation .
Thus LaTeXMLMath and LaTeXMLMath are equivalent for any ( log , LaTeXMLMath ) -convex function LaTeXMLMath .
If , in addition , LaTeXMLMath is increasing , then LaTeXMLMath and LaTeXMLMath are equivalent by Theorem LaTeXMLRef .
It follows that LaTeXMLMath and LaTeXMLMath are equivalent for such a function LaTeXMLMath .
The next Lemma LaTeXMLRef can be obtained from Theorem LaTeXMLRef and Lemma LaTeXMLRef .
Suppose LaTeXMLMath is increasing and ( log , LaTeXMLMath ) -convex .
Then for any LaTeXMLMath , we have LaTeXMLEquation .
Before going to the main theorem , we need to introduce another equivalent family of norms on LaTeXMLMath , i.e. , LaTeXMLMath .
This family of norms is intrinsic in the sense that LaTeXMLMath is defined directly in terms of the analyticity and growth condition of LaTeXMLMath .
First , it is well-known that each test function LaTeXMLMath in LaTeXMLMath has a unique analytic extension ( see §6.3 of LaTeXMLCite ) given by LaTeXMLEquation where LaTeXMLMath is the unique linear operator taking LaTeXMLMath into LaTeXMLMath for all LaTeXMLMath .
By Theorem 6.2 in LaTeXMLCite with minor modifications , LaTeXMLMath is shown to be a continuous linear operator from LaTeXMLMath into itself .
Note that we still assume conditions ( U0 ) , ( U2 ) and ( U3 ) on LaTeXMLMath given in section LaTeXMLRef .
Now , let LaTeXMLMath be any fixed number .
Choose LaTeXMLMath such that LaTeXMLMath .
Then use Equations ( LaTeXMLRef ) , ( LaTeXMLRef ) and Theorem LaTeXMLRef to get LaTeXMLEquation .
Note that LaTeXMLMath by the above choice of LaTeXMLMath .
Since LaTeXMLMath is an increasing function , we see that LaTeXMLEquation .
But LaTeXMLMath is a continuous linear operator from LaTeXMLMath into itself .
Hence there exist positive constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Therefore , LaTeXMLEquation where LaTeXMLMath .
This is the growth condition for test functions .
Being motivated by Equation ( LaTeXMLRef ) , we define LaTeXMLEquation .
Obviously , LaTeXMLMath is a norm on LaTeXMLMath for each LaTeXMLMath .
Suppose LaTeXMLMath satisfies conditions ( U1 ) ( U2 ) ( U3 ) .
Then the families of norms LaTeXMLMath and LaTeXMLMath are equivalent , i.e. , they generate the same topology on LaTeXMLMath .
Remark .
This theorem gives an alternative construction of test functions .
This idea is due to Lee LaTeXMLCite , see also §15.2 of LaTeXMLCite .
For LaTeXMLMath , let LaTeXMLMath consist of all functions LaTeXMLMath on LaTeXMLMath satisfying the conditions : LaTeXMLMath is an analytic function on LaTeXMLMath .
There exists a constant LaTeXMLMath such that LaTeXMLEquation .
For each LaTeXMLMath , define LaTeXMLMath by Equation ( LaTeXMLRef ) .
Then LaTeXMLMath is a Banach space with norm LaTeXMLMath .
Let LaTeXMLMath be the projective limit of LaTeXMLMath .
We can use the above theorem to conclude that LaTeXMLMath as vector spaces with the same topology .
Here the equality LaTeXMLMath requires the use of analytic extensions of test functions in LaTeXMLMath , which exists in view of Equation ( LaTeXMLRef ) .
Let LaTeXMLMath be any given number .
We have already shown that there exist constants LaTeXMLMath and LaTeXMLMath such that Equation ( LaTeXMLRef ) holds .
It follows that LaTeXMLEquation .
Hence for any LaTeXMLMath , there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
To show the converse , first note that by condition ( U2 ) there exist constants LaTeXMLMath such that LaTeXMLMath .
Next note that by Fernique ’ s theorem ( see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) we have LaTeXMLEquation .
Now , let LaTeXMLMath be any given number .
Choose LaTeXMLMath large enough such that LaTeXMLEquation .
With this choice of LaTeXMLMath we will show below that LaTeXMLEquation where LaTeXMLMath is the constant given by LaTeXMLEquation .
Observe that the theorem follows from Equations ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Finally , we prove Equation ( LaTeXMLRef ) .
Let LaTeXMLMath .
Then we can use an integral form of S-transform ( see LaTeXMLCite ) given by LaTeXMLEquation .
Hence for the above choice of LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Here by condition ( U1 ) , we have LaTeXMLMath for all LaTeXMLMath .
Therefore , LaTeXMLEquation .
By condition ( U3 ) , we have LaTeXMLEquation .
Put LaTeXMLMath and LaTeXMLMath to get LaTeXMLEquation .
LaTeXMLEquation Then integrate over LaTeXMLMath to obtain the inequality : LaTeXMLEquation .
Put Equation ( LaTeXMLRef ) into Equation ( LaTeXMLRef ) to get LaTeXMLEquation .
Now , by the inequality LaTeXMLMath , we have LaTeXMLEquation which is finite by the choice of LaTeXMLMath in Equation ( LaTeXMLRef ) .
From Equations ( LaTeXMLRef ) and ( LaTeXMLRef ) , we see that LaTeXMLEquation .
With this inequality and the choice of LaTeXMLMath in Equation ( LaTeXMLRef ) we can apply Lemma LaTeXMLRef ( see below ) and Equation ( LaTeXMLRef ) to show that for any LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath is given by Equation ( LaTeXMLRef ) .
Thus the inequality in Equation ( LaTeXMLRef ) holds and so the proof is completed .
∎ In the proof of the prevous theorem , we have used the next lemma from LaTeXMLCite .
Suppose LaTeXMLMath satisfies conditions ( U1 ) ( U2 ) ( U3 ) .
Let LaTeXMLMath be a complex-valued function on LaTeXMLMath satisfying the conditions : For any LaTeXMLMath , the function LaTeXMLMath is an entire function of LaTeXMLMath .
There exist constants LaTeXMLMath such that LaTeXMLEquation .
Let LaTeXMLMath be a number such that LaTeXMLMath .
Then there exist functions LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
A measure LaTeXMLMath on LaTeXMLMath is called a Hida measure associated with LaTeXMLMath if LaTeXMLMath and the linear functional LaTeXMLMath is continuous on LaTeXMLMath .
In this case , LaTeXMLMath induces a generalized function , denoted by LaTeXMLMath , in LaTeXMLMath such that LaTeXMLEquation .
Suppose LaTeXMLMath satisfies conditions ( U1 ) ( U2 ) ( U3 ) .
Then a measure LaTeXMLMath on LaTeXMLMath is a Hida measure with LaTeXMLMath if and only if LaTeXMLMath is supported by LaTeXMLMath for some LaTeXMLMath and LaTeXMLEquation .
Remarks .
( a ) The integrability condition in the theorem can be replaced by LaTeXMLEquation .
To verify this fact , just note that LaTeXMLMath and LaTeXMLMath are equivalent ( from the Remark of Lemma LaTeXMLRef ) and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
( b ) This theorem is due to Lee LaTeXMLCite for the case LaTeXMLMath .
See §15.2 of the book LaTeXMLCite for the case LaTeXMLMath .
In the case of LaTeXMLMath , we need special treatment since our Legendre transform method should be modified .
In order to take care of LaTeXMLMath case , we have to remove the assumption LaTeXMLEquation on LaTeXMLMath introduced in §2 , for example .
It will be discussed in the future .
On the other hand , there are references LaTeXMLCite , LaTeXMLCite discussed this case with a diffrent way from our point of view .
To prove the sufficiency , suppose LaTeXMLMath is supported by LaTeXMLMath for some LaTeXMLMath and Equation ( LaTeXMLRef ) holds .
Then for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By Theorem LaTeXMLRef , LaTeXMLMath and LaTeXMLMath are equivalent .
Hence Equation ( LaTeXMLRef ) implies that LaTeXMLMath and the linear functional LaTeXMLEquation is continuous on LaTeXMLMath .
Thus LaTeXMLMath is a Hida measure with LaTeXMLMath in LaTeXMLMath .
To prove the necessity , suppose LaTeXMLMath is a Hida measure inducing a generalized function LaTeXMLMath .
Then for all LaTeXMLMath , LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLMath are equivalent , the linear functional LaTeXMLMath is continuous with respect to LaTeXMLMath .
Hence there exist constants LaTeXMLMath such that for all LaTeXMLMath , LaTeXMLEquation .
Note that by continuity , Equations ( LaTeXMLRef ) and ( LaTeXMLRef ) also hold for all LaTeXMLMath defined in the Remark of Theorem LaTeXMLRef .
Now , with this LaTeXMLMath , we define a function LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath is the bilinear pairing on LaTeXMLMath .
Obviously , LaTeXMLMath is analytic on LaTeXMLMath .
On the other hand , apply Lemma LaTeXMLRef with LaTeXMLMath to get LaTeXMLEquation .
This shows that LaTeXMLMath and we have LaTeXMLEquation .
Then apply Equation ( LaTeXMLRef ) to the function LaTeXMLMath , LaTeXMLEquation .
Therefore , from Equation ( LaTeXMLRef ) with LaTeXMLMath we conclude that LaTeXMLEquation .
Note that LaTeXMLMath for LaTeXMLMath .
Hence Equation ( LaTeXMLRef ) implies that LaTeXMLEquation .
But LaTeXMLMath from Theorem LaTeXMLRef ( 2 ) with LaTeXMLMath .
Therefore , LaTeXMLEquation .
Now , choose LaTeXMLMath large enough such that LaTeXMLMath .
Then LaTeXMLMath .
Recall that LaTeXMLMath is increasing .
Hence LaTeXMLEquation .
Note that LaTeXMLMath and so LaTeXMLMath .
Thus we conclude that LaTeXMLEquation .
This inequality implies that LaTeXMLMath is supported by LaTeXMLMath and Equation ( LaTeXMLRef ) holds .
∎ ( Poisson noise measure ) Let LaTeXMLMath be the Poisson measure on LaTeXMLMath given by LaTeXMLEquation .
It has been shown LaTeXMLCite that the Poisson noise measure induces a generalized function in LaTeXMLMath .
Thus by Theorem LaTeXMLRef and Example LaTeXMLRef we have the integrability condition LaTeXMLEquation for some LaTeXMLMath .
( Grey noise measure ) Let LaTeXMLMath .
The grey noise measure on LaTeXMLMath is the measure LaTeXMLMath having the characteristic function LaTeXMLEquation where LaTeXMLMath is the Mittag-Leffler function with parameter LaTeXMLMath ; LaTeXMLEquation .
Here LaTeXMLMath is the Gamma function .
This measure was introduced by Schneider LaTeXMLCite .
It is shown in LaTeXMLCite that LaTeXMLMath is a Hida measure which induces a generalized function LaTeXMLMath in LaTeXMLMath .
Therefore by Theorem LaTeXMLRef and Example LaTeXMLRef the grey noise measure LaTeXMLMath satisfies LaTeXMLEquation for some LaTeXMLMath .
Throughout this paper we study the existence of irreducible curves LaTeXMLMath on smooth projective surfaces LaTeXMLMath with singular points of prescribed topological types LaTeXMLMath .
There are necessary conditions for the existence of the type LaTeXMLMath for some fixed divisor LaTeXMLMath on LaTeXMLMath and suitable coefficients LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , and the main sufficient condition that we find is of the same type , saying it is asymptotically optimal .
An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up LaTeXMLMath of the form LaTeXMLMath , deduced from the Kawamata-Vieweg Vanishing Theorem .
A large part of the paper is devoted to the investigation of our conditions on ruled surfaces , products of elliptic curves , surfaces in LaTeXMLMath , and K3-surfaces .
cm12 [ General Assumptions and Notations ] Throughout this paper LaTeXMLMath will be a smooth projective surface over LaTeXMLMath .
Given distinct points LaTeXMLMath , we denote by LaTeXMLMath the blow up of LaTeXMLMath in LaTeXMLMath , and the exceptional divisors LaTeXMLMath will be denoted by LaTeXMLMath , LaTeXMLMath .
We shall write LaTeXMLMath for the strict transform of a curve LaTeXMLMath .
For any smooth surface LaTeXMLMath we will denote by LaTeXMLMath the group of divisors on LaTeXMLMath and by LaTeXMLMath its canonical divisor .
If LaTeXMLMath is any divisor on LaTeXMLMath , LaTeXMLMath shall be a corresponding invertible sheaf .
LaTeXMLMath denotes the system of curves linearly equivalent to LaTeXMLMath , while we use the notation LaTeXMLMath for the system of curves algebraically equivalent to LaTeXMLMath ( cf .
LaTeXMLCite Ex .
V.1.7 ) , that is the reduction of the connected component of LaTeXMLMath , the Hilbert scheme of all curves on LaTeXMLMath , containing any curve algebraically equivalent to LaTeXMLMath ( cf .
LaTeXMLCite Chapter 15 ) .
We will use the notation LaTeXMLMath for the Picard group of LaTeXMLMath , that is LaTeXMLMath modulo linear equivalence ( denoted by LaTeXMLMath ) , LaTeXMLMath for the Néron-Severi group , that is LaTeXMLMath modulo algebraic equivalence ( denoted by LaTeXMLMath ) , and LaTeXMLMath for LaTeXMLMath modulo numerical equivalence ( denoted by LaTeXMLMath ) .
Note that for all examples of surfaces LaTeXMLMath which we consider in Section LaTeXMLRef LaTeXMLMath and LaTeXMLMath coincide .
Given a curve LaTeXMLMath we will write LaTeXMLMath for its arithmetical genus and LaTeXMLMath for the geometrical one .
Let LaTeXMLMath be a Zariski topological space .
We say a subset LaTeXMLMath is very general if it is an at most countable intersection of open dense subsets of LaTeXMLMath .
Some statement is said to hold for points LaTeXMLMath ( or LaTeXMLMath ) in very general position if there is a suitable very general subset LaTeXMLMath , contained in the complement of the closed subvariety LaTeXMLMath of LaTeXMLMath , such that the statement holds for all LaTeXMLMath .
The main results of this paper will only be valid for points in very general position .
Given distinct points LaTeXMLMath and non-negative integers LaTeXMLMath we denote by LaTeXMLMath the zero-dimensional subscheme of LaTeXMLMath defined by the ideal sheaf LaTeXMLMath with stalks LaTeXMLEquation .
We call a scheme of the type LaTeXMLMath a generic fat point scheme .
For a reduced curve LaTeXMLMath we define the zero-dimensional subscheme LaTeXMLMath of LaTeXMLMath via the ideal sheaf LaTeXMLMath with stalks LaTeXMLEquation where LaTeXMLMath is a local equation of LaTeXMLMath at LaTeXMLMath .
LaTeXMLMath is called the equisingularity ideal of the singularity LaTeXMLMath , and it is of course LaTeXMLMath whenever LaTeXMLMath is a smooth point .
If LaTeXMLMath are local coordinates of LaTeXMLMath at LaTeXMLMath , then LaTeXMLMath can be identified with the tangent space of the equisingular stratum in the semiuniversal deformation of LaTeXMLMath .
( cf .
LaTeXMLCite , LaTeXMLCite , and Definition LaTeXMLRef ) We call LaTeXMLMath the equisingularity scheme of LaTeXMLMath .
If LaTeXMLMath is any zero-dimensional scheme with ideal sheaf LaTeXMLMath and if LaTeXMLMath is any curve with ideal sheaf LaTeXMLMath , we define the residue scheme LaTeXMLMath by the ideal sheaf LaTeXMLMath with stalks LaTeXMLEquation where LaTeXMLMath is a local equation for LaTeXMLMath and “ LaTeXMLMath ” denotes the ideal quotient .
This naturally leads to the definition of the trace scheme LaTeXMLMath via the ideal sheaf LaTeXMLMath given by the exact sequence LaTeXMLEquation .
Given topological singularity types LaTeXMLCite 1.2 .
LaTeXMLMath and a divisor LaTeXMLMath , we denote by LaTeXMLMath the locally closed subspace of LaTeXMLMath of reduced curves in the linear system LaTeXMLMath having precisely LaTeXMLMath singular points of types LaTeXMLMath .
Analogously , LaTeXMLMath denotes the locally closed subspace of LaTeXMLMath of reduced curves having precisely LaTeXMLMath ordinary singular points of multiplicities LaTeXMLMath .
( cf .
LaTeXMLCite 1.3.2 ) The spaces LaTeXMLMath respectively LaTeXMLMath are the main objects of interest of this paper .
We say LaTeXMLMath is T-smooth at LaTeXMLMath if the germ LaTeXMLMath is smooth of the ( expected ) dimension LaTeXMLMath , where LaTeXMLMath respectively LaTeXMLMath with LaTeXMLMath .
By LaTeXMLCite Proposition 2.1 T-smoothness of LaTeXMLMath at LaTeXMLMath is implied by the vanishing of LaTeXMLMath .
It is the aim of this paper to give sufficient conditions for the non-emptiness of LaTeXMLMath in terms of the linear system LaTeXMLMath and invariants of the imposed singularities .
The results are generalisations of known results for LaTeXMLMath , and for an overview on these we refer to LaTeXMLCite Chapter 4 .
We basically follow the ideas described in LaTeXMLCite 4.1.2 .
The case of ordinary singularities ( Corollary LaTeXMLRef ) is treated by applying a vanishing theorem for generic fat point schemes ( Theorem LaTeXMLRef ) , and the more interesting case of prescribed topological types LaTeXMLMath is then dealt with by gluing local equations into a curve with ordinary singularities .
Upper bounds for the minimal possible degrees of these local equations can be taken from the LaTeXMLMath -case ( cf .
LaTeXMLCite Theorem 4.2 ) .
Thus the main results of this paper are the following theorems and their corollaries Corollary LaTeXMLRef and Corollary LaTeXMLRef .
Let LaTeXMLMath be non-negative integers , LaTeXMLMath with LaTeXMLMath , LaTeXMLMath and let LaTeXMLMath be a divisor satisfying the following three conditions LaTeXMLMath , LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath is nef .
Then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
In particular , LaTeXMLEquation .
Given LaTeXMLMath , not all zero , and LaTeXMLMath , LaTeXMLMath , in very general position .
Let LaTeXMLMath be very ample over LaTeXMLMath , and let LaTeXMLMath be such that LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath .
Then there exists a curve LaTeXMLMath with ordinary singular points of multiplicity LaTeXMLMath at LaTeXMLMath for LaTeXMLMath and no other singular points .
Furthermore , LaTeXMLEquation and in particular , LaTeXMLMath is T-smooth at LaTeXMLMath .
If in addition ( LaTeXMLRef ) LaTeXMLMath , then LaTeXMLMath can be chosen to be irreducible and reduced .
[ Existence ] Let LaTeXMLMath be singularity types , and suppose there exists an irreducible curve LaTeXMLMath with LaTeXMLMath ordinary singular points LaTeXMLMath of multiplicities LaTeXMLMath respectively as its only singularities such that LaTeXMLMath , for LaTeXMLMath , and LaTeXMLEquation .
Then there exists an irreducible curve LaTeXMLMath with LaTeXMLMath singular points of types LaTeXMLMath and LaTeXMLMath ordinary singular points of multiplicities LaTeXMLMath as its only singularities .
Here , of course , LaTeXMLMath and LaTeXMLMath .
See Definition LaTeXMLRef for the definition of LaTeXMLMath .
Of course , combining the vanishing theorem Theorem LaTeXMLRef with the existence theorems Theorem LaTeXMLRef and Theorem LaTeXMLRef we get sufficient numerical conditions for the existence of curves with certain singularities ( see Corollaries LaTeXMLRef and LaTeXMLRef , and see Section LaTeXMLRef for special surfaces ) .
Given any scheme LaTeXMLMath and any coherent sheaf LaTeXMLMath on LaTeXMLMath , we will often write LaTeXMLMath instead of LaTeXMLMath when no ambiguity can arise .
Moreover , if LaTeXMLMath is the invertible sheaf corresponding to a divisor LaTeXMLMath , we will usually use the notation LaTeXMLMath instead of LaTeXMLMath .
Similarly when considering tensor products over the structure sheaf of some scheme LaTeXMLMath we may sometimes just write LaTeXMLMath instead of LaTeXMLMath .
Section LaTeXMLRef is devoted to the proof of the vanishing theorem Theorem LaTeXMLRef , and Section LaTeXMLRef provides an important ingredient in this proof .
The following sections Section LaTeXMLRef and Section LaTeXMLRef are concerned with the existence theorems Theorem LaTeXMLRef and Theorem LaTeXMLRef , while in Section LaTeXMLRef we calculate the conditions which we have found in the case of ruled surfaces , products of elliptic curves , surfaces in LaTeXMLMath , and K3-surfaces .
Finally , in the appendix we gather some well known respectively fairly easy facts used throughout the paper for the convenience of the reader .
Let LaTeXMLMath be non-negative integers , LaTeXMLMath with LaTeXMLMath , LaTeXMLMath and let LaTeXMLMath be a divisor satisfying the following three conditions LaTeXMLMath , LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath is nef .
Then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
In particular , LaTeXMLEquation .
By the Kawamata–Viehweg Vanishing Theorem ( cf .
LaTeXMLCite and LaTeXMLCite ) it suffices to show that LaTeXMLMath is big and nef , i. e. we have to show : LaTeXMLMath , and LaTeXMLMath for any irreducible curve LaTeXMLMath in LaTeXMLMath .
Note that LaTeXMLMath , and thus by Hypothesis ( LaTeXMLRef ) we have LaTeXMLEquation which gives condition ( a ) .
For condition ( b ) we observe that an irreducible curve LaTeXMLMath on LaTeXMLMath is either the strict transform of an irreducible curve LaTeXMLMath in LaTeXMLMath or is one of the exceptional curves LaTeXMLMath .
In the latter case we have LaTeXMLEquation .
We may , therefore , assume that LaTeXMLMath is the strict transform of an irreducible curve LaTeXMLMath on LaTeXMLMath having multiplicity LaTeXMLMath at LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation and thus condition ( b ) is equivalent to LaTeXMLMath .
Since LaTeXMLMath is in very general position Lemma LaTeXMLRef applies in view of Corollary LaTeXMLRef .
Using the Hodge Index Theorem , Hypothesis ( LaTeXMLRef ) , Lemma LaTeXMLRef , and the Cauchy-Schwarz Inequality we get the following sequence of inequalities : LaTeXMLEquation where LaTeXMLMath is such that LaTeXMLMath .
Since LaTeXMLMath is nef , condition ( b ’ ) is satisfied as soon as we have LaTeXMLEquation .
If this is not fulfilled , then LaTeXMLMath for all LaTeXMLMath , and thus LaTeXMLEquation .
Hence , for the remaining considerations ( b ’ ) may be replaced by the worst case LaTeXMLEquation .
Note that since the LaTeXMLMath are in very general position and LaTeXMLMath we have that LaTeXMLMath and LaTeXMLMath ( cf .
Corollary LaTeXMLRef ) .
If LaTeXMLMath then we are done by the Hodge Index Theorem and Hypothesis ( LaTeXMLRef ) , since LaTeXMLMath is nef : LaTeXMLEquation .
It remains to consider the case LaTeXMLMath which is covered by Hypothesis ( LaTeXMLRef ) .
For the “ in particular ” part we just note that LaTeXMLEquation .
LaTeXMLEquation ∎ Choosing the constant LaTeXMLMath in Theorem LaTeXMLRef , then LaTeXMLMath and thus LaTeXMLMath .
We therefore get the following corollary , which has the advantage that the conditions look simpler , and that the hypotheses on the “ exceptional ” curves are not too hard .
Let LaTeXMLMath , and LaTeXMLMath be a divisor satisfying the following three conditions LaTeXMLMath , LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath is nef .
Then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
In particular , LaTeXMLEquation .
Condition ( LaTeXMLRef ) respectively Condition ( LaTeXMLRef ) are in several respects “ expectable ” .
First , Theorem LaTeXMLRef is a corollary of the Kawamata–Viehweg Vanishing Theorem , and if we take all LaTeXMLMath to be zero , our assumptions should basically be the same , i. e. LaTeXMLMath nef and big .
The latter is more or less just ( LaTeXMLRef ) respectively ( LaTeXMLRef ) .
Secondly , we want to apply the theorem to an existence problem .
A divisor being nef means it is somehow close to being effective , or better its linear system is close to being non-empty .
If we want that some linear system LaTeXMLMath contains a curve with certain properties , then it seems not to be so unreasonable to restrict to systems where already LaTeXMLMath , or even LaTeXMLMath with LaTeXMLMath some fixed divisor , is of positive dimension , thus nef .
In many interesting examples , such as LaTeXMLMath , Condition ( LaTeXMLRef ) respectively ( LaTeXMLRef ) turn out to be obsolete or easy to handle .
So finally the most restrictive obstruction seems to be ( LaTeXMLRef ) respectively ( LaTeXMLRef ) .
If we consider the situation where the largest multiplicity LaTeXMLMath occurs in a large number , more precisely , if LaTeXMLMath with LaTeXMLMath , then Condition ( LaTeXMLRef ) comes down to LaTeXMLMath .
Even though we said that Condition ( LaTeXMLRef ) was the really restrictive condition we would like to understand better what Condition ( LaTeXMLRef ) means .
We therefore show in Appendix LaTeXMLRef that an algebraic system LaTeXMLMath of dimension greater than zero with LaTeXMLMath irreducible and LaTeXMLMath gives rise to a fibration LaTeXMLMath of LaTeXMLMath over a smooth projective curve LaTeXMLMath whose fibres are just the elements of LaTeXMLMath .
Let LaTeXMLMath be in very general position , LaTeXMLMath , and let LaTeXMLMath be an irreducible curve with LaTeXMLMath , then LaTeXMLEquation .
A proof for the above lemma in the case LaTeXMLMath can be found in LaTeXMLCite and in the case LaTeXMLMath in LaTeXMLCite .
Here we just extend the arguments given there to the slightly more general situation .
For better estimates of the self intersection number of curves where one has some knowledge on equisingular deformations inside the algebraic system see LaTeXMLCite .
With the notation of Lemma LaTeXMLRef respectively Corollary LaTeXMLRef the assumption in Lemma LaTeXMLRef could be formulated more precisely as “ let LaTeXMLMath be an irreducible curve such that LaTeXMLMath ” , or “ let LaTeXMLMath ” .
Note , that one can not expect to get rid of the “ LaTeXMLMath ” .
E. g. LaTeXMLMath , the projective plane blown up in a point LaTeXMLMath , and LaTeXMLMath the strict transform of a line through LaTeXMLMath .
Let now LaTeXMLMath , LaTeXMLMath and LaTeXMLMath be any point .
Then there is of course a ( unique ) curve LaTeXMLMath through LaTeXMLMath , but LaTeXMLMath .
[ Idea of the proof ] Set LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , where w. l. o. g. LaTeXMLMath .
By assumption there is a family LaTeXMLMath in LaTeXMLMath satisfying the requirements of Lemma LaTeXMLRef .
Setting LaTeXMLMath the proof is done in three steps : LaTeXMLMath .
( Lemma LaTeXMLRef ) LaTeXMLMath .
( Lemma LaTeXMLRef ) LaTeXMLMath , but this degree is just LaTeXMLMath .
LaTeXMLMath Given LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be a non-trivial family of curves in LaTeXMLMath such that LaTeXMLMath is a smooth curve , LaTeXMLMath for all LaTeXMLMath , LaTeXMLMath for any LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
Then for LaTeXMLMath LaTeXMLEquation i. e. there is a non-trivial section of the normal bundle of LaTeXMLMath , vanishing at LaTeXMLMath to the order of at least LaTeXMLMath for LaTeXMLMath .
We stick to the convention LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , and we set LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a small disc around LaTeXMLMath with coordinate LaTeXMLMath , and choose coordinates LaTeXMLMath on LaTeXMLMath around LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , where LaTeXMLMath locally at LaTeXMLMath ( for LaTeXMLMath ) .
We view LaTeXMLMath as a non-trivial deformation of LaTeXMLMath , which implies that the image of LaTeXMLMath under the Kodaira-Spencer map is a non-zero section LaTeXMLMath of LaTeXMLMath .
LaTeXMLMath is locally at LaTeXMLMath given by LaTeXMLMath .
Show that LaTeXMLMath , which are the stalks of LaTeXMLMath at the LaTeXMLMath , and hence LaTeXMLMath is actually a global section of the subsheaf LaTeXMLMath .
Set LaTeXMLMath .
By assumption for any LaTeXMLMath the multiplicity of LaTeXMLMath at LaTeXMLMath is at least LaTeXMLMath , i. e. LaTeXMLMath for every fixed complex number LaTeXMLMath .
Hence , LaTeXMLMath for every LaTeXMLMath .
LaTeXMLRef .
On the other hand we have LaTeXMLEquation .
Since LaTeXMLMath , we have LaTeXMLMath , and hence LaTeXMLMath .
For this note that LaTeXMLMath , if LaTeXMLMath .
∎ Given LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
The canonical morphism LaTeXMLMath induces a surjective morphism LaTeXMLMath on the level of global sections .
If LaTeXMLMath , but not in LaTeXMLMath , then LaTeXMLMath induces a non-zero section LaTeXMLMath in LaTeXMLMath .
Set LaTeXMLMath .
We start with the structure sequence for LaTeXMLMath : LaTeXMLEquation .
Tensoring with the locally free sheaf LaTeXMLMath and then applying LaTeXMLMath we get a morphism : LaTeXMLEquation .
Now tensoring by LaTeXMLMath over LaTeXMLMath we have an exact sequence : LaTeXMLEquation .
And finally taking global sections , we end up with : LaTeXMLEquation .
Since the sheaves we look at are actually LaTeXMLMath -sheaves and since LaTeXMLMath is a closed subscheme of LaTeXMLMath , the global sections of the sheaves as sheaves on LaTeXMLMath and as sheaves on LaTeXMLMath coincide ( cf .
LaTeXMLCite III.2.10 - for more details , see Corollary LaTeXMLRef ) .
Furthermore , LaTeXMLMath .
Thus it suffices to show that LaTeXMLMath .
Since LaTeXMLMath is an isomorphism , we have that LaTeXMLMath is finite .
LaTeXMLRef .
Hence , by Lemma LaTeXMLRef , LaTeXMLEquation .
LaTeXMLEquation Let now LaTeXMLMath be given .
We have to show that LaTeXMLMath , i. e. LaTeXMLMath for every LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
Thus we may assume LaTeXMLMath .
As we have shown , LaTeXMLEquation where LaTeXMLMath is a local equation of LaTeXMLMath at LaTeXMLMath .
Therefore , there exists a LaTeXMLMath such that LaTeXMLMath ( note that LaTeXMLMath ! ) .
But then LaTeXMLMath is just the residue class of LaTeXMLMath in LaTeXMLMath , and is thus zero .
∎ Using the notation of the idea of the proof given on page LaTeXMLRef , we have , by Lemma LaTeXMLRef , a non-zero section LaTeXMLMath .
This lifts under the surjection LaTeXMLMath to a section LaTeXMLMath which is not in the kernel of LaTeXMLMath .
Again setting LaTeXMLMath , by Lemma LaTeXMLRef , we have a non-zero section LaTeXMLMath , where by the projection formula the latter is just LaTeXMLMath .
Since LaTeXMLMath has a global section and since LaTeXMLMath is irreducible and reduced , we get by Lemma LaTeXMLRef : LaTeXMLEquation ∎ Given LaTeXMLMath , not all zero , and LaTeXMLMath , LaTeXMLMath , in very general position .
Let LaTeXMLMath be very ample over LaTeXMLMath , and let LaTeXMLMath be such that LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath .
Then there exists a curve LaTeXMLMath with ordinary singular points of multiplicity LaTeXMLMath at LaTeXMLMath for LaTeXMLMath and no other singular points .
Furthermore , LaTeXMLEquation and in particular , LaTeXMLMath is T-smooth at LaTeXMLMath .
If in addition ( 4.1 ) LaTeXMLMath , then LaTeXMLMath can be chosen to be irreducible and reduced .
[ Idea of the proof ] For each LaTeXMLMath find a curve LaTeXMLMath with an ordinary singular point of multiplicity LaTeXMLMath and show that this linear system has no other base points than LaTeXMLMath .
Then the generic element is smooth outside LaTeXMLMath and has an ordinary singularity of multiplicity LaTeXMLMath in LaTeXMLMath .
W. l. o. g. LaTeXMLMath for all LaTeXMLMath .
For the convenience of notation we set LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is very ample , we may choose smooth curves LaTeXMLMath through LaTeXMLMath and LaTeXMLMath for LaTeXMLMath ( cf .
Lemma LaTeXMLRef ) .
Writing LaTeXMLMath for LaTeXMLMath we introduce zero-dimensional schemes LaTeXMLMath for LaTeXMLMath by LaTeXMLEquation [ Step 1 ] LaTeXMLMath .
By Condition ( LaTeXMLRef ) we get LaTeXMLEquation and the exact sequence LaTeXMLEquation implies with the aid of ( LaTeXMLRef ) LaTeXMLEquation ( LaTeXMLRef ) and ( LaTeXMLRef ) allow us to apply Lemma LaTeXMLRef in order to obtain LaTeXMLEquation [ Step 2 ] For each LaTeXMLMath there exists a curve LaTeXMLMath with an ordinary singular point of multiplicity LaTeXMLMath at LaTeXMLMath and with LaTeXMLMath for LaTeXMLMath .
Consider the exact sequence LaTeXMLEquation twisted by LaTeXMLMath and the corresponding long exact cohomology sequence LaTeXMLEquation .
Thus we may choose the LaTeXMLMath to be given by a section in LaTeXMLMath where the LaTeXMLMath tangent directions at LaTeXMLMath are all different .
[ Step 3 ] The base locus of LaTeXMLMath is LaTeXMLMath .
Suppose LaTeXMLMath was an additional base point and define the zero-dimensional scheme LaTeXMLMath by LaTeXMLEquation .
Choosing a generic , and thus smooth , curve LaTeXMLMath through LaTeXMLMath we may deduce as in Step 1 LaTeXMLEquation and thus as in Step 2 LaTeXMLEquation .
But by assumption LaTeXMLMath is a base point , and thus LaTeXMLEquation which gives us the desired contradiction .
[ Step 4 ] LaTeXMLMath with an ordinary singular point of multiplicity LaTeXMLMath at LaTeXMLMath for LaTeXMLMath and no other singular points .
Because of Step 2 the generic element in LaTeXMLMath has an ordinary singular point of multiplicity LaTeXMLMath at LaTeXMLMath and is by Bertini ’ s Theorem ( cf .
LaTeXMLCite III.10.9.2 ) smooth outside its base locus .
For two generic curves LaTeXMLMath the intersection multiplicity in LaTeXMLMath is LaTeXMLMath .
Thus , if Condition ( LaTeXMLRef ) is fulfilled then LaTeXMLMath and LaTeXMLMath have an additional intersection point outside the base locus of LaTeXMLMath , and Bertini ’ s Theorem ( cf .
LaTeXMLCite §47 , Satz 4 ) implies that the generic curve in LaTeXMLMath is irreducible .
[ Step 5 ] LaTeXMLMath , by Equation ( LaTeXMLRef ) .
[ Step 6 ] LaTeXMLMath is T-smooth at LaTeXMLMath .
By LaTeXMLCite , Lemma 2.7 , we have LaTeXMLEquation and thus by Step 5 LaTeXMLEquation which proves the claim .
∎ Let LaTeXMLMath , not all zero , LaTeXMLMath , and let LaTeXMLMath be very ample over LaTeXMLMath .
Suppose LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is nef , and LaTeXMLMath for all LaTeXMLMath .
Then for LaTeXMLMath in very general position there exists a curve LaTeXMLMath with ordinary singular points of multiplicity LaTeXMLMath at LaTeXMLMath for LaTeXMLMath and no other singular points .
Furthermore , LaTeXMLEquation and in particular , LaTeXMLMath is T-smooth in LaTeXMLMath .
If in addition ( 4.5 ) LaTeXMLMath , then LaTeXMLMath can be chosen to be irreducible and reduced .
Follows from Theorem LaTeXMLRef and Corollary LaTeXMLRef .
∎ In view of Condition ( LaTeXMLRef ) Condition ( LaTeXMLRef ) is satisfied if the following condition is fulfilled : LaTeXMLEquation .
Suppose ( LaTeXMLRef ) was not satisfied , then LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLEquation which implies that ( LaTeXMLRef ) is sufficient .
∎ [ Notation ] In the following we will denote by LaTeXMLMath , respectively by LaTeXMLMath the LaTeXMLMath -vector spaces of polynomials of degree LaTeXMLMath , respectively of degree at most LaTeXMLMath .
If LaTeXMLMath we denote by LaTeXMLMath for LaTeXMLMath the homogeneous part of degree LaTeXMLMath of LaTeXMLMath , and thus LaTeXMLMath .
By LaTeXMLMath we will denote the coordinates of LaTeXMLMath with respect to the basis LaTeXMLMath .
For any LaTeXMLMath the tautological family LaTeXMLEquation induces a deformation of the plane curve singularity LaTeXMLMath whose base space is the germ LaTeXMLMath of LaTeXMLMath at LaTeXMLMath .
Given any deformation LaTeXMLMath of a plane curve singularity LaTeXMLMath , we will denote by LaTeXMLMath the germ of the equisingular stratum of LaTeXMLMath .
Thus , fixed an LaTeXMLMath , LaTeXMLMath is the ( local ) equisingular stratum of LaTeXMLMath at LaTeXMLMath .
We say the family LaTeXMLMath is T-smooth at LaTeXMLMath if for any LaTeXMLMath there exists a LaTeXMLMath with LaTeXMLMath such that LaTeXMLMath is given by equations LaTeXMLEquation with LaTeXMLMath where LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , and where LaTeXMLMath is the codimension of the equisingular stratum in the base space of the semiuniversal deformation of LaTeXMLMath .
A polynomial LaTeXMLMath is said to be a good representative of the singularity type LaTeXMLMath in LaTeXMLMath if it meets the following conditions : LaTeXMLMath , LaTeXMLMath , LaTeXMLMath is reduced , and LaTeXMLMath is T-smooth at LaTeXMLMath .
Given a singularity type LaTeXMLMath we define LaTeXMLMath to be the minimal number LaTeXMLMath such that LaTeXMLMath has a good representative of degree LaTeXMLMath .
The condition for T-smoothness just means that for any LaTeXMLMath the equisingular stratum LaTeXMLMath is smooth at the point LaTeXMLMath of the expected codimension in LaTeXMLMath .
Note that for a polynomial of degree LaTeXMLMath the highest homogeneous part LaTeXMLMath defines the normal cone , i. e. the intersection of the curve LaTeXMLMath with the line at infinity in LaTeXMLMath , where LaTeXMLMath is the homogenisation of LaTeXMLMath .
Thus the condition “ LaTeXMLMath reduced ” in the definition of a good representative just means that the line at infinity intersects the curve transversally in LaTeXMLMath different points .
If LaTeXMLMath is an irreducible polynomial such that LaTeXMLMath is the only singular point of the plane curve LaTeXMLMath , then a linear change of coordinates of the type LaTeXMLMath will ensure that the dehomogenisation LaTeXMLMath of LaTeXMLMath satisfies “ LaTeXMLMath reduced ” .
Note for this that the coordinate change corresponds to choosing a line in LaTeXMLMath , not passing through LaTeXMLMath and meeting the curve in LaTeXMLMath distinct points .
Therefore , the bounds for LaTeXMLMath given in LaTeXMLCite Theorem 4.2 and Remark 4.3 do apply here .
For refined results using the techniques of the following proof we refer to LaTeXMLCite .
[ Existence ] Let LaTeXMLMath be singularity types , and suppose there exists an irreducible curve LaTeXMLMath with LaTeXMLMath ordinary singular points LaTeXMLMath of multiplicities LaTeXMLMath respectively as its only singularities such that LaTeXMLMath , for LaTeXMLMath , and LaTeXMLEquation .
Then there exists an irreducible curve LaTeXMLMath with LaTeXMLMath singular points of types LaTeXMLMath and LaTeXMLMath ordinary singular points of multiplicities LaTeXMLMath as its only singularities .
LaTeXMLMath and LaTeXMLMath .
[ Idea of the proof ] The basic idea is to glue locally at the LaTeXMLMath equations of good representatives for the LaTeXMLMath into the curve LaTeXMLMath .
Let us now explain more detailed what we mean by this .
If LaTeXMLMath , LaTeXMLMath , are good representatives of the LaTeXMLMath , then we are looking for a family LaTeXMLMath , LaTeXMLMath , in LaTeXMLMath which in local coordinates LaTeXMLMath at LaTeXMLMath looks like LaTeXMLEquation where the LaTeXMLMath should be convergent power series in LaTeXMLMath with LaTeXMLMath .
Replacing LaTeXMLMath by some arbitrarily small multiple LaTeXMLMath the curve defined by LaTeXMLMath is an arbitrarily small deformation of LaTeXMLMath inside some suitable linear system , thus it is smooth outside LaTeXMLMath and has ordinary singular points in LaTeXMLMath .
For LaTeXMLMath , on the other hand , LaTeXMLMath can be transformed , by LaTeXMLMath , into a member of some family LaTeXMLEquation with LaTeXMLEquation .
Using now the T-smoothness property of LaTeXMLMath , LaTeXMLMath , we can choose the LaTeXMLMath such that this family is equisingular .
Hence , for small LaTeXMLMath , the curve given by LaTeXMLMath will have the right singularities at the LaTeXMLMath .
Finally , the knowledge on the singularities of the curve defined by LaTeXMLMath and the conservation of Milnor numbers will ensure that the curve given by LaTeXMLMath has no further singularities , for LaTeXMLMath sufficiently small .
The proof will be done in several steps .
First of all we are going to fix some notation by choosing a basis of LaTeXMLMath which reflects the “ independence ” of the coordinates at the different LaTeXMLMath ensured by LaTeXMLMath ( Step 1.1 ) , and by choosing good representatives for the LaTeXMLMath ( Step 1.2 ) .
In a second step we are making an “ Ansatz ” for the family LaTeXMLMath , and , for the local investigation of the singularity type , we are switching to some other families LaTeXMLMath , LaTeXMLMath ( Step 2.1 ) .
We , then , reduce the problem of LaTeXMLMath , for LaTeXMLMath small , having the right singularities to a question about the equisingular strata of some families of polynomials ( Step 2.2 ) , which in Step 2.3 will be solved .
The final step serves to show that the curves LaTeXMLMath have only the singularities which we controlled in the previous steps .
[ Step 1.1 ] Parametrise LaTeXMLMath .
Consider the following exact sequence : LaTeXMLEquation .
Since LaTeXMLMath , the long exact cohomology sequence gives LaTeXMLEquation where LaTeXMLMath are local coordinates of LaTeXMLMath .
We , therefore , can find a basis LaTeXMLMath of LaTeXMLMath , with LaTeXMLMath , such that LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath is the curve defined by LaTeXMLMath , LaTeXMLMath for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath Let us now denote the coordinates of LaTeXMLMath w. r. t. this basis by LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Thus the family LaTeXMLEquation parametrises LaTeXMLMath .
[ Step 1.2 ] By the definition of LaTeXMLMath and since LaTeXMLMath , we may choose good representatives LaTeXMLEquation for the LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
We should remark here that for any LaTeXMLMath the polynomial LaTeXMLMath is also a good representative , and thus , replacing LaTeXMLMath by LaTeXMLMath , we may assume that the LaTeXMLMath are arbitrarily close to LaTeXMLMath .
[ Step 2 ] We are going to glue the good representatives for the LaTeXMLMath into the curve LaTeXMLMath .
More precisely , we are constructing a subfamily LaTeXMLMath , LaTeXMLMath , in LaTeXMLMath such that , if LaTeXMLMath denotes the curve defined by LaTeXMLMath , LaTeXMLMath are the only singular points of the irreducible reduced curve LaTeXMLMath , and they are ordinary singularities of multiplicities LaTeXMLMath , for LaTeXMLMath , and LaTeXMLMath , for LaTeXMLMath respectively , locally in LaTeXMLMath , LaTeXMLMath , the LaTeXMLMath , for small LaTeXMLMath , can be transformed into members of a fixed LaTeXMLMath -equisingular family , while for LaTeXMLMath and LaTeXMLMath small LaTeXMLMath has an ordinary singularity of multiplicity LaTeXMLMath in LaTeXMLMath .
[ Step 2.1 ] “ Ansatz ” and first reduction for a local investigation .
Let us make the following “ Ansatz ” : LaTeXMLEquation .
LaTeXMLEquation This gives rise to a family LaTeXMLEquation with LaTeXMLMath and LaTeXMLMath where LaTeXMLMath with LaTeXMLMath .
Fixing LaTeXMLMath , in local coordinates at LaTeXMLMath the family looks like LaTeXMLEquation with LaTeXMLEquation .
For LaTeXMLMath the transformation LaTeXMLMath is indeed a coordinate transformation , and thus LaTeXMLMath is contact equivalent LaTeXMLMath be two convergent power series in LaTeXMLMath indeterminates .
We call LaTeXMLMath and LaTeXMLMath contact equivalent , if LaTeXMLMath , and we write in this case LaTeXMLMath .
Equivalently , we could ask the germs LaTeXMLMath and LaTeXMLMath to be isomorphic , that is , ask the singularities to be analytically equivalent .
C. f. LaTeXMLCite Definition 9.1.1 and Definition 3.4.19. to LaTeXMLEquation .
Note that for this new family in LaTeXMLMath we have LaTeXMLEquation and hence it gives rise to a deformation of LaTeXMLMath .
[ Step 2.2 ] Reduction to the investigation of the equisingular strata of certain families of polynomials .
It is basically our aim to verify the LaTeXMLMath as convergent power series in LaTeXMLMath such that the corresponding family is equisingular .
However , since the LaTeXMLMath are power series in LaTeXMLMath and LaTeXMLMath , we can not right away apply the T-smoothness property of LaTeXMLMath , but we rather have to reduce to polynomials .
For this let LaTeXMLMath be the determinacy bound A power series LaTeXMLMath ( respectively the singularity LaTeXMLMath defined by LaTeXMLMath ) is said to be finitely determined with respect to some equivalence relation LaTeXMLMath if there exists some positive integer LaTeXMLMath such that LaTeXMLMath whenever LaTeXMLMath and LaTeXMLMath have the same LaTeXMLMath -jet .
If LaTeXMLMath is finitely determined , the smallest possible LaTeXMLMath is called the determinacy bound .
Isolated singularities are finitely determined with respect to analytical equivalence and hence , for LaTeXMLMath , as well with respect to topological equivalence .
C. f. LaTeXMLCite Theorem 9.1.3 and Footnote LaTeXMLRef .
of LaTeXMLMath and define LaTeXMLEquation .
Thus LaTeXMLMath is a family in LaTeXMLMath , and still LaTeXMLEquation .
We claim that it suffices to find LaTeXMLMath with LaTeXMLMath , such that the families LaTeXMLMath , LaTeXMLMath , are in the equisingular strata LaTeXMLMath , for LaTeXMLMath .
Since then we have , for small LaTeXMLMath , for two convergent power series LaTeXMLMath , means that the singularities LaTeXMLMath and LaTeXMLMath are topologically equivalent , that is , there exists a homeomorphism LaTeXMLMath with LaTeXMLMath , which of course means , that this is correct for suitably chosen representatives .
Note that if LaTeXMLMath and LaTeXMLMath are contact equivalent , then there exists even an analytic coordinate change LaTeXMLMath , that is , LaTeXMLMath implies LaTeXMLMath .
LaTeXMLMath , LaTeXMLEquation by the LaTeXMLMath -determinacy and since LaTeXMLMath is a coordinate change for LaTeXMLMath , which proves condition ( 2 ) .
Note that the singular points LaTeXMLMath will move with LaTeXMLMath .
It remains to verify conditions ( 1 ) and ( 3 ) .
Setting LaTeXMLMath , for LaTeXMLMath , we find that LaTeXMLEquation is an element inside the linear system LaTeXMLMath , where LaTeXMLMath .
Locally at LaTeXMLMath , LaTeXMLMath , LaTeXMLMath induces a deformation of LaTeXMLMath with equations LaTeXMLEquation and LaTeXMLEquation .
LaTeXMLEquation respectively .
Thus any element of LaTeXMLMath has ordinary singularities of multiplicity LaTeXMLMath at LaTeXMLMath for LaTeXMLMath , and since LaTeXMLMath has an ordinary singularity of multiplicity LaTeXMLMath at LaTeXMLMath for LaTeXMLMath , so has a generic element of LaTeXMLMath .
Moreover , a generic element of LaTeXMLMath has not more singular points than the special element LaTeXMLMath and has thus singularities precisely in LaTeXMLMath .
Replacing the LaTeXMLMath by some suitable multiples , we may assume that the curve defined by LaTeXMLMath is a generic element of LaTeXMLMath , which proves ( 1 ) .
Similarly , we note that LaTeXMLMath in local coordinates at LaTeXMLMath , for LaTeXMLMath , looks like LaTeXMLEquation .
LaTeXMLEquation and thus , for LaTeXMLMath sufficiently small , the singularity of LaTeXMLMath at LaTeXMLMath will be an ordinary singularity of multiplicity LaTeXMLMath , which gives ( 3 ) .
[ Step 2.3 ] Find LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , such that the families LaTeXMLMath , LaTeXMLMath , are in the equisingular strata LaTeXMLMath , for LaTeXMLMath .
In the sequel we adopt the notation of definition LaTeXMLRef adding indices LaTeXMLMath in the obvious way .
Since LaTeXMLMath is T-smooth at LaTeXMLMath , for LaTeXMLMath , there exist LaTeXMLMath and power series LaTeXMLMath , for LaTeXMLMath , such that the equisingular stratum LaTeXMLMath is given by the LaTeXMLMath equations LaTeXMLEquation .
Setting LaTeXMLMath we use the notation LaTeXMLMath and , similarly LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Moreover , setting LaTeXMLMath , we define an analytic map germ LaTeXMLEquation by LaTeXMLEquation and we consider the system of equations LaTeXMLEquation .
One easily verifies that LaTeXMLEquation .
Thus by the Inverse Function Theorem there exist LaTeXMLMath with LaTeXMLMath such that LaTeXMLEquation .
Now , setting LaTeXMLMath , the families LaTeXMLMath are in the equisingular strata LaTeXMLMath , for LaTeXMLMath .
[ Step 3 ] It finally remains to show that LaTeXMLMath , for small LaTeXMLMath , has no other singular points than LaTeXMLMath .
Since for any LaTeXMLMath the family LaTeXMLMath , LaTeXMLMath , induces a deformation of the singularity LaTeXMLMath there are , by the Conservation of Milnor Numbers LaTeXMLMath respectively of LaTeXMLMath at a point LaTeXMLMath : LaTeXMLMath ( cf .
LaTeXMLCite , Chapter 6 ) , ( Euclidean ) open neighbourhoods LaTeXMLMath and LaTeXMLMath such that for any LaTeXMLMath LaTeXMLMath , i. e. singular points of LaTeXMLMath come from singular points of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
For LaTeXMLMath condition ( LaTeXMLRef ) implies LaTeXMLEquation and thus LaTeXMLMath must be the only critical point of LaTeXMLMath in LaTeXMLMath , in particular , LaTeXMLEquation .
Let now LaTeXMLMath .
For LaTeXMLMath fixed , we consider the transformation defined by the coordinate change LaTeXMLMath , LaTeXMLEquation and the transformed equations LaTeXMLEquation .
Condition ( LaTeXMLRef ) then implies , LaTeXMLEquation .
For LaTeXMLMath very small LaTeXMLMath becomes very large , so that , by shrinking LaTeXMLMath we may suppose that for any LaTeXMLMath LaTeXMLEquation and that for any LaTeXMLMath there is an open neighbourhood LaTeXMLMath such that LaTeXMLEquation .
If we now take into account that LaTeXMLMath has precisely one critical point , LaTeXMLMath , on its zero level , and that the critical points on the zero level of LaTeXMLMath all contribute to the Milnor number LaTeXMLMath , then we get the following sequence of inequalities : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Hence all inequalities must have been equalities , and , in particular , LaTeXMLEquation which in view of Condition ( LaTeXMLRef ) finishes the proof .
Note that LaTeXMLMath , being a small deformation of the irreducible reduced curve LaTeXMLMath , will again be irreducible and reduced .
∎ Let LaTeXMLMath be very ample over LaTeXMLMath .
Suppose that LaTeXMLMath and LaTeXMLMath are topological singularity types with LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is nef , LaTeXMLMath LaTeXMLMath , then there is an irreducible reduced curve LaTeXMLMath in LaTeXMLMath with LaTeXMLMath singular points of topological types LaTeXMLMath as its only singularities .
This follows right away from Corollary LaTeXMLRef , Theorem LaTeXMLRef , and LaTeXMLCite Theorem 4.2 .
∎ One could easily simplify the above formulae by not distinguishing the cases LaTeXMLMath and LaTeXMLMath .
However , one would loose information .
On the other hand , knowing something more about the singularity type one could achieve much better results , applying the corresponding bounds for the LaTeXMLMath .
We leave it to the reader to apply the bounds .
( Cf .
LaTeXMLCite Remarks 4.3 , 4.8 , and 4.15 ) As we have already mentioned earlier the most restrictive of the above sufficient conditions is ( LaTeXMLRef ) , which could be characterised as a condition of the type LaTeXMLEquation where LaTeXMLMath is some fixed divisor class , LaTeXMLMath and LaTeXMLMath are some constants .
There are also necessary conditions of this type , e. g. LaTeXMLEquation which follows from the genus formula .
If LaTeXMLMath is an irreducible curve with precisely LaTeXMLMath singular points of types LaTeXMLMath and LaTeXMLMath its normalisation , then LaTeXMLMath , where LaTeXMLMath is the delta-invariant of LaTeXMLMath ( cf .
LaTeXMLCite II.11 ) .
Moreover , by definition LaTeXMLMath where LaTeXMLMath is the local delta-invariant at LaTeXMLMath , and it is well known that LaTeXMLMath , where LaTeXMLMath is the number of branches of the curve singularity LaTeXMLMath and LaTeXMLMath is its Milnor number ( cf .
LaTeXMLCite Chapter 10 ) .
Using now the genus formula we get LaTeXMLEquation .
LaTeXMLEquation See LaTeXMLCite Section 4.1 for considerations on the asymptotical optimality of the constant LaTeXMLMath .
In this section we are going to examine the conditions in the vanishing theorem ( Corollary LaTeXMLRef ) and in the corresponding existence results for various types of surfaces .
Unless otherwise stated , LaTeXMLMath is a positive integer , and LaTeXMLMath are non-negative , while at least one LaTeXMLMath is positive whenever we consider conditions for existence theorems .
Since in LaTeXMLMath there are no irreducible curves of self-intersection number zero , Condition ( LaTeXMLRef ) is redundant .
Moreover , Condition ( LaTeXMLRef ) takes in view of ( LaTeXMLRef ) the form LaTeXMLMath .
Corollary LaTeXMLRef thus takes the following form , where LaTeXMLMath is a generic line .
Let LaTeXMLMath be any integer such that LaTeXMLMath , LaTeXMLMath .
Then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
Now turning to the existence theorem Corollary LaTeXMLRef for generic fat point schemes , we , of course , find that Condition ( LaTeXMLRef ) is obsolete , and so is ( LaTeXMLRef ) , taking into account that ( LaTeXMLRef ) implies LaTeXMLMath .
But then Conditions ( LaTeXMLRef ) and ( LaTeXMLRef ) become also redundant in view of Condition ( LaTeXMLRef ) and equation ( LaTeXMLRef ) .
Thus the conditions in Corollary LaTeXMLRef reduce to LaTeXMLMath and LaTeXMLMath , and , similarly , the conditions in Corollary LaTeXMLRef reduce to LaTeXMLMath and LaTeXMLMath .
These results are much weaker than the previously known ones ( e. g. LaTeXMLCite Proposition 4.11 , where the factor LaTeXMLMath is replaced by LaTeXMLMath ) which use the Vanishing Theorem of Geng Xu ( cf .
LaTeXMLCite Theorem 3 ) , particularly designed for LaTeXMLMath .
– Using LaTeXMLMath with LaTeXMLMath instead of LaTeXMLMath in Corollary LaTeXMLRef does not improve the conditions .
Let LaTeXMLMath be a geometrically ruled surface with normalised bundle LaTeXMLMath ( in the sense of LaTeXMLCite V.2.8.1 ) .
The Néron-Severi group of LaTeXMLMath is LaTeXMLEquation with intersection matrix LaTeXMLEquation where LaTeXMLMath is a fibre of LaTeXMLMath , LaTeXMLMath a section of LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath .
LaTeXMLCite Theorem 1 there is some section LaTeXMLMath with LaTeXMLMath .
Since LaTeXMLMath is irreducible , by LaTeXMLCite V.2.20/21 LaTeXMLMath , and thus LaTeXMLMath .
For the canonical divisor we have LaTeXMLEquation where LaTeXMLMath is the genus of the base curve LaTeXMLMath .
In order to understand Condition ( LaTeXMLRef ) we have to examine special irreducible curves on LaTeXMLMath .
Let LaTeXMLMath be an irreducible curve with LaTeXMLMath and LaTeXMLMath .
Then we are in one of the following cases LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Moreover , if LaTeXMLMath , then LaTeXMLMath .
Since LaTeXMLMath is irreducible , we have LaTeXMLEquation .
If LaTeXMLMath , then LaTeXMLMath , but since the general element of LaTeXMLMath is irreducible , LaTeXMLMath has to be one , and we are in case ( LaTeXMLRef ) .
We , therefore , may assume that LaTeXMLMath .
Since LaTeXMLMath we have LaTeXMLEquation .
Combining this with ( LaTeXMLRef ) we get LaTeXMLMath .
Moreover , if LaTeXMLMath , then of course LaTeXMLMath , while , if LaTeXMLMath , then LaTeXMLMath by LaTeXMLCite V.2.21 , since otherwise LaTeXMLMath would have to be non-negative .
This brings us down to the cases ( LaTeXMLRef ) and ( LaTeXMLRef ) .
It remains to show , that LaTeXMLMath implies LaTeXMLMath .
But by assumption the elements of LaTeXMLMath are disjoint sections of the fibration LaTeXMLMath .
Thus , by Lemma LaTeXMLRef , LaTeXMLMath .
∎ If LaTeXMLMath has three disjoint sections , then LaTeXMLMath is isomorphic to LaTeXMLMath as a ruled surface , i. e. there is an isomorphism LaTeXMLMath such that the following diagram is commutative : LaTeXMLEquation .
See LaTeXMLCite p. 229 .
LaTeXMLMath is a locally trivial LaTeXMLMath -bundle , thus LaTeXMLMath is covered by a finite number of open affine subsets LaTeXMLMath with trivialisations LaTeXMLEquation which are linear on the fibres .
The three disjoint sections on LaTeXMLMath , say LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , give rise to three sections LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath on LaTeXMLMath .
For each point LaTeXMLMath there is a unique linear projectivity on the fibre LaTeXMLMath mapping the three points LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath to the standard basis LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath of LaTeXMLMath .
If LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , the projectivity is given by the matrix LaTeXMLEquation whose entries are rational functions in the coordinates of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Inserting for the coordinates local equations of the sections , LaTeXMLMath finally gives rise to an isomorphism of LaTeXMLMath -bundles LaTeXMLEquation mapping the sections LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath to the trivial sections .
The transition maps LaTeXMLEquation with LaTeXMLMath , are linear on the fibres and fix the three trivial sections .
Thus they must be the identity maps , which implies that the LaTeXMLMath , LaTeXMLMath , glue together to an isomorphism of ruled surfaces : LaTeXMLEquation ∎ Knowing the algebraic equivalence classes of irreducible curves in LaTeXMLMath which satisfy the assumptions in Condition ( LaTeXMLRef ) we can give a better formulation of the vanishing theorem in the case of geometrically ruled surfaces .
In order to do the same for the existence theorems , we have to study very ample divisors on LaTeXMLMath .
These , however , depend very much on the structure of the base curve LaTeXMLMath , and the general results which we give may be not the best possible .
Only in the case LaTeXMLMath we can give a complete investigation .
The geometrically ruled surfaces with base curve LaTeXMLMath are , up to isomorphism , just the Hirzebruch surfaces LaTeXMLMath , LaTeXMLMath .
Note that LaTeXMLMath , that is , algebraic equivalence and linear equivalence coincide .
Moreover , by LaTeXMLCite V.2.18 a divisor class LaTeXMLMath is very ample over LaTeXMLMath if and only if LaTeXMLMath and LaTeXMLMath .
The conditions throughout the existence theorems turn out to be optimal if we work with LaTeXMLMath , while for other choices of LaTeXMLMath they become more restrictive .
Let LaTeXMLMath , then LaTeXMLMath , and thus the optimality of the conditions follows from LaTeXMLMath , LaTeXMLMath , and for LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath implies LaTeXMLMath .
In the case LaTeXMLMath , we may choose an integer LaTeXMLMath such that the algebraic equivalence class LaTeXMLMath contains a very ample divisor LaTeXMLMath , e. g. LaTeXMLMath will do , if LaTeXMLMath is an elliptic curve .
LaTeXMLMath will be the degree of a suitable very ample divisor LaTeXMLMath on LaTeXMLMath .
Now LaTeXMLMath defines an embedding of LaTeXMLMath into some LaTeXMLMath such that the degree of the image LaTeXMLMath is just LaTeXMLMath .
Therefore LaTeXMLMath , unless LaTeXMLMath is linear , which implies LaTeXMLMath .
In particular , LaTeXMLMath as soon as LaTeXMLMath .
With the above choice of LaTeXMLMath we have LaTeXMLMath , and hence the generic curve in LaTeXMLMath is a smooth curve whose genus equals the genus of the base curve .
Given two integers LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , if LaTeXMLMath , LaTeXMLMath , if LaTeXMLMath , and LaTeXMLMath , if LaTeXMLMath .
For LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
Note that if the invariant LaTeXMLMath is non-positive , then LaTeXMLMath implies LaTeXMLEquation so that this inequality is fulfilled for any choice of LaTeXMLMath .
Setting LaTeXMLMath we have LaTeXMLEquation which is just ( LaTeXMLRef ) .
Similarly , by ( LaTeXMLRef b ) and Lemma LaTeXMLRef Condition ( LaTeXMLRef .i/ii/iii ) is satisfied .
LaTeXMLMath be an irreducible curve with LaTeXMLMath .
Then by Lemma LaTeXMLRef either LaTeXMLMath and LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
In the first case , LaTeXMLMath by ( LaTeXMLRef b.i ) .
In the second case , LaTeXMLMath by ( LaTeXMLRef b.ii ) .
And finally , in the third case , we have LaTeXMLMath by ( LaTeXMLRef b.iii ) .
Finally Condition ( LaTeXMLRef b ) implies that LaTeXMLMath is nef .
In order to see the last statement , we have to consider two cases .
LaTeXMLMath .
If LaTeXMLMath is irreducible , then we are in one of the following situations , by LaTeXMLCite V.2.20 : LaTeXMLMath and LaTeXMLMath , which , considering ( LaTeXMLRef b.i ) , implies LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath , which by ( LaTeXMLRef ) leads to LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath , which in view of ( LaTeXMLRef b.i ) , ( LaTeXMLRef ) , and LaTeXMLMath gives LaTeXMLEquation .
Hence , LaTeXMLMath is nef .
LaTeXMLMath .
In this case we may apply LaTeXMLCite V.2.21 and find that if LaTeXMLMath is irreducible , then we are in one of the following situations : LaTeXMLMath and LaTeXMLMath , which is treated as in Case 1 .
LaTeXMLMath and LaTeXMLMath , which , considering ( LaTeXMLRef b.i ) and ( LaTeXMLRef ) , implies LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath , which in view of ( LaTeXMLRef b.iii ) leads to LaTeXMLEquation .
LaTeXMLEquation Hence , LaTeXMLMath is nef .
∎ In order to obtain nice formulae we considered LaTeXMLMath in the formulation of the vanishing theorem .
For the existence theorems it turns out that the formulae look best if we work with LaTeXMLMath instead .
In the case of Hirzebruch surfaces this is just LaTeXMLMath .
Given integers LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , if LaTeXMLMath , LaTeXMLMath , if LaTeXMLMath , and LaTeXMLMath , if LaTeXMLMath , then for LaTeXMLMath in very general position there is an irreducible reduced curve LaTeXMLMath with ordinary singularities of multiplicities LaTeXMLMath at the LaTeXMLMath as only singularities .
Moreover , LaTeXMLMath is T-smooth at LaTeXMLMath .
Note that by ( LaTeXMLRef b ) and ( LaTeXMLRef b.i ) LaTeXMLMath , if LaTeXMLMath , and thus the inequality LaTeXMLEquation is fulfilled no matter what LaTeXMLMath is .
Noting that LaTeXMLMath , it is in view of Lemma LaTeXMLRef clear , that the Conditions ( LaTeXMLRef ) and ( LaTeXMLRef ) take the form ( LaTeXMLRef b ) respectively ( LaTeXMLRef b ) .
It , therefore , remains to show that ( LaTeXMLRef ) and ( LaTeXMLRef ) are obsolete , and that ( LaTeXMLRef ) takes the form ( LaTeXMLRef b ) , which in particular means that it is obsolete in the case LaTeXMLMath .
[ Step 1 ] ( LaTeXMLRef ) is obsolete .
If LaTeXMLMath , then LaTeXMLMath .
Since , moreover , LaTeXMLMath and LaTeXMLMath , Condition ( LaTeXMLRef b.i ) and ( LaTeXMLRef ) imply ( LaTeXMLRef ) , i. e. for all LaTeXMLMath LaTeXMLEquation [ Step 2 ] ( LaTeXMLRef ) takes the form ( LaTeXMLRef b ) .
We have to consider two cases .
LaTeXMLMath .
If LaTeXMLMath is irreducible , then we are in one of the following situations , by LaTeXMLCite V.2.20 : LaTeXMLMath and LaTeXMLMath , which , considering ( LaTeXMLRef b.i ) , implies LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath , which by ( LaTeXMLRef ) leads to LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath , which in view of ( LaTeXMLRef b.i ) , ( LaTeXMLRef ) , and LaTeXMLMath gives LaTeXMLEquation .
Hence , LaTeXMLMath is nef .
LaTeXMLMath .
In this case we may apply LaTeXMLCite V.2.21 and find that if LaTeXMLMath is irreducible , then we are in one of the following situations : LaTeXMLMath and LaTeXMLMath , which is treated as in Case 1 .
LaTeXMLMath and LaTeXMLMath , which , considering ( LaTeXMLRef b.i ) and ( LaTeXMLRef ) , implies LaTeXMLEquation .
LaTeXMLMath and LaTeXMLMath , which in view of ( LaTeXMLRef b.iii ) leads to LaTeXMLEquation .
Hence , LaTeXMLMath is nef .
[ Step 3 ] ( LaTeXMLRef ) is satisfied , and thus ( LaTeXMLRef ) is obsolete .
We have LaTeXMLEquation and LaTeXMLEquation .
Hence Condition ( LaTeXMLRef ) is equivalent to LaTeXMLEquation .
If LaTeXMLMath , then the situation is symmetric and we may w. l. o. g. assume that LaTeXMLMath .
Since by ( LaTeXMLRef b.i ) LaTeXMLMath we have to consider the following cases : Then LaTeXMLMath , and by assumption LaTeXMLMath and LaTeXMLMath .
We thus have LaTeXMLMath and LaTeXMLMath , which implies ( LaTeXMLRef ) .
By ( LaTeXMLRef ) we get LaTeXMLMath .
Then LaTeXMLMath , and hence LaTeXMLMath and LaTeXMLMath .
Thus LaTeXMLMath , which implies LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath , and thus LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
In any of the above cases ( LaTeXMLRef ) is satisfied .
Then LaTeXMLMath and LaTeXMLMath .
We therefore consider the following cases : Thus LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
By ( LaTeXMLRef b.i ) and ( LaTeXMLRef b.iii ) LaTeXMLMath , and of course LaTeXMLMath .
Moreover , since LaTeXMLMath , we have LaTeXMLMath .
And finally , LaTeXMLMath and LaTeXMLMath .
These considerations together ensure that ( LaTeXMLRef ) is fulfilled .
In this case ( LaTeXMLRef ) comes down to LaTeXMLEquation .
Then LaTeXMLMath .
Thus LaTeXMLMath , so that the inequality ( LaTeXMLRef ) is certainly satisfied .
Then LaTeXMLMath and LaTeXMLMath , so that again the inequality ( LaTeXMLRef ) is fulfilled .
( LaTeXMLRef ) reads just LaTeXMLEquation .
Then LaTeXMLMath , and thus LaTeXMLMath , which implies ( LaTeXMLRef ) .
Then LaTeXMLMath , and hence ( LaTeXMLRef ) is fulfilled .
∎ With the same LaTeXMLMath and LaTeXMLMath as above the conditions in the existence theorem Corollary LaTeXMLRef reduce to LaTeXMLMath , LaTeXMLMath LaTeXMLMath if LaTeXMLMath , LaTeXMLMath if LaTeXMLMath , and LaTeXMLMath , if LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be two smooth projective curves of genuses LaTeXMLMath and LaTeXMLMath respectively .
The surface LaTeXMLMath is naturally equipped with two fibrations LaTeXMLMath , LaTeXMLMath , and by abuse of notation we denote two generic fibres LaTeXMLMath resp .
LaTeXMLMath again by LaTeXMLMath resp .
LaTeXMLMath .
One can show that for a generic choice of the curves LaTeXMLMath and LaTeXMLMath the Neron-Severi group LaTeXMLMath of LaTeXMLMath is two-dimensional LaTeXMLMath and LaTeXMLMath are elliptic curves , generic means precisely , that they are not isogenous - see Section LaTeXMLRef .
For a further investigation of the Neron-Severi group of a product of two curves we refer to Appendix LaTeXMLRef .
with intersection matrix LaTeXMLEquation .
Thus , the only irreducible curves LaTeXMLMath with selfintersection LaTeXMLMath are the fibres LaTeXMLMath and LaTeXMLMath , and for any irreducible curve LaTeXMLMath the coefficients LaTeXMLMath and LaTeXMLMath must be non-negative .
Taking into account that LaTeXMLMath Corollary LaTeXMLRef comes down to the following .
Let LaTeXMLMath and LaTeXMLMath be two generic curves with LaTeXMLMath , LaTeXMLMath , and let LaTeXMLMath be integers satisfying LaTeXMLMath , and LaTeXMLMath , then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
We know that LaTeXMLMath has positive self-intersection and intersects any irreducible curve positive , is thus ample by Nakai-Moishezon .
But then we may find some integer LaTeXMLMath such that LaTeXMLMath is very ample .
We choose LaTeXMLMath minimal with this property for the existence theorem Corollary LaTeXMLRef , and we claim that the Conditions ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) become obsolete , while ( LaTeXMLRef ) and ( LaTeXMLRef ) take the form LaTeXMLMath , and LaTeXMLMath .
That is , under these hypotheses there is an irreducible curve in LaTeXMLMath , for any LaTeXMLMath , with precisely LaTeXMLMath ordinary singular points of multiplicities LaTeXMLMath .
( LaTeXMLRef ) becomes redundant in view of ( LaTeXMLRef c ) and since an irreducible curve LaTeXMLMath has non-negative coefficients LaTeXMLMath and LaTeXMLMath .
For ( LaTeXMLRef ) we look at ( LaTeXMLRef ) , which in this case takes the form LaTeXMLEquation .
LaTeXMLEquation However , in view of ( LaTeXMLRef c ) the factors and summands on the left-hand side are all positive , so that the inequality is fulfilled .
It remains to show that LaTeXMLMath for all LaTeXMLMath .
However , by the adjunction formula LaTeXMLMath , and by ( LaTeXMLRef c ) LaTeXMLMath .
Thus the claim is proved .
From these considerations we at once deduce the conditions for the existence of an irreducible curve in LaTeXMLMath , LaTeXMLMath , with prescribed singularities of arbitrary type , i. e. the conditions in Corollary LaTeXMLRef .
They come down to LaTeXMLMath , and LaTeXMLMath Let LaTeXMLMath and LaTeXMLMath be two elliptic curves , where LaTeXMLMath is a lattice and LaTeXMLMath is in the upper half plane of LaTeXMLMath .
We denote the natural group structure on each of the LaTeXMLMath by LaTeXMLMath and the neutral element by LaTeXMLMath .
Our interest lies in the study of the surface LaTeXMLMath .
This surface is naturally equipped with two fibrations LaTeXMLMath , LaTeXMLMath , and by abuse of notation we denote the fibres LaTeXMLMath resp .
LaTeXMLMath again by LaTeXMLMath resp .
LaTeXMLMath .
The group structures on LaTeXMLMath and LaTeXMLMath extend to LaTeXMLMath so that LaTeXMLMath itself is an abelian variety .
Moreover , for LaTeXMLMath the mapping LaTeXMLMath is an automorphism of abelian varieties .
Due to these translation morphisms we know that for any curve LaTeXMLMath the algebraic family of curves LaTeXMLMath covers the whole of LaTeXMLMath , and in particular LaTeXMLMath .
This also implies LaTeXMLMath .
Since LaTeXMLMath is an abelian surface , LaTeXMLMath , LaTeXMLMath , and the Picard number LaTeXMLMath ( cf .
LaTeXMLCite 4.11.2 and Ex .
2.5 ) .
But the Néron-Severi group of LaTeXMLMath contains the two independent elements LaTeXMLMath and LaTeXMLMath , so that LaTeXMLMath .
The general case LaTeXMLMath possessing a principle polarisation are parametrized by a countable number of surfaces in a three-dimensional space , and the Picard number of such an abelian surface is two unless it is contained the intersection of two or three of these surfaces ( cf .
LaTeXMLCite 11.2 ) .
See also LaTeXMLCite p. 286 and Proposition LaTeXMLRef .
is indeed LaTeXMLMath , however LaTeXMLMath might also be larger ( see Example LaTeXMLRef ) , in which case the additional generators may be chosen to be graphs of surjective morphisms from LaTeXMLMath to LaTeXMLMath ( cf .
LaTeXMLCite 3.2 Example 3 ) .
That is , LaTeXMLMath if and only if LaTeXMLMath and LaTeXMLMath are not isogenous .
Let LaTeXMLMath be an irreducible curve , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath is smooth , LaTeXMLMath , and LaTeXMLMath is an unramified covering of degree LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath for any LaTeXMLMath , and if LaTeXMLMath , then LaTeXMLMath .
If LaTeXMLMath , then the base curve LaTeXMLMath in the fibration LaTeXMLMath with fibre LaTeXMLMath , which exists according to Proposition LaTeXMLRef , is an elliptic curve .
If LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
If LaTeXMLMath is the graph of a morphism LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
The adjunction formula gives LaTeXMLEquation .
Since LaTeXMLMath covers the whole of LaTeXMLMath and LaTeXMLMath , the two irreducible curves LaTeXMLMath and LaTeXMLMath must intersect properly , that is , LaTeXMLMath is not a fibre of LaTeXMLMath .
But then the mapping LaTeXMLMath is a finite surjective morphism of degree LaTeXMLMath .
If LaTeXMLMath was a singular curve its normalisation would have to have arithmetical genus LaTeXMLMath and the composition of the normalisation with LaTeXMLMath would give rise to a surjective morphism from LaTeXMLMath to an elliptic curve , contradicting Hurwitz ’ s formula .
Hence , LaTeXMLMath is smooth and LaTeXMLMath .
We thus may apply the formula of Hurwitz to LaTeXMLMath and the degree of the ramification divisor LaTeXMLMath turns out to be LaTeXMLEquation .
The remaining case is treated analogously .
W. l. o. g. LaTeXMLMath .
For LaTeXMLMath we have LaTeXMLMath is a fibre of LaTeXMLMath , and since LaTeXMLMath is unramified , LaTeXMLMath .
Suppose LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLMath , and hence LaTeXMLMath , which contradicts the assumption LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLCite Lemma I.3.18 and Proposition I.3.22 imply that LaTeXMLMath .
W. l. o. g. LaTeXMLMath .
Let LaTeXMLMath .
We claim that LaTeXMLMath , and hence LaTeXMLMath .
Suppose LaTeXMLMath , then there is an LaTeXMLMath such that LaTeXMLMath , i. e. LaTeXMLMath and LaTeXMLMath .
Hence , LaTeXMLMath .
But , LaTeXMLMath , and thus LaTeXMLMath in contradiction to the choice of LaTeXMLMath .
LaTeXMLMath via LaTeXMLMath follows from ( i ) .
By ( iv ) we have LaTeXMLMath .
LaTeXMLMath is an isomorphism , and has thus degree one .
But LaTeXMLMath .
Thus we are done with ( iv ) .
∎ Let LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath .
The Picard number LaTeXMLMath is then either three or four , depending on whether the group LaTeXMLMath of endomorphisms of LaTeXMLMath fixing LaTeXMLMath is just LaTeXMLMath or larger .
Using LaTeXMLCite Theorem IV.4.19 and Exercise IV.4.11 we find the following classification .
LaTeXMLMath such that LaTeXMLMath , i. e. LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath where LaTeXMLMath is the diagonal in LaTeXMLMath and LaTeXMLMath is the graph of the morphism LaTeXMLMath of degree LaTeXMLMath , where LaTeXMLMath minimal with LaTeXMLMath and LaTeXMLMath .
Setting LaTeXMLMath , the intersection matrix is LaTeXMLEquation .
If e. g. LaTeXMLMath , then LaTeXMLMath and LaTeXMLEquation [ Case 2 ] LaTeXMLMath such that LaTeXMLMath , i. e. LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath where again LaTeXMLMath is the diagonal in LaTeXMLMath .
The intersection matrix in this case is LaTeXMLEquation .
Let LaTeXMLMath and LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath .
We consider the surjective morphisms LaTeXMLMath , LaTeXMLMath , induced by multiplication with the complex numbers LaTeXMLMath and LaTeXMLMath respectively .
Denoting by LaTeXMLMath the graph of LaTeXMLMath , we claim , LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath being an unramified covering , we can calculate its degree by counting the preimages of LaTeXMLMath .
If LaTeXMLMath with LaTeXMLMath , then LaTeXMLEquation and LaTeXMLEquation .
Moreover , the graphs LaTeXMLMath and LaTeXMLMath intersect only in the point LaTeXMLMath and the intersection is obviously transversal , so LaTeXMLMath .
Thus LaTeXMLMath is an example for a product of non-isomorphic elliptic curves with LaTeXMLMath , LaTeXMLMath , and intersection matrix LaTeXMLEquation .
See LaTeXMLCite p. 4 for examples LaTeXMLMath with LaTeXMLMath and intersection matrix LaTeXMLEquation .
If LaTeXMLMath and LaTeXMLMath are isogenous , then there are irreducible curves LaTeXMLMath with LaTeXMLMath arbitrarily large .
For this just note , that we have a curve LaTeXMLMath which is the graph of an isogeny LaTeXMLMath .
Denoting by LaTeXMLMath the morphism induced by the multiplication with LaTeXMLMath , we have a morphism LaTeXMLMath whose degree is just LaTeXMLMath .
But the degree is the intersection number of the graph with LaTeXMLMath .
The dual morphism of LaTeXMLMath has the the same degree , which then is the intersection multiplicity of its graph with LaTeXMLMath .
( cf .
LaTeXMLCite Ex .
IV.4.7 ) If LaTeXMLMath and LaTeXMLMath are isogenous , then LaTeXMLMath might very well contain smooth irreducible elliptic curves LaTeXMLMath which are neither isomorphic to LaTeXMLMath nor to LaTeXMLMath , and hence can not be the graph of an isogeny between LaTeXMLMath and LaTeXMLMath .
But being an elliptic curve we have LaTeXMLMath by the adjunction formula .
If now LaTeXMLMath , where the additional generators are graphs , then LaTeXMLMath with some LaTeXMLMath .
( cf .
LaTeXMLCite Ex .
10.6 ) Throughout the remaining part of the subsection we will restrict our attention to the general case , that is that LaTeXMLMath and LaTeXMLMath are not isogenous .
This makes the formulae look much nicer , since then LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be non-isogenous elliptic curves , LaTeXMLMath with LaTeXMLMath .
LaTeXMLMath if and only if LaTeXMLMath or LaTeXMLMath .
If LaTeXMLMath is an irreducible curve , then we are in one of the following cases : LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , and if we are in one of these cases , then there is an irreducible curve algebraically equivalent to LaTeXMLMath .
If LaTeXMLMath is an irreducible curve and LaTeXMLMath , then either LaTeXMLMath or LaTeXMLMath .
LaTeXMLMath is nef if and only if LaTeXMLMath .
LaTeXMLMath is ample if and only if LaTeXMLMath .
LaTeXMLMath is very ample if and only if LaTeXMLMath .
LaTeXMLMath if and only if LaTeXMLMath or LaTeXMLMath .
Let us first consider the case that LaTeXMLMath is irreducible .
If LaTeXMLMath or LaTeXMLMath , then LaTeXMLMath is algebraically equivalent to a multiple of a fibre of one of the projections LaTeXMLMath , LaTeXMLMath .
In this situation LaTeXMLMath and thus the irreducible curve LaTeXMLMath does not intersect any of the fibres properly .
Hence it must be a union of several fibres , and being irreducible it must be a fibre .
That is we are in one of the first two cases .
Suppose now that LaTeXMLMath .
Thus LaTeXMLMath intersects LaTeXMLMath properly , and LaTeXMLMath and LaTeXMLMath .
It now remains to show that the mentioned algebraic systems contain irreducible curves , which is clear for the first two of them .
Let therefore LaTeXMLMath and LaTeXMLMath be positive .
Then obviously the linear system LaTeXMLMath contains no fixed component , and being ample by ( v ) its general element is irreducible according to LaTeXMLCite Theorem 4.3.5 .
Follows from ( i ) and ( ii ) .
By definition LaTeXMLMath is nef if and only if LaTeXMLMath for every irreducible curve LaTeXMLMath .
Thus the claim is an immediate consequence of ( ii ) .
Since by the Nakai-Moishezon-Criterion ampleness depends only on the numerical class of a divisor , we may assume that LaTeXMLMath .
Moreover , by LaTeXMLCite Proposition 4.5.2 LaTeXMLMath is ample if and only if LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath and the effective divisor LaTeXMLMath , thus LaTeXMLMath is ample .
Conversely , if LaTeXMLMath is ample , then LaTeXMLMath and LaTeXMLMath , thus LaTeXMLMath .
By LaTeXMLCite Corollary 4.5.3 and ( v ) LaTeXMLMath is very ample .
If LaTeXMLMath , then the system LaTeXMLMath is basepoint free , which is an immediate consequence of the existence of the translation morphisms LaTeXMLMath , LaTeXMLMath .
But then LaTeXMLMath is globally generated and LaTeXMLMath is very ample .
Conversely , if LaTeXMLMath , then LaTeXMLMath is a divisor of degree LaTeXMLMath on the elliptic curve LaTeXMLMath and hence not very ample ( cf .
LaTeXMLCite Example IV.3.3.3 ) .
But then LaTeXMLMath is not very ample .
Analogously if LaTeXMLMath .
∎ In view of ( LaTeXMLRef d ) and Lemma LaTeXMLRef ( iv ) the Condition ( LaTeXMLRef ) becomes obsolete , and Corollary LaTeXMLRef has the following form , taking Lemma LaTeXMLRef ( iii ) and LaTeXMLMath into account .
Let LaTeXMLMath and LaTeXMLMath be two non-isogenous elliptic curves , LaTeXMLMath be integers satisfying LaTeXMLMath , and LaTeXMLMath , then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
As for the existence theorem Corollary LaTeXMLRef we work with the very ample divisor class LaTeXMLMath , and we claim that the Conditions ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) become obsolete , while , in view of Lemma LaTeXMLRef ( iii ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) take the form LaTeXMLMath , and LaTeXMLMath .
That is , under these hypotheses there is an irreducible curve in LaTeXMLMath , for any LaTeXMLMath , with precisely LaTeXMLMath ordinary singular points of multiplicities LaTeXMLMath .
( LaTeXMLRef ) becomes redundant in view of ( LaTeXMLRef d ) and Lemma LaTeXMLRef ( iv ) , while ( LaTeXMLRef ) is fulfilled in view of ( LaTeXMLRef ) and LaTeXMLMath .
It remains to show that LaTeXMLMath for all LaTeXMLMath .
However , by the adjunction formula LaTeXMLMath , and by ( LaTeXMLRef d ) LaTeXMLMath .
Thus the claim is proved .
From these considerations we at once deduce the conditions for the existence of an irreducible curve in LaTeXMLMath , LaTeXMLMath , with prescribed singularities of arbitrary type , i. e. the conditions in Corollary LaTeXMLRef .
They come down to LaTeXMLMath , and LaTeXMLMath A smooth projective surface LaTeXMLMath in LaTeXMLMath is given by a single equation LaTeXMLMath with LaTeXMLMath homogeneous , and by definition the degree of LaTeXMLMath , say LaTeXMLMath , is just the degree of LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath , for LaTeXMLMath , LaTeXMLMath , and for LaTeXMLMath , LaTeXMLMath is isomorphic to LaTeXMLMath blown up in six points in general position .
Thus the Picard number LaTeXMLMath , i. e. the rank of the Néron-Severi group , in these cases is LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath respectively .
Note that these are also precisely the cases where LaTeXMLMath is rational .
In general the Picard number LaTeXMLMath of a surface in LaTeXMLMath may be arbitrarily large , E. g. the LaTeXMLMath -th Fermat surface , given by LaTeXMLMath has Picard number LaTeXMLMath , with equality if LaTeXMLMath .
( cf .
LaTeXMLCite Theorem 7 , see also LaTeXMLCite pp .
1f .
and LaTeXMLCite p. 146 ) but the Néron-Severi group always contains a very special member , namely the class LaTeXMLMath of a hyperplane section with LaTeXMLMath .
And the class of the canonical divisor is then just LaTeXMLMath .
Moreover , if the degree of LaTeXMLMath is at least four , that is , if LaTeXMLMath is not rational , then it is likely that LaTeXMLMath .
More precisely , if LaTeXMLMath , Noether ’ s Theorem says that LaTeXMLMath is a very general subset of the projective space of projective surfaces in LaTeXMLMath of fixed degree LaTeXMLMath , i. e. it ’ s complement is an at most countable union of lower dimensional subvarieties .
( cf .
LaTeXMLCite Corollary 3.5 or LaTeXMLCite p. 146 ) Since we consider the case of rational surfaces separately the following considerations thus give a full answer for the “ general case ” of a surface in LaTeXMLMath .
Let LaTeXMLMath be a surface in LaTeXMLMath of degree LaTeXMLMath , LaTeXMLMath be the algebraic class of a hyperplane section , and LaTeXMLMath an integer satisfying LaTeXMLMath , and LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath , then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
If LaTeXMLMath , then ( LaTeXMLRef e ) is redundant , since there are no irreducible curves LaTeXMLMath with LaTeXMLMath .
Otherwise we would have LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath would imply LaTeXMLMath , but then LaTeXMLMath in contradiction to LaTeXMLMath being ample .
However , a quadric in LaTeXMLMath or the K3-surface given by LaTeXMLMath contain irreducible curves of self-intersection zero .
If LaTeXMLMath for all LaTeXMLMath then again ( LaTeXMLRef e ) becomes obsolete in view of ( LaTeXMLRef e ) , since LaTeXMLMath anyway .
The above inequality is , for instance , fulfilled if the highest multiplicity occurs at least LaTeXMLMath times .
In the existence theorems the condition depending on curves of self-intersection will vanish in any case .
As for Corollary LaTeXMLRef we claim that if LaTeXMLMath , then LaTeXMLMath , ensures the existence of an irreducible curve LaTeXMLMath with precisely LaTeXMLMath ordinary singular points of multiplicities LaTeXMLMath and LaTeXMLMath .
The role of the very ample divisor LaTeXMLMath is filled by a hyperplane section , and thus LaTeXMLMath .
Therefore , ( LaTeXMLRef e ) obviously implies ( LaTeXMLRef ) , and ( LaTeXMLRef ) takes the form LaTeXMLEquation .
However , from ( LaTeXMLRef e ) we deduce for any LaTeXMLMath LaTeXMLEquation unless LaTeXMLMath , in which case we are done by the assumption LaTeXMLMath .
Thus ( LaTeXMLRef ) is redundant .
Moreover , there are no curves of self-intersection zero on LaTeXMLMath , and it thus remains to verify ( LaTeXMLRef ) , which in this situation takes the form LaTeXMLEquation and follows at once from ( LaTeXMLRef ) .
With the aid of this result the conditions of Corollary LaTeXMLRef for the existence of an irreducible curve LaTeXMLMath with prescribed singularities LaTeXMLMath in this situation therefore reduce to LaTeXMLMath , and We note that if LaTeXMLMath is a K3-surface then the Néron-Severi group LaTeXMLMath and the Picard group LaTeXMLMath of LaTeXMLMath coincide , i. e. LaTeXMLMath for every divisor LaTeXMLMath on LaTeXMLMath .
Moreover , an irreducible curve LaTeXMLMath has self-intersection LaTeXMLMath if and only if the arithmetical genus of LaTeXMLMath is one .
In that case LaTeXMLMath is a pencil of elliptic curves without base points endowing LaTeXMLMath with the structure of an elliptic fibration over LaTeXMLMath .
( cf .
LaTeXMLCite or Proposition LaTeXMLRef ) We , therefore , distinguish two cases .
Since a generic K3-surface does not possess an elliptic fibration the following version of Corollary LaTeXMLRef applies for generic K3-surfaces .
( cf .
LaTeXMLCite I.1.3.7 ) Let LaTeXMLMath be a K3-surface which is not elliptic , and let LaTeXMLMath a divisor on LaTeXMLMath satisfying LaTeXMLMath , and LaTeXMLMath nef , then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
In view of equation ( LaTeXMLRef ) the conditions in Corollary LaTeXMLRef reduce to LaTeXMLMath , LaTeXMLMath nef , and LaTeXMLMath for all LaTeXMLMath , and , analogously , the conditions in Corollary LaTeXMLRef reduce to ( LaTeXMLRef ) , LaTeXMLMath , and LaTeXMLMath nef .
The hypersurface in LaTeXMLMath given by the equation LaTeXMLMath is an example of a K3-surface which is endowed with an elliptic fibration .
Among the elliptic K3-surfaces the general one will possess a unique elliptic fibration while there are examples with infinitely many different such fibrations .
( cf .
LaTeXMLCite I.1.3.7 ) Let LaTeXMLMath be a K3-surface which possesses an elliptic fibration , and let LaTeXMLMath be a divisor on LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath nef , then for LaTeXMLMath in very general position and LaTeXMLMath LaTeXMLEquation .
If LaTeXMLMath is generic among the elliptic K3-surfaces , i. e. admits exactly one elliptic fibration , then Condition ( LaTeXMLRef f ) means that a curve in LaTeXMLMath meets a general fibre in at least LaTeXMLMath distinct points .
The conditions in Corollary LaTeXMLRef then reduce to ( LaTeXMLRef f ) , ( LaTeXMLRef f ) , ( LaTeXMLRef f ) , and LaTeXMLMath for any curve LaTeXMLMath with LaTeXMLMath .
Similarly , the conditions in Corollary LaTeXMLRef reduce to ( LaTeXMLRef f ) , ( LaTeXMLRef f ) , ( LaTeXMLRef ) , and LaTeXMLMath for any irreducible curve LaTeXMLMath with LaTeXMLMath .
It is our first aim to show that if there is a curve passing through points LaTeXMLMath in very general position with multiplicities LaTeXMLMath then it can be equimultiply deformed in its algebraic system in a good way - i. e. suitable for Lemma LaTeXMLRef .
Let LaTeXMLMath be a curve , and LaTeXMLMath .
Then LaTeXMLEquation is a closed subset of LaTeXMLMath .
[ Step 1 ] Show first that for LaTeXMLMath LaTeXMLEquation is a closed subset of LaTeXMLMath , where LaTeXMLMath .
Being the reduction of a connected component of the Hilbert scheme LaTeXMLMath , LaTeXMLMath is a projective variety endowed with a universal family of curves , giving rise to the following diagram of morphisms LaTeXMLEquation where LaTeXMLMath is an effective Cartier divisor on LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be a global section defining LaTeXMLMath .
Then LaTeXMLEquation .
We may consider a finite open affine covering of LaTeXMLMath of the form LaTeXMLMath , LaTeXMLMath and LaTeXMLMath open , such that LaTeXMLMath is locally on LaTeXMLMath given by one polynomial equation , say LaTeXMLEquation .
It suffices to show that LaTeXMLMath is closed in LaTeXMLMath for all LaTeXMLMath .
However , for LaTeXMLMath we have LaTeXMLEquation if and only if LaTeXMLEquation .
LaTeXMLEquation Thus , LaTeXMLEquation is a closed subvariety of LaTeXMLMath .
[ Step 2 ] LaTeXMLMath is a closed subset of LaTeXMLMath .
By Step 1 for LaTeXMLMath the set LaTeXMLEquation is a closed subset of LaTeXMLMath .
Considering now LaTeXMLEquation we find that LaTeXMLMath , being the image of a closed subset under a morphism between projective varieties , is a closed subset of LaTeXMLMath ( cf .
LaTeXMLCite Ex .
II.4.4 ) .
∎ Then the complement of the set LaTeXMLEquation is very general , where LaTeXMLMath is the Hilbert scheme of curves on LaTeXMLMath .
In particular , there is a very general subset LaTeXMLMath such that if for some LaTeXMLMath there is a curve LaTeXMLMath with LaTeXMLMath for LaTeXMLMath , then for any LaTeXMLMath there is a curve LaTeXMLMath with LaTeXMLMath .
Fixing some embedding LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is a projective variety and has thus only finitely many connected components .
Thus the Hilbert scheme LaTeXMLMath has only a countable number of connected components , and we have only a countable number of different LaTeXMLMath , where LaTeXMLMath runs through LaTeXMLMath and LaTeXMLMath through LaTeXMLMath .
By Lemma LaTeXMLRef the sets LaTeXMLMath are closed , hence their complements LaTeXMLMath are open .
But then LaTeXMLEquation is an at most countable intersection of open dense subsets of LaTeXMLMath , and is hence very general .
∎ In the proof of Theorem LaTeXMLRef we use at some place the result of Corollary LaTeXMLRef .
We could instead use Corollary LaTeXMLRef .
However , since the results are quite nice and simple to prove we just give them .
The number of curves LaTeXMLMath in LaTeXMLMath with LaTeXMLMath is at most countable .
The number of exceptional curves in LaTeXMLMath ( i. e. curves with negative self intersection ) is at most countable .
There is a very general subset LaTeXMLMath of LaTeXMLMath , LaTeXMLMath , such that for LaTeXMLMath no LaTeXMLMath belongs to a curve LaTeXMLMath with LaTeXMLMath , in particular to no exceptional curve .
By definition LaTeXMLMath is a connected component of LaTeXMLMath , whose number is at most countable .
If in addition LaTeXMLMath , then LaTeXMLMath which proves the claim .
Curves of negative self-intersection are not algebraically equivalent to any other curve ( cf .
LaTeXMLCite p. 153 ) .
Follows from ( i ) .
∎ [ Kodaira ] Let LaTeXMLMath be in very general position To be precise , no three of the nine points should be collinear , and after any finite number of quadratic Cremona transformations centred at the LaTeXMLMath ( respectively the newly obtained centres ) still no three should be collinear .
Thus the admissible tuples in LaTeXMLMath form a very general set , cf .
LaTeXMLCite Ex .
V.4.15 .
and let LaTeXMLMath be the blow up of LaTeXMLMath in LaTeXMLMath .
Then LaTeXMLMath contains infinitely many irreducible smooth rational LaTeXMLMath -curves , i. e. exceptional curves of the first kind .
It suffices to find an infinite number of irreducible curves LaTeXMLMath in LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath , since the expression in ( LaTeXMLRef ) is the self intersection of the strict transform LaTeXMLMath of LaTeXMLMath and ( LaTeXMLRef ) gives its arithmetical genus .
In particular LaTeXMLMath can not contain any singularities , since they would contribute to the arithmetical genus , and , being irreducible anyway , LaTeXMLMath is an exceptional curve of the first kind .
We are going to deduce the existence of these curves with the aid of quadratic Cremona transformations .
[ Claim ] If for some LaTeXMLMath and LaTeXMLMath with LaTeXMLMath there is an irreducible curve LaTeXMLMath , then LaTeXMLMath is an irreducible curve , where LaTeXMLMath are such that LaTeXMLMath , LaTeXMLMath is the quadratic Cremona transformation at LaTeXMLMath , LaTeXMLMath LaTeXMLMath LaTeXMLMath Note that , LaTeXMLMath , i. e. we may iterate the process since the hypothesis of the claim will be preserved .
Since LaTeXMLMath , there must be a triple LaTeXMLMath such that LaTeXMLMath .
Let us now consider the following diagram LaTeXMLEquation and let us denote the exceptional divisors of LaTeXMLMath by LaTeXMLMath and those of LaTeXMLMath by LaTeXMLMath .
Moreover , let LaTeXMLMath be the strict transform of LaTeXMLMath under LaTeXMLMath and let LaTeXMLMath be the strict transform of LaTeXMLMath under LaTeXMLMath .
Then of course LaTeXMLMath , and LaTeXMLMath , being the projection LaTeXMLMath of the strict transform LaTeXMLMath of the irreducible curve LaTeXMLMath , is of course an irreducible curve .
Note that the condition LaTeXMLMath ensures that LaTeXMLMath is not one of the curves which are contracted .
It thus suffices to verify LaTeXMLEquation and LaTeXMLEquation .
Since outside the lines LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath the transformation LaTeXMLMath is an isomorphism and since by hypothesis none of the remaining LaTeXMLMath belongs to one of these lines we clearly have LaTeXMLMath for LaTeXMLMath .
Moreover , we have LaTeXMLEquation .
Analogously for LaTeXMLMath and LaTeXMLMath .
Finally we find LaTeXMLEquation .
This proves the claim .
Let us now show by induction that for any LaTeXMLMath there is an irreducible curve LaTeXMLMath of degree LaTeXMLMath satisfying ( LaTeXMLRef ) and ( LaTeXMLRef ) .
For LaTeXMLMath the line LaTeXMLMath through LaTeXMLMath and LaTeXMLMath gives the induction start .
Given some suitable curve of degree LaTeXMLMath the above claim then ensures that through points in very general position there is an irreducible curve of higher degree satisfying ( LaTeXMLRef ) and ( LaTeXMLRef ) , since LaTeXMLMath .
Thus the induction step is done .
∎ The example shows that a smooth projective surface LaTeXMLMath may indeed carry an infinite number of exceptional curves - even of the first kind .
According to Nagata ( LaTeXMLCite Theorem 4a , p. 283 ) the example is due to Kodaira .
For further references on the example see LaTeXMLCite Ex .
V.4.15 , LaTeXMLCite Example 4.2.7 , or LaTeXMLCite .
LaTeXMLCite p. 198 Example 3 shows that also LaTeXMLMath blown up in the nine intersection points of two plane cubics carries infinitely many exceptional curves of the first kind .
Suppose that LaTeXMLMath is an irreducible curve with LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath is an irreducible projective curve , and there is a fibration LaTeXMLMath whose fibres are just the elements of LaTeXMLMath , and LaTeXMLMath is the normalisation of LaTeXMLMath .
We are proving the proposition in several steps .
Let LaTeXMLMath be a finite flat morphism of noetherian schemes with LaTeXMLMath irreducible such that for some point LaTeXMLMath the fibre LaTeXMLMath is a single reduced point .
Then the structure map LaTeXMLMath is an isomorphism , and hence so is LaTeXMLMath .
Since there is at least one connected reduced fibre LaTeXMLMath , semicontinuity of flat , proper morphisms in the version LaTeXMLCite IV.12.2.4 ( vi ) implies that there is an open dense subset LaTeXMLMath such that LaTeXMLMath is connected and reduced , hence a single reduced point , LaTeXMLMath .
( LaTeXMLMath dense in LaTeXMLMath is due to the fact that LaTeXMLMath is irreducible . )
Thus the assumptions are stable under restriction to open subschemes of LaTeXMLMath , and since the claim that we have to show is local on LaTeXMLMath , we may assume that LaTeXMLMath is affine .
Moreover , LaTeXMLMath being finite , thus affine , we have LaTeXMLMath is also affine .
Since LaTeXMLMath is flat it is open and hence dominates the irreducible affine variety LaTeXMLMath and , therefore , induces an inclusion of rings LaTeXMLMath .
It now suffices to show : LaTeXMLMath is an isomorphism .
By assumption there exists a LaTeXMLMath such that LaTeXMLMath is a single point with reduced structure .
In particular we have for the multiplicity of LaTeXMLMath over LaTeXMLMath LaTeXMLEquation which implies that LaTeXMLEquation is an isomorphism .
Hence by Nakayama ’ s Lemma also LaTeXMLEquation is an isomorphism , that is , LaTeXMLMath is free of rank 1 over LaTeXMLMath .
LaTeXMLMath being locally free over LaTeXMLMath , with LaTeXMLMath an integral domain , thus fulfils LaTeXMLEquation is an isomorphism for all LaTeXMLMath , and hence the claim follows .
∎ [ Principle of Connectedness ] Let LaTeXMLMath and LaTeXMLMath be noetherian schemes , LaTeXMLMath connected , and let LaTeXMLMath be a flat projective morphism such that for some LaTeXMLMath the fibre LaTeXMLMath is reduced and connected .
Then for all LaTeXMLMath the fibre LaTeXMLMath is connected .
Considering points in the intersections of the finite number of irreducible components of LaTeXMLMath we can reduce to the case LaTeXMLMath irreducible .
Stein Factorisation ( cf .
LaTeXMLCite III.4.3.3 ) gives a factorisation of LaTeXMLMath of the form LaTeXMLEquation with LaTeXMLMath connected ( i. e. its fibres are connected ) , LaTeXMLMath finite , LaTeXMLMath locally free over LaTeXMLMath , since LaTeXMLMath is flat , and LaTeXMLMath is connected and reduced , i. e. a single reduced point .
Because of ( 1 ) it suffices to show that LaTeXMLMath is connected , and we claim that they are reduced as well .
Since LaTeXMLMath is finite ( 3 ) is equivalent to saying that LaTeXMLMath is flat .
Hence LaTeXMLMath fulfils the assumptions of Proposition LaTeXMLRef , and we conclude that LaTeXMLMath and the proposition follows from LaTeXMLCite III.11.3 .
Alternatively , from LaTeXMLCite IV.15.5.9 ( ii ) it follows that there is an open dense subset LaTeXMLMath such that LaTeXMLMath is connected for all LaTeXMLMath .
Since , moreover , by the same theorem the number of connected components of the fibres is a lower semi-continuous function on LaTeXMLMath the special fibres can not have more connected components than the general ones , that is , all fibres are connected .
∎ Under the hypotheses of Proposition LaTeXMLRef let LaTeXMLMath then LaTeXMLMath is connected .
Consider the universal family LaTeXMLEquation over the connected projective scheme LaTeXMLMath .
Since the projection LaTeXMLMath is a flat projective morphism , and since the fibre LaTeXMLMath is connected and reduced , the result follows from Proposition LaTeXMLRef .
∎ Under the hypotheses of Proposition LaTeXMLRef let LaTeXMLMath with LaTeXMLMath , then LaTeXMLMath .
Suppose LaTeXMLMath , then the Hilbert polynomials of LaTeXMLMath and LaTeXMLMath are different in contradiction to LaTeXMLMath .
∎ Under the hypotheses of Proposition LaTeXMLRef let LaTeXMLMath with LaTeXMLMath , then LaTeXMLMath .
Since LaTeXMLMath is irreducible by Lemma LaTeXMLRef LaTeXMLMath and LaTeXMLMath do not have a common component .
Suppose LaTeXMLMath , then LaTeXMLMath in contradiction to LaTeXMLMath .
∎ [ Zariski ’ s Lemma ] Under the hypotheses of Proposition LaTeXMLRef let LaTeXMLMath , where the LaTeXMLMath are pairwise different irreducible curves , LaTeXMLMath for LaTeXMLMath .
Then the intersection matrix LaTeXMLMath is negative semi-definite , and , moreover , LaTeXMLMath , considered as an element of the vectorspace LaTeXMLMath , generates the annihilator of LaTeXMLMath .
In particular , LaTeXMLMath for all curves LaTeXMLMath , and , moreover , LaTeXMLMath if and only if LaTeXMLMath .
By Lemma LaTeXMLRef LaTeXMLMath is connected .
We are going to apply LaTeXMLCite I.2.10 , and thus we have to verify three conditions .
LaTeXMLMath for all LaTeXMLMath by Lemma LaTeXMLRef .
Thus LaTeXMLMath is an element of the annihilator of LaTeXMLMath with LaTeXMLMath for all LaTeXMLMath .
LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath is connected there is no non-trivial partition LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath .
Thus LaTeXMLCite I.2.10 implies that LaTeXMLMath is positive semi-definite .
∎ Under the hypotheses of Proposition LaTeXMLRef let LaTeXMLMath be two distinct curves , then LaTeXMLMath .
Suppose LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath have no common component .
We have LaTeXMLEquation and thus LaTeXMLEquation which implies that LaTeXMLEquation where each summand on the left hand side is less than or equal to zero by Proposition LaTeXMLRef , and the right hand side is greater than or equal to zero , since the curves LaTeXMLMath and LaTeXMLMath have no common component .
We thus conclude LaTeXMLEquation .
But then again Proposition LaTeXMLRef implies that LaTeXMLMath and LaTeXMLMath , that is , LaTeXMLMath and LaTeXMLMath have no common component .
Suppose LaTeXMLMath , then LaTeXMLMath would be a contradiction to LaTeXMLMath .
Hence , LaTeXMLMath .
∎ Under the hypotheses of Proposition LaTeXMLRef consider once more the universal family ( LaTeXMLRef ) together with its projection onto LaTeXMLMath , LaTeXMLEquation .
Then LaTeXMLMath is an irreducible projective surface , LaTeXMLMath is an irreducible curve , and LaTeXMLMath is surjective .
[ Step 1 ] LaTeXMLMath is an irreducible projective surface and LaTeXMLMath is surjective .
The universal property of LaTeXMLMath implies that LaTeXMLMath is an effective Cartier divisor of LaTeXMLMath , and thus in particular projective of dimension at least LaTeXMLMath .
Since LaTeXMLMath is projective , its image is closed in LaTeXMLMath and of dimension 2 , hence it is the whole of LaTeXMLMath , since LaTeXMLMath is irreducible .
By Lemma LaTeXMLRef the fibres of LaTeXMLMath are all single points , and thus , by LaTeXMLCite Theorem 11.14 , LaTeXMLMath is irreducible .
Moreover , LaTeXMLEquation [ Step 2 ] LaTeXMLMath [ Step 3 ] LaTeXMLMath is irreducible .
Let LaTeXMLMath be any irreducible component of LaTeXMLMath of dimension one , then we have a universal family over LaTeXMLMath and the analogue of Step 1 for LaTeXMLMath shows that the curves in LaTeXMLMath cover LaTeXMLMath .
But then by Lemma LaTeXMLRef there can be no further curve in LaTeXMLMath , since any further curve would necessarily have a non-empty intersection with one of the curves in LaTeXMLMath .
∎ Let ’ s consider the following commutative diagram of projective morphisms LaTeXMLEquation .
The map LaTeXMLMath is birational .
Since LaTeXMLMath and LaTeXMLMath are irreducible and reduced , and since LaTeXMLMath is surjective , we may apply LaTeXMLCite III.10.5 , and thus there is an open dense subset LaTeXMLMath such that LaTeXMLMath is smooth .
Hence , in particular LaTeXMLMath is flat and the fibres are single reduced points .
Since LaTeXMLMath is projective and quasi-finite , it is finite ( cf .
LaTeXMLCite Ex .
III.11.2 ) , and it follows from Proposition LaTeXMLRef that LaTeXMLMath is an isomorphism onto its image , i. e. LaTeXMLMath is birational .
∎ If LaTeXMLMath denotes the rational inverse of the map LaTeXMLMath in ( LaTeXMLRef ) , then LaTeXMLMath is indeed a morphism , i. e. LaTeXMLMath is an isomorphism .
By Lemma LaTeXMLRef the fibres of LaTeXMLMath over the possible points of indeterminacy of LaTeXMLMath are just points , and thus the result follows from LaTeXMLCite Lemma II.9 .
∎ The map LaTeXMLMath assigning to each point LaTeXMLMath the unique curve LaTeXMLMath with LaTeXMLMath is a morphism , and is thus a fibration whose fibres are the curves in LaTeXMLMath .
We just have LaTeXMLMath .
∎ Let LaTeXMLMath be the normalisation of the irreducible curve LaTeXMLMath .
Then LaTeXMLMath is a smooth irreducible curve .
Moreover , since LaTeXMLMath is irreducible and smooth , and since LaTeXMLMath is surjective , LaTeXMLMath factorises over LaTeXMLMath , i. e. we have the following commutative diagram LaTeXMLEquation .
Then LaTeXMLMath is the desired fibration .
∎ In this section we are , in particular , writing down some identifications of certain sheaves respectively of their global sections .
Doing this we try to be very formal .
However , in a situation of the kind LaTeXMLMath we usually do not distinguish between LaTeXMLMath and LaTeXMLMath , or between LaTeXMLMath and any restriction of LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath with LaTeXMLMath for every fixed LaTeXMLMath in some small disc LaTeXMLMath around LaTeXMLMath .
Then LaTeXMLMath for every LaTeXMLMath .
We write the power series as LaTeXMLMath .
LaTeXMLMath for every LaTeXMLMath implies LaTeXMLEquation .
The identity theorem for power series in LaTeXMLMath then implies that LaTeXMLEquation ∎ Let LaTeXMLMath be a noetherian scheme , LaTeXMLMath a closed subscheme , LaTeXMLMath a sheaf of modules on LaTeXMLMath , and LaTeXMLMath a sheaf of modules on LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
For LaTeXMLMath open , we define LaTeXMLEquation .
This morphism induces on the stalks the isomorphism LaTeXMLEquation where LaTeXMLMath is the ideal defining LaTeXMLMath in LaTeXMLMath locally at LaTeXMLMath .
The identification ( LaTeXMLRef ) together with LaTeXMLCite III.2.10 gives : LaTeXMLEquation .
The adjoint property of LaTeXMLMath and LaTeXMLMath together with LaTeXMLMath gives rise to the following isomorphisms : LaTeXMLEquation .
LaTeXMLEquation That means , that the identity morphism on LaTeXMLMath must correspond to an isomorphism from LaTeXMLMath to LaTeXMLMath via these identifications .
follows from ( LaTeXMLRef ) and once more LaTeXMLCite III.2.10 .
In the situation of Lemma LaTeXMLRef we have : LaTeXMLMath , and LaTeXMLMath .
We denote by LaTeXMLMath and LaTeXMLMath respectively the given embeddings .
By ( LaTeXMLRef ) in Lemma LaTeXMLRef we have : LaTeXMLEquation .
By the projection formula this is just equal to : LaTeXMLEquation .
LaTeXMLEquation Using ( LaTeXMLRef ) in Lemma LaTeXMLRef we get : LaTeXMLEquation .
LaTeXMLEquation ∎ With the notation of Lemma LaTeXMLRef we show that LaTeXMLMath .
Since LaTeXMLMath is an isomorphism , we have for any sheaf LaTeXMLMath of LaTeXMLMath -modules and LaTeXMLMath : LaTeXMLEquation .
In particular , LaTeXMLEquation and LaTeXMLEquation .
Moreover , the morphism LaTeXMLMath becomes under these identifications just the morphism given by LaTeXMLMath , which is injective .
Thus , LaTeXMLMath , and LaTeXMLMath .
∎ Let LaTeXMLMath be an irreducible noetherian scheme , LaTeXMLMath a coherent sheaf on LaTeXMLMath , and LaTeXMLMath such that LaTeXMLMath .
Then LaTeXMLMath .
The multiplication by LaTeXMLMath gives rise to the following exact sequence : LaTeXMLEquation .
Since LaTeXMLMath and LaTeXMLMath are coherent , so is LaTeXMLMath , and hence LaTeXMLMath is closed in LaTeXMLMath .
Now , LaTeXMLEquation .
LaTeXMLEquation But then the complement LaTeXMLMath is open and is contained in LaTeXMLMath ( since LaTeXMLMath implies that LaTeXMLMath ) , and is thus empty since LaTeXMLMath is irreducible and LaTeXMLMath of lower dimension .
∎ Let LaTeXMLMath be a reduced curve on a smooth projective surface LaTeXMLMath over LaTeXMLMath , where the LaTeXMLMath are irreducible , and let LaTeXMLMath be a line bundle on LaTeXMLMath .
Then we define the degree of LaTeXMLMath with the aid of the normalisation LaTeXMLMath .
We have LaTeXMLMath , and thus the image of LaTeXMLMath in LaTeXMLMath , which is the first Chern class of LaTeXMLMath , can be viewed as a vector LaTeXMLMath of integers , and we may define the degree of LaTeXMLMath by LaTeXMLEquation .
In particular , if LaTeXMLMath is irreducible , we get : LaTeXMLEquation .
Since LaTeXMLMath implies that LaTeXMLMath , and since the existence of a non-vanishing global section of LaTeXMLMath on the smooth curve LaTeXMLMath implies that the corresponding divisor is effective , we get the following lemma .
( cf .
LaTeXMLCite Section II.2 ) Let LaTeXMLMath be an irreducible reduced curve on a smooth projective surface LaTeXMLMath , and let LaTeXMLMath be a line bundle on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath .
Let LaTeXMLMath be very ample over LaTeXMLMath on the smooth projective surface LaTeXMLMath , and let LaTeXMLMath be two distinct points .
Then there is a smooth curve through LaTeXMLMath and LaTeXMLMath in LaTeXMLMath .
Considering the embedding into LaTeXMLMath defined by LaTeXMLMath there is an LaTeXMLMath -dimensional family of hyperplane sections going through two fixed points of LaTeXMLMath , which in local coordinates w. l. o. g. is given by the family of equations LaTeXMLMath .
Since the local analytic rings of LaTeXMLMath in every point are smooth , hence , in particular complete intersections , they are given as LaTeXMLMath modulo some ideal generated by LaTeXMLMath power series LaTeXMLMath forming a regular sequence .
Thus , having LaTeXMLMath free indeterminates in our family LaTeXMLMath of equations , a generic equation LaTeXMLMath will lead to a regular sequence LaTeXMLMath , i. e. the hyperplane section defined by LaTeXMLMath is smooth in each of the two points , and thus everywhere .
∎ Let LaTeXMLMath be a smooth curve and LaTeXMLMath a zero-dimensional scheme .
If LaTeXMLMath such that LaTeXMLMath , and LaTeXMLMath , then LaTeXMLEquation .
Condition ( LaTeXMLRef ) ) implies LaTeXMLEquation .
LaTeXMLEquation and thus by Riemann-Roch ( cf .
LaTeXMLCite IV.1.3.4 ) LaTeXMLEquation .
Consider now the exact sequence LaTeXMLEquation .
The result then follows from the corresponding long exact cohomology sequence LaTeXMLEquation ∎ Througout this section we stick to the notation of Section LaTeXMLRef and LaTeXMLRef .
Let LaTeXMLMath and LaTeXMLMath be two smooth projective curves of genus LaTeXMLMath and LaTeXMLMath respectively , and let LaTeXMLMath .
Supposed that one of the curves is rational , the surface is geometrically ruled and the Picard number of LaTeXMLMath is two .
Whereas , if both LaTeXMLMath and LaTeXMLMath are of strictly positive genus , this need no longer be the case as we have seen in Remark LaTeXMLRef .
Thus the following proposition is the best we may expect .
For a generic choice of smooth projective curves LaTeXMLMath and LaTeXMLMath the Neron-Severi group of LaTeXMLMath is LaTeXMLMath .
More precisely , fixing LaTeXMLMath and LaTeXMLMath there is a very general subset LaTeXMLMath such that for any LaTeXMLMath the Picard number of LaTeXMLMath is two , where LaTeXMLMath denotes the moduli space of smooth projective curves of genus LaTeXMLMath , LaTeXMLMath .
As already mentioned , if either LaTeXMLMath or LaTeXMLMath is zero , then we may take LaTeXMLMath .
Suppose that LaTeXMLMath .
Given an elliptic curve LaTeXMLMath there is a countable union LaTeXMLMath of proper subvarieties of LaTeXMLMath such that for any LaTeXMLMath the Picard number of LaTeXMLMath is two - namely , if LaTeXMLMath and LaTeXMLMath denote the periods as in Section LaTeXMLRef , then we have to require that there exists no invertible integer matrix LaTeXMLMath such that LaTeXMLMath .
( Compare also LaTeXMLCite p .
286 . )
We , therefore , may assume that LaTeXMLMath and LaTeXMLMath .
The claim then follows from Lemma LaTeXMLRef , which is due to Denis Gaitsgory .
∎ [ Denis Gaitsgory ] Let LaTeXMLMath be any smooth projective curve of genus LaTeXMLMath .
Then for any LaTeXMLMath there is a very general subset LaTeXMLMath of the moduli space LaTeXMLMath of smooth projective curves of genus LaTeXMLMath such that the Picard number of LaTeXMLMath is two for any LaTeXMLMath .
We note that a curve LaTeXMLMath with LaTeXMLMath induces a non-trivial morphism LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath , denote the canonical projections .
It thus makes sense to study the moduli problem of ( non-trivial ) maps from curves of genus LaTeXMLMath into LaTeXMLMath .
More precisely , let LaTeXMLMath and let LaTeXMLMath be given , where LaTeXMLMath is the Picard variety of divisors of degree LaTeXMLMath on LaTeXMLMath .
Following the notation of LaTeXMLCite we denote by LaTeXMLMath the moduli space of pairs LaTeXMLMath , where LaTeXMLMath is a smooth projective curve of genus LaTeXMLMath and LaTeXMLMath a morphism with LaTeXMLMath .
We then have the canonical morphism LaTeXMLEquation just forgetting the map LaTeXMLMath , and the proposition reduces to the following claim : For no choice of LaTeXMLMath and LaTeXMLMath the morphism LaTeXMLMath is dominant .
Let LaTeXMLMath be any morphism with LaTeXMLMath .
Then LaTeXMLMath is not a contraction and the image of LaTeXMLMath is a projective curve in the abelian variety LaTeXMLMath .
Moreover , we have the following exact sequence of sheaves LaTeXMLEquation .
Since LaTeXMLMath is a non-zero inclusion , its dual LaTeXMLMath is not zero on global sections , that is LaTeXMLEquation is not the zero map .
Since LaTeXMLMath we have LaTeXMLMath , and thus LaTeXMLMath has global sections .
Therefore , the induced map LaTeXMLMath is not the zero map , which by Serre duality gives that the map LaTeXMLEquation from the long exact cohomology sequence of ( LaTeXMLRef ) is not zero .
Hence the coboundary map LaTeXMLEquation can not be surjective .
According to LaTeXMLCite p. 96 we have LaTeXMLEquation .
But if the differential of LaTeXMLMath is not surjective , then LaTeXMLMath itself can not be dominant .
∎ LaTeXMLCite LaTeXMLCite LaTeXMLCite LaTeXMLCite LaTeXMLCite LaTeXMLCite LaTeXMLCite LaTeXMLCite LaTeXMLCite
In global seismology Earth ’ s properties of fractal nature occur .
Zygmund classes appear as the most appropriate and systematic way to measure this local fractality .
For the purpose of seismic wave propagation , we model the Earth ’ s properties as Colombeau generalized functions .
In one spatial dimension , we have a precise characterization of Zygmund regularity in Colombeau algebras .
This is made possible via a relation between mollifiers and wavelets .
Wave propagation in highly irregular media .
In global seismology , ( hyperbolic ) partial differential equations the coefficients of which have to be considered generalized functions ; in addition , the source mechanisms in such application are highly singular in nature .
The coefficients model the ( elastic ) properties of the Earth , and their singularity structure arises from geological and physical processes .
These processes are believed to reflect themselves in a multi-fractal behavior of the Earth ’ s properties .
Zygmund classes appear as the most appropriate and systematic way to measure this local fractality ( cf .
LaTeXMLCite ) .
The modelling process and Colombeau algebras .
In the seismic transmission problem , the diagonalization of the first order system of partial differential equations and the transformation to the second order wave equation requires differentiation of the coefficients .
Therefore , highly discontinuous coefficients will appear naturally although the original model medium varies continuously .
However , embedding the fractal coefficient first into the Colombeau algebra ensures the equivalence after transformation and yields unique solvability if the regularization scaling LaTeXMLMath is chosen appropriately ( cf .
LaTeXMLCite ) .
We use the framework and notation ( in particular , LaTeXMLMath for the algebra and LaTeXMLMath for the mollifier sets ) of Colombeau algebras as presented in LaTeXMLCite .
An interesting aspect of the use of Colombeau theory in wave propagation is that it leads to a natural control over and understanding of ‘ scale ’ .
In this paper , we focus on this modelling process .
We briefly review homogeneous and inhomogeneous Zygmund spaces , LaTeXMLMath and LaTeXMLMath , via a characterization in pseudodifferential operator style which follows essentially the presentation in LaTeXMLCite , Sect .
8.6 .
Alternatively , for practical and implementation issues one may prefer the characterization via growth properties of the discrete wavelet transform using orthonormal wavelets ( cf .
LaTeXMLCite ) .
Classically , the Zygmund spaces were defined as extension of Hölder spaces by boundedness properties of difference quotients .
Within the systematic and unified approach of Triebel ( cf .
LaTeXMLCite ) we can simply identify the Zygmund spaces in a scale of inhomogeneous and homogeneous ( quasi ) Banach spaces , LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath ) , by LaTeXMLMath and LaTeXMLMath .
Both LaTeXMLMath and LaTeXMLMath are Banach spaces .
To emphasize the close relation with mollifiers we describe a characterization of Zygmund spaces in pseudodifferential operator style in more detail .
Let LaTeXMLMath and choose LaTeXMLMath , LaTeXMLMath symmetric and positive , LaTeXMLMath if LaTeXMLMath , LaTeXMLMath if LaTeXMLMath , and LaTeXMLMath strictly decreasing in the interval LaTeXMLMath .
Putting LaTeXMLMath for LaTeXMLMath then defines a function LaTeXMLMath .
Finally we set LaTeXMLEquation and note that if LaTeXMLMath then LaTeXMLMath .
We denote by LaTeXMLMath the set of all pairs LaTeXMLMath that are constructed as above ( we usually suppress the dependence of LaTeXMLMath on LaTeXMLMath and LaTeXMLMath in the notation ) .
We are now in aposition to state the characterization theorem for the inhomogeneous Zygmund spaces as subspaces of LaTeXMLMath .
It follows from LaTeXMLCite , Sec .
2.3 , Thm .
3 or , alternatively , from LaTeXMLCite , Sec .
8.6 .
Note that all appearing pseudodifferential operators in the following have LaTeXMLMath -independent symbols and are thus given simply by convolutions .
Assume that LaTeXMLMath and LaTeXMLMath and choose LaTeXMLMath arbitrary .
Let LaTeXMLMath then LaTeXMLMath belongs to the inhomogeneous Zygmund space of order LaTeXMLMath LaTeXMLMath if and only if LaTeXMLEquation ( Note that we made use of the modification for LaTeXMLMath in LaTeXMLCite , equ .
( 82 ) . )
LaTeXMLMath defines an equivalent norm on LaTeXMLMath .
In fact that all norms defined as above by some LaTeXMLMath are equivalent can be seen as in LaTeXMLCite , Lemma 8.6.5 .
If LaTeXMLMath then LaTeXMLMath is the classical Hölder space of regularity LaTeXMLMath .
Denoting by LaTeXMLMath the greatest integer less than LaTeXMLMath it consists of all LaTeXMLMath times continuously differentiable functions LaTeXMLMath such that LaTeXMLMath is bounded when LaTeXMLMath and globally Hölder continuous with exponent LaTeXMLMath if LaTeXMLMath .
Due to the term LaTeXMLMath the norm LaTeXMLMath is not homogeneous with respect to a scale change in the argument of LaTeXMLMath .
If LaTeXMLMath then ( cf .
LaTeXMLCite , Sect .
8.6 ) LaTeXMLEquation .
Using LaTeXMLMath this can be rewritten in the form LaTeXMLMath and resembles Calderon ’ s classical identity in terms of a continuous wavelet transform ( cf .
LaTeXMLCite , Ch .
1 , ( 5.9 ) and ( 5.10 ) ) .
In a similar way one can characterize the homogeneous Zygmund spaces as subspaces of LaTeXMLMath modulo the polynomials LaTeXMLMath .
A proof can be found in LaTeXMLCite , Sec .
3.1 , Thm .
1 .
We may identify LaTeXMLMath with the dual space LaTeXMLMath of LaTeXMLMath , the Schwartz functions with vanishing moments , by mapping the class LaTeXMLMath with representative LaTeXMLMath to LaTeXMLMath .
Assume that LaTeXMLMath and LaTeXMLMath and choose LaTeXMLMath as constructed above and let LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath belongs to the homogeneous Zygmund space LaTeXMLMath of order LaTeXMLMath if and only if LaTeXMLEquation ( Note that we use the modification for LaTeXMLMath in LaTeXMLCite , equ .
( 16 ) . )
Following LaTeXMLCite we call a function LaTeXMLMath with LaTeXMLMath a wavelet .
We shall say that it is a wavelet of order LaTeXMLMath ( LaTeXMLMath ) if the moments up to order LaTeXMLMath vanish , i.e. , LaTeXMLMath for LaTeXMLMath .
The ( continuous ) wavelet transform is defined for LaTeXMLMath ( LaTeXMLMath ) by ( LaTeXMLMath ) LaTeXMLEquation where we have used the notation LaTeXMLMath and LaTeXMLMath .
By Young ’ s inequality LaTeXMLMath is in LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath defines a continuous operator on this space for each LaTeXMLMath .
If LaTeXMLMath we can define LaTeXMLMath for LaTeXMLMath directly by the same formula ( LaTeXMLRef ) .
If LaTeXMLMath then LaTeXMLMath can be extended to LaTeXMLMath as the adjoint of the wavelet synthesis ( cf .
LaTeXMLCite , Ch .
1 , Sects .
24 , 25 , and 30 ) or directly by LaTeXMLMath - LaTeXMLMath -convolution in formula ( LaTeXMLRef ) .
If LaTeXMLMath is a polynomial and LaTeXMLMath it is easy to see that LaTeXMLMath .
In fact , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are in LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is in LaTeXMLMath the Fourier transform LaTeXMLMath is smooth and vanishes of infinite order at LaTeXMLMath .
But LaTeXMLMath has to be a linear combination of derivatives of LaTeXMLMath implying LaTeXMLMath .
Therefore the wavelet transform ‘ is blind to polynomial parts ’ of the analyzed function ( or distribution ) LaTeXMLMath .
In terms of geophysical modelling this means that a polynomially varying background medium is filtered out automatically .
The Zygmund class characterization in Theorem LaTeXMLRef ( and remark LaTeXMLRef , ( v ) ) used asymptotic estimates of scaled smoothings of the distribution which resembles typical mollifier constructions in Colombeau theory .
In this subsection we relate this in turn directly to the wavelet transform obtaining the well-known wavelet characterization of Zygmund spaces .
Let LaTeXMLMath with LaTeXMLMath and define the function LaTeXMLMath by LaTeXMLEquation .
Then LaTeXMLMath is in LaTeXMLMath and is a wavelet since a simple integration by parts shows that LaTeXMLEquation .
LaTeXMLMath and if LaTeXMLMath we have LaTeXMLMath if and only if LaTeXMLMath .
Therefore LaTeXMLMath defined by ( LaTeXMLRef ) is a wavelet of order LaTeXMLMath if and only if the mollifier LaTeXMLMath has vanishing moments of order LaTeXMLMath .
Furthermore , by straightforward computation , we have LaTeXMLEquation yielding an alternative of ( LaTeXMLRef ) in the form LaTeXMLMath .
If LaTeXMLMath arbitrary and LaTeXMLMath , LaTeXMLMath are the unique Schwartz functions such that LaTeXMLMath and LaTeXMLMath , then straightforward computation shows that LaTeXMLMath and LaTeXMLMath satisfy the relation ( LaTeXMLRef ) .
Therefore since LaTeXMLMath is then a real valued and even wavelet we have for LaTeXMLMath LaTeXMLEquation .
Hence the distributions LaTeXMLMath in the Zygmund class LaTeXMLMath can be characterized in terms of a wavelet transform and a smoothing pseudodifferential operator by LaTeXMLMath and LaTeXMLMath .
We have shown Let LaTeXMLMath .
A distribution LaTeXMLMath belongs to the Zygmund class LaTeXMLMath if and only if LaTeXMLEquation .
Observe that the condition on LaTeXMLMath implies that LaTeXMLMath and hence LaTeXMLMath can never have compact support .
If this characterization is to be used in a theory of Zygmund regularity detection within Colombeau algebras one has to allow for mollifiers of this kind in the corresponding embedding procedures .
This is the issue of the following subsection .
Nevertheless we note here that according to remarks in LaTeXMLCite and , more precisely , in LaTeXMLCite the restrictions on the wavelet itself in a characterization of type ( LaTeXMLRef ) may be considerably relaxed — depending on the generality one wishes to allow for the analyzed distribution LaTeXMLMath .
However , in case LaTeXMLMath and LaTeXMLMath a function a flexible and direct characterization ( due to Holschneider and Tchamitchian ) can be found in LaTeXMLCite , Sect .
2.9 , or LaTeXMLCite , Sect .
4.2 .
There are more refined results in the spirit of the above theorem describing local Hölder ( Zygmund ) regularity by growth properties of the wavelet transform ( cf .
in particular LaTeXMLCite , Sect .
4.2 , LaTeXMLCite , and LaTeXMLCite ) .
The counterpart of ( LaTeXMLRef ) for LaTeXMLMath -functions in terms of ( discrete ) multiresolution approximations is LaTeXMLCite , Sect .
6.4 , Thm .
5 .
We consider a variant of the Colombeau embedding LaTeXMLMath that was discussed in LaTeXMLCite , subsect .
3.2 .
As indicated in remark LaTeXMLRef , ( i ) we need to allow for mollifiers with noncompact support in order to gain the flexibility of using wavelet-type arguments for the extraction of regularity properties from asymptotic estimates .
On the side of the embedded distributions this forces us to restrict to LaTeXMLMath , a space still large enough for the geophysically motivated coefficients in model PDEs .
Recall ( LaTeXMLCite , Def .
11 ) that an admissible scaling is defined to be a continuous function LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath if LaTeXMLMath ( fixed ) as LaTeXMLMath .
Let LaTeXMLMath be an admissible scaling , LaTeXMLMath with LaTeXMLMath , then we define LaTeXMLMath by LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLMath is well-defined since LaTeXMLMath is clearly moderate and negligibility is preserved under this scaled convolution .
By abuse of notation we will write LaTeXMLMath for the standard representative of LaTeXMLMath .
The following statements describe properties of such a modelling procedure resembling the original properties used by M. Oberguggenberger in LaTeXMLCite , Prop.1.5 , to ensure unique solvability of symmetric hyperbolic systems of PDEs ( cf .
LaTeXMLCite ) .
The definition of Colombeau functions of logarithmic and bounded type is given in LaTeXMLCite , Def .
19.2 , the variation used below is an obvious extension .
LaTeXMLMath is linear , injective , and commutes with partial derivatives .
LaTeXMLMath : LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath is of LaTeXMLMath -type , i.e. , there is LaTeXMLMath such that for all LaTeXMLMath there exist LaTeXMLMath and LaTeXMLMath : LaTeXMLEquation .
If LaTeXMLMath then LaTeXMLMath is of bounded type and its first order derivatives are of LaTeXMLMath -type .
ad ( i ) , ( ii ) : Is clear from LaTeXMLMath in LaTeXMLMath as LaTeXMLMath and the convolution formula .
ad ( iii ) : Although this involves only marginal changes in the proof of LaTeXMLCite , Prop .
1.5 ( i ) , we recall it here to make the presentation more self-contained .
Let LaTeXMLMath with LaTeXMLMath ( LaTeXMLMath ) then with LaTeXMLMath LaTeXMLEquation where the expression within brackets on the r.h.s .
is bounded by some constant LaTeXMLMath , dependent on LaTeXMLMath and LaTeXMLMath only but independent of LaTeXMLMath , as soon as LaTeXMLMath with LaTeXMLMath chosen appropriately ( and dependent on LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath ) .
Therefore the assertion is proved by putting LaTeXMLMath and LaTeXMLMath .
ad ( iv ) : is proved by similar reasoning ∎ In particular , we can model a fairly large class of distributions as Colombeau functions of logarithmic growth ( or log-type ) thereby ensuring unique solvability of hyperbolic PDEs incorporating such as coefficients .
If LaTeXMLMath then LaTeXMLMath and LaTeXMLEquation .
If LaTeXMLMath for LaTeXMLMath then LaTeXMLMath is of LaTeXMLMath -type .
In particular , there is an admissible scaling LaTeXMLMath such that LaTeXMLMath and all first order derivatives LaTeXMLMath ( LaTeXMLMath ) are of log-type .
One of the most important properties of the embedding procedure introduced in LaTeXMLCite was its faithfulness with respect to the microlocal properties if ‘ appropriately measured ’ in terms of the set of LaTeXMLMath -regular Colombeau functions LaTeXMLMath ( LaTeXMLCite , Def .
11 ) .
But there the proof of this microlocal invariance property heavily used the compact support property of the standard mollifier LaTeXMLMath which is no longer true in the current situation .
In this subsection we show how to extend the invariance result to the new embedding procedure defined above .
Let LaTeXMLMath , LaTeXMLMath an admissible scaling , and LaTeXMLMath with LaTeXMLMath then LaTeXMLEquation .
The necessary changes in the proof of LaTeXMLCite , Thm .
15 , are minimal once we established the following If LaTeXMLMath and LaTeXMLMath with LaTeXMLMath then LaTeXMLMath .
Using the short-hand notation LaTeXMLMath and LaTeXMLMath we have LaTeXMLEquation .
Hence we need to estimate terms of the form LaTeXMLMath when LaTeXMLMath .
Let LaTeXMLMath be a closed set satisfying LaTeXMLMath and put LaTeXMLMath .
Since LaTeXMLMath is a temperate distribution there is LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLMath implies that each term in the sum on the right-hand side can be estimated for arbitrary LaTeXMLMath by LaTeXMLEquation if LaTeXMLMath varies in LaTeXMLMath .
Since LaTeXMLMath we obtain LaTeXMLEquation with a constant LaTeXMLMath depending on LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath but LaTeXMLMath still arbitrary .
Choosing LaTeXMLMath , for example , we conclude that LaTeXMLMath has a uniform LaTeXMLMath -growth over all orders of derivatives .
Hence it is a LaTeXMLMath -regular Colombeau function .
∎ Referring to the proof ( and the notation ) of LaTeXMLCite , Thm .
15 , we may now finish the proof of the theorem simply by carrying out the following slight changes in the two steps of that proof .
Ad step 1 : Choose LaTeXMLMath such that LaTeXMLMath in a neighborhood of LaTeXMLMath and write LaTeXMLEquation .
The first term on the right can be estimated by the same methods as in LaTeXMLCite and the second term is LaTeXMLMath -regular by the lemma above .
Ad step 2 : Rewrite LaTeXMLEquation and observe that the reasoning of LaTeXMLCite is applicable since LaTeXMLMath by the above lemma .
∎ Simple wavelet-mollifier correspondences as in subsection 2.3 allow us to rewrite the Colombeau modelling procedure and hence prepare for the detection of original Zygmund regularity in terms of growth properties in the scaling parameters .
A first version describes directly LaTeXMLMath but involves an additional nonhomogeneous term .
If LaTeXMLMath has the properties LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) then LaTeXMLMath defines a wavelet of order LaTeXMLMath and we have for any LaTeXMLMath LaTeXMLEquation .
Let LaTeXMLMath then eq .
( LaTeXMLRef ) implies LaTeXMLMath and integration with respect to LaTeXMLMath from LaTeXMLMath to LaTeXMLMath yields LaTeXMLEquation ∎ A more direct mollifier wavelet correspondence is possible via derivatives of LaTeXMLMath instead .
If LaTeXMLMath with LaTeXMLMath then for any LaTeXMLMath with LaTeXMLMath LaTeXMLEquation is a wavelet of order LaTeXMLMath and for any LaTeXMLMath we have LaTeXMLEquation .
Let LaTeXMLMath then LaTeXMLMath which proves the first assertion .
The second assertion follows from LaTeXMLEquation with the short-hand notation LaTeXMLMath .
∎ Both lemmas LaTeXMLRef and LaTeXMLRef may be used to translate ( global ) Zygmund regularity of the modeled ( embedded ) distribution LaTeXMLMath via Thm .
LaTeXMLRef into asymptotic growth properties with respect to the regularization parameter .
To what extent this can be utilized to develop a faithful and completely intrinsic Zygmund regularity theory of Colombeau functions may be subject of future research .
If we combine the basic ideas of the Zygmund class characterization in 2.3 with the simple observations in 3.3 we are naturally lead to define a corresponding regularity notion intrinsically in Colombeau algebras as follows .
Let LaTeXMLMath be an admissible scaling function and LaTeXMLMath be a real number .
A Colombeau function LaTeXMLMath is said to be globally of LaTeXMLMath -Zygmund regularity LaTeXMLMath if for all LaTeXMLMath there is LaTeXMLMath such that for all LaTeXMLMath we can find positive constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
The set of all ( globally ) LaTeXMLMath -Zygmund regular Colombeau functions of order LaTeXMLMath will be denoted by LaTeXMLMath .
A detailed analysis of LaTeXMLMath in arbitrary space dimensions and not necessarily positive regularity LaTeXMLMath will appear elsewhere .
Here , as an illustration , we briefly study the case LaTeXMLMath and LaTeXMLMath in some detail .
Concerning applications to PDEs this would mean that we are allowing for media of typical fractal nature varying continuously in one space dimension .
For example one may think of a coefficient function LaTeXMLMath in LaTeXMLMath to appear in the following ways .
Let LaTeXMLMath be constant outside some interval LaTeXMLMath and equal to a typical trajectory of Brownian motion in LaTeXMLMath ; it is well-known that with probability LaTeXMLMath those trajectories are in LaTeXMLMath whenever LaTeXMLMath .
This is proved , e.g. , in LaTeXMLCite , Sect .
4.4 , elegantly by wavelet transform methods .
We refer to LaTeXMLCite , Sect .
V.3 , for notions and notation in this example .
Then similarly to the above one can set LaTeXMLMath in LaTeXMLMath , LaTeXMLMath in LaTeXMLMath and in LaTeXMLMath let LaTeXMLMath be Lebesgue ’ s singular function associated with a Cantor-type set of order LaTeXMLMath with ( constant ) dissection ratio LaTeXMLMath .
Then LaTeXMLMath belongs to LaTeXMLMath with LaTeXMLMath .
( The classical triadic Cantor set corresponds to the case LaTeXMLMath and LaTeXMLMath . )
We have already seen that the Colombeau embedding does not change the microlocal structure ( i.e. , the LaTeXMLMath -wave front set ) of the original distribution .
We will show now that also the refined Zygmund regularity information is accurately preserved .
If LaTeXMLMath we denote by LaTeXMLMath the set of all LaTeXMLMath times continuously differentiable functions with the derivatives up to order LaTeXMLMath bounded .
Note that LaTeXMLMath is a strict superset of LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
Define LaTeXMLMath such that LaTeXMLMath then we have LaTeXMLEquation .
In other words , in case LaTeXMLMath we can precisely identify those Colombeau functions that arise from the Zygmund class of order LaTeXMLMath within all embedded bounded continuous functions .
We use the characterizations in LaTeXMLCite , Thms .
2.9.1 and 2.9.2 and the remarks on p. 48 following those ; choosing a smooth compactly supported wavelet LaTeXMLMath of order LaTeXMLMath we may therefore state the following LaTeXMLMath -spaces here : LaTeXMLMath belongs to LaTeXMLMath if and only if there is LaTeXMLMath such that LaTeXMLEquation .
Now the proof is straightforward .
First let LaTeXMLMath .
If LaTeXMLMath then LaTeXMLMath by Young ’ s inequality .
If LaTeXMLMath we use lemma LaTeXMLRef and set LaTeXMLMath to obtain LaTeXMLEquation where LaTeXMLMath is a wavelet of order at least LaTeXMLMath .
Hence ( LaTeXMLRef ) gives an upper bound LaTeXMLMath uniformly in LaTeXMLMath .
Hence ( LaTeXMLRef ) follows .
Finally , if we know that LaTeXMLMath and LaTeXMLMath then combination of ( LaTeXMLRef ) and lemma LaTeXMLRef gives if LaTeXMLMath LaTeXMLEquation uniformly in LaTeXMLMath .
Hence another application of ( LaTeXMLRef ) proves the assertion .
∎
math-ph/0104009 math-ph/0104009 A Transfer Matrix approach to the Enumeration of Colored Links A Transfer Matrix approach to the Enumeration of Colored Links J. L. Jacobsen and P. Zinn-Justin Laboratoire de Physique Théorique et Modèles Statistiques Université Paris-Sud , Bâtiment 100 91405 Orsay Cedex , France We propose a transfer matrix algorithm for the enumeration of alternating link and tangle diagrams , giving a weight LaTeXMLMath to each connected component .
Considering more general tetravalent diagrams with self-intersections and tangencies allows us to treat topological ( flype ) equivalences .
This is done by means of a finite renormalization scheme for an associated matrix model .
We give results , expressed as polynomials in LaTeXMLMath , for the various generating functions up to order 19 ( 2-legged tangle diagrams ) , 15 ( 4-legged tangles ) and 11 ( 6-legged tangles ) crossings .
The limit LaTeXMLMath is solved explicitly .
We then analyze the large-order asymptotics of the generating functions .
For LaTeXMLMath good agreement is found with a conjecture for the critical exponent , based on the KPZ relation .
04/2000 1 .
Introduction It is well-known that a LaTeXMLMath -dimensional system in statistical mechanics can be conceived as a LaTeXMLMath -dimensional quantum field theory , by distinguishing one of the spatial coordinates as the direction of time .
This correspondence lies at the heart of the transfer matrix formalism , where a linear operator is used to describe the discrete time evolution of the corresponding quantum system .
More generally , transfer matrices have numerous applications for the combinatorial enumeration of discrete objects for which a definite direction ( the transfer , or time , direction ) can be singled out .
Recently , this combinatorial aspect has come into focus through the enumeration of various objects pertaining to two-dimensional quantum gravity [ [ 1 ] 1 , , [ 2 ] 2 , , [ 3 ] 3 , , [ 4 ] 4 ] , such as plane meanders .
Common to these examples is the existence of a preferred direction ( e.g .
the river , in the case of meanders ) , which can be straightforwardly promoted to the time direction .
In a previous paper [ 4 ] we have shown how this scheme also applies to the enumeration of alternating knot diagrams .
Here the knot itself defines the transfer direction , since the algorithm essentially consists in reading the knot starting from one “ ingoing ” leg and ending at the other “ outgoing ” leg .
It is however far from obvious how this principle may generalize to the case of link diagrams with more than one connected component .
This is the purpose of the present paper .
More specifically , the final goal is to count alternating tangles at fixed number of connected components ; it is therefore a generalization of the counting of alternating tangles with minimum number of components done in [ 4 ] , but also of the counting of alternating tangles of [ [ 5 ] 5 , , [ 6 ] 6 ] and of oriented alternating tangles of [ 7 ] .
We also present some results for tangles with a higher number of outgoing strings ( “ external legs ” ) , instead of just four as in the publications mentioned above .
We shall in what follows present not just one , but two rather different transfer matrices addressing this enumeration problem .
After the definitions in section 2 , which include various intermediate generating functions needed in the calculation , we shall present in section 3 the basic ideas behind the two proposed transfer matrices for alternating tangle diagrams .
Section 4 is devoted to more technical details on the actual implementation of these ideas on a computer , and section 5 gives the numerical results and their analysis .
Finally , in Section 6 , we discuss how our algorithms may be adapted to various other problems of interest in graph theory and statistical physics .
2 .
Definitions of the generating functions The objects we want to consider are tangles with LaTeXMLMath “ external legs ” , that is roughly speaking the data of LaTeXMLMath intervals embedded in a ball LaTeXMLMath and whose endpoints are given distinct points on the boundary LaTeXMLMath , plus an arbitrary number of ( unoriented ) circles embedded in LaTeXMLMath , all intertwined , and considered up to orientation preserving homeomorphisms of LaTeXMLMath that reduce to the identity on LaTeXMLMath .
Tangles with LaTeXMLMath external legs will be simply called tangles .
The rest of the basic definitions is identical to those given in [ 4 ] .
We represent these objects using diagrams , and restrict ourselves to alternating diagrams .
This implies in particular that tangles can be considered as flype equivalence classes of diagrams [ 8 ] .
Our goal is to count the number of prime tangles with a certain number of external legs and connected components .
We shall relate in this section their generating functions to a simpler , more directly computable quantity , which is the following triple generating function LaTeXMLEquation where LaTeXMLMath is the number of topologically inequivalent open curves in the plane going from LaTeXMLMath to LaTeXMLMath together with LaTeXMLMath circles , connected together by LaTeXMLMath regular intersections and LaTeXMLMath tangencies , see Fig .
1 .
Here LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are formal parameters , which can be evaluated at arbitrary complex values ; however , it is natural to identify LaTeXMLMath with a number of colors one can assign to any of the closed loops of the diagram , so that the factor LaTeXMLMath correctly counts the total number of possible colorings of the diagram ( assuming the external legs to carry a fixed color ) .
The coefficient LaTeXMLMath of the double generating function LaTeXMLMath possesses the following interpretation : it is the number of alternating tangle diagrams with LaTeXMLMath external legs , LaTeXMLMath circles ( i.e .
LaTeXMLMath connected components ) and LaTeXMLMath crossings .
The general coefficients do not possess such a clear knot-theoretic interpretation ; however , they are needed to take into account the flyping equivalence ( see [ [ 9 ] 9 , , [ 10 ] 10 ] ) .
Fig .
1 : Open curve and a circle with intersections ( green dots ) and tangencies ( red dots ) .
Fig .
2 : Tangles of types 1 and 2 are distinguished by the two ways of connecting their external legs .
Next we define the generating functions of tangles LaTeXMLMath and LaTeXMLMath ( see Fig .
2 ) , which are necessary for the flyping equivalence .
They are given by : LaTeXMLEquation via intermediate functions LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Note that inverting Eqs .
LaTeXMLMath requires LaTeXMLMath .
These special values of LaTeXMLMath will be investigated in detail in [ 10 ] .
As in [ 4 ] , we introduce an extra parameter to count edges of the diagram , according to the following definitions : LaTeXMLMath and LaTeXMLMath .
The parameters LaTeXMLMath , LaTeXMLMath and LaTeXMLMath must then be chosen as a function of LaTeXMLMath and LaTeXMLMath according to the following renormalization procedure ( see [ 10 ] ) : LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are auxiliary quantities defined by : LaTeXMLEquation .
These equations are independent only for LaTeXMLMath , but have a smooth LaTeXMLMath limit which is given in [ 4 ] .
LaTeXMLMath and LaTeXMLMath , once equations LaTeXMLMath are solved , are the desired generating functions for the number of prime alternating tangles of types 1 and 2 respectively [ 7 ] ( see Fig .
2 ) .
The total number of tangles is given by LaTeXMLMath .
Similarly , one can define more general generating functions in the variables LaTeXMLMath , LaTeXMLMath and LaTeXMLMath which count tangles with more external legs ; an explicit example will be given in Section 5 .
3 .
The transfer matrices for alternating tangles We now turn to the description of the two transfer matrices .
The basic idea is common to both of them : starting from an initial state consisting of all external legs ( i.e .
two in the case under consideration ) , the system is time evolved through the addition of LaTeXMLMath intersections , until an empty final state is obtained .
Supposing the tangle diagram oriented from left to right , the first algorithm procedes by always evolving the uppermost vertex .
We describe the details of this “ single-step ” algorithm in Section 3.1 .
The second algorithm , on the contrary , evolves all parts of the diagram simultaneously , adding one vertex to each of them in a given time step .
In this way , the time can be defined as the geodesic distance from the pair of external legs .
The details of this “ geodesic ” algorithm can be found in Section 3.2 .
As in [ 4 ] , we shall first concentrate on the enumeration of ( prime , alternating ) tangle diagrams with two external legs , which are related to the generating function LaTeXMLMath of Section 2 .
Adding tangencies , which is needed to take into account the flyping equivalence , will be discussed in Section 3.3 , since it is an elementary extension of the algorithms .
Another minor modification of the algorithms will enable us to enumerate diagrams with more than two external legs ; we shall develop this point in Section 3.4 .
In both algorithms , the needs for CPU-time and memory increase exponentially with the system size LaTeXMLMath , though mercifully much more slowly than the number of knot diagrams actually being enumerated .
As will become clear shortly , the single-step algorithm favors speed at the expense of memory consumption , while for the geodesic algorithm it is the other way around .
However , since in practice both of these parameters are limiting factors for the maximally obtainable system size , it is a priori not clear which of the algorithms performs best .
We defer a detailed comparison to Section 4.3 , and it turns out that the single-step algorithm comes out as the winner .
Incidentally , even in the case of knots ( one connected component ) , it performs slightly better than the algorithm described in [ 4 ] .
3.1 .
The single-step algorithm Let us briefly recall the working principle behind the knot enumeration algorithm presented in [ 4 ] .
Reading the two-legged knot diagram from the first “ ingoing ” to the second “ outgoing ” leg , and calling at any instant the edge being read the “ active line ” , there are two possibilities at each time step : 1 ) The active line is crossed by a line segment with edge labels that have not previously been encountered .
We then add the new line segment to the current state .
2 ) The active line is identified with one of the endpoints of a line segment previously encountered .
We then join the active line to that endpoint , wind around the line segment in question , and identify the new position of the active line with the opposite endpoint .
Fig .
3 : Working principle of the single-step algorithm .
a ) A two-legged knot diagram with LaTeXMLMath intersections and LaTeXMLMath connected components .
The edges are labelled from A to M. b ) The same diagram in the time-slice representation .
For reasons of clarity , the time slices are not drawn in chronological order .
The single-step algorithm can be viewed as a generalization of this principle .
Let us , for the sake of illustration , consider the tangle diagram shown in Fig .
3 .
Since there is in general more than one connected component , clearly the concept of a unique “ active line ” no longer applies .
Let us instead start from an initial state given by both external legs ( edges A and G ) .
Moving along either of the edges A or G , a new line segment ( DE resp .
HI ) is encountered .
The question then arises which of these to process first .
We resolve this ambiguity by stipulating that in any given state , we evolve the line which at that instant is uppermost .
LaTeXMLMath With the optimization to be discussed in Sec .
4.2 we shall permit certain permutations of the lines .
However , the line being evolved is in all cases the uppermost in the given state , though not necessarily in the corresponding time-slice representation .
At time LaTeXMLMath , the edge A thus becomes B , and the new line segment DE is added .
The edge D is now the new top line .
Analogously , at the instants LaTeXMLMath and LaTeXMLMath , the top line ( D resp .
K ) crosses a new line segment , which is then added to the current state .
We can formalize this by stating the transformation rule shown in Fig .
4.1 .
This generalizes the corresponding rule of [ 4 ] , except that the “ active line ” is now replaced by the uppermost line .
At time LaTeXMLMath , the new top line carries the label B , which was however already produced by the transformation acting at LaTeXMLMath .
We therefore proceed , at LaTeXMLMath , to the identification of the two “ copies ” of B , joining them through an arch .
This is an example of the general transformation rule shown in Fig .
4.2 .
The addition of an arch means that the lines intermediate between the two instances of B ( at positions LaTeXMLMath and LaTeXMLMath on Fig .
4.2 ) can henceforth not communicate with the lines at the exterior of the arch .
These “ trapped lines ” must therefore eventually evolve to the empty state ( vacuum ) , independently of the rest of the diagram .
This observation has two implications : First , since both transformations conserve the parity ( even/odd ) of the number of lines , LaTeXMLMath must be odd .
Second , due to the above-mentioned rule that the current top line must always be treated first , the evolution of a possible set of “ trapped lines ” must take place at a later time .
This means that when illustrating the sequence of moves on Fig .
3 , we can not draw the time-slices in chronological order .
Note however , that the time ordering of the transformations is given by the time labels shown in the bottom of Fig .
3 .
Fig .
4 : The two types of transformations in the single-step algorithm .
1 ) Addition of a new line segment .
2 ) Identification of the top line ( at position LaTeXMLMath ) with another line ( at position LaTeXMLMath ) , accompanied with the creation of a new block .
Several remarks apply to the relative positions of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath ( see text ) .
The existence of a number of trapped lines is visualized on Fig .
4.2 by a delimiter ( shown as a gray rectangle ) , which seperates the remaining lines into two blocks .
Lines in different blocks can not communicate , and must eventually evolve to the vacuum separately .
In particular this means that the transformation 2 ) only applies when LaTeXMLMath and LaTeXMLMath belong to the same block .
Conversely , LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath may very well be separated by one or more delimiters ( not shown ) .
Also , although we have illustrated the case LaTeXMLMath , we may as well have LaTeXMLMath .
Of the thirteen transformations shown on Fig .
3 , number 1 , 2 , 3 , 5 , 9 , and 10 are of type 1 , and the rest are of type 2 .
Clearly , the two types of transformations increase ( resp .
decrease ) the number of lines by unity .
Since the initial state consists of one line , the number of type 2 transformations must therefore exceed the number of type 1 transformations by one .
The purpose of the transfer matrix is not only to count the total number of tangle diagrams , but to do so for any fixed number of connected components .
In particular , when performing a type 2 transformation , we need to know whether the points LaTeXMLMath and LaTeXMLMath were already connected though an arbitrary number of edges at an earlier time .
On Fig .
4 we have represented this information by a number of lines on the left , connecting the points at a given instant into pairs .
It may thus happen that on Fig .
4.2 , LaTeXMLMath and LaTeXMLMath .
In this case , the type 2 transformation marks the completion of one connected component in the tangle diagram .
Fig .
5 : Intermediate states produced by applying the single-step algorithm to the tangle diagram shown in Fig .
3 .
We are now ready to define the set of states on which the transfer matrix acts .
A state is defined by an even number of points ( represented on Fig .
4 as black rectangles ) , connected into pairs by means of the edges encountered at previous times .
In addition , the points are divided into LaTeXMLMath blocks by means of LaTeXMLMath delimiters .
Note that points in different blocks can very well be connected , since any connection made beforehand persists after the addition of a delimiter .
On Fig .
5 we show the set of intermediate states corresponding to the time-slice representation of the tangle of Fig .
3 .
The initial state of a two-legged tangle is given by a pair of points ( the exterior legs ) , implicitly connected at the point at infinity .
It can be noted that the same state may occur at different instants of the transfer process .
Also , any given state is not necessarily allowed at all instants .
After each type 2 transformation one may be able to simplify the set of delimiters .
Namely , a delimiter may be eliminated if it is adjacent to another delimiter , or if it precedes the first point or succedes the last point of a state .
On Fig .
5 we have implicitly assumed that such simplications have been carried out .
Finally , we must define the transfer matrix LaTeXMLMath which counts all tangle diagrams with LaTeXMLMath vertices and LaTeXMLMath connected components .
Its entries LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are two basis states of the kind just defined , are LaTeXMLMath unless LaTeXMLMath is a descendant of LaTeXMLMath .
An allowed state LaTeXMLMath is a descendant of LaTeXMLMath if it can be obtained via a transformation of one of the two types shown on Fig .
4 ( for an arbitrary even LaTeXMLMath belonging to the same block as LaTeXMLMath ) , followed by an arbitrary number of simplifications .
LaTeXMLMath is then the sum over all transformations from LaTeXMLMath to LaTeXMLMath of the corresponding weight : LaTeXMLMath or LaTeXMLMath depending on whether one closes a connected component or not ( LaTeXMLMath can be either a given number , or a formal parameter , with a space of states enlarged by polynomials in LaTeXMLMath in the latter case ) .
For the moment the simplications are just the elimination of superfluous delimiters .
But we shall later ( in Sec .
4.2 ) show that hitherto different states are equivalent by means of suitable transformations of the blocks , and of the points within each block , such that a given state may be brought into a normal form .
LaTeXMLMath then acts on the space of such normal forms .
Apart from considerably reducing the dimension of the state space , these additional simplifications greatly enhance the efficiency of the algorithm .
As an example , we give in Appendix A the complete set of intermediate states with their corresponding weights for the counting of tangle diagrams up to four crossings .
3.2 .
The geodesic algorithm We now turn to the description of our second algorithm .
Apart from providing a highly non-trivial check of our results , our motivation for developing this alternative algorithm was to try to limit the number of intermediate states and thus lower the memory needs of the program .
We still define the initial state as the set of external legs , but we redefine the chronological order of the tangle diagram by taking the time coordinate to be the geodesic distance to the set of external legs .
Roughly speaking , in each time step we apply one of the transformations shown in Fig .
4 to each of the lines present in the state at that instant .
However , in order to introduce a valid time ordering of the diagram this rough idea needs to be refined .
Fig .
6 : The geodesic time-slice representation of the tangle diagram given in Fig .
3 .
To progress , let us again consider the sample tangle diagram of Fig .
3 .
In Fig .
6 we show its new time-slice representation , this time using the above geodesic definition of time .
At time LaTeXMLMath the edges A and G are both subject to the same transformation , in which three new edge labels ( D , B , E resp .
H , F , I ) are encountered : as a short-hand notation we shall refer to this transformation as LaTeXMLMath .
It closely resembles the type 1 transformation in the single-step algorithm .
When LaTeXMLMath the two upper edges ( D and B ) again undergo a LaTeXMLMath transformation , whereas the four lower edges ( E , H , F , and I ) annihilate at a common vertex : this is the LaTeXMLMath transformation .
At the instant LaTeXMLMath , it is recognized that two edge pairs ( K and C ) created at LaTeXMLMath carry the same label and thus must be identified .
This transformation is reminiscent of the type 2 transformation in the single-step algorithm , and we shall here tag it LaTeXMLMath .
At the same time , the two edges J and L cross so as to become a new pair of edges , which are incidentally both labelled M. This is yet another transformation , the LaTeXMLMath .
Fig .
7 : Intermediate states used by the geodesic algorithm .
The intermediate states produced by this time-slice representation of this example are listed in Fig .
7 .
When comparing with the single-step algorithm ( see Fig .
5 ) we note a considerable simplification .
Turning now to the general case , we see that apart from the LaTeXMLMath move , the transformations discussed above simply express the various ways of “ time ordering ” a tetravalent vertex , i.e .
to assign the label LaTeXMLMath to at least one of its incident edges , and the label LaTeXMLMath to the remaining edges .
The possibility LaTeXMLMath is excluded , as it would lead to the creation of disconnected diagrams .
This leaves us with the transformations LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
But due to the planarity of the diagrams , one also needs to take into account that some of these transformations exist in several variants .
For example , the “ tadpole ” labelled M on Fig .
6 is situated to the left of its adjacent vertex ; however , a different diagram exists in which it is situated to the right .
One must therefore accept that the transformation LaTeXMLMath comes in ( at least ) two guises : in the first , the outgoing edges bend backwards to the left , in the other they continue to the right .
Fig .
8 : Schematic transformation rules for the geodesic transfer matrix .
In Fig .
8 we show schematically the complete set of transformation rules .
The black rectangles represent points at time LaTeXMLMath , and the white ones points at LaTeXMLMath .
The solid lines indicate the action of the transfer matrix at time LaTeXMLMath .
The first eight transformations are simply the topologically inequivalent ways of presenting a tetravalent vertex with at least one point labelled by LaTeXMLMath .
In particular we note that the transformations of type LaTeXMLMath and LaTeXMLMath each occur in three different variants .
The more exotic possibilities 3 , 4 ( resp .
5 ) start contributing to tangle diagrams with at least 4 ( resp .
5 ) crossings .
Finally , the ninth transformation is simply the LaTeXMLMath .
Fig .
9 : Exact tranformation rules for the geodesic transfer matrix .
We still need to transcribe these rules in terms of the states previously defined .
In general , a given transformation leads to the creation of several new enclosed regions ( blocks ) , and to represent the latter in terms of delimiters one needs to make use of the fact that cyclically permuting the points within a given block yields a topologically equivalent state ( we shall come back to this point later , in Sec .
4.2 ) .
Also , since each block must evolve separately to the vacuum there are various parity constraints on the positions of the points entering a given transformation .
By convention , we shall use capital letters to designate ( possibly empty ) blocks of points .
Subscripts LaTeXMLMath and LaTeXMLMath indicate that the number of points must be even ( resp .
odd ) .
Blocks with no subscript may have any parity : they are however subject to the global constraint that the total number of points must be even .
Represented in this way , the exact transformation rules are given in Fig .
9 .
Just like in the single-step algorithm , a transformation may be followed by an arbitrary number of simplifications ( see Sec .
3.1 ) .
3.3 .
Tangencies Until now we have been discussing the enumeration of tangle diagrams in which every vertex represents a crossing .
However , to account for the flype equivalence we need to enumerate more general diagrams with LaTeXMLMath intersections and LaTeXMLMath tangencies , as discussed in Section 2 .
Fortunately , this is a very simple extension of either of our algorithms .
Let us for simplicity consider the case of the single-step algorithm .
Adding a tangency rather than an intersection is obtained by modifying the transformation if Fig .
4.1 , so that the two points added at time LaTeXMLMath are both immediately above ( resp .
immediately below ) the uppermost point at time LaTeXMLMath .
Calling these variants respectively transformation 1a and 1b , a knot diagram with intersections and tangencies is then generated by acting on the initial state with a sequence of transformations 1 , 1a , 1b and 2 .
We also need to add to the characterization of each state a variable that , at any given time , specifies how many tangency transformations ( type 1a or 1b ) were used prior to that instant .
The desired diagrams are then generated by sequences of LaTeXMLMath transformations of type 1 , LaTeXMLMath of type 1a or 1b , and LaTeXMLMath transformations of type 2 , so that no intermediate state is empty .
Omitting the details , we notice that it is equally straightforward to include tangencies in the geodesic algorithm by obvious modifications of the eight first transformations of Fig .
8 .
3.4 .
More external legs Another extension of our algorithms consists in the enumeration of diagrams with LaTeXMLMath external legs , LaTeXMLMath .
To this end we simply start from an appropriate initial state comprising LaTeXMLMath line segments , instead of just one , and we demand that the total number of type 2 transformations exceed the total number of type 1 ( i.e .
1 , 1a , or 1b ) transformations by LaTeXMLMath .
For LaTeXMLMath , such diagrams come in several types , corresponding to the number of ways of pairwise connecting the set of external legs at infinity .
More precisely , given an ordered set of LaTeXMLMath points LaTeXMLMath , the number of types equals the number of ways to divide the set LaTeXMLMath into pairs , considered up to the action of the dihedral group LaTeXMLMath on LaTeXMLMath .
LaTeXMLMath This also has a group-theoretic interpretation , in terms of number of LaTeXMLMath -invariants in the tensor product of LaTeXMLMath fundamental representations of LaTeXMLMath , for generic LaTeXMLMath .
In particular , there are two types of ( four-legged ) tangles ( see Fig .
2 ) , and five types of tangles with LaTeXMLMath external legs ( see Section 5 ) .
The general integer sequence LaTeXMLMath thus defined is discussed in [ 11 ] .
4 .
Implementational details Although both of the algorithms described in the previous section are operational ( as the reader may verify by studying Appendix A ) , we still need to give various details relative to their implementations on a computer .
In particular , it is not clear how states of the type shown in Fig .
5 may be conveniently represented and manipulated .
We shall address this question in Section 4.1 .
Another important observation is that states which until now have appeared to be different are in fact topologically equivalent .
We shall discuss this point in Section 4.2 and demonstrate how it can be used to improve the efficiency of both algorithms .
Finally , we compare the performances of the two different algorithms ( single-step and geodesic ) in Section 4.3 .
4.1 .
Representation of the states In order to render the information contained in states of the type shown in Fig .
5 machine recognizable we shall represent each of them by an ordered list of non-negative integers .
The length of the list representing a given state equals the number of points in the state plus the number of delimiters , and the order of its elements is given simply by reading the state from top to bottom .
Each delimiter is represented by the digit zero .
The other points each correspond to a positive integer , with the convention that two points are connected if and only if they are represented by the same integer .
Clearly , this convention is not unique : for instance , LaTeXMLMath and LaTeXMLMath both describe the same state .
To get rid of this ambiguity we shall stipulate that each consecutive digit , starting from the left , be chosen as small as possible , consistent with the above rules .
Thus , LaTeXMLMath is the unique normal form of our sample state .
The sequence of states shown in Fig .
5 can then be transcribed as follows : LaTeXMLEquation .
At a given stage in the transfer process we need to run through the states present at time LaTeXMLMath , apply the transformation rules described in Section 3.1–3.2 , and produce the set of descendant states ( the states at time LaTeXMLMath ) with their respective weight .
The first time a given descendant state is produced , it must be inserted in a suitable data structure along with is weight .
If subsequently the same state is produced again as a descendant of another parent state , rather than inserting it again we need to retrieve it in the data structure and update its weight .
In order for the algorithm to be efficient , the operations of insertion and retrieval must be accomplished in constant time ( i.e .
in a time that does not depend on the number of states accomodated by the data structure ) .
These demands are fulfilled by a standard data structure known as a hash table [ 12 ] .
It relies on the fact that to each state LaTeXMLMath we can assign a unique integer LaTeXMLMath ( the hash key ) , and devise a function LaTeXMLMath ( the hash function ) that distributes the set of LaTeXMLMath ’ s more-or-less uniformly on the set LaTeXMLMath .
By inserting the states LaTeXMLMath into an array of noded lists indexed by LaTeXMLMath , we can retrieve any given state in a time proportional to the mean length of one of the pointer lists , LaTeXMLMath , where LaTeXMLMath is the total number of entries .
In practice we choose LaTeXMLMath to be a large prime such that LaTeXMLMath , and we use the hash function LaTeXMLMath .
A convenient key LaTeXMLMath can be defined by concatenating the list of “ digits ” entering the normal form of the state LaTeXMLMath into one large integer .
To find the minimum number of bits required to store one digit , we remark that for the counting of tangles with LaTeXMLMath intersections the digits are all LaTeXMLMath .
In the case at hand this means that we need to use at least five bits per digit ; in practice we have however chosen to use eight bits , in order to profit from standard routines for handling character strings .
4.2 .
Equivalences between states As has already been mentioned , it is true for either of the two tangle enumeration algorithms that some of the states generated at a given stage in the transfer process are topologically equivalent .
Clearly , it is of the utmost interest to factor out as many topological equivalences as possible from the state space , since the memory demands as well as the time consumption of the algorithm are roughly proportional to the number of states being treated .
A first such equivalence is due to the fact that any two different blocks of points must evolve separately to the vacuum , without any mutual interaction .
The relative position of the blocks is thus immaterial .
The standard version of either algorithm ( say , version 1 ) can thus be ameliorated by introducing a standard order among the blocks before inserting a given state in the hash table ( version 2 ) .
We have done so by simply sorting the blocks according to their size .
In the special case of the single-step algorithm it is advantageous to place the smallest blocks at the top of the state , since such blocks will then be evolved to the vacuum before touching any other block .
The small block being eliminated , the remainder of the state will be smaller and can thus be processed more expeditiously .
For the geodesic algorithm , the choice betwen ascending and descending ordering is irrelevant .
We take the convention of not changing the relative order of two equally sized blocks .
LaTeXMLMath Inspecting by hand some modestly sized systems reveals that changing the order of equally sized blocks will only lead to a very small further gain .
We have nevertheless made various attempts of imposing a more unique way of arranging the blocks , but since such transformations tend to break down a certain regularity in the connectivities which is imposed by the type 1 transformation , these attempts actually resulted in a slight increase in the number of states .
After the permutation of the blocks , the representation of the state in terms of a list of integers is brought back to its normal form ( see Sec .
4.1 ) .
In Table 1 we illustrate the resulting decrease in the number of states inserted in the hash table .
It seems clear that not only is the number of states much smaller , but it even grows with a smaller exponent .
LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Tab .
1 : Maximal number of intermediate states in the transfer process for three different versions ( see text ) of the single-step algorithm for tangle diagrams with LaTeXMLMath self-intersections and no tangencies .
Second , the states are equivalent upon cyclic rotations ( eventually combined with a reflection ) of the points within any given block .
Since , once again , the blocks are independent , these dihedral transformations can be performed independently within each block .
The ultimate way of implementing this equivalence would be the following : before inserting a state in the hash table , subject it to all possible dihedral transformations , and check whether any of the transformed states is already present in the table .
Unfortunately , the number of transformations increases faster than exponentially with the size of the state , and this exhaustive search would quickly end up usurping the majority of the CPU time .
We have therefore opted for a less perfect but much faster alternative ( version 3 ) .
LaTeXMLMath Comparing with the exhaustive method applied to some modestly sized systems shows that , once again , the number of topological equivalences not detected by the “ approximate ” method is negligible when compared to the number of states which are in fact topologically distinct .
Having sorted the blocks according to their size , we consider in turn all possible dihedral transformations on the points in the first block , keeping fixed the positions of the points in the other blocks .
After each transformation we bring the integer representation of the state into its normal form .
We then identify ( one of ) the normal form ( s ) which lexicographically precedes all the others , and lock the points of the first block into their corresponding positions .
Leaving the first block locked , we procede to apply the same procedure to the second block .
We continue this way until all blocks have been locked , and only then the resulting representation of the state is inserted into the hash table .
The gain of version 3 over version 2 is comparable to the gain of version 2 over version 1 , as witnessed by Table 1 .
In the following we shall therefore exclusively understand version 3 when referring to any one of the two algorithms ( single-step or geodesic ) .
4.3 .
Comparing the two algorithms A first striking difference between the single-step and the geodesic algorithm can be observed by comparing how their respective number of intermediate states ( and thus the memory needs ) evolve as a function of the “ time ” defined by the transfer process .
Fig .
10 : Memory profile of the single-step algorithm ( version 3 , cf .
Sec .
4.2 ) .
The curves represent knot diagrams with LaTeXMLMath crossings and no tangencies .
Fig .
11 : Memory profile of the geodesic algorithm ( version 2 , cf .
Sec .
4.2 ) .
The curves represent knot diagrams with LaTeXMLMath crossings and no tangencies .
In both cases , the number of states grows exponentially in the beginning , decreases exponentially towards the end , and reaches a maximum somewhere in between .
However , for the single-step algorithm this maximum is reached at roughly LaTeXMLMath ( where LaTeXMLMath is the total number of time steps ) , whereas for the geodesic algorithm the maximum is situated around LaTeXMLMath .
The reason for this difference is that the geodesic algorithm will produce the majority of its states by applying the LaTeXMLMath rule as often as possible in the beginning of the process .
LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath Tab .
2 : Maximal number of intermediate states in the transfer process for three different versions ( see Sec .
4.2 ) of the geodesic algorithm for tangle diagrams with LaTeXMLMath self-intersections and no tangencies .
To estimate the actual memory needs of the algorithms , it is instructive to compare the maximal number of intermediate states for the three different versions defined in Sec .
4.2 .
For the single-step algorithm the data were given in Tab .
1 ; we show the corresponding numbers for the geodesic algorithm in Tab .
2 .
As expected , version 1 of the geodesic algorithm employs considerably fewer states than version 1 of the single-step algorithm .
However , quite surprisingly , the ameliorations implied by version 2 and version 3 lead to an enormous gain in the single-step case , but only a modest one in the geodesic case .
Thus , in version 3 the asymptotic growth of the number of states is significantly slower in the single-step algorithm than in the geodesic one , even though the latter was explicitly designed to use fewer states !
Although a qualitative explanation of this phenomenon can be given by inquiring into the structure of a typical state we refrain from doing this here .
Rather , let us simply accept the efficiency of the single-step algorithm as a remarkable fact .
Even when discarding the issue of memory , the geodesic algorithm has a serious drawback compared with its single-step counterpart as far as time consumption is concerned .
Namely , the single-step algorithm processes each state in a time that grows roughly linearly with its size , whereas for the geodesic algorithm this time grows exponentially .
To see this , consider the intermediate state which is obtained from the initial state by performing LaTeXMLMath transformations of type LaTeXMLMath .
To turn this state into tangle diagrams with exactly LaTeXMLMath crossings , one needs to complete it with LaTeXMLMath transformation of type LaTeXMLMath .
Supposing LaTeXMLMath chosen so that the intermediate state has a complete number of time slices , a total of LaTeXMLMath diagrams will be recursively generated , where LaTeXMLMath are the Catalan numbers .
The geodesic algorithm will therefore ( asymptotically ) spend the majority of the CPU time closing up this “ maximally opened state ” .
We have therefore used the geodesic algorithm as a highly non-trivial check of our numerical results , but the data for large system sizes are generated exclusively by the single-step algorithm .
To conclude this section , let us briefly discuss the time complexity of our best algorithm ( single-step , version 3 ) .
Based on the data in Tab .
1 , we infer that both time and memory needs grow asymptotically as LaTeXMLMath , with LaTeXMLMath .
5 .
Numerical results We now present the numerical results that we obtained using the single-step algorithm ( version 3 ) .
Due to the enormous amount of data gathered we shall only give the main results .
The first data are obtained by running a program that implements the single-step algorithm without any tangencies .
This corresponds to the generating function LaTeXMLMath in the notation of Section 2 .
Its coefficients LaTeXMLMath are given in Table 3 up to LaTeXMLMath .
The computation took a few hours on a 1 GHz single-processor work station with 1 GByte of memory .
LaTeXMLMath 0 1 2 3 4 5 6 7 8 9 0 1 1 2 2 8 1 3 42 12 4 260 114 4 5 1796 1030 90 6 13396 9290 1349 22 7 105706 84840 17220 728 8 870772 787082 203568 14884 140 9 7420836 7415814 2312094 244908 6120 10 65004584 70867212 25691670 3575045 158354 969 11 582521748 685839770 282000444 48517524 3185314 52668 12 5320936416 6712285600 3074136464 628013796 55273668 1647728 7084 13 49402687392 66349573368 33387698708 7871666088 871779428 39142116 460460 14 465189744448 661680191832 361969672904 96451145091 12876308613 786444610 16890227 53820 15 4434492302426 6651030871168 3921901043440 1162484964230 181430681094 14126467392 462455640 4071600 16 42731740126228 67329662060890 42499598861832 13840075278704 2468480436152 234358127880 10552931952 171277860 420732 17 415736458808868 685953949494774 460831546801414 163246693686684 32699872694298 3666111325052 212581611050 5308497112 36312408 18 4079436831493480 7028941367108708 5001468564165262 1911737961254907 424232095742826 54835331971380 3912429396360 135564649071 1722788176 3362260 19 40338413922226212 72403769391718890 54341248085414380 22262254374655710 5413174461572394 791922013806504 67266181855770 3025712334552 59605106568 326023280 Tab .
3 : Table of the number of alternating tangle diagrams with LaTeXMLMath external legs .
Let us first note that there are various quantities which can be extracted from this table and which are known exactly in an independent way .
They provide a number of non-trivial checks .
Let us define the number LaTeXMLMath of diagrams at fixed number of colors LaTeXMLMath LaTeXMLEquation .
The simplest choice is to set LaTeXMLMath , that is to consider the sum of each row of the table .
This series of numbers is known exactly ( see [ 13 ] for a purely combinatorial argument , and [ 14 ] for a field theoretic one ) LaTeXMLEquation and the corresponding generating function is LaTeXMLEquation .
It is perhaps less well known that for LaTeXMLMath , one also has an exact expression , in terms of elliptic integrals [ 15 ] : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the complete elliptic integrals of the first and second kinds .
We note that the generating function is non-algebraic , and it is not known how to find it by direct combinatorial arguments .
There exists a similar , although more complicated , formula for the case LaTeXMLMath which will be presented elsewhere [ 10 ] .
One can also find expressions for the last non-zero element of each row , which formally corresponds to LaTeXMLMath ( with LaTeXMLMath fixed ) .
However there is a parity effect which forces us to redefine separately odd and even generating functions : LaTeXMLEquation .
Fig .
12 : a ) Equation satisfied by LaTeXMLMath .
b ) Recursive definition of a circle diagram and its interpretation in terms of a rooted tree .
For the even case , a general diagram has the form of successive insertions of circles in the bare propagator , which leads to the equation , depicted on Fig .
12 a ) , LaTeXMLEquation .
This can be described more explicitly in terms of rooted trees .
Each insertion of a circle requires two additional intersections , and it leads to the creation of four new edges in which new circles can be inserted .
On Fig .
12 b ) we have labelled two successive generations of edges as LaTeXMLMath and LaTeXMLMath respectively .
Clearly , this reduces the problem to that of enumerating rooted trees in which each node ( resp .
the root ) can have degree 1 or 5 ( resp .
0 or 4 ) .
This is a simple example of a rather broad class of rooted trees discussed by Takács [ 16 ] .
From Eq .
( 5.6 ) we infer that LaTeXMLEquation .
The proof of the formula for LaTeXMLMath is left as an exercize to the reader : LaTeXMLEquation .
We infer that LaTeXMLEquation .
Finally , the first column in Table 3 reproduces the knot diagrams discussed in [ 4 ] , of course .
Next we want to deduce some properties of the asymptotic behavior of these series from the numerical data .
The first quantity one can extract is the “ bulk entropy ” of alternating tangles and links .
LaTeXMLMath The bulk behavior should be independent of the number of external legs , and in particular be the same for links and tangles .
At fixed LaTeXMLMath it is defined by the leading exponential behavior of LaTeXMLMath : LaTeXMLEquation .
It would however be more natural to consider alternating links/tangles at fixed number of connected components .
This requires making an appropriate scaling ansatz for the coefficients LaTeXMLMath , which turns out to be LaTeXMLEquation where LaTeXMLMath is the bulk entropy at fixed ratio LaTeXMLMath of the number of connected components by the number of crossings .
It is clear from Eq .
( 5.1 ) that the two entropies defined above are related to each other by a Legendre transform ; namely , if one defines the average ratio LaTeXMLMath at fixed LaTeXMLMath : LaTeXMLEquation then the following relation holds : LaTeXMLEquation so that we also have the dual equation of ( 5.12 ) LaTeXMLEquation .
Fig .
13 : The bulk entropy LaTeXMLMath .
On Fig .
13 the behavior of LaTeXMLMath is shown for LaTeXMLMath .
The various exact solutions mentioned above correspond to the following known values of LaTeXMLMath : LaTeXMLEquation .
In [ 4 ] , the following numerical value was given : LaTeXMLMath , which is confirmed here .
Fig .
14 : The bulk entropy LaTeXMLMath .
Even more interesting is the curve LaTeXMLMath shown on Fig .
14 for LaTeXMLMath .
It displays a clear maximum at a value which is nothing but LaTeXMLMath , cf Eq .
( 5.14 ) .
Numerically we find : LaTeXMLEquation .
This number has the following significance : at fixed number of crossings LaTeXMLMath , a typical tangle/link diagram will have LaTeXMLMath connected components when LaTeXMLMath goes to infinity .
Note that this number is extremely small : any average made over ( equally weighted ) alternating link diagrams will be dominated by objects with few connected components .
Let us note that the bulk quantities are affected by the various renormalizations of Eq .
LaTeXMLMath , i.e .
restriction to 2PI diagrams and inclusion of the flype equivalence .
However , it is expected that the qualitative properties ( and in particular the maximum of the entropy for a very small value of LaTeXMLMath ) are unchanged .
For example , if one considers 2PI diagrams ( which corresponds to the counting of reduced prime alternating link/tangle diagrams ) , one finds a maximum at LaTeXMLMath instead .
The discussion of the exponent associated to the subdominant power-law behavior of the series LaTeXMLMath ( or , equivalently , LaTeXMLMath ) is much more involved .
We define the critical exponent LaTeXMLEquation .
Let us first recall the conjecture made in [ 9 ] , which relies on several hypotheses : a ) the asymptotic behavior of LaTeXMLMath is related to a singularity of the corresponding generating function LaTeXMLMath which has the physical meaning of singularity of 2D quantum gravity , i.e .
large link diagrams behave as continuum random surfaces for which conformal field theory techniques apply ( KPZ formula [ 17 ] ) ; b ) the model describing link diagrams with LaTeXMLMath colors is in the same universality class as the usual LaTeXMLMath model of dense loops [ 18 ] , which relies on the assumption that there is no phase transition in the generalized LaTeXMLMath matrix model .
For LaTeXMLMath , LaTeXMLMath ( LaTeXMLMath ) , this implies that LaTeXMLEquation .
Let us now discuss separately various regions of LaTeXMLMath and the corresponding numerical analysis .
LaTeXMLMath For LaTeXMLMath , a difficulty arises in that coefficients LaTeXMLMath do not have a fixed sign .
This implies in particular that the dominant singularity of the generating function LaTeXMLMath is not necessarily on the real positive axis , as would be implied by hypothesis a ) above .
Numerically it seems that pairs of complex conjugated singularities do occur and become dominant in a large region of LaTeXMLMath which includes at least part of the interval LaTeXMLMath , thus invalidating conjecture ( 5.18 ) in this region .
The analysis of such behavior is fairly involved and we leave it to future work .
LaTeXMLMath LaTeXMLMath : at LaTeXMLMath it was suggested in [ 4 ] that even though conjecture ( 5.18 ) is correct ( LaTeXMLMath ) , there might be a logarithmic correction which spoils the asymptotic behavior of the coefficients .
For LaTeXMLMath small , we expect several singularities extremely close to the dominant singularity making any analysis difficult .
Estimates of the critical exponent do not contradict ( 5.18 ) , but they have very low accuracy ; for example , LaTeXMLEquation .
LaTeXMLMath LaTeXMLMath : we can first extract from the exact solutions ( Eqs .
( 5.2 ) , LaTeXMLMath and LaTeXMLMath ) the asymptotics LaTeXMLEquation .
They are of course compatible with ( 5.18 ) ; however we note a logarithmic correction in LaTeXMLMath which comes from inverting the singularity of Eq .
LaTeXMLMath : LaTeXMLMath .
At this point it becomes clear that in order to remove the LaTeXMLMath factor , one just needs to perform an appropriate functional inversion on the generating series LaTeXMLMath .
Applying the same procedure to the numerical data of LaTeXMLMath for LaTeXMLMath , one can then use standard convergence acceleration methods and obtain precise estimates of the critical exponent .
They are in good agreement with ( 5.18 ) ; for example , LaTeXMLEquation .
LaTeXMLMath LaTeXMLMath : let us first recall that we have found exact expressions at LaTeXMLMath for odd and even coefficients separately ( Eqs .
( 5.7 ) and ( 5.9 ) ) .
Asymptotically , LaTeXMLEquation i.e .
the bulk terms are identical but the critical exponents are different .
This can be understood easily since in the odd case there is one “ defect ” in the sequence of circles which corresponds to marking one connected component , that is multiplying by LaTeXMLMath ( cf the differentiation in Eq .
( 5.8 ) ) .
Numerically , it is clear that for all LaTeXMLMath odd and even series behave differently .
However once can not perform any serious analysis on these series since they are too short .
It may be that the exponents of Eq .
LaTeXMLMath are preserved for any LaTeXMLMath , or finite values of LaTeXMLMath might smooth the difference between odd and even series ; the data we possess are unconclusive on this issue .
Let us end this analysis by noting that contrary to the bulk terms , critical exponents are expected to be independent of the various renormalizations of Eq .
LaTeXMLMath , due to universality arguments .
We now turn to the data obtained by inclusion of tangencies .
For reasons of conciseness , we here refrain from displaying the three-dimensional array of coefficients LaTeXMLMath ; these are electronically available from the authors upon request .
Even the final results are fairly cumbersome to treat and display , so that we only show the results for the number of prime alternating tangles up to LaTeXMLMath ( even though they can be easily obtained for LaTeXMLMath up to LaTeXMLMath or LaTeXMLMath , as in [ 4 ] , on a work station , and probably a bit further using larger computers ) .
LaTeXMLMath LaTeXMLMath LaTeXMLMath 0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 1 0 2 0 1 3 2 1 4 2 3 1 5 6 3 9 1 6 30 2 21 11 1 7 62 40 2 101 32 1 8 382 106 2 346 153 24 1 9 1338 548 83 2 1576 747 68 1 10 6216 2968 194 2 7040 3162 562 43 1 11 29656 11966 2160 124 2 31556 17188 2671 121 1 12 131316 71422 9554 316 2 153916 80490 15295 1484 69 1 13 669138 328376 58985 5189 184 2 724758 425381 87865 6991 194 1 14 3156172 1796974 347038 22454 478 2 3610768 2176099 471620 52231 3280 103 1 15 16032652 9298054 1864884 193658 10428 260 2 17853814 11376072 2768255 308697 15431 290 1 Tab .
4 : Table of the number of prime alternating tangles .
These data satisfy once more various non-trivial checks , including the comparison with the table in the appendix of [ 5 ] ( for LaTeXMLMath ) , of the Tables 1 and 2 in [ 7 ] ( for LaTeXMLMath ) , and Table 3 of [ 4 ] ( for LaTeXMLMath ) .
Fig .
15 : Large LaTeXMLMath expansion of tangles .
The last term in each column can also be verified by means of the large LaTeXMLMath expansion .
It is easy to convince oneself that in general the 2PI diagrams with the largest possible number of connected components is obtained by decorating the bare tangles by means of a festoon of chained circles , as shown on Fig .
15 .
In the case of LaTeXMLMath ( resp .
LaTeXMLMath ) there are two ( resp .
one ) flype-inequivalent ways of doing so .
As it stands , this argument holds true for odd LaTeXMLMath ( resp .
even LaTeXMLMath ) in the case of LaTeXMLMath ( resp .
LaTeXMLMath ) , but a similar reasoning holds true for the opposite parity .
We conclude that the last term in each row of Tab .
4 should be LaTeXMLMath ( for LaTeXMLMath and LaTeXMLMath ) resp .
1 ( for LaTeXMLMath and LaTeXMLMath ) , as is indeed observed .
The next to leading terms should be obtainable in a similar fashion .
Finally , we demonstrate the power of our method by applying it to objects with more external legs .
One may for example ask how many ways there are to intertwine three strings , and not just two as in the case of tangles .
One must first establish the different ways the strings are coming out , which leads to Fig .
16 .
Fig .
16 : The five types of tangles with 6 external legs .
We only consider configurations such that no strings can be pulled out altogether ( “ connected ” correlation functions in the language of quantum field theory ) .
Tab .
5 provides the first few orders of the series of the numbers of such objects .
LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath LaTeXMLMath 0 1 2 3 0 1 2 3 0 1 2 3 4 0 1 2 3 0 1 2 3 2 0 1 0 0 0 3 2 0 2 0 0 4 0 7 2 4 3 5 18 6 16 2 8 9 6 18 53 8 42 2 42 7 41 7 7 156 24 154 6 171 44 2 156 14 168 21 8 516 18 609 181 6 748 114 2 608 153 10 663 165 12 9 2016 598 18 2956 422 6 2877 858 81 2 2850 586 20 3072 740 36 10 10608 1428 18 11203 3498 318 6 14037 3752 213 2 11918 3445 364 13 13347 3966 438 18 11 40428 12318 1062 18 57664 15330 738 6 61028 19757 2511 131 2 57602 17558 1406 26 63393 20994 2040 54 Tab .
5 : Table of the number of prime alternating tangles with LaTeXMLMath external legs .
The lowest order in LaTeXMLMath is explicited in Fig .
16 .
It is again relatively straightforward to check the correctness of the last entry in each row of Tab .
5 by considering the LaTeXMLMath limit .
In the cases of [ LaTeXMLMath with LaTeXMLMath ] , [ LaTeXMLMath with LaTeXMLMath ] , and [ LaTeXMLMath with LaTeXMLMath ] it is straightforward to show that the diagrams having the highest power of LaTeXMLMath are just the trivial diagrams decorated by festoons ( as in Fig .
15 ) , meaning that the last entry in the corresponding rows should be respectively 18 , 6 and 2 .
For the cases [ LaTeXMLMath with LaTeXMLMath ] resp .
[ LaTeXMLMath with LaTeXMLMath ] we conjecture that the last entry of each row LaTeXMLMath should read , for LaTeXMLMath even , LaTeXMLEquation .
For LaTeXMLMath odd we have LaTeXMLMath resp .
LaTeXMLMath .
6 .
Discussion and outlook In this paper we have shown how to efficiently enumerate alternating tangle diagrams with a given number of connected components and external legs , and we have explained how the flypes can be easily incorporated into our algorithm to count only topologically inequivalent objects .
We have illustrated our method with numerical data that are in agreement with the exact solutions at LaTeXMLMath [ [ 5 ] 5 , , [ 6 ] 6 , , [ 10 ] 10 ] , LaTeXMLMath [ 7 ] , LaTeXMLMath [ 10 ] , and the limit LaTeXMLMath , and that surpass the general results of [ 7 ] by several orders .
From a computational point of view , we note that the time required to compute order LaTeXMLMath grows exponentially with LaTeXMLMath , but much more slowly than the number of diagrams counted ( LaTeXMLMath compared to the number of diagrams LaTeXMLMath ) .
A remarkable advantage of our transfer matrix algorithms is that they allow to generate planar diagrams with external legs , even in the absence of a line or closed circuit defining an obvious transfer direction .
It should be noticed that our algorithms can be straightforwardly adapted to graphs of any coordination number LaTeXMLMath .
For the single-step algorithm , it suffices to modify the type 1 transformation so as to insert LaTeXMLMath new points , instead of just two .
For odd LaTeXMLMath , the parity constraints on transformation 2 no longer apply .
The geodesic algorithm can be similarly generalized .
For example , on Fig .
8 , changing to trivalent vertices would simply imply having five possible transformation rules instead of nine .
These generalizations open several interesting perspectives .
One obvious possibility would be to numerically study matter theories defined on random graphs , by exact evaluation of correlation functions [ 19 ] .
Appendix A. Tangle diagrams up to LaTeXMLMath crossings .
As an illustration we show on Fig .
17 the nine first iterations of the transfer matrix .
We restrict ourselves to states which generate diagrams with at most four crossings .
The states have all been simplified , as described in Sec .
4 .
The weight of the trivial state ( represented by a cross on the figure ) after step LaTeXMLMath gives the number of two-legged tangle diagrams with exactly LaTeXMLMath crossings , each connected component being weighed by a factor of LaTeXMLMath .
LaTeXMLMath By convention , we also give a weight LaTeXMLMath to components connecting a pair of external legs , though from the point of view of diagrammatic perturbation theory this is , strictly speaking , not correct .
Fig .
17 : The list of all intermediate states for tangle diagrams up to four crossings .
Below each state we indicate its weight .
References [ 1 ] % [ 2 ] % [ 3 ] % [ 4 ] % [ 5 ] % [ 6 ] % [ 7 ] % [ 8 ] % [ 9 ] % [ 10 ] % [ 11 ] % [ 12 ] % [ 13 ] % [ 14 ] % [ 15 ] % [ 16 ] % [ 17 ] % [ 18 ] % [ 19 ] %
By using anholonomic frames in ( pseudo ) Riemannian spaces we define anisotropic extensions of Euclidean Taub–NUT spaces .
With respect to coordinate frames such spaces are described by off-diagonal metrics which could be diagonalized by corresponding anholonomic transforms .
We define the conditions when the 5D vacuum Einstein equations have as solutions anisotropic Taub–NUT spaces .
The generalized Killing equations for the configuration space of anisotropically spinning particles ( anisotropic spinning space ) are analyzed .
Simple solutions of the homogeneous part of these equations are expressed in terms of some anisotropically modified Killing-Yano tensors .
The general results are applied to the case of the four-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature .
We emphasize that all constructions are for ( pseudo ) Riemannian spaces defined by vacuum soltions , with generic anisotropy , of 5D Einstein equations , the solutions being generated by applying the moving frame method .
Pacs 12.10.-g,12.90.+b , 02.40.+m , 04.20.Me ; MSC numbers : 83E15 , 83E99 Much attention has been paid to off–diagonal metrics in higher dimensional gravity beginning the Salam , Strathee and Perracci work LaTeXMLCite which showed that including off–diagonal components in higher dimensional metrics is equivalent to including LaTeXMLMath and LaTeXMLMath gauge fields .
The approach was developed by construction of various locally isotropic solutions of vacuum 5D Einstein equations describing 4D wormholes and/or flux tube gravitational–electromagnetic configurations ( see Refs .
LaTeXMLCite ) .
Recently , the off–diagonal metrics were considered in a new fashion by applying the method of anholonomic frames with associated nonlinear connections LaTeXMLCite which allowed to construct new classes of solutions of Einstein ’ s equations in three ( 3D ) , four ( 4D ) and five ( 5D ) dimensions which had generic local anisotropy , e.g .
static black hole and cosmological solutions with ellipsoidal or toroidal symmetry , various soliton–dilaton 2D and 3D configurations in 4D gravity , and wormhole and flux tubes with anisotropic polarizations and/or running constants with different extensions to backgrounds of rotation ellipsoids , elliptic cylinders , bipolar and toroidal symmetry and anisotropy .
Another class of 4D metrics induced from 5D Kaluza–Klein theory is connected with the Euclidean Taub–NUT metric which is involved in many modern studies of physics , for instance , in definition of the gravitational analogue of the Yang–Mills instantons LaTeXMLCite and of Kaluza–Klein monopole LaTeXMLCite related with geodesic geometry of higher dimension ( pseudo ) Riemannian spaces LaTeXMLCite ( see a recent review and original results in LaTeXMLCite ) .
The construction of monopole and instanton solutions , with deformed symmetries , in modern string theory , extra dimensional gravity and quantum chromodynamics is of fundamental importance in understanding these theories ( especially their non-perturbative aspects ) .
Such solutions are difficult to find , and the solutions which are known usually have a high degree of symmetry .
In this work we apply the method of anholonomic frames to construct the general form anholonomically constrained Taub NUT metrics in 5D Kaluza-Klein theory .
These solutions have local anisotropy which would make their study using holonomic frames difficult .
This helps to demonstrate the usefulness of the anholonomic frames method in studying anisotropic solutions .
Most physical situations do not possess a high degree of symmetry , and so the anholonomic frames method provides a useful mathematical framework for studying these less symmetric configurations .
We emphasize that the anholonomic moving frame method works effectively in construction of anisotropic mass hierarchies with running of constants in modern brane physics LaTeXMLCite ( on new directions in extra dimension gravity see Refs .
LaTeXMLCite .
This allows us to approach a task of of primordial importance of definition of non–perturbative models and finding of exact solutions in higher dimension field theory describing anisotropic monopole/instanton configurations with running constants .
The metrics considered for both wormhole and Taub-NUT geometry and physics could be given by 5D line elements with 3D spherical coordinates LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation where the metric coefficients and constants LaTeXMLMath have to be correspondingly parametrized in order to select two particular cases : We must put LaTeXMLMath LaTeXMLEquation and to impose on the fifth coordinate the condition LaTeXMLMath LaTeXMLMath if we want to obtain the Taub-NUT metric connected with the gauge field LaTeXMLMath of a monopole LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath denotes a three-vector LaTeXMLMath the so called NUT singularity is absent if LaTeXMLMath is periodic with period LaTeXMLMath LaTeXMLCite .
The wormhole / flux tube metrics LaTeXMLCite are parametrized if we put LaTeXMLMath and LaTeXMLMath for LaTeXMLMath LaTeXMLMath LaTeXMLMath all functions LaTeXMLMath and LaTeXMLMath are taken to be even functions of LaTeXMLMath satisfying the conditions LaTeXMLMath .
The coefficient LaTeXMLMath is treated as the LaTeXMLMath –component of the electromagnetic potential and LaTeXMLMath is the LaTeXMLMath -component .
These electromagnetic potentials lead to radial Kaluza-Klein ‘ electrical ’ LaTeXMLMath and ‘ magnetic ’ LaTeXMLMath fields : LaTeXMLEquation with the ‘ electric ’ charge LaTeXMLMath and LaTeXMLEquation with ‘ magnetic ’ charge LaTeXMLMath The solution in LaTeXMLCite satisfied the boundary conditions LaTeXMLMath where it was proved that the free parameters of the metric are varied there are five classes of wormhole /flux tube solutions .
We note that the metric ( LaTeXMLRef ) defines solutions of vacuum Einstein equations only for particular parametrizations of type 1 or 2 ; it is not a vacuum solution for arbitrary values of coefficients .
The main results of works LaTeXMLCite were obtained by applying the anholonomic frame method which allowed to construct off–diagonal metrics describing wormhole / flux tube configurations with anisotropic varying on the coordinates LaTeXMLMath or LaTeXMLMath given by some higher dimension renormalizations of the constants LaTeXMLMath ( or , inversely , LaTeXMLMath and/or LaTeXMLMath ( or , inversely , LaTeXMLMath The purpose of this paper is to construct Taub-NUT like metrics with anisotropic variations of the constant LaTeXMLMath when LaTeXMLMath or LaTeXMLMath We note that such anisotropic metrics are given by off–diagonal coefficients which define solutions of the 5D vacuum Einstein equations and generalize the constructions from LaTeXMLCite to Taub–NUT locally anisotropic gravitational instantons embedded into anisotropic variants of Kaluza–Klein monopoles ( the first anisotropic instanton solutions were proposed in in Refs .
LaTeXMLCite for the so–called generalized Finsler–Kaluza–Klein spaces and locally anisotropic gauge gravity , here we note that in this paper we shall not concern topics relating generalized Lagrange and Finsler ( super ) spaces LaTeXMLCite ) ; by using anholonomic frames we can model anisotropic instanton configurations in usual Riemannian spaces .
The anisotropic metrics are defined as ( pseudo ) Riemannian ones which admit a diagonalization with respect to some anholonomic frame bases with associated nonlinear connection structures .
Such spacetimes , provided with metrics with generic anisotropy and anholonomic frame structure , are called as anisotropic spaces-time .
Let us introduce a new 5th coordinate LaTeXMLEquation for which LaTeXMLMath and LaTeXMLEquation if the factor LaTeXMLMath is taken , for instance , LaTeXMLEquation .
With respect to the new extradimensional coordinate LaTeXMLMath the component LaTeXMLMath of the electromagnetic potential is removed into the component LaTeXMLMath this will allow us to treat the coordinates LaTeXMLMath as holonomic coordinates but LaTeXMLMath as anholonomic ones .
For our further considerations it is convenient to use a conformally transformed ( multiplied on the factor LaTeXMLMath ) Taub NUT metric with the 5th coordinate LaTeXMLMath LaTeXMLEquation which will be used for generalizations in order to obtain new solutions of the vacuum Einstein equations , being anisotropic on coordinates LaTeXMLMath This metric generates a monopole configuration ( LaTeXMLRef ) .
The paper is organized as follow : Section 2 outlines the geometry of anholonomic frames with associated nonlinear connections on ( pseudo ) Riemannian spaces .
The metric ansatz for anisotropic solutions is introduced .
In Section 3 , there are analyzed the basic properties of solutions of vacuum Einstein equations with mixed holonomic and anholonomic variables .
The method of construction of exact solutions with generic local anisotropy is developed .
In Section 4 , we construct three classes of generalized anisotropic Taub NUT metrics , being solutions of the vacuum Einstein equations , which posses anisotropies of parameter LaTeXMLMath on angular coordinate LaTeXMLMath or contains a running constant LaTeXMLMath and/or are elliptically polarized on angular coordinate LaTeXMLMath Section 5 is devoted to a new exact 5D vacuum solution for anisotropic Taub NUT wormholes obtained as a nonlinear superposition of the running on extra dimension coordinate Taub NUT metric and a background metric describing locally isotropic wormhole / flux tube configurations .
Section 6 elucidates the problem of definition of integrals of motion for anholonomic spinning of particles in anisotropic spaces .
There are introduced Killing , energy and momentum and Runge–Lenz vectors with respect to anholonomic bases with associated nonlinear connection structures defined by anisotropic solutions of vacuum Einstein equations .
There are proposed and analyzed the action for anisotropic spinning of particles , defined the Poisson–Dirac brackets on anisotropic spaces .
We consider anisotropic Killing equations and discuss the problem of construction their generic solutions and non–generic solutions with anholonomic Killing–Yano tensors .
In Section 7 , we approach the problem of definition of Killing–Yano tensors for anisotropic Taub NUT spinning spaces and construct the corresponding Lie algebra with anisotropic variation of constants .
Finally , in Section 8 , some conclusion remarks are presented .
In this section we outline the basic formulas on anholonomic frames with mixed holonomic–anholonomic components ( variables ) and associated nonlinear connection structures in Riemannian spaces .
Let us consider a 5D pseudo–Riemannian spacetime of signature LaTeXMLMath LaTeXMLMath and denote the local coordinates LaTeXMLEquation where LaTeXMLMath or , inversely , LaTeXMLMath – or more compactly LaTeXMLMath – where the Greek indices are conventionally split into two subsets LaTeXMLMath and LaTeXMLMath labeled , respectively , by Latin indices of type LaTeXMLMath and LaTeXMLMath The local coordinate bases , LaTeXMLMath and their duals , LaTeXMLMath are written respectively as LaTeXMLEquation and LaTeXMLEquation .
The 5D ( pseudo ) Riemannian squared linear interval LaTeXMLEquation is given by the metric coefficients LaTeXMLMath ( a matrix ansatz definded with respect to the coordinate frame base ( LaTeXMLRef ) ) in the form LaTeXMLEquation where the coefficients are some necessary smoothly class functions of type : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation one considers dependencies of the coefficients of metric on two so–called isotropic variables LaTeXMLMath and on one anisotropic variable LaTeXMLMath ( in similar fashions we can alternatively consider dependencies on arbitrary couples of LaTeXMLMath –coordinates completed with one LaTeXMLMath –coordinate , for instance , LaTeXMLMath and LaTeXMLMath .
The metric ( LaTeXMLRef ) with coefficients ( LaTeXMLRef ) can be equivalently rewritten in the form LaTeXMLEquation with diagonal coefficients LaTeXMLEquation if instead the coordinate bases ( LaTeXMLRef ) and ( LaTeXMLRef ) one introduces the anholonomic frames ( anisotropic bases ) LaTeXMLEquation and LaTeXMLEquation where the LaTeXMLMath –coefficients are parametrized LaTeXMLEquation ( they define an associated to some anholonomic frames ( LaTeXMLRef ) and ( LaTeXMLRef ) , nonlinear connection , N–connection , structure , see details in Refs LaTeXMLCite ) .
A N–connection induces a global decomposition of the 5D pseudo–Riemannian spacetime into holonomic ( horizontal , h ) and anholonomic ( vertical , v ) directions .
In a preliminary form the concept of N–connections was applied by E. Cartan in his approach to Finsler geometry LaTeXMLCite and a rigorous definition was given by Barthel LaTeXMLCite ( Ref .
LaTeXMLCite gives a modern approach to the geometry of N–connections , and to generalized Lagrange and Finsler geometry , see also Ref .
LaTeXMLCite for applications of N–connection formalism in supergravity and superstring theory ) .
As a particular case one obtains the linear connections if LaTeXMLMath A quite surprising result is that the N–connection structures can be naturally defined on ( pseudo ) Riemannian spacetimes LaTeXMLCite by associating them with some anholonomic frame fields ( vielbeins ) of type ( LaTeXMLRef ) satisfying the relations LaTeXMLMath with nontrivial anholonomy coefficients LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation is the nonlinear connection curvature ( N–curvature ) .
One says that the N–connection coefficients model a locally anisotropic structure on spacetime ( a locally anisotropic spacetime ) when the partial derivative operators and coordinate differentials , ( LaTeXMLRef ) and ( LaTeXMLRef ) , are respectively changed into N–elongated operators ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Conventionally , the N–coefficients decompose the spacetime values ( tensors , spinors and connections ) into sets of mixed holonomic–anholonomic variables ( coordinates ) provided respectively with ’ holonomic ’ indices of type LaTeXMLMath and with ’ anholonomic ’ indices of type LaTeXMLMath .
Tensors , metrics and linear connections with coefficients defined with respect to anholonomic frames ( LaTeXMLRef ) and ( LaTeXMLRef ) are distinguished ( d ) by N–coefficients into holonomic and anholonomic subsets and called , in brief , d–tensors , d–metrics and d–connections .
On ( pseudo ) –Riemannian spacetimes the associated N–connection structure can be treated as a ” pure ” anholonomic frame effect which is induced if we are dealing with mixed sets of holonomic–anholonomic basis vectors .
When we are transferring our considerations only to coordinate frames ( LaTeXMLRef ) and ( LaTeXMLRef ) the N–connection coefficients are removed into both off–diagonal and diagonal components of the metric like in ( LaTeXMLRef ) .
In some cases the N–connection ( anholonomic ) structure is to be stated in a non–dynamical form by definition of some initial ( boundary ) conditions for the frame structure , following some prescribed symmetries of the gravitational–matter field interactions , or , in another cases , a subset of N–coefficients have to be treated as some dynamical variables defined as to satisfy the Einstein equations .
A metric of type ( LaTeXMLRef ) , in general , with arbitrary coefficients LaTeXMLMath and LaTeXMLMath defined with respect to a N–elongated basis ( LaTeXMLRef ) is called a d–metric .
A linear connection LaTeXMLMath associated to an operator of covariant derivation LaTeXMLMath is compatible with a metric LaTeXMLMath and N–connection structure on a 5D pseudo–Riemannian spacetime if LaTeXMLMath The linear d–connection is parametrized by irreducible h–v–components , LaTeXMLMath where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
This defines a canonical linear connection ( distinguished by a N–connection ) which is similar to the metric connection introduced by Christoffel symbols in the case of holonomic bases .
The anholonomic coefficients LaTeXMLMath and N–elongated derivatives give nontrivial coefficients for the torsion tensor , LaTeXMLMath where LaTeXMLEquation and for the curvature tensor , LaTeXMLMath where LaTeXMLEquation .
We emphasize that the torsion tensor on ( pseudo ) Riemannian spacetimes is induced by anholonomic frames , whereas its components vanish with respect to holonomic frames .
All tensors are distinguished ( d ) by the N–connection structure into irreducible h–v–components , and are called d–tensors .
For instance , the torsion , d–tensor has the following irreducible , nonvanishing , h–v–components , LaTeXMLMath where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation ( the d–torsion is computed by substituting the h–v–components of the canonical d–connection ( LaTeXMLRef ) and anholonomic coefficients ( LaTeXMLRef ) into the formula for the torsion coefficients ( LaTeXMLRef ) ) .
The curvature d-tensor has the following irreducible , non-vanishing , h–v–components LaTeXMLMath LaTeXMLMath where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation ( the d–curvature components are computed in a similar fashion by using the formula for curvature coefficients ( LaTeXMLRef ) ) .
In this section we write and analyze the Einstein equations on 5D ( pseudo ) Riemannian spacetimes provided with anholonomic frame structures and associated N–connections .
The Ricci tensor LaTeXMLMath has the d–components LaTeXMLEquation .
LaTeXMLEquation In general , since LaTeXMLMath , the Ricci d-tensor is non-symmetric ( this could be with respect to anholonomic frames of reference ) .
The scalar curvature of the metric d–connection , LaTeXMLMath is computed as LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath By substituting ( LaTeXMLRef ) and ( LaTeXMLRef ) into the 5D Einstein equations LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are respectively the coupling constant and the energy–momentum tensor .
The definition of matter sources with respect to anholonomic frames is considered in Refs .
LaTeXMLCite .
In this paper we deal only with vacuum 5D , locally , anisotropic gravitational equations which in invariant h– v–components are written LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The main ‘ trick ’ of the anholonomic frames method for integrating the Einstein equations in general relativity and various ( super ) string and higher / lower dimension gravitational theories is to find the coefficients LaTeXMLMath such that the block matrices LaTeXMLMath and LaTeXMLMath are diagonalized LaTeXMLCite .
This greatly simplifies computations .
With respect to such anholonomic frames the partial derivatives are N–elongated ( locally anisotropic ) .
The metric ( LaTeXMLRef ) with coefficients ( LaTeXMLRef ) ( equivalently , the d–metric ( LaTeXMLRef ) with coefficients ( LaTeXMLRef ) ) is assumed to solve the 5D Einstein vacuum equations LaTeXMLMath which are distinguished in h– and v–components as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation for further applications we gave the formulas with respect to LaTeXMLMath coefficients of metric as well with respect to LaTeXMLMath , see ( LaTeXMLRef ) , and , for simplicity , the partial derivatives are denoted LaTeXMLMath and LaTeXMLMath It was possible to construct very general classes of solutions for such equations LaTeXMLCite describing locally anisotropic soliton , black hole , black tori and wormhole objects .
In the vacuum case the equations ( LaTeXMLRef ) , ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) form a very simplified system of equations with separations of variables which can be solved consequently for every couples of d–metric coefficients , LaTeXMLMath and N–connection coefficients LaTeXMLMath and LaTeXMLMath ( see Refs LaTeXMLCite on the main teorems and methods of constructing exact solutions ) : The equation ( LaTeXMLRef ) relates two functions LaTeXMLMath and LaTeXMLMath and their partial derivatives on ’ isotropic ’ coordinates LaTeXMLMath and LaTeXMLMath The solution is trivial if we chose LaTeXMLMath and LaTeXMLMath in order to reduce the coefficients from ( LaTeXMLRef ) , respectively , to those from ( LaTeXMLRef ) .
The equation ( LaTeXMLRef ) contains partial derivatives only on anisotropic coordinate LaTeXMLMath and relates two functions LaTeXMLMath and LaTeXMLMath This equation is satisfied by arbitrary two functions LaTeXMLMath and LaTeXMLMath for which LaTeXMLMath If the condition LaTeXMLMath is satisfied , we can write ( LaTeXMLRef ) , in LaTeXMLMath –variables ( see ( LaTeXMLRef ) ) , as LaTeXMLEquation which is solved by arbitrary functions LaTeXMLMath and LaTeXMLEquation where LaTeXMLMath The general solution of ( LaTeXMLRef ) expressing LaTeXMLMath via LaTeXMLMath is LaTeXMLEquation for some functions LaTeXMLMath and LaTeXMLMath stated by boundary conditions and locally anisotropic limits as well from the conditions that the equations ( LaTeXMLRef ) and ( LaTeXMLRef ) are compatible .
Inversely , for a prescribed value of LaTeXMLMath the general solution of ( LaTeXMLRef ) is ( LaTeXMLRef ) which can be rewritten with respect to variables LaTeXMLMath LaTeXMLEquation .
If the functions LaTeXMLMath and LaTeXMLMath were defined , the equations ( LaTeXMLRef ) can be solved as independent linear algebraic equations for LaTeXMLMath LaTeXMLMath For zero matter sources this is a trivial result because in this case the conditions LaTeXMLMath and LaTeXMLMath ( see the formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) ) are automatically fulfilled .
In consequence , the resulting sourceless equations ( LaTeXMLRef ) became some trivial equations admitting arbitrary values of functions LaTeXMLMath such functions can be associated to some coordinate transforms for vanishing anholonomy coefficients LaTeXMLMath see ( LaTeXMLRef ) , or to some anholonomy coefficients , in such cases being not contained in the vacuum Einstein equations , which must be stated by some boundary and symmetry conditions .
The equations ( LaTeXMLRef ) can be solved in general form if the functions LaTeXMLMath and LaTeXMLMath ( and , in consequence , the coefficient LaTeXMLMath from ( LaTeXMLRef ) ) are known , LaTeXMLEquation .
LaTeXMLEquation where the functions LaTeXMLMath and LaTeXMLMath should be defined from some boundary conditions .
The conformally transformed Taub NUT metric ( LaTeXMLRef ) can be considered as a locally isotropic background with trivial vanishing local anisotropies .
By coordinate transforms of the 5th coordinate and a conformal transform on two holonomic coordinates ( here we mention that in two dimensions the coordinate and conformal transforms are equivalent ) the isotropic background metric can be transformed into a form parameterizing the usual Taub NUT solution of the vacuum Einstein equations .
The aim of this section is to construct and analyze three types of anisotropic generalizations of the Taub NUT solution .
The data parameterizing a metric ( LaTeXMLRef ) ( equivalently , a d–metric ( LaTeXMLRef ) , or ( LaTeXMLRef ) ) are LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation which defines a trivial ( locally isotropic ) solution of the vacuum Einstein equations ( LaTeXMLRef ) – ( LaTeXMLRef ) satisfying the conditions LaTeXMLMath The simplest way to obtain anisotropic Taub NUT like solutions , is to follow the approach developed for generation anisotropic black hole LaTeXMLCite and wormhole / flux tube solutions LaTeXMLCite when the constants like mass and charges are considered to be effectively anisotropically polarized by some anholonomic ( anisotropic ) higher dimension interactions ; in our case is to consider that the parameter LaTeXMLMath from ( LaTeXMLRef ) ( see also ( LaTeXMLRef ) ) is not a constant but a renormalized value LaTeXMLMath We generate from the isotropic solution ( LaTeXMLRef ) a new one , anisotropic , in this manner : Let us consider the case when LaTeXMLMath when LaTeXMLMath but LaTeXMLMath Following the solution ( LaTeXMLRef ) we parameterize LaTeXMLEquation and LaTeXMLEquation where LaTeXMLMath is just the function ( LaTeXMLRef ) but defined by a renormalized value LaTeXMLMath The conditions of vanishing of constants ( LaTeXMLRef ) and ( LaTeXMLRef ) are also satisfied if LaTeXMLMath For simplicity , in this subsection , we shall consider solutions with ” pure ” running of the constant LaTeXMLMath on LaTeXMLMath and the function LaTeXMLEquation .
Because ( in this case ) the coefficients LaTeXMLMath and LaTeXMLMath and in consequence .
LaTeXMLMath could be arbitrary functions ( see formulas ( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) and equation ( LaTeXMLRef ) ) ; in the locally isotropic limit , LaTeXMLMath we put that LaTeXMLMath .
The values LaTeXMLMath depends on anisotropic variable LaTeXMLMath as follow from the solution ( LaTeXMLRef ) with LaTeXMLMath We obtain the locally isotropic limit ( LaTeXMLRef ) , for LaTeXMLMath if we fix the boundary conditions with LaTeXMLMath but LaTeXMLMath So , a parametrization of the ansatz ( LaTeXMLRef ) , LaTeXMLEquation defines a locally anisotropic solution of the vacuum Einstein equations ( LaTeXMLRef ) – ( LaTeXMLRef ) generalizing the Taub NUT solution ( LaTeXMLRef ) .
We can treat the solution ( LaTeXMLRef ) as describing an anisotropic Kaluza–Klein monopole with running constant ( on extra dimension coordinate ) obtained by embedding the anisotropic Taub NUT gravitational instanton into 5D theory , adding the coordinate in a way as to be compatible with running of constant of effective magnetic configurations ( in brief , we shall call such solutions as LaTeXMLMath –solutions ) .
We conclude that the solutions describing gravitational monopoles and instantons LaTeXMLCite can be generalized to some anisotropic configurations with running constants .
In a similar fashion we can consider anisotropic ( angular ) dependencies of constants with LaTeXMLMath ( in brief , we call such solutions as LaTeXMLMath –solutions ) .
The simplest way is to take LaTeXMLMath but LaTeXMLMath i.e to define a solution with LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the function ( LaTeXMLRef ) but defined by the renormalized parameter LaTeXMLMath LaTeXMLEquation we can take arbitrary values of LaTeXMLMath because LaTeXMLMath and LaTeXMLMath from the equations ( LaTeXMLRef ) vanish .
For small , constant , polarizations we can approximate LaTeXMLEquation and consider only anisotropic angular variations of the constant , LaTeXMLMath The d–metric LaTeXMLEquation .
LaTeXMLEquation models a locally anisotropic generalization of the solution ( LaTeXMLRef ) for anisotropic dependencies of the constant LaTeXMLMath on angle LaTeXMLMath which describe a 5D Kaluza Klein monopole with angular anisotropic constant obtained by embedding the anisotropic Taub NUT gravitational instanton into 5D theory .
So , in this subsection , we constructed two classes of generalized , anisotropic , Taub NUT solutions of the 5D vacuum Einstein equations : the first class is for LaTeXMLMath ( i.e .
anisotropic polarizations ) and the second is for LaTeXMLMath ( i.e .
with dependence of the constant LaTeXMLMath on the fifth coordinate ) .
The metric ( LaTeXMLRef ) , describing these two classes of solutions , can be written with respect to a coordinate frame ( LaTeXMLRef ) where the existence of non–diagonal terms is emphasized .
This subsection is devoted to such generalizations of the Taub NUT metrics when the anisotropies of constant are modeled by elliptic polarizations on the angle LaTeXMLMath i. e. the constant is renormalized as LaTeXMLMath LaTeXMLMath or LaTeXMLMath where LaTeXMLMath is the eccentricity of an ellipse and LaTeXMLMath ( or LaTeXMLMath is defined as for the solution ( LaTeXMLRef ) , in brief , for LaTeXMLMath –solutions ( or as for the solution ( LaTeXMLRef ) , in brief , for LaTeXMLMath –solution ) .
Because the method of construction of such solutions is very similar to that considered in the previous subsection , we shall omit the details on integration of equations and present only the final values of the d–metric and N–connection coefficients .
The results can be verified by straightforward calculations .
The simplest way of definition of such polarizations for LaTeXMLMath –solutions ( LaTeXMLRef ) and ( LaTeXMLRef ) is to consider the off–diagonal metric LaTeXMLEquation the function LaTeXMLMath is the function ( LaTeXMLRef ) redefined by an ellipsoidally renormalized coefficient LaTeXMLMath LaTeXMLEquation where there is also a linear dependence on extradimension coordinate LaTeXMLMath for simplicity , we can choose LaTeXMLMath Such LaTeXMLMath –solutions with elliptic variations of the constant LaTeXMLMath on angle LaTeXMLMath are distinguished by the metric LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath generalize the functions ( LaTeXMLRef ) , ( LaTeXMLRef ) for anisotropic dependencies like LaTeXMLEquation .
LaTeXMLEquation The constructed solutions of 5D vacuum Einstein equations , ( LaTeXMLRef ) and ( LaTeXMLRef ) contain some additional elliptic polarizations comparing with respective solutions ( LaTeXMLRef ) and ( LaTeXMLRef ) .
Such 5D Kaluza–Klein monopoles induced from 4D Taub NUT instantons behave theirselves as some objects with running on LaTeXMLMath or anisotropic on LaTeXMLMath constant LaTeXMLMath which is also elliptically polarized on the angle LaTeXMLMath By applying the method of anholonomic frames we can construct nonlinear anisotropic superpositions of the Taub NUT metric with some metrics defining wormhole / flux tube configurations : this way one defines anisotropic generalizations of the metric ( LaTeXMLRef ) which are called anisotropic Taub NUT wormholes .
For simplicity , in this paper , we consider only a LaTeXMLMath –solution for the Taub NUT configurations with the wormhole background chosen as to be locally isotropic ( see Refs LaTeXMLCite , on isotropic wormholes and flux tubes , and LaTeXMLCite , on locally anisotropic wormhole solutions ) .
For our purpose the ansatz ( LaTeXMLRef ) is generalized by introducing the coefficient LaTeXMLMath of the nonlinear connection and , for simplicity , we consider LaTeXMLMath and LaTeXMLMath LaTeXMLEquation which results in a diagonal d–metric LaTeXMLEquation .
LaTeXMLEquation where the coefficients LaTeXMLMath and LaTeXMLMath and the constant LaTeXMLMath are introduced for the LaTeXMLMath and LaTeXMLMath components of the wormhole electromagnetic potential like in ( LaTeXMLRef ) .
The data defining a vacuum LaTeXMLMath –solution for the ( LaTeXMLRef ) , including the solution ( LaTeXMLRef ) into a wormhole background are given by the coefficients of the d–metric LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are respectively those from ( LaTeXMLRef ) and ( LaTeXMLRef ) and the constant LaTeXMLMath and term LaTeXMLMath were introduced as some multiples defining the wormhole / flux tube configuration .
The proprieties of the Taub NUT monopole with running constant LaTeXMLMath are the same as those stated for the solution ( LaTeXMLRef ) , resulting in a similar to ( LaTeXMLRef ) magnetic field components LaTeXMLEquation .
LaTeXMLEquation embedded into a static background of wormhole / flux tube configurations defined by additional components of electric , LaTeXMLMath and magnetic , LaTeXMLMath fields as it is defined respectively by the formulas ( LaTeXMLRef ) and ( LaTeXMLRef ) .
As the free parameters of the wormhole background ( see Ref LaTeXMLCite ) are varied there are five classes of solutions with the properties : LaTeXMLMath or LaTeXMLMath , a wormhole–like ‘ electric ’ object ; LaTeXMLMath or LaTeXMLMath , a finite ‘ magnetic ’ flux tube ; LaTeXMLMath , an infinite ‘ electromagnetic ’ flux tube ; LaTeXMLMath , a wormhole–like ‘ electromagnetic ’ object ; LaTeXMLMath , a finite , ‘ magnetic–electric ’ flux tube .
The number wormhole / flux tube classes must be extended to new configurations which arise in the presence of the magnetic field of the Taub NUT monopole , for instance , by considering the 6th class for the pure monopole configuration ; we must take into account possible contribution of the monopole magnetic field to the structure of magnetic flux tubes , wormhole–like electromagnetic object and/or magnetic–electric flux tubes by analyzing the total magnetic field LaTeXMLMath with possible ( elliptic and another type ) vacuum gravitational polarizations .
Finally , in this section , we remark that in a similar fashion we can construct LaTeXMLMath –solutions with an anisotropic parameter LaTeXMLMath describing an anisotropic Taub NUT monopole embedded into a wormhole / flux tube background and to generalized both LaTeXMLMath – and LaTeXMLMath –solutions for various configurations with elliptic polarizations and rotation hypersurface symmetries ( like rotation ellipsoids , elliptic cylinders , bipolars and tori ) as we constructed exact solutions of 3D-5D Einstein equations describing anisotropic black holes/tori and anisotropic wormholes LaTeXMLCite .
In the previous two sections we proved that the Taub NUT and wormhole metrics admit various type of anisotropic generalizations modeled by anholonomic frames with associated N–connection structure .
It is of interest the investigation of symmetries of such anisotropic spaces and definition of corresponding invariants of motion of spinning particles .
The general rules for developing of corresponding geodesic calculus and definition of generalized anisotropic Killing vectors are : We have to change the partial derivatives and differentials into N–elongated ones ( LaTeXMLRef ) and ( LaTeXMLRef ) by redefinition of usual formulas for developing a formalism of differential , integral and variational calculus on anisotropic spaces .
The metric , linear connection , curvature and Ricci tensors have to be changed into respective d–objects ( d–metric ( LaTeXMLRef ) , d–connection ( LaTeXMLRef ) and corresponding curvature and Ricci d–tensors ) ; the d–torsion on ( pseudo ) Riemannian spaces should be treated as an anholonomic frame effect which vanishes with respect to coordinate bases .
The Greek indices LaTeXMLMath should be split into horizontal LaTeXMLMath LaTeXMLMath LaTeXMLMath and vertical LaTeXMLMath ones which on necessity will point to some holonomic–anholonomic character of variables .
By using d–metrics and d–connections the differential , integral and variational calculus on Riemannian manifolds is adapted to the anholonomic frame structure with associated N–connection .
With respect to N–adapted frames ( LaTeXMLRef ) and ( LaTeXMLRef ) the geometry is similar to the usual Riemannian one when the anholonomic ( constrained dynamics ) is coded into the coefficients of N–connection modeling a local anisotropy .
As a matter of principle , all constructions defined with respect to anholonomic bases can be removed with respect to usual coordinate frames , but in this case the metrics became generically off–diagonal and a number of symmetries ( and their constraints ) of manifolds are hidden in rather sophisticate structures and relations for redefined holonomic objects .
The geodesic motion of a spinless particle of unit mass moving into a background stated by a d–metric LaTeXMLMath see ( LaTeXMLRef ) ) can be derived from the action : LaTeXMLEquation where LaTeXMLMath is a parameter .
The invariance of d–metrics defining anisotropic generalizations of Taub NUT metrics under spatial rotations and LaTeXMLMath translations is generated by four Killing d–vectors which are obtained by anholonomic transforms of the usual Killing vectors into corresponding one with elongated partial derivatives LaTeXMLMath where the partial derivative on the new 5th coordinate , are LaTeXMLMath see ( LaTeXMLRef ) .
We write the Killing d–vectors as LaTeXMLEquation or , in details , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation following general considerations we give the formulas for arbitrary transforms of the 5th coordinate , LaTeXMLMath and elongations of derivatives of type LaTeXMLMath We know that in the purely locally isotropic bosonic case such invariances correspond to conservation of angular momentum and ” relative electric charge ” LaTeXMLCite .
For anisotropic Taub NUTS we can define similar objects by using anholonomic transforms of the values given with respect to coordinate bases into the corresponding values with anisotropic coefficients and variables given with respect to anholonomic frames : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the ” mechanical momentum ” which is only part of the momentum canonically conjugate to LaTeXMLMath ( in anisotropization of the momentum formula we do not write the multiple LaTeXMLMath because the d–metric used for anisotropic constructions has its locally isotropic limit being multiplied on the LaTeXMLMath as a conformal factor , see ( LaTeXMLRef ) ) .
The energy is defined by LaTeXMLEquation is also conserved , LaTeXMLMath being the covariant momentum d–vector ; this value variates with respect to coordinate frames but behaves itselve as a usual energy with respect to N–adapted frames .
There is a conserved vector analogous to the Runge- Lenz vector of the Coulomb problem in the locally isotropic case LaTeXMLCite , which with respect to anholonomic frames with anisotropic variables in spite of the fact of complexity of anholonomic motion in the anisotropic Taub- NUT spaces defined in previous Sections , LaTeXMLEquation which implies that the trajectories are anisotropic deformations of conic sections .
The pseudo-classical limit of the Dirac theory of a spin 1/2 fermion in curved spacetime is described by the supersymmetric extension of the usual relativistic point-particle LaTeXMLCite ( the theory of spinors on spaces with generic local anisotropy was developed in Refs .
LaTeXMLCite , see also some models of locally anisotropic supergravity and superstring theories in Refs .
LaTeXMLCite ) .
In this work , the configuration space of spinning particles in anisotropic space ( anisotropic spinning space ) is an extension of an ordinary Riemannian manifold provided with an anholonomic frame and associated N–connection structure , parametrized by local coordinates LaTeXMLMath , to a graded manifold parametrized by local coordinates LaTeXMLMath , with the first set of variables being Grassmann-even ( commuting ) and the second set Grassmann-odd ( anti-commuting ) .
We emphasize that in anholonomic spaces distinguished by a N–connection structure we must define spinor and Grassmann variables separately on h–subspace ( with holonomic variables ) and on v–subspace ( with anholonomic variables ) .
The equation of motion of an anisotropically spinning particle on a autoparallel ( geodesic ) is derived from the action : LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath The corresponding world-line anholonomic Hamiltonian is given by : LaTeXMLEquation where LaTeXMLMath is the covariant momentum d–vector .
For any integral ( we use the term integral instead of the usual one , constant , because on anisotropic spaces we can define any conservations lows with respect to N–adapted anholonomic frames ; the invariants of such conservations laws are not constant with respect to coordinate frames ) of anholonomic motion LaTeXMLMath , the bracket with LaTeXMLMath vanishes , LaTeXMLMath where the Poisson-Dirac brackets for functions of the covariant phase-space variables LaTeXMLMath is defined LaTeXMLEquation .
In definition of LaTeXMLMath there are used the operators LaTeXMLEquation .
LaTeXMLEquation where on anisotropic spaces LaTeXMLMath is the canonical d–connection ( LaTeXMLRef ) , LaTeXMLMath is the curvature d–tensor ( LaTeXMLRef ) with components ( LaTeXMLRef ) and LaTeXMLMath is the Grassmann parity of LaTeXMLMath : LaTeXMLMath for LaTeXMLMath = ( even , odd ) .
Expanding LaTeXMLMath in a power series on the canonical momentum , LaTeXMLMath we conclude that the bracket LaTeXMLMath vanishes for arbitrary LaTeXMLMath if and only if the components of LaTeXMLMath satisfy the generalized anisotropic Killing equations LaTeXMLCite : LaTeXMLEquation where the round brackets LaTeXMLMath denote full symmetrization over the indices enclosed and the covariant derivation ” ; ” is defined by the canonical d–connection ( LaTeXMLRef ) and one should be emphasized that every Greek index split into horizontal and vertical groups like LaTeXMLMath which results that this equation will contain both ” pure ” horizontal or vertical terms as well terms of ” mixed ” character , like LaTeXMLMath The type of solutions of the generalized anisotropic Killing equations ( LaTeXMLRef ) is defined by two classes from the locally isotropic limit LaTeXMLCite : the first class of solutions are generic ones , which exists for any spinning particle model and the second class of solutions are non-generic ones , which depend on the specific background space and anisotropy considered .
The proper-time translations and supersymmetry are generated by the Hamiltonian and supercharge LaTeXMLEquation and belong to the first class .
There is also an additional ” chiral ” symmetry generated by the chiral charge LaTeXMLEquation and a dual supersymmetry with generator LaTeXMLEquation where LaTeXMLMath is the dimension of spacetime .
The non-generic conserved quantities depend on the explicit form of the metric LaTeXMLMath and , in our case , on N–connection structure .
Following Ref .
LaTeXMLCite , generalizing the constructions for anisotropic spaces , we introduce the Killing-Yano d–tensors as objects generating non-generic N–distinguished supersymmetries .
A d–tensor LaTeXMLMath is called Killing-Yano of valence LaTeXMLMath if it is totally antisymmetric and satisfies the equation LaTeXMLMath The method of solution of the system of coupled differential equations ( LaTeXMLRef ) is similar to the method developed for locally isotropic spaces LaTeXMLCite , that why here we present only the key results which have to be split on h– and v– indices if we need explicit formulas for holonomic–anholonomic components .
We start with a LaTeXMLMath solution of the homogeneous equation : LaTeXMLEquation .
This solution is introduced in the r.h.s .
of ( LaTeXMLRef ) for LaTeXMLMath and the iteration is carried on to LaTeXMLMath .
For the bosonic case the first equation shows that LaTeXMLMath is a trivial constant , the next one is the equation for the Killing d–vectors and so on .
In general , the homogeneous equation for a given LaTeXMLMath defines a Killing d–tensor LaTeXMLMath for which LaTeXMLMath is a first integral of the geodesic equation LaTeXMLCite .
This does not hold for the spinning particles .
Let us consider the case LaTeXMLMath , when LaTeXMLMath is a solution if LaTeXMLMath is a Killing-Yano d–tensor –covariantly constant , i. e. LaTeXMLMath is a separately conserved quantity .
For LaTeXMLMath , the natural solution is : LaTeXMLMath where LaTeXMLMath is a Killing d–vector ( LaTeXMLMath ) and LaTeXMLMath is a Killing-Yano d–tensor d–covariantly constant .
Introducing this solution in the r.h.s .
of the equation ( LaTeXMLRef ) with LaTeXMLMath , we get LaTeXMLMath where the square bracket denotes the antisymmetrization with norm one .
We define a new integral of anholonomic motion which is peculiar to the spinning case and has its analogous in the locally isotropic limit : LaTeXMLEquation .
We can generate another LaTeXMLMath -dependent solution of the LaTeXMLMath by starting from a Killing-Yano d–tensor with LaTeXMLMath indices , LaTeXMLMath or , following the above prescription , we express LaTeXMLEquation stating that the integral of motion corresponding to these solutions of the Killing equations is : LaTeXMLEquation .
We conclude that the existence of a Killing-Yano d–tensor with LaTeXMLMath indices is equivalent to the existence of a supersymmetry for the spinning space with supercharge LaTeXMLMath which anticommutes with LaTeXMLMath , such constructions are anholonomic and distinguished by the N–connection structure .
There are four Killing-Yano tensors in the usual , locally isotropic , Taub-NUT geometry LaTeXMLCite which for anisotropic spaces are transformed into corresponding d–tensors for anisotropic Taub NUT spaces , LaTeXMLEquation which are d–covariantly constant and the fourth Killing-Yano d–tensor is LaTeXMLEquation having only one non-vanishing component of the field strength LaTeXMLMath where LaTeXMLMath and the values LaTeXMLMath are N–elongated for the v–components .
The corresponding supercharges constructed from the Killing-Yano d-tensors are LaTeXMLMath and LaTeXMLMath .
The supercharges LaTeXMLMath together with LaTeXMLMath from ( LaTeXMLRef ) realize the N=4 supersymmetry algebra LaTeXMLCite , in our case distinguished by the N–connection structure LaTeXMLCite : LaTeXMLEquation making manifest the link between the existence of the Killing-Yano d–tensors and the hyper-Kähler d–geometry of the anisotropic Taub-NUT manifold .
Such distinguished manifolds and geometries are constructed as the usual ones but with respect to N–connection decompositions on holonomic–anholonomic variables .
Starting with these results from the bosonic sector of the Taub-NUT space one can proceed with the spin contributions .
The first generalized Killing equation ( LaTeXMLRef ) shows that with each Killing vector LaTeXMLMath there is an associated Killing scalar LaTeXMLMath The expression for the Killing scalar is taken as in Ref .
LaTeXMLCite : LaTeXMLEquation with that modification that we use a d–covariant derivation which gives that the total angular momentum and ” relative electric charge ” become in the anisotropic spinning case LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are given with respect to anholonomic bases .
These integrals of motion are superinvariant : LaTeXMLMath We can introduce a Lie algebra with anholonomic constraints defined by the Killing d–vectors and realized by the integral of motion through the Poisson-Dirac brackets .
Now , introducing the Killing d–tensors LaTeXMLMath into the generalized anisotropic Killing equation ( LaTeXMLRef ) we obtain that the corresponding Killing d–vectors LaTeXMLMath have a spin dependent part LaTeXMLMath as in the locally isotropic case LaTeXMLCite , LaTeXMLMath where LaTeXMLMath are the standard Killing d–vectors .
The LaTeXMLMath -dependent parts of the Killing d–vectors LaTeXMLMath contributes to the Runge-Lenz d–vector for the anisotropic spinning space LaTeXMLEquation or in terms of the supercharges LaTeXMLMath and LaTeXMLMath , LaTeXMLEquation we are using underlined Latin indices like LaTeXMLMath enumerating the number of supercharges , in order to distinguish them from the h–indices LaTeXMLMath used for holonomic variables on anisotropic spacetime .
The non-vanishing Poisson brackets are ( after some algebra ) : LaTeXMLEquation .
LaTeXMLEquation where the values have anisotropic dependencies .
Following the fact of existence of the Killing-Yano covariantly constants d–tensors LaTeXMLMath , we can define three integrals of motion : LaTeXMLMath which realize an LaTeXMLMath Lie–algebra similar to that of the angular momentum : LaTeXMLMath These components of the spin are separately conserved , do not depend on the frame of reference , holonomic or anholonomic , and can be combined with the angular momentum LaTeXMLMath in order to get a new improved form of the angular momentum LaTeXMLMath with the property that one preserves the algebra LaTeXMLMath and that it commutes with the LaTeXMLMath algebra generated by the spin LaTeXMLMath LaTeXMLMath Let us consider the Dirac brackets of LaTeXMLMath with supercharges LaTeXMLMath We can combine the above presented two LaTeXMLMath algebras to obtain the generators of a conserved LaTeXMLMath symmetry among the constants of motion , a standard basis for which is spanned by LaTeXMLMath There are also possible the combinations LaTeXMLCite : LaTeXMLEquation .
LaTeXMLEquation So , there are a sort of Runge-Lenz d–vectors involving only Grassmann components : LaTeXMLEquation with the commutation relations : LaTeXMLEquation .
Finally , we note the following Dirac brackets of LaTeXMLMath with supercharges : LaTeXMLEquation .
LaTeXMLEquation and emphasize that the presented algebraic relations hold true for anisotropic ( as on some parameters ) dependencies of the constant LaTeXMLMath because we are working with respect to anholonomic frames locally adapted to the N–connection structure .
In this paper , we have extended the method of construction of new exact solutions , with generic local anisotropy , of the Einstein equations by using anholonomic frames with associated nonlinear connection structure ( the method was proposed and developed in Refs .
LaTeXMLCite ) in order to generate vacuum metrics defining locally anisotropic Taub NUT instanton and Kaluza Klein monopoles .
Such metrics are off–diagonal with respect to usual coordinate bases and reflect possible interactions of gravitational fields with gauge fields , induced from higher dimension gravity , in a new fashion when the constants interactions could run on the 5th dimension and/or polarized to some anisotropic configurations on angular coordinates .
The first key result of this paper is the proof that the introduced ansatz for the metric and anholonomy coefficients substantially simplifies the structure of resulting vacuum gravitational field equations , the variables being separated and the system of nonlinear partial equations admitting exact solutions .
In consequence , a straightforward definition of new classes of anisotropic Taub NUT metrics with the effective constant LaTeXMLMath varying on the 5th coordinate , as well with anisotropies and elliptic polarizations on angular coordinates , was possible .
There were emphasized classes of anisotropic Taub NUT wormhole solutions which can be generalized to various type of rotation hypersurface backgrounds and deformations .
The second key result is connected with the definition of integrals of motion of scalar and spinning particles in curved spacetimes provided with anholonomic frame structure .
We proved that the symmetries of such generalized anisotropic Taub NUT spaces are connected with anholonomic Killing vectors and tensors which are subjected to some anholonomic conservation laws .
The problem of generation of non–generic anisotropic supersymmetries was solved by introducing Killing–Yano tensors adapted to the anholonomic spacetime structure .
Finally , we note that the results of this paper were extended for 3D solitonic configurations LaTeXMLCite .
The authors thank M. Visinescu for hospitality , substantial support and discussion of results .
S. V. is grateful to D. Singleton , E. Gaburov and D. Gontsa for collaboration and discussing of results .
He also thanks V. Manu for support and help .
The work is supported both by ” The 2000–2001 California State University Legislative Award ” and a NATO/Portugal fellowship for the Instituto Superior Tecnico , Lisabon .
Traces on Noncommutative Homogeneous Spaces Magnus B. Landstad The noncommutative Heisenberg manifolds constructed by Rieffel in [ R2 ] have turned out to provide interesting examples of LaTeXMLMath -algebras which are similar , but not isomorphic to irrational rotation algebras as shown by Abadie in [ A2 ] .
It was shown in [ LR2 ] that these algebras are special cases of a more general construction giving deformations of LaTeXMLMath for a compact homogeneous space LaTeXMLMath .
The LaTeXMLMath -algebras obtained were denoted LaTeXMLMath and their ideal structure was determined in [ LR2 ] , in this follow-up we shall describe the algebras more closely : ( 1 ) The center LaTeXMLMath is isomorphic to LaTeXMLMath for a certain subgroup LaTeXMLMath of LaTeXMLMath , and ( 2 ) there is a conditional expectation LaTeXMLMath and therefore a 1-1 correspondence between normalised traces on LaTeXMLMath and probability measures on LaTeXMLMath .
This is used to show that LaTeXMLMath also can be represented over LaTeXMLMath just as in the nondeformed case .
The results here generalise some of those in [ A1-3 ] and should provide useful tools for extending other results such as the determination of the ( ordered ) K-theory and the noncommutative metrics of these algebras .
1 .
Preliminaries .
The deformations of LaTeXMLMath constructed in [ LR2 ] are based on the following standard assumptions : ( S1 ) There is a compact abelian subgroup LaTeXMLMath of LaTeXMLMath and a homomorphism LaTeXMLMath such that each LaTeXMLMath commutes with LaTeXMLMath , ( S2 ) LaTeXMLMath is a closed subgroup of LaTeXMLMath , each LaTeXMLMath commutes with LaTeXMLMath and satisfies LaTeXMLEquation ( S3 ) LaTeXMLMath is compact and LaTeXMLMath .
In [ LR2 , 4.8-11 ] it is then explained how one obtains a closed subgroup LaTeXMLMath of LaTeXMLMath by LaTeXMLMath and a homomorphism LaTeXMLMath .
The subgroup LaTeXMLMath of LaTeXMLMath is then defined as the closure of LaTeXMLEquation and it is shown that LaTeXMLMath is a closed normal subgroup of LaTeXMLMath with LaTeXMLMath a compact abelian group .
We define the partial Fourier transform by LaTeXMLMath for LaTeXMLMath as in [ LR2 ] and LaTeXMLEquation .
The space of functions we shall work with is LaTeXMLMath .
With the operations given by LaTeXMLMath and LaTeXMLEquation we have a Banach LaTeXMLMath -algebra denoted LaTeXMLMath .
Its regular representation LaTeXMLMath over LaTeXMLMath is described in [ LR2 , Section 1 ] , and LaTeXMLMath is the LaTeXMLMath -closure of LaTeXMLMath .
Note that this definition is closely related to the Fell bundle approach in [ AE1 ] .
We refer to [ LR2 ] for more details and other concepts not explained here , in fact this article is unreadable without [ LR1-2 ] .
2 .
The center of LaTeXMLMath .
It was shown in [ LR2 , Theorem 4.15 ] that LaTeXMLMath with the product ( 1.1 ) is a dense subalgebra of the center LaTeXMLMath .
However , this product will not be the pointwise product for functions in LaTeXMLMath .
We shall see that the center actually is isomorphic to LaTeXMLMath where LaTeXMLMath is another subgroup of LaTeXMLMath .
We thank the referee for discovering an error in our description , this means that Remark 4.16 in [ LR2 ] is incorrect .
We shall also need some other subgroups of LaTeXMLMath , and a guiding example in this section is to take the Heisenberg manifolds as described in [ LR2 , Section 3 ] with LaTeXMLMath and LaTeXMLMath rational .
All concepts in [ LR2 , Section 4 ] are used without further explanation .
For the first new subgroup note that LaTeXMLEquation .
The following should then be obvious : Take LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLEquation .
Furthermore , all LaTeXMLMath satisfies LaTeXMLMath for LaTeXMLMath .
In particular this means that we have LaTeXMLMath for all LaTeXMLMath , so there is a function LaTeXMLMath such that LaTeXMLEquation .
There is a function LaTeXMLMath such that LaTeXMLEquation .
Proof .
Pick one LaTeXMLMath from each equivalence class in LaTeXMLMath and define LaTeXMLMath by LaTeXMLEquation .
It is then straightforward to check that ( 2.2 ) holds .
For LaTeXMLMath define LaTeXMLEquation .
Then LaTeXMLMath is a 1-1 LaTeXMLMath -homomorphism from LaTeXMLMath with the product ( 1.1 ) into LaTeXMLMath equipped with the usual pointwise operations .
Proof .
For LaTeXMLMath we have by ( 1.1 ) and the definition of LaTeXMLMath that LaTeXMLEquation .
The LaTeXMLMath -operation is complex conjugation in both algebras , so LaTeXMLEquation .
In order to define the subgroup LaTeXMLMath we need the following construction : There is a continuous homomorphism LaTeXMLMath satisfying LaTeXMLMath and LaTeXMLEquation .
Proof .
If LaTeXMLMath or LaTeXMLMath then LaTeXMLMath , so we have LaTeXMLMath .
It follows from ( S1-S3 ) that for LaTeXMLMath and LaTeXMLMath that LaTeXMLEquation .
For LaTeXMLMath and LaTeXMLMath we proved in [ LR2 , Lemma 4.11 ] that LaTeXMLMath .
So if we look at the function LaTeXMLMath we have for LaTeXMLMath that LaTeXMLEquation .
So there is an element LaTeXMLMath such that LaTeXMLEquation .
This holds for all LaTeXMLMath , and for a fixed LaTeXMLMath this expression is continuous in LaTeXMLMath .
So ( 2.3-5 ) together with LaTeXMLMath implies that LaTeXMLMath is a continuous homomorphism with LaTeXMLMath and therefore LaTeXMLMath in LaTeXMLMath .
LaTeXMLMath Note the map LaTeXMLMath can be defined the same way on the group LaTeXMLMath defined in [ LR2 , Lemma 4.7 ] , but this is not needed here .
Define LaTeXMLEquation .
Then the homomorphism LaTeXMLMath in Lemma 2.3 has dense image in LaTeXMLMath .
Proof .
If LaTeXMLMath , then LaTeXMLMath for LaTeXMLMath .
So from the definition of LaTeXMLMath we have LaTeXMLEquation .
Thus LaTeXMLMath .
LaTeXMLMath is a bijection between LaTeXMLMath and LaTeXMLMath , so the image is dense .
LaTeXMLMath We want to show that LaTeXMLMath extends to an isomorphism between LaTeXMLMath and LaTeXMLMath , and since LaTeXMLMath is defined by using the regular representation LaTeXMLMath this will follow from : For LaTeXMLMath the unitary operator LaTeXMLMath satisfies LaTeXMLEquation .
Proof .
From Propositions 1.3 and ( 1.11 ) in [ LR2 ] and Lemma 2.2 above we have LaTeXMLEquation .
Note that every LaTeXMLMath satisfies ( S2 ) , so by [ LR2 , Theorem 4.3 ] LaTeXMLMath defined by LaTeXMLMath extends to a LaTeXMLMath -automorphism of LaTeXMLMath .
If we also look at part ( 1 ) and ( 2 ) of the proof of [ LR2 , Theorem 4.15 ] , we see that it can be rephrased as LaTeXMLEquation .
The map LaTeXMLMath extends to a LaTeXMLMath -isomorphism between LaTeXMLMath and LaTeXMLMath with the pointwise product .
Proof .
This now follows , just note that in part ( 2 ) of the proof of [ LR2 , Theorem 4.15 ] it was shown that LaTeXMLMath is dense in the center of LaTeXMLMath .
Remarks 2.8 .
The map LaTeXMLMath is an isomorphism between the groups LaTeXMLMath and LaTeXMLMath .
However , this will not imply that the groups LaTeXMLMath and LaTeXMLMath themselves are isomorphic or that LaTeXMLMath is homeomorphic to LaTeXMLMath .
Also note that LaTeXMLMath which again by [ LR2 , Theorem 4.15 ] is equivalent to LaTeXMLMath being simple .
We shall see in Section 5 that LaTeXMLMath can be nontrivial and LaTeXMLMath .
3 .
The conditional expectation onto the center and traces on LaTeXMLMath .
We have now proved that LaTeXMLMath is generated by LaTeXMLMath and is via the map LaTeXMLMath isomorphic to LaTeXMLMath .
From ( 2.6 ) we get the natural conditional expectation LaTeXMLMath of LaTeXMLMath onto its center by LaTeXMLEquation .
Note that LaTeXMLMath is not necessarily a group , but as noted in the preliminaries LaTeXMLMath is a closed normal subgroup of LaTeXMLMath with LaTeXMLMath a compact abelian group .
( The same is true if we replace LaTeXMLMath with LaTeXMLMath . )
This means that the map LaTeXMLMath is given by LaTeXMLEquation .
LaTeXMLMath defines a conditional expectation from LaTeXMLMath onto LaTeXMLMath .
For LaTeXMLMath we have LaTeXMLEquation .
Proof .
With LaTeXMLMath we have LaTeXMLMath , so LaTeXMLEquation .
Since LaTeXMLMath we have LaTeXMLEquation .
From Lemma 2.4 we have LaTeXMLMath , so LaTeXMLEquation .
For LaTeXMLMath we have LaTeXMLMath .
Proof .
There are no problems in interchanging integrals and sums here , so for LaTeXMLMath LaTeXMLEquation .
Here we used the substitution LaTeXMLMath , we continue with the substitution LaTeXMLMath to get LaTeXMLEquation .
From ( 4.9 ) in [ LR2 ] we have that LaTeXMLMath for LaTeXMLMath , which together with LaTeXMLMath gives LaTeXMLEquation .
This holds for all LaTeXMLMath , so from ( 3.2 ) we have LaTeXMLMath .
LaTeXMLMath We can now prove the following generalisation of [ A3 , Corollary 3.11 ] : The conditional expectation from LaTeXMLMath onto LaTeXMLMath given by LaTeXMLMath satisfies LaTeXMLMath and LaTeXMLMath .
There is a 1-1 correspondence between normalised traces LaTeXMLMath on LaTeXMLMath and probability measures LaTeXMLMath on LaTeXMLMath given by LaTeXMLMath .
LaTeXMLMath is faithful if and only if LaTeXMLMath is .
Proof .
The LaTeXMLMath -invariance is obvious .
Since LaTeXMLMath is dense in LaTeXMLMath , the first part follows from Lemma 3.2 .
Hence LaTeXMLMath is a normalised trace for all probability measures LaTeXMLMath on LaTeXMLMath .
Since LaTeXMLMath is faithful , it is immediate that LaTeXMLMath is faithful if and only if LaTeXMLMath is .
Conversely , if LaTeXMLMath is a normalised trace on LaTeXMLMath , it follows from [ LR2 , Lemma 4.12 ] that LaTeXMLMath for LaTeXMLMath .
So LaTeXMLMath where LaTeXMLMath .
From [ LR2 , Lemma 4.14 ] it follows that LaTeXMLMath for LaTeXMLMath , so LaTeXMLEquation .
Let LaTeXMLMath be the measure on LaTeXMLMath given by LaTeXMLMath for LaTeXMLMath , then LaTeXMLMath .
LaTeXMLMath From [ LR2 , Theorem 4.15 ] we now have If LaTeXMLMath ( which is equivalent to LaTeXMLMath ) , there is a unique trace on the simple LaTeXMLMath -algebra LaTeXMLMath .
4 .
Quasi-invariant measures on LaTeXMLMath and representations over LaTeXMLMath .
In this section we shall look at traces on LaTeXMLMath obtained from a LaTeXMLMath -quasi-invariant measure LaTeXMLMath on LaTeXMLMath .
We shall see that the corresponding GNS-representation can be realised over LaTeXMLMath .
If LaTeXMLMath is a LaTeXMLMath -quasi-invariant measure on LaTeXMLMath , there is a function LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath , in fact LaTeXMLMath where LaTeXMLMath is a continuous rho-function on LaTeXMLMath corresponding to LaTeXMLMath .
If we take LaTeXMLMath and use Theorem 3.3 on such measures , we get If LaTeXMLMath is a LaTeXMLMath -quasi-invariant probability measure on LaTeXMLMath and LaTeXMLMath , there is a continuous function LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath has a LaTeXMLMath -invariant probability measure and LaTeXMLMath is compact , it follows that LaTeXMLMath has a LaTeXMLMath -invariant probability measure if and only if LaTeXMLMath has one .
And if so , it is unique .
However , note that the compactness of LaTeXMLMath does not imply the existence of a LaTeXMLMath -invariant probability measure .
If LaTeXMLMath has a LaTeXMLMath -invariant probability measure , then there is a unique normalised LaTeXMLMath -invariant trace on LaTeXMLMath .
The regular representation of LaTeXMLMath is over LaTeXMLMath , but it seems natural to also represent it over LaTeXMLMath .
This is obtained by using the GNS-representation obtained from LaTeXMLMath as in Corollary 4.1 .
LaTeXMLMath has a LaTeXMLMath -quasi-invariant probability measure LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath .
Proof .
First we shall need the closed subgroup LaTeXMLMath of LaTeXMLMath .
Exactly as in [ LR2 , Lemma 4.7 ] one proves that LaTeXMLMath is a closed , normal subgroup of LaTeXMLMath , and LaTeXMLMath is a compact abelian group .
If we now take a LaTeXMLMath -quasi-invariant probability measure on LaTeXMLMath , Haar-measures on LaTeXMLMath and LaTeXMLMath , then LaTeXMLEquation defines a LaTeXMLMath -quasi-invariant probability measure on LaTeXMLMath .
So for LaTeXMLMath we have LaTeXMLEquation .
Let LaTeXMLMath be a LaTeXMLMath -quasi-invariant probability measure on LaTeXMLMath .
Then there is a function LaTeXMLMath such that for all LaTeXMLMath LaTeXMLEquation .
Proof .
Let LaTeXMLMath be a continuous rho-function as in [ FD , Section III.13.2 ] or [ LR1 , Section 2 ] , we may assume that LaTeXMLMath for LaTeXMLMath .
So for LaTeXMLMath LaTeXMLEquation .
Since LaTeXMLMath is compact , there is a function LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLMath is compact and commutes with LaTeXMLMath , so we may assume that LaTeXMLMath for LaTeXMLMath .
Now take LaTeXMLMath .
Since LaTeXMLMath , we get from [ LR2 , Proposition 1.3 ] that LaTeXMLEquation .
These two lemmas show that LaTeXMLMath – which was defined over LaTeXMLMath – also can be represented over LaTeXMLMath .
This is done as follows : Define LaTeXMLMath by LaTeXMLMath .
Then LaTeXMLMath , so LaTeXMLMath extends to a partial isometry from LaTeXMLMath into LaTeXMLMath with LaTeXMLMath .
We also have LaTeXMLEquation .
Thus the GNS-representation of LaTeXMLMath corresponding to LaTeXMLMath is really over LaTeXMLMath and LaTeXMLMath sets up an equivalence with a sub-representation of the regular representation LaTeXMLMath .
It is faithful , since LaTeXMLMath and LaTeXMLMath is faithful .
Thus we have shown : If LaTeXMLMath is a LaTeXMLMath -quasi-invariant probability measure on LaTeXMLMath as in Lemma 4.3 , then the GNS-representation of LaTeXMLMath corresponding to LaTeXMLMath is a faithful representation over LaTeXMLMath and equivalent to a sub-representation of the regular representation LaTeXMLMath .
5 .
The Heisenberg manifolds revisited .
Let us go back to the Heisenberg manifolds in [ R2 ] , we shall use our description in [ LR2 , Section 3 ] ; so LaTeXMLMath in the terminology of [ LR2 ] and [ A1-3 ] , we only look at the case with LaTeXMLMath .
We only state the results and leave the computations to the reader .
If LaTeXMLMath one finds that LaTeXMLMath , where LaTeXMLMath is the smallest integer LaTeXMLMath with both LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath for LaTeXMLMath , so LaTeXMLMath .
For any LaTeXMLMath define LaTeXMLMath and LaTeXMLMath satisfies ( 2.3 ) for LaTeXMLMath .
If e.g .
LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath , but LaTeXMLMath is not trivial on LaTeXMLMath , so LaTeXMLMath .
However , since LaTeXMLMath is defined on the whole group LaTeXMLMath , we have LaTeXMLMath and LaTeXMLMath is homeomorphic to LaTeXMLMath .
On the other hand if LaTeXMLMath or LaTeXMLMath is irrational , then LaTeXMLMath , so LaTeXMLMath and is a normal subgroup of LaTeXMLMath with LaTeXMLMath where LaTeXMLMath is the closed subgroup generated by LaTeXMLMath .
So Theorem 3.3 is our version of [ A3 , Corollary 3.11 ] .
Many of the structural results about LaTeXMLMath in [ A1-3 ] can be proved using the present description .
Let us briefly illustrate this by showing the existence of projections á la Rieffel using only functions in LaTeXMLMath .
Let LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are continuous functions satisfying a slightly modified version of [ R1 , Theorem 1.1 ( 1-3 ) ] .
In particular we need LaTeXMLMath in order to make LaTeXMLMath and LaTeXMLMath continuous .
Computations as in [ LR2 , Proposition 3.6 ] show that LaTeXMLMath and LaTeXMLMath are projections in LaTeXMLMath with LaTeXMLMath and LaTeXMLMath for any normalised trace LaTeXMLMath .
This is used in [ A1 , Theorem 1 ] to show that LaTeXMLMath .
To show equality Abadie proves in [ A3 ] that LaTeXMLMath can be embedded in an AF-algebra .
This can also be done using only functions in LaTeXMLMath .
The function LaTeXMLEquation is in LaTeXMLMath , but is not continuous .
Since LaTeXMLMath , the regular representation in [ LR2 , Proposition 1.3 ] can still be used on LaTeXMLMath to get a unitary operator LaTeXMLMath .
The LaTeXMLMath -algebra LaTeXMLMath generated by LaTeXMLMath and LaTeXMLMath is invariant under the automorphisms LaTeXMLMath and we have LaTeXMLMath for all LaTeXMLMath .
It is then standard to show that LaTeXMLMath is in fact the crossed product of a norm-closed subalgebra of LaTeXMLMath with LaTeXMLMath as in [ A3 , Theorem 2.3 ] , and by classical results by Pimsner in [ P ] it follows that LaTeXMLMath can be embedded in an AF-algebra .
Abadie uses this to determine the ordered K-theory of LaTeXMLMath and to describe when two such algebras are isomorphic : In most cases LaTeXMLMath if and only if LaTeXMLMath and LaTeXMLMath belong to the same orbit under the natural action of GL ( 2 , LaTeXMLMath ) on LaTeXMLMath , see [ AE2 , Theorem 2.2 ] and [ A3 , Corollary 3.17 ] .
Note that the functions above ( except LaTeXMLMath ) can be taken to be LaTeXMLMath -functions , so also the cyclic cohomology of LaTeXMLMath can be studied .
It should then be possible to extend the results for the Heisenberg manifolds to the more general algebras LaTeXMLMath described here by finding functions in LaTeXMLMath having the right properties .
It is our belief that the presentation of these algebras given in [ LR1-2 ] and here will be useful for such constructions .
The noncommutative metrics studied by Rieffel and Weaver in [ R3 ] and [ W ] are examples of this .
References .
[ A1 ] B. Abadie , ” Vector bundles ” over quantum Heisenberg manifolds .
Algebraic methods in operator theory , 307–315 , Birkhäuser Boston , Boston , ( 1994 ) .
[ A2 ] B. Abadie , Generalized fixed-point algebras of certain actions on crossed products , Pacific J .
Math .
171 ( 1995 ) , 1–21 .
[ A3 ] B. Abadie , The range of traces on quantum Heisenberg manifolds , Trans .
Amer .
Math .
Soc .
352 ( 2000 ) , 5767–5780 ( electronic ) .
[ AE1 ] B. Abadie and R. Exel , Deformation quantization via Fell Bundles , Math .
Scand .
( to appear ) .
[ AE2 ] B. Abadie and R. Exel , Hilbert LaTeXMLMath -bimodules over commutative LaTeXMLMath -algebras and an isomorphism condition for quantum Heisenberg manifolds .
Rev .
Math .
Phys .
9 ( 1997 ) , 411–423 .
[ FD ] J. M. G. Fell and R. Doran , Representations of LaTeXMLMath -algebras , locally compact groups , and Banach LaTeXMLMath -algebraic bundles , Academic Press , ( 1988 ) .
[ LR1 ] M. B. Landstad and I. Raeburn , Twisted dual-group algebras : Equivariant deformations of LaTeXMLMath , J. Funct .
Anal .
132 ( 1995 ) , 43–85 .
[ LR2 ] M. B. Landstad and I. Raeburn , Equivariant deformations of homogeneous spaces , J. Funct .
Anal .
148 ( 1997 ) , 480–507 .
[ P ] M. V. Pimsner , Embedding some transformation group LaTeXMLMath -algebras into AF-algebras , Ergodic Theory Dynamical Systems 3 ( 1983 ) , 613–626 .
[ R1 ] M. A. Rieffel , LaTeXMLMath -algebras associated with irrational rotations .
Pacific J .
Math .
91 ( 1981 ) , 415–429 .
[ R2 ] M. A. Rieffel , Deformation quantization of Heisenberg manifolds , Comm .
Math .
Phys .
122 ( 1989 ) , 531–562 .
[ R3 ] M. A. Rieffel , Metrics on states from actions of compact groups .
Doc .
Math .
3 ( 1998 ) , 215–229 ( electronic ) .
[ W ] N. Weaver , Sub-Riemannian metrics for quantum Heisenberg manifolds , J .
Operator Theory 43 ( 2000 ) , 223–242 .
Department of Mathematical Sciences The Norwegian University of Science and Technology N-7491 Trondheim Norway
We show that we can skip the skew-symmetry assumption in the definition of Nambu-Poisson brackets .
In other words , a LaTeXMLMath -ary bracket on the algebra of smooth functions which satisfies the Leibniz rule and a LaTeXMLMath -ary version of the Jacobi identity must be skew-symmetric .
A similar result holds for a non-antisymmetric version of Lie algebroids .
Two main directions have been suggested for the generalization of the notion of a Lie algebra .
First , Filippov developed a proposal for brackets with more than two arguments , i.e. , LaTeXMLMath -ary brackets .
In LaTeXMLCite he proposed a definition of such structures ( which we shall call Filippov algebras ) with a version of the Jacobi identity for LaTeXMLMath -arguments ( we shall call it Filippov identity , shortly FI ) : LaTeXMLEquation .
Note that in the binary case ( LaTeXMLMath ) , the Filippov identity coincides with the Jacobi identity .
Independently , Nambu LaTeXMLCite , looking for generalized formulations of Hamiltonian Mechanics , found LaTeXMLMath -ary analogs of Poisson brackets and then Takhtajan LaTeXMLCite rediscovered the Filippov identity ( and called it Fundamental Identity ) for them .
The Filippov brackets are assumed to be LaTeXMLMath -linear and skew-symmetric and Nambu-Poisson brackets , defined on algebras of smooth functions , satisfy additionally the Leibniz rule : LaTeXMLEquation .
On the other hand , Loday ( cf .
LaTeXMLCite ) , while studying relations between Hochschild and cyclic homology in the context of searching for obstructions to the periodicity of algebraic K-theory , discovered that one can skip the skew-symmetry assumption in the definition of a Lie algebra , still having a possibility to define appropriate ( co ) homology ( see LaTeXMLCite and LaTeXMLCite , Chapter 10.6 ) .
His Jacobi identity for such structures was formally the same as the classical Jacobi identity in the form of ( LaTeXMLRef ) for LaTeXMLMath : LaTeXMLEquation .
This time , however , this is no longer equivalent to LaTeXMLEquation since we have no skew-symmetry .
Loday called such structures Leibniz algebras but , since we have already associated the name of Leibniz with the Leibniz identity , we shall call them Loday algebras .
This is in accordance with the terminology of LaTeXMLCite , where analogous structures in the graded case are defined .
Of course , there is no particular reason not to define Loday algebras by means of ( LaTeXMLRef ) instead of ( LaTeXMLRef ) ( and in fact , it was the original Loday definition ) , but both categories are equivalent via transposition of arguments .
Similarly , for associative algebras we can obtain associated algebras by transposing arguments , but in this case we still get associative algebras .
It is interesting that Nambu-Poisson brackets lead to some Loday algebras and hence to the corresponding ( co ) homology ( see LaTeXMLCite ) .
It is now clear that we can combine both generalizations and define Filippov-Loday algebras as those which are equipped with LaTeXMLMath -ary brackets , not skew-symmetric in general , but satisfying the Filippov identity .
We can also define a Loday version of Nambu-Poisson algebras or rings ( we shall call them Nambu-Poisson-Loday , or simply Nambu-Loday , algebras ( or rings ) ) , assuming additionally that a Filippov-Loday structure is defined on a commutative associative algebra ( resp .
ring ) and satisfies the Leibniz rule ( with respect to all arguments separately , since we have no skew-symmetry ) .
In this short note we first deal with the problem of finding examples of new , i.e. , non antisymmetric , Nambu-Poisson-Loday brackets .
The result is , to some extend , unexpected .
We show that for a wide variety of associative commutative algebras , including algebras of smooth functions , we get nothing more than what we already know , since Nambu-Loday algebras have to be skew-symmetric .
In particular , we can skip the skew-symmetry axiom in the standard definition of Poisson bracket .
We obtain a similar negative result for a Loday-type generalization of Lie algebroids : they are locally , in principle , skew-symmetric , or they are bundles of Loday algebras .
Definition .
Let LaTeXMLMath be an associative commutative algebra .
Let LaTeXMLMath be a LaTeXMLMath -ary bracket on LaTeXMLMath , i.e. , an operation with LaTeXMLMath -arguments .
LaTeXMLEquation which is linear with respect to all arguments : LaTeXMLEquation .
We shall call such a bracket a Nambu-Loday bracket , if it satisfies the following two conditions : ( i ) the Leibniz rule ( with respect to each argument ) : LaTeXMLEquation for all LaTeXMLMath , and ( ii ) the Filippov identity : LaTeXMLEquation .
The commutative algebra LaTeXMLMath , equipped with a Nambu-Loday bracket , will be called Nambu-Loday algebra .
For Nambu-Loday algebras we have no direct inductive characterization as that for Nambu-Poisson and Nambu-Jacobi brackets LaTeXMLCite , since the property saying that fixing an argument we get a bracket satisfying FI , but of one argument less , is based on skew-symmetry .
However , we can prove the following .
If LaTeXMLMath is an associative commutative algebra over a field of characteristic 0 and LaTeXMLMath contains no nilpotents , then every Nambu-Loday bracket on LaTeXMLMath is skew-symmetric .
Proof .
Let us assume that we have fixed a Nambu-Loday bracket on LaTeXMLMath .
First , observe that the skew-symmetry property is equivalent to the fact that the bracket vanishes , if only two arguments are the same .
Explicitly , if LaTeXMLEquation then writing LaTeXMLMath and using ( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath , we get LaTeXMLEquation .
LaTeXMLEquation Second , since we can get the skew-symmetry ( LaTeXMLRef ) with respect to the transposition LaTeXMLMath composing transpositions LaTeXMLMath and LaTeXMLMath again , it is sufficient to prove ( LaTeXMLRef ) ( or ( LaTeXMLRef ) ) for LaTeXMLMath .
Fix LaTeXMLMath .
Replacing LaTeXMLMath in ( LaTeXMLRef ) by LaTeXMLMath , we get , due to the Leibniz rule , LaTeXMLEquation .
Subtracting from ( LaTeXMLRef ) the Filippov identity ( LaTeXMLRef ) multiplied by LaTeXMLMath , we get LaTeXMLEquation which holds for all LaTeXMLMath Now , for LaTeXMLMath , we shall show inductively the following : LaTeXMLEquation .
Of course , LaTeXMLMath tells us just that the bracket is skew-symmetric with respect to all transpositions LaTeXMLMath , so it is totally skew-symmetric , according to the previous remarks .
We start with LaTeXMLMath .
Putting in ( LaTeXMLRef ) all LaTeXMLMath ’ s and LaTeXMLMath ’ s equal to LaTeXMLMath , we get LaTeXMLMath , which gives us LaTeXMLMath , since there are no nilpotents in LaTeXMLMath , so the induction starts .
To prove the inductive step , assume ( LaTeXMLMath ) for some LaTeXMLMath .
We shall show ( LaTeXMLMath ) .
Take LaTeXMLMath such that LaTeXMLMath for LaTeXMLMath from a subset LaTeXMLMath of LaTeXMLMath with LaTeXMLMath elements .
Put LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
For a fixed LaTeXMLMath , we have ( i ) if LaTeXMLMath , then LaTeXMLMath by the inductive assumption , and ( ii ) if LaTeXMLMath , then LaTeXMLEquation .
This implies that ( LaTeXMLRef ) reads in this case LaTeXMLEquation which gives LaTeXMLEquation for any LaTeXMLMath such that LaTeXMLMath of them equal LaTeXMLMath .
LaTeXMLMath Remark .
The assumption that there are no nilpotents in LaTeXMLMath is essential .
To see this , consider the commutative associative algebra LaTeXMLMath over LaTeXMLMath freely generated by LaTeXMLMath , with the constraint LaTeXMLMath , i.e. , LaTeXMLMath .
It is easy to see that the bracket LaTeXMLEquation satisfies the Leibniz rule and the Jacobi identity , but it is symmetric .
Every LaTeXMLMath -ary bracket on the algebra LaTeXMLMath of smooth functions on a manifold LaTeXMLMath , which satisfies the Leibniz rule , is associated with a LaTeXMLMath -contravariant tensor LaTeXMLMath according to LaTeXMLEquation .
The vector fields LaTeXMLEquation we can call ( left ) Hamiltonian vector fields of LaTeXMLMath .
It is easy to see that the Filippov identity for the LaTeXMLMath -bracket is in this case equivalent to the fact that the Hamiltonian vector fields preserve the tensor LaTeXMLMath , i.e. , LaTeXMLEquation where LaTeXMLMath stands for the Lie derivative .
Theorem 1 can be formulated in this case as follows .
If a LaTeXMLMath -contravariant tensor field is preserved by its Hamiltonian vector fields , then it is skew-symmetric .
It is well known that with a LaTeXMLMath -ary bracket on a finite-dimensional vector space LaTeXMLMath ( over LaTeXMLMath ) we canonically associate a linear contravariant LaTeXMLMath -tensor LaTeXMLMath on the dual space LaTeXMLMath such that ( LaTeXMLRef ) is satisfied for linear functions on LaTeXMLMath ( thus elements of LaTeXMLMath ) .
Explicitly , if LaTeXMLMath is a basis of LaTeXMLMath ( thus a coordinate system of LaTeXMLMath ) , then LaTeXMLEquation .
Lie algebras correspond in this way to linear Poisson tensors .
This can be generalized to vector bundles as follows .
By linear functions on a vector bundle LaTeXMLMath over a manifold LaTeXMLMath we understand the functions we get from sections of the dual bundle LaTeXMLMath by contraction , i.e. , the linear function LaTeXMLMath associated with a section LaTeXMLMath of LaTeXMLMath is given by LaTeXMLMath , where LaTeXMLMath for LaTeXMLMath .
We say that a LaTeXMLMath -tensor LaTeXMLMath on LaTeXMLMath is linear if linear functions on LaTeXMLMath are closed with respect to the LaTeXMLMath -ary bracket LaTeXMLMath generated by LaTeXMLMath .
Hence , we can define a LaTeXMLMath -ary operation LaTeXMLMath on sections of the bundle LaTeXMLMath by LaTeXMLEquation .
In LaTeXMLCite this idea was used to define general ( binary ) algebroid structures , and in LaTeXMLCite to define LaTeXMLMath -ary Lie algebroids .
Let us concentrate now on the binary case and let us recall from LaTeXMLCite the following definition .
Let LaTeXMLMath be a manifold .
An algebroid on LaTeXMLMath is a vector bundle LaTeXMLMath , together with a bracket LaTeXMLMath on the module LaTeXMLMath of global sections of LaTeXMLMath , and two vector bundle morphism LaTeXMLMath , over the identity on LaTeXMLMath , from LaTeXMLMath to the tangent bundle LaTeXMLMath , called the anchors of the Lie algebroid ( left and right ) , such that LaTeXMLEquation for all LaTeXMLMath and all LaTeXMLMath .
It is clear that any finite-dimensional algebra structure can be viewed as an algebroid structure on a bundle over a single point .
Note that in the case when the algebroid bracket is a Lie bracket , we have LaTeXMLMath and LaTeXMLMath for all LaTeXMLMath .
Such structures are called Lie algebroids .
They were introduced by Pradines LaTeXMLCite as infinitesimal objects for differentiable groupoids , but one can find similar notions proposed by several authors in increasing number of papers ( which proves their importance and naturalness ) .
For basic properties and the literature on the subject we refer to the survey article by Mackenzie LaTeXMLCite .
( LaTeXMLCite ) There is a one-one correspondence between linear 2-contravariant tensors LaTeXMLMath on the dual bundle LaTeXMLMath and algebroid brackets LaTeXMLMath on LaTeXMLMath .
Note that , equivalently , we can think of algebroid structures on the vector bundle LaTeXMLMath as morphisms of double vector bundles LaTeXMLMath ( cf .
LaTeXMLCite ) .
We can speak about Loday algebroids when we impose the Jacobi identity ( LaTeXMLRef ) but we skip the skew-symmetry assumption .
One can think that imposing the Jacobi identity for an algebroid , we get the Jacobi identity for the bracket LaTeXMLMath of functions defined by the corresponding tensor LaTeXMLMath on LaTeXMLMath and , in view of Theorem 1 , that this implies that LaTeXMLMath is a Poisson tensor , so our algebroid is a Lie algebroid .
This reasoning , however , is wrong , since the Jacobi identity on sections of LaTeXMLMath forces the Jacobi identity for the bracket LaTeXMLMath only for linear functions .
Such tensors may be non-skew-symmetric , i.e. , clearly , Loday algebras do exist .
A simple example is the following .
Example 1 .
Consider the 2-tensor on LaTeXMLMath given by LaTeXMLEquation .
It is easy to see that the Hamiltonian vector fields of linear functions preserve LaTeXMLMath , so we have the Jacobi identity for the associated bracket : LaTeXMLEquation where we assume the missing brackets to be zero .
This example is also an example of a Loday algebroid over a single point , but we can obtain a Loday algebroid over LaTeXMLMath just tensoring the above algebra with LaTeXMLMath .
The anchors of the Loday algebroids from the above example are trivial .
We shall show that this is not incidental and Loday algebroids can be reduced to Lie algebroids and bundles of Loday algebras .
For any Loday algebroid bracket the left anchor is the same as the right anchor and the bracket is skew-symmetric at points where they do not vanish .
Proof .
Let LaTeXMLMath be a Loday algebroid bracket on the space LaTeXMLMath of sections of a vector bundle LaTeXMLMath over LaTeXMLMath .
The Jacobi identity implies immediately LaTeXMLEquation for all LaTeXMLMath .
Putting LaTeXMLMath in ( LaTeXMLRef ) , we get LaTeXMLEquation .
LaTeXMLEquation for all LaTeXMLMath and all LaTeXMLMath .
Suppose that at LaTeXMLMath the right anchor does not vanish , i.e. , there are LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
We can additionally assume that LaTeXMLMath and then ( LaTeXMLRef ) implies that LaTeXMLMath for all LaTeXMLMath .
Hence the vector LaTeXMLMath annihilates any covector from LaTeXMLMath not annihilated by LaTeXMLMath , thus it is zero .
But if LaTeXMLMath does not vanish at LaTeXMLMath , then it does not vanish in a neighborhood of LaTeXMLMath , so LaTeXMLMath in a neighborhood of LaTeXMLMath and ( LaTeXMLRef ) implies now that in this neighborhood LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is nontrivial in this neighborhood , this in turn implies LaTeXMLMath , i.e. , the bracket is skew-symmetric .
In particular , the left anchor equals the right one .
Assume now that the right anchor vanishes at LaTeXMLMath .
By ( LaTeXMLRef ) we obtain now LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath , so LaTeXMLEquation .
Replacing LaTeXMLMath in ( LaTeXMLRef ) by LaTeXMLMath , we get LaTeXMLEquation .
Multiplying the above equation by LaTeXMLMath and taking into account ( LaTeXMLRef ) , we get LaTeXMLEquation for all LaTeXMLMath and LaTeXMLMath which clearly implies that the left anchor vanishes at LaTeXMLMath , since , if the bracket is trivial at LaTeXMLMath , then both anchors are trivial at LaTeXMLMath .
Hence , the right anchor is the same as the left anchor and the bracket is skew-symmetric at points where they do not vanish .
LaTeXMLMath Poisson and Lie algebroid brackets are ones of the most fundamental algebraic structures in Classical and Quantum Physics .
We have composed the two ways of generalizing Poisson bracket : the Nambu ’ s idea of LaTeXMLMath -ary bracket and the Loday ’ s observation that skipping the skew-symmetry assumption in the definition of a Lie algebra we still have a ( co ) homology theory .
What we get is that no new structures appear in this way , since the Leibniz rule and the Filippov identity imply the skew-symmetry .
A similar phenomena we find out when looking for a non-skew version of a Lie algebroid .
This shows that skew-symmetry is in fact forced by other properties of these important algebraic structures .
It would be interesting to know whether the same is true for more general brackets , like Nambu-Jacobi brackets or brackets acting as multidifferential operators .
If we skip skew-symmetry , then it is even not clear if the last ones have to be of first order .
We can prove the skew-symmetry for binary Nambu-Jacobi-Loday brackets and hope that the methods used in the proof of an algebraic version of the well-known Kirillov ’ s theorem on local Lie algebras ( LaTeXMLCite , Theorem 4.2 ) can be of some help in proving a general result .
We postpone these studies to a separate paper .
Let LaTeXMLMath be the space of tensor densities on LaTeXMLMath of degree LaTeXMLMath ( or , equivalently , of conformal densities of degree LaTeXMLMath ) considered as a module over the Lie algebra LaTeXMLMath .
We classify LaTeXMLMath -invariant bilinear differential operators from LaTeXMLMath to LaTeXMLMath .
The classification of linear LaTeXMLMath -invariant differential operators from LaTeXMLMath to LaTeXMLMath already known in the literature ( see LaTeXMLCite ) is obtained in a different manner .
Keywords : Conformal structures , modules of differential operators , tensor densities , invariant differential operators .
In the one-dimensional case , the problem of classification of LaTeXMLMath -invariant ( bi ) linear differential operators has already been treated in the classical literature .
Consider the action of LaTeXMLMath on the space of functions in one variable , say on LaTeXMLMath given by LaTeXMLEquation depending on a parameter LaTeXMLMath ( or LaTeXMLMath ) .
This LaTeXMLMath -module is called the space of tensor densities of degree LaTeXMLMath and denoted LaTeXMLMath .
The classification of LaTeXMLMath -invariant linear differential operators from LaTeXMLMath to LaTeXMLMath ( i.e .
of the operators commuting with the action ( LaTeXMLRef ) ) was obtained in classical works on projective differential geometry .
The result is as follows : there exists a unique ( up to a constant ) LaTeXMLMath -invariant linear differential operator LaTeXMLEquation where LaTeXMLMath , and there are no LaTeXMLMath -invariant operators from LaTeXMLMath to LaTeXMLMath for any other values of LaTeXMLMath and LaTeXMLMath .
Each operator LaTeXMLMath is of order LaTeXMLMath and , for any choice of the coordinate LaTeXMLMath such that the LaTeXMLMath -action is as in ( LaTeXMLRef ) , is plainly given by the LaTeXMLMath -th order derivative , LaTeXMLMath .
The bilinear LaTeXMLMath -invariant differential operators from LaTeXMLMath to LaTeXMLMath were already classified by Gordan LaTeXMLCite .
For generic values of LaTeXMLMath and LaTeXMLMath ( more precisely , for LaTeXMLMath ) , there exists a unique ( up to a constant ) LaTeXMLMath -invariant bilinear differential operator LaTeXMLEquation where LaTeXMLMath , and there are no LaTeXMLMath -invariant operators for any other value of LaTeXMLMath .
This differential operator is given by the formula LaTeXMLEquation and is called the transvectant .
In the multi-dimensional case , one has to distinguish the conformally flat case , that can be reduced to LaTeXMLMath endowed with the standard LaTeXMLMath -action , where LaTeXMLMath , and the curved ( generic ) case of an arbitrary manifold LaTeXMLMath endowed with a conformal structure .
In the conformally flat case , the analogues of the operators ( LaTeXMLRef ) was classified in LaTeXMLCite .
The result is as follows .
( LaTeXMLCite ) There exists a unique ( up to a constant ) LaTeXMLMath -invariant linear differential operator LaTeXMLEquation where LaTeXMLMath , and there are no LaTeXMLMath -invariant operators from LaTeXMLMath to LaTeXMLMath for any other values of LaTeXMLMath and LaTeXMLMath ( or LaTeXMLMath ) .
In the adopted coordinate system ( corresponding to the chosen conformally flat structure ) , the explicit expressions of the operators LaTeXMLMath are LaTeXMLEquation .
In the generic ( curved ) case , the situation is much more complicated , see LaTeXMLCite .
( We also refer to LaTeXMLCite for a recent study of conformally invariant differential operators on tensor fields , and a complete list of references . )
The purpose of this note is to extend the classical Gordan result to the multi-dimensional case .
We will classify LaTeXMLMath -invariant bilinear differential operators consider differential operators on tensor densities on LaTeXMLMath , where LaTeXMLMath .
The results can be also generalized for an arbitrary manifold LaTeXMLMath endowed with a conformally flat structure ( e.g .
upon a pseudo-Riemannian manifold LaTeXMLMath with a conformally flat metric ) .
We will also give a simple direct proof of Theorem LaTeXMLRef .
We will not consider the curved case and state here a problem of existence of bilinear conformally invariant differential operators for an arbitrary conformal structure .
There are other ways to generalize LaTeXMLMath -symmetries in the multi-dimensional case .
For instance , one can consider the LaTeXMLMath -action on LaTeXMLMath ; this is related to projective differential geometry .
Let LaTeXMLMath be the space of tensor densities of degree LaTeXMLMath on LaTeXMLMath , i.e .
of smooth section of the line bundle LaTeXMLMath over LaTeXMLMath .
We will be considering the space LaTeXMLMath of linear differential operators from LaTeXMLMath to LaTeXMLMath and the space LaTeXMLMath of bilinear differential operators from LaTeXMLMath to LaTeXMLMath .
These spaces of differential operators are naturally LaTeXMLMath - and LaTeXMLMath -modules .
Note that the modules LaTeXMLMath have been studied in a series of recent papers ( see LaTeXMLCite and references therein ) .
Denote LaTeXMLMath the standard quadratic form on LaTeXMLMath of signature LaTeXMLMath , where LaTeXMLMath .
The Lie algebra of infinitesimal conformal transformations is generated by the vector fields LaTeXMLEquation where LaTeXMLMath are coordinates on LaTeXMLMath and LaTeXMLMath , throughout this paper , sum over repeated indices is understood .
Let us also consider the following Lie subalgebras LaTeXMLEquation where LaTeXMLMath is generated by the LaTeXMLMath and the Euclidean subalgebra LaTeXMLMath by LaTeXMLMath and LaTeXMLMath .
We will study the spaces LaTeXMLMath and LaTeXMLMath as LaTeXMLMath -modules and classify the differential operators commuting with the LaTeXMLMath -action .
It should be stressed that the classification of differential operators invariant with respect to the Lie subalgebras LaTeXMLMath is the classical result of the Weyl theory of invariants LaTeXMLCite ( see also LaTeXMLCite for the case of LaTeXMLMath ) .
We will use the Weyl classification in our work .
It is worth noticing that the conformal Lie algebra LaTeXMLMath is maximal in the class of finite-dimensional subalgebras of LaTeXMLMath , that is , any bigger subalgebra of LaTeXMLMath is infinite-dimensional ( see LaTeXMLCite for a simple proof ) .
The multi-dimensional analogues of the transvectants ( LaTeXMLRef ) are described in the following For every LaTeXMLMath , there exists a unique ( up to a constant ) LaTeXMLMath -invariant bilinear differential operator LaTeXMLEquation where LaTeXMLMath , and there are no LaTeXMLMath -invariant operators from LaTeXMLMath to LaTeXMLMath for any other value of LaTeXMLMath .
The explicit formula for the operators LaTeXMLMath is complicated and is known only in some particular cases .
The differential operators LaTeXMLMath and LaTeXMLMath are of order LaTeXMLMath ; comparing with the one-dimensional case , one has twice less invariant differential operators .
Note that if one takes semi-integer LaTeXMLMath in ( LaTeXMLRef ) , then the corresponding operator is pseudo-differential .
It would be interesting to obtain a complete classification of LaTeXMLMath -invariant bilinear differential operators ( see LaTeXMLCite for the one-dimensional case ) .
We will start the proof with classical results of the theory of invariants and describe the differential operators invariant with respect to the action of the Lie algebra LaTeXMLMath .
We refer LaTeXMLCite as a classical source and LaTeXMLCite for the description of the Euclidean invariants .
Using the standard affine connection on LaTeXMLMath , one identifies the space of linear differential operators on LaTeXMLMath with the corresponding space of symbols , i.e. , with the space of smooth functions on LaTeXMLMath polynomial on LaTeXMLMath .
This identification is an isomorphism of modules over the algebra of affine transformations and allows us to apply the theory of invariants .
Moreover , choosing a ( dense ) subspace of symbols which are also polynomials on the first summand , one reduces the classification of LaTeXMLMath -invariant differential operators from LaTeXMLMath to the classification of LaTeXMLMath -invariant polynomials in the space LaTeXMLMath , where LaTeXMLMath are the coordinates on LaTeXMLMath dual to LaTeXMLMath .
Consider first invariants with respect to LaTeXMLMath .
It is well-known ( see LaTeXMLCite ) that the algebra of LaTeXMLMath -invariant polynomials is generated by three elements LaTeXMLEquation .
Second , taking into account the invariance with respect to translations in LaTeXMLMath , any LaTeXMLMath -invariant polynomial LaTeXMLMath satisfies LaTeXMLMath .
The only remaining generator is LaTeXMLMath and , therefore , LaTeXMLMath -invariant linear differential operators from LaTeXMLMath to LaTeXMLMath are linear combinations of operators ( LaTeXMLRef ) .
Note that the obtained result is , of course , independent on LaTeXMLMath and LaTeXMLMath since the degree of tensor densities does not intervene in the LaTeXMLMath -action .
We must check now for which values of LaTeXMLMath and LaTeXMLMath the operators ( LaTeXMLRef ) from LaTeXMLMath to LaTeXMLMath are invariant with respect to the action of the full conformal algebra LaTeXMLMath .
By definition , the action of a vector field LaTeXMLMath on an element LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath is the operator of Lie derivative of LaTeXMLMath -density , namely LaTeXMLEquation for any coordinate system .
Consider the action of the generator LaTeXMLMath in ( LaTeXMLRef ) on the operator LaTeXMLMath .
Using the preceding expressions , one readily gets LaTeXMLEquation where LaTeXMLMath .
Thus , the invariance condition LaTeXMLMath is satisfied if and only if for each LaTeXMLMath in the above sum either LaTeXMLMath or LaTeXMLMath ; and one gets the values of the shift LaTeXMLMath in accordance with ( LaTeXMLRef ) .
Consider , at last , the action of the generators LaTeXMLMath ( with LaTeXMLMath ) .
After the identification of the differential operators with polynomials one has the following result from LaTeXMLCite .
The action of the generator LaTeXMLMath on LaTeXMLMath is as follows LaTeXMLEquation where LaTeXMLEquation is the cotangent lift , and where LaTeXMLMath is the trace and LaTeXMLMath the Euler operator .
Applying LaTeXMLMath to the operator LaTeXMLMath one then obtains LaTeXMLEquation .
The first term in this expression vanishes for LaTeXMLMath =0 , this condition is precisely the preceding one ; the second term vanishes if and only if LaTeXMLEquation .
Hence the result .
As in Section LaTeXMLRef , let us first consider the operators invariant with respect to the Lie algebra LaTeXMLMath .
Again , identifying the bilinear differential operators with their symbols , one is led to study the algebra of LaTeXMLMath -invariant polynomials in the space LaTeXMLMath .
The Weyl invariant theory just applied guarantees that there are three generators : LaTeXMLEquation .
Any LaTeXMLMath -invariant bilinear differential operator is then of the form LaTeXMLEquation where , to simplify the notations , we put LaTeXMLMath The action of a vector field LaTeXMLMath on a bilinear operator LaTeXMLMath is defined as follows LaTeXMLEquation .
Let us apply the generator LaTeXMLMath to the operator LaTeXMLMath , one has LaTeXMLEquation where LaTeXMLMath .
The equation LaTeXMLMath leads to the homogeneity condition LaTeXMLEquation .
The general expression for LaTeXMLMath retains the form LaTeXMLEquation .
The operator LaTeXMLMath is of order LaTeXMLMath and LaTeXMLMath .
Now , to determine the coefficients LaTeXMLMath , one has to apply the generators LaTeXMLMath .
One has the following analog of Proposition LaTeXMLRef .
The action of the generator LaTeXMLMath on LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLEquation is just the natural lift of LaTeXMLMath to LaTeXMLMath .
Applying LaTeXMLMath to each monomial , LaTeXMLMath , in the operator LaTeXMLMath , one immediately gets LaTeXMLEquation .
At last , applying LaTeXMLMath to the operator LaTeXMLMath written in the form ( LaTeXMLRef ) and collecting the terms , one readily gets the following recurrent system of two linear equations LaTeXMLEquation and LaTeXMLEquation for the coefficients .
For LaTeXMLMath this system has a unique ( up to a constant ) solution .
Indeed , choosing LaTeXMLMath as a parameter , one uses the first equation to express LaTeXMLMath from LaTeXMLMath and the second one to express LaTeXMLMath from LaTeXMLMath .
Theorem LaTeXMLRef is proved .
The differential operators ( LaTeXMLRef ) and ( LaTeXMLRef ) play important rôle in the theory of modular functions ( see LaTeXMLCite ) , in projective differential geometry ( see LaTeXMLCite ) and in the representation theory of LaTeXMLMath .
The transvectants have been recently used in LaTeXMLCite to construct LaTeXMLMath -invariant star-products on LaTeXMLMath .
We plan to discuss the relation of the conformally invariant operators described in this note to the representation theory in a subsequent paper .
Let us give here explicit formulæ for the bilinear differential operators ( LaTeXMLRef ) with LaTeXMLMath .
Using ( LaTeXMLRef ) and ( LaTeXMLRef ) one has : LaTeXMLEquation and LaTeXMLEquation .
The expressions of further orders operators are much more complicated , and we do not have an explicit general formula , except for some particular coefficients LaTeXMLMath .
A direct computation using equations leads to LaTeXMLEquation and LaTeXMLEquation .
Acknowledgments : We are indebted to C. Duval , A .
A. Kirillov , Y. Kosmann-Schwarzbach and P. Lecomte for enlightening discussions .
Brooks and Makover introduced an approach to random Riemann surfaces based on associating a dense set of them – Belyi surfaces – with random cubic graphs .
In this paper , using Bollobas model for random regular graphs , we examine the topological structure of these surfaces , obtaining in particular an estimate for the expected value of their genus .
In LaTeXMLCite Brooks and Makover constructed Riemann surfaces from oriented cubic graphs .
For each orientated graph LaTeXMLMath they associate two Riemann surfaces , LaTeXMLMath a finite area noncompact surface , and LaTeXMLMath a compact surface .
The surface LaTeXMLMath is an orbifold cover of LaTeXMLMath and therefore shares some of the global geometric properties with the graph LaTeXMLMath .
The compact surface LaTeXMLMath is a conformal compactification of LaTeXMLMath ; Brooks and Makover proved that almost always the global geometry of LaTeXMLMath is controlled by the geometry of LaTeXMLMath .
Moreover , according to Belyi theorem LaTeXMLCite the surfaces LaTeXMLMath are precisely the Riemann surfaces which can be defined over some number field and form a dense set in the space of all Riemann surfaces .
Therefore these surfaces , called Belyi surfaces , can be used to model a process of picking random Riemann surfaces , by picking a random graph with a random orientation .
In their work , Brooks and Makover use the graphs to get information on the global geometry of the surfaces LaTeXMLMath , but they do not have control on the distribution of the surfaces in the Teichmüller spaces .
In this paper we investigate the topology of Belyi surfaces , which is the first step in understanding the distribution of LaTeXMLMath in different Teichmüller spaces .
Our main result ( with LaTeXMLMath denoting the number of vertices in a cubic graph ) is the following theorem : LaTeXMLEquation .
The genera of these surfaces can be calculated by the Euler formula where the number of vertices and edges is determined by the size of the graph LaTeXMLMath .
The faces , which correspond to the cusps of LaTeXMLMath can be described in a purely combinatorial way from the oriented graph LaTeXMLMath .
Using Bollobas model for random regular graphs LaTeXMLCite LaTeXMLCite , we estimate the number of faces : There exist constants LaTeXMLMath and LaTeXMLMath such that , for a large enough LaTeXMLMath : LaTeXMLEquation .
Where LaTeXMLMath is the number of faces of the oriented graph .
The Euler formula then gives our main result .
We will start in section LaTeXMLRef in a review on the construction of the Belyi surfaces from cubic graphs , and we will show how to calculate the genus of the surfaces .
In section LaTeXMLRef we will review the Bollobas model and use it to calculate the expected value of the number of faces in Belyi surfaces .
Note : Our main interest is the construction of the Belyi surfaces , this is why we work with cubic graphs .
Other authors LaTeXMLCite work on the average value of the genus of a given graph : our result can be easily generalized to LaTeXMLMath -regular graphs .
An orientation LaTeXMLMath on a graph LaTeXMLMath is a function which associates to each vertex LaTeXMLMath a cyclic ordering of the edges emanating from LaTeXMLMath .
We build the surface LaTeXMLMath from an oriented graph as follows : we take the ideal hyperbolic triangle LaTeXMLMath with vertices LaTeXMLMath , and LaTeXMLMath shown in Figure LaTeXMLRef .
The solid lines in Figure LaTeXMLRef are geodesics joining the points LaTeXMLMath , and LaTeXMLMath with the point LaTeXMLMath , while the dotted lines are horocycles joining pairs of points from the set LaTeXMLMath .
We may think of these points as “ midpoints ” of the corresponding sides of the ideal triangles , even though the sides are of infinite length .
We may also think of the three solid lines as edges of a graph emanating from a vertex .
We may then give them the cyclic ordering LaTeXMLMath .
Given a cubic graph with orientation LaTeXMLMath we build a non-compact Riemann surface denoted by LaTeXMLMath , by associating to each vertex an ideal triangle , and gluing neighboring triangles .
We glue two copies of LaTeXMLMath along the corresponding sides , subject to the following two conditions : the “ midpoints ” of the two sides are glued together , and the gluing preserves the orientation of the two copies of LaTeXMLMath .
The conditions ( a ) and ( b ) determine the gluing uniquely .
It is easily seen that the surface LaTeXMLMath is a complete Riemann surface with a finite area equal to LaTeXMLMath , where LaTeXMLMath is the number of vertices of LaTeXMLMath , and that the horocycles of the copies of LaTeXMLMath fit together to give closed horocycles about the cusps of LaTeXMLMath .
We denote by LaTeXMLMath the conformal compactification of LaTeXMLMath .
Using the results about random cubic graphs Brooks and Makover proved LaTeXMLCite : There exist constants LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath such that , as LaTeXMLMath : The first eigenvalue LaTeXMLMath satisfies LaTeXMLEquation .
The Cheeger constant LaTeXMLMath satisfies LaTeXMLEquation .
The shortest geodesic LaTeXMLMath satisfies LaTeXMLEquation .
The diameter LaTeXMLMath satisfies LaTeXMLEquation .
The topology of the surface can be read from the oriented graph using left-hand-turn paths .
A l eft-hand-turn path on LaTeXMLMath is a closed path on LaTeXMLMath such that , at each vertex , the path turns left in the orientation LaTeXMLMath .
On LaTeXMLMath a left-hand path describe a closed path around a cusp .
Let LaTeXMLMath and LaTeXMLMath be the number of disjoint left-hand paths .
For a cubic graph the number of edges is LaTeXMLMath and the number of faces is LaTeXMLMath .
Therefore we can write the Euler formula : LaTeXMLEquation .
To estimate the genus of a random surface we need to estimate the number of faces ( or left-hand paths of the oriented graph ) .
It is important to observe that a left-hand path is not necessarily a simple closed path on the graph .
For example , if we take the 1-skeleton of the cube , with the usual orientation ( Figure LaTeXMLRef A ) , we have LaTeXMLMath , and all the faces are simple paths of the graph .
In Figure LaTeXMLRef B we changed the orientation of the right upper vertex .
With the new orientation Now if we change the orientation of the right upper vertex , the three simple faces that were adjacent to the upper right vertex before we changed orientation are now joined to one composite face while the other three faces are unchanged hence LaTeXMLMath .
In section LaTeXMLRef we obtain an upper and lower bound on the number of faces given in theorem LaTeXMLRef .
The lower bound is obtained by counting the number of faces that are simple closed paths on the graph .
The upper bound on the number of faces is obtained by first dividing the faces into two groups : Small faces LaTeXMLMath Large faces LaTeXMLMath The number of large faces is easily bounded : since the total number of edges is LaTeXMLMath and each edge has two sides , we obtain that LaTeXMLMath .
To estimate the number of small faces we first introduce the notion of root .
A root in LaTeXMLMath is a simple closed path in LaTeXMLMath , such that the orientation LaTeXMLMath agrees with the root in all but maybe one vertex .
Each face contains at least one root : Proof : Pick a vertex LaTeXMLMath and start walking along a left hand path until the path intersects itself , i.e. , LaTeXMLMath LaTeXMLMath is the first intersection , then the cycle LaTeXMLMath is a root ( in all the vertices LaTeXMLMath LaTeXMLMath the orientation agrees with the cycle ) .
In section LaTeXMLRef we will prove that the expected number of roots of length less than LaTeXMLMath , and therefore the total number of small faces , is bounded by LaTeXMLMath .
By adding the number of short and long faces we get that the total number of faces is LaTeXMLMath ; from equation LaTeXMLRef we deduce our main result LaTeXMLRef .
This section is devoted to the proof of theorem LaTeXMLRef .
Let LaTeXMLMath be the space of LaTeXMLMath -regular graphs with LaTeXMLMath vertices .
We shall , following Bollobas , represent our graphs as the images of so-called configurations .
Let LaTeXMLMath be a fixed set of LaTeXMLMath vertices , where LaTeXMLMath .
A configuration LaTeXMLMath is a partition of LaTeXMLMath into LaTeXMLMath pairs of vertices called edges of LaTeXMLMath .
Clearly there are LaTeXMLEquation configurations .
( We write LaTeXMLMath ) .
Let LaTeXMLMath be a set of configurations .
We now define a map LaTeXMLMath as follows .
Given a configuration LaTeXMLMath let LaTeXMLMath be the graph with vertex set LaTeXMLMath in which LaTeXMLMath is an edge iff LaTeXMLMath has a pair with one end in LaTeXMLMath and the other in LaTeXMLMath .
Every LaTeXMLMath is the image of LaTeXMLMath for LaTeXMLMath configurations .
The number of configurations containing a given fixed set of LaTeXMLMath edges is LaTeXMLEquation .
From ( LaTeXMLRef ) and ( LaTeXMLRef ) , the probability that a configuration contains a given set of LaTeXMLMath edges is LaTeXMLEquation where we use LaTeXMLEquation .
For LaTeXMLMath a k-cycle of a configuration is a set of LaTeXMLMath edges , say LaTeXMLMath such that for some LaTeXMLMath distinct groups LaTeXMLMath the edge LaTeXMLMath joins LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath .
A 1-cycle is said to be a loop and a 2-cycle is a coupling .
Given a configuration LaTeXMLMath , denote by LaTeXMLMath the number of k-cycles .
If we are to restrict to graphs in LaTeXMLMath to have no loops and no multiple edges , then not every LaTeXMLMath belongs to LaTeXMLMath but only those satisfying LaTeXMLMath ; such graphs are called simple .
( In order to restrict to graphs with LaTeXMLMath we have to multiply all the ensuing estimates by LaTeXMLMath . )
Let LaTeXMLMath be the number of sets of pairs of vertices in LaTeXMLMath which can be k-cycles of configurations .
By elementary counting : LaTeXMLEquation .
Using ( LaTeXMLRef ) and ( LaTeXMLRef ) we get the following expression for the expected number of LaTeXMLMath -cycles : LaTeXMLEquation .
To obtain a lower bound on the number of faces LaTeXMLMath , we count the number of faces that are simple closed paths .
For each simple closed path of length LaTeXMLMath the probability of correct orientation is LaTeXMLMath , consequently we have : LaTeXMLEquation where LaTeXMLEquation .
Now using Stirling ’ s formula ( with LaTeXMLMath positive real ) , LaTeXMLEquation we have for LaTeXMLMath LaTeXMLEquation and similarly LaTeXMLEquation .
Recalling that LaTeXMLMath , we have LaTeXMLEquation .
The estimate ( LaTeXMLRef ) and the fact that LaTeXMLMath are monotonic decreasing in LaTeXMLMath , show that if we let LaTeXMLMath range from LaTeXMLMath to LaTeXMLMath where LaTeXMLMath , the remaining terms in ( LaTeXMLRef ) are negligible .
With that in mind , let LaTeXMLMath be an arbitrary positive number independent of LaTeXMLMath .
If LaTeXMLMath , the estimate in ( LaTeXMLRef ) is between LaTeXMLMath and LaTeXMLMath .
Using LaTeXMLEquation we get that the asymptotic estimate for that part of sum in ( LaTeXMLRef ) with LaTeXMLMath is between LaTeXMLMath and LaTeXMLMath .
For the remainder of the sum , LaTeXMLMath , and the sum of LaTeXMLMath is asymptotic to LaTeXMLMath times an integral of LaTeXMLMath .
It follows that for any value of LaTeXMLMath , the contribution of this part of sum is bounded as LaTeXMLMath , and so ( LaTeXMLRef ) is asymptotic to a function of LaTeXMLMath which lies between LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is arbitrary , LaTeXMLMath can be made as close to 1 as desired ; hence we obtain that the sum in ( LaTeXMLRef ) is asymptotic to LaTeXMLMath .
Now we turn to estimating the number of faces from above .
Recall that the number of large faces is bounded from above by LaTeXMLMath , so we need to estimate from above the number of small faces .
As their length is less than LaTeXMLMath , so is the length of their roots ; thus the estimate on the number of roots of length less than LaTeXMLMath will suffice .
Now given a cycle of length LaTeXMLMath , we can turn it into a root in LaTeXMLMath ways : one simple face and LaTeXMLMath proper roots .
Furthermore , the probability of correct orientation is LaTeXMLMath .
So we have the following estimate ( with LaTeXMLMath given in LaTeXMLRef ) : LaTeXMLEquation .
The second sum is clearly dominated by the sum in ( LaTeXMLRef ) , which we established to be of order LaTeXMLMath .
For the first sum , using ( LaTeXMLRef ) , we obtain LaTeXMLEquation .
We have therefore established upper and lower bound of LaTeXMLMath and LaTeXMLMath respectively for the number of faces , proving the theorem LaTeXMLRef .
We give a new characterization of Lusztig ’ s canonical quotient , a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra .
This group plays an important role in the classification of unipotent representations of finite groups of Lie type .
We also define a duality map .
To each pair of a nilpotent orbit and a conjugacy class in its fundamental group , the map assigns a nilpotent orbit in the Langlands dual Lie algebra .
This map is surjective and is related to a map introduced by Lusztig ( and studied by Spaltenstein ) .
When the conjugacy class is trivial , our duality map is just the one studied by Spaltenstein and by Barbasch and Vogan which has image consisting of the special nilpotent orbits .
To each special nilpotent orbit in a simple Lie algebra over the complex numbers , Lusztig has assigned a finite group ( called Lusztig ’ s canonical quotient ) which is naturally a quotient of the fundamental group of the orbit .
This group is defined using the Springer correspondence and the generic degrees of Hecke algebra representations ; it plays an important role in Lusztig ’ s work on parametrizing unipotent representations of a finite group of Lie type LaTeXMLCite .
In this paper we give a description of the canonical quotient which uses the description of the component group of a nilpotent element given in LaTeXMLCite when LaTeXMLMath is of adjoint type .
To do this we assign to each conjugacy class in the component group a numerical value ( called the LaTeXMLMath -value ) .
The definition of the LaTeXMLMath -value makes use of the duality map of LaTeXMLCite and involves the dimension of a Springer fiber ; it is related to the usual LaTeXMLMath -value of a Springer representation ( the smallest degree in which the representation appears in the harmonic polynomials on the Cartan subalgebra ) .
Our observation is that the kernel of the quotient map from the component group to the canonical quotient consists of all conjugacy classes whose LaTeXMLMath -value is equal to the LaTeXMLMath -value of the trivial conjugacy class .
In the second part of the paper we define a surjective map from the set of pairs consisting of a nilpotent orbit and a conjugacy class in its fundamental group to the set of nilpotent orbits in the Langlands dual Lie algebra .
This extends the duality of LaTeXMLCite ( which corresponds to the situation when the conjugacy class is trivial ) .
In fact the map depends only on the image of the conjugacy class in Lusztig ’ s canonical quotient .
Our map is defined by combining the duality of LaTeXMLCite and a map defined by Lusztig LaTeXMLCite .
Lusztig ’ s map can be thought of as a generalization of induction LaTeXMLCite ; in his book , Spaltenstein also studied this generalization of induction LaTeXMLCite .
I would like to thank G. Lusztig , D. Vogan , P. Achar , P. Trapa , and D. Barbasch for helpful conversations .
I gratefully acknowledge the support of NSF grant DMS-0070674 and the hospitality of H. Kraft and the Mathematisches Institut at the Universität Basel in June , 1999 .
Throughout the paper , LaTeXMLMath is a connected simple algebraic group over the complex numbers LaTeXMLMath .
In LaTeXMLMath we fix a maximal torus LaTeXMLMath contained in a Borel subgroup LaTeXMLMath .
Let the character group of LaTeXMLMath be LaTeXMLMath .
We use LaTeXMLMath for the Lie algebra of LaTeXMLMath .
Let LaTeXMLMath be the roots of LaTeXMLMath and let LaTeXMLMath be the Weyl group of LaTeXMLMath ( with respect to LaTeXMLMath ) .
The group LaTeXMLMath acts on LaTeXMLMath via the adjoint action .
If LaTeXMLMath is an element of LaTeXMLMath , we denote by LaTeXMLMath the orbit of LaTeXMLMath through LaTeXMLMath under the action of LaTeXMLMath .
If LaTeXMLMath is a nilpotent element , we call LaTeXMLMath a nilpotent orbit .
We denote by LaTeXMLMath the set of nilpotent orbits in LaTeXMLMath .
This set is partially ordered by the relation LaTeXMLMath whenever LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath , let LaTeXMLMath be the component group LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath may be identified with LaTeXMLMath and so we write LaTeXMLMath for this finite group .
Furthermore , we may speak in a well-defined manner of the conjugacy classes of LaTeXMLMath .
When LaTeXMLMath is simply-connected ( and we pass to the analytic topology ) , LaTeXMLMath is just the fundamental group of LaTeXMLMath .
We denote by LaTeXMLMath the irreducible representations of LaTeXMLMath and use the notion of irreducible local system on LaTeXMLMath interchangeably with the notion of irreducible representation of LaTeXMLMath , or LaTeXMLMath for any LaTeXMLMath .
We denote by LaTeXMLMath the set of pairs LaTeXMLMath consisting of an orbit LaTeXMLMath and a conjugacy class LaTeXMLMath .
The set of special orbits in LaTeXMLMath ( see LaTeXMLCite ) will be denoted by LaTeXMLMath .
There is an order-reversing duality map LaTeXMLMath studied by Spaltenstein such that LaTeXMLMath is the identity on LaTeXMLMath LaTeXMLCite .
This map is already implicit in LaTeXMLCite , so we refer to LaTeXMLMath henceforth as Lusztig-Spaltenstein duality .
In the classical groups it will be helpful to have a description of the elements of LaTeXMLMath and LaTeXMLMath and the map LaTeXMLMath in terms of partitions .
We introduce that notation now ( roughly following the references LaTeXMLCite , LaTeXMLCite , and LaTeXMLCite ) .
Let LaTeXMLMath denote the set of partitions LaTeXMLMath of LaTeXMLMath ( we assume that LaTeXMLMath ) .
For a part LaTeXMLMath of LaTeXMLMath , we call LaTeXMLMath the position of LaTeXMLMath in LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath .
For LaTeXMLMath let LaTeXMLMath be the set of partitions LaTeXMLMath of LaTeXMLMath such that LaTeXMLMath whenever LaTeXMLMath ( all congruences are modulo LaTeXMLMath ) .
It is well known that LaTeXMLMath is in bijection with LaTeXMLMath when LaTeXMLMath is of type LaTeXMLMath ; with LaTeXMLMath when LaTeXMLMath is of type LaTeXMLMath ; with LaTeXMLMath when LaTeXMLMath is of type LaTeXMLMath ; and with LaTeXMLMath when LaTeXMLMath is of type LaTeXMLMath , except that those partitions with all even parts correspond to two orbits in LaTeXMLMath ( called the very even orbits ) .
In what follows we will never have a need to address separately the very even orbits , so we will not bother to introduce notation to distinguish between very even orbits .
We will also refer to LaTeXMLMath as LaTeXMLMath ; to LaTeXMLMath as LaTeXMLMath ; and to LaTeXMLMath as LaTeXMLMath .
The set LaTeXMLMath is partially ordered by the usual partial ordering on partitions .
This induces a partial ordering on the sets LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath and these partial orderings coincide with the partial ordering on nilpotent orbits given by inclusion of closures .
We will refer to nilpotent orbits and partitions interchangeable in the classical groups ( with the caveat mentioned earlier for the very even orbits in type LaTeXMLMath ) .
We can also represent elements of LaTeXMLMath in terms of partitions in the classical groups ( see the last section of LaTeXMLCite ) .
Given LaTeXMLMath let LaTeXMLMath and let LaTeXMLMath be a semisimple element whose image in LaTeXMLMath belongs to LaTeXMLMath .
Set LaTeXMLMath .
We may as well assume that LaTeXMLMath contains Lie ( LaTeXMLMath ) ; then the subalgebra LaTeXMLMath of LaTeXMLMath ( which we called a pseudo-Levi subalgebra in LaTeXMLCite ) corresponds to a proper subset of the extended Dynkin diagram of LaTeXMLMath .
It is always possible to choose LaTeXMLMath so that LaTeXMLMath has semisimple rank equal to LaTeXMLMath , which we will do .
Next we write LaTeXMLMath , where LaTeXMLMath is a semisimple subalgebra of the same type as LaTeXMLMath and LaTeXMLMath is a simple Lie algebra ( or possibly zero ) which contains the root space corresponding to the lowest root of LaTeXMLMath ( the extra node in the extended Dynkin diagram ) .
Then LaTeXMLMath ( if non-zero ) is of type LaTeXMLMath or LaTeXMLMath depending on whether LaTeXMLMath is of type LaTeXMLMath or LaTeXMLMath , respectively .
Write LaTeXMLMath where LaTeXMLMath and LaTeXMLMath .
Finally , we may modify the choice of LaTeXMLMath so that LaTeXMLMath is a distinguished nilpotent element in LaTeXMLMath .
Then we can attach to LaTeXMLMath a pair of partitions LaTeXMLMath where LaTeXMLMath is the partition of LaTeXMLMath in LaTeXMLMath and LaTeXMLMath is the partition of LaTeXMLMath in LaTeXMLMath .
We have LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath is a distinguished partition , meaning LaTeXMLMath for all LaTeXMLMath ( this necessarily forces LaTeXMLMath whenever LaTeXMLMath ) .
The partition LaTeXMLMath of LaTeXMLMath is just the partition consisting of all the parts in LaTeXMLMath and LaTeXMLMath .
We denote this partition by LaTeXMLMath and write LaTeXMLMath .
In types LaTeXMLMath and LaTeXMLMath we make the further assumption that the largest LaTeXMLMath with LaTeXMLMath is not a part of LaTeXMLMath .
These assumptions guarantee that when LaTeXMLMath is of adjoint type there is a bijection between LaTeXMLMath and the pairs LaTeXMLMath specified above ( if LaTeXMLMath is very even in type LaTeXMLMath , the component group LaTeXMLMath is trivial when LaTeXMLMath is adjoint ; so there are no additional complications beyond the one mentioned earlier ) .
We note that LaTeXMLMath will be the empty partition ( that is , LaTeXMLMath in equation ( LaTeXMLRef ) ) if and only if LaTeXMLMath is the trivial conjugacy class .
For example , let LaTeXMLMath be the orbit with partition LaTeXMLMath in LaTeXMLMath .
Then we have LaTeXMLEquation are the four elements of LaTeXMLMath corresponding to the four conjugacy classes of LaTeXMLMath when LaTeXMLMath is of adjoint type , with the first pair corresponding to the trivial conjugacy class .
The dual partition of LaTeXMLMath , denoted LaTeXMLMath , is defined by LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
We define the LaTeXMLMath -collapse of a partition LaTeXMLMath where LaTeXMLMath is even when LaTeXMLMath or LaTeXMLMath and LaTeXMLMath is odd when LaTeXMLMath .
The LaTeXMLMath -collapse of LaTeXMLMath is the partition LaTeXMLMath such that LaTeXMLMath and such that LaTeXMLMath whenever LaTeXMLMath and LaTeXMLMath .
The LaTeXMLMath -collapse of LaTeXMLMath is denoted LaTeXMLMath .
It is well-defined and unique .
The duality map LaTeXMLMath can now be expressed as follows in the classical groups .
In type LaTeXMLMath , we have LaTeXMLMath and in type LaTeXMLMath ( for LaTeXMLMath ) we have LaTeXMLMath .
The set of special nilpotent orbits correspond to the partitions in the image of LaTeXMLMath ( in type LaTeXMLMath all very even orbits are special ) .
Hence in type LaTeXMLMath all nilpotent orbits are special .
In the other types it is known that LaTeXMLMath is special if and only if LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In this section we assign to each pair LaTeXMLMath a natural number which we will call the LaTeXMLMath -value of LaTeXMLMath and which will be denoted by LaTeXMLMath .
Given LaTeXMLMath let LaTeXMLMath and LaTeXMLMath be a semisimple element whose image in LaTeXMLMath belongs to LaTeXMLMath .
The group LaTeXMLMath has as its Lie algebra LaTeXMLMath and clearly LaTeXMLMath .
Let LaTeXMLMath denote the orbit in LaTeXMLMath through LaTeXMLMath under the action of LaTeXMLMath .
Applying Lusztig-Spaltenstein duality LaTeXMLMath with respect to LaTeXMLMath , we obtain the orbit LaTeXMLMath .
We define LaTeXMLMath .
Equivalently , if LaTeXMLMath is the variety of Borel subalgebras of LaTeXMLMath which contain LaTeXMLMath , then LaTeXMLMath .
We have the following proposition whose proof we give after the proof of proposition LaTeXMLRef .
The number LaTeXMLMath is well-defined .
In other words , it is independent of the choices made for LaTeXMLMath and LaTeXMLMath .
By results in LaTeXMLCite it is always possible to choose LaTeXMLMath so that LaTeXMLMath is distinguished in LaTeXMLMath .
By the previous proposition , in order to compute LaTeXMLMath it is enough to determine a pair LaTeXMLMath attached to LaTeXMLMath where LaTeXMLMath is distinguished in LaTeXMLMath ( this is done in LaTeXMLCite ) and to compute the LaTeXMLMath -value of LaTeXMLMath with respect to LaTeXMLMath ( with the trivial conjugacy class LaTeXMLMath of LaTeXMLMath ) .
Hence it suffices to compute LaTeXMLMath for each distinguished orbit LaTeXMLMath in each simple Lie algebra .
We now record LaTeXMLMath for the distinguished orbits in the exceptional groups .
We now record LaTeXMLMath for the distinguished orbits in the classical groups .
In fact , it is no harder to record the formula for LaTeXMLMath for any orbit LaTeXMLMath , whether distinguished or not .
Let LaTeXMLMath be the partition for LaTeXMLMath in the appropriate classical group .
Then LaTeXMLMath equals LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
We omit the calculations ( which are easy in all types except type LaTeXMLMath ) since they will follow quickly from later work .
If LaTeXMLMath is the Richardson orbit for the parabolic subgroup LaTeXMLMath whose Levi subgroup has root system isomorphic to LaTeXMLMath ( LaTeXMLMath -times ) , we have observed that LaTeXMLMath .
Here , LaTeXMLMath is the rank of LaTeXMLMath , LaTeXMLMath is the number of positive roots of LaTeXMLMath , and LaTeXMLMath is the LaTeXMLMath -th exponent of LaTeXMLMath when the exponents are listed in increasing order .
This offers the hope that there is a way to determine the LaTeXMLMath -value of any distinguished orbit from standard data arising purely from the root system .
The next proposition is analogous to a statement ( due to Lusztig in LaTeXMLCite ) about LaTeXMLMath -values of Springer representations which we will recall in the next section .
Let LaTeXMLMath be the trivial conjugacy and let LaTeXMLMath be any conjugacy class in LaTeXMLMath .
Then LaTeXMLMath .
In the exceptional groups , we have verified this directly .
In the classical groups , we must show that for any LaTeXMLMath and LaTeXMLMath with LaTeXMLMath that we have LaTeXMLMath where the LaTeXMLMath -value is computed with respect to the appropriate subalgebras for LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
In type LaTeXMLMath , LaTeXMLMath .
For any LaTeXMLMath with LaTeXMLMath , we must use both the LaTeXMLMath -value formula in type LaTeXMLMath for LaTeXMLMath and in type LaTeXMLMath for LaTeXMLMath .
It is clear that LaTeXMLMath minus various LaTeXMLMath .
It turns out that that for each LaTeXMLMath either LaTeXMLMath or LaTeXMLMath will be subtracted in the formula , but not both , as we now show .
There are four possibilities for each LaTeXMLMath .
If both LaTeXMLMath and LaTeXMLMath belong to LaTeXMLMath or both belong to LaTeXMLMath , then LaTeXMLMath will have a term LaTeXMLMath or LaTeXMLMath , but not both , since the formulas for LaTeXMLMath and LaTeXMLMath select every other LaTeXMLMath to subtract .
If on the other hand , LaTeXMLMath belongs to LaTeXMLMath and LaTeXMLMath belongs to LaTeXMLMath , then again LaTeXMLMath will have a term LaTeXMLMath or LaTeXMLMath , but not both .
This is because the parity of the position of LaTeXMLMath in LaTeXMLMath will be the same as the parity of the position of LaTeXMLMath in LaTeXMLMath ( as the position of LaTeXMLMath in LaTeXMLMath is an even number , namely LaTeXMLMath ) .
But the formula for LaTeXMLMath and LaTeXMLMath choose to subtract parts whose positions have opposite parity .
Hence , only one can be selected in the formula for LaTeXMLMath .
The result is the same if LaTeXMLMath belongs to LaTeXMLMath and LaTeXMLMath belongs to LaTeXMLMath .
Thus in either of the four cases , the effect is to subtract a number which is less than or equal to LaTeXMLMath since LaTeXMLMath , hence the inequality LaTeXMLMath .
In types LaTeXMLMath and LaTeXMLMath , a similar argument holds , except we look at the consecutive parts LaTeXMLMath and LaTeXMLMath of LaTeXMLMath ( the part LaTeXMLMath will never be subtracted in the formulas for the LaTeXMLMath -value ) .
One or the other , but not both , of these parts will be subtracted in the formula for LaTeXMLMath demonstrating the inequality .
∎ We recall some notation from LaTeXMLCite in the classical groups LaTeXMLEquation .
LaTeXMLEquation List the elements of LaTeXMLMath in decreasing order LaTeXMLMath .
Assume that LaTeXMLMath is even in type LaTeXMLMath by setting LaTeXMLMath if necessary ( LaTeXMLMath is automatically odd in type LaTeXMLMath and automatically even in type LaTeXMLMath ) .
An element LaTeXMLMath determines two sets LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath coming from the parts of LaTeXMLMath .
Namely , the parts of LaTeXMLMath ( which each occur with multiplicity one since LaTeXMLMath is distinguished ) consist precisely of the elements in LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLMath if LaTeXMLMath and LaTeXMLMath if LaTeXMLMath .
Note we are assuming that LaTeXMLMath in types LaTeXMLMath and LaTeXMLMath .
We define subsets LaTeXMLMath of LaTeXMLMath as follows : let LaTeXMLMath consist of those LaTeXMLMath such that LaTeXMLMath and define LaTeXMLMath to be the cardinality of LaTeXMLMath .
The next proposition follows easily from the previous proposition and the formulas for the LaTeXMLMath -value .
In the classical groups ( not of type LaTeXMLMath ) , the equality LaTeXMLMath holds if and only if LaTeXMLMath when LaTeXMLMath is even and LaTeXMLMath is even when LaTeXMLMath is odd .
Our first main result is a new description of Lusztig ’ s canonical quotient LaTeXMLMath , which is a quotient of LaTeXMLMath .
These finite groups play an important role in Lusztig ’ s classification of unipotent representations of finite groups of Lie type .
Namely , the set of unipotent representations for LaTeXMLMath ( split , with connected center , over a finite field ) is parametrized by the following data : a special nilpotent orbit LaTeXMLMath , an element LaTeXMLMath , and an irreducible representation of the centralizer of LaTeXMLMath in LaTeXMLMath ( all up to the appropriate conjugation ) .
We hope to obtain a better understanding of this parametrization by having such an explicit description of the canonical quotient and its conjugacy classes .
Let us recall Lusztig ’ s definition of LaTeXMLMath .
Although Lusztig assumed LaTeXMLMath is special , his definition remains valid even if LaTeXMLMath is not special , so in what follows we do not assume LaTeXMLMath is special .
Recall that for each nilpotent element LaTeXMLMath and local system LaTeXMLMath on LaTeXMLMath , Springer has defined a representation LaTeXMLMath which ( if non-zero ) is an irreducible representation of LaTeXMLMath .
Recall also that each irreducible representation LaTeXMLMath of LaTeXMLMath comes with two important numerical invariants .
One comes from the generic degree of LaTeXMLMath ( the LaTeXMLMath -value ) and one comes from the fake degree of LaTeXMLMath ( the LaTeXMLMath -value ) .
We refer to LaTeXMLCite for the definitions .
Note that our notation is consistent with LaTeXMLCite , but is not consistent with LaTeXMLCite or LaTeXMLCite .
In those sources , our LaTeXMLMath -value is their LaTeXMLMath -value and our LaTeXMLMath -value is their LaTeXMLMath -value .
The original definition of the canonical quotient of LaTeXMLMath is as follows .
Given LaTeXMLMath , consider the set LaTeXMLEquation where LaTeXMLMath is the set of irreducible representations of LaTeXMLMath and LaTeXMLMath denotes the Springer representation associated to the trivial representation of LaTeXMLMath .
Let LaTeXMLMath be the intersection of all the kernels of the representations in LaTeXMLMath .
Then LaTeXMLMath is defined to be the quotient LaTeXMLMath LaTeXMLCite .
If a local system LaTeXMLMath is not equivariant for LaTeXMLMath when LaTeXMLMath is of adjoint type , then LaTeXMLMath will be zero .
Hence , the canonical quotient is the same for groups in the same isogeny class of LaTeXMLMath and so we assume LaTeXMLMath is of adjoint type in what follows .
Lusztig has observed that for LaTeXMLMath and LaTeXMLMath that LaTeXMLMath .
Compare this with proposition LaTeXMLRef .
In light of Lusztig ’ s original definition and proposition LaTeXMLRef we consider all conjugacy classes LaTeXMLMath in LaTeXMLMath with the property that LaTeXMLMath .
Let LaTeXMLMath be the union of all such conjugacy classes .
The set LaTeXMLMath coincides with LaTeXMLMath , so that LaTeXMLMath .
Moreover , the LaTeXMLMath -value is constant on the cosets of LaTeXMLMath in LaTeXMLMath ( which are always a union of conjugacy classes ) .
We are assuming that LaTeXMLMath is of adjoint type ( the result is still valid for any LaTeXMLMath ) .
In the exceptional groups we verified the results directly using the tables for LaTeXMLMath -values and Springer representations in LaTeXMLCite , the tables of conjugacy classes in LaTeXMLMath in LaTeXMLCite , and knowledge of the LaTeXMLMath -values for distinguished orbits given above .
In the classical groups we need to do some work in order to understand which local systems appear in the set LaTeXMLMath .
We illustrate the situation in type LaTeXMLMath .
Let LaTeXMLMath have partition LaTeXMLMath .
As before , LaTeXMLMath is the number of parts of LaTeXMLMath of size LaTeXMLMath .
So LaTeXMLMath is even whenever LaTeXMLMath is even as we are in type LaTeXMLMath .
We associate to LaTeXMLMath a symbol as in chapter 13.3 of LaTeXMLCite .
It consists of LaTeXMLMath elements .
Let LaTeXMLMath .
Each odd LaTeXMLMath contributes to the symbol the interval of length LaTeXMLMath LaTeXMLEquation and each even LaTeXMLMath contributes to the symbol the LaTeXMLMath numbers LaTeXMLEquation each repeated twice in weakly increasing order .
Consider the elementary LaTeXMLMath -group with basis consisting of elements LaTeXMLMath , one for each odd LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLMath is the subgroup of this group consisting of elements expressible as a sum of an even number of basis elements ( this is because we are working in the special orthogonal group and not in the full orthogonal group ) .
Representations LaTeXMLMath of LaTeXMLMath are thus specified by their values ( of LaTeXMLMath ) on the LaTeXMLMath ’ s .
The representations LaTeXMLMath for which LaTeXMLMath is non-zero are those with the property that LaTeXMLEquation .
If LaTeXMLMath is even , LaTeXMLMath may have the value LaTeXMLMath or LaTeXMLMath on LaTeXMLMath .
To determine the set LaTeXMLMath it is necessary to compute the LaTeXMLMath -value of each non-zero representation LaTeXMLMath ( see chapter 11.4 of LaTeXMLCite ) .
We have to convert between two different notations for symbols .
This is a bit of a pain , but the work is greatly simplified since we are only interested in when the LaTeXMLMath -value of LaTeXMLMath equals the LaTeXMLMath -value of LaTeXMLMath .
We find that the set LaTeXMLMath consists of the following LaTeXMLMath .
Above we listed those odd LaTeXMLMath with LaTeXMLMath odd ( which was denoted LaTeXMLMath above ) in decreasing order as LaTeXMLMath ( note that LaTeXMLMath must be odd ) .
If LaTeXMLMath , we must have LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be odd and LaTeXMLMath even .
If LaTeXMLMath for some LaTeXMLMath , then LaTeXMLMath may take values LaTeXMLMath on LaTeXMLMath .
If , on the other hand , LaTeXMLMath , then LaTeXMLMath .
Since LaTeXMLMath is abelian , each element forms its own conjugacy class .
We need to relate our two descriptions of conjugacy classes in LaTeXMLMath .
Given LaTeXMLMath write LaTeXMLMath where LaTeXMLMath ( the usual classical description ) .
Then set LaTeXMLMath and define LaTeXMLMath by LaTeXMLMath .
Then LaTeXMLMath exactly corresponds to the conjugacy class of LaTeXMLMath ( see LaTeXMLCite ) .
Using our previous notation of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , we see that LaTeXMLMath consists of those conjugacy classes where LaTeXMLMath when LaTeXMLMath is even and LaTeXMLMath is even when LaTeXMLMath is odd .
By proposition LaTeXMLRef this is exactly the condition that the conjugacy class LaTeXMLMath belongs to LaTeXMLMath , showing that LaTeXMLMath .
Finally , the statement that LaTeXMLMath is constant on the cosets on LaTeXMLMath is an easy computation similar to the one done to find which classes belonged to LaTeXMLMath .
A similar proof holds in type LaTeXMLMath and LaTeXMLMath which we omit .
∎ For completeness we record LaTeXMLMath in the classical groups .
In the exceptional groups , LaTeXMLMath is listed in the last section .
In type LaTeXMLMath , LaTeXMLMath is trivial .
In the other classical types , LaTeXMLMath is an elementary LaTeXMLMath -group .
Assuming LaTeXMLMath is the special orthogonal group in types LaTeXMLMath and LaTeXMLMath and LaTeXMLMath is the symplectic group in type LaTeXMLMath , we describe a subgroup LaTeXMLMath of LaTeXMLMath which maps bijectively onto LaTeXMLMath .
In types LaTeXMLMath and LaTeXMLMath , consider the subgroup LaTeXMLMath of LaTeXMLMath consisting of all elements expressible as a sum of an even number of LaTeXMLMath where LaTeXMLMath is equal to some LaTeXMLMath for LaTeXMLMath odd or LaTeXMLMath for LaTeXMLMath even .
In type LaTeXMLMath , consider the subgroup LaTeXMLMath of LaTeXMLMath consisting of all elements expressible as a sum of LaTeXMLMath where LaTeXMLMath is equal to some LaTeXMLMath for LaTeXMLMath odd or LaTeXMLMath for LaTeXMLMath even .
Then LaTeXMLMath .
In other words , the LaTeXMLMath in question correspond , in type LaTeXMLMath , to corners of the Young diagram of LaTeXMLMath which have odd length and odd height ( LaTeXMLMath is odd and LaTeXMLMath is odd ) ; in type LaTeXMLMath , to corners of the Young diagram of LaTeXMLMath which have odd length and even height ( LaTeXMLMath is odd and LaTeXMLMath is even ) ; in type LaTeXMLMath , to corners of the Young diagram of LaTeXMLMath which have even length and even height ( LaTeXMLMath is even and LaTeXMLMath is even ) .
Let LaTeXMLMath be the Langlands dual group of LaTeXMLMath with Lie algebra LaTeXMLMath ( in other words , the root data of LaTeXMLMath and LaTeXMLMath are dual ) .
Denote by LaTeXMLMath the set of nilpotent orbits in LaTeXMLMath and LaTeXMLMath the set of special nilpotent orbits in LaTeXMLMath .
There is a natural order-preserving ( and dimension-preserving ) bijection between LaTeXMLMath and LaTeXMLMath .
Therefore we can also view Lusztig-Spaltenstein duality as a map LaTeXMLMath .
It is in this context that Barbasch and Vogan have given a representation-theoretic description of Lusztig-Spaltenstein duality using primitive ideals LaTeXMLCite .
To distinguish between the duality which stays within LaTeXMLMath and the version which passes to the dual group , we use the notation LaTeXMLMath to mean the dual orbit within LaTeXMLMath and we use the notation LaTeXMLMath to mean the dual orbit within LaTeXMLMath .
This is consistent with later ( and earlier ) notation .
We will now define a map LaTeXMLMath with the property that it extends Lusztig-Spaltenstein duality .
It will turn out that LaTeXMLMath is surjective and depends only on the image of LaTeXMLMath in the canonical quotient LaTeXMLMath .
Moreover , if LaTeXMLMath , then we will have LaTeXMLMath where LaTeXMLMath denotes the flag variety of LaTeXMLMath .
The definition of LaTeXMLMath is as follows .
Given LaTeXMLMath , pick LaTeXMLMath and a semisimple element LaTeXMLMath whose image in LaTeXMLMath belongs to LaTeXMLMath .
The group LaTeXMLMath has as its Lie algebra LaTeXMLMath and clearly LaTeXMLMath .
Let LaTeXMLMath denote the orbit in LaTeXMLMath through LaTeXMLMath under the action of LaTeXMLMath .
Applying Lusztig-Spaltenstein duality LaTeXMLMath with respect to LaTeXMLMath ( we assume here that LaTeXMLMath stays within LaTeXMLMath and we don ’ t pass to the dual group ) , we obtain the orbit LaTeXMLMath .
At this point we simply apply Lusztig ’ s map from Chapter 13 of LaTeXMLCite .
But first we recall the following result due to Joseph and proved uniformly in LaTeXMLCite .
Every Weyl group representation of the form LaTeXMLMath possesses property ( B ) of LaTeXMLCite with respect to LaTeXMLMath .
That is , the LaTeXMLMath -value of LaTeXMLMath coincides with LaTeXMLMath and moreover the multiplicity of LaTeXMLMath in the harmonic polynomials of degree LaTeXMLMath on a Cartan subalgebra of LaTeXMLMath is exactly one .
The Springer correspondence for LaTeXMLMath produces an irreducible representation LaTeXMLMath ( always non-zero ) of LaTeXMLMath , the Weyl group of LaTeXMLMath .
The second part of the statement of property ( B ) means that we can apply the operation of truncated induction to LaTeXMLMath ( see LaTeXMLCite ) .
So let LaTeXMLMath be the representation of LaTeXMLMath obtained by truncated induction of LaTeXMLMath from LaTeXMLMath to LaTeXMLMath .
Then LaTeXMLMath has the property that LaTeXMLMath from the definition of truncated induction .
Now consider LaTeXMLMath as a representation of LaTeXMLMath , the Weyl group of LaTeXMLMath ( since LaTeXMLMath is isomorphic to LaTeXMLMath via the involution which interchanges long and short roots ) .
Lusztig has observed that LaTeXMLMath is always of the form LaTeXMLMath for some nilpotent element LaTeXMLMath in LaTeXMLMath ( this will be re-verified explicitly below ) .
We then define LaTeXMLMath to be the orbit in LaTeXMLMath through LaTeXMLMath .
It is clear from the definition ( assuming it is well-defined ) and the first part of the statement of property ( B ) ( applied twice , once in LaTeXMLMath and once in LaTeXMLMath ) that LaTeXMLMath as promised above .
The duality map is well-defined ; that is , it is independent of the choices made for LaTeXMLMath and LaTeXMLMath .
We show that the representation LaTeXMLMath constructed above is independent of LaTeXMLMath ( it is clearly independent of LaTeXMLMath , since Springer representations only depend on the orbit through LaTeXMLMath ) .
One of the main properties of Lusztig-Spaltenstein duality is that if LaTeXMLMath where LaTeXMLMath is a Levi subalgebra of LaTeXMLMath , then LaTeXMLEquation .
The notation on the right-side is Lusztig-Spaltenstein induction .
According to LaTeXMLCite and the validity of property ( B ) , one therefore has that LaTeXMLEquation where LaTeXMLMath is the Weyl group of LaTeXMLMath .
Now let LaTeXMLMath be a maximal torus in LaTeXMLMath .
Then LaTeXMLMath is a Levi subalgebra of LaTeXMLMath and LaTeXMLMath .
Moreover LaTeXMLMath is of the form LaTeXMLMath for a semisimple element LaTeXMLMath and the image of LaTeXMLMath in LaTeXMLMath necessarily belongs to LaTeXMLMath .
By the transitivity of truncated induction applied to the sequence LaTeXMLMath , we see that the representation LaTeXMLMath is the same whether we work with respect to LaTeXMLMath or LaTeXMLMath , i.e .
whether we work with LaTeXMLMath or LaTeXMLMath .
The main result of LaTeXMLCite is that the pair LaTeXMLMath ( up to simultaneous conjugation by elements in LaTeXMLMath ) is determined by ( and determines in the case when LaTeXMLMath is of adjoint type ) the conjugacy class LaTeXMLMath .
In other words , LaTeXMLMath ( up to simultaneous conjugation by elements in LaTeXMLMath ) is independent of the choice of LaTeXMLMath and thus the construction of LaTeXMLMath is independent of the choices made for LaTeXMLMath and LaTeXMLMath .
∎ Proposition LaTeXMLRef now follows from the above proof either by invoking property ( B ) in LaTeXMLMath , or more simply by applying the dimension formula LaTeXMLMath for induction of orbits from the Levi subalgebra LaTeXMLMath of LaTeXMLMath .
We now calculate the duality map for the classical groups not of type LaTeXMLMath ( there is nothing new here in type LaTeXMLMath ) .
First , we assign to a partition LaTeXMLMath ( which may or may not correspond to a nilpotent orbit in that classical group ) a representation LaTeXMLMath of the Weyl group of that group .
When LaTeXMLMath corresponds to an actual nilpotent orbit , LaTeXMLMath is just the Springer representation LaTeXMLMath for the orbit with trivial local system .
Let LaTeXMLMath where LaTeXMLMath is even in types LaTeXMLMath and odd in type LaTeXMLMath .
Form the dual partition LaTeXMLMath .
Separate the parts of LaTeXMLMath into its odd parts LaTeXMLMath and its even parts LaTeXMLMath .
We then associate to LaTeXMLMath the representation LaTeXMLMath where LaTeXMLMath is the Weyl group of the subsystem of type LaTeXMLEquation and LaTeXMLMath is the sign representation of LaTeXMLMath .
In type LaTeXMLMath we are thinking of the representation as truncated induction in type LaTeXMLMath followed by restriction to LaTeXMLMath which is known to produce an irreducible representation as long as LaTeXMLMath is not very even ( we ignore the case where LaTeXMLMath is very even since we never need to consider it in what follows ) .
When LaTeXMLMath corresponds to a nilpotent orbit in the appropriate classical group , LaTeXMLMath is the Springer representation of this orbit with the trivial local system .
This is shown by following Lusztig ’ s version of Shoji ’ s algorithm ( see chapter 13.3 of LaTeXMLCite ) .
∎ For two partitions LaTeXMLMath , LaTeXMLMath with the same LaTeXMLMath -collapse where LaTeXMLMath or LaTeXMLMath , we have LaTeXMLMath as representations of LaTeXMLMath .
We give the proof in type LaTeXMLMath , the other types being similar .
Assume LaTeXMLMath , but LaTeXMLMath .
List all the even parts of LaTeXMLMath in decreasing order as LaTeXMLMath .
Then there exists an LaTeXMLMath such that LaTeXMLMath since LaTeXMLMath .
Let LaTeXMLMath be the partition obtained from LaTeXMLMath by replacing the part LaTeXMLMath by LaTeXMLMath and the part LaTeXMLMath by LaTeXMLMath and leaving all other parts of LaTeXMLMath unchanged .
This is the basic LaTeXMLMath -collapsing move ( see LaTeXMLCite ) and it suffices to show that LaTeXMLMath .
The dual partition LaTeXMLMath equals LaTeXMLMath except that LaTeXMLMath and LaTeXMLMath .
Now write LaTeXMLMath where LaTeXMLMath and LaTeXMLMath consist respectively of the odd and even parts of LaTeXMLMath .
Given a part LaTeXMLMath of LaTeXMLMath , recall that we are calling LaTeXMLMath the position of LaTeXMLMath in LaTeXMLMath .
Now LaTeXMLMath will occur as a part of either LaTeXMLMath or LaTeXMLMath .
We refer to its position in whichever partition LaTeXMLMath or LaTeXMLMath it occurs in as its parity position .
We use similar language for LaTeXMLMath .
Assume first that LaTeXMLMath is even .
Then it is possible to show that the parity position of LaTeXMLMath is odd and the parity position of LaTeXMLMath is also odd .
These parts will each contribute a root subsystem of type LaTeXMLMath in the definition of LaTeXMLMath or LaTeXMLMath , respectively .
On the other hand , if LaTeXMLMath is odd the parity position of these parts is both even and they will each contribute a root subsystem of type LaTeXMLMath in the definition of LaTeXMLMath or LaTeXMLMath , respectively .
Since there are no even parts of LaTeXMLMath between LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath for LaTeXMLMath .
Since the same equalities hold for LaTeXMLMath , we see that these parts ( which come in pairs ) contribute the same terms to LaTeXMLMath and LaTeXMLMath .
Finally consider LaTeXMLMath .
If LaTeXMLMath is even , then the parity position of both LaTeXMLMath and LaTeXMLMath is even , so they contribute a term LaTeXMLMath in the definition of LaTeXMLMath or LaTeXMLMath , respectively .
And if LaTeXMLMath is odd , then they both have odd parity position and their contribution is LaTeXMLMath .
Hence a basic collapsing move does not affect the attached representation and the result is proved .
∎ The formulas for the LaTeXMLMath -values in the classical groups are a consequence of the previous propositions and the validity of property ( B ) .
We now explain the bijection between LaTeXMLMath and LaTeXMLMath explicitly in terms of partitions ( see LaTeXMLCite , LaTeXMLCite ) .
Given LaTeXMLMath , let LaTeXMLMath and set LaTeXMLMath .
Then LaTeXMLMath .
Similarly given LaTeXMLMath , let LaTeXMLMath and LaTeXMLMath and set LaTeXMLMath .
Then LaTeXMLMath and moreover , LaTeXMLMath .
For LaTeXMLMath , we have LaTeXMLMath ; in particular , if LaTeXMLMath is special , LaTeXMLMath .
Similarly for LaTeXMLMath , we have LaTeXMLMath ; in particular , if LaTeXMLMath is special , LaTeXMLMath .
In what follows we identify representations of LaTeXMLMath and LaTeXMLMath via the isomorphism of these two Coxeter groups which corresponds to interchanging long and short roots .
For LaTeXMLMath we have LaTeXMLMath , where the left side of the identity is computed in type LaTeXMLMath and the right in type LaTeXMLMath .
For LaTeXMLMath we have LaTeXMLMath , where the left side of the identity is computed in type LaTeXMLMath and the right in type LaTeXMLMath .
We prove the first isomorphism .
We noted above that LaTeXMLMath and so in type LaTeXMLMath , LaTeXMLMath since by the previous proposition we can omit the LaTeXMLMath -collapse on the right side .
To prove the desired identity we must study the odd and even parts of LaTeXMLMath ( in type LaTeXMLMath ) and LaTeXMLMath ( in type LaTeXMLMath ) .
These partitions are the same except that the latter has an extra part equal to LaTeXMLMath at the end .
Now because LaTeXMLMath the definition of the subsystem in equation ( LaTeXMLRef ) for LaTeXMLMath in type LaTeXMLMath and for LaTeXMLMath in type LaTeXMLMath coincide ( with the extra part in LaTeXMLMath playing no role at all ) .
We now prove the second isomorphism .
The first isomorphism implies that LaTeXMLMath , where the left side is in type LaTeXMLMath and the right in type LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLMath and thus LaTeXMLMath .
The last equality holds since in type LaTeXMLMath we can omit the LaTeXMLMath -collapse .
∎ Our duality map LaTeXMLMath sends the pair LaTeXMLMath to the orbit LaTeXMLMath according to the following recipe : LaTeXMLEquation .
Note that the case of LaTeXMLMath equal to the empty partition corresponds to Lusztig-Spaltenstein duality .
In type LaTeXMLMath if LaTeXMLMath is non-empty , our assumptions about LaTeXMLMath ensure that LaTeXMLMath is not very even .
We may choose LaTeXMLMath representing LaTeXMLMath so that LaTeXMLMath has semisimple rank equal to the rank of LaTeXMLMath .
Then LaTeXMLMath is specified by the pair of partitions LaTeXMLMath .
Our first step is to compute the Springer representation LaTeXMLMath of LaTeXMLMath associated to LaTeXMLMath .
The pair of partitions associated to LaTeXMLMath is LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
By lemmas LaTeXMLRef and LaTeXMLRef , the associated Springer representation LaTeXMLMath of LaTeXMLMath is LaTeXMLMath .
Consider this now as a representation of LaTeXMLMath in type LaTeXMLMath , LaTeXMLMath in type LaTeXMLMath , and LaTeXMLMath in type LaTeXMLMath ( there is no change in type LaTeXMLMath ) .
Then by applying lemma LaTeXMLRef in types LaTeXMLMath and LaTeXMLMath and lemma LaTeXMLRef again in type LaTeXMLMath , this representation can be described as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
These representations possess property ( B ) as they possess property ( B ) in each simple component .
Hence we can apply truncated induction up to LaTeXMLMath .
Then by transitivity of induction we claim that we arrive at the representation LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where the first is a representation of LaTeXMLMath , the second of LaTeXMLMath , and the third of LaTeXMLMath .
In type LaTeXMLMath , we use the fact that LaTeXMLMath belongs to LaTeXMLMath .
Therefore in all types if the multiplicity of LaTeXMLMath in LaTeXMLMath is odd , then the multiplicity of LaTeXMLMath in LaTeXMLMath or LaTeXMLMath , respectively , is even .
Then the validity of equation ( LaTeXMLRef ) follows from the definition of LaTeXMLMath in equation ( LaTeXMLRef ) .
The proof is completed by applying lemma LaTeXMLRef in LaTeXMLMath .
∎ Our duality map is surjective .
We verified this case-by-case ( Lusztig already did this in his work with his original map although the details are not recorded ) .
In fact , we will try to exhibit canonical elements LaTeXMLMath of LaTeXMLMath which map bijectively to LaTeXMLMath .
These are denoted by a star ( * ) in the tables for the exceptional groups and we now explain their construction in the classical groups .
Assume LaTeXMLMath is of type LaTeXMLMath where LaTeXMLMath and LaTeXMLMath where LaTeXMLMath is even or odd depending on LaTeXMLMath .
Consider LaTeXMLMath .
We ask whether LaTeXMLMath belongs to LaTeXMLMath , or LaTeXMLMath depending on whether LaTeXMLMath is LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , respectively .
In other words , we ask whether LaTeXMLMath is special ( note the funny situation in type LaTeXMLMath ) .
If not , we may uniquely write LaTeXMLMath where LaTeXMLMath is distinguished of type LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath , and where LaTeXMLMath belongs to LaTeXMLMath , or LaTeXMLMath , for some LaTeXMLMath , depending on whether LaTeXMLMath is LaTeXMLMath , LaTeXMLMath , or LaTeXMLMath .
We now show that LaTeXMLMath has the property that LaTeXMLMath .
This is because LaTeXMLMath and the process of forming LaTeXMLMath amounts to taking LaTeXMLMath and diminishing some of its parts ; however , parts not congruent to LaTeXMLMath will only be diminished by an even number , so the resulting partition remains of the same type as LaTeXMLMath .
Hence LaTeXMLMath for some LaTeXMLMath .
It follows that LaTeXMLMath since LaTeXMLMath belonged to LaTeXMLMath , or LaTeXMLMath depending on LaTeXMLMath .
It is also true that LaTeXMLMath is itself special in types LaTeXMLMath and LaTeXMLMath .
We can now define LaTeXMLMath .
In type LaTeXMLMath we set LaTeXMLMath ; in type LaTeXMLMath we set LaTeXMLMath ; and in type LaTeXMLMath we set LaTeXMLMath .
Now because LaTeXMLMath is special ( and LaTeXMLMath is special in types LaTeXMLMath and LaTeXMLMath ) , we have LaTeXMLMath in type LaTeXMLMath ; LaTeXMLMath in type LaTeXMLMath ; and LaTeXMLMath in type LaTeXMLMath .
The second equality in type LaTeXMLMath follows since applying Lusztig-Spaltenstein duality twice ( to LaTeXMLMath in this case ) is the identity on special orbits .
Thus in all types LaTeXMLMath where the last equality holds since LaTeXMLMath .
We conclude in all types that LaTeXMLMath has the property that LaTeXMLMath .
∎ These canonical elements LaTeXMLMath that we have listed which surject onto LaTeXMLMath have the property that LaTeXMLMath is always special .
In fact , all orbits LaTeXMLMath of LaTeXMLMath in the same special piece are affiliated with the same special orbit LaTeXMLMath of LaTeXMLMath and in fact LaTeXMLMath ( nilpotent orbits are in the same special piece exactly when their dual orbits are the same ) .
Hence for each orbit LaTeXMLMath in LaTeXMLMath we get a conjugacy class in LaTeXMLMath where LaTeXMLMath .
This should be the same conjugacy class that Lusztig attaches to orbits in LaTeXMLCite under the identification LaTeXMLMath .
Let LaTeXMLMath be conjugacy classes in LaTeXMLMath whose image in LaTeXMLMath coincide , then LaTeXMLMath .
Again we checked this on a case-by-case basis .
In the exceptional groups , this amounts to a quick glance at the tables which follow .
In the classical groups , it requires attention to the computations in the proof of theorem LaTeXMLRef , together with the explicit description of the canonical quotient .
We omit the details .
∎ To compute the duality in the exceptional groups we used knowledge of Lusztig-Spaltenstein duality and the Springer correspondence ( see LaTeXMLCite ) .
Furthermore , we computed truncated induction by using the induce/restrict tables of Alvis LaTeXMLCite .
We have listed only those orbits with non-trivial component groups when LaTeXMLMath is of adjoint type since these are the only orbits for which we are saying something new .
The stars ( * ) refer to the ( putatively ) canonical pair LaTeXMLMath which maps to a given orbit in the dual Lie algebra as is done in proposition LaTeXMLRef for the classical groups .
We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields , i.e. , the logarithm of the ( regularized ) determinant of a Dirac operator on flat LaTeXMLMath twisted by generalized Yang-Mills fields .
According to physics folklore , the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action .
We present an explicit computation proving this fact , generalized to the chiral case .
We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension .
MSC-class : 81T13 ; 58J42 ; 35S99 April 8 , 2001 Generalized Yang-Mills actions from Dirac operator determinants Edwin Langmann Theoretical Physics , Royal Institute of Technology , SE-10044 Stockholm , Sweden Determinants of differential operators arise as ( exponentials of ) effective actions in quantum field theory .
The precise definition and investigation of such objects is an interesting and challenging mathematical problem which has lead to an active and fruitful interplay between mathematics and physics .
In this paper we compute the logarithmic divergent part , LaTeXMLMath , of the logarithm of the regularized determinant for Dirac operators LaTeXMLMath describing Dirac fermions coupled to a generalized Yang-Mills field LaTeXMLMath on four dimensional spacetime .
For simplicity we assume spacetime to be flat LaTeXMLMath with Euclidean signature and the natural spin structure .
The Yang-Mills fields we consider contain , besides the vector part LaTeXMLMath , also a chiral ( axial ) part LaTeXMLMath ( for precise definitions see Eq .
( LaTeXMLRef ) ff .
below ) ; we write LaTeXMLMath .
Our definition of LaTeXMLMath is motivated by physical considerations and will be explained further below .
To indicate the mathematical significance of our calculation , we note that LaTeXMLMath is ( essentially ) the noncommutative residue LaTeXMLCite of the logarithm of LaTeXMLMath ( see Eq .
( LaTeXMLRef ) for the precise statement ) .
A main motivation for this work is to present a computation method for effective fermion actions which at the same time is mathematically rigorous , close to standard Feynman diagram computations in quantum field theory ( see , e.g. , LaTeXMLCite ) , and simple to use .
We believe that this method is a useful alternative to other methods like the LaTeXMLMath -function regularizations or the heat kernel expansions ( see , e.g. , LaTeXMLCite ) .
We therefore made some effort to present this method in a self-contained way , in the hope that this is useful also for readers who are mainly interested in learning how to compute effective actions .
We now discuss our computation method ( parts of this method were used previously by us in LaTeXMLCite ) .
We regard the Dirac operator LaTeXMLMath as a PSDO ( pseudo-differential operator ) on a Hilbert space of square-integrable functions on LaTeXMLMath .
Our starting point is the following definition for the regularized effective fermion action , LaTeXMLEquation where LaTeXMLMath is a real parameter which has the physical interpretation of a fermion mass , and LaTeXMLMath is a positive regularization parameter which we call UV ( ultra-violet ) cutoff .
The role of the non-zero and complex parameter LaTeXMLMath is two-fold .
Firstly , it makes the argument of the logarithm dimensionless , and secondly , setting LaTeXMLMath avoids possible ambiguities due to the branch cuts of the logarithm which otherwise can arise .
LaTeXMLMath , and this is a useful check .
This definition above has three ingredients .
Firstly , a definition of the LaTeXMLMath of an operator LaTeXMLMath as an integral of the resolvent of LaTeXMLMath .
Secondly , some basic facts about PSDO which imply a simple and powerful formula for the symbol of the resolvent of the Dirac operator LaTeXMLMath .
And thirdly , a definition of a regularized Hilbert space trace LaTeXMLMath ( where removing the regularization corresponds to the limit LaTeXMLMath ) .
Combining these ingredients we obtain an expansion of LaTeXMLMath in local and quasi-gauge invariant polynomials of decreasing scaling dimension .
We find LaTeXMLEquation and this provides our definition of LaTeXMLMath .
Our results for LaTeXMLMath and LaTeXMLMath will be presented in the next Section .
We shall also demonstrate on our way that LaTeXMLMath is proportional to the noncommutative residue LaTeXMLCite of the logarithm of the Dirac operator LaTeXMLMath , LaTeXMLEquation .
The logarithm of the regularized trace of the determinant of the Dirac operator can then be defined as LaTeXMLEquation where LaTeXMLMath is the renormalized trace which we will define , and we will provide all mathematical tools necessary for computing LaTeXMLMath explicitly .
We note that our computation method is closely related to methods which have been used in the physics literature for a long time ( see , e.g. , LaTeXMLCite ) .
The regularization we use is simple and close to how regularizations are often done in Feynman diagram computations , i.e. , by introducing a sharp UV cutoff ( see Eq .
( LaTeXMLRef ) ) .
We believe , however , that we can offer some improvements in detail which make computations easier , more transparent in structure , but nevertheless such that each step can be easily justified with mathematical rigor .
We now discuss some motivation for our computation from a quantum field theory point of view .
As was known already to Schwinger for the Abelian case , the effective action of fermions coupled to a Yang-Mills field LaTeXMLMath ( i.e. , LaTeXMLMath ) contains a logarithmic divergence , LaTeXMLMath , and LaTeXMLMath ( for LaTeXMLMath ) is proportional to the usual Yang-Mills action LaTeXMLEquation ( see , e.g. , LaTeXMLCite , Eq .
( 12.123 ) where LaTeXMLMath corresponds to LaTeXMLMath ) .
This is important since it implies that a change in the cutoff in the gauge theory , LaTeXMLMath , leads to a finite change of the effective fermion action which can be absorbed by changing the Yang-Mills coupled constant , LaTeXMLMath .
The logarithmic dependence of the Yang-Mills coupling constant on the UV cutoff is remarkable and distinguishes four spacetime dimensions from all others .
Our computation is closely related to more recent ideas which have lead to a deeper geometric understanding of the standard model of elementary particle physics ( including Higgs sector ) .
This approach is based on Connes ’ NCG ( noncommutative geometry ; textbooks on this subject are , e.g. , LaTeXMLCite ) .
One important ingredient of this approach is to define a generalized Dirac operator LaTeXMLMath , and this Dirac operator not only specifies the fermion part of the action of the model but also the Yang-Mills part LaTeXMLMath : there is a definition of LaTeXMLMath in terms of LaTeXMLMath ( see LaTeXMLCite and references therein ) .
Our discussion above suggests a simple physical interpretation of this spectral action principle LaTeXMLCite : the logarithmic divergence of the fermion effective action is potentially ‘ dangerous ’ since it can make the model ambiguous : there is no preferred choice for the cut-off , and changing it generates a term proportional to LaTeXMLMath .
However , the fact that LaTeXMLMath is proportional to the Yang-Mills action resolves this problem for the standard ( purely vector ) Yang-Mills theory on LaTeXMLMath , as discussed above .
It therefore is natural to require that the Yang-Mills action is proportional to the logarithmic divergent part of the fermion effective action in any gauge theory models .
In particular this suggests the following definition of the generalized ( vector and chiral ) Yang-Mills action in terms of the generalized Dirac operators LaTeXMLMath , LaTeXMLEquation ( for one fermion flavor LaTeXMLMath ) .
Eq .
( LaTeXMLRef ) shows that for flat Euclidean space LaTeXMLMath , this definition is equivalent to the one given in LaTeXMLCite .
We conjecture that this is true for other four dimensional spin manifolds as well .
The plan of this paper is as follows .
We summarize our notation and results in Section 2 .
Section 3 contains a summary of the mathematical prerequisites , i.e. , the three ingredients of our method mentioned above .
The computations of LaTeXMLMath is presented in Section 4 with some computation details deferred to Appendix B .
We conclude with some remarks in Section 5 .
Appendix A contains some discussion on regularized traces and the noncommutative residue .
Notation : We write LaTeXMLMath for the complex LaTeXMLMath matrices and LaTeXMLMath for the invertible matrices in LaTeXMLMath .
We sometimes write LaTeXMLMath or LaTeXMLMath for the identity operator on a vector space LaTeXMLMath but often abuse notation and do not distinguish between LaTeXMLMath and LaTeXMLMath for complex numbers .
For LaTeXMLMath , LaTeXMLMath vector spaces and LaTeXMLMath an operator on LaTeXMLMath , we often use the same symbols LaTeXMLMath to also denote the corresponding operator LaTeXMLMath and LaTeXMLMath on LaTeXMLMath and LaTeXMLMath , respectively .
The real part of a complex number LaTeXMLMath is denoted as LaTeXMLMath .
For simplicity we assume spacetime LaTeXMLMath with Euclidean signature ( the extension of our calculation to other four–dimensional spin manifolds should be possible using symbol calculus of pseudo–differential operators LaTeXMLCite ) .
We consider the Hilbert LaTeXMLEquation which has the physical interpretation as space of the 1–particle states of the fermions .
We also introduce the space LaTeXMLMath of functions in LaTeXMLMath which are smooth ( i.e. , LaTeXMLMath ) and LaTeXMLMath ; LaTeXMLMath is a convenient dense domain in LaTeXMLMath .
The Dirac operators of interest to us are of the form LaTeXMLEquation where LaTeXMLMath ( repeated indices LaTeXMLMath are summed over ; LaTeXMLMath ) , with LaTeXMLMath and LaTeXMLMath the Dirac spin matrices acting on LaTeXMLMath and obeying LaTeXMLEquation for LaTeXMLMath , where LaTeXMLMath is the metric tensor , and LaTeXMLEquation as usual ( for the convenience of the reader , explicit formulas for these matrices are given in Appendix A.1 ) .
For simplicity we assume that the functions LaTeXMLMath and LaTeXMLMath LaTeXMLMath are regular , i.e. , they are LaTeXMLMath and vanish like LaTeXMLMath , for some LaTeXMLMath , as LaTeXMLMath ( the latter condition is to ensure that integrals of regular functions over LaTeXMLMath absolutely converge ) .
In particular , the free Dirac operator is defined by the differential operator LaTeXMLEquation .
We define the gauge group LaTeXMLMath as follows .
Let LaTeXMLMath be the group of all invertible matrices in LaTeXMLMath .
Then LaTeXMLMath is the group of all LaTeXMLMath -valued functions LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath is a regular function .
Note that one can write LaTeXMLEquation where LaTeXMLMath are the chiral components of the gauge field .
This representation shows that it is natural to consider two kinds of gauge transformations , LaTeXMLEquation .
For LaTeXMLMath we denote these as vector gauge transformation , otherwise as chiral gauge transformation .
Note that LaTeXMLMath in Eq .
( LaTeXMLRef ) is well-defined on the domain LaTeXMLMath , and we find it useful to distinguish this formally self-adjoint differential operator in notation from the corresponding self-adjoint extension on LaTeXMLMath which we denote as LaTeXMLMath , i.e. , LaTeXMLMath for all LaTeXMLMath .
We also write LaTeXMLEquation where LaTeXMLMath is the free Dirac operator ( i.e. , self-adjoint extension of LaTeXMLMath ) and LaTeXMLMath the operator defined by multiplication with the generalized Yang-Mills field LaTeXMLEquation .
We will compute the fermion effective action LaTeXMLMath defined in Eq .
( LaTeXMLRef ) , and we will show that it can be expanded as in Eq .
( LaTeXMLRef ) .
As discussed , LaTeXMLMath is a Hilbert space trace with an ultraviolet ( UV ) cutoff LaTeXMLMath , and LaTeXMLMath is an arbitrary , in general complex , parameter makes the argument of the logarithm dimensionless .
Moreover , the real ( positive or negative ) parameter LaTeXMLMath corresponds to a fermion mass and serves as an infrared ( IR ) regulator in our computation .
Our main result is an explicit formula for LaTeXMLMath .
Proposition : The logarithmic divergent piece LaTeXMLMath of the logarithm of the ( regularized ) determinant of the Dirac operator LaTeXMLMath equals LaTeXMLEquation where LaTeXMLMath is the usual matrix trace in LaTeXMLMath and LaTeXMLMath LaTeXMLEquation is the curvature associated with the chiral component LaTeXMLMath of the Yang-Mills field .
( Proof in Section LaTeXMLRef with some details deferred to Appendix B . )
For LaTeXMLMath ( no chiral field ) we obtain LaTeXMLEquation with LaTeXMLEquation which is the standard Yang-Mills action .
Note that LaTeXMLMath with LaTeXMLEquation the covariant derivative , and similarly , LaTeXMLEquation .
It is important to note that for LaTeXMLMath , LaTeXMLMath in Eq .
( LaTeXMLRef ) this is manifestly invariant under all gauge transformations Eq .
( LaTeXMLRef ) .
For LaTeXMLMath , there is also a mass term LaTeXMLMath for the chiral gauge field which is only invariant under vector gauge transformations , i.e. , only the transformations Eq .
( LaTeXMLRef ) with LaTeXMLMath .
The parameter in front of this term is fixed by the fermion mass .
There is no similar term for the vector gauge field ( note that such a term would spoil vector gauge invariance ) .
It is interesting to note that the result of our computation in Section LaTeXMLRef suggests that for manifolds LaTeXMLMath with boundary LaTeXMLMath , LaTeXMLMath has an additional contribution LaTeXMLEquation with LaTeXMLEquation .
This is a boundary term ( by Stokes ’ s theorem ) .
Note that this term is also invariant under vector gauge transformations , and it vanishes if the axial Yang-Mills field LaTeXMLMath is zero .
It is also worth noting that , as a by-product , we also obtain the explicit expression for the quadratic divergent part of the effective action , LaTeXMLEquation .
In contrast to LaTeXMLMath this term is not gauge invariant ( as already mentioned , the term LaTeXMLMath spoils vector gauge invariance ) !
This highlights the fact that the regularization procedure we use it not manifestly gauge invariant but only quasi-gauge invariant .
It shows that the vector gauge invariance of our result for LaTeXMLMath somewhat remarkable .
It is also interesting to note that for LaTeXMLMath , LaTeXMLMath .
In this Section we collect the mathematical prerequisites for our computation .
We will explain the three ingredients for our method : Firstly , a definition of the logarithm of operators LaTeXMLMath in terms of an integral of the resolvent of LaTeXMLMath .
Secondly , a few basic definitions for PSDO which imply a simple and elegant formula for the symbol of the resolvent of Dirac operators LaTeXMLMath .
And finally , a definition of a regularized Hilbert space trace LaTeXMLMath ( corresponding to introducing an UV cutoff LaTeXMLMath ) .
In the next Section we will put these ingredients together and obtain an expansion of the effective action as described in the Introduction .
1 .
The logarithm of operators .
Let LaTeXMLMath be a bounded operator on a Hilbert space LaTeXMLMath with norm less then one .
Then ( LaTeXMLMath is the identity operator ) LaTeXMLEquation as can be seen by a Taylor expansion , LaTeXMLEquation interchanging summation and integration , and using the geometric series .
We take this as a motivation to define LaTeXMLEquation where LaTeXMLMath is a some complex number .
This representation of the logarithm as integral of a resolvent will be convenient for us since there is a simple formula for the resolvent of ( generalized ) Dirac operators , as discussed below .
2.A .
Pseudo–differential operators .
Generalities .
We summarize some basic facts about pseudo–differential operators ( PSDO ) on LaTeXMLMath ( a discussion for general manifolds can be found , e.g. , in LaTeXMLCite ) .
We consider PSDO LaTeXMLMath on LaTeXMLMath which can be represented by their symbol LaTeXMLMath , i.e. , a LaTeXMLMath –valued functions on phase space LaTeXMLMath defined such that LaTeXMLCite LaTeXMLEquation for all LaTeXMLMath ( matrix multiplication is understood ; LaTeXMLMath ) .
In particular , LaTeXMLMath and LaTeXMLMath are PSDO with symbols LaTeXMLEquation .
Note that Eq .
( LaTeXMLRef ) implies the following equation which encodes the product of operators in terms of their symbols , LaTeXMLEquation .
We will encounter PSDO LaTeXMLMath which allow an asymptotic expansion LaTeXMLEquation where LaTeXMLMath is homogeneous of degree LaTeXMLMath in LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath and goes to zero like LaTeXMLMath for LaTeXMLMath ( LaTeXMLMath ) .
We write LaTeXMLEquation for all integers LaTeXMLMath .
Eq .
( LaTeXMLRef ) implies , LaTeXMLEquation .
This equation allows to determine the asymptotic expansions of LaTeXMLMath and LaTeXMLMath from the ones of LaTeXMLMath and LaTeXMLMath .
2.B .
The symbol of the resolvent .
Eq .
( LaTeXMLRef ) expresses LaTeXMLMath as an integral of resolvents of the Dirac operator LaTeXMLMath , i.e. , of operators LaTeXMLMath with LaTeXMLMath complex numbers .
We will therefore need the symbol of such a resolvent .
To determine this we note that LaTeXMLEquation .
We then could use Eq .
( LaTeXMLRef ) to find the expansion for LaTeXMLMath .
We now present a useful result summarizing this expansion in a simple formula .
Lemma : The following holds for all LaTeXMLMath , LaTeXMLEquation .
Remark : The proper interpretation of this equation is as follows , LaTeXMLEquation where the differential operators LaTeXMLMath in LaTeXMLMath act to the right on the functions LaTeXMLMath according to the Leibniz rule .
We note that we will need this equation only for LaTeXMLMath .
Proof of the Lemma : One can check Eq .
( LaTeXMLRef ) by using Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , taking LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , and inserting Eq .
( LaTeXMLRef ) .
A simpler argument avoiding tedious expansions is as follows : Note that by definition , LaTeXMLMath for all LaTeXMLMath , thus LaTeXMLEquation .
LaTeXMLEquation where we used Eq .
( LaTeXMLRef ) and the Leibniz rule .
Replacing LaTeXMLMath in this equation by LaTeXMLMath , we see that this is equivalent to Eq .
( LaTeXMLRef ) .
( Note that this argument implies the interpretation of Eq .
( LaTeXMLRef ) as given above ! )
LaTeXMLMath Remark : We believe that our expansion in powers of the differential operator LaTeXMLMath is very natural for at least two reasons .
Firstly , since under a vector gauge transformation , LaTeXMLMath , such an expansion is close to being manifestly gauge invariant ( we will discuss this point in more detail below ) .
Secondly , it is natural from the point of view of power counting : in contrast to an expansion in LaTeXMLMath , the LaTeXMLMath -th order term in our expansion includes precisely those local polynomials LaTeXMLMath in LaTeXMLMath and LaTeXMLMath ( and derivatives thereof ) which all have the same scaling behavior LaTeXMLMath under LaTeXMLMath .
Remark : Loosely speaking , PSDO are useful since they allow to interpolate between Fourier- and position space : generically in quantum theory one deals with operators LaTeXMLMath on some Hilbert space of LaTeXMLMath -functions on LaTeXMLMath which are a sum of a free part LaTeXMLMath diagonal in Fourier space , LaTeXMLMath denotes the Fourier transform .
LaTeXMLMath , and a potential term LaTeXMLMath diagonal in position space , LaTeXMLMath .
The symbol of LaTeXMLMath is then simply the sum of LaTeXMLMath and LaTeXMLMath , which is an attractive feature .
The price one has to pay is that the symbol of ( ‘ nice ’ ) functions LaTeXMLMath of LaTeXMLMath are somewhat complicated : in a first approximation , LaTeXMLMath , but there are correction terms LaTeXMLMath depending on derivatives .
The Lemma above is a special case of the following formula , LaTeXMLEquation nicely summarizing the systematic derivative expansion of functions of LaTeXMLMath .
3 .
Regularized traces and the noncommutative residue .
We now define the regularized trace which we will use .
We first note that due to our technical assumptions on the gauge fields all operators LaTeXMLMath which we will encounter are PSDO which have symbols LaTeXMLMath which go at least like LaTeXMLMath , some LaTeXMLMath , for fixed LaTeXMLMath and LaTeXMLMath , and are finite for finite LaTeXMLMath .
Thus LaTeXMLEquation where LaTeXMLMath is the full matrix trace , LaTeXMLMath in LaTeXMLMath and the trace LaTeXMLMath in LaTeXMLMath .
is well-defined for LaTeXMLMath , and this defines a regularized Hilbert space trace : If LaTeXMLMath is a trace–class operator then LaTeXMLMath has a well–defined limit LaTeXMLMath which is equal to the Hilbert space trace of LaTeXMLMath LaTeXMLCite .
More generally one can consider PSDO LaTeXMLMath for which LaTeXMLMath can be expanded as LaTeXMLEquation .
LaTeXMLEquation with LaTeXMLMath some non–negative integer .
We recall that the noncommutative residue LaTeXMLCite of a PSDO LaTeXMLMath with an asymptotic expansion as in Eq .
( LaTeXMLRef ) can be defined as ( see , e.g. , Eq .
( 2.7 ) in Ref .
LaTeXMLCite ) LaTeXMLEquation and for PSDO LaTeXMLMath as above , LaTeXMLEquation i.e. , the residue is equal , up to a constant , to the logarithmic divergent part of the regularized trace of LaTeXMLMath .
( An elementary proof of this latter fact is outlined in Appendix A . )
Remark : In our definition Eq .
( LaTeXMLRef ) of LaTeXMLMath we use a sharp cutoff , i.e. , LaTeXMLEquation where LaTeXMLMath equals the Heaviside step function LaTeXMLMath .
In principle one could define a regularized trace using Eq .
( LaTeXMLRef ) and choosing any non-negative , piece-wise smooth , function LaTeXMLMath which vanishes exponentially fast for LaTeXMLMath and is such that LaTeXMLMath .
For example , the choice LaTeXMLMath would correspond to the standard heat kernel regularization .
We will show in Appendix A that LaTeXMLMath is in fact independent of LaTeXMLMath .
Using any such regularization one can define the renormalized trace as the finite part of the regularized trace , LaTeXMLEquation but this is not quite independent of the regularization : as also discussed in Appendix A , changing the regularization function LaTeXMLMath amounts to changing LaTeXMLEquation with some constant LaTeXMLMath depending on LaTeXMLMath and LaTeXMLMath : the logarithmic divergent piece accounts for the regularization dependence of the renormalized trace , and this is the reason for our interest in it , as discussed in the Introduction .
Remark : We note Eq .
( LaTeXMLRef ) is equivalent to LaTeXMLEquation ( using the spectral theorem for self-adjoint operators ) .
This naturally extends the definition of LaTeXMLMath from PSDO to a large class of operators on LaTeXMLMath .
More generally , one could change the regularization by changing LaTeXMLMath in the definition of LaTeXMLMath , for some fixed Yang-Mills field LaTeXMLMath .
One can show that this would change LaTeXMLMath by a term proportional to LaTeXMLMath ( see , e.g. , Eq .
( 1.6 ) in LaTeXMLCite ) .
It would be interesting to explore this possibility in more detail .
In this Section we present the explicit computation of the effective fermion action and thus prove the proposition in Section LaTeXMLRef .
Our computation amounts to a quasi-gauge invariant gradient expansion , which is essentially an expansion in powers of the UV cutoff LaTeXMLMath .
This allows us to extract , in a simple manner , the quadratic and logarithmic divergent pieces which is what we are interested in .
1 .
Quasi-gauge covariant expansion .
We write LaTeXMLEquation where LaTeXMLMath is obtained by computing the symbol of the operator LaTeXMLMath as explained in the last Section , i.e. , LaTeXMLEquation .
LaTeXMLEquation we used Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) , introduced the convenient short-hand notion , LaTeXMLEquation and changed integration variables , LaTeXMLMath .
The LaTeXMLMath on the r.h.s .
of Eq .
( LaTeXMLRef ) is the symbol of the identity operator .
As explained in more detail below , LaTeXMLMath here is to be regarded as a differential operators acting on LaTeXMLMath .
It is straightforward to expand the integrand in this equation in powers of LaTeXMLMath , LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation is a remainder term .
In the following we find it convenient to use the short-hand notation and write LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
We then define LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
This allows us to write LaTeXMLEquation here and in the following , LaTeXMLMath is short for LaTeXMLMath .
The following Lemma simplifies the computation significantly : it implies that the LaTeXMLMath for odd integers LaTeXMLMath all vanish , and that an series expansion in the mass LaTeXMLMath only has non-zero even powers .
Lemma : The coefficients LaTeXMLMath in Eq .
( LaTeXMLRef ) are non-zero only for even integers LaTeXMLMath , and they are invariant under LaTeXMLMath , i.e. , they are independent of the sign of the mass .
( Proof in Appendix B . )
Remark : We now can explain why we denote our expansion quasi-gauge invariant .
This is because the operators LaTeXMLMath transform gauge covariantly under a vector gauge transformation LaTeXMLMath , LaTeXMLMath .
This implies that the differential operators defined in Eq .
( LaTeXMLRef ) all are gauge invariant .
However , the action is a polynomial which is obtained by applying these differentiation operators to LaTeXMLMath ( cf .
Eqs .
( LaTeXMLRef ) ) using Leibniz rule and LaTeXMLMath , e.g. , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation etc .
The result will be gauge invariant only if the differential operator in Eq .
( LaTeXMLRef ) is already a polynomial .
This happens , e.g. , if the differential operators LaTeXMLMath only appear in combinations LaTeXMLMath and LaTeXMLMath .
This is not obvious .
However , we will see below that this happens for the terms leading to logarithmic divergent part .
2 .
Expansion in powers of the UV cutoff .
We now show that our expansion above is essentially an expansion in powers of the UV cutoff LaTeXMLMath .
Our computation can be simplified by the following argument ( this argument is refined and justified in detail in Appendix B ) .
As mentioned , LaTeXMLMath serves as a particular IR cutoff for momentum integrals .
We expect that our result is independent of the precise form the IR regularization .
Thus we use instead the following , simpler one : we set LaTeXMLMath in LaTeXMLMath but restrict integrations over LaTeXMLMath to LaTeXMLMath .
We stress that we use this simplification in the main text only to ease our presentation , and that it is appropriate only for computing the diverging contributions to the regularized determinant : the computation of the finite part should be done with the method explained in Appendix B .
Below we shall see that this simplified procedure gives a IR regularization provided we also set LaTeXMLMath ( a justification of this can be also found in the Appendix B ) .
Using then LaTeXMLEquation and rescaling LaTeXMLMath we see that LaTeXMLMath in Eq .
( LaTeXMLRef ) becomes LaTeXMLMath to indicate that these numbers are obtained with a simplified IR regularization .
LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation we used LaTeXMLMath with LaTeXMLEquation the angular average ( i.e. , integration over the unit sphere in LaTeXMLMath ) .
We now see that LaTeXMLMath is needed to specify how to treat the singularity in the LaTeXMLMath -integral .
These LaTeXMLMath -integrals are then finite ( see Eqs .
( LaTeXMLRef ) and ( LaTeXMLRef ) below ) .
The result we get is independent of LaTeXMLMath , as expected .
It shows explicitly that our expansion leads to an expansion of the action in powers of LaTeXMLMath .
We are interested in LaTeXMLMath .
In this limit , LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath : the former terms are divergent in the UV ( i.e. , for LaTeXMLMath , the latter in the IR ( i.e. , for LaTeXMLMath ) .
It is precisely the ‘ boundary case ’ LaTeXMLMath which gives rise to the logarithmic divergence .
This result is obtained with the simplified IR treatment is correct only in leading order in LaTeXMLMath .
In Appendix B we show how to do the computation without this simplification , and that LaTeXMLEquation showing that the simplified IR treatment gives the correct result for the diverging terms for all LaTeXMLMath but LaTeXMLMath .
For LaTeXMLMath there are corrections LaTeXMLMath which contribute to LaTeXMLMath and which we therefore have to compute exactly .
3 .
Computation of diverging parts of the effective action .
We now proceed to compute the coefficients LaTeXMLMath Eq .
( LaTeXMLRef ) for those terms we are interested in , i.e. , for LaTeXMLMath .
Using Eq .
( LaTeXMLRef ) this is straightforward : one only needs to evaluate the integrals LaTeXMLEquation the angular averages LaTeXMLMath , and traces of products of Dirac matrices .
The integrals in Eq .
( LaTeXMLRef ) are ( cf. , e.g. , Eq .
3.251 ( 11 . )
in LaTeXMLCite ) LaTeXMLEquation where LaTeXMLMath .
We only need LaTeXMLEquation .
LaTeXMLEquation The computation of the traces of Dirac matrices is simplified using the following relations LaTeXMLEquation .
LaTeXMLEquation which follow from Eq .
( LaTeXMLRef ) .
We also need LaTeXMLEquation .
LaTeXMLEquation and that the angular average for a product of an odd number of components LaTeXMLMath is zero .
Moreover , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is the completely antisymmetric symbol with LaTeXMLMath .
Note that LaTeXMLMath always .
It is now easy to see that LaTeXMLMath thus LaTeXMLEquation .
The simplest non-zero terms are for LaTeXMLMath .
Combining the formulas given above it is easy to see that LaTeXMLEquation .
LaTeXMLEquation Thus LaTeXMLEquation .
This is a gauge invariant differential operator .
When acting on LaTeXMLMath ( cf .
Eq .
( LaTeXMLRef ) ) we obtain the quadratic divergent part of the effective action Eq .
( LaTeXMLRef ) which is not gauge invariant .
As mentioned , LaTeXMLMath is only the leading order contribution to LaTeXMLMath .
A more careful computation without the simplified IR regularization gives ( see Appendix B ) , LaTeXMLEquation where ‘ LaTeXMLMath ’ are terms which remain finite for LaTeXMLMath .
We see that the subleading term which was missed by the naive IR regularization contributes to LaTeXMLMath .
As discussed in Section LaTeXMLRef , this term is gauge invariant .
We now turn to the case LaTeXMLMath which leads to the logarithmic divergence .
All relations needed to compute the LaTeXMLMath were listed above .
The result can be written as follows LaTeXMLEquation where LaTeXMLMath .
The numbers LaTeXMLMath are all given in Table LaTeXMLRef .
( We have checked this result extensively using the symbolic programming language MAPLE . )
We note that the numbers LaTeXMLMath ( LaTeXMLMath ) all are real ( purely imaginary ) and non-zero only if an even ( odd ) number of the LaTeXMLMath equal LaTeXMLMath .
Combining these results we find LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation with the coefficients given in Table LaTeXMLRef .
LaTeXMLMath is a sum of 19 non-zero terms .
We now claim that it is possible to write LaTeXMLMath where LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation with LaTeXMLMath given in Eq .
( LaTeXMLRef ) .
Similarly , LaTeXMLEquation ( The proof of Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) are straightforward calculation which we skip . )
We see that , LaTeXMLMath equals LaTeXMLMath with LaTeXMLMath defined in Eq .
( LaTeXMLRef ) .
The remaining terms are linear combinations of commutators !
Using the cyclicity of the matrix trace we thus obtain LaTeXMLEquation .
This implies Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) and completes our computation .
LaTeXMLMath Remark : Note that LaTeXMLMath and LaTeXMLMath are not differential operators but polynomials ( i.e. , there are no terms LaTeXMLMath ) .
This implies that both these terms are gauge covariant which , as we believe , is remarkable .
The regularization which we used was simple but not manifestly gauge invariant .
For the result computed in this paper the latter property is irrelevant : since the logarithmic divergence is regularization dependent one can compute it using any regularization .
However , we believe that our method is useful even for computing the finite part of the effective action , i.e. , LaTeXMLMath in Eq .
( LaTeXMLRef ) .
We stress again that the simplified IR regularization used in the main text is not appropriate in this computation but the formulas given in Appendix B should be used .
We conjecture that LaTeXMLMath computed in this way is gauge invariant .
As mentioned in the Remark at the end of Section LaTeXMLRef , we defined a renormalized trace LaTeXMLMath using the free Dirac operator LaTeXMLMath .
More general we could use the Dirac operator LaTeXMLMath with some fixed non-trivial Yang-Mills field LaTeXMLMath .
In particular , we expect that the standard LaTeXMLMath -function regularization of the logarithm of the determinant of LaTeXMLMath should be identical with LaTeXMLEquation were the regularization function is LaTeXMLMath .
The latter definition has the advantage that it is manifestly gauge invariant , but it seems less easy to use for explicit computations as ours .
It is natural to expect that the difference between the latter definition and LaTeXMLMath in Eq .
( LaTeXMLRef ) is also proportional to LaTeXMLMath .
Effective action computations are used in many applications of quantum field theory .
We believe that the methods which we presented should be useful in other such contexts as well .
Acknowledgment : I would like to thank A. Laptev , J. Mickelsson , S. Paycha , F. Scheck and K. Wojciechowski for their interest and helpful discussions and S. Paycha for comments on the manuscript .
This work was supported by the Swedish Natural Science Research Council ( NFR ) .
In this Appendix we outline elementary proofs of some facts about regularized traces stated in the main text .
The logarithmic divergence .
We compute the regularized trace in Eq .
( LaTeXMLRef ) for an operator LaTeXMLMath with a symbol allowing for an asymptotic expansion as in Eq .
( LaTeXMLRef ) .
It is easy to see that the contribution of LaTeXMLMath to LaTeXMLMath is LaTeXMLEquation where we used the homogeneity of LaTeXMLMath .
Changing variables , LaTeXMLMath , and comparing with Eq .
( LaTeXMLRef ) we see that for all LaTeXMLMath , LaTeXMLEquation with LaTeXMLMath constants depending on LaTeXMLMath .
For LaTeXMLMath the computation above does not make sense ( the constant LaTeXMLMath diverges ) , but we can compute LaTeXMLMath as follows .
We first subtract from the symbol of LaTeXMLMath the diverging part which we already accounted for and define , LaTeXMLEquation .
Eq .
( LaTeXMLRef ) then suggests that LaTeXMLEquation .
Computing this using L ’ Hospital ’ s rule we obtain LaTeXMLEquation .
LaTeXMLEquation Changing variables etc .
as above and using LaTeXMLMath ( independent of LaTeXMLMath ! )
we obtain LaTeXMLEquation .
Recalling Eq .
( LaTeXMLRef ) we obtain Eq .
( LaTeXMLRef ) .
LaTeXMLMath Renormalized traces .
It is obvious that changing the regularization functions LaTeXMLMath for some fixed LaTeXMLMath , amounts to changing LaTeXMLMath , and thus changes LaTeXMLMath .
Thus ( LaTeXMLRef ) is obvious for this special case .
For more general changes LaTeXMLMath of the regularization function , Eq .
( LaTeXMLRef ) can be shown using LaTeXMLEquation which follows from our discussion above .
In this Appendix we present some details concerning our computations discussed in the main text .
In particular , we give explicit formulas for the Dirac matrices , and we also show show how to compute the structure constants LaTeXMLMath in Eq .
( LaTeXMLRef ) exactly , i.e. , without the simplified IR regularization .
We also prove the Lemma in Section 4.1 and Eq .
( LaTeXMLRef ) , and we give some details about the computation yielding Eq .
( LaTeXMLRef ) .
A convenient representation for the Dirac matrices is as follows , LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the LaTeXMLMath unit- and zero matrices and LaTeXMLEquation the Pauli sigma matrices as usual .
We start by rewriting the LaTeXMLMath in a convenient form .
We define LaTeXMLEquation which are orthogonal projections , LaTeXMLMath and LaTeXMLMath , satisfying LaTeXMLMath .
We then can write LaTeXMLEquation which we insert LaTeXMLMath times in Eq .
( LaTeXMLRef ) , LaTeXMLEquation .
LaTeXMLEquation We thus obtain LaTeXMLEquation with LaTeXMLEquation and LaTeXMLEquation .
Rescaling LaTeXMLMath and introducing LaTeXMLMath yields LaTeXMLEquation .
Proof of the Lemma in Section 4.1 .
We note that LaTeXMLEquation is invariant under LaTeXMLMath ( since the latter transformation amounts to the variable change LaTeXMLMath in the integral Eq .
( LaTeXMLRef ) defining the angular average ) .
Moreover , the cyclicity of trace and LaTeXMLMath implies that LaTeXMLMath does not change if we replace all LaTeXMLMath and LaTeXMLMath by LaTeXMLMath and LaTeXMLMath , respectively .
Using LaTeXMLMath and LaTeXMLMath we obtain LaTeXMLMath and using LaTeXMLMath this proves that LaTeXMLMath — and thus LaTeXMLMath in Eq .
( LaTeXMLRef ) — is non-zero only for even LaTeXMLMath .
From Eq .
( LaTeXMLRef ) it is obvious that LaTeXMLMath corresponds to to LaTeXMLMath , and thus LaTeXMLMath implies that we can replace LaTeXMLMath by LaTeXMLMath in Eq .
( LaTeXMLRef ) .
We can write the latter as a sum of the terms which are even and odd under the change the sign of the mass LaTeXMLMath .
A simple change of variables shows that LaTeXMLMath -integrals in the odd term LaTeXMLEquation can be written as follows ( LaTeXMLMath even ) , LaTeXMLEquation plus the same integral but with LaTeXMLMath and LaTeXMLMath interchanged .
The latter integrals can be computed using Cauchy ’ s theorem : the poles of the integrand are in LaTeXMLMath and LaTeXMLMath and thus both always in the same half of the complex LaTeXMLMath -plane ( upper or lower , depending on the sign of LaTeXMLMath ) .
Computing the integral by closing the integration path in the half plane where the integrand is analytic ( which is possible since the integrand vanishes like LaTeXMLMath for LaTeXMLMath ) one sees that the integral is zero .
This implies LaTeXMLMath .
LaTeXMLMath Proof of Eq .
( LaTeXMLRef ) : Our discussion above implies that we can replace LaTeXMLMath in Eq .
( LaTeXMLRef ) by LaTeXMLMath .
We are interested in the terms which diverge for LaTeXMLMath .
To isolate them it is convenient to determine LaTeXMLMath .
We thus compute LaTeXMLEquation where we introduced the functions LaTeXMLEquation .
Note that the functions LaTeXMLMath are well-defined for all real LaTeXMLMath , have a finite limit LaTeXMLMath as LaTeXMLMath , and they have series expansions in LaTeXMLMath .
LaTeXMLMath .
It is easy to see that with the simplified regularization used in the main text we can obtain a formula for LaTeXMLMath as in Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) but with LaTeXMLMath replaced by LaTeXMLEquation .
We thus get LaTeXMLEquation which proves Eq .
( LaTeXMLRef ) .
LaTeXMLMath Remark : We now can explain the reason for our choice LaTeXMLMath in the main text : this yields a regularization specifying the otherwise undefined integrals LaTeXMLMath , and from Eq .
( LaTeXMLRef ) it is clear that this is the regularization yielding a result identical with the one obtained with the proper regularization , up to lower order terms .
Computation of LaTeXMLMath .
For LaTeXMLMath we need compute LaTeXMLMath in Eq .
( LaTeXMLRef ) exactly , using the formulas given above .
Similarly as explained in the main text we compute ( cf .
Eq .
( LaTeXMLRef ) ) LaTeXMLEquation .
Moreover , the integrals defined in Eq .
( LaTeXMLRef ) for LaTeXMLMath and LaTeXMLMath are , LaTeXMLEquation .
LaTeXMLEquation and with Eqs .
( LaTeXMLRef ) , ( LaTeXMLRef ) and ( LaTeXMLRef ) we can compute LaTeXMLMath .
Straightforward computations yield LaTeXMLEquation .
LaTeXMLEquation and with Eq .
( LaTeXMLRef ) we obtain Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) .
Let LaTeXMLMath be an associative simple ( central ) superalgebra over LaTeXMLMath and LaTeXMLMath an invariant linear functional on it ( trace ) .
Let LaTeXMLMath be an antiautomorphism of LaTeXMLMath such that LaTeXMLMath , where LaTeXMLMath is the parity of LaTeXMLMath , and let LaTeXMLMath .
Then LaTeXMLMath admits a nondegenerate supersymmetric invariant bilinear form LaTeXMLMath .
For LaTeXMLMath , where LaTeXMLMath is any maximal ideal of LaTeXMLMath , Leites and I have constructed orthogonal basis in LaTeXMLMath whose elements turned out to be , essentially , Chebyshev ( Hahn ) polynomials in one discrete variable .
Here I take LaTeXMLMath for any maximal ideal LaTeXMLMath and apply a similar procedure .
As a result we obtain either Hahn polynomials over LaTeXMLMath , where LaTeXMLMath , or a particular case of Meixner polynomials , or — when LaTeXMLMath — dual Hahn polynomials of even degree , or their ( hopefully , new ) analogs of odd degree .
Observe that the nondegenerate bilinear forms we consider for orthogonality are , as a rule , not sign definite .
822001 LaTeXMLRef – LaTeXMLRef Article 2001A Sergeev Enveloping Superalgebra LaTeXMLMath and Orthogonal Polynomials in Discrete Indeterminate A SERGEEV ( Correspondence ) D Leites , Department of Mathematics , University of Stockholm Roslagsv .
101 , Kräftriket hus 6 , SE-106 91 , Stockholm , Sweden E-mail : mleites @ matematik.su.se On leave of absence from Balakovo Institute of Technique of Technology and Control , Branch of Saratov Technical University , Balakovo , Saratov Region , Russia Received May 6 , 2000 ; Accepted November 18 , 2000 Classically , orthogonal polynomials were considered with respect to a sign definite bilinear form .
Lately we encounter the growth of interest to the study of orthogonal polynomials relative an arbitrary ( but still symmetric and nondegenerate ) form , cf .
LaTeXMLCite and references therein .
In these approaches , however , the bilinear forms are introduced “ by hands ” and the differential or difference equations the orthogonal polynomials satisfy are of high degree .
We would like to point out that traces and supertraces on associative algebras and superalgebras are natural sources of bilinear symmetric forms which are seldom sign-definite .
The Lie structure on the algebras obtained from these associative algebras and superalgebras is more adapted to the study of orthogonal polynomials .
In particular , the eigenvalue problem for the Casimir operator — the quadratic element of the center with respect to the Lie structure — naturally provides with a 2nd degree difference equation for the polynomials orthogonal relative the above ( super ) traces .
Let LaTeXMLMath be represented as LaTeXMLMath subject to relations LaTeXMLEquation .
The quadratic Casimir operator of LaTeXMLMath LaTeXMLEquation lies in the center of LaTeXMLMath .
Let LaTeXMLMath be the two-sided ideal in the associative algebra LaTeXMLMath generated by LaTeXMLMath .
It turns out that the associative algebra LaTeXMLMath is simple for LaTeXMLMath , otherwise LaTeXMLMath contains an ideal such that the quotient is isomorphic to the matrix algebra LaTeXMLMath .
Set ( LaTeXMLCite ) LaTeXMLEquation .
Clearly , LaTeXMLMath .
As associative algebra , LaTeXMLMath is generated by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath subject to relations LaTeXMLEquation and one more relation for integer values of LaTeXMLMath : LaTeXMLEquation .
It is also known that LaTeXMLMath possesses an antiautomorphism LaTeXMLMath given on generators by the formula LaTeXMLEquation .
In LaTeXMLCite we have shown that on LaTeXMLMath there exists a unique , up to a constant factor , nontrivial linear functional LaTeXMLMath , which for positive integer LaTeXMLMath ’ s is the usual trace and which satisfies LaTeXMLMath .
By means of this functional we define an invariant symmetric bilinear form LaTeXMLMath on LaTeXMLMath , by setting LaTeXMLMath .
The form LaTeXMLMath is nondegerate and symmetric .
Now , consider LaTeXMLMath as an LaTeXMLMath -module with respect to the adjoint representation .
We have LaTeXMLEquation where LaTeXMLMath is the irreducible finite dimensional LaTeXMLMath -module with highest weight LaTeXMLMath ( cf .
LaTeXMLCite ) .
Clearly , LaTeXMLMath arranges a LaTeXMLMath -grading on LaTeXMLMath , namely LaTeXMLEquation .
For any LaTeXMLMath and LaTeXMLMath set LaTeXMLEquation .
Denote : LaTeXMLEquation .
Set LaTeXMLEquation further set LaTeXMLEquation .
Theorem 1.1 LaTeXMLCite .
LaTeXMLMath for LaTeXMLMath .
For LaTeXMLMath the polynomials LaTeXMLMath are of degree LaTeXMLMath , they are orthogonal relative to the form LaTeXMLMath .
For LaTeXMLMath the polynomials LaTeXMLMath are of degree LaTeXMLMath , they are orthogonal relative to the form LaTeXMLMath .
The polynomials LaTeXMLMath satisfy the following difference equation : LaTeXMLEquation .
Explicitly we have LaTeXMLEquation where LaTeXMLEquation is a generalized hypergeometric function , LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Our goal is to generalize this theorem by replacing LaTeXMLMath with LaTeXMLMath .
The main result obtained is the union of Theorems 2.2 , 2.3 and 2.4 .
We select the following basis in LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation The defining relations ( we give only the ones with nonzero values in the right hand side ) are LaTeXMLEquation .
For convenience we add also the following corollaries LaTeXMLEquation .
These relations immediately imply that LaTeXMLMath is generated , as associative superalgebra , by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Set LaTeXMLEquation .
Lemma 2.1 .
The element LaTeXMLMath belongs to the supercenter of LaTeXMLMath , i.e. , it commutes with the even elements and anticommutes with the odd ones .
The centralizer of the Cartan subalgebra of LaTeXMLMath in LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath is the quadratic Casimir element of LaTeXMLMath .
LaTeXMLMath is the quadratic Casimir element of LaTeXMLMath .
Proof .
i ) ( the proof of the fact that LaTeXMLMath is similar ) : LaTeXMLEquation ii ) It is easy to verify that LaTeXMLEquation .
Now observe that any element of the centralizer is a linear combination of the elements LaTeXMLMath for LaTeXMLMath .
Headings iii ) and iv ) are subject to a similar direct verification .
A theorem of Pinczon .
Pinczon LaTeXMLCite described the maximal two-sided ideals of LaTeXMLMath .
Let us formulate his results in a form convenient to us .
Theorem 2.1 .
Every maximal two-sided ideal of LaTeXMLMath is of the form LaTeXMLMath , where LaTeXMLMath is : generated by LaTeXMLMath for LaTeXMLMath ; the kernel of the finite dimensional representation with highest weight LaTeXMLMath for LaTeXMLMath ; generated by LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath .
Then If LaTeXMLMath , then LaTeXMLMath is generated by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath subject to the relations LaTeXMLEquation .
The superalgebra LaTeXMLMath is isomorphic to the Weil algebra LaTeXMLMath considered as superalgebra when generators are considered to be odd ( recall that the defining relations in LaTeXMLMath are LaTeXMLMath .
LaTeXMLMath for LaTeXMLMath .
On the structure of LaTeXMLMath .
Recall that an antiautomorphism of super algebra LaTeXMLMath is an even linear map LaTeXMLMath for LaTeXMLMath such that LaTeXMLMath .
Define an antiautomorphism of LaTeXMLMath by setting LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
Clearly , this antiautomorphism induces an antiautomorphism of LaTeXMLMath for every LaTeXMLMath .
Later on , I will show that on LaTeXMLMath exists a unique , up to a scalar factor , nontrivial invariant linear functional — the supertrace str .
So the form LaTeXMLMath determines an invariant supersymmetric bilinear form on LaTeXMLMath .
The lack of nonzero two-sided ideals guarantees the non-degeneracy of the form .
For LaTeXMLMath the algebra LaTeXMLMath possesses a LaTeXMLMath -grading of the form LaTeXMLMath , where LaTeXMLEquation .
For LaTeXMLMath and LaTeXMLMath set LaTeXMLEquation .
Recall that LaTeXMLMath and introduce a LaTeXMLMath -grading in LaTeXMLMath by setting LaTeXMLMath , LaTeXMLMath .
Now it is not difficult to verify that for any LaTeXMLMath there exists a basis LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath if LaTeXMLMath .
In what follows the basis elements with such properties will be called orthogonal polynomials in LaTeXMLMath and LaTeXMLMath .
Set also LaTeXMLEquation and extend the action of the operators LaTeXMLMath and LaTeXMLMath onto LaTeXMLMath by setting LaTeXMLEquation .
Besides , set LaTeXMLEquation and extent the action of the operators LaTeXMLMath and LaTeXMLMath onto LaTeXMLMath by formulas similar to ( LaTeXMLRef ) .
Define now polynomials LaTeXMLMath for LaTeXMLMath by setting LaTeXMLEquation and LaTeXMLEquation .
Theorem 2.2 .
LaTeXMLMath for LaTeXMLMath .
Polynomials LaTeXMLMath are orthogonal relative to the form LaTeXMLMath ; the polynomial LaTeXMLMath is a degree LaTeXMLMath polynomial in LaTeXMLMath and LaTeXMLMath .
Polynomial LaTeXMLMath satisfies the difference equation LaTeXMLEquation .
Polynomial LaTeXMLMath satisfies the difference equation LaTeXMLEquation .
Polynomials LaTeXMLMath are orthogonal relative to the form LaTeXMLMath ; it is a degree LaTeXMLMath polynomial in LaTeXMLMath and LaTeXMLMath .
Polynomial LaTeXMLMath satisfies the difference equation LaTeXMLEquation .
Polynomial LaTeXMLMath satisfies the difference equation LaTeXMLEquation .
Polynomial LaTeXMLMath can be expressed via Hahn polynomials with parameter LaTeXMLMath , namely LaTeXMLEquation .
Polynomials LaTeXMLMath can be expressed via Hahn polynomials with parameter LaTeXMLMath , namely LaTeXMLEquation .
Another theorem of Pinczon .
Recall ( Theorem 2.1 , B ) ) that the superalgebra LaTeXMLMath is isomorphic to the Weyl algebra LaTeXMLMath considered as super algebra with generators LaTeXMLMath , LaTeXMLMath and relations LaTeXMLMath .
The corresponding isomorphism LaTeXMLMath is given by the formulas LaTeXMLEquation .
As is easy to verify , LaTeXMLMath .
There is a LaTeXMLMath -grading of LaTeXMLMath such that ( having identified LaTeXMLMath with LaTeXMLMath ) LaTeXMLEquation .
For LaTeXMLMath and LaTeXMLMath set LaTeXMLEquation where LaTeXMLMath is the bilinear form on LaTeXMLMath defined in Section 4 .
Now , for LaTeXMLMath define the polynomials LaTeXMLMath from the equations LaTeXMLEquation .
Let us endow the algebra LaTeXMLMath with a grading by setting LaTeXMLMath .
Theorem 2.3 .
LaTeXMLMath for LaTeXMLMath .
LaTeXMLMath are polynomials in LaTeXMLMath and LaTeXMLMath of degree LaTeXMLMath orthogonal with respect to the form LaTeXMLMath .
LaTeXMLMath are polynomials in LaTeXMLMath and LaTeXMLMath of degree LaTeXMLMath orthogonal with respect to the form LaTeXMLMath .
LaTeXMLMath satisfies the difference equation LaTeXMLEquation .
LaTeXMLMath satisfies the difference equation LaTeXMLEquation .
The polynomials LaTeXMLMath can be expressed via Meixner polynomials : LaTeXMLEquation where LaTeXMLEquation .
The polynomials LaTeXMLMath can be expressed via Meixner polynomials : LaTeXMLEquation .
The case of LaTeXMLMath for LaTeXMLMath In this case LaTeXMLMath and the image of LaTeXMLMath under the natural homomorphism LaTeXMLMath is a polynomial in LaTeXMLMath .
Therefore , having applied the arguments after Theorem 2.1 ( on the structure of LaTeXMLMath ) we obtain an orthogonal basis distinct from the basis of orthogonal polynomials .
To construct orthogonal polynomials , set LaTeXMLEquation .
It is easy to verify that LaTeXMLEquation .
Relations ( LaTeXMLRef ) mean that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath generate in LaTeXMLMath a subalgebra isomorphic to LaTeXMLMath considered as a superalgebra such that LaTeXMLMath , LaTeXMLMath .
Observe also that the images of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath in LaTeXMLMath generate LaTeXMLMath and are subject to relations LaTeXMLEquation ( we have identified LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath with their images in LaTeXMLMath ) .
The superalgebra LaTeXMLMath is LaTeXMLMath -graded LaTeXMLMath , where LaTeXMLEquation .
Being a matrix superalgebra , LaTeXMLMath possesses an antiautomorphism , the supertransposition , which in terms of the generators is given by the formula LaTeXMLEquation .
The supertrace gives rise to a bilinear form LaTeXMLMath on LaTeXMLMath .
For LaTeXMLMath and LaTeXMLMath define the bilinear forms LaTeXMLEquation .
Further on , for LaTeXMLMath set LaTeXMLEquation .
For LaTeXMLMath set LaTeXMLEquation and define : LaTeXMLEquation .
Further on , set LaTeXMLEquation and LaTeXMLEquation .
Theorem 2.4 .
LaTeXMLMath for LaTeXMLMath .
LaTeXMLMath are orthogonal polynomials of degree LaTeXMLMath with respect to the form LaTeXMLMath .
Polynomials LaTeXMLMath satisfy the difference equation LaTeXMLEquation .
Polynomials LaTeXMLMath satisfy the difference equation LaTeXMLEquation .
Polynomials LaTeXMLMath are orthogonal with respect to the form LaTeXMLMath .
Polynomials LaTeXMLMath are of degree LaTeXMLMath and satisfy the following relations LaTeXMLEquation .
Polynomials LaTeXMLMath are of degree LaTeXMLMath and satisfy the following relations LaTeXMLEquation .
Polynomials LaTeXMLMath satisfy the difference equation LaTeXMLEquation .
Polynomials LaTeXMLMath can be expressed via the dual Hahn polynomials : LaTeXMLEquation .
Lemma 3.1 .
Let LaTeXMLMath be an associative superalgebra generated by a set LaTeXMLMath .
Denote by LaTeXMLMath the set of linear combinations of the form LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath .
Proof .
Let us apply the identity LaTeXMLCite LaTeXMLEquation where LaTeXMLMath .
Namely , let LaTeXMLMath ; let us perform induction on LaTeXMLMath to prove that LaTeXMLMath .
For LaTeXMLMath the statement is obvious .
If LaTeXMLMath , then LaTeXMLMath , where LaTeXMLMath and due to ( LaTeXMLRef ) we have LaTeXMLEquation .
Lemma 3.2 .
Let LaTeXMLMath be an associative superalgebra and LaTeXMLMath be its antiautomorphism ( supertransposition , i.e. , it satisfies LaTeXMLMath and LaTeXMLMath ) .
Let LaTeXMLMath be an even invariant functional on LaTeXMLMath ( like supertrace , i.e. , LaTeXMLMath ) such that LaTeXMLMath for any LaTeXMLMath .
Define the bilinear form on LaTeXMLMath by setting LaTeXMLEquation .
Then LaTeXMLMath and LaTeXMLEquation .
Proof .
Observe first that LaTeXMLEquation .
Since LaTeXMLMath is even , we see that LaTeXMLMath only if LaTeXMLMath .
But in this case LaTeXMLMath LaTeXMLMath .
Further on : LaTeXMLEquation .
But LaTeXMLMath .
Therefore , LaTeXMLEquation .
Lemma 3.3 .
Set LaTeXMLMath and let LaTeXMLMath be the bilinear form as in Lemma 3.2 .
Then for LaTeXMLMath we have LaTeXMLEquation .
Proof .
It is not difficult to verify the following identities : LaTeXMLEquation .
They imply LaTeXMLEquation .
Lemma 3.4 .
On LaTeXMLMath , there exists a unique , up to a scalar factor , invariant linear functional , LaTeXMLMath .
It is uniquely determined by its restriction onto LaTeXMLMath .
To every functional LaTeXMLMath on LaTeXMLMath assign its generating function LaTeXMLMath .
Then for a constant LaTeXMLMath we have LaTeXMLEquation .
Proof .
By Theorem 2.1 the superalgebra LaTeXMLMath is generated by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath subject to relations LaTeXMLEquation .
Recall that LaTeXMLMath for LaTeXMLMath .
Then for LaTeXMLMath we have LaTeXMLEquation .
Therefore , LaTeXMLMath if LaTeXMLMath , so any trace LaTeXMLMath is only nonzero on LaTeXMLMath .
To this restriction assign the generating function LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
The following statements are easy to verify : i ) If LaTeXMLMath is an automorphism of LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , then LaTeXMLMath , where LaTeXMLMath for each LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are formal power series in LaTeXMLMath .
ii ) If LaTeXMLMath , where LaTeXMLMath , then LaTeXMLEquation iii ) If LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath , then LaTeXMLEquation .
Making use of these statements , let us calculate the generating function for the restriction of the functional LaTeXMLMath onto LaTeXMLMath .
For LaTeXMLMath we have LaTeXMLEquation .
Hence , LaTeXMLMath .
Therefore , thanks to i ) –iii ) we have LaTeXMLEquation .
For LaTeXMLMath we obtain a system : LaTeXMLEquation .
Hence , LaTeXMLEquation or , even simpler , LaTeXMLEquation .
This implies that LaTeXMLEquation .
Since LaTeXMLMath is uniquely recovered from LaTeXMLMath , we see that LaTeXMLMath is uniquely recovered by its restriction onto LaTeXMLMath .
This proves uniqueness .
Let us prove existence of LaTeXMLMath .
It suffices to prove that LaTeXMLMath .
Indeed , by Lemma 3.1 LaTeXMLEquation .
Hence , LaTeXMLEquation for any LaTeXMLMath .
But LaTeXMLMath ( recall that LaTeXMLMath , LaTeXMLMath ) , so LaTeXMLMath .
Lemma 3.5 .
Let LaTeXMLMath be a linear functional on LaTeXMLMath determined in Lemma 3.4 and normed so that LaTeXMLMath .
Define an antiautomorphism of LaTeXMLMath by setting LaTeXMLEquation .
Then LaTeXMLMath for any LaTeXMLMath ; LaTeXMLMath for any LaTeXMLMath .
Proof .
i ) Induction on LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath for each LaTeXMLMath .
For LaTeXMLMath the statement is obvious .
Let LaTeXMLMath and let for LaTeXMLMath and for LaTeXMLMath the statement be true .
Then LaTeXMLEquation ii ) Let us represent LaTeXMLMath in the form LaTeXMLMath , where LaTeXMLMath .
But we know that LaTeXMLMath .
So LaTeXMLEquation .
Hence , LaTeXMLMath and LaTeXMLMath .
Thus , LaTeXMLMath .
Therefore , we can define a bilinear form LaTeXMLMath on LaTeXMLMath .
By Lemma 3.2 we have : LaTeXMLMath and LaTeXMLEquation .
Proof of heading 1 of Theorem 2.2 .
Let LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation .
Therefore , if LaTeXMLMath , then LaTeXMLMath .
Proof of heading 2 of Theorem 2.2 .
By LaTeXMLCite ( see also LaTeXMLCite ) , there is an expansion LaTeXMLMath , where LaTeXMLMath is an irreducible highest weight module over LaTeXMLMath with even highest weight vector , LaTeXMLMath is the change of parity functor and where LaTeXMLMath is considered as LaTeXMLMath -module with respect to the adjoint representation .
It is easy to verify that LaTeXMLMath is generated by the highest weight vector LaTeXMLMath , whereas LaTeXMLMath is generated by LaTeXMLMath .
Hence , LaTeXMLMath .
Making use of Lemma 3.5 , it is not difficult to verify that LaTeXMLEquation where LaTeXMLMath is defined in Lemma 2.1 and LaTeXMLMath denotes the adjoint action .
This immediately implies that LaTeXMLMath if LaTeXMLMath and , therefore , LaTeXMLMath if LaTeXMLMath .
Let us show now that LaTeXMLMath .
( Recall again that LaTeXMLMath , LaTeXMLMath . )
It is easy to verify that LaTeXMLEquation .
This implies LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
Formulas ( LaTeXMLRef ) imply that LaTeXMLMath , hence , LaTeXMLMath .
We similarly prove that LaTeXMLMath if LaTeXMLMath and LaTeXMLMath .
Proof of heading 3 of Theorem 2.2 .
For any LaTeXMLMath set LaTeXMLEquation .
The following identities are easy to check LaTeXMLEquation .
Moreover , LaTeXMLEquation .
Let us calculate the results of the adjoint action of LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath .
From the explicit expression of LaTeXMLMath ( Lemma 2.2 ) we deduce LaTeXMLEquation .
On the other hand , if LaTeXMLMath , then LaTeXMLMath because in any irreducible LaTeXMLMath -module LaTeXMLMath acts as a scalar multiple of the parity operator LaTeXMLMath , i.e. , an operator such that LaTeXMLMath for any LaTeXMLMath .
Operator LaTeXMLMath acts on the highest weight vector of LaTeXMLMath as multiplication by LaTeXMLMath .
Observe that LaTeXMLMath if LaTeXMLMath is even and LaTeXMLMath if LaTeXMLMath is odd .
Therefore , LaTeXMLEquation .
Proof of heading 4 of Theorem 2.2 .
Let us calculate the results of the adjoint action of LaTeXMLMath on LaTeXMLMath , where LaTeXMLMath .
We obtain LaTeXMLEquation .
LaTeXMLEquation On the other hand , as in the proof of heading 3 , we see that LaTeXMLEquation .
But LaTeXMLMath and the parity of LaTeXMLMath coincides with that of the highest weight vector of LaTeXMLMath if LaTeXMLMath is even , and is opposite if LaTeXMLMath is odd ; so LaTeXMLEquation .
Proof of heading 5 of Theorem 2.2 .
Let LaTeXMLMath be an automorphism of LaTeXMLMath given on generators as follows : LaTeXMLEquation .
Let LaTeXMLMath be a functional on LaTeXMLMath defined in Lemma 3.4 .
Since LaTeXMLMath is unique , up to a scalar factor , invariant linear functional on LaTeXMLMath , it follows that LaTeXMLMath .
Hence , LaTeXMLMath and LaTeXMLMath , if LaTeXMLMath .
Therefore , LaTeXMLEquation .
But LaTeXMLMath and LaTeXMLEquation .
Proof of heading 6 and 7 of Theorem 2.2 .
Recall that LaTeXMLMath and LaTeXMLMath .
Moreover , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath span a Lie algebra isomorphic to LaTeXMLMath and the following relations hold LaTeXMLEquation .
It is easy to verify that for LaTeXMLMath we have LaTeXMLEquation .
Let us compute the result of the adjoint action of the Casimir operator LaTeXMLMath from LaTeXMLMath on LaTeXMLMath .
We have LaTeXMLEquation .
As LaTeXMLMath -module , LaTeXMLMath , where LaTeXMLMath is the irreducible ( finite dimensional ) LaTeXMLMath -module with highest weight LaTeXMLMath .
As is easy to calculate , LaTeXMLMath acts on LaTeXMLMath as multiplication by LaTeXMLMath .
We have LaTeXMLEquation .
Now , let us compute the action of LaTeXMLMath on LaTeXMLMath : LaTeXMLEquation .
Proof of heading 8 and 9 of Theorem 2.2 .
Since LaTeXMLMath , we can express the left hand side of the equation of heading 6 ) as LaTeXMLEquation .
Making the change LaTeXMLMath we reduce the above equation to the form LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Thanks to LaTeXMLCite , we know that one of the solutions of the above equation is equal to LaTeXMLMath .
( In LaTeXMLCite it is supposed that LaTeXMLMath is a positive integer and LaTeXMLMath is real , but one can clearly assume that LaTeXMLMath and LaTeXMLMath belong to any commutative ring . )
Thus , LaTeXMLEquation .
To calculate the exact value of the constant LaTeXMLMath , it suffices to compute the leading coefficient of the polynomial LaTeXMLMath .
Formula ( LaTeXMLRef ) implies that LaTeXMLEquation .
In particular , LaTeXMLEquation .
Let LaTeXMLMath , where LaTeXMLMath denotes the integer part of LaTeXMLMath .
Then ( LaTeXMLRef ) implies that LaTeXMLEquation .
Formula ( LaTeXMLRef ) implies that LaTeXMLEquation .
Since the coefficient of the leading power of LaTeXMLMath in LaTeXMLMath is equal to LaTeXMLEquation we deduce that LaTeXMLEquation which leads to formulas of heading 8 .
Similar calculations show that LaTeXMLEquation .
This leads to formulas of heading 9 .
We will stick to notations introduced after Theorem 2.2 concerning “ another theorem of Pinczon ” .
Lemma 4.1 .
On Weyl algebra LaTeXMLMath , there exists a unique up to a constant factor invariant linear functional LaTeXMLMath .
It is uniquely determined by its restriction onto LaTeXMLMath .
The generating function of LaTeXMLMath is of the form LaTeXMLEquation .
Proof .
Since LaTeXMLMath for LaTeXMLMath , we see that LaTeXMLMath for LaTeXMLMath .
So LaTeXMLMath is uniquely determined by its restrictions onto LaTeXMLMath .
Further , LaTeXMLEquation where LaTeXMLMath .
But LaTeXMLEquation .
And , since LaTeXMLMath ( here we assume that LaTeXMLMath ) , it follows that LaTeXMLMath which proves the existence of LaTeXMLMath .
Further on , LaTeXMLEquation wherefrom , as in Lemma 3.4 , we deduce that LaTeXMLEquation and the desired form of LaTeXMLMath .
Lemma 4.2 .
Define an automorphism of LaTeXMLMath by setting LaTeXMLEquation .
Then i ) LaTeXMLMath ; ii ) LaTeXMLMath , where LaTeXMLMath .
Proof is similar to that of Lemma 3.5 .
Thus , the form LaTeXMLMath is supersymmetric and invariant .
Proof of Theorem 2.3 .
Heading 1 is proved as heading 1 of Theorem 2.2 .
Heading 2 is proved as heading 2 of Theorem 2.2 with the help of decomposition LaTeXMLMath , where LaTeXMLMath is an irreducible highest weight module over LaTeXMLMath with even highest weight vector .
Heading 3 is proved as heading 5 of Theorem 2.2 with the help of automorphism ( LaTeXMLRef ) , where LaTeXMLMath and LaTeXMLMath .
The difference equations for LaTeXMLMath follows from the study of the result of application of LaTeXMLMath to LaTeXMLMath under the adjoint action of LaTeXMLMath on LaTeXMLMath and arguments similar to those from the proof of heading 3 of Theorem 2.2 .
Statements of headings 6 and 7 are results of comparison of difference equations in headings 4 and 5 with corresponding equations in LaTeXMLCite and calculation of the leading terms .
As was observed in proof after formula ( 2.13 ) , the elements LaTeXMLMath and LaTeXMLMath generate in LaTeXMLMath a subalgebra isomorphic to LaTeXMLMath , considered as a superalgebra with nontrivial odd part .
It is also convenient to consider LaTeXMLMath per se , not as a subalgebra of LaTeXMLMath .
We mean the following .
Let LaTeXMLMath with relations ( LaTeXMLRef ) .
Consider LaTeXMLMath as a superalgebra with parity given by the formula LaTeXMLMath ( hence , LaTeXMLMath ) .
Set LaTeXMLEquation we , clearly , have LaTeXMLEquation .
The Casimir operator , being even , remains the same : LaTeXMLEquation .
Therefore , LaTeXMLEquation ( Recall that LaTeXMLMath . )
Let LaTeXMLMath be a quotient of LaTeXMLMath , as in ( LaTeXMLRef ) .
Then LaTeXMLMath for any LaTeXMLMath .
Set LaTeXMLMath for any LaTeXMLMath ; in other words , LaTeXMLMath is generated by odd indeterminates LaTeXMLMath and LaTeXMLMath subject to relations LaTeXMLEquation and one more relation : LaTeXMLEquation .
In what follows we will assume that LaTeXMLMath is considered for LaTeXMLMath because all the proofs hold for such LaTeXMLMath , not only for LaTeXMLMath .
Formulas ( LaTeXMLRef ) easily imply that by setting LaTeXMLEquation we determine an antiautomorphism of superalgebra LaTeXMLMath , i.e. , LaTeXMLMath for any LaTeXMLMath .
Lemma 5.1 .
On LaTeXMLMath , there exists a unique , up to a scalar multiple , invariant linear functional LaTeXMLMath , such that LaTeXMLMath and LaTeXMLMath for any LaTeXMLMath .
The functional LaTeXMLMath is uniquely determined by its restriction onto LaTeXMLMath and its generating function is LaTeXMLEquation .
Proof .
We have LaTeXMLEquation where LaTeXMLMath for LaTeXMLMath .
Hence , LaTeXMLMath unless LaTeXMLMath .
Further on , LaTeXMLEquation and LaTeXMLMath .
Therefore , as in Lemma 3.4 , we obtain LaTeXMLEquation and LaTeXMLEquation .
The condition LaTeXMLMath implies that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath otherwise .
This proves the uniqueness .
Let us prove the existence .
By Lemma 3.1 we have LaTeXMLEquation and LaTeXMLEquation .
Hence , LaTeXMLMath .
Hence , LaTeXMLMath and LaTeXMLMath is the linear span of the images of 1 and LaTeXMLMath .
This proves the existence of LaTeXMLMath .
Let LaTeXMLMath .
Then LaTeXMLMath is invariant and LaTeXMLMath .
The uniqueness implies that LaTeXMLMath .
Lemma 5.2 .
Let LaTeXMLMath .
We have if LaTeXMLMath , then LaTeXMLMath ; if LaTeXMLMath , then LaTeXMLMath .
Proof .
i ) Determine an automorphism of LaTeXMLMath by setting LaTeXMLEquation and set LaTeXMLEquation .
Then LaTeXMLEquation .
In other words , LaTeXMLMath is an odd polynomial .
By Lemma 5.1 the generating function for LaTeXMLMath is an even one , so LaTeXMLMath .
Further on , LaTeXMLEquation ii ) Set LaTeXMLEquation .
Then LaTeXMLEquation .
Therefore , LaTeXMLMath .
Further on , LaTeXMLEquation hence , LaTeXMLMath .
Lemma 5.3 .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
We have : If LaTeXMLMath , then the condition LaTeXMLMath is equivalent to the condition LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then the condition LaTeXMLMath is equivalent to the condition LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then the condition LaTeXMLMath is equivalent to the condition LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath , then the condition LaTeXMLMath is equivalent to the condition LaTeXMLMath and LaTeXMLMath .
Proof .
i ) The condition LaTeXMLMath is equivalent to the equation LaTeXMLEquation where LaTeXMLMath .
This implies LaTeXMLEquation .
Conversely , let LaTeXMLMath .
Set LaTeXMLMath .
Then ( LaTeXMLRef ) implies LaTeXMLEquation .
If LaTeXMLMath , then polynomial LaTeXMLMath has a root LaTeXMLMath such that LaTeXMLMath is not a root of LaTeXMLMath for any LaTeXMLMath .
Select this root LaTeXMLMath .
Then equation ( LaTeXMLRef ) implies LaTeXMLMath , but then LaTeXMLMath and , therefore , LaTeXMLMath .
So LaTeXMLMath for any LaTeXMLMath .
Thus , LaTeXMLMath .
Headings ii ) –iv ) are similarly proved .
Proof of heading 2 of Theorem 2.4 .
Let LaTeXMLMath .
Let us prove by induction on LaTeXMLMath that LaTeXMLMath is an orthogonal polynomials of degree LaTeXMLMath with respect to the form LaTeXMLMath .
Let LaTeXMLEquation and LaTeXMLEquation .
Then Lemma 5.2 implies that the spaces LaTeXMLMath and LaTeXMLMath are orthogonal with respect to the form LaTeXMLMath .
So it suffices to prove that LaTeXMLMath for LaTeXMLMath and LaTeXMLMath .
Let us induct on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLEquation .
Let LaTeXMLMath ; then equations ( LaTeXMLRef ) imply that LaTeXMLEquation .
This equation and Lemma 5.3 imply that LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLEquation .
LaTeXMLEquation since LaTeXMLMath .
Proof of orthogonality of polynomials LaTeXMLMath is similarly performed with appellation to Lemma 3.3 .
Proof of headings 3 and 4 of Theorem 2.4 .
Define an operator LaTeXMLMath by setting LaTeXMLEquation where LaTeXMLMath satisfies LaTeXMLMath .
Let us show that LaTeXMLMath is well defined .
Indeed , LaTeXMLEquation .
Set LaTeXMLMath .
We have LaTeXMLEquation .
Therefore , LaTeXMLMath by Lemma 5.3 .
Let us show now that LaTeXMLMath is selfadjoint with respect to the form LaTeXMLMath .
Indeed , let LaTeXMLMath and LaTeXMLMath be as in ( 5.6 ) – ( 5.7 ) .
We have shown that LaTeXMLMath sends LaTeXMLMath into itself .
Let LaTeXMLMath ; then LaTeXMLEquation .
Hence , LaTeXMLMath if LaTeXMLMath and LaTeXMLMath .
The uniqueness of the orthogonal polynomial of given degree implies that LaTeXMLMath .
Furthermore , LaTeXMLEquation .
In other words , LaTeXMLEquation .
By calculating the leading coefficient of LaTeXMLMath leads us to the equation of heading 3 .
Equation of heading 4 is similarly obtained by considering operator LaTeXMLEquation .
Proof of headings 5–7 of Theorem 2.4 .
Statements of heading 5 follow from the study of automorphism LaTeXMLMath given by formulas LaTeXMLEquation in the same way as in heading 2 of Theorem 2.4 .
To prove statements of heading 6 , consider the following subspaces of LaTeXMLMath : LaTeXMLEquation .
By Lemma 5.2 these subspaces are completely isotropic with respect to the form LaTeXMLMath .
Let us show that the form LaTeXMLMath , while the remaining scalar products vanish .
Indeed , equations ( LaTeXMLRef ) – ( LaTeXMLRef ) imply LaTeXMLEquation and LaTeXMLEquation .
Hence , LaTeXMLEquation where LaTeXMLEquation and is well defined thanks to Lemma 5.3i ) .
It is easy to show that LaTeXMLMath is selfadjoint and LaTeXMLMath is its eigenfunction corresponding to a nonzero eigenvalue LaTeXMLMath .
So LaTeXMLEquation .
This proves statement of heading 6 .
Statements of heading 7 are proved similar to those of heading 5 .
Proof of headings 1 , 8 , 9 of Theorem 2.4 .
Proof of heading 1 is similar to that of heading 1 of Theorem 2.2 .
To prove heading 8 , consider the operator LaTeXMLEquation .
It is easy to verify that LaTeXMLMath and , the other way round , if LaTeXMLMath , then LaTeXMLMath .
Moreover , LaTeXMLMath , i.e. , LaTeXMLMath is selfadjoint and LaTeXMLMath if LaTeXMLMath .
Hence , LaTeXMLEquation .
So LaTeXMLMath .
Having calculated LaTeXMLMath we obtain the statement of heading 8 .
Proof of heading 9 is similar to arguments from the proof of of headings 8 and 9 of Theorem 2.2 .
I am thankful to D Leites for encouragement and help and to ESI , Vienna , for hospitality and support .
Characterization of Product Measures by Integrability Condition Nobuhiro ASAI International Institute for Advanced Studies Kizu , Kyoto 619-0225 , Japan .
asai @ iias.or.jp Let LaTeXMLMath be the real Gaussian space , where LaTeXMLMath is the space of tempered distributions and LaTeXMLMath be the standard Gaussian measure on LaTeXMLMath .
In the recent papers LaTeXMLCite , Asai , Kubo and Kuo ( AKK for short ) have shown that in order to construct the Gel ’ fand triple LaTeXMLMath associated with a growth function LaTeXMLMath , essential conditions on LaTeXMLMath are ( U0 ) ( U2 ) ( U3 ) stated in Section LaTeXMLRef .
Legendre transform and dual Legendre transform ( Section LaTeXMLRef ) play important roles to get this result .
We note that Gannoun et al .
LaTeXMLCite have obtained similar results independently .
Some relationships with LaTeXMLCite are discussed in Section LaTeXMLRef .
In addition , the intrinsic topology for LaTeXMLMath has been given and the characterization theorem for positive Radon measures on LaTeXMLMath has also been proved by considering an integrability condition LaTeXMLCite .
Now it is natural to ask whether “ positivity ” of white noise operators can be discussed in some sense and characterized .
To answer this question , we consider the Gel ’ fand triple over the Complex Gaussian space LaTeXMLMath , i.e .
LaTeXMLMath equipped with the product measure LaTeXMLMath where LaTeXMLMath is the Gaussian measure on LaTeXMLMath with variance LaTeXMLMath ( Section LaTeXMLRef ) .
Following AKK ’ s Legendre transform technique , we have LaTeXMLMath for functions LaTeXMLMath satisfying ( U0 ) ( U2 ) ( U3 ) .
Several examples for LaTeXMLMath are given in Section LaTeXMLRef .
We remark that Ouerdiane LaTeXMLCite studied a special case LaTeXMLMath , where LaTeXMLMath .
In Section LaTeXMLRef , the characterization theorem for measures can be extended to the case of positive product Radon measures on LaTeXMLMath .
In addition , the notion of pseudo-positive operators is naturally introduced via kernel theorem and characterized by an integrability condition .
Lemma LaTeXMLRef plays crucial roles in Section LaTeXMLRef .
In this section we introduce the Legendre transform and dual Legendre transform which will be used for the constructions of the Gel ’ fand triples over the real and complex Gaussian spaces .
First , let us define two kinds of convex functions .
A positive function LaTeXMLMath on LaTeXMLMath is called ( log , exp ) -convex if the function LaTeXMLMath is convex on LaTeXMLMath ; ( log , LaTeXMLMath ) -convex if the function LaTeXMLMath is convex on LaTeXMLMath .
Here LaTeXMLMath .
Let LaTeXMLMath denote the collection of all positive continuous functions LaTeXMLMath on LaTeXMLMath satisfying the condition : LaTeXMLEquation .
The Legendre transform LaTeXMLMath of LaTeXMLMath is defined to be the function LaTeXMLEquation .
Let LaTeXMLMath denote the collection of all positive continuous functions LaTeXMLMath on LaTeXMLMath satisfying the condition : LaTeXMLEquation .
The dual Legendre transform LaTeXMLMath of LaTeXMLMath is defined to be the function LaTeXMLEquation .
Note that LaTeXMLMath .
Assume that LaTeXMLMath and LaTeXMLMath .
We define the LaTeXMLMath - function LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation .
For discussions in the rest of the paper , we will need the following facts in LaTeXMLCite .
See also LaTeXMLCite .
( 1 ) Let LaTeXMLMath be ( log , exp ) -convex .
Then its LaTeXMLMath -function LaTeXMLMath is also ( log , exp ) -convex and for any LaTeXMLMath , LaTeXMLEquation ( 2 ) Let LaTeXMLMath be increasing and ( log , LaTeXMLMath ) -convex .
Then there exists a constant LaTeXMLMath , independent of LaTeXMLMath , such that LaTeXMLEquation ( 3 ) Let LaTeXMLMath be increasing and ( log , LaTeXMLMath ) -convex .
Then for any LaTeXMLMath , we have LaTeXMLEquation .
If LaTeXMLMath is ( log , LaTeXMLMath ) -convex , then the Legendre transform LaTeXMLMath of LaTeXMLMath is given by LaTeXMLEquation .
Let us start with taking a special choice of a Gel ’ fand triple : LaTeXMLEquation just for convenience where LaTeXMLMath is the Schwarz space of rapidly decreasing functions and LaTeXMLMath is the space of tempered distributions .
Consult LaTeXMLCite for more general setting .
Let LaTeXMLMath be a positive self-adjoint operator in LaTeXMLMath .
So there exists an orthonormal basis LaTeXMLMath for LaTeXMLMath satisfying LaTeXMLMath .
LaTeXMLMath denotes the norm of LaTeXMLMath .
For each LaTeXMLMath we define LaTeXMLMath and let LaTeXMLMath .
Note that LaTeXMLMath is the completion of LaTeXMLMath with respect to the norm LaTeXMLMath .
Moreover , LaTeXMLEquation for any LaTeXMLMath .
Then the projective limit space LaTeXMLMath of LaTeXMLMath as LaTeXMLMath is a nuclear space and the dual space of LaTeXMLMath is nothing but the inductive limit space LaTeXMLMath .
Hence we have the following continuous inclusions : LaTeXMLEquation where the norm on LaTeXMLMath is given by LaTeXMLMath .
Throughout this paper , we denote the complexification of a real space LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath be the Gaussian measure on LaTeXMLMath with variance LaTeXMLMath , namely , a probability measure on LaTeXMLMath given by the characteristic function : LaTeXMLEquation .
Due to the topological isomorphism LaTeXMLMath , we can define a probability measure LaTeXMLMath on LaTeXMLMath .
The probability space LaTeXMLMath is called the complex Gaussian space , see LaTeXMLCite .
We denote by LaTeXMLMath the space of LaTeXMLMath -square integrable functions on LaTeXMLMath .
We should note that LaTeXMLMath .
Let LaTeXMLMath be the Gaussian measure on LaTeXMLMath with variance LaTeXMLMath and LaTeXMLMath be the real Gaussian space .
The next Fact LaTeXMLRef has been obtained in LaTeXMLCite for the Gel ’ fand triple LaTeXMLMath associated with a growth function LaTeXMLMath .
This triple is refered to as the CKS-space with a weight sequence LaTeXMLMath .
For more precise discussion , we will need the following conditions on LaTeXMLMath : LaTeXMLMath .
LaTeXMLMath is increasing and LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath is ( log , LaTeXMLMath ) -convex .
Then we have Suppose LaTeXMLMath satisfies conditions ( U0 ) ( U2 ) ( U3 ) .
Then the CKS-space with a weight sequence LaTeXMLMath can be constructed .
Moreover , characterization theorems hold .
Remark .
( 1 ) We refer the reader to the papers of Asai et al .
LaTeXMLCite for details .
We cite papers LaTeXMLCite for characterizaion theorems on papticular cases .
( 2 ) We should mention here that our formulation has some links with a recent work by Gannoun et al .
LaTeXMLCite .
So let us explain some of them .
Essential relationships are LaTeXMLEquation where LaTeXMLMath is adopted in LaTeXMLCite .
In the following table we give the correspondence between our LaTeXMLMath -conditions and LaTeXMLMath -conditions .
Next let us consider the Gel ’ fand triple over the complex white noise space LaTeXMLMath for our purpose .
LaTeXMLMath -norm LaTeXMLMath of LaTeXMLMath is given by LaTeXMLEquation .
In order to define norms in the spaces of test and generalized functions , we need a notation .
For LaTeXMLMath , we put LaTeXMLEquation .
For LaTeXMLMath and given functions LaTeXMLMath satisfying conditions LaTeXMLMath , define the norm by LaTeXMLEquation .
Let LaTeXMLMath .
Define the space LaTeXMLMath on LaTeXMLMath to be the projective limit of LaTeXMLMath as LaTeXMLMath .
For abbreviation , we put LaTeXMLMath and LaTeXMLMath for the dual space .
For given functions LaTeXMLMath satisfying LaTeXMLMath , we have the following continuous inclusions : LaTeXMLEquation where LaTeXMLMath is the dual space of LaTeXMLMath .
In general , LaTeXMLMath and LaTeXMLMath are not necessarily the same functions .
A Gel ’ fand triple LaTeXMLMath is refered to as a CKS-space with a weight sequence LaTeXMLMath .
The bilinear form on LaTeXMLMath is denoted by LaTeXMLMath .
Then LaTeXMLEquation and it holds that LaTeXMLEquation where LaTeXMLEquation .
The combinations of two functions out of following examples are applicable to our setting .
Consider LaTeXMLEquation .
Then it is obvious to check that consditions ( U0 ) ( U2 ) ( U3 ) are satisfied .
This example produces to the Hida-Kubo-Takenaka space over the real Gaussian space .
See LaTeXMLCite .
For LaTeXMLMath , let LaTeXMLMath be the function defined by LaTeXMLEquation .
It is easy to check that LaTeXMLMath belongs to LaTeXMLMath and satisfies conditions ( U0 ) ( U2 ) ( U3 ) .
By Example 4.3 in LaTeXMLCite , the dual Legendre transform LaTeXMLMath of LaTeXMLMath is given by LaTeXMLEquation .
This example is for the construction of the Kodratiev-Streit space over the real Gaussian space .
See LaTeXMLCite Consider the function LaTeXMLMath .
Obviously , LaTeXMLMath .
Let LaTeXMLMath be the dual Legendre transform of LaTeXMLMath .
Then LaTeXMLMath and it can be shown that LaTeXMLMath belongs to LaTeXMLMath and is an increasing ( log , LaTeXMLMath ) -convex function on LaTeXMLMath ( See LaTeXMLCite ) .
Hence LaTeXMLMath satisfies conditions ( U1 ) and ( U3 ) .
It is shown in Example 4.4 in LaTeXMLCite that LaTeXMLMath is equivalent to the function LaTeXMLEquation ( “ LaTeXMLMath is equivalent to LaTeXMLMath ” means that there exist constants LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath . )
Obviously , LaTeXMLMath satisfies condition ( U2 ) and so LaTeXMLMath also satisfies condition ( U2 ) .
On the other hand , we have the involution property LaTeXMLMath .
This example can be applied to the Gel ’ fand triple LaTeXMLMath for the following pair of functions : LaTeXMLEquation .
In general , we can consider the follwing general pair of functions : LaTeXMLEquation .
We refer the reader to papers LaTeXMLCite .
We shall define another norm as follows .
Let LaTeXMLMath for LaTeXMLMath be the space of all functions LaTeXMLMath on LaTeXMLMath satisfying the following conditions : ( L1 ) LaTeXMLMath is an analytic function on LaTeXMLMath .
( L2 ) There exists a nonnegative constant LaTeXMLMath such that LaTeXMLEquation .
For LaTeXMLMath , its norm is defined by LaTeXMLEquation for a function LaTeXMLMath .
Define the space LaTeXMLMath of test functions on LaTeXMLMath to be the projective limit of LaTeXMLMath as LaTeXMLMath .
Let LaTeXMLMath be the dual space of LaTeXMLMath .
Remark .
This construction is motivated by Lee LaTeXMLCite and Asai et al .
LaTeXMLCite .
See also LaTeXMLCite and references cited therein .
Asai et al .
LaTeXMLCite and Gannoun et al .
LaTeXMLCite have considered the case of LaTeXMLMath , independently .
In addition , Ouerdiane studied similar situations and the case LaTeXMLMath where LaTeXMLMath .
If LaTeXMLMath and an entire function LaTeXMLMath on LaTeXMLMath satisfies the growth condition LaTeXMLEquation for a fixed positive LaTeXMLMath , then for LaTeXMLMath with LaTeXMLMath , there exists a kernel LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation .
Consider an entire function on LaTeXMLMath LaTeXMLEquation .
Define an LaTeXMLMath -linear functional LaTeXMLMath on LaTeXMLMath LaTeXMLEquation .
Taking LaTeXMLMath and LaTeXMLMath we get LaTeXMLEquation by ( LaTeXMLRef ) .
Minimizing the right term , we have LaTeXMLEquation .
This shows that LaTeXMLMath can be expressed in the form LaTeXMLEquation with LaTeXMLMath , LaTeXMLEquation with finite Hilbert-Schmidt norm LaTeXMLMath for any LaTeXMLMath .
Therefore we derive LaTeXMLEquation ∎ Similarly , we have If LaTeXMLMath and an entire function LaTeXMLMath on LaTeXMLMath satisfies the growth condition LaTeXMLEquation for any LaTeXMLMath and some LaTeXMLMath , then there exists a kernel LaTeXMLMath such that LaTeXMLEquation and LaTeXMLEquation for any LaTeXMLMath satisfying LaTeXMLMath .
We can prove this Lemma with modifications of the proof of the previous Lemma .
Therefore , we omit the proof .
∎ Remark .
Lemma LaTeXMLRef will be expected to characterize the continuous linear operator from LaTeXMLMath into LaTeXMLMath and expand it in terms of integral kernel operators LaTeXMLCite .
To our best knowledge , such consideratoins have not been done in any literature .
For LaTeXMLMath , its multiple S-transform LaTeXMLMath is defined to be the function LaTeXMLEquation .
For LaTeXMLMath , we have an integral representation of the multiple S-transform as LaTeXMLEquation .
Note that the multiple S-transform is nothing but a symbol of operators frequently used in white noise operator theory LaTeXMLCite .
Lemma LaTeXMLRef , Equation ( LaTeXMLRef ) and LaTeXMLMath ( Lemma LaTeXMLRef , Equation ( LaTeXMLRef ) and LaTeXMLMath ) give us the characterization theorem for LaTeXMLMath ( LaTeXMLMath ) , respectively , which generalize recent results in LaTeXMLCite .
Suppose LaTeXMLMath satisfy LaTeXMLMath LaTeXMLMath LaTeXMLMath .
Then the families of norms LaTeXMLMath and LaTeXMLMath are equivalent .
First , we will show that for any LaTeXMLMath , LaTeXMLMath , there exist LaTeXMLMath and LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , be given .
Since it has been proved LaTeXMLCite ( see also LaTeXMLCite ) that every test function in LaTeXMLMath has an analytic extention , there exist LaTeXMLMath and LaTeXMLMath such that for any LaTeXMLMath LaTeXMLEquation for any LaTeXMLMath .
Hence it is derived by ( LaTeXMLRef ) and ( LaTeXMLRef ) that LaTeXMLEquation .
LaTeXMLEquation To prove the converse , The multiple S-transform of LaTeXMLMath is given by LaTeXMLEquation .
Then observe that for LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation By the condition LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath .
Therefore , LaTeXMLEquation .
By the condition LaTeXMLMath , we have LaTeXMLEquation .
LaTeXMLEquation Thus , it is easy to get LaTeXMLEquation where LaTeXMLEquation ( Note that finiteness concerning LaTeXMLMath can be shown easily by simple estimation and the Fernique theorem LaTeXMLCite . )
Then applying Lemma LaTeXMLRef with ( LaTeXMLRef ) , we have LaTeXMLEquation .
We complete the proof .
∎ A generalized function LaTeXMLMath is called positive if LaTeXMLMath for all nonnegative test functions LaTeXMLMath .
Remark .
Positivity of generalized functions in white noise context has been studied by Yokoi LaTeXMLCite .
It is possible to give an alternative definition to Definition LaTeXMLRef as follows by the kernel theorem LaTeXMLCite .
An operator LaTeXMLMath is called positive in the sense of distributions if LaTeXMLMath for all nonnegative test functions LaTeXMLMath .
We call such an operator pseudo-positive operator Notation .
For locally convex spaces LaTeXMLMath , let LaTeXMLMath denote the space of all continuous operators from LaTeXMLMath into LaTeXMLMath equipped with the topology of uniform convergence on every bounded subset .
Suppose LaTeXMLMath satisfy LaTeXMLMath .
A measure LaTeXMLMath on LaTeXMLMath is a positive product Radon measure inducing a positive generalized function LaTeXMLMath if and only if LaTeXMLMath is supported in LaTeXMLMath for some LaTeXMLMath and LaTeXMLEquation .
First we shall prove sufficiency .
Suppose that LaTeXMLMath is supported in LaTeXMLMath for some LaTeXMLMath and Equation ( LaTeXMLRef ) holds .
Then for any LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation With the help of Proposition LaTeXMLRef , LaTeXMLMath and LaTeXMLEquation is a continuous linear functional on LaTeXMLMath .
Therefore , LaTeXMLMath is a positive product Radon measure which induces a positive generalized function LaTeXMLMath in LaTeXMLMath .
Conversely , suppose that LaTeXMLMath is a positive product Radon measure .
Then for all LaTeXMLMath , LaTeXMLEquation is a continuous linear functional with respect to LaTeXMLMath by Proposition LaTeXMLRef .
Thus there exist constants LaTeXMLMath such that for all LaTeXMLMath LaTeXMLEquation .
Let us define an analytic function LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath is the bilinear pairing on LaTeXMLMath .
On the other hand , Fact LaTeXMLRef ( 3 ) implies that LaTeXMLEquation .
LaTeXMLEquation ( It is easy to find an increasing function LaTeXMLMath equivalent to a function LaTeXMLMath . )
This implies that LaTeXMLMath .
Thus , from Equation ( LaTeXMLRef ) with LaTeXMLMath we obtain that LaTeXMLEquation .
Due to Equation ( LaTeXMLRef ) , we have LaTeXMLEquation .
However Fact LaTeXMLRef ( 2 ) says that LaTeXMLMath .
Therefore , LaTeXMLEquation .
By choosing an appropriate LaTeXMLMath satisfying LaTeXMLMath , so that LaTeXMLMath .
Therefore we get the assertion .
∎ Suppose LaTeXMLMath satisfy LaTeXMLMath .
A measure LaTeXMLMath on LaTeXMLMath is a positive product Radon measure inducing a pseudo-positive operator LaTeXMLMath if and only if LaTeXMLMath is supported in LaTeXMLMath for some LaTeXMLMath and LaTeXMLEquation
Let LaTeXMLMath be a pattern and let LaTeXMLMath be the number of LaTeXMLMath -permutations having exactly LaTeXMLMath copies of LaTeXMLMath .
We investigate when the sequence LaTeXMLMath has internal zeros .
If LaTeXMLMath is a monotone pattern it turns out that , except for LaTeXMLMath or LaTeXMLMath , the nontrivial sequences ( those where LaTeXMLMath is at least the length of LaTeXMLMath ) always have internal zeros .
For the pattern LaTeXMLMath there are infinitely many sequences which contain internal zeros and when LaTeXMLMath there are also infinitely many which do not .
In the latter case , the only possible places for internal zeros are the next-to-last or the second-to-last positions .
Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most three .
Let LaTeXMLMath be a permutation in the symmetric group LaTeXMLMath .
We call LaTeXMLMath the length of LaTeXMLMath .
We say that the permutation LaTeXMLMath contains a LaTeXMLMath -pattern if and only if there is a subsequence LaTeXMLMath of LaTeXMLMath whose elements are in the same relative order as those in LaTeXMLMath , i.e. , LaTeXMLEquation whenever LaTeXMLMath .
For example , 41523 contains exactly two 132-patterns , namely 152 and 153 .
We let LaTeXMLEquation so that LaTeXMLMath .
Permutations containing a given number of LaTeXMLMath -patterns have been extensively studied recently [ 1–11 ] .
In this paper , we consider permutations with a given number of LaTeXMLMath -patterns from a new angle .
Let LaTeXMLEquation .
For LaTeXMLMath and LaTeXMLMath fixed , the sequence LaTeXMLMath is called the frequency sequence of the pattern LaTeXMLMath for LaTeXMLMath .
Clearly this sequence consists entirely of zeros if LaTeXMLMath is less than the length of LaTeXMLMath and so we call these sequences trivial and all others nontrivial .
We also say that an LaTeXMLMath -permutation LaTeXMLMath is LaTeXMLMath -optimal if there is no LaTeXMLMath -permutation with more copies of LaTeXMLMath than LaTeXMLMath , and let LaTeXMLEquation .
The only LaTeXMLMath for which the frequency sequence is well understood is LaTeXMLMath ( or equivalently LaTeXMLMath ) .
Occurences of this pattern are called inversions .
It is well known LaTeXMLCite that for all LaTeXMLMath , the frequency sequence of inversions is log-concave , and so is unimodal and has no internal zeros .
When LaTeXMLMath is has length greater than 2 , numerical evidence suggests that the frequency sequence of LaTeXMLMath will no longer be unimodal , let alone log-concave .
In fact , internal zeros seem to be present in most frequency sequences .
An integer LaTeXMLMath is called an internal zero of the sequence LaTeXMLMath if for some LaTeXMLMath we have LaTeXMLMath , but there exist LaTeXMLMath and LaTeXMLMath with LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
In the rest of this paper we study the frequency sequences of the monotone pattern LaTeXMLMath and the pattern LaTeXMLMath .
We will show that in the first case , when LaTeXMLMath ( the case LaTeXMLMath has already been mentioned ) the nontrivial sequences always have internal zeros .
For LaTeXMLMath -patterns there are infinitely many LaTeXMLMath where the sequence has internal zeros .
For the LaTeXMLMath -pattern there are also infinitely many LaTeXMLMath where the sequence has no internal zeros .
And internal zeros can only appear in positions LaTeXMLMath or LaTeXMLMath .
We will now consider the sequence LaTeXMLMath where LaTeXMLMath .
For later reference , we single out the known case when LaTeXMLMath discussed in the introduction .
The sequence LaTeXMLMath has no internal zeros ( and is , in fact , log concave ) .
The unique optimal permutation is LaTeXMLMath with LaTeXMLEquation .
It turns out that this is the only monotone pattern ( aside from 21 ) whose sequence has no internal zeros .
To prove this result , define an inversion ( respectively , noninversion ) in LaTeXMLMath to be a pair LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath ( respectively , LaTeXMLMath ) .
Let LaTeXMLMath where LaTeXMLMath .
Then in LaTeXMLMath , the unique optimal permutation is LaTeXMLMath and LaTeXMLEquation .
The set of permutations having the next greatest number of copies of LaTeXMLMath are those obtained from LaTeXMLMath by an adjacent transposition and this number of copies is LaTeXMLEquation .
Proof : Consider any LaTeXMLMath different from LaTeXMLMath .
Then LaTeXMLMath has an inversion LaTeXMLMath .
So the number of copies of LaTeXMLMath in LaTeXMLMath is the number not containing LaTeXMLMath plus the number which do contain LaTeXMLMath .
The permutations in the latter case can not contain LaTeXMLMath .
So ( LaTeXMLRef ) gives an upper bound for the number of copies of LaTeXMLMath which is strict unless LaTeXMLMath has exactly one inversion .
The theorem follows .
LaTeXMLMath Let LaTeXMLMath where LaTeXMLMath .
Then for LaTeXMLMath the sequence LaTeXMLMath has internal zeros .
Proof : From the previous theorem , we see that the number of zeros directly before LaTeXMLMath is LaTeXMLEquation since LaTeXMLMath .
LaTeXMLMath For use in the 132 case , we record the following observation .
For any integer LaTeXMLMath with LaTeXMLMath there is a permutation LaTeXMLMath having LaTeXMLMath copies of the pattern LaTeXMLMath and no copies of LaTeXMLMath .
Proof : We induct on LaTeXMLMath .
The result is clearly true if LaTeXMLMath .
Assuming it is true for LaTeXMLMath , first consider LaTeXMLMath and let LaTeXMLMath satisfy the lemma .
Then the concatenation LaTeXMLMath works for such LaTeXMLMath .
On the other hand , if LaTeXMLMath then consider LaTeXMLMath .
Pick LaTeXMLMath with LaTeXMLMath copies of LaTeXMLMath and none of LaTeXMLMath .
Then LaTeXMLMath is the desired permutation .
LaTeXMLMath The rest of this paper is devoted to the study of the frequency sequences of the patterns LaTeXMLMath for LaTeXMLMath .
To simplify notation , and write LaTeXMLMath for the sequence LaTeXMLMath .
One crucial property of these patterns is that they are layered .
This section gives an overview of some important results on layered patterns .
A pattern is layered if it is the concatenation of subwords ( the layers ) where the entries decrease within each layer , and increase between the layers .
For example , LaTeXMLMath is a layered pattern with layers LaTeXMLMath , and 9 .
Layered patterns are examined in Stromquist ’ s work LaTeXMLCite and in Price ’ s thesis LaTeXMLCite .
The most important result for our current purposes is the following theorem .
Let LaTeXMLMath be a layered pattern .
Then the set of LaTeXMLMath -optimal LaTeXMLMath -permutations contains at least one layered permutation .
Layered LaTeXMLMath -optimal permutations have a simple recursive structure .
This comes from the fact , which we will use many times , that to form a LaTeXMLMath pattern in a layered permutation one must take a single element from some layer and LaTeXMLMath elements from a subsequent layer Let LaTeXMLMath be a layered LaTeXMLMath -optimal LaTeXMLMath -permutation whose last layer is of length LaTeXMLMath .
Then the leftmost LaTeXMLMath elements of LaTeXMLMath form a LaTeXMLMath -optimal LaTeXMLMath -permutation .
Proof : Let LaTeXMLMath be the number of LaTeXMLMath -copies of LaTeXMLMath that are disjoint from the last layer .
The number of LaTeXMLMath -copies of LaTeXMLMath is clearly LaTeXMLMath So once LaTeXMLMath is chosen , LaTeXMLMath will have the maximum number of copies only if LaTeXMLMath is maximal .
LaTeXMLMath We point out that the proof of this proposition uses the fact that LaTeXMLMath has only two layers , the first of which is a singleton .
Let LaTeXMLMath .
Then the previous proposition implies that LaTeXMLEquation .
The integer LaTeXMLMath for which the right hand side attains its maximum will play a crucial role throughout this paper .
Therefore , we introduce specific notation for it .
For any positive integer LaTeXMLMath , let LaTeXMLMath be the positive integer for which LaTeXMLMath is maximal .
If there are several integers with this property , then let LaTeXMLMath be the largest among them .
In other words , LaTeXMLMath is the largest possible length of the remaining permutation after removing the last layer of a LaTeXMLMath -optimal LaTeXMLMath -permutation LaTeXMLMath .
When there is no danger of confusion , we will only write LaTeXMLMath to simplify notation .
We will also always use LaTeXMLMath to denote the length of the last layer of LaTeXMLMath .
We will first show that if LaTeXMLMath then there are infinitely many integers LaTeXMLMath such that LaTeXMLMath does not have internal zeros .
We will call such an integer , or its corresponding sequence , NIZ ( no internal zero ) , and otherwise IZ .
Our strategy is recursive : We will show that if LaTeXMLMath is NIZ , then so is LaTeXMLMath .
As LaTeXMLMath , this will lead to an infinite sequence of NIZ integers .
There is a problem , however .
In order for this strategy to work , we must ensure that given LaTeXMLMath , then there is an LaTeXMLMath such that LaTeXMLMath .
This is the purpose of the following theorem which is in fact true for the general pattern LaTeXMLMath .
For LaTeXMLMath , the sequence LaTeXMLMath diverges to infinity and satisfies LaTeXMLEquation for all LaTeXMLMath .
So , since LaTeXMLMath , for all positive integers LaTeXMLMath there is a positive integer LaTeXMLMath so that LaTeXMLMath .
The next section is devoted to a proof of this theorem .
We suggest that the reader assume the result now and continue with this section to preserve continuity .
We now consider the case LaTeXMLMath which behaves differently from LaTeXMLMath for LaTeXMLMath .
This is essentially due to the difference between the patterns LaTeXMLMath and LaTeXMLMath for LaTeXMLMath as seen in Proposition LaTeXMLRef and Theorem LaTeXMLRef .
First we note the useful fact that LaTeXMLEquation which follows by considering the permutation LaTeXMLMath .
For LaTeXMLMath There are infinitely many NIZ integers .
Proof : It is easy to verify that LaTeXMLMath is NIZ .
So , by Theorem LaTeXMLRef , it suffices to show that if LaTeXMLMath is NIZ then so is LaTeXMLMath .
To simplify notation in the two proofs which follow , we will write LaTeXMLMath for LaTeXMLMath , LaTeXMLMath for LaTeXMLMath , and so forth .
Now given LaTeXMLMath with LaTeXMLMath we will construct a permutation LaTeXMLMath having LaTeXMLMath copies of 132 .
Because of ( LaTeXMLRef ) and LaTeXMLMath we have LaTeXMLMath .
So it is possible to write LaTeXMLMath ( not necessarily uniquely ) as LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is NIZ , there is a permutation LaTeXMLMath with LaTeXMLMath .
Also , by Lemma LaTeXMLRef , there is a permutation in LaTeXMLMath with no copies of 132 and LaTeXMLMath copies of 21 .
Let LaTeXMLMath be the result of adding LaTeXMLMath to every element of that permutation .
Then , by construction , LaTeXMLMath and LaTeXMLMath as desired .
LaTeXMLMath One can modify the proof of the previous theorem to locate precisely where the internal zeros could be for an IZ sequence .
We will need the fact ( established by computer ) that for LaTeXMLMath the only IZ integers were 6 , 8 , and 9 , and that they all satisfied the following result .
For any positive integer LaTeXMLMath , the sequence LaTeXMLMath does not have internal zeros , except possibly for LaTeXMLMath or LaTeXMLMath , but not both .
Proof : We prove this theorem by induction on LaTeXMLMath .
As previously remarked , it is true if LaTeXMLMath .
Now suppose we know the statement for all integers smaller than LaTeXMLMath , and prove it for LaTeXMLMath .
If LaTeXMLMath is NIZ , then we are done .
If LaTeXMLMath is IZ then , by the proof of Theorem LaTeXMLRef , LaTeXMLMath is IZ .
So LaTeXMLMath and we have LaTeXMLMath by ( LaTeXMLRef ) .
Now take LaTeXMLMath with LaTeXMLMath so that we can write LaTeXMLMath with LaTeXMLMath and LaTeXMLMath .
Since the portion of LaTeXMLMath up to LaTeXMLMath has no internal zeros by induction , we can use the same technique as in the previous theorem to construct a permutation LaTeXMLMath with LaTeXMLMath for LaTeXMLMath in the given range .
Furthermore , this construction shows that if LaTeXMLMath for LaTeXMLMath or LaTeXMLMath then LaTeXMLMath .
This completes the proof .
LaTeXMLMath For the rest of this paper , all invariants will refer to the pattern LaTeXMLMath unless explicitly stated otherwise .
In order to prove Theorem LaTeXMLRef , we first need a lemma about the lengths of various parts of a LaTeXMLMath -optimal permutation LaTeXMLMath .
In all that follows , we use the notation LaTeXMLEquation .
LaTeXMLEquation Also observe that the sequence LaTeXMLMath is strictly increasing .
This is because when LaTeXMLMath , any layered LaTeXMLMath -optimal permutation LaTeXMLMath contains at least one copy of LaTeXMLMath .
So inserting LaTeXMLMath in front of any layer contributing to the LaTeXMLMath portion of some copy results in a permutation with more LaTeXMLMath -patterns than LaTeXMLMath .
It follows from ( LaTeXMLRef ) that LaTeXMLMath for LaTeXMLMath , a fact that will be useful in proving the following result .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then we have the following inequalities LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , so in particular LaTeXMLMath , LaTeXMLMath .
Proof : The basic idea behind all four of the inequalities is as follows .
Let LaTeXMLMath be the permutation obtained from our LaTeXMLMath - optimal permutation LaTeXMLMath by replacing its last two layers with a last layer of length LaTeXMLMath and a next-to-last layer of length LaTeXMLMath .
Then in passing from LaTeXMLMath to LaTeXMLMath we lose some LaTeXMLMath -patterns and gain some .
Since LaTeXMLMath was optimal , the number lost must be at least as large as the number gained .
And this inequality can be manipulated to give the one desired .
For the details , the following chart gives the relevant information to describe LaTeXMLMath for each of the four inequalities .
In the second case , the last two layers of LaTeXMLMath are combined into one , so the value of LaTeXMLMath is irrelevant .
LaTeXMLEquation .
Now ( i ) follows easily by cancelling LaTeXMLMath from the inequality in the first row of the table .
From the second line of the table , we have LaTeXMLEquation and cancelling LaTeXMLMath , which is not zero becase LaTeXMLMath , gives us ( ii ) .
To prove ( iii ) we induct on LaTeXMLMath .
If LaTeXMLMath , then we must have LaTeXMLMath , so LaTeXMLMath .
Now we assume LaTeXMLMath .
If LaTeXMLMath , then the leftmost LaTeXMLMath elements of LaTeXMLMath contain no copies of LaTeXMLMath , so we may replace them with any LaTeXMLMath -permutation and still have LaTeXMLMath optimal .
Therefore we may pick LaTeXMLMath and LaTeXMLMath , and thus the second row of the table shows LaTeXMLEquation so LaTeXMLMath , as desired .
If LaTeXMLMath , recall that from Proposition LaTeXMLRef , the leftmost LaTeXMLMath elements of LaTeXMLMath form a LaTeXMLMath -optimal permutation , so we may , without loss , choose LaTeXMLMath maximal and thus assume that LaTeXMLMath .
From the third line of the chart , we have LaTeXMLEquation .
Using ( i ) we get that LaTeXMLMath .
Substituting this in the previous equation , cancelling LaTeXMLMath , and solving for LaTeXMLMath gives LaTeXMLEquation .
Since LaTeXMLMath , we have by induction that LaTeXMLMath .
Substituting and solving for LaTeXMLMath again and then cancelling LaTeXMLMath , we get LaTeXMLMath .
A final substitution of LaTeXMLMath results in ( iii ) .
For ( iv ) , notice that the last row of the table gives LaTeXMLEquation so cancelling LaTeXMLMath gives LaTeXMLMath , which can be converted to the desired inequality .
LaTeXMLMath We now turn to the proof of Theorem LaTeXMLRef .
First note that , by Lemma LaTeXMLRef ( iv ) , we have LaTeXMLEquation .
So LaTeXMLMath clearly diverges to infinity .
For our next step , we prove that LaTeXMLMath is monotonically weakly increasing .
Let LaTeXMLMath denote an LaTeXMLMath -permutation whose last layer is of length LaTeXMLMath , and whose leftmost LaTeXMLMath elements form a LaTeXMLMath -optimal LaTeXMLMath -permutation , and let LaTeXMLMath .
Clearly LaTeXMLEquation .
For LaTeXMLMath and all integers LaTeXMLMath , we have LaTeXMLMath .
Proof : It suffices to show that LaTeXMLMath for all LaTeXMLMath .
This is equivalent to showing that LaTeXMLEquation .
However , by definition of LaTeXMLMath , we know that for all LaTeXMLMath , LaTeXMLEquation .
Subtracting ( LaTeXMLRef ) from ( LaTeXMLRef ) , we are reduced to proving LaTeXMLMath .
We will induct on LaTeXMLMath .
If LaTeXMLMath , then we would like to show that LaTeXMLEquation so it suffices to show that LaTeXMLMath , which follows from Lemma LaTeXMLRef ( iii ) .
For LaTeXMLMath we have , by induction , that LaTeXMLMath , so it suffices to show that LaTeXMLEquation which simplifies to LaTeXMLMath , and this is is true because LaTeXMLMath .
LaTeXMLMath The proof of the upper bound on LaTeXMLMath is a bit more involved but follows the same general lines as the previous demonstration .
Note that this will finish the proof of Theorem LaTeXMLRef .
For LaTeXMLMath and all integers LaTeXMLMath , we have LaTeXMLMath .
Proof : Induct on LaTeXMLMath .
The lemma is true for LaTeXMLMath since LaTeXMLMath .
Suppose the lemma is true for integers smaller than or equal to LaTeXMLMath , and prove it for LaTeXMLMath .
For simplicity , let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Since we have already proved the lower bound , it suffices to show that LaTeXMLEquation .
Note that we do not have to consider LaTeXMLMath because of Lemma LaTeXMLRef ( iii ) .
We prove ( LaTeXMLRef ) by induction on LaTeXMLMath .
For the base case , LaTeXMLMath , we wish to show LaTeXMLEquation .
But since LaTeXMLMath is optimal by assumption , we have LaTeXMLEquation .
Subtracting ( LaTeXMLRef ) from ( LaTeXMLRef ) and rearranging terms , it suffices to prove LaTeXMLEquation .
First , if LaTeXMLMath , then ( LaTeXMLRef ) is easy to verify using Lemma LaTeXMLRef ( iii ) and the values LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath .
Therefore we may assume that LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath be layered LaTeXMLMath -optimal permutations having last layer lengths LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , respectively , as short as possible .
Also let LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
We would like to be able to assume the lemma holds for these permutations , and thus we would like to have LaTeXMLMath .
But by Lemma LaTeXMLRef ( iii ) we have LaTeXMLMath if LaTeXMLMath .
Since LaTeXMLMath this holds for LaTeXMLMath and the case LaTeXMLMath is easy to check directly .
Therefore we may assume that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath all satisfy the lemma .
If LaTeXMLMath then let LaTeXMLMath be the largest element in the last layer of LaTeXMLMath ( namely LaTeXMLMath ) .
Otherwise , LaTeXMLMath and removing the last layer of both LaTeXMLMath and LaTeXMLMath leaves permutations in LaTeXMLMath and LaTeXMLMath , respectively .
So we can iterate this process until we find the single layer where LaTeXMLMath and LaTeXMLMath have different lengths ( those lengths must differ by 1 ) and let LaTeXMLMath be the largest element in that layer of LaTeXMLMath .
Similarly we can find the element LaTeXMLMath which is largest in the unique layer were LaTeXMLMath and LaTeXMLMath have different lengths .
Now let LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation Note that there is a bijection between the LaTeXMLMath -patterns of LaTeXMLMath not containing LaTeXMLMath and the LaTeXMLMath -patterns of LaTeXMLMath .
A similar statement holds for LaTeXMLMath and LaTeXMLMath .
So LaTeXMLEquation .
Note also that LaTeXMLMath because increasing the length of the layer of LaTeXMLMath results in the most number of LaTeXMLMath -patterns being added to LaTeXMLMath .
It follows that LaTeXMLMath .
By Lemma LaTeXMLRef ( iii ) , LaTeXMLMath , so to obtain ( LaTeXMLRef ) it suffices to show that LaTeXMLMath .
But LaTeXMLMath is the total number of subsequences of LaTeXMLMath having length LaTeXMLMath and containing LaTeXMLMath and LaTeXMLMath .
So the inequality follows .
The proof of the induction step is similar .
Assume that ( LaTeXMLRef ) is true for LaTeXMLMath so that LaTeXMLEquation where LaTeXMLMath .
We wish to prove LaTeXMLEquation .
Subtracting as usual and simplifying , we need to show LaTeXMLEquation .
Proceeding exactly as in the base case , we will be done if we can show that LaTeXMLEquation .
Because LaTeXMLMath we have LaTeXMLMath , so it suffices to show that LaTeXMLEquation .
This simplifies to showing that LaTeXMLMath , and this is guaranteed by our choice of LaTeXMLMath .
LaTeXMLMath The following lemma contains two inequalities essentially shown in the proof of Lemma LaTeXMLRef which we will need to use again .
If LaTeXMLMath then LaTeXMLMath .
Proof : For the upper bound , recall that LaTeXMLMath is the total number of subsequences of LaTeXMLMath of length LaTeXMLMath containing LaTeXMLMath and LaTeXMLMath while the double difference just counts those subsequences corresponding to the pattern LaTeXMLMath .
For the lower bound , we showed that LaTeXMLEquation .
Recall that LaTeXMLMath is the total contribution of LaTeXMLMath in LaTeXMLMath , and LaTeXMLMath is the total contribution of LaTeXMLMath in LaTeXMLMath .
Therefore LaTeXMLMath , as otherwise one could create a permutation with more LaTeXMLMath -patterns than LaTeXMLMath by inserting a new element in the same layer as LaTeXMLMath LaTeXMLMath Now that we have completed the proof of Theorem LaTeXMLRef , we turn our attention to the tools which will enable us to show that there are infinitely many IZ integers .
As before , all invariants are for LaTeXMLMath unless otherwise stated .
For LaTeXMLMath , we will need the following lemma .
For all LaTeXMLMath , we have LaTeXMLMath .
Proof : Let LaTeXMLMath .
We induct on LaTeXMLMath .
It is easy to check the base cases LaTeXMLMath .
Note that by Theorem LaTeXMLRef , either LaTeXMLMath or LaTeXMLMath .
If LaTeXMLMath , then we have LaTeXMLEquation and maximizing this as a function of LaTeXMLMath gives LaTeXMLEquation .
If LaTeXMLMath , then we have LaTeXMLEquation .
By induction , we have LaTeXMLMath , and thus we have that LaTeXMLEquation .
By Lemma LaTeXMLRef ( iii ) and ( iv ) , this function is to be maximized on the interval LaTeXMLMath and for LaTeXMLMath this maximum occurs at LaTeXMLMath .
So LaTeXMLEquation as desired .
LaTeXMLMath For LaTeXMLMath and any positive integer LaTeXMLMath , let LaTeXMLMath be the least integer greater than LaTeXMLMath such that LaTeXMLMath .
If there is no integer with this property , let LaTeXMLMath .
Do not confuse LaTeXMLMath , which will always be subscripted , with the length-related parameter LaTeXMLMath , which will never be .
Our next result shows that the sequence LaTeXMLMath is “ bimodal ” with a maximum at LaTeXMLMath and a minimum at LaTeXMLMath .
For LaTeXMLMath and all positive integers LaTeXMLMath , we have the following three results about the shape of LaTeXMLMath LaTeXMLMath for all LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath .
Proof : For ( i ) we induct on LaTeXMLMath .
The claim is true trivially for LaTeXMLMath since then LaTeXMLMath for all LaTeXMLMath , so we will assume LaTeXMLMath .
If LaTeXMLMath then the claim is true by definition .
If LaTeXMLMath then LaTeXMLMath by Theorem LaTeXMLRef and we are able to apply induction .
We would like to show that LaTeXMLEquation and we know by induction that LaTeXMLEquation .
Subtracting as usual , we are reduced to showing that LaTeXMLMath .
This further reduces to LaTeXMLMath which is true by Lemma LaTeXMLRef ( iii ) and the fact that LaTeXMLMath .
Statement ( ii ) is implied by the definition of LaTeXMLMath , so we are left with ( iii ) .
By the definition of LaTeXMLMath we have that LaTeXMLMath , so it suffices to show that for all LaTeXMLMath , if LaTeXMLMath then LaTeXMLMath .
Subtracting in the usual way , we are reduced to showing that LaTeXMLEquation .
Since we know that LaTeXMLMath by Lemma LaTeXMLRef , our approach will be to show that LaTeXMLMath for LaTeXMLMath by showing that LaTeXMLEquation .
Before we prove ( LaTeXMLRef ) , we will need the following two facts .
LaTeXMLMath and LaTeXMLMath .
The first fact follows from our proof of Lemma LaTeXMLRef , in which we showed that LaTeXMLMath for LaTeXMLMath .
So to prove the second fact , it suffices to show that LaTeXMLMath implies LaTeXMLMath for LaTeXMLMath .
This is proved in exactly the same way as ( i ) with all the inequalities reversed .
Now we are ready to prove ( LaTeXMLRef ) .
First we tackle the case where LaTeXMLMath by induction .
If LaTeXMLMath then LaTeXMLMath and we are done .
So suppose LaTeXMLMath .
If LaTeXMLMath , then since LaTeXMLMath and LaTeXMLMath we have LaTeXMLMath as desired .
Hence we may assume that LaTeXMLMath .
In this case we claim that LaTeXMLMath , which will imply ( LaTeXMLRef ) by induction .
Let LaTeXMLMath .
We want to show that LaTeXMLEquation and we have LaTeXMLEquation .
Subtracting , it suffices to show that LaTeXMLEquation .
By Lemma LaTeXMLRef , LaTeXMLMath , so it suffices to show that LaTeXMLEquation .
Since LaTeXMLMath , we have that LaTeXMLMath , and since LaTeXMLMath , we have that LaTeXMLMath , so ( LaTeXMLRef ) is true , and thus ( LaTeXMLRef ) holds .
For the case where LaTeXMLMath , we examine the quadratics LaTeXMLEquation which agree with LaTeXMLMath , wherever both LaTeXMLMath and LaTeXMLMath are defined .
We will also need to refer to the roots of LaTeXMLMath , which occur at LaTeXMLEquation .
LaTeXMLEquation Lemma LaTeXMLRef gives us that LaTeXMLEquation so LaTeXMLMath and LaTeXMLMath are real numbers and for LaTeXMLMath , LaTeXMLMath .
These roots are important in our situation for the following reasons : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Statement ( LaTeXMLRef ) is easily verified .
Assume to the contrary that the forward direction of ( LaTeXMLRef ) is not true , and thus LaTeXMLMath but LaTeXMLMath .
Let LaTeXMLMath be such that LaTeXMLMath .
By Proposition LaTeXMLRef , we have that LaTeXMLMath , and thus LaTeXMLMath by ( LaTeXMLRef ) .
However because LaTeXMLMath , we have that LaTeXMLMath , a contradiction .
To prove the reverse direction of ( LaTeXMLRef ) , notice that if LaTeXMLMath then by ( i ) and the definition of LaTeXMLMath , we must have that LaTeXMLMath .
Therefore by ( LaTeXMLRef ) , either LaTeXMLMath ( as we would like ) or LaTeXMLMath , and by ( LaTeXMLRef ) , it can not be the case that LaTeXMLMath , as that would imply that LaTeXMLMath if LaTeXMLMath , contradicting Lemma LaTeXMLRef ( iii ) .
To prove ( LaTeXMLRef ) , note that by ( LaTeXMLRef ) we can not have LaTeXMLMath as then we would have LaTeXMLMath , contradicting the definition of LaTeXMLMath .
Also , we can not have LaTeXMLMath as then we would have LaTeXMLMath by ( LaTeXMLRef ) , again contradicting the definition of LaTeXMLMath .
Hence we must have ( LaTeXMLRef ) .
With these tools , ( LaTeXMLRef ) is easy to prove ; we have LaTeXMLMath for LaTeXMLMath , and thus LaTeXMLMath , as desired .
It is easily checked that LaTeXMLMath for LaTeXMLMath .
LaTeXMLMath We will depend on the following lemma to find integers LaTeXMLMath with an internal zero at LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath and all LaTeXMLMath , if LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath for all LaTeXMLMath , so in particular , LaTeXMLMath .
Proof : By Theorem LaTeXMLRef it suffices to show the following inequalities : LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation .
Statement ( LaTeXMLRef ) is clear for LaTeXMLMath because LaTeXMLMath , LaTeXMLMath for all LaTeXMLMath by Lemma LaTeXMLRef , and LaTeXMLMath .
We prove ( LaTeXMLRef ) by induction on LaTeXMLMath .
First , if LaTeXMLMath , then LaTeXMLMath , so it suffices to show that LaTeXMLEquation and since LaTeXMLMath , we have LaTeXMLEquation .
Subtracting that latter from the former , it suffices to show that LaTeXMLEquation .
So we ’ re done in this case since LaTeXMLMath which follows from LaTeXMLMath and LaTeXMLMath .
Now assume that LaTeXMLMath , so we may prove ( LaTeXMLRef ) by showing the stronger statement that LaTeXMLEquation and thus we would like to show that LaTeXMLEquation and as LaTeXMLMath , we have LaTeXMLEquation .
Subtracting as usual , we are reduced to showing LaTeXMLEquation .
By Lemma LaTeXMLRef ( iii ) LaTeXMLEquation .
The upper bound in Lemma LaTeXMLRef now completes the proof of ( LaTeXMLRef ) .
To prove ( LaTeXMLRef ) , we want to show LaTeXMLEquation and we are given LaTeXMLEquation .
Subtracting as usual , we are reduced to showing that LaTeXMLEquation .
Cancelling LaTeXMLMath and simplifying , it suffices to show that LaTeXMLEquation .
By Lemma LaTeXMLRef ( iii ) , LaTeXMLMath , so it suffices to show that LaTeXMLEquation which is true for LaTeXMLMath .
For LaTeXMLMath , note that proving ( LaTeXMLRef ) reduces to showing LaTeXMLMath which we will prove by induction on LaTeXMLMath .
Checking the base cases LaTeXMLMath is easy .
Also note that ( LaTeXMLRef ) holds for LaTeXMLMath if we make the strict inequality weak , so we still can conclude the LaTeXMLMath part of the Lemma .
There are now two cases .
If LaTeXMLMath then by induction LaTeXMLMath .
By Theorem LaTeXMLRef and the part of the Lemma that we ’ ve already proved , the only other possibility is LaTeXMLMath and LaTeXMLMath .
But then LaTeXMLMath which is equivalent to the desired inequality .
LaTeXMLMath There is an intimate connection between partially ordered sets , called posets for short , and permutations .
Using this connection , we will provide characterizations of all LaTeXMLMath -permutations LaTeXMLMath which have LaTeXMLMath for LaTeXMLMath .
This will provide us with the tools we need to show that there are an infinite number of IZ sequences for each of these patterns .
Any necessary definitions from the theory of posets that are not given here will be found in Stanley ’ s text LaTeXMLCite .
If LaTeXMLMath is a poset such that any two distinct elements of LaTeXMLMath are incomparable we say that LaTeXMLMath is an antichain .
Since there is a unique unlabelled antichain on LaTeXMLMath elements , we denote this poset by LaTeXMLMath .
Given posets LaTeXMLMath and LaTeXMLMath , the ordinal sum of LaTeXMLMath and LaTeXMLMath , denoted LaTeXMLMath , is the unique poset on the elements LaTeXMLMath where LaTeXMLMath in LaTeXMLMath if either LaTeXMLMath with LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , or LaTeXMLMath and LaTeXMLMath .
A poset LaTeXMLMath is layered if it is an ordinal sum of antichains , i.e .
if LaTeXMLMath for some LaTeXMLMath .
To introduce a related notion , let LaTeXMLMath denote the set of maximal elements of LaTeXMLMath and LaTeXMLMath .
Then LaTeXMLMath is LOT ( layered on top ) if LaTeXMLMath .
Note that if LaTeXMLMath is layered then it is LOT , but not conversely .
If LaTeXMLMath is a permutation , then the corresponding poset LaTeXMLMath has elements LaTeXMLMath with partial order LaTeXMLMath if LaTeXMLMath is a noninversion in LaTeXMLMath .
So , for example , LaTeXMLMath is a chain , LaTeXMLMath and LaTeXMLMath .
Clearly not every poset is of the form LaTeXMLMath for some LaTeXMLMath .
In fact , the LaTeXMLMath are exactly the posets of dimension at most 2 , being the intersection of the total orders LaTeXMLMath and LaTeXMLMath .
Given posets LaTeXMLMath and LaTeXMLMath let LaTeXMLEquation .
Now given permutations LaTeXMLMath with corresponding posets LaTeXMLMath , we have LaTeXMLMath since the elements of each copy of LaTeXMLMath in LaTeXMLMath form a subposet of LaTeXMLMath isomorphic to LaTeXMLMath .
If LaTeXMLMath then let LaTeXMLEquation .
LaTeXMLEquation We will freely combine these notations and eliminate the subscript when talking about a fixed poset LaTeXMLMath .
We will also abbreviate LaTeXMLMath to LaTeXMLMath and LaTeXMLMath to LaTeXMLMath .
As with permutations , for any non-negative integer LaTeXMLMath we will let LaTeXMLMath .
We will say a poset LaTeXMLMath is LaTeXMLMath -optimal if LaTeXMLMath .
Stromquist proved Theorem LaTeXMLRef by first demonstrating the following stronger result .
If LaTeXMLMath is a LOT pattern , then there is some LaTeXMLMath -optimal LOT poset LaTeXMLMath .
The same holds with “ LOT ” replaced by “ layered. ” To show that the sequences of the patterns LaTeXMLMath , for LaTeXMLMath , have infinitely many IZ integers , we will need to know more about LaTeXMLMath -optimal posets .
The best possible case would be if all ( sufficiently large ) LaTeXMLMath -optimal posets were layered .
This is true for the pattern LaTeXMLMath , but not in general .
For example , it can be computed that LaTeXMLMath is LaTeXMLMath -optimal , but LaTeXMLMath is not layered .
Fortunately , we are able to show that all LaTeXMLMath -optimal posets are of the following slightly more general form .
We say LaTeXMLMath is an LaTeXMLMath -decomposition of LaTeXMLMath if LaTeXMLMath is layered and for all LaTeXMLMath with LaTeXMLMath we have LaTeXMLMath .
The first part of this section concerns the proof of the following theorem .
If LaTeXMLMath is an LaTeXMLMath -optimal poset then LaTeXMLMath has an LaTeXMLMath -decomposition .
After this proof we will investigate ‘ almost ’ LaTeXMLMath -optimal posets , that is , posets LaTeXMLMath with LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are permutations , it is generally not the case that LaTeXMLMath .
For example , LaTeXMLMath and thus LaTeXMLMath , but LaTeXMLMath .
However , there is an important case in which we do get equality .
If LaTeXMLMath and LaTeXMLMath are permutations then LaTeXMLMath .
Furthermore , if either LaTeXMLMath or LaTeXMLMath is layered then LaTeXMLMath .
Proof : The inequality follows from the fact that each copy of LaTeXMLMath in LaTeXMLMath gives rise to a copy of LaTeXMLMath in LaTeXMLMath .
For the equality , if LaTeXMLMath is layered then it is the unique permutation giving rise to the poset LaTeXMLMath .
So every copy of LaTeXMLMath in LaTeXMLMath corresponds to a copy of LaTeXMLMath in LaTeXMLMath and we are done .
The only other case we need to consider is if LaTeXMLMath is layered and LaTeXMLMath is not .
But then both sides of the equality are zero .
LaTeXMLMath This lemma and the preceeding theorems imply several important features about the connection between pattern matching in posets and permutations .
Given any pattern LaTeXMLMath , the first statement in Lemma LaTeXMLRef implies that LaTeXMLMath for all LaTeXMLMath .
If LaTeXMLMath is layered , then by Theorem LaTeXMLRef there is a layered LaTeXMLMath -optimal poset LaTeXMLMath for some positive integers LaTeXMLMath .
It follows that there is a layered permutation LaTeXMLMath such that LaTeXMLMath , namely LaTeXMLMath is the permutation whose layer lengths from left to right are LaTeXMLMath .
By the preceeding lemma , LaTeXMLMath , so LaTeXMLMath .
For all patterns LaTeXMLMath , the sequence LaTeXMLMath is positive and strictly increasing .
Proof : We will write LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Given LaTeXMLMath , it is easy to construct a poset LaTeXMLMath with LaTeXMLMath .
So let LaTeXMLMath be a LaTeXMLMath -optimal poset .
Now there must be some LaTeXMLMath with LaTeXMLMath .
Now adjoin an element LaTeXMLMath to LaTeXMLMath to form a poset LaTeXMLMath with LaTeXMLMath in LaTeXMLMath if either LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , LaTeXMLMath with LaTeXMLMath , or LaTeXMLMath , LaTeXMLMath with LaTeXMLMath .
Then LaTeXMLEquation so LaTeXMLMath .
LaTeXMLMath We now begin the proof of Theorem LaTeXMLRef by making a few definitions .
If LaTeXMLMath is a poset and LaTeXMLMath then the open down-set generated by LaTeXMLMath is LaTeXMLEquation .
If LaTeXMLMath then let LaTeXMLMath be the unique poset on the same set of elements which satisfies LaTeXMLEquation .
Note that LaTeXMLMath .
The following lemma is essentially in Stromquist LaTeXMLCite , but is not explicitly proved there .
So we will provide a demonstration .
Let LaTeXMLMath be a LOT pattern and LaTeXMLMath be any poset with LaTeXMLMath .
Then LaTeXMLEquation .
Proof : As before , we write LaTeXMLMath for LaTeXMLMath .
Since LaTeXMLEquation .
LaTeXMLEquation it is enough to show that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
First , ( LaTeXMLRef ) is clear since LaTeXMLMath and LaTeXMLMath agree on all subsets not including LaTeXMLMath .
Next , notice that LaTeXMLEquation and thus to prove ( LaTeXMLRef ) , it suffices to show that LaTeXMLMath , but this is easy .
Let LaTeXMLMath with LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then LaTeXMLMath is an occurance of LaTeXMLMath in LaTeXMLMath , i.e. , LaTeXMLMath , so ( LaTeXMLRef ) is proved .
Finally , to prove ( LaTeXMLRef ) , let LaTeXMLMath be an occurance of LaTeXMLMath in LaTeXMLMath which contains LaTeXMLMath and LaTeXMLMath , i.e. , LaTeXMLMath .
Then we have that LaTeXMLMath as well .
This is because LaTeXMLMath in LaTeXMLMath since LaTeXMLMath are maximal and LaTeXMLMath is LOT .
So LaTeXMLMath forms an occurance of LaTeXMLMath in LaTeXMLMath , and thus ( LaTeXMLRef ) is proven .
LaTeXMLMath For the rest of this section , let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a poset such that LaTeXMLMath .
If for some LaTeXMLMath and LaTeXMLMath we have LaTeXMLMath is LOT , LaTeXMLMath , and LaTeXMLMath for all LaTeXMLMath , then LaTeXMLMath is LOT ( and thus LaTeXMLMath is actually LaTeXMLMath ) .
Proof : Choose LaTeXMLMath with LaTeXMLMath , let LaTeXMLMath and LaTeXMLMath .
First consider what happens when LaTeXMLMath .
Then ( a ) implies that LaTeXMLMath for all LaTeXMLMath .
This forces LaTeXMLMath by ( c ) .
Now ( b ) yields LaTeXMLMath , contradicting Lemma LaTeXMLRef .
So we may assume LaTeXMLMath .
Note that LaTeXMLMath , and since LaTeXMLMath we get that LaTeXMLEquation .
Furthermore , since LaTeXMLMath is LOT we get that LaTeXMLEquation and LaTeXMLEquation .
Also , since LaTeXMLMath , we have that LaTeXMLEquation .
LaTeXMLEquation Furthermore , since LaTeXMLMath is LOT , LaTeXMLMath is LOT , so we have LaTeXMLEquation .
Therefore LaTeXMLEquation .
Furthermore , by Lemma LaTeXMLRef and assumptions ( b ) and ( c ) we have that LaTeXMLMath .
So we must have LaTeXMLMath and , by ( b ) again , LaTeXMLMath .
It follows that LaTeXMLMath .
Therefore since LaTeXMLMath we have LaTeXMLMath and so LaTeXMLMath .
Also , because LaTeXMLMath , we have LaTeXMLMath and thus LaTeXMLMath is LOT , as desired .
LaTeXMLMath For any poset LaTeXMLMath , let LaTeXMLMath be defined by LaTeXMLEquation .
Clearly LaTeXMLMath , with equality if and only if LaTeXMLMath is LOT .
It turns out that LaTeXMLMath is a useful statistic for induction .
We now have all the necessary tools to prove Theorem LaTeXMLRef .
Proof of Theorem LaTeXMLRef : Notice that the claim is trivial for LaTeXMLMath as all posets on less than LaTeXMLMath elements can not have any LaTeXMLMath -patterns and thus they have the trivial LaTeXMLMath - decomposition LaTeXMLMath .
Assume to the contrary that the claim is not true and let LaTeXMLMath be an LaTeXMLMath - optimal poset of least cardinality that does not have a LOT LaTeXMLMath -decomposition with LaTeXMLMath maximal over all such choices of LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the set from Definition LaTeXMLRef , LaTeXMLMath and LaTeXMLMath .
First , we claim that LaTeXMLMath is LOT .
If not , then there is some element , say LaTeXMLMath .
Also let LaTeXMLMath .
If LaTeXMLMath , then by Lemma LaTeXMLRef either LaTeXMLMath or LaTeXMLMath , both contradictions , so LaTeXMLMath and LaTeXMLMath is LaTeXMLMath -optimal .
Since LaTeXMLMath , by our choice of LaTeXMLMath we know that LaTeXMLMath has an LaTeXMLMath -decomposition LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath , so by Lemma LaTeXMLRef , LaTeXMLMath ( because LaTeXMLMath ) , a contradiction to our choice of LaTeXMLMath .
Hence we may assume that LaTeXMLMath , so LaTeXMLMath is LOT .
As the only element LaTeXMLMath and LaTeXMLMath disagree on is LaTeXMLMath , we have that LaTeXMLMath is LOT .
Hence by Lemma LaTeXMLRef , LaTeXMLMath is also LOT .
Now that we know that LaTeXMLMath is LOT , we get that LaTeXMLMath , so LaTeXMLMath is LaTeXMLMath - optimal .
By induction , LaTeXMLMath has an LaTeXMLMath -decomposition LaTeXMLMath and thus LaTeXMLMath is an LaTeXMLMath - decomposition for LaTeXMLMath .
LaTeXMLMath Note that by using the ideas in the last paragraph of this proof one may show that if LaTeXMLMath is an LaTeXMLMath -decomposition for an LaTeXMLMath -optimal poset LaTeXMLMath then LaTeXMLMath .
Hence because all posets on less than three elements are layered , all LaTeXMLMath -optimal posets ( and thus LaTeXMLMath -optimal permutations ) are layered .
This observation will be useful in the following proof .
If LaTeXMLMath is such that LaTeXMLMath then there is a poset LaTeXMLMath with LaTeXMLMath and one of the following : LaTeXMLMath and LaTeXMLMath is LOT , or LaTeXMLMath is LaTeXMLMath -optimal and LaTeXMLMath , or LaTeXMLMath and LaTeXMLMath Proof : Assume that ( i ) does not hold and choose LaTeXMLMath with LaTeXMLMath and LaTeXMLMath maximal over all such choices .
Let LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
We must have LaTeXMLMath for all LaTeXMLMath as otherwise by Lemma LaTeXMLRef we would have either LaTeXMLMath or LaTeXMLMath , a contradiction .
Hence we have LaTeXMLEquation .
First we tackle the easier case , where LaTeXMLMath .
Pick two maximal elements of LaTeXMLMath , say LaTeXMLMath , so that LaTeXMLMath .
By Lemma LaTeXMLRef we have that LaTeXMLMath , and thus by Theorem LaTeXMLRef we know LaTeXMLMath has an LaTeXMLMath -decomposition LaTeXMLMath .
Since LaTeXMLMath , we must have LaTeXMLMath , so we also have that LaTeXMLMath and LaTeXMLMath .
Therefore LaTeXMLMath and consequently LaTeXMLMath are LOT .
Hence by Lemma LaTeXMLRef , LaTeXMLMath is LOT , a contradiction .
Now assume LaTeXMLMath .
Let LaTeXMLMath be as in Definition LaTeXMLRef , pick LaTeXMLMath ( LaTeXMLMath must exist as LaTeXMLMath is not LOT ) and LaTeXMLMath .
Now LaTeXMLMath and thus LaTeXMLMath by Lemma LaTeXMLRef .
However if LaTeXMLMath then we have contradicted our choice of LaTeXMLMath as LaTeXMLMath .
Therefore LaTeXMLMath so by Theorem LaTeXMLRef , LaTeXMLMath has an LaTeXMLMath -decomposition LaTeXMLMath .
By the same reasoning as the previous case , LaTeXMLMath , so again LaTeXMLMath and LaTeXMLMath are both LOT .
Although we can not apply Lemma LaTeXMLRef in this case , ( LaTeXMLRef ) still holds for LaTeXMLMath with LaTeXMLMath , so LaTeXMLEquation .
Therefore we must have LaTeXMLMath .
If LaTeXMLMath , this implies that LaTeXMLMath , so ( ii ) is true with LaTeXMLMath .
If LaTeXMLMath then we must have LaTeXMLMath , so there is precisely one element , say LaTeXMLMath .
Since LaTeXMLMath is LOT , LaTeXMLMath must lie in LaTeXMLMath .
Let LaTeXMLMath .
Then we have LaTeXMLEquation .
Because LaTeXMLMath is LOT , we have that LaTeXMLMath , and because LaTeXMLMath is LOT we have that LaTeXMLMath .
Notice that because LaTeXMLMath is LaTeXMLMath -optimal , by the comment after the proof of Theorem LaTeXMLRef , LaTeXMLMath is layered , and thus LaTeXMLMath is layered .
Since the LaTeXMLMath -patterns in LaTeXMLMath containing both LaTeXMLMath and LaTeXMLMath are formed with exactly one element which lies in LaTeXMLMath , LaTeXMLMath .
Finally , LaTeXMLMath .
Now combining all these LaTeXMLMath -values with equation ( LaTeXMLRef ) gives LaTeXMLEquation so LaTeXMLMath .
We have by Lemma LaTeXMLRef ( iii ) that LaTeXMLMath and LaTeXMLMath ( this follows from the fact that LaTeXMLMath is layered and LaTeXMLMath -optimal ) , which forces LaTeXMLMath .
This in turn implies LaTeXMLMath .
Now it can be checked by direct computation that for LaTeXMLMath in this range either the theorem is true vacuously or one of ( i ) to ( iii ) holds .
LaTeXMLMath If there is an LaTeXMLMath -poset LaTeXMLMath with LaTeXMLMath then there is an LaTeXMLMath -poset LaTeXMLMath with LaTeXMLMath and if LaTeXMLMath then LaTeXMLMath is layered , or if LaTeXMLMath then LaTeXMLMath where LaTeXMLMath and LaTeXMLMath is layered .
Furthermore , in either case LaTeXMLMath for some permutation LaTeXMLMath and integer LaTeXMLMath which is positive unless LaTeXMLMath and LaTeXMLMath .
Proof : Induct on LaTeXMLMath .
If LaTeXMLMath , then LaTeXMLMath , so the theorem is true vacuously .
If LaTeXMLMath , then LaTeXMLMath and LaTeXMLMath .
Hence we may assume that LaTeXMLMath .
If case ( ii ) of Theorem LaTeXMLRef is true , let LaTeXMLMath be the poset guaranteed there , LaTeXMLMath and LaTeXMLMath .
Then by Lemma LaTeXMLRef ( iii ) , LaTeXMLMath , so LaTeXMLMath , a case we have already dealt with .
It is routine to check that the poset LaTeXMLMath satisfies case ( ii ) of this theorem if case ( iii ) of Theorem LaTeXMLRef is true .
Therefore we may assume that case ( i ) of Theorem LaTeXMLRef is true , and thus there is a LOT LaTeXMLMath -poset LaTeXMLMath so that LaTeXMLMath .
Since LaTeXMLMath is LOT , LaTeXMLMath .
As LaTeXMLMath , we must have LaTeXMLMath .
If LaTeXMLMath , then by Theorem LaTeXMLRef , there is some layered LaTeXMLMath -poset LaTeXMLMath so that LaTeXMLMath , and thus LaTeXMLMath is layered , LaTeXMLMath and LaTeXMLMath for some LaTeXMLMath .
If LaTeXMLMath , then by induction , there is some poset LaTeXMLMath , LaTeXMLMath , which satisfies this theorem .
So LaTeXMLMath is the desired poset .
LaTeXMLMath For the pattern LaTeXMLMath , there are infinitely many IZ integers .
Proof : Assume that the theorem is false .
Since LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , there must be some maximal LaTeXMLMath so that LaTeXMLMath .
By Theorem LaTeXMLRef , there is some LaTeXMLMath so that LaTeXMLMath and LaTeXMLMath .
Also note that since LaTeXMLMath , by Lemma LaTeXMLRef ( iii ) we have LaTeXMLMath , so we may apply Lemma LaTeXMLRef to see that LaTeXMLMath .
By our choice of LaTeXMLMath , LaTeXMLMath , so there is some LaTeXMLMath so that LaTeXMLMath .
By Lemma LaTeXMLRef , LaTeXMLMath , and thus Theorem LaTeXMLRef produces a poset LaTeXMLMath for some LaTeXMLMath and integer LaTeXMLMath which is positive since LaTeXMLMath .
Let LaTeXMLMath .
By Theorem LaTeXMLRef , there is a layered LaTeXMLMath -optimal LaTeXMLMath -poset LaTeXMLMath , and so we must have LaTeXMLMath .
Therefore , by Lemma LaTeXMLRef , we have LaTeXMLMath , and thus the inequality in Lemma LaTeXMLRef implies that LaTeXMLMath .
However , if LaTeXMLMath then we have LaTeXMLMath , contradicting our choice of LaTeXMLMath .
LaTeXMLMath Numerical evidence and the contrast between Proposition LaTeXMLRef and Theorem LaTeXMLRef amkes us suspect that Theorem LaTeXMLRef is not true for LaTeXMLMath , LaTeXMLMath .
In fact , we believe the following is true .
The frequency sequence for LaTeXMLMath , LaTeXMLMath has internal zeros for all LaTeXMLMath .
It would be interesting to find a proof of this conjecture .
Perhaps a first step would be to find a simpler proof of Theorem LaTeXMLRef .
Acknowledgement .
This paper was written , in part , when Bruce Sagan was resident at the Isaac Newton Institute for Mathematical Sciences .
He would like to express his appreciation for the support of the Institute during this period .
G. Papadopoulos and P.K .
Townsend , Compactification of LaTeXMLMath Supergravity on Spaces of Exceptional Holonomy , Phys .
Lett B357 ( 1995 ) 300 : hep-th/9506150 .
Superpotentials and Membrane Instantons hep-th/9907026 .
Moduli Space of Calabi-Yau Manifolds , Nucl .
Phys .
B355 ( 1991 ) 455 .
Rolling Among LaTeXMLMath Vacua , JHEP 0103:005 , 2001 ; hep-th/0011130 .
Compact Riemannian 7-Manifolds with Holonomy LaTeXMLMath , I , II , J. Diff .
Geom .
43 ( 1996 ) 291 and 329 .
D-Brane Approach to Black Hole Quantum Mechanics , Nucl .
Phys .
B472 ( 1996 ) 591 ; hep-th/9602043 .
LaTeXMLMath Extremal Black Holes , Phys .
Rev .
D52 ( 1995 ) 5412 ; hep-th/9508072 .
Attractors and Arithmetic , hep-th/9807056 .
The geometry of three-forms in six and seven dimensions , math.DG/0010054 .
Bogomolny Bounds for Cosmic Strings Nucl.Phys .
B299 :719 , 1988 .
Stringy Cosmic Strings and Noncompact Calabi-Yau Manifolds , Nucl.Phys .
B337 :1 , 1990 .
Intersecting M-branes , Phys .
Lett .
B380 ( 1996 ) 273 ; hep-th/9603087 .
Non-extreme Black Holes from Non-extreme Intersecting M-Branes , Nucl .
Phys .
B478 ( 1996 ) 181 ; hep-th/9606033 .
Yang-Mills Theories with Local Supersymmetry : Lagrangian , Transformation Laws and Superhiggs Effect , Nucl.Phys .
B212 :413,1983 .
Quantization of Newton ’ s Constant in Certain Supergravity Theories , Phys.Lett .
B115 :202,1982 .
Supersymmetry and Supergravity , Princeton University Press , ( 1992 ) .
The Geometry of Duality , Phys .
Lett .
168B ( 1986 ) 98 .
Supergravity in Eleven-Dimensions Phys .
Lett .
B76 ( 1978 ) 403 .
Solitons , Superpotentials and Calibrations , Nucl .
Phys .
B574 ( 2000 ) 169 ; hep-th/9911011 .
Calibrated Geometries and Non Perturbative Superpotentials in M-Theory , Eur .
Phys .
J. C18 ( 2001 ) 619 ; hep-th/9912022 .
Flux , Supersymmetry and M Theory on 7-Manifolds , hep-th/0007213 .
Compactifying M-theory to Four-Dimensions , hep-th/0010282 .
Calibrated Geometries , Acta Mathematica 148 ( 1982 ) 47 .
Five-Branes , Membranes and Non-Perturbative String Theory , Nucl .
Phys .
B456 ( 1995 ) 130 ; hep-th/9507158 .
Kappa Symmetry , Supersymmetry and Intersecting Branes , Nucl .
Phys .
B502 ( 1997 ) 149 ; hep-th/9705040 .
Deformations of Calibrated Submanifolds , Commun .
in Analysis and Geometry , 6 ( 1998 ) 705 .
Compact Manifolds with Special Holonomy Oxford University Press , ( 2000 ) .
Calibrations and Intersecting Branes , Commun .
Math .
Phys .
202 ( 1999 ) 593 ; hep-th/9803163 .
Branes and Calibrated Geometries , Commun .
Math .
Phys .
202 ( 1999 ) 571 ; hep-th/9803216 .
Magnetic Cosmic Strings of N=1 , D = 4 Supergravity with Cosmological Constant : hep-th/0102165 .
Evidence for Heterotic - Type I String Duality , Nucl.Phys .
B460 :525-540,1996 ; hep-th/9510169 .
Duality of Type II 7 Branes and 8 Branes , Nucl.Phys .
B470 :113-135,1996 ; hep-th/9601150 .
Lectures on Harmonic Maps Conference Proceedings and Lecture Notes in Geometry and Topology , Vol II , International Press ( 1997 ) .
Supergravity Domain Walls , Phys .
Rep. 282 ( 1997 ) 159 ; hep-th/9604090 .
Moduli Space Metric for Maximally-Charged Dilaton Black Holes , Nucl .
Phys .
B402 ( 1993 ) 399 .
October , 2001 LaTeXMLMath Manifolds We investigate the local geometry on the moduli space of LaTeXMLMath structures that arises in compactifications of M-theory on holonomy LaTeXMLMath manifolds .
In particular , we determine the homogeneity properties of couplings of the associated LaTeXMLMath , LaTeXMLMath supergravity under the scaling of moduli space coordinates .
We then find some brane solitons of LaTeXMLMath , LaTeXMLMath supergravity that are associated with wrapping M-branes on cycles of the compact space .
These include cosmic strings and domain walls that preserve LaTeXMLMath of supersymmetry of the four-dimensional theory , and non-supersymmetric electrically and magnetically charged black holes .
The geometry of some of the black holes is that of non-extreme M-brane configurations reduced to four-dimensions on a seven torus .
=2 Introduction Compactifications of M-theory on manifolds with LaTeXMLMath holonomy provide a way of constructing four-dimensional effective theories which have a realistic amount of supersymmetry .
These effective theories are LaTeXMLMath supergravities with field content which is determined by the Betti numbers of the compact space .
In particular it has been shown that the associated LaTeXMLMath supergravity has LaTeXMLMath vector and LaTeXMLMath chiral multiplets [ LaTeXMLMath supergravity on compact LaTeXMLMath holonomy manifolds , one can also determine the couplings of the four-dimensional theory as a function of the various moduli fields [ LaTeXMLMath structures in a similar way to that of Calabi-Yau compactifications of string theory , see for example [ LaTeXMLMath manifolds have been investigated , see for example [ LaTeXMLMath manifolds constructed by Joyce in [ Some of the solutions of D=4 and D=5 supergravity theories which arise from compactifications of strings and M-theory , like black holes , strings and domain walls , can be associated with branes wrapped on the homology cycles of the compact manifold .
This correspondence between solutions of lower dimensional supergravity theories and ten- and eleven-dimensional brane configurations has been very fruitful , for example it has led to the microscopic computation of the black hole entropy for a certain class of extreme black holes [ LaTeXMLMath supergravity theories [ In this paper , we shall examine the couplings of the LaTeXMLMath supergravity theories that arise from compactifications of LaTeXMLMath supergravity on compact holonomy LaTeXMLMath manifolds .
In particular we shall show that the components of the metric of the sigma model manifold , which is LaTeXMLMath , of LaTeXMLMath theory are homogeneous of degree LaTeXMLMath under the scaling of certain coordinates of the moduli space LaTeXMLMath of LaTeXMLMath structures .
For this we shall use a result obtained by Hitchin [ LaTeXMLMath manifold is homogeneous of degree LaTeXMLMath under the scaling of some coordinates of LaTeXMLMath .
It turns out that the metric on the moduli space of LaTeXMLMath structures is invariant under this scaling transformation ; the isometry group of the sigma model metric on LaTeXMLMath is generated by the same scaling transformation and the translations along the fibres .
In addition , we shall show that the Kähler potential of the sigma model manifold can be expressed in terms of the logarithm of the volume of the compact holonomy LaTeXMLMath manifold , see also [ Having established the homogeneity properties of the couplings of LaTeXMLMath supergravity associated with compactifications of LaTeXMLMath supergravity on holonomy LaTeXMLMath manifolds , we shall explore the various solutions of the four-dimensional theory that arise by wrapping M-branes on the homology cycles of the compact manifold .
We shall find that the LaTeXMLMath supergravity admits string solutions which preserve LaTeXMLMath of supersymmetry .
These are associated with M5-branes wrapped on coassociative cycles of the compact manifold .
The form of these string solutions is that of cosmic string solutions of [ LaTeXMLMath compactifications have infinite tension because the sigma model manifold LaTeXMLMath is non-compact .
We shall also describe the M-theory origin of domain wall solutions of LaTeXMLMath supergravity which preserve LaTeXMLMath of the supersymmetry .
The Killing spinor equations for such domain walls are given in an appendix .
Next we shall explore the electric and magnetic black hole solutions of LaTeXMLMath supergravity that arise from wrapping M2-branes and M5-branes on 2- and 5-cycles of LaTeXMLMath , respectively .
We shall show that such solutions are not supersymmetric as expected .
We shall mainly focus in the case where the only non-vanishing modulus field is that corresponding to the overall scale of the moduli space coordinates .
We shall call such solutions “ dilatonic ” ; we justify this terminology in an appendix .
We shall find a class of extreme dilatonic solutions of the LaTeXMLMath theory which have the same spacetime geometry as two supersymmetric orthogonally intersecting M-branes , eg two M2-branes intersecting on a LaTeXMLMath -brane or two M5-branes intersecting on a 3-brane [ We remark that our dilatonic domain wall and black hole solutions depend only on the homogeneity properties of the couplings of the LaTeXMLMath supergravity .
So they will remain solutions of the effective theory of LaTeXMLMath compactifications after perturbative or non-perturbative corrections to the couplings are taken into account which preserve these homogeneity properties .
The organization of this paper is as follows : In section two , we present the action , Killing spinor equations and the geometry associated with the couplings LaTeXMLMath supergravity with scalar and vector multiplets .
In section three , we give the couplings of LaTeXMLMath supergravity that arise from the compactification of LaTeXMLMath supergravity on holonomy LaTeXMLMath manifolds .
We then present two approaches in the investigation of the geometry of the moduli space of LaTeXMLMath structures .
One is based on the Kähler geometry and the other is based on the symplectic geometry .
We also express the metric on the LaTeXMLMath moduli space , that arises in these compactifications , in terms of the volume of the compact holonomy LaTeXMLMath manifold .
In section four , we summarize some of the results on calibrating cycles in holonomy LaTeXMLMath manifolds .
We also give the number of supersymmetries preserved by M-brane probes wrapping such cycles .
In section five , we present our string solutions of LaTeXMLMath supergravity associated with LaTeXMLMath compactification .
In section six , we describe various domain walls .
In section seven , we give various black hole solutions associated with M2- and M5-branes wrapped on homology 2- and 5-cycles of the compact manifold .
Finally in the appendices , we give our spinor conventions , analyze the Killing spinor equations of LaTeXMLMath supergravity in connection to strings and domain walls that arise in LaTeXMLMath compactifications , and give the bosonic action that describes the dilatonic black hole system .
LaTeXMLMath LaTeXMLMath Supergravity Supergravity Action and Killing Spinor Equations The geometric data that determine the couplings of LaTeXMLMath supergravity in four-dimensions with LaTeXMLMath vector and LaTeXMLMath chiral multiplets that we shall use in this paper are the following : ( i ) A Kähler-Hodge manifold LaTeXMLMath of complex dimension LaTeXMLMath with Kähler potential LaTeXMLMath .
( ii ) A vector bundle LaTeXMLMath over LaTeXMLMath of rank LaTeXMLMath for which its complexified symmetric product admits a holomorphic section LaTeXMLMath .
( iii ) A locally defined holomorphic function LaTeXMLMath on LaTeXMLMath .
( iv ) Sigma model maps , LaTeXMLMath , from the four-dimensional spacetime LaTeXMLMath into the manifold LaTeXMLMath .
( v ) A principal bundle LaTeXMLMath on the four-dimensional spacetime LaTeXMLMath with fibre the abelian group LaTeXMLMath such that the pull back of LaTeXMLMath with respect to LaTeXMLMath is isomorphic to LaTeXMLMath , where LaTeXMLMath is the Lie algebra of LaTeXMLMath .
Given these data , the bosonic part of LaTeXMLMath supergravity action [ LaTeXMLEquation where LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLMath are LaTeXMLMath ( Maxwell ) gauge potentials and the LaTeXMLMath are constants associated to a Fayet-Iliopoulos term .
We remark that the gauge indices LaTeXMLMath are raised and lowered with LaTeXMLMath ; LaTeXMLMath and LaTeXMLMath are holomorphic sigma model manifold and spacetime indices , respectively .
Clearly LaTeXMLMath are the gauge couplings , LaTeXMLMath is a superpotential and LaTeXMLMath is the sigma model manifold .
The above action may also describe the coupling of LaTeXMLMath linear multiplets to LaTeXMLMath , LaTeXMLMath supergravity [ This is because the two-form gauge potentials of the linear multiplets can be dualized to scalars in four dimensions .
The resulting action depends only on the spacetime derivatives of dual scalars .
Therefore it is invariant under LaTeXMLMath acting on these scalars with constant shifts .
In what follows some of the solutions of the LaTeXMLMath supergravity that we shall consider will be supersymmetric .
To explore their properties , we need the Killing spinor equations of LaTeXMLMath as LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLEquation where underlined indices LaTeXMLMath denote tangent frame indices and LaTeXMLMath .
For our spinor conventions see the appendix .
The field equations of the supergravity action ( 1 ) The Einstein equations are : LaTeXMLEquation ( 2 ) The Maxwell field equations are : LaTeXMLEquation ( 3 ) The scalar equations ; varying LaTeXMLMath gives the equation LaTeXMLEquation where LaTeXMLMath is the covariant derivative with respect to the Levi-Civita connection of the spacetime metric and LaTeXMLEquation .
Taking the conjugate of this equation , one obtains the field equation for LaTeXMLMath .
Supergravity Actions from LaTeXMLMath Compactifications Compactification Ansatz The bosonic part of the LaTeXMLMath supergravity Lagrangian [ LaTeXMLEquation where LaTeXMLMath is the eleven-dimensional metric , LaTeXMLMath is the 4-form LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
field strength and LaTeXMLMath is the 3-form gauge potential , LaTeXMLMath .
To derive the LaTeXMLMath supergravity action that arises from the compactification of LaTeXMLMath supergravity on a holonomy LaTeXMLMath manifold LaTeXMLMath , we introduce a basis of harmonic forms LaTeXMLMath in LaTeXMLMath and similarly a basis LaTeXMLMath in LaTeXMLMath .
Repeating the analysis in [ LaTeXMLMath and the three-form gauge potential LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath is the metric on LaTeXMLMath with holonomy LaTeXMLMath depending on the real coordinates LaTeXMLMath of the moduli space LaTeXMLMath of the LaTeXMLMath structures which have been promoted to four-dimensional scalar fields ; LaTeXMLMath .
In addition , LaTeXMLMath and LaTeXMLMath are the one-form gauge potentials and real scalars of the four-dimensional theory , respectively .
The fields LaTeXMLMath describe the small fluctuations of the LaTeXMLMath manifold LaTeXMLMath within the LaTeXMLMath supergravity .
To solve the field equations of LaTeXMLMath supergravity at the linearized level , the basis of harmonic forms LaTeXMLMath in LaTeXMLMath is chosen with respect to the LaTeXMLMath metric and similarly for the basis LaTeXMLMath in LaTeXMLMath .
So far the coordinates LaTeXMLMath on the moduli space LaTeXMLMath have been chosen in an arbitrary way .
However below for the investigation of the couplings of the LaTeXMLMath supergravity theory a special choice will be made .
The compactification of LaTeXMLMath supergravity on holonomy LaTeXMLMath manifolds preserves four real supercharges .
So it is expected that the four-dimensional action that describes the dynamics of the small fluctuations of such background will exhibit LaTeXMLMath supersymmetry .
Some of the couplings of the four-dimensional effective action can be easily deduced from the eleven dimensional supergravity action and are as follows : LaTeXMLEquation where LaTeXMLMath is the parallel 3-form associated with the LaTeXMLMath structure on LaTeXMLMath and we have used LaTeXMLMath to also denote the determinant of the metric LaTeXMLMath with holonomy LaTeXMLMath .
In an adapted frame LaTeXMLMath of the LaTeXMLMath structure the 3-form LaTeXMLMath can be written as LaTeXMLEquation .
The last equality in the third equation in LaTeXMLMath representation theory , see for example [ LaTeXMLMath are topological and so they do not depend on the moduli space coordinates LaTeXMLMath of LaTeXMLMath structures or the choice of harmonic representatives .
In addition , we remark that the couplings in not depend on the harmonic representatives chosen for the basis LaTeXMLMath in LaTeXMLMath and the basis LaTeXMLMath in LaTeXMLMath .
To express the four-dimensional couplings as above , we have rescaled the four-dimensional metric LaTeXMLMath with the volume LaTeXMLMath of the compact space in order to bring the LaTeXMLMath action in the Einstein frame .
Observe that the sigma model metric LaTeXMLMath above is invariant under the action of LaTeXMLMath acting with constant shifts on LaTeXMLMath .
Since the LaTeXMLMath compactifications preserve four real supercharges , the effective theory has LaTeXMLMath supersymmetry .
In particular the couplings the previous section for the couplings of LaTeXMLMath supergravity .
There are two ways to describe this depending on the way we choose coordinates on the moduli space which we shall now describe below .
LaTeXMLMath Moduli Space : A Kähler Approach The sigma model metric LaTeXMLMath in LaTeXMLMath supersymmetry to be Kähler .
In addition the action of the group LaTeXMLMath on LaTeXMLMath leaves the sigma model metric LaTeXMLMath invariant and also commutes with the supersymmetry transformations of the scalars .
This is because , as for the D=4 effective action , the supersymmetry transformations depend only on the spacetime derivatives of the fields LaTeXMLMath .
This implies that LaTeXMLMath acts with holomorphic isometries on the sigma model target space LaTeXMLMath which can be identified with the tangent space LaTeXMLMath of the moduli space LaTeXMLMath of LaTeXMLMath structures .
The typical fibre of LaTeXMLMath is LaTeXMLMath .
To continue the investigation of the moduli space geometry , it is convenient to choose coordinates on the moduli space LaTeXMLMath so that the parallel form is LaTeXMLEquation .
In fact , the basis LaTeXMLMath of harmonic 3-form with respect to the LaTeXMLMath metric depends on the choice of LaTeXMLMath structure and so on the coordinates LaTeXMLMath .
We take the origin of the coordinate system to be LaTeXMLMath .
However this dependence does not contribute in the couplings LaTeXMLMath .
Next one can introduce holomorphic coordinates LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath acts on LaTeXMLMath with shifts along the imaginary directions .
In these coordinates , the sigma model metric on LaTeXMLMath is LaTeXMLEquation and the Kähler form is LaTeXMLEquation .
Comparing the metric LaTeXMLEquation where LaTeXMLMath is the Kähler potential .
We turn now to investigate the couplings of the vector multiplets .
Using the holomorphicity of LaTeXMLMath required by supersymmetry and the expression given in LaTeXMLEquation .
The coupling LaTeXMLMath can be though as a holomorphic section of a bundle with fibre LaTeXMLMath over the sigma model manifold LaTeXMLMath .
It remains now to find the Kähler potential of the metric on the sigma model manifold LaTeXMLMath .
We shall show that the Kähler potential LaTeXMLEquation .
For this , we shall use the relation shown by Hitchin in [ LaTeXMLEquation where LaTeXMLMath is a top-form constructed taking the determinant of LaTeXMLEquation .
It is clear from this that LaTeXMLMath is homogeneous of degree LaTeXMLMath in the LaTeXMLMath coordinates LaTeXMLMath is taken to be the Hessian of LaTeXMLMath which differs from the metric that arises in the compactifications we are investigating .
The metric associated with the Hessian of LaTeXMLMath has Lorentzian signature .
.
The metric on the LaTeXMLMath holonomy manifold is given by LaTeXMLMath .
Using this one can show that LaTeXMLEquation .
To proceed we remark that the LaTeXMLMath representation of LaTeXMLMath of dimension LaTeXMLMath can be decomposed under the action of LaTeXMLMath as LaTeXMLMath .
The first is the direction along the LaTeXMLMath invariant form LaTeXMLMath .
The representation LaTeXMLMath is associated with the vector arising from the inner product of a 3-forms with LaTeXMLMath .
If the three-form is harmonic with respect to the LaTeXMLMath metric , as it is the case here , then this part vanishes due to a standard argument about harmonic one-forms on irreducible Ricci-flat spaces .
Therefore LaTeXMLMath can be written as LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are the obvious projections .
In addition it is known [ LaTeXMLEquation .
Using LaTeXMLEquation .
The numerical normalization factors that appear in the expressions for the moduli metric and Kähler potential are important in the investigation of the various brane solitons that arise in these compactifications .
We remark that the components LaTeXMLMath and LaTeXMLMath of the sigma model metric are homogeneous of degree LaTeXMLMath under the scaling of the LaTeXMLMath coordinates of the moduli space LaTeXMLMath .
This follows in a straightforward manner from the homogeneity properties of the volume of LaTeXMLMath manifolds that we have explained above .
We find that LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath .
So this scaling transformation is an isometry .
The isometry group of the metric LaTeXMLMath is the semi-direct product of LaTeXMLMath with LaTeXMLMath the group of translations along the LaTeXMLMath coordinates .
LaTeXMLMath Moduli Space : A symplectic Approach An alternative way to describe the geometry of the moduli space is to use symplectic geometry .
For this we consider the cotangent bundle LaTeXMLMath of the moduli space LaTeXMLMath , a typical fibre of which can be identified with LaTeXMLMath .
There is a symplectic pairing between LaTeXMLMath and LaTeXMLMath given by Poincaré duality .
Of course LaTeXMLMath is isomorphic to LaTeXMLMath .
Choose now coordinates on LaTeXMLMath such that the Kähler form is LaTeXMLEquation .
Next we write the metric on LaTeXMLMath as LaTeXMLEquation .
Given the symplectic form and the metric , one can introduce the ( almost ) complex structure LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are the inverse matrices of LaTeXMLMath and LaTeXMLMath , respectively .
Requiring that LaTeXMLMath , we find that LaTeXMLEquation .
It remains to investigate the integrability of the above complex structure .
For this define the ( 1,0 ) forms LaTeXMLEquation .
Observe that LaTeXMLMath .
Requiring that LaTeXMLMath and LaTeXMLMath do not contain a ( 0,2 ) part , we find that LaTeXMLEquation which in turn implies that LaTeXMLEquation for some function LaTeXMLMath .
A set of complex coordinates with respect to the above complex structure is LaTeXMLEquation .
To make connection with the Kähler approach to the geometry of the moduli space , define the coordinates LaTeXMLEquation .
Then the sigma model metric becomes LaTeXMLEquation .
Next observe from LaTeXMLMath .
Taking the exterior derivative of this equation , we find that LaTeXMLEquation which in turn implies that LaTeXMLMath can be expressed as two LaTeXMLMath -derivatives on a scalar .
Setting LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , we establish the relation between the Kähler and symplectic approaches to the geometry of LaTeXMLMath .
The Kähler geometry on LaTeXMLMath will be used in the sections below to construct solutions for the LaTeXMLMath , LaTeXMLMath supergravity that arise from compactifications of M-theory on holonomy LaTeXMLMath manifolds and are associated with M-branes wrapped on cycles in the compact space .
Potentials As we have seen , potentials do not arise in the four-dimensional effective theory associated with the ( direct ) compactification of LaTeXMLMath supergravity on LaTeXMLMath holonomy manifolds .
However several mechanisms have been proposed for generating a potential .
One such mechanism involves compactifications in the presence of a non-vanishing 4-form field strength LaTeXMLMath along the directions of the compact manifold .
This is a Scherk-Schwarz type of mechanism which has been recently adapted in the context of LaTeXMLMath compactifications of string theory in [ LaTeXMLMath in the context of LaTeXMLMath compactifications of M-theory is not consistent with the compactness of the internal manifold [ LaTeXMLMath associated with such LaTeXMLMath is LaTeXMLEquation .
The imaginary part of LaTeXMLMath is determined by holomorphicity .
Other mechanisms of generating a potential in the low energy effective action in four dimensions involve instanton effects which arise from wrapping M2-branes on associative 3-cycles of the LaTeXMLMath manifold .
Such cycles exists in some LaTeXMLMath holonomy manifolds and the associated instantons induce a scalar potential .
In particular it has been found in [ LaTeXMLMath generates the superpotential LaTeXMLEquation where LaTeXMLMath is a constant and LaTeXMLMath .
Such cycles exist in special holonomy LaTeXMLMath manifolds but they may not exist for generic ones ( see next section ) .
In the case that such a superpotential LaTeXMLMath .
The theory may have other vacua but they are not supersymmetric .
Because of the above mentioned difficulties for generating a potential for generic LaTeXMLMath compactifications of M-theory , we shall mostly focus in the investigation of the solutions of the four-dimensional action with couplings described in section three and without a potential .
However from the perspective of the general LaTeXMLMath , LaTeXMLMath supergravity , one can do a more general analysis which we shall present in an appendix .
Cycles and Wrapped Branes Calibrations and Supersymmetric Cycles On LaTeXMLMath holonomy manifolds there are two calibrations that are associated with supersymmetric cycles .
One is of degree three ( associative ) calibration and the other of degree four ( coassociative ) calibration associated with the parallel three- and four-forms on these manifolds [ LaTeXMLMath -manifolds but they will not be supersymmetric .
To see this , the supersymmetry condition which is deduced from LaTeXMLMath -symmetry is [ LaTeXMLEquation where LaTeXMLMath is the LaTeXMLMath -symmetry projector and LaTeXMLMath is the LaTeXMLMath supersymmetry parameter which should be parallel with respect to the Levi-Civita connection of the compact LaTeXMLMath holonomy manifold .
Since from such a parallel spinor LaTeXMLMath , one can construct the parallel three- and four-forms , only calibrations associated to these forms are supersymmetric .
Even though supersymmetric calibration forms exist on LaTeXMLMath manifolds , it is not apparent that there always exist ( calibrated ) supersymmetric representatives of the homology 3- and 4- classes of the LaTeXMLMath manifold , respectively .
It is known that if supersymmetric ( associative ) 3-cycles exist , they do not have moduli [ LaTeXMLMath manifolds , though one expects to find them in special cases [ LaTeXMLMath , have moduli in LaTeXMLMath manifolds with dimension LaTeXMLMath [ There are several homology cycles on LaTeXMLMath manifolds on which one can wrap M-theory branes .
We shall be mainly concerned with two-cycles , three-cycles , four-cycles and five-cycles .
As we have seen , two- and five- cycles can not be supersymmetric .
This however does not mean that none of them is calibrated , with respect to a non-supersymmetric calibration , or there is no minimal submanifold in the homology class of these cycles .
Three- and four-cycles can be supersymmetric , but as it has been mentioned above this does not necessarily imply that every three- and four-cycle has a supersymmetric calibrated submanifold representing its homology class .
Wrapping M-branes on Homology Cycles The brane solitons in four dimensions that one expects to find by wrapping M-branes on homology cycles of LaTeXMLMath holonomy manifold LaTeXMLMath which are represented by a minimal submanifold are as follows : ( i ) Wrapping M2-branes on two-cycles leads to non-supersymmetric 0-branes in four dimensions .
( ii ) Wrapping M5-brane on two-cycles , three-cycles , four-cycles and five-cycles leads to non-supersymmetric spacetime filling 3-branes , supersymmetric 2-branes , supersymmetric 1-branes and non-supersymmetric 0-branes , respectively .
All brane configurations above that arise from wrapping M-branes to four-dimensions should be described by solutions of the effective LaTeXMLMath effective supergravity theory of this compactification .
Since spacetime filling supersymmetric or non-supersymmetric 3-branes are characterized by LaTeXMLMath -dimensional Poincaré invariance , the associated supergravity solutions are those of flat Minkowski spacetime with vanishing gauge potentials and constant scalars .
Such solutions are of course the ( supersymmetric ) vacua of the theory .
Some non-supersymmetric 0-brane solutions can be identified with the black hole solutions of the supergravity theory .
Typically the electrically charged black holes are associated with wrapped M2-branes , and magnetically charged ones with wrapped M5-branes .
The 2-cycles and 5-cycles in the LaTeXMLMath holonomy manifold are Poincaré dual to each other .
It is well known that the electrically and magnetically charged black holes in four dimensions are dual to each other via electromagnetic duality .
So one can view the electro-magnetic duality in four-dimensions as consequence of the Poincaré duality on LaTeXMLMath manifolds .
The 1-brane configurations can be identified with strings .
As we shall see the solutions are in fact similar to those of cosmic strings [ LaTeXMLMath supergravity theory .
Strings Coassociative Cycles and M5-branes As we have mentioned the string solutions of LaTeXMLMath , LaTeXMLMath supergravity can be thought off as M5-branes wrapped on coassociative cycles of the LaTeXMLMath manifold .
It is known that the supersymmetry conditions [ LaTeXMLMath are LaTeXMLEquation .
Now suppose that we place a M5-brane extended in the directions LaTeXMLMath associated with the projection LaTeXMLEquation .
The string directions are taken to be LaTeXMLMath .
It is easy to see that the above projectors lead to a configuration that preserves two supersymmetries , ie it preserves LaTeXMLMath of supersymmetry of LaTeXMLMath , LaTeXMLMath theory .
There are two simple cases of coassociative cycles to consider .
One is that of coassociative cycles with the topology of the torus LaTeXMLMath and the other is of coassociative cycles with the topology of a LaTeXMLMath surface .
In both cases the dimension of the moduli space is three .
So one expects to find string solutions of LaTeXMLMath supergravity associated with the wrapping of M5-branes on these coassociative cycles with topology LaTeXMLMath and LaTeXMLMath .
The tension of corresponding strings will be equal to the tension of the M5-brane times the volume of the coassociative cycles .
LaTeXMLMath Strings To investigate the string solutions to the supergravity field equations it is convenient to use the Kähler parameterisation of the moduli space of LaTeXMLMath structures .
The LaTeXMLMath strings are a special case of the cosmic strings for which the sigma model manifold is the space LaTeXMLMath .
The solution is LaTeXMLEquation where LaTeXMLMath is a complex coordinate of spacetime , and the Kähler potential LaTeXMLMath and the complex coordinates LaTeXMLMath are given in section three .
The LaTeXMLMath strings above do not have finite tension because LaTeXMLMath is not compact .
However it is known that the fibre directions of LaTeXMLMath can be compactified to a torus by dividing with LaTeXMLMath which is thought of as a group of large gauge transformations .
This leaves the base LaTeXMLMath of LaTeXMLMath which is an open set in LaTeXMLMath .
It is expected that certain points of LaTeXMLMath should be identified by large diffeomorphisms of the compact LaTeXMLMath holonomy manifold LaTeXMLMath .
However to our knowledge it is not known how such large diffeomorphisms act on LaTeXMLMath and whether they are sufficient to allow for string solutions with finite tension adapting a similar construction in [ LaTeXMLMath strings with wrapped M5-branes on coassociative cycles provides some indirect evidence that LaTeXMLMath strings with finite tension exist in the case when coassociative cycles are present .
We note that there are supersymmetric string solutions even in the presence of a Fayet-Iliopoulos term [ LaTeXMLMath Domain Walls The supersymmetry projections of domain walls that arise from a M2-brane in the directions LaTeXMLMath in the background of the LaTeXMLMath manifold in the directions LaTeXMLMath , are those of LaTeXMLEquation .
So the supersymmetry preserved is LaTeXMLMath of M-theory , ie LaTeXMLMath of that of LaTeXMLMath supergravity .
The domain wall is in the directions LaTeXMLMath .
It is straightforward to write the supergravity solution of a brane that is located in a Ricci-flat manifold .
For the case of interest here , the transverse space of the M2-brane is LaTeXMLMath , where LaTeXMLMath is the holonomy LaTeXMLMath compact space .
The solution is LaTeXMLEquation where LaTeXMLMath is harmonic in LaTeXMLMath .
Additional fluxes LaTeXMLMath can be added in the solution along LaTeXMLMath .
However in this case LaTeXMLMath is not harmonic but rather obeys the equation LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath can be chosen to be harmonic in ℝ , ie LaTeXMLMath is piece-wise linear function of the coordinate LaTeXMLMath , see [ ansatz of section three .
We shall not pursue this point further here .
Alternatively , domain walls can arise by wrapping M5-branes on associative 3-cycles of the compact space .
If the LaTeXMLMath manifold is in the directions LaTeXMLMath and the M5-brane is in the directions LaTeXMLMath , then the projections are as in LaTeXMLEquation .
These lead again to a configuration preserving LaTeXMLMath of M-theory supersymmetry , ie LaTeXMLMath of supersymmetry of LaTeXMLMath supergravity .
From the perspective of LaTeXMLMath supergravity , these domain walls may arise from a superpotential generated by a M2-brane instanton wrapping the associative cycle .
The investigation of the Killing spinor equations of LaTeXMLMath domain walls that preserve LaTeXMLMath of supersymmetry will be given in an appendix .
Black Holes Black Holes as Wrapped M2-branes As we have mentioned the electrically charged black holes of LaTeXMLMath , LaTeXMLMath supergravity action that arise from compactifying M-theory on holonomy LaTeXMLMath manifolds can be viewed as wrapped M2-branes on the two-cycles of the compact space LaTeXMLMath .
It is known that there are minimal sphere representatives of every homotopy class LaTeXMLMath ( see for example chapter VI [ LaTeXMLMath and LaTeXMLMath , respectively , where LaTeXMLMath is the M2-brane tension and LaTeXMLMath is the two-cycle .
Since two cycles in holonomy LaTeXMLMath manifolds are not supersymmetric , it is not expected to find relation between the mass and the charges of the black holes .
This is despite the fact that the mass and the charge per-unit volume of the associated M2-brane are equal .
Although it is not apparent that there are non-supersymmetric degree two calibrations on holonomy LaTeXMLMath manifolds , suppose that there is one associated with the two-form LaTeXMLMath .
Then LaTeXMLMath , for some constants LaTeXMLMath , and LaTeXMLMath which can be interpreted as an extremality relation for the black hole .
As we shall see there are extreme solutions LaTeXMLMath supergravity associated with LaTeXMLMath compactifications which however exhibit a naked singularity .
Electric LaTeXMLMath Black Holes Ansatz and Field Equations In order to find black hole solutions of LaTeXMLMath supergravity associated with LaTeXMLMath compactifications of M-theory , we consider the ansatz LaTeXMLEquation .
We recall from section 3 that LaTeXMLMath LaTeXMLMath with LaTeXMLMath LaTeXMLMath , where LaTeXMLMath , LaTeXMLMath is the volume of the compact space and LaTeXMLMath ; so LaTeXMLMath is homogeneous of degree LaTeXMLMath in LaTeXMLMath .
Furthermore we shall take the scalar potential of LaTeXMLMath supergravity to vanish LaTeXMLMath .
It is straightforward to observe from the Killing spinor equations that all electrically charged solutions can not be supersymmetric .
Substituting the ansatz LaTeXMLEquation where LaTeXMLMath are constants .
To obtain the scalar equations we vary LaTeXMLMath and LaTeXMLMath .
Recalling that LaTeXMLMath and LaTeXMLMath , we find that the field equations for LaTeXMLMath and LaTeXMLMath are LaTeXMLEquation and LaTeXMLEquation respectively .
It is convenient to define LaTeXMLEquation .
Then the Maxwell equations may be rewritten as LaTeXMLEquation .
On substituting the black hole ansatz into the scalar equations , and eliminating LaTeXMLMath using LaTeXMLEquation .
Lastly we consider the Einstein equations .
We adopt the notation LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
The non-vanishing components of the Einstein tensor are given by LaTeXMLEquation .
Eliminating LaTeXMLMath from the above equations using LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
A useful identity implied by the scalar equations ( or the Einstein equations ) is LaTeXMLEquation .
Using the Einstein equation LaTeXMLMath as LaTeXMLEquation for constants LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation .
In addition , from the LaTeXMLMath scalar equation , LaTeXMLMath must satisfy LaTeXMLEquation for some real constants LaTeXMLMath , and here LaTeXMLMath is the inverse of the Hessian of LaTeXMLMath .
A solution to the above system of equations is the Schwarzschild black hole .
In this case the Maxwell gauge potential vanishes and the scalars are constant .
This black hole can not be thought of as a M2-brane wrapped on a homology 2-cycle of the compact LaTeXMLMath holonomy manifold because it does not carry electric charges .
Generically , to find electrically charged black holes , one has to take some of the scalars to be non constant LaTeXMLMath but we shall not pursue this further here..
So we take LaTeXMLMath .
On eliminating LaTeXMLMath from the scalar equation for LaTeXMLMath and making use of the fact that LaTeXMLMath is the Hessian of LaTeXMLMath , one obtains LaTeXMLEquation .
To solve the field equations , we shall make use of the homogeneity properties of LaTeXMLMath .
In particular , contracting LaTeXMLMath and setting LaTeXMLMath , one finds the identity LaTeXMLEquation .
From this it follows that in order to have charged solutions , we shall require LaTeXMLMath .
Substituting LaTeXMLMath for LaTeXMLMath , one obtains LaTeXMLEquation for constants LaTeXMLMath and LaTeXMLMath .
In the special case when LaTeXMLMath one obtains LaTeXMLEquation .
The only remaining independent equations are the Einstein equation Dilatonic LaTeXMLMath Electric Black Holes To find an explicit solution to the equation LaTeXMLEquation for some constants LaTeXMLMath .
Then the homogeneity of LaTeXMLMath implies that LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are constants related by LaTeXMLEquation .
It is also convenient to define LaTeXMLMath so that LaTeXMLMath .
Then the LaTeXMLMath scalar equation LaTeXMLEquation for some constant LaTeXMLMath , and LaTeXMLEquation .
Note that on contracting LaTeXMLMath and using the homogeneity properties of LaTeXMLMath , one obtains LaTeXMLEquation .
Then using the above identities , it follows that for LaTeXMLMath the scalar equations are implied by the Einstein equations .
It therefore suffices to solve LaTeXMLEquation .
To find solutions to these equations which are asymptotically Minkowski , we shall set LaTeXMLMath , see LaTeXMLMath from the first equation by making use of the second equation .
This equation for LaTeXMLMath may be simplified further by defining LaTeXMLMath according to LaTeXMLEquation .
Then LaTeXMLMath must satisfy LaTeXMLEquation .
We discard the solution LaTeXMLMath because it is inconsistent with LaTeXMLMath .
To simplify this equation even further , define LaTeXMLMath and define implicitly LaTeXMLEquation .
Then LaTeXMLEquation where LaTeXMLMath .
Substituting these expressions into LaTeXMLEquation .
On setting LaTeXMLMath , this simplifies to give LaTeXMLEquation .
There are two possible solutions to this equation , LaTeXMLEquation or LaTeXMLEquation for some constant LaTeXMLMath .
However , LaTeXMLMath leads to LaTeXMLMath constant and so the Schwarzschild black hole .
We therefore take LaTeXMLEquation .
Hence LaTeXMLMath must satisfy LaTeXMLEquation .
For LaTeXMLMath there are three cases to consider , according as to whether LaTeXMLMath , LaTeXMLMath or LaTeXMLMath .
We shall focus on the cases LaTeXMLMath and LaTeXMLMath .
For LaTeXMLMath , we find that LaTeXMLEquation for constant LaTeXMLMath .
This solution is asymptotically Minkowski , and the charges , mass and asymptotic values of the scalar are LaTeXMLEquation respectively .
The spacetime geometry of the solution LaTeXMLMath -brane [ LaTeXMLMath .
For LaTeXMLMath , we find that LaTeXMLEquation which in turn implies that the solution is LaTeXMLEquation where LaTeXMLEquation for constant LaTeXMLMath , and we have set LaTeXMLMath in This solution is again asymptotically Minkowski as LaTeXMLMath , and the electric charges LaTeXMLMath , the mass LaTeXMLMath and asymptotic value LaTeXMLMath of the moduli scalar are LaTeXMLEquation respectively .
To examine this spacetime geometry it is particularly useful to consider the metric in the neighbourhood of LaTeXMLMath .
We define LaTeXMLMath .
Then as LaTeXMLMath , LaTeXMLEquation where LaTeXMLMath is determined by LaTeXMLEquation .
The leading order behaviour of the metric in the neighbourhood of LaTeXMLMath is given by LaTeXMLEquation .
From this metric it is straightforward to show that if there exist geodesics along which in-falling particles may pass through LaTeXMLMath in a finite proper time then it is necessary to impose the constraint LaTeXMLMath .
This condition is sufficient to ensure the positivity of the mass given in LaTeXMLEquation where LaTeXMLMath .
In order for this to remain bounded as LaTeXMLMath , one must impose the condition LaTeXMLMath , i.e .
LaTeXMLMath .
Hence , in order for there to be a horizon at LaTeXMLMath one requires LaTeXMLMath .
For this special case , it is most convenient to change co-ordinates by setting LaTeXMLEquation so that the metric can be written as LaTeXMLEquation where LaTeXMLEquation and LaTeXMLEquation .
This four-dimensional spacetime geometry has been considered before in [ Magnetic LaTeXMLMath Black Holes To find magnetic black hole solutions which are associated with M5-branes wrapped on 5-cycles of the compact space , we take the ansatz for the metric and moduli scalars as in the electric case LaTeXMLEquation where LaTeXMLMath are constants and LaTeXMLMath are the standard angular coordinates on a two sphere .
We shall see that the electric and the magnetic solutions are related as expected because of electro-magnetic duality in four-dimensions .
So we shall not elaborate in the description of this case .
First it is straightforward to see that the gauge field equations are satisfied provided that LaTeXMLMath .
Taking this to be the case , the scalar equations of LaTeXMLMath are automatically satisfied .
Given this , the LaTeXMLMath scalar equations are given by LaTeXMLEquation where LaTeXMLMath are defined as in the electric case .
The Einstein field equations with vanishing scalar potential imply LaTeXMLEquation .
The first two of these equations are identical to the electric black hole case .
In particular , the first equation implies LaTeXMLMath for real constants LaTeXMLMath , LaTeXMLMath .
Next write LaTeXMLMath for some constants LaTeXMLMath .
Using LaTeXMLMath as in the electric case , the LaTeXMLMath scalar equation implies that LaTeXMLEquation for some constant LaTeXMLMath .
So it follows that LaTeXMLMath .
Then just as for the electric case , by making use of the identity LaTeXMLEquation obtained from the Einstein equations , one may see that the scalar equations are implied by the Einstein equations .
Hence , it suffices to solve the remaining Einstein equations LaTeXMLEquation .
Clearly , these equations are equivalent to LaTeXMLMath .
Hence it follows that the spacetime geometry of these magnetic black holes is identical to the electric black hole case , and LaTeXMLMath .
For example the M-theory interpretation of the analogue of the solution Acknowledgments : G.P .
thanks ITP and the Physics Department of University of Stanford for hospitality .
G.P .
is supported by a University Research Fellowship from the Royal Society .
J.G .
is supported by a EPSRC postdoctoral grant .
This work is partially supported by SPG grant PPA/G/S/1998/00613 .
AA : Spinor Notation It is most convenient to present the supersymmetry transformations in terms of a 4-component Majorana spinor LaTeXMLMath with real components .
We take LaTeXMLMath to be the Pauli matrices ; LaTeXMLEquation .
We set LaTeXMLMath , LaTeXMLMath , and to perform the supersymmetry calculations we define explicitly LaTeXMLEquation so that LaTeXMLEquation .
With these definitions , the gamma matrices satisfy the Clifford algebra LaTeXMLEquation .
BB : String Solitons and Supersymmetry In order to examine string and domain wall solutions , we shall consider the following ansatz ; LaTeXMLEquation where LaTeXMLMath is a metric on the manifold spanned by LaTeXMLMath where LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Without loss of generality , we shall take LaTeXMLEquation to be diagonal , using the fact that any metric on a Riemann surface is locally conformally flat .
Substituting this ansatz into the Killing spinor equations , we find that LaTeXMLEquation together with LaTeXMLEquation and LaTeXMLEquation .
There are two cases to consider .
Firstly , if the the scalar potential vanishes , then the first Killing spinor equation above with LaTeXMLMath implies that LaTeXMLMath is constant , and we set LaTeXMLMath .
The remaining Killing spinor equations can be solved by taking LaTeXMLEquation i.e .
the LaTeXMLMath are holomorphic .
The solution preserves LaTeXMLMath of the supersymmetry ; LaTeXMLMath is a constant spinor satisfying LaTeXMLEquation .
More generally , one may construct solutions for which LaTeXMLMath .
In particular , we may begin by examining LaTeXMLEquation with LaTeXMLMath , LaTeXMLMath real ; then LaTeXMLEquation where LaTeXMLMath .
Suppose now LaTeXMLMath such that LaTeXMLMath .
Then for these LaTeXMLMath , these equations may be solved by taking LaTeXMLMath constant .
Alternatively , one may have LaTeXMLMath non-constant holomorphic with LaTeXMLMath ; or LaTeXMLMath non-constant anti-holomorphic with LaTeXMLMath ( however if there exists more that one value of LaTeXMLMath such that LaTeXMLMath then one can not have a supersymmetric solution with a mixture of corresponding non-constant holomorphic and anti-holomorphic complex scalars ) .
Suppose now we consider LaTeXMLMath for which LaTeXMLMath .
Define LaTeXMLEquation .
Then one requires for these LaTeXMLMath ; LaTeXMLEquation .
There are several possibilities .
Firstly , note that one can not have a supersymmetric solution with both LaTeXMLMath .
If LaTeXMLMath then it turns out that LaTeXMLMath .
If LaTeXMLMath , however , then LaTeXMLMath .
Alternatively , one may have LaTeXMLMath , LaTeXMLMath both nonzero .
It turns out that if both LaTeXMLMath and LaTeXMLMath then the solution can not be supersymmetric .
If however , LaTeXMLMath but LaTeXMLMath then one has LaTeXMLMath .
Another possibility is to take LaTeXMLMath and LaTeXMLMath ; then LaTeXMLMath .
We shall however concentrate on the remaining possibility , in which we take LaTeXMLMath ( but we do not not necessarily require LaTeXMLMath ) .
Writing then LaTeXMLMath , LaTeXMLMath for real LaTeXMLMath , LaTeXMLMath , the supersymmetry constraint LaTeXMLMath and LaTeXMLMath is satisfied by taking LaTeXMLEquation .
Analogous reasoning may be used to consider LaTeXMLEquation .
Defining LaTeXMLEquation we note that LaTeXMLEquation .
Hence the reasoning used to determine the various possible values of LaTeXMLMath , LaTeXMLMath also applies to LaTeXMLMath and LaTeXMLMath .
To summarize then , neglecting the cases for which LaTeXMLMath , LaTeXMLMath .
Furthermore , if LaTeXMLMath then LaTeXMLMath is constant , and if LaTeXMLMath then LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
LaTeXMLMath is given by It is also necessary to examine LaTeXMLEquation where LaTeXMLMath .
In this case LaTeXMLMath and LaTeXMLMath imply ( for LaTeXMLMath ) LaTeXMLEquation and we solve the supersymmetry constraints by taking LaTeXMLMath , LaTeXMLMath , for LaTeXMLMath , LaTeXMLMath with LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLEquation .
This is solved by taking LaTeXMLEquation for constant LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are determined by LaTeXMLEquation .
We note that these solutions generically preserve LaTeXMLMath of the supersymmetry .
It is straightforward to check that these conditions ensure that the scalar and Einstein field equations hold .
CC : LaTeXMLMath Domain Walls = Ansatz and Killing Spinor Equations To find domain wall solutions of LaTeXMLMath supergravity , we shall use the ansatz LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath will be determined by the field equations .
Properties of domain walls in supergravity have been reviewed in [ LaTeXMLMath which we shall not specify .
Substituting this ansatz to the Killing spinor equations of section three , one finds LaTeXMLEquation .
We are seeking solutions that preserve two real supercharges .
This leads to consider a set of Killing spinor equations which arise as a special case of the analysis presented in the previous appendix .
In particular for the domain wall solution we take ( for those LaTeXMLMath directions such that LaTeXMLMath ) LaTeXMLMath with LaTeXMLMath .
The angles LaTeXMLMath , LaTeXMLMath defined in the appendix therefore satisfy LaTeXMLMath .
Then the Killing spinors are given by LaTeXMLEquation for real constants LaTeXMLMath , LaTeXMLMath .
Clearly the solution preserves LaTeXMLMath of the supersymmetry of LaTeXMLMath theory .
The conditions which are implied from the Killing spinor equations on the fields are LaTeXMLMath , i.e .
LaTeXMLEquation together with LaTeXMLEquation and LaTeXMLEquation .
Dilatonic LaTeXMLMath Domain Walls So far we have considered the general case of domain walls of LaTeXMLMath supergravity associated with a superpotential LaTeXMLMath which preserve LaTeXMLMath of supersymmetry .
Now we shall consider the special case of domain walls in the context of LaTeXMLMath compactifications .
In the beginning of the analysis , we shall keep the superpotential LaTeXMLMath arbitrary but to give some explicit solutions of domain walls , we shall later consider some special cases .
We recall that the Kähler potential is given by LaTeXMLMath , where LaTeXMLMath is the volume of the compact LaTeXMLMath -manifold , and LaTeXMLMath is homogeneous of degree LaTeXMLMath in LaTeXMLMath coordinates of the moduli space LaTeXMLMath .
To proceed motivated by LaTeXMLEquation for some constants LaTeXMLMath , and LaTeXMLMath , ie the only modulus field is the volume of the compact manifold .
Then LaTeXMLMath , where LaTeXMLMath is constant .
Defining LaTeXMLMath , we may write LaTeXMLEquation and LaTeXMLEquation respectively .
Furthermore because LaTeXMLMath is homogeneous of degree LaTeXMLMath , we write LaTeXMLEquation for some constants LaTeXMLMath .
But we also know from section three that LaTeXMLEquation .
It then follows that LaTeXMLMath .
Now LaTeXMLEquation .
Motivated by this we shall consider LaTeXMLMath of the form LaTeXMLMath , so that LaTeXMLEquation where LaTeXMLMath denotes differentiation with respect to LaTeXMLMath .
So LaTeXMLEquation .
In addition , because LaTeXMLMath is homogeneous of degree LaTeXMLMath in LaTeXMLMath we have LaTeXMLEquation for constant LaTeXMLMath .
Hence LaTeXMLMath .
Eliminating LaTeXMLMath from LaTeXMLMath of the metric in terms of the superpotential as LaTeXMLEquation where LaTeXMLMath .
To find the full solution , it remains to substitute LaTeXMLMath .
However the resulting equation is rather involved for a general superpotential .
To find explicit solutions additional information is needed to describe the superpotential .
DD : LaTeXMLMath and Dilatonic Black Holes It is convenient to make an explicit connection between the LaTeXMLMath , LaTeXMLMath supergravity theory with couplings fixed by LaTeXMLMath compactification of LaTeXMLMath supergravity and the standard form of Einstein-Maxwell-Dilaton supergravity such as that used in [ LaTeXMLEquation where LaTeXMLMath is the dilaton , LaTeXMLMath is the constant dilaton coupling , LaTeXMLMath is the Maxwell field strength , and the Chern-Simons term has been neglected as we are considering only purely electrically ( or magnetically ) charged solutions .
We recall that the portion of the LaTeXMLMath , LaTeXMLMath supergravity action containing the curvature and scalar terms LaTeXMLMath is LaTeXMLEquation .
Suppose we consider the special case where the only modulus field is the volume of the compactified LaTeXMLMath manifold , so that LaTeXMLMath for constants LaTeXMLMath .
Then the truncated theory has LaTeXMLEquation .
Then setting LaTeXMLMath in LaTeXMLMath , LaTeXMLMath supergravity action with LaTeXMLMath , LaTeXMLMath supergravity action is equivalent to LaTeXMLMath .
The Minimal Resolution Conjecture for points in projective space has attracted considerable attention in recent years , starting with the original LaTeXMLCite , LaTeXMLCite and continuing most notably with LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
The purpose of this paper is to explain how a completely analogous problem can be formulated for sets of points on arbitrary varieties embedded in projective space , and then study in detail the case of curves .
Similarly to the well-known analysis of syzygies of curves carried out by Green and Lazarsfeld ( LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) , we divide our work into a study of resolutions of points on canonical curves and on curves of large degree .
The central result of the paper states that the Minimal Resolution Conjecture is true on any canonical curve .
In contrast , it always fails for curves embedded with large degree , although a weaker result , called the Ideal Generation Conjecture , holds also in this case .
These results turn out to have surprisingly deep connections with the geometry of difference varieties in Jacobians , special divisors on moduli spaces of curves with marked points , and moduli spaces of stable bundles .
Let LaTeXMLMath be a projective variety over an algebraically closed field , embedded by a ( not necessarily complete ) linear series .
We begin by formulating a general version of the Minimal Resolution Conjecture ( MRC ) , in analogy with the case of LaTeXMLMath , predicting how the Betti numbers of a general subset of points of LaTeXMLMath in the given embedding are related to the Betti numbers of LaTeXMLMath itself .
More precisely ( cf .
Theorem LaTeXMLRef below ) , for a large enough general set of points LaTeXMLMath on LaTeXMLMath , the Betti diagram consisting of the graded Betti numbers LaTeXMLMath is obtained from the Betti diagram of LaTeXMLMath by adding two more nontrivial rows , at places well determined by the length of LaTeXMLMath .
Recalling that the Betti diagram has the Betti number LaTeXMLMath in the LaTeXMLMath -th position , and assuming that the two extra rows are indexed by LaTeXMLMath and LaTeXMLMath , for some integer LaTeXMLMath , the MRC predicts that LaTeXMLEquation i.e .
at least one of the two Betti numbers on any ” diagonal ” is zero .
As the difference LaTeXMLMath can be computed exactly , this implies a precise knowledge of the Betti numbers in these two rows .
Summing up , knowing the Betti diagram of LaTeXMLMath would be the same as knowing the Betti diagram of LaTeXMLMath .
A subtle question is however to understand how the shape of the Betti diagram of LaTeXMLMath influences whether MRC is satisfied for points on LaTeXMLMath .
An example illustrating this is given at the end of Section 1 .
The Minimal Resolution Conjecture has been extensively studied in the case LaTeXMLMath .
The conjecture holds for LaTeXMLMath by results of Gaeta , Ballico and Geramita , and Walter ( see LaTeXMLCite , LaTeXMLCite and LaTeXMLCite , respectively ) .
Moreover , Hirschowitz and Simpson proved in LaTeXMLCite that it holds if the number of points is large enough with respect to LaTeXMLMath .
However , the conjecture does not hold in general : it fails for every LaTeXMLMath , LaTeXMLMath for almost LaTeXMLMath values of the number of points , by a result of Eisenbud , Popescu , Schreyer and Walter ( see LaTeXMLCite ) .
We refer to LaTeXMLCite and LaTeXMLCite for a nice introduction and an account of the present status of the problem in this case .
The main body of the paper is dedicated to a detailed study of MRC in the case of curves .
We will simply say that a curve satisfies MRC in a given embedding if MRC is satisfied by a general set of points LaTeXMLMath of any sufficiently large degree ( for the precise numerical statements see Section 1 ) .
We will also sometimes say that MRC holds for a line bundle LaTeXMLMath if it holds for LaTeXMLMath in the embedding given by LaTeXMLMath .
Our main result says that MRC holds in the most significant case , namely the case of canonically embedded nonhyperelliptic curves .
If LaTeXMLMath is a canonical curve , then LaTeXMLMath satisfies MRC .
In contrast , under very mild assumptions on the genus , the MRC always fails in the case of curves of large degree , at well-determined spots in the Betti diagram ( cf .
Section 2 for precise details ) .
The statement LaTeXMLMath , i.e .
the case LaTeXMLMath , does hold though ; this is precisely the Ideal Generation Conjecture , saying that the minimal number of generators of LaTeXMLMath is as small as possible .
( a ) If LaTeXMLMath is a very ample line bundle of degree LaTeXMLMath , then LaTeXMLMath holds for LaTeXMLMath .
( b ) If LaTeXMLMath and LaTeXMLMath is a line bundle of degree LaTeXMLMath , then there exists a value of LaTeXMLMath such that LaTeXMLMath does not satisfy MRC for LaTeXMLMath .
The same holds if LaTeXMLMath and LaTeXMLMath .
It is interesting to note that by the ” periodicity ” property of Betti diagrams of general points on curves ( see LaTeXMLCite §2 ) , the theorem above implies that on curves of high degree , MRC fails for sets of points of arbitrarily large length .
This provides a very different picture from the case of projective space ( cf .
LaTeXMLCite ) , where asymptotically the situation is as nice as possible .
We explain the strategy involved in the proof of these results in some detail , as it appeals to some new geometric techniques in the study of syzygy related questions .
For simplicity we assume here that LaTeXMLMath is a smooth curve embedded in projective space by means of a complete linear series corresponding to a very ample line bundle LaTeXMLMath ( but see §2 for more general statements ) .
A well-known geometric approach , developed by Green and Lazarsfeld in the study of syzygies of curves ( see LaTeXMLCite for a survey ) , is to find vector bundle statements equivalent to the algebraic ones , via Koszul cohomology .
This program can be carried out completely in the case of MRC , and for curves we get a particularly clean statement .
Assume that LaTeXMLMath is the kernel of the evaluation map LaTeXMLMath and LaTeXMLMath .
Then ( cf .
Corollary LaTeXMLRef below ) MRC holds for a collection of LaTeXMLMath general points on LaTeXMLMath if and only if the following is true : LaTeXMLEquation .
Condition LaTeXMLMath above is essentially the condition studied by Raynaud LaTeXMLCite , related to the existence of theta divisors for semistable vector bundles .
In the particular situation of LaTeXMLMath , with LaTeXMLMath a line bundle of large degree , it has been considered in LaTeXMLCite in order to produce base points for the determinant linear series on the moduli spaces LaTeXMLMath of semistable bundles of rank LaTeXMLMath and trivial determinant .
A similar approach shows here the failure of condition LaTeXMLMath ( and so of MRC ) for LaTeXMLMath .
On the other hand , the fact that IGC holds is a rather elementary application of the Base Point Free Pencil Trick LaTeXMLCite III §3 .
The case of canonical curves is substantially more involved , but in the end one is rewarded with a positive answer .
As above , it turns out that MRC is equivalent to the vanishing : LaTeXMLEquation where LaTeXMLMath is the dual of the bundle LaTeXMLMath defined by the evaluation sequence : LaTeXMLEquation .
As the slope of LaTeXMLMath is LaTeXMLMath , this is in turn equivalent to the fact that LaTeXMLMath has a theta divisor LaTeXMLMath .
On a fixed curve , if indeed a divisor , LaTeXMLMath will be identified as being precisely the difference variety LaTeXMLMath ( cf .
LaTeXMLCite Ch.V.D ) , where LaTeXMLMath is the LaTeXMLMath -th symmetric product of LaTeXMLMath .
This is achieved via a filtration argument and a cohomology class calculation similar to the classical Poincaré theorem ( cf .
Proposition LaTeXMLRef ) .
A priori though , on an arbitrary curve the nonvanishing locus LaTeXMLMath may be the whole LaTeXMLMath , in which case this identification is meaningless .
We overcome this problem by working with all curves at once , that is by setting up a similar universal construction on the moduli space of curves with marked points LaTeXMLMath .
Here we slightly oversimplify the exposition in order to present the main idea , but for the precise technical details see Section 3 .
We essentially consider the ” universal nonvanishing locus ” in LaTeXMLMath : LaTeXMLEquation .
The underlying idea is that the difference line bundles LaTeXMLMath in fact cover the whole LaTeXMLMath ( i.e .
LaTeXMLMath ) , and so for any given curve LaTeXMLMath , LaTeXMLMath is precisely the nonvanishing locus described above .
The advantage of writing it in this form is that we are led to performing a computation on LaTeXMLMath rather than on a universal Picard , where for example one does not have a canonical choice of generators for the Picard group .
A “ deformation to hyperelliptic curves ” argument easily implies that MRC holds for general canonical curves , so LaTeXMLMath is certainly a divisor .
We then show that LaTeXMLMath is the degeneracy locus of a morphism of vector bundles of the same rank and compute its class using a Grothendieck-Riemann-Roch argument ( cf .
Proposition LaTeXMLRef ) .
On the other hand , one can define an ( a priori different ) divisor LaTeXMLMath in LaTeXMLMath which is a global analogue of the preimage of LaTeXMLMath in LaTeXMLMath via the difference map .
It is convenient to see LaTeXMLMath as the locus of curves with marked points LaTeXMLMath having a LaTeXMLMath which contains LaTeXMLMath in a fiber and LaTeXMLMath in a different fiber .
An equivalent formulation of the discussion above is that LaTeXMLMath , and in order for MRC to hold for all canonical curves one should have precisely LaTeXMLMath .
As we show that LaTeXMLMath is reduced ( cf .
Proposition LaTeXMLRef ) , it suffices then to prove that the class of LaTeXMLMath coincides with that of LaTeXMLMath .
To this end we consider the closure of LaTeXMLMath in the compactification LaTeXMLMath , where the corresponding boundary condition is defined by means of limit linear series .
The computation of the class of LaTeXMLMath via this closure is essentially independent of the rest of the paper .
It relies on degeneration and enumerative techniques in the spirit of LaTeXMLCite and LaTeXMLCite .
The results of both this and the computation of the class of LaTeXMLMath are summarized in the following theorem .
For the statement , we recall that LaTeXMLMath is generated by the class LaTeXMLMath of the Hodge bundle and the classes LaTeXMLMath , LaTeXMLMath , where LaTeXMLMath , with LaTeXMLMath the relative dualizing sheaf on the universal curve LaTeXMLMath and LaTeXMLMath the projection onto the LaTeXMLMath -th factor .
The divisors LaTeXMLMath and LaTeXMLMath defined above have the same class in LaTeXMLMath , namely LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
In particular LaTeXMLMath .
As mentioned above , this implies that LaTeXMLMath always has a theta divisor , for all LaTeXMLMath , so equivalently that MRC holds for an arbitrary canonical curve .
We record the more precise identification of this theta divisor , which now follows in general .
For any nonhyperelliptic curve LaTeXMLMath , LaTeXMLMath .
In this particular form , our result answers positively a conjecture of R. Lazarsfeld .
It is worth mentioning that it also answers negatively a question that was raised in connection with LaTeXMLCite , namely if LaTeXMLMath provide base points for determinant linear series on appropriate moduli spaces of vector bundles .
The paper is structured as follows .
In Section 1 we give some equivalent formulations of the Minimal Resolution Conjecture and we describe the vector bundle setup used in the rest of the paper .
In Section 2 we treat the case of curves embedded with large degree , proving IGC and showing that MRC fails .
Section 3 is devoted to the main result , namely the proof of MRC for canonical curves , and here is where we look at the relationship with difference varieties and moduli spaces of curves with marked points .
The divisor class computation in LaTeXMLMath , on which part of the proof relies , is carried out in Section 4 by means of limit linear series .
Acknowledgments .
We would like to thank D. Eisenbud and R. Lazarsfeld for sharing with us numerous useful ideas on the subject .
We work over an algebraically closed field LaTeXMLMath which , unless explicitly mentioned otherwise , has arbitrary characteristic .
Let LaTeXMLMath be a vector space over LaTeXMLMath with LaTeXMLMath and LaTeXMLMath the homogeneous coordinate ring of the corresponding projective space LaTeXMLMath .
For a finitely generated graded LaTeXMLMath -module LaTeXMLMath , the Betti numbers LaTeXMLMath of LaTeXMLMath are defined from the minimal free resolution LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation .
The Betti diagram of LaTeXMLMath has in the LaTeXMLMath -th position the Betti number LaTeXMLMath .
The regularity LaTeXMLMath of LaTeXMLMath can be defined as the index of the last nontrivial row in the Betti diagram of LaTeXMLMath ( see LaTeXMLCite , 20.5 for the connection with the cohomological definition ) .
We will use the computation of Betti numbers via Koszul cohomology : LaTeXMLMath is the dimension over LaTeXMLMath of the cohomology of the following piece of the Koszul complex : LaTeXMLEquation ( see LaTeXMLCite for details ) .
For an arbitrary subscheme LaTeXMLMath , we denote by LaTeXMLMath its saturated ideal and let LaTeXMLMath .
We denote by LaTeXMLMath and LaTeXMLMath the Hilbert polynomial and Hilbert function of LaTeXMLMath , respectively .
The regularity LaTeXMLMath of LaTeXMLMath is defined to be the regularity of LaTeXMLMath , if LaTeXMLMath , and LaTeXMLMath otherwise .
Notice that with this convention , in the Betti diagram of LaTeXMLMath , which by definition is the Betti diagram of LaTeXMLMath , the last nontrivial row is always indexed by LaTeXMLMath .
For a projective variety LaTeXMLMath , a line bundle LaTeXMLMath on LaTeXMLMath , and a linear series LaTeXMLMath which generates LaTeXMLMath , we denote by LaTeXMLMath the vector bundle which is the kernel of the evaluation map LaTeXMLEquation .
When LaTeXMLMath we use the notation LaTeXMLMath .
If LaTeXMLMath is a smooth curve of genus LaTeXMLMath , and LaTeXMLMath is the canonical line bundle , then LaTeXMLMath denotes the vector bundle LaTeXMLMath .
The dual vector bundles will be denoted by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , respectively .
Whenever there is no risk of confusion , we will simply write LaTeXMLMath and LaTeXMLMath , instead of LaTeXMLMath and LaTeXMLMath .
In this section LaTeXMLMath is a fixed irreducible projective variety of positive dimension .
We study the Betti numbers of a general set of LaTeXMLMath points LaTeXMLMath .
Since the Betti numbers are upper semicontinuous functions , for every positive integer LaTeXMLMath , there is an open subset LaTeXMLMath of LaTeXMLMath such that for all LaTeXMLMath and LaTeXMLMath , LaTeXMLMath takes its minimum value for LaTeXMLMath .
Notice that as the regularity is bounded in terms of LaTeXMLMath , we are concerned with finitely many Betti numbers .
From now on , LaTeXMLMath general means LaTeXMLMath .
It is easy to determine the Hilbert function of a general set of points LaTeXMLMath in terms of the Hilbert function of LaTeXMLMath ( see LaTeXMLCite ) .
We have the following : If LaTeXMLMath is a general set of LaTeXMLMath points , then LaTeXMLEquation .
To determine the Betti numbers of a general set of points LaTeXMLMath is a much more subtle problem .
If LaTeXMLMath is large enough , then the Betti diagram of LaTeXMLMath looks as follows : in the upper part we have the Betti diagram of LaTeXMLMath and there are two extra nontrivial rows at the bottom .
Moreover , the formula in Proposition LaTeXMLRef gives an expression for the differences of the Betti numbers in these last two rows .
We record the formal statement in the following theorem and for the proof we refer to LaTeXMLCite .
Assume that LaTeXMLMath is a general set of LaTeXMLMath points , with LaTeXMLMath for some LaTeXMLMath , where LaTeXMLMath .
( i ) For every LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath .
( ii ) LaTeXMLMath , for LaTeXMLMath and there is an LaTeXMLMath such that LaTeXMLMath .
( iii ) For every LaTeXMLMath , we have LaTeXMLEquation ( iv ) If LaTeXMLMath , then for every LaTeXMLMath , we have LaTeXMLMath , where LaTeXMLEquation .
We will focus our attention on the Betti numbers in the bottom two rows in the Betti diagram of LaTeXMLMath .
The equation in Theorem LaTeXMLRef ( iv ) gives lower bounds for these numbers , namely LaTeXMLMath and LaTeXMLMath .
In analogy with the case LaTeXMLMath ( see LaTeXMLCite and LaTeXMLCite ) , we say that the Minimal Resolution Conjecture ( to which we refer from now on as MRC ) holds for a fixed value of LaTeXMLMath as above if for every LaTeXMLMath and every general set LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Equivalently , it says that LaTeXMLEquation .
This conjecture has been extensively studied in the case LaTeXMLMath , LaTeXMLMath .
It is known to hold for small values of LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath or LaTeXMLMath ) and for large values of LaTeXMLMath , depending on LaTeXMLMath , but not in general .
In fact , it has been shown that for every LaTeXMLMath , LaTeXMLMath , MRC fails for almost LaTeXMLMath values of LaTeXMLMath ( see LaTeXMLCite , where one can find also a detailed account of the problem ) .
Note that the assertion in MRC holds obviously for LaTeXMLMath .
The first nontrivial case LaTeXMLMath is equivalent by Theorem LaTeXMLRef to saying that the minimal number of generators of LaTeXMLMath is as small as possible .
This suggests the following : We say that the Ideal Generation Conjecture ( IGC , for short ) holds for LaTeXMLMath as above if for a general set of points LaTeXMLMath of cardinality LaTeXMLMath , we have LaTeXMLMath .
( LaTeXMLCite ) MRC holds for every LaTeXMLMath when LaTeXMLMath , since in this case LaTeXMLMath for every LaTeXMLMath .
Similarly , MRC holds for every LaTeXMLMath when LaTeXMLMath , since in this case LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
We derive now a cohomological interpretation of MRC .
From now on we assume that LaTeXMLMath is nondegenerate , so that we have LaTeXMLMath .
Using a standard Koszul cohomology argument , we can express the Betti numbers in the last two rows of the Betti diagram of LaTeXMLMath as follows .
With the above notation , we have in general for every LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation We compute the Betti numbers via Koszul cohomology , using the formula in Theorem LaTeXMLRef ( iii ) .
Consider the complex : LaTeXMLEquation .
Since LaTeXMLMath , it follows that LaTeXMLMath and LaTeXMLMath .
The exact sequence LaTeXMLEquation induces long exact sequences LaTeXMLEquation .
By tensoring with LaTeXMLMath and taking global sections , we get the exact sequence LaTeXMLEquation .
This proves the first assertion in the proposition .
We have a similar exact sequence : LaTeXMLEquation .
Therefore LaTeXMLMath is the dimension over LaTeXMLMath of the cokernel of LaTeXMLEquation .
Using again the exact sequence LaTeXMLMath , by tensoring with LaTeXMLMath and taking a suitable part of the long exact sequence , we get : LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath , we have LaTeXMLMath and therefore LaTeXMLMath .
From the above exact sequence we see that LaTeXMLMath , which proves the second assertion of the proposition .
∎ The higher cohomology groups LaTeXMLMath , LaTeXMLMath , always vanish .
Indeed , using the exact sequences in the proof of the proposition , we get LaTeXMLEquation .
Therefore we have LaTeXMLMath and MRC can be interpreted as saying that for general LaTeXMLMath , the cohomology of LaTeXMLMath is supported in cohomological degree either zero or one .
In the case of a curve LaTeXMLMath , MRC can be reformulated using Proposition LaTeXMLRef in terms of general line bundles on LaTeXMLMath .
We will denote by LaTeXMLMath and LaTeXMLMath the integers defined by LaTeXMLMath and LaTeXMLMath .
Suppose that LaTeXMLMath is a nondegenerate , integral curve of arithmetic genus LaTeXMLMath and degree LaTeXMLMath .
We consider the following two statements : ( i ) For every LaTeXMLMath and for a general line bundle LaTeXMLMath , where LaTeXMLMath , we have LaTeXMLMath .
( ii ) For every LaTeXMLMath and for a general line bundle LaTeXMLMath , where LaTeXMLMath , we have LaTeXMLMath .
Then MRC holds for LaTeXMLMath for every LaTeXMLMath if and only if both ( i ) and ( ii ) are true .
Moreover , if LaTeXMLMath is locally Gorenstein , then ( i ) and ( ii ) are equivalent .
If LaTeXMLMath , then for a general set LaTeXMLMath of LaTeXMLMath points , LaTeXMLMath is a general line bundle on LaTeXMLMath of degree LaTeXMLMath .
Since in this case LaTeXMLMath is a general line bundle of degree LaTeXMLMath and LaTeXMLMath , Proposition LaTeXMLRef says that MRC holds for every LaTeXMLMath if and only if for every LaTeXMLMath such that LaTeXMLMath and for a general line bundle LaTeXMLMath , either LaTeXMLMath or LaTeXMLMath .
Since LaTeXMLMath , we have LaTeXMLMath .
It follows immediately that LaTeXMLMath if and only if LaTeXMLMath .
The first statement of the corollary follows now from the fact that if LaTeXMLMath is a vector bundle on a curve and LaTeXMLMath is a point , then LaTeXMLMath implies LaTeXMLMath and LaTeXMLMath implies LaTeXMLMath .
The last statement follows from Serre duality and the isomorphism LaTeXMLMath .
∎ The corresponding assertion for IGC says that LaTeXMLMath satisfies IGC for every LaTeXMLMath if and only if both ( i ) and ( ii ) are true for LaTeXMLMath .
Note that if LaTeXMLMath is locally Gorenstein , then by Serre duality condition ( ii ) for LaTeXMLMath is equivalent to condition ( i ) for LaTeXMLMath .
If LaTeXMLMath is a locally Gorenstein integral curve such that LaTeXMLMath , then in order to check MRC for all LaTeXMLMath , it is enough to check condition ( i ) in Corollary LaTeXMLRef only for LaTeXMLMath .
Indeed , using Serre duality and Riemann-Roch , we see that the conditions for LaTeXMLMath and LaTeXMLMath are equivalent .
In light of Corollary LaTeXMLRef , we make the following : If LaTeXMLMath is a nondegenerate integral curve of arithmetic genus LaTeXMLMath and regularity LaTeXMLMath , we say that LaTeXMLMath satisfies MRC if a general set of LaTeXMLMath points on LaTeXMLMath satisfies MRC for every LaTeXMLMath .
If LaTeXMLMath is a very ample line bundle on a curve LaTeXMLMath as before , we say that LaTeXMLMath satisfies MRC if LaTeXMLMath satisfies MRC .
Analogous definitions are made for IGC .
( Rational quintics in LaTeXMLMath . )
We illustrate the above discussion in the case of smooth rational quintic curves in LaTeXMLMath .
We consider two explicit examples , the first when the curve lies on a ( smooth ) quadric and the second when it does not .
Let LaTeXMLMath be given parametrically by LaTeXMLMath , so that it lies on the quadric LaTeXMLMath .
The Betti diagram of LaTeXMLMath is and if LaTeXMLMath is a set of LaTeXMLMath points , then the Betti diagram of LaTeXMLMath is As LaTeXMLMath and LaTeXMLMath , we see that MRC is not satisfied by LaTeXMLMath for this number of points .
Let now LaTeXMLMath be the curve given parametrically by LaTeXMLMath .
In this case LaTeXMLMath does not lie on a quadric , and in fact , its Betti diagram is given by If LaTeXMLMath is a set of LaTeXMLMath points , then the Betti diagram of LaTeXMLMath is which shows that MRC is satisfied for LaTeXMLMath and this number of points .
These two examples show the possible behavior with respect to the MRC for smooth rational quintics in LaTeXMLMath .
The geometric condition of lying on a quadric translates into a condition on the splitting type of LaTeXMLMath .
More precisely , it is proved in LaTeXMLCite that if LaTeXMLMath is a smooth rational quintic curve , then LaTeXMLMath lies on a quadric if and only if we have LaTeXMLMath ( the other possibility , which is satisfied by a general such quintic , is that LaTeXMLMath ) .
Corollary LaTeXMLRef explains therefore the behaviour with respect to MRC in the above examples .
In this section we assume that LaTeXMLMath is a smooth projective curve of genus LaTeXMLMath and LaTeXMLMath is a very ample line bundle on LaTeXMLMath .
Our aim is to investigate whether LaTeXMLMath satisfies MRC , or at least IGC , for every LaTeXMLMath , in the embedding given by the complete linear series LaTeXMLMath .
As before , LaTeXMLMath will denote the regularity of LaTeXMLMath .
If LaTeXMLMath or LaTeXMLMath , then LaTeXMLMath satisfies MRC for all LaTeXMLMath in every embedding given by a complete linear series ( see LaTeXMLCite , Proposition 3.1 ) .
In higher genus we will concentrate on the study of MRC for canonical curves and curves embedded with high degree , in direct analogy with the syzygy questions of Green-Lazarsfeld ( cf .
LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
The main conclusion of this section will be that , while IGC is satisfied in both situations , the high-degree embeddings always fail to satisfy MRC at a well-specified spot in the Betti diagram .
This is in contrast with our main result , proved in §3 , that MRC always holds for canonical curves , and the arguments involved here provide an introduction to that section .
The common theme of the proofs is the vector bundle interpretation of MRC described in §1 .
Review of filtrations for LaTeXMLMath and LaTeXMLMath LaTeXMLCite .
Here we recall a basic property of the vector bundles LaTeXMLMath which will be essential for our arguments .
Let LaTeXMLMath be a very ample line bundle on LaTeXMLMath of degree LaTeXMLMath , and recall from §1 that LaTeXMLMath is given by the defining sequence LaTeXMLEquation .
Assume first that LaTeXMLMath is non-special and LaTeXMLMath are the points of a general hyperplane section of LaTeXMLMath .
One shows ( see e.g .
LaTeXMLCite §1.4 ) that there exists an exact sequence LaTeXMLEquation .
On the other hand , assuming that LaTeXMLMath is nonhyperelliptic and LaTeXMLMath , if LaTeXMLMath are the points of a general hyperplane section , the analogous sequence reads : LaTeXMLEquation .
We start by looking at the case of curves embedded with large degree .
The main results are summarized in the following : ( a ) If LaTeXMLMath is a very ample line bundle of degree LaTeXMLMath , then LaTeXMLMath holds for LaTeXMLMath .
( b ) If LaTeXMLMath and LaTeXMLMath is a line bundle of degree LaTeXMLMath , then there exists a value of LaTeXMLMath such that LaTeXMLMath does not satisfy MRC for LaTeXMLMath .
The same holds if LaTeXMLMath and LaTeXMLMath .
( a ) Let LaTeXMLMath be a very ample line bundle of degree LaTeXMLMath .
By Corollary LaTeXMLRef and Serre duality , it is easy to see that IGC holds for LaTeXMLMath if : ( i ) LaTeXMLMath for LaTeXMLMath general and ( ii ) LaTeXMLMath for LaTeXMLMath general .
Condition ( i ) is a simple consequence of the filtration ( LaTeXMLRef ) .
More precisely , if LaTeXMLMath are the points of a general hyperplane section of LaTeXMLMath , from the exact sequence LaTeXMLEquation we conclude that it would be enough to prove : LaTeXMLEquation for LaTeXMLMath general .
Now for every LaTeXMLMath , LaTeXMLMath is a general line bundle of degree LaTeXMLMath , so LaTeXMLMath .
On the other hand LaTeXMLMath , so clearly LaTeXMLMath .
For condition ( ii ) one needs a different argument .
By twisting the defining sequence of LaTeXMLMath : LaTeXMLEquation by LaTeXMLMath general and taking cohomology , we see that ( ii ) holds if and only if the map LaTeXMLEquation is injective , or dually if and only if the cup-product map LaTeXMLEquation is surjective .
We make the following : Claim .
LaTeXMLMath is a base point free pencil .
Assuming this for the time being , one can apply the Base Point Free Pencil Trick ( see LaTeXMLCite III §3 ) to conclude that LaTeXMLEquation .
But LaTeXMLMath is a general line bundle of degree LaTeXMLMath and so LaTeXMLMath .
By Riemann-Roch this means LaTeXMLMath .
On the other hand LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , so LaTeXMLMath must be surjective .
We are only left with proving the claim .
Since LaTeXMLMath is general , LaTeXMLMath , and so we easily get : LaTeXMLEquation .
Also , for every LaTeXMLMath , LaTeXMLMath is general , hence still noneffective .
Thus : LaTeXMLEquation .
This implies that LaTeXMLMath is base point free .
( b ) Here we follow an argument in LaTeXMLCite leading to the required nonvanishing statement .
First note that it is clear from ( LaTeXMLRef ) that for every LaTeXMLMath with LaTeXMLMath there is an inclusion LaTeXMLEquation where LaTeXMLMath are general points on LaTeXMLMath .
This immediately implies that LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are general effective divisors on LaTeXMLMath of degree LaTeXMLMath .
On the other hand we use the fact ( see e.g .
LaTeXMLCite Ex .
V. D ) that every line bundle LaTeXMLMath can be written as a difference LaTeXMLEquation which means that LaTeXMLEquation .
Now by Serre duality : LaTeXMLEquation so that Corollary LaTeXMLRef easily implies that LaTeXMLMath does not satisfy MRC for LaTeXMLMath as long as LaTeXMLMath .
A simple computation gives then the stated conclusion .
∎ Motivation for the argument in ( b ) above was quite surprisingly provided by the study LaTeXMLCite of the base locus of the determinant linear series on the moduli space LaTeXMLMath of semistable bundles of rank LaTeXMLMath and trivial determinant on a curve LaTeXMLMath .
In fact this argument produces explicit base points for the determinant linear series under appropriate numerical conditions .
The technique in Theorem LaTeXMLRef ( b ) can be extended to produce examples of higher dimensional varieties for which appropriate choices of LaTeXMLMath force the failure of MRC for general sets of LaTeXMLMath points .
More precisely , the varieties in question are projective bundles LaTeXMLMath over a curve LaTeXMLMath , associated to very ample vector bundles LaTeXMLMath on LaTeXMLMath of arbitrary rank and large degree , containing sub-line bundles of large degree .
Using the interpretation given in Proposition LaTeXMLRef , the problem is reduced to a cohomological question about the exterior powers LaTeXMLMath , where LaTeXMLMath is defined analogously as the kernel of the evaluation map LaTeXMLEquation .
This question is then treated essentially as above , and we do not enter into details .
Unfortunately once a bundle LaTeXMLMath of higher rank is fixed , this technique does not seem to produce couterexamples for arbitrarily large values of LaTeXMLMath , as in the case of line bundles .
Such examples would be very interesting , in light of the asymptotically nice behavior of general points in LaTeXMLMath ( cf .
LaTeXMLCite ) .
Finally we turn to the case of canonical curves with the goal of providing an introduction to the main result in Section 3 .
Let LaTeXMLMath be a nonhyperelliptic curve of genus LaTeXMLMath , LaTeXMLMath and LaTeXMLMath the canonical embedding .
We note here that an argument similar to Theorem LaTeXMLRef ( a ) immediately implies IGC for LaTeXMLMath .
This will be later subsumed in the general Theorem LaTeXMLRef .
IGC holds for the canonical curve LaTeXMLMath .
The argument is similar ( and in fact simpler ) to the proof of ( ii ) in Theorem LaTeXMLRef ( a ) .
In this case , again by interpreting Proposition LaTeXMLRef ( se Remark LaTeXMLRef IGC holds if and only if LaTeXMLEquation .
This is in turn equivalent to the surjectivity of the multiplication map : LaTeXMLEquation which is again a quick application of the Base Point Free Pencil Trick .
∎ The geometric picture in the present case of canonical curves can be described a little more precisely .
In fact , for LaTeXMLMath , we have LaTeXMLEquation where LaTeXMLMath denotes in general the slope of the vector bundle LaTeXMLMath .
By standard determinantal results , the subset LaTeXMLEquation is either a divisor or the whole variety .
The statement of IGC is then equivalent to saying that LaTeXMLMath is indeed a divisor in LaTeXMLMath ( one says that LaTeXMLMath has a theta divisor ) .
A simple filtration argument based on the sequence ( 3 ) above shows that in fact LaTeXMLEquation which has already been observed by Paranjape and Ramanan in LaTeXMLCite A generalization of this observation to the higher exterior powers LaTeXMLMath will be the starting point for our approach to proving MRC for canonical curves in what follows .
In this section LaTeXMLMath will be a canonical curve , i.e .
a smooth curve of genus LaTeXMLMath embedded in LaTeXMLMath by the canonical linear series LaTeXMLMath ( in particular LaTeXMLMath is not hyperelliptic ) .
Our goal is to prove the following : If LaTeXMLMath is a canonical curve , then LaTeXMLMath satisfies MRC .
In fact , since LaTeXMLMath is canonically embedded , its regularity is LaTeXMLMath , and as LaTeXMLMath we always have LaTeXMLMath .
Thus the statement means that MRC holds for every LaTeXMLMath .
The general condition required for a curve to satisfy MRC which was stated in Corollary LaTeXMLRef ( see also Remark LaTeXMLRef ) takes a particularly clean form in the case of canonical embeddings .
We restate it for further use .
Let LaTeXMLMath be a canonical curve .
Then LaTeXMLMath satisfies MRC if and only if , for all LaTeXMLMath we have LaTeXMLEquation or equivalently LaTeXMLEquation .
Note that LaTeXMLMath , so LaTeXMLMath .
This means that the condition LaTeXMLMath in Lemma LaTeXMLRef is equivalent to saying that LaTeXMLMath has a theta divisor ( in LaTeXMLMath ) , which we denote LaTeXMLMath .
In other words , the set defined by LaTeXMLEquation with the scheme structure of a degeneracy locus of a map of vector bundles of the same rank is an actual divisor as expected ( cf .
LaTeXMLCite II §4 ) .
Note that the statement LaTeXMLMath in Lemma LaTeXMLRef makes sense even for hyperelliptic curves .
Again LaTeXMLMath is the dual of LaTeXMLMath , where LaTeXMLMath is the kernel of the evaluation map for the canonical line bundle .
Therefore we will say slightly abusively that MRC is satisfied for some smooth curve of genus LaTeXMLMath if LaTeXMLMath is satisfied for all LaTeXMLMath , LaTeXMLMath .
In fact , the hyperelliptic case is the only one for which we can give a direct argument .
MRC holds for hyperelliptic curves .
We show that for every LaTeXMLMath , LaTeXMLMath , if LaTeXMLMath is general .
Since LaTeXMLMath is hyperelliptic , we have a degree two morphism LaTeXMLMath and if LaTeXMLMath , then LaTeXMLMath .
Therefore the morphism LaTeXMLMath defined by LaTeXMLMath is the composition of the Veronese embedding LaTeXMLMath with LaTeXMLMath .
Note that we have LaTeXMLMath .
Since on LaTeXMLMath we have the exact sequence : LaTeXMLEquation we get LaTeXMLMath .
Therefore for every LaTeXMLMath , we have LaTeXMLEquation .
Now if LaTeXMLMath is general , then LaTeXMLMath is a general line bundle of degree LaTeXMLMath and so LaTeXMLMath .
∎ We noted above that MRC is satisfied for LaTeXMLMath if and only if LaTeXMLMath is a divisor .
We now identify precisely what the divisor should be , assuming that this happens .
( At the end of the day this will hold for all canonical curves . )
Recall that by general theory , whenever a divisor , LaTeXMLMath belongs to the linear series LaTeXMLMath , where we slightly abusively denote by LaTeXMLMath a certain theta divisor on LaTeXMLMath ( more precisely LaTeXMLMath , where LaTeXMLMath is a LaTeXMLMath -th root of LaTeXMLMath ) .
From now on we always assume that we are in this situation .
The Picard variety LaTeXMLMath contains a difference subvariety LaTeXMLMath defined as the image of the difference map LaTeXMLEquation .
LaTeXMLEquation The geometry of the difference varieties has interesting links with the geometry of the curve LaTeXMLCite and LaTeXMLCite ( see below ) .
The key observation is that our theta divisor is nothing else but the difference variety above .
For every smooth curve LaTeXMLMath of genus LaTeXMLMath , we have LaTeXMLEquation .
Moreover , if LaTeXMLMath is nonhyperelliptic and LaTeXMLMath is a divisor , then the above inclusion is an equality .
We start with a few properties of the difference varieties , which for instance easily imply that LaTeXMLMath is a divisor .
More generally , we study the difference variety LaTeXMLMath , LaTeXMLMath , defined analogously .
Note that this study is suggested in a series of exercises in [ ACGH ] Ch.V.D and Ch.VI.A in the case LaTeXMLMath , but the formula in V.D-3 there giving the cohomology class of LaTeXMLMath is unfortunately incorrect , as we first learned from R. Lazarsfeld .
The results we need are collected in the following : ( a ) Assume that LaTeXMLMath .
Then the difference map : LaTeXMLEquation is birational onto its image if LaTeXMLMath is nonhyperelliptic .
When LaTeXMLMath is hyperelliptic , LaTeXMLMath has degree LaTeXMLMath onto its image .
( b ) If LaTeXMLMath is nonhyperelliptic , the cohomology class LaTeXMLMath of LaTeXMLMath in LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath is the class of a theta divisor .
Assuming this , the particular case LaTeXMLMath and LaTeXMLMath quickly implies the main result .
( of Proposition LaTeXMLRef ) From Proposition LaTeXMLRef ( b ) we see that if LaTeXMLMath is nonhyperelliptic , then the class of LaTeXMLMath is given by : LaTeXMLEquation .
On the other hand , as LaTeXMLMath is associated to the vector bundle LaTeXMLMath , if it is a divisor , then its cohomology class is LaTeXMLMath ( recall that LaTeXMLMath has the same class as LaTeXMLMath ) .
As in this case both LaTeXMLMath and LaTeXMLMath are divisors , in order to finish the proof of the proposition it is enough to prove the first statement .
To this end , we follow almost verbatim the argument in Theorem LaTeXMLRef ( b ) .
Namely , the filtration ( 2 ) in §2 implies that for every LaTeXMLMath there is an inclusion : LaTeXMLEquation where LaTeXMLMath are general points on LaTeXMLMath .
This means that LaTeXMLEquation for all general effective divisors LaTeXMLMath of degree LaTeXMLMath and LaTeXMLMath of degree LaTeXMLMath , which gives the desired inclusion .
∎ We are left with proving Proposition LaTeXMLRef .
This follows by more or less standard arguments in the study of Abel maps and Poincaré formulas for cohomology classes of images of symmetric products .
( of Proposition LaTeXMLRef ) ( a ) This is certainly well known ( cf .
LaTeXMLCite Ch.V.D ) , and we do not reproduce the proof here .
( b ) Assume now that LaTeXMLMath is nonhyperelliptic , so that LaTeXMLEquation is birational onto its image .
For simplicity we will map everything to the Jacobian of LaTeXMLMath , so fix a point LaTeXMLMath and consider the commutative diagram : LaTeXMLEquation
We show that the relativistic analogue of the two types of time translation in a non-relativistic history theory is the existence of two distinct Poincaré groups .
The ‘ internal ’ Poincaré group is analogous to the one that arises in the standard canonical quantisation scheme ; the ‘ external ’ Poincaré group is similar to the group that arises in a Lagrangian description of the standard theory .
In particular , it performs explicit changes of the spacetime foliation that is implicitly assumed in standard canonical field theory .
The generalisation of continuous-time history theory to include relativistic quantum fields raises some subtle issues that tend to be hidden in the normal canonical treatment of a quantum field .
The standard canonical quantisation of a relativistic field requires the choice of a Lorentzian foliation on the background spacetime : the Hamiltonian is then defined with respect to this foliation .
There exist many unitarily inequivalent representations of the canonical commutation relations for this quantum field theory : the physically appropriate one is chosen by requiring that the Hamiltonian exists as a well-defined self-adjoint operator .
In this sense—like the Hamiltonian itself—the physically appropriate representation is foliation-dependent .
Relativistic covariance is then implemented by seeking a representation of the Poincaré group on the resulting Hilbert space .
However , the Poincaré group thus constructed does not explicitly perform a change of the foliation .
The HPO continuous-time histories approach to quantum theory LaTeXMLCite is particularly suited to deal with systems that have a non-trivial temporal structure , and therefore it should be able to provide a significant clarification of this point .
Specifically , we will show that the relativistic analogue of the two types of time translation that arise in a non-relativistic history theory is the existence of two distinct Poincaré groups .
The ‘ internal ’ Poincaré group is analogous to the one that arises in the standard canonical quantisation scheme as sketched above .
However , the ‘ external ’ one is a novel object : it is similar to the group that arises in the Lagrangian description of the field theory .
In particular , it explicitly performs changes of the foliation .
This arises from the striking property that HPO theories admit two distinct types of time transformation , each representing a distinct quality of time LaTeXMLCite .
The first corresponds to time considered purely as a kinematical parameter of a physical system , with respect to which a history is defined as a succession of possible events .
It is strongly connected with the temporal-logical structure of the theory and is related to the view of time as a parameter that determines the ordering of events .
The second corresponds to the dynamical evolution generated by the Hamiltonian .
For a detailed presentation of the HPO continuous-time programme see LaTeXMLCite .
As we shall see , one of the important results of the formalism as applied to a field theory is that , even though the representations of the history algebra are foliation dependent , the physical quantities ( probabilities ) are not .
In section 2 , we shall give a brief description of the underlying concepts of the continuous-histories programme : this is necessary for establishing the framework of the ensuing work .
In section 3 , we present the histories version of a classical scalar field theory : in particular , we show how two Poincaré groups arise as an analogue of the two types of time transformation in the non-relativistic history theory .
The free quantum scalar field theory is presented in section 4 .
We show that due to the histories temporal structure previously introduced in LaTeXMLCite , manifest Poincaré invariance is possible .
Specifically , we show how different representations of the history algebra—corresponding to different choices of foliation—are realised on the same Fock space ( notwithstanding the fact that the different representations are unitarily inequivalent ) , and we show that they are related in a certain way with Poincaré transformations .
The History Projection Operator ( the , so-called , ‘ HPO ’ approach ) theory was a development LaTeXMLCite ( emphasizing quantum temporal logic ) of the consistent-histories approach to quantum theory inaugurated by Griffiths , Omnés , Gell-Mann and Hartle LaTeXMLCite .
However , the novel temporal structure introduced in LaTeXMLCite led to a departure from the original ideas on decoherence .
In particular , in our approach , emphasis is placed on the distinction between ( i ) the temporal logic structure of the theory ; and ( ii ) the dynamics LaTeXMLCite .
In consistent-histories theory , a history is defined as a sequence of time-ordered propositions about properties of a physical system , each of which can be represented , as usual , by a projection operator .
In normal quantum theory , it is not possible to assign a probability measure to the set of all histories .
However , when a certain ‘ decoherence condition ’ is satisfied by a set of histories , the elements of this set can be given probabilities .
The probability information of the theory is encoded in the decoherence functional : a complex function of pairs of histories which—in the original approach of Griffiths et al —can be written as LaTeXMLEquation where LaTeXMLMath is the initial density-matrix , and where the class operator LaTeXMLMath is defined in terms of the standard Schrödinger-picture projection operators LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath is the unitary time-evolution operator from time LaTeXMLMath to LaTeXMLMath .
Each projection operator LaTeXMLMath represents a proposition about the system at time LaTeXMLMath , and the class operator LaTeXMLMath represents the composite history proposition “ LaTeXMLMath is true at time LaTeXMLMath , and then LaTeXMLMath is true at time LaTeXMLMath , and then … , and then LaTeXMLMath is true at time LaTeXMLMath ” .
Isham and Linden developed the consistent-histories formalism further , concentrating on its temporal quantum logic structure LaTeXMLCite .
They showed that propositions about the histories of a system could be represented by projection operators on a new , ‘ history ’ Hilbert space .
In particular , the history proposition “ LaTeXMLMath is true at time LaTeXMLMath , and then LaTeXMLMath is true at time LaTeXMLMath , and then … , and then LaTeXMLMath is true at time LaTeXMLMath ” is represented by the tensor product LaTeXMLMath which , unlike LaTeXMLMath , is a genuine projection operator , that is defined on the tensor product of copies of the standard Hilbert space LaTeXMLMath .
Hence the ‘ History Projection Operator ’ formalism extends to multiple times , the quantum logic of single-time quantum theory .
An important way of understanding the history Hilbert space LaTeXMLMath is in terms of the representations of the ‘ history group ’ —in elementary systems this is the history analogue of the canonical group LaTeXMLCite .
For example , for the simple case of a point particle moving on a line , the Lie algebra of the history group for a continuous time parameter LaTeXMLMath is described by the history commutation relations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath .
It is important to note that these operators are in the Schrödinger picture , and that the history algebra is invariant under translations of the time index of these operators .
The choice of the Dirac delta-function in the right hand side of Eq .
( LaTeXMLRef ) is associated with the requirement that time be treated as a continuous variable .
One important consequence is the fact that the observables can not be defined at sharp moments of time but rather appear naturally as time-averaged .
A unique representation of this algebra can be found by requiring the existence of an operator analogue of a time-averaged Hamiltonian LaTeXMLMath , where LaTeXMLMath is the standard Hamiltonian defined at a moment of time LaTeXMLMath LaTeXMLCite .
One of the original problems in the development of the HPO theory was the lack of a clear notion of time evolution , in the sense that , there was no natural way to express the time translations from one time slot—that refers to one copy of the Hilbert space LaTeXMLMath —to another one , that refers to another copy LaTeXMLMath .
The situation changed with the introduction of the ‘ action ’ operator S .
Indeed , the crucial step for constructing the temporal structure of the theory was the definition in LaTeXMLCite of the action operator LaTeXMLMath —a quantum analogue of the Hamilton-Jacobi functional LaTeXMLCite , written as LaTeXMLEquation where LaTeXMLMath is an appropriate test function .
The first term of the action operator LaTeXMLMath Eq .
( LaTeXMLRef ) is identical to the kinematical part of the classical phase space action functional .
This ‘ Liouville ’ operator is formally written as LaTeXMLEquation so that LaTeXMLEquation .
A fundamental property of the HPO form of history theory is that the Liouville operator LaTeXMLMath and the Hamiltonian operator LaTeXMLMath generate two distinct types of time transformation .
The Liouville operator LaTeXMLMath relates the Schrödinger-picture operators associated with different time- LaTeXMLMath labels , whereas LaTeXMLMath is associated with internal dynamical changes at the fixed time LaTeXMLMath ( with an analogous statement for the smeared operator LaTeXMLMath ) .
The action operator LaTeXMLMath is thus the generator of both types of time translation LaTeXMLCite .
More precisely , it was shown that there exist two distinct types of time transformation .
One—generated by the Liouville operator LaTeXMLMath —refers to time as it appears in temporal logic , and it is related to LaTeXMLMath -label in Eqs .
( LaTeXMLRef – LaTeXMLRef ) .
The other—generated by the Hamiltonian—refers to time as it appears in the implementation of dynamical laws , and it is related to the label LaTeXMLMath in the ‘ history Heisenberg picture ’ operator , that is hence defined in accord to the novel conceptual issues introduced with the ‘ two modes of time ’ LaTeXMLEquation where LaTeXMLMath is defined to be LaTeXMLMath with LaTeXMLMath set equal to LaTeXMLMath .
We will use the notation LaTeXMLMath for these history Heisenberg-picture operators smeared with respect to the time label LaTeXMLMath , and we notice from Eq .
( LaTeXMLRef ) that these quantities behave like standard Heisenberg-picture operators with a time parameter LaTeXMLMath .
For any specific physical system these two transformations are intertwined with the aid of the action operator LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath , and where LaTeXMLMath means LaTeXMLMath with LaTeXMLMath .
The continuous-time histories description has a natural analogue for classical histories LaTeXMLCite .
In this scheme , the basic mathematical entity is the space LaTeXMLMath of differentiable paths taking their value in the manifold LaTeXMLMath of classical states .
Hence an element of LaTeXMLMath is a smooth path LaTeXMLMath .
In effect , we associate a copy of the classical state space with each moment of time , and employ differentiable sections of the ensuing bundle over LaTeXMLMath .
The key idea in this approach to classical histories is contained in the symplectic structure on this space of temporal paths LaTeXMLMath .
For example , for a particle moving in one dimension ( with configuration coordinate LaTeXMLMath and momentum coordinate LaTeXMLMath ) , the history space LaTeXMLMath is equipped with a symplectic form LaTeXMLEquation which generates the history Poisson brackets LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
In general , given a function LaTeXMLMath on LaTeXMLMath we can define an associated family LaTeXMLMath of functions on LaTeXMLMath as LaTeXMLEquation .
In this way , all transformations implemented through the Poisson bracket in the normal canonical theory , correspond to transformations in the history theory that preserve the time label LaTeXMLMath .
Indeed , for two families of functions LaTeXMLMath and LaTeXMLMath defined through ( LaTeXMLRef ) we have LaTeXMLEquation where LaTeXMLMath corresponds to the function LaTeXMLMath on LaTeXMLMath LaTeXMLEquation .
In this way all relevant structures of the canonical theory can be naturally transferred to the histories framework LaTeXMLCite .
The Liouville , Hamilton and action functionals on LaTeXMLMath are defined respectively as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is the velocity at the time point LaTeXMLMath of the path LaTeXMLMath .
These definitions are crucial for the dynamics of the theory .
In particular , LaTeXMLMath and LaTeXMLMath are the classical analogues of the generators of the two types of time transformation in the history quantum theory LaTeXMLCite .
The crucial result of classical histories theory is that one may deduce the equations of motion in the following way LaTeXMLCite : a classical history LaTeXMLMath is the realised path of the system— i.e .
a solution of the equations of motion of the system—if it satisfies the equations LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the path LaTeXMLMath , and LaTeXMLMath is the position coordinate of the realised path LaTeXMLMath at the time point LaTeXMLMath .
The above equations ( LaTeXMLRef – LaTeXMLRef ) are the history equivalent of the canonical equations of motion .
In particular , the symplectic transformation generated by the history action functional LaTeXMLMath leaves invariant the paths that are classical solutions of the system : LaTeXMLEquation .
LaTeXMLEquation More generally , any function LaTeXMLMath on LaTeXMLMath satisfies the equation LaTeXMLEquation .
This is the way in which equations of motion appear in the classical history theory .
Notice that the role of the action as the generator of time transformations emerges naturally in this classical case .
Furthermore , the condition ( LaTeXMLRef ) above emphasises the role of the Hamiltonian and Liouville functionals in histories theory as generators of different types of time transformation .
It also clarifies the new temporal structure that arises in history theory when compared with the standard classical theory .
This result is of particular importance in the case of parameterised systems , where the notion of time is recovered after the phase space reduction LaTeXMLCite .
In the Hamiltonian description of a free scalar field LaTeXMLMath with mass LaTeXMLMath on Minkowski spacetime , the first step is to choose a spacelike foliation , which can be specified by its normal—a unit time-like vector LaTeXMLMath .
We shall take the signature of the Minkowski metric LaTeXMLMath to be LaTeXMLMath .
The first step is to select a specific foliation , and to choose a reference leaf LaTeXMLMath that is characterised by LaTeXMLMath , where LaTeXMLMath is the natural time label associated with the foliation .
The corresponding configuration space is the space LaTeXMLMath of all smooth scalar functions LaTeXMLMath on LaTeXMLMath , while the phase space LaTeXMLMath is its cotangent bundle LaTeXMLMath defined in an appropriate way LaTeXMLMath .
However , we do not need to become involved in such complexities here : for our purposes it suffices to postulate the basic Poisson algebra relations ( LaTeXMLRef – LaTeXMLRef ) that follow .
.
The key point about this structure is that the state space of fields is equipped with the Poisson brackets LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The relativistic scalar field theory is covariant under the action of the Poincaré group LaTeXMLCite .
For a free massive scalar field , the generators of time-translations LaTeXMLMath , space translations LaTeXMLMath , spatial rotations LaTeXMLMath and Lorentz boosts LaTeXMLMath are respectively LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where we note that the sub/superscripts LaTeXMLMath refer to coordinates in the surface LaTeXMLMath that is spatial with respect to the chosen foliation vector LaTeXMLMath .
Similarly , the integrals above are all defined over LaTeXMLMath .
If we define the partial differential operator LaTeXMLEquation we can write the convenient expressions for the Hamiltonian and the boosts generator as LaTeXMLEquation .
LaTeXMLEquation In the histories formalism of a scalar field , the space of phase-space histories LaTeXMLMath is an appropriate subset of the continuous Cartesian product LaTeXMLMath of copies of the standard state space LaTeXMLMath , each labeled by the time parameter LaTeXMLMath .
The choice of LaTeXMLMath depends on the choice of a foliation vector LaTeXMLMath , hence the space of histories also has an implicit dependence on LaTeXMLMath and should therefore be written as LaTeXMLMath .
Furthermore , we write LaTeXMLMath , the space-like surface LaTeXMLMath defined with respect to its normal vector LaTeXMLMath , and labeled by the parameter LaTeXMLMath .
To be more precise , for each space-like surface LaTeXMLMath we consider the state space LaTeXMLMath .
Then we define the fiber bundle with basis LaTeXMLMath and fiber LaTeXMLMath , at each LaTeXMLMath .
Histories are defined as the cross-sections of the ensuing bundle , and the history space LaTeXMLMath is the space of all smooth cross-sections of this bundle .
The Poisson algebra relations of the history theory are LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are space-time points .
The field LaTeXMLMath and its conjugate momentum LaTeXMLMath are implicitly defined with respect to the foliation vector LaTeXMLMath .
The definitions of the action LaTeXMLMath , Liouville LaTeXMLMath and ‘ Hamiltonian ’ LaTeXMLMath functionals are LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation respectively , where again there is an implicit LaTeXMLMath label on these three quantities ; and where LaTeXMLMath is the differential operator LaTeXMLEquation introduced above .
As we explained earlier , the variation of LaTeXMLMath leaves invariant the paths LaTeXMLMath that are classical solutions of the system : LaTeXMLEquation .
LaTeXMLEquation As we shall now see , LaTeXMLMath is the generator to the time averaged internal Poincaré group .
The Poincaré group is the group of isometries of the Minkowski metric .
Hence , any field theory in Minkowski space-time needs to be covariant under the action of the Poincaré group .
As we shall now see , in a history theory—because of its augmented temporal structure—the associated group theory leads to a particular interesting result : namely , there are two distinct Poincaré groups that act on the history space .
One significant feature of histories theory is that it gives a representation of the temporal logic of the system that is independent of the dynamics involved .
Hence , propositions about the state of the system at different times are represented by appropriate subsets of the space of paths .
In the context of symmetries , however , the temporal logic structure entails the following .
For each copy LaTeXMLMath of the standard state space , there exists a Poincarè group symmetry of the type one would expect in a canonical treatment of relativistic field theory .
On the other hand , in the history theory the state space is heuristically the Cartesian product of such copies , and all physical quantities in the standard treatment now appear as naturally time-averaged LaTeXMLCite .
Hence one may write time-averaged generators of the internal Poincaré groups , in a covariant-like notation as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is an ‘ LaTeXMLMath -spacelike ’ vector , i.e .
one such that LaTeXMLMath .
Of special interest are the groups of canonical transformations generated by the Hamiltonian generator LaTeXMLMath and the boosts generator LaTeXMLMath .
Note that a space-time point LaTeXMLMath can be associated with the pair LaTeXMLMath , as LaTeXMLMath , where the three-vector LaTeXMLMath has been associated with a corresponding LaTeXMLMath -spatial four-vector LaTeXMLMath ( i.e. , LaTeXMLMath ) ; note that LaTeXMLMath .
Then we define the classical analogue of the Heisenberg picture fields as LaTeXMLEquation or LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath .
The square-root operator LaTeXMLMath , and functions thereof , can be defined rigorously using the spectral theory of the self-adjoint , partial differential operator LaTeXMLMath on the Hilbert space LaTeXMLMath .
Notice also that the time label LaTeXMLMath is not affected by this transformation since LaTeXMLMath .
For a fixed value of time LaTeXMLMath , the field LaTeXMLMath is the ‘ Heisenberg-picture ’ field of the standard canonical treatment .
The action of boost transformations is best shown upon objects LaTeXMLMath as LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are related by the Lorentz boost parametrised by LaTeXMLMath as LaTeXMLEquation .
LaTeXMLEquation where , as above , LaTeXMLMath is the spatial part of LaTeXMLMath with respect to LaTeXMLMath , so that LaTeXMLMath and LaTeXMLMath .
Hence , for each copy of the standard classical state space , there exists an ‘ internal ’ Poincarè group that acts on the copy of standard canonical field theory that is labeled by the same LaTeXMLMath -time label .
For each fixed LaTeXMLMath , there also exists an ‘ external ’ Poincaré group with generators LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath generate spacetime translations .
The LaTeXMLMath -spatial parts of the tensor LaTeXMLMath generate spatial rotations ; the time parts generates boosts .
The space translations and rotations are identical to those of the internal Poincaré group .
However the time translation and the boosts differ .
Indeed , under LaTeXMLMath we have LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the time translation generated by LaTeXMLMath .
Thus , what we have shown here is that the time-translation generator for the ‘ external ’ Poincaré group is the Liouville functional LaTeXMLMath .On the other hand , the boost generator LaTeXMLMath generates Lorentz transformations of the type LaTeXMLEquation .
LaTeXMLEquation where for future convenience we write as LaTeXMLMath the element of the Lorentz group obtained by exponentiation of the boost parameterised by LaTeXMLMath .
Furthermore , under the action of this external group , the generators of the internal Poincaré group transform as follows LaTeXMLEquation .
LaTeXMLEquation where we have now attached the explicit LaTeXMLMath labels that were implicit in our previous notation for these quantities .
The action functional transforms in the same way LaTeXMLEquation .
Note that the action of the two groups coincides on classical solutions LaTeXMLMath : LaTeXMLEquation .
LaTeXMLEquation We must emphasise again that the definition of LaTeXMLMath depends on the foliation vector .
Hence , so will the action of the Poincaré group .
Here we deal with the scalar field , for which this dependence is not explicit .
However , this dependence , and analogue of the Poincaré group action is a major feature in systems where there is an explicit foliation dependence .
For example , this is the case of general relativity which is discussed in LaTeXMLCite .
Canonical quantisation proceeds by looking for a representation of the canonical commutation relations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation on a Hilbert space which , in practice , is selected by requiring the existence of the Hamiltonian as a genuine ( essentially ) self-adjoint operator .
For a free field , such a representation can be found on the Fock space LaTeXMLMath on which the fields can be written in terms of the creation and annihilation operators LaTeXMLMath and LaTeXMLMath that define LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLEquation .
The ( normal-ordered ) Hamiltonian then reads LaTeXMLMath and LaTeXMLMath from the well known relation LaTeXMLMath .
LaTeXMLEquation .
A representation of the full Poincaré group exists on this Hilbert space .
The starting point is the generators of the classical theory , suitably normal-ordered to correspond to well-defined operators .
Substituting the fields in terms of creation and annihilation operators , the generators can be written as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
These generators , together with LaTeXMLMath defined in Eq .
( LaTeXMLRef ) , satisfy the Lie algebra relations of the Poincaré group .
In the canonical picture , the covariant fields are obtained by the Heisenberg equations of motion LaTeXMLEquation .
LaTeXMLEquation The explicit automorphisms generated by the boosts may easily be calculated for the Heisenberg picture creation and annihilation operators LaTeXMLEquation and they give LaTeXMLEquation where the transformation LaTeXMLMath is given by Eq .
( LaTeXMLRef ) , so that we can write LaTeXMLEquation .
From this , one can write the explicit transformation laws for the Heisenberg fields LaTeXMLMath and LaTeXMLMath .
The first question that arises in this standard treatment is whether the Poincaré transformations are associated with any changes of foliation .
Working canonically there is no trace of the foliation vector on the Fock space defined by Eq .
( LaTeXMLRef ) , so this question can not readily be answered .
Being able to talk about foliations is a necessary step if we are to elucidate the spacetime character of a quantum theory , in which the parameter LaTeXMLMath of the Heisenberg picture objects corresponds to the foliation time parameter in spacetime .
For example , the physical meaning of the parameter LaTeXMLMath of the Heisenberg objects depends on the choice of foliation vector .
As we have already mentioned in section 2 , the introduction of the history group LaTeXMLCite as an analogue of the canonical group relates the spectral projectors of the generators of its Lie algebra with propositions about history phase space quantities .
This algebra is infinite-dimensional and therefore there exist infinitely many representations .
However the physically appropriate representation of the smeared history algebra can be uniquely selected by the requirement that the time-averaged energy exists as a proper self-adjoint operator LaTeXMLCite .
The resulting Hilbert space has a natural interpretation as a continuous-tensor product : hence by this means we also gain a natural mathematical implementation of the concept of ‘ continuous ’ temporal logic .
We shall now apply the histories ideas to relativistic quantum field theory on Minkowski space-time .
The representation of the history algebra is to be selected by requiring that the time-averaged energy LaTeXMLMath , ( which is associated with history propositions about temporal averages of the energy ) exists as a proper essentially self-adjoint operator LaTeXMLCite .
In what follows , for the sake of typographical simplicity we will no longer use hats to indicate quantum operators .
We start with the abstract algebra LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are spacetime points .
In order to find suitable representations of this algebra we start with the Fock space LaTeXMLMath in which there is a natural definition of creation and annihilation operators LaTeXMLMath and LaTeXMLMath that satisfy the commutation relations LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
An appropriate representation of the Poincaré group can be defined by requiring LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the cyclic ’ vacuum ’ state for the theory .
Then clearly history fields can be defined by LaTeXMLEquation .
LaTeXMLEquation and satisfy Eqs .
( LaTeXMLRef – LaTeXMLRef ) .
They also transform in the obvious covariant way under the operators LaTeXMLMath introduced above .
It should be emphasized that the fields LaTeXMLMath and LaTeXMLMath thus defined do not have any foliation vector dependence .
However , an operator LaTeXMLMath of the time-averaged energy of the system can not be well defined so that it depends functionally on these fields in the usual way .
Hence we must seek a different , and more physically appropriate representation , for the history algebra on the history Hilbert space LaTeXMLMath .
We start by making a fixed choice of a unit time-like vector LaTeXMLMath which we use to foliate the four-dimensional Minkowski space-time .
It is clear that the average-energy operator is itself dependent upon the choice of foliation LaTeXMLMath , and therefore this must also be true for the elements of the history algebra .
Hence to emphasise that the physically appropriate representation depends on LaTeXMLMath we rewrite the history commutation relations as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are spacetime points .
The dependence of the representation of the history algebra on the choice of the time-like foliation vector LaTeXMLMath is indicated by the upper left symbol for the field LaTeXMLMath and its ‘ conjugate ’ LaTeXMLMath .
One may also write the canonical version of the history algebra .
Notice that—as in the discussion above of classical history theory—in relating Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) with the canonical version of the history algebra the three-vector LaTeXMLMath may be equated with a four-vector LaTeXMLMath that satisfies LaTeXMLMath ( the dot product is taken with respect to the Minkowski metric LaTeXMLMath ) so that the pair LaTeXMLMath is associated with the spacetime point LaTeXMLMath ( in particular , LaTeXMLMath ) .
The canonical history commutation relations can be written therefore as LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where , for each LaTeXMLMath , the fields LaTeXMLMath and LaTeXMLMath are associated with the spacelike hypersurface LaTeXMLMath , characterised by the normal vector LaTeXMLMath and by the foliation parameter LaTeXMLMath .
In particular , the three-vector LaTeXMLMath in LaTeXMLMath or in LaTeXMLMath denotes a vector in this space .
A central feature of the approach that is followed in this work for the histories quantum field theory , is that for all foliation vectors LaTeXMLMath , the corresponding foliation-dependent representations of the history algebra Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) can all be realised on the same Fock space LaTeXMLMath that also carries the ‘ covariant ’ fields LaTeXMLMath and LaTeXMLMath defined in Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) .
The foliation-dependent fields LaTeXMLMath and LaTeXMLMath are expressed in terms of the covariant creation and annihilation operators of LaTeXMLMath , and the related covariant fields LaTeXMLMath and LaTeXMLMath of Eqs .
( LaTeXMLRef – LaTeXMLRef ) , as LaTeXMLEquation .
LaTeXMLEquation and conversely , LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath denotes the partial differential operator defined in Eq .
( LaTeXMLRef ) on the Hilbert space LaTeXMLMath .
For a fixed foliation vector LaTeXMLMath , we seek a family of ‘ internal ’ Hamiltonians LaTeXMLMath , LaTeXMLMath , whose explicit formal expression may be deduced from the standard quantum field theory expression to be LaTeXMLEquation .
The corresponding smeared expression ( which must be normal-ordered to be well-defined ) is LaTeXMLEquation where LaTeXMLMath is a real-valued test function .
We next augment the history algebra with the following commutation relations that would be satisfied by the operators LaTeXMLMath , if they existed , LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
If the operators LaTeXMLMath existed , the above commutation relations would give rise to the transformations LaTeXMLEquation .
LaTeXMLEquation Note that the expression LaTeXMLMath is unambiguous since , viewed as an operator on LaTeXMLMath , multiplication by LaTeXMLMath commutes with LaTeXMLMath .
The right hand side of Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) defines an automorphism of the history algebra Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) , and all that remains is to show that these automorphisms are unitarily implementable in this representation .
To this end , we use Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) to prove that LaTeXMLEquation .
However , the operator defined on LaTeXMLMath by LaTeXMLEquation is easily seen to be unitary , and hence we conclude LaTeXMLCite that the desired quantities LaTeXMLMath exist as self-adjoint operators on the Fock space LaTeXMLMath associated with the creation and annihilation operators LaTeXMLMath and LaTeXMLMath .
The spectral projectors of these operators LaTeXMLMath represent propositions about the time-averaged value of the energy in the spacetime foliation determined by LaTeXMLMath .
To conclude : for each fixed choice of a foliation vector LaTeXMLMath , we have a physically meaningful representation of the history algebra Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) on the Hilbert space LaTeXMLMath .
Thus the same Hilbert space LaTeXMLMath carries all different representations—for different choices of LaTeXMLMath —of the quantum field theory history algebra .
We now define the action LaTeXMLMath and the Liouville LaTeXMLMath operators as normal-ordered versions of their classical analogues LaTeXMLEquation .
LaTeXMLEquation The automorphisms of the history algebra that are generated by the action and Liouville operators are LaTeXMLEquation .
LaTeXMLEquation and are easily shown to be unitarily implementable .
In what follows , the real-valued smearing function LaTeXMLMath is set equal to LaTeXMLMath for every LaTeXMLMath .
A significant feature of the histories formalism is the temporal structure of the theory .
It introduces a new approach to the concept of time , in which time is distinguished as an ordering parameter ( logical structure ) , and as an evolution parameter ( dynamics ) .
In particular , as we have already shown in non-relativistic quantum mechanics LaTeXMLCite , the Liouville operator LaTeXMLMath generates time translations with respect to the ‘ external ’ LaTeXMLMath -time parameter , and the Hamiltonian operator LaTeXMLMath generates time translations with respect to the ‘ internal ’ evolution LaTeXMLMath -time parameter .
The action operator LaTeXMLMath generates both types of time transformations ; it is the time generator for the histories theory for solutions of the equations of motion LaTeXMLMath ; both Liouville LaTeXMLMath and Hamiltonian LaTeXMLMath operators are time translation generators that correspond to two different aspects ( two modes ) of the notion of time .
However , only LaTeXMLMath is related to the actual physical time parameter , in analogy with the standard theory where the Hamiltonian LaTeXMLMath is the time translation generator .
.
The same construction is true for a histories quantum field theory .
The invariance of standard quantum field theory under the Poincaré group , has been a difficult issue to address for many years .
In a canonical treatment of quantum field theory , the Schrödinger-picture fields depend on the reference frame ( i.e .
, choice of foliation ) .
In order to demonstrate manifest independence of this choice with the aid of Heisenberg-picture fields , one still has to contend with the foliation-dependence of the Hamiltonian that generates the Heisenberg fields .
In histories theory , the enhanced temporal structure enables the study of a Poincaré group transformation between different foliations .
In particular we will show that different representations corresponding to different foliation vectors LaTeXMLMath , are related by Lorentz boosts of the ‘ external ’ Poincaré group : LaTeXMLEquation and where the time translations generator is closely related to the ‘ Liouville ’ operator LaTeXMLMath .
We first define the Heisenberg-picture analogue of the scalar field , to illustrate the different time translations associated with the two time labels .
We use a similar notation to that in the classical case : i.e .
, the Heisenberg-picture field is written as LaTeXMLMath , where the space-time point LaTeXMLMath is expressed in coordinates adapted to LaTeXMLMath .
Thus LaTeXMLEquation .
LaTeXMLEquation The different types of time translation are particularly easy to see by studying the action of the Liouville LaTeXMLMath and action LaTeXMLMath operators on the Heisenberg-picture fields LaTeXMLMath LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
The label LaTeXMLMath corresponds to the ‘ internal ’ time of the unitary Hamiltonian time evolution , while LaTeXMLMath corresponds to the ‘ external ’ time that labels the time-ordering of events in a history for the Shrödinger-picture operators .
As we showed previously , each fixed choice of foliation vector LaTeXMLMath corresponds to a different representation of the history algebra on the same Fock space LaTeXMLMath .
Hence , we may heuristically say , that , for a given vector LaTeXMLMath , and for each value of the associated time LaTeXMLMath , there will be a Hilbert space LaTeXMLMath that carries an independent copy of the standard quantum field theory .
In particular , there exists a representation of the Poincaré group associated with each spacelike slice LaTeXMLMath , where LaTeXMLMath .
In what follows , a particularly important role will be assigned to the averaged ‘ internal ’ Poincaré group .
For example , we define the averaged energy LaTeXMLMath that generates translations on the LaTeXMLMath -time parameter of the Heisenberg-picture fields LaTeXMLMath , without affecting the ‘ external ’ LaTeXMLMath -time parameter : LaTeXMLEquation .
LaTeXMLEquation The expressions for the ‘ internal ’ Poincaré generators of spatial translations LaTeXMLMath , and rotations LaTeXMLMath can be written in direct analogy with the expressions Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) of the classical case .
We use the normal-ordered expressions LaTeXMLEquation .
LaTeXMLEquation We have used an ‘ pseudo-covariant ’ notation by employing a LaTeXMLMath -spacelike vector LaTeXMLMath ( i.e. , such that ( LaTeXMLMath ) .
Note that the terms involving a pair of creation operators , or a pair of annihilation operators , can be shown to vanish through integration by parts .
Of particular interest , is the action of the boost generator LaTeXMLMath defined as LaTeXMLEquation .
LaTeXMLEquation The key feature of the boost generator LaTeXMLMath is that it mixes the LaTeXMLMath -time parameter with the three-vectors x .
The action of these boost transformations is most clearly seen on the Heisenberg objects LaTeXMLMath LaTeXMLEquation where LaTeXMLMath is the unitary operator that generates Lorentz transformations , and LaTeXMLMath is the Lorentz transformation generated by LaTeXMLMath .
At this point we note the action of the internal Poincaré group on the action LaTeXMLMath , Hamiltonian LaTeXMLMath and Liouville LaTeXMLMath operators respectively : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
As we would expect from standard canonical quantum field theory , we see that the above operators remain invariant under the ‘ internal ’ Lorentz transformations .
A key result in histories classical field theory is that there also exists a second—the ‘ external ’ —Poincaré group symmetry of the theory , with generators LaTeXMLEquation .
LaTeXMLEquation Note that these definitions use the covariant fields LaTeXMLMath and LaTeXMLMath that satisfy the algebra Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) rather than the foliation-dependent fields LaTeXMLMath and LaTeXMLMath of Eqs .
( LaTeXMLRef ) – ( LaTeXMLRef ) .
However , many of the generators of the external Poincaré group are exactly the same whether one uses covariant fields expressions or foliation-dependent ones : they differ only for the case of the boosts generators LaTeXMLMath .
In particular , the Liouville operator LaTeXMLMath , given by the expression LaTeXMLEquation generates translations on the time label LaTeXMLMath .
The space translations and rotation generators are identical to those of the internal Poincaré group Eqs .
( LaTeXMLRef – LaTeXMLRef ) .
However the external boost generator LaTeXMLMath differs from the internal one LaTeXMLMath , and hence it is of particular interest to study the action of the former .
The generator of time-translations LaTeXMLMath acts on Schrödinger picture objects as LaTeXMLEquation .
LaTeXMLEquation The ‘ external ’ boost generator LaTeXMLMath is LaTeXMLEquation .
LaTeXMLEquation where we define the operator LaTeXMLMath as LaTeXMLEquation and LaTeXMLMath .
Then the boost generator LaTeXMLMath acts on the fields LaTeXMLMath as LaTeXMLEquation and it mixes the LaTeXMLMath -time parameter with the three-vector x .
However , the crucial point is that LaTeXMLMath generates Lorentz transformations on the foliation vector LaTeXMLMath as well .
This can be viewed as a demonstration of explicit Poincaré covariance , as we can see from the action of the external Lorentz transformations on the Heisenberg-picture fields LaTeXMLMath as LaTeXMLEquation .
The generators of the internal Poincaré group transform under the action of the external Poincaré group as LaTeXMLEquation .
LaTeXMLEquation Of considerable importance is the fact that the action operator LaTeXMLMath transforms in the same way : LaTeXMLEquation .
Hence the action of the external Poincaré group relates representations of the theory that differ with respect to the foliation vector LaTeXMLMath .
As we shall see in the following section , this is crucial when we discuss the Poincaré invariance of probabilities .
In summary , we have showed that the history version of quantum field theory carries representations of two Poincaré groups .
The ‘ internal ’ Poincaré group is defined in analogy to the one in the standard canonical treatment of the theory .
It corresponds to time-translations with respect to the ‘ internal ’ LaTeXMLMath -time parameter of histories theory .
The Lorentz part of the ‘ external ’ Poincaré group intertwines representations of the theory associated with different choices of foliation , all of which however are realised on the same Fock space LaTeXMLMath .
It corresponds to time-translations with respect to the ‘ external ’ LaTeXMLMath -time parameter .
The translation parts of these two types of Poincaré transformation—corresponding to the relations between the LaTeXMLMath time parameter and kinematics , and the LaTeXMLMath time parameter and dynamics—have very significant analogues in the case of the histories version of general relativity LaTeXMLCite .
In LaTeXMLCite , we showed how a classical-quantum relation can be nicely described in histories theory by using the history analogue of coherent states .
In the histories formalism , a non-normalised coherent state vector is written as LaTeXMLCite LaTeXMLEquation .
The corresponding normalised coherent states can be obtained by unitary transformations of the vacuum state as LaTeXMLEquation where LaTeXMLMath is the Weyl operator defined as LaTeXMLEquation and LaTeXMLMath and LaTeXMLMath are smearing functions that belong to LaTeXMLMath .
We write the normalised coherent state LaTeXMLMath corresponding to the pair LaTeXMLMath as LaTeXMLMath .
In this context we know that LaTeXMLMath and LaTeXMLMath correspond to classical values and therefore correspond to a path on classical phase space .
In this correspondence , the functions LaTeXMLMath and LaTeXMLMath are the classical values of the field LaTeXMLMath and its conjugate momenta LaTeXMLMath , respectively .
The set of all coherent states is independent of the choice of foliation since these coherent states are eigenstates of the annihilation operator LaTeXMLMath , which is foliation independent .
However , the physical identification of the vector LaTeXMLMath with a phase space path is foliation-dependent since it depends on the Weyl operator , which itself depends on the choice of the representation of the history algebra on the Fock space LaTeXMLMath LaTeXMLMath , the classical phase space path LaTeXMLMath is defined by the foliation-dependent expression LaTeXMLMath .
One should recall that the space of classical histories LaTeXMLMath is itself dependent on the choice of foliation .
So far our discussion of the histories version of quantum field theory has been at the level of field algebras and group transformations .
However , in histories formalism physically crucial ‘ probabilistic ’ information is contained in the decoherence functional .
In this HPO formalism , the most general form for the decoherence functional of a pair of history propositions LaTeXMLMath , LaTeXMLMath is LaTeXMLEquation in terms of an operator LaTeXMLMath on LaTeXMLMath LaTeXMLCite .
In our case , the operator LaTeXMLMath reads LaTeXMLEquation in terms of the operator LaTeXMLMath that we proved in LaTeXMLCite that it is an implicit function of the action operator : therefore there is an implicit dependence of LaTeXMLMath on the foliation vector LaTeXMLMath .
The matrix elements of LaTeXMLMath in a coherent state basis can written in terms of the classical action functional LaTeXMLMath as LaTeXMLEquation .
The explicit relation of LaTeXMLMath with the action operator LaTeXMLMath is as follows .
For a general operator LaTeXMLMath on LaTeXMLMath one can define an operator LaTeXMLMath on LaTeXMLMath as LaTeXMLEquation .
In our case we have LaTeXMLEquation .
LaTeXMLEquation in terms of the operator LaTeXMLMath .
Hence , the decoherence functional depends on the representation through the phase space action LaTeXMLMath .
This raises the critical issue of the physical meaning of the fact that the formalism appears to depend on a specific choice of the foliation vector LaTeXMLMath .
We have seen above that the representation of the phase space quantities by Hilbert space operators depends on LaTeXMLMath , and that there exist unitary intertwiners between different representations given by the boosts of the external Poincaré group .
As has been discussed in LaTeXMLCite , a transformation law for the observables by means of a unitary operator LaTeXMLMath LaTeXMLEquation implies that the operator LaTeXMLMath of the decoherence functional , carrying a label for the foliation dependence LaTeXMLMath , ought to transform as LaTeXMLEquation so that the values of the decoherence functional ( corresponding to probabilities and correlation functions of the theory ) are representation-independent LaTeXMLEquation where LaTeXMLMath is the decoherence functional defined with reference to the operator LaTeXMLMath .
In our case we have LaTeXMLMath .
This changes the foliation dependence of the fundamental fields LaTeXMLMath and LaTeXMLMath , and hence of any observable LaTeXMLMath that depends upon them LaTeXMLEquation .
Some physically interesting examples of observables , in this sense , are integrals LaTeXMLMath of fields LaTeXMLMath , smeared with appropriate test functions LaTeXMLMath , that satisfy LaTeXMLMath ; another example is any space-time average of the normal-ordered polynomial functions of these fields .
In order to see , how the boosts generator acts on LaTeXMLMath , it suffices to check its action on LaTeXMLMath .
This is LaTeXMLEquation .
Consequently the operator LaTeXMLMath transforms as LaTeXMLMath .
Hence the values of the decoherence functional are foliation independent LaTeXMLEquation .
We have studied both the classical and the quantum history versions of scalar field theory .
We have showed that , in both cases , the crucial feature of the history field theory is the appearance of two Poincaré groups , in direct analogy to the two types of time transformation that characterizes the history formalism .
The internal Poincaré group is related to time as an ordering parameter ( the Hamiltonian LaTeXMLMath is the time translations generator ) , and it is in analogy to the Poincaré group of standard field theory .
On the other hand , the external Poincaré group is related to time as a parameter of evolution ( the Liouville LaTeXMLMath is the time translations generator ) , and it is of particular interest for the quantum case , as it relates representations of the quantum field theory , for different choices of foliation , with Poincaré transformations .
These results will be proved of great importance in the study of history general relativity theory in LaTeXMLCite .
In particular , the histories formalism is suitable to deal with issues that lie at the level of the interplay between quantum theory and the spacetime structure .
The present work focuses on quantum field theory in a fixed spacetime , however the techniques involved and the concepts introduced , have been able to precisely identify the relation between the quantum mechanical observables and the necessary notion of the spacetime foliation .
Many issues are raised at the level of the meaning of reference frames in quantum theory—a foliation corresponds to a reference frame—and more importantly at the level of quantum gravity .
The latter is eventually the aim of the histories programme , and this involves a further elucidation of the meaning of spacetime in a quantum theory .
What strikes us as relevant at present is that , one might have to disentangle between the two different views of spacetime transformations : the passive and the active view .
This is subtly hinted by the fact that the transformations generated by the external Poincaré group should be viewed in the passive sense , since the argument LaTeXMLMath can not be identified with a fixed , absolute spacetime point in all representations .
In order to successfully address the above issues we must first study the history version of general relativity ; this is the context of the forthcoming paper LaTeXMLCite .
Acknowledgements I would like to thank Charis Anastopoulos and Chris Isham for very helpful discussions .
I gratefully acknowledge support from the L.D .
Rope Third Charitable Settlement and from the EPSRC GR/R36572 grant .
Abstract The transition from phase chaos to defect chaos in the complex Ginzburg-Landau equation ( CGLE ) is related to saddle-node bifurcations of modulated amplitude waves ( MAWs ) .
First , the spatial period LaTeXMLMath of MAWs is shown to be limited by a maximum LaTeXMLMath which depends on the CGLE coefficients ; MAW-like structures with period larger than LaTeXMLMath evolve to defects .
Second , slowly evolving near-MAWs with average phase gradients LaTeXMLMath and various periods occur naturally in phase chaotic states of the CGLE .
As a measure for these periods , we study the distributions of spacings LaTeXMLMath between neighboring peaks of the phase gradient .
A systematic comparison of LaTeXMLMath and LaTeXMLMath as a function of coefficients of the CGLE shows that defects are generated at locations where LaTeXMLMath becomes larger than LaTeXMLMath .
In other words , MAWs with period LaTeXMLMath represent “ critical nuclei ” for the formation of defects in phase chaos and may trigger the transition to defect chaos .
Since rare events where LaTeXMLMath becomes sufficiently large to lead to defect formation may only occur after a long transient , the coefficients where the transition to defect chaos seems to occur depend on system size and integration time .
We conjecture that in the regime where the maximum period LaTeXMLMath has diverged , phase chaos persists in the thermodynamic limit .
PACS : 05.45.Jn 03.40.Kf ; 05.45.-a ; Keywords : Phase chaos , Defect chaos , Complex Ginzburg-Landau equation , Coherent structures The transition from phase to defect chaos for the one dimensional complex Ginzburg-Landau equation ( CGLE ) was recently related to the bifurcation properties of a family of coherent structures called modulated amplitude waves ( MAWs ) LaTeXMLCite .
In this paper the relationship between MAWs and large scale chaos is studied in detail , providing a comprehensive description of various aspects of the CGLE chaotic dynamics .
When a spatially extended system is driven sufficiently far away from equilibrium , patterns can eventually form LaTeXMLCite .
In many cases these patterns show an erratic behavior in space and time : such behavior is commonly referred to as spatiotemporal chaos LaTeXMLCite .
Examples of extended systems displaying such chaotic dynamics in one spatial dimension include : heated wire convection LaTeXMLCite , printers instability and film drag experiments LaTeXMLCite , eutectic growth LaTeXMLCite , binary convection LaTeXMLCite , sidewall convection LaTeXMLCite , the far field of spiral waves in the Belousov-Zhabotinsky reaction LaTeXMLCite , the Taylor-Dean system LaTeXMLCite , hydrothermal LaTeXMLCite and internal LaTeXMLCite waves excited by the Marangoni effect and the oscillatory instability of a Rayleigh-Bénard convection pattern LaTeXMLCite .
Near the pattern forming threshold , the dynamics of such systems can often be described by so-called amplitude equations .
When the pattern forming bifurcation from the homogeneous state is a forward Hopf bifurcation , the appropriate amplitude equation is the CGLE LaTeXMLCite , which in one spatial dimension reads as : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are real coefficients and the field LaTeXMLMath has complex values .
For different choices of the coefficients numerical investigations of the CGLE have revealed the existence of various steady and spatiotemporally chaotic states LaTeXMLCite .
Many of these states appear to consist of individual structures with well defined propagation and interaction properties .
It is thus tempting to use these structures as building blocks for a better understanding of spatiotemporal chaos .
In this paper we will essentially follow such an approach .
As a function of the coefficients LaTeXMLMath and LaTeXMLMath , the CGLE ( LaTeXMLRef ) can exhibit two qualitatively different spatiotemporal chaotic states known as phase chaos ( when the modulus LaTeXMLMath is at any time bounded away from zero ) and defect chaos ( when LaTeXMLMath can vanish leading to phase singularities ) .
It is under dispute whether the transition from phase to defect chaos is sharp or not , and if a pure phase-chaotic , ( i.e. , defect-free ) state can persist in the thermodynamic limit LaTeXMLCite .
We will address these issues by suggesting a mechanism for the formation of defects related to the range of existence of MAWs .
The main points of this paper are outlined in the following and illustrated in Figs .
LaTeXMLRef , LaTeXMLRef .
( i ) Our investigation starts with the study of MAWs , which are uniformly propagating , spatially periodic solutions of the CGLE .
These MAWs are parameterized by the average phase gradient LaTeXMLMath and their spatial period LaTeXMLMath .
Our study is confined to the case LaTeXMLMath for reasons specified below .
Spatial profiles and the stable propagation of a particular MAW are presented in Fig .
LaTeXMLRef a-c. Isolated MAW structures consisting of just one spatial period LaTeXMLMath play an important role in defect formation .
In particular , for fixed CGLE coefficients the range of existence of coherent MAWs is limited by a saddle-node ( LaTeXMLMath ) bifurcation which occurs when LaTeXMLMath reaches a maximal period LaTeXMLMath .
( ii ) If the MAWs are driven into conditions with LaTeXMLMath a dynamical instability occurs leading to the formation of defects ( Fig .
LaTeXMLRef d ) .
( iii ) Slowly evolving structures reminiscent of MAWs ( “ near-MAWs ” ) are observed in the phase chaotic regime ( Fig .
LaTeXMLRef e , f ) .
In order to characterize such states , we have examined the distribution LaTeXMLMath of spacings LaTeXMLMath between neighboring peaks of the phase-gradient profile .
In particular for sufficiently long spacing LaTeXMLMath , the observed phase chaos structures are often very similar to a single period of a coherent MAW ( Fig .
LaTeXMLRef f ) .
( iv ) When a phase chaotic state displays spacings LaTeXMLMath larger than LaTeXMLMath , phase chaos breaks down and defects are formed ( e.g .
at LaTeXMLMath in Fig .
LaTeXMLRef i ) .
Thus , the MAW with LaTeXMLMath may be viewed as a “ critical nucleus ” for the creation of defects .
In phase chaos defect formation is similar to the dynamical process by which isolated MAW structures generate defects ( Fig .
LaTeXMLRef d ) .
Therefore purely phase chaotic states are those for which LaTeXMLMath remains bounded below LaTeXMLMath ( Fig .
LaTeXMLRef g ) , while defect chaos can occur when LaTeXMLMath becomes larger than LaTeXMLMath ( Fig .
LaTeXMLRef i ) .
( v ) A more detailed study of the probability distribution of the LaTeXMLMath ’ s shows that for large LaTeXMLMath the probability decays exponentially ( Fig .
LaTeXMLRef h , j ) .
As long as LaTeXMLMath has a finite value , we expect that , possibly after a very long transient time , defects will be generated .
( vi ) However , in a finite domain of the phase chaotic region , MAWs of arbitrarily large LaTeXMLMath exist : we expect that in this region , even in the thermodynamic limit , phase chaos will persist .
Fig .
LaTeXMLRef shows the main quadrant of the CGLE coefficient space .
The region of persistent phase chaos is bounded by the Benjamin-Feir-Newell curve ( thin dot-dashed ) and the curve along which LaTeXMLMath ( full curve in Fig .
LaTeXMLRef ) .
The outline of this paper is as follows : Section LaTeXMLRef is devoted to the study of the coherent MAW structures .
In section LaTeXMLRef we study the bifurcation diagram of the MAWs , starting from the homogeneous oscillation .
In section LaTeXMLRef the incoherent dynamics of near-MAW structures is presented .
We show that for LaTeXMLMath , i.e. , beyond the saddle-node bifurcation , near-MAWs evolve to defects .
To illustrate the origin of the saddle-node bifurcations in section LaTeXMLRef we compare bifurcation diagrams of coherent structures for different phase gradient expansions of the CGLE .
For the lowest order expansion ( known as the Kuramoto-Sivashinsky equation LaTeXMLCite ) the saddle-node bifurcation is absent while it is captured by expansions of higher order .
This explains why the divergence of the phase gradient was exclusively observed in simulations LaTeXMLCite of higher order expansions .
In section LaTeXMLRef we study various aspects of spatiotemporal chaos in the CGLE , and relate the observed continuous ( LaTeXMLMath ) and discontinuous ( LaTeXMLMath ) transitions ( see Fig .
LaTeXMLRef ) to properties of the MAWs .
The transition to defect chaos takes place when near-MAWs with periods larger than LaTeXMLMath occur in a phase chaotic state .
In section LaTeXMLRef the typical values of LaTeXMLMath in the phase chaotic regime are related to the competition of two instabilities of the MAWs , and it is possible to give a good estimate for the numerically measured transition from phase to defect chaos from these considerations .
A discussion of the presented results and some final remarks are reported in section LaTeXMLRef .
In this section we study the main properties of modulated amplitude waves ( MAWs ) LaTeXMLCite .
First , in section LaTeXMLRef the coherent structure framework that we use to describe the MAWs is introduced .
The bifurcation diagram of MAWs is explored in section LaTeXMLRef , with a particular focus on the saddle-node bifurcations that limit the range of existence of MAWs .
In section LaTeXMLRef we study the nonlinear evolution of near-MAWs that are “ pushed ” beyond their saddle-node bifurcation and show that this leads to the formation of defects .
Finally , in section LaTeXMLRef a bifurcation analysis of MAW-like coherent structures is performed in various phase equations that have been proposed as approximated models for the phase chaotic dynamics of the CGLE , and we show that only higher order phase equations reproduce the saddle-node bifurcation .
Coherent structures in the CGLE are uniformly propagating structures of the form LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are real-valued functions of LaTeXMLMath .
Coherent structures have been studied extensively LaTeXMLCite and play an important role in various regimes of the CGLE LaTeXMLCite .
The restriction to uniformly propagating structures reduces the CGLE to a set of three coupled ordinary differential equations ( ODEs ) LaTeXMLCite .
These ODEs are readily found by substitution of Ansatz ( LaTeXMLRef ) into the CGLE ( LaTeXMLRef ) and read as : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Solutions of the ODEs ( LaTeXMLRef ) correspond to coherent structures of the CGLE .
The simplest relevant solutions of these ODEs are the fixed points given by LaTeXMLMath ; these correspond to plane wave solutions of the CGLE where LaTeXMLMath and LaTeXMLMath .
An example of more complex solutions of the ODEs ( LaTeXMLRef ) are heteroclinic orbits which correspond to coherent structures that asymptotically connect different states .
Examples of such structures are fronts that connect nonlinear plane waves to the homogeneous state LaTeXMLMath LaTeXMLCite and Nozaki-Bekki holes that connect plane waves of different wavenumber LaTeXMLMath LaTeXMLCite .
Here we present an extensive study of the structures that are associated with the limit cycles of the ODEs ( LaTeXMLRef ) LaTeXMLCite .
These limit cycles correspond to spatially periodic solutions of the CGLE that we have already referred to as MAWs ( Fig .
LaTeXMLRef ) .
For appropriate choices of LaTeXMLMath and LaTeXMLMath , the period LaTeXMLMath of these MAWs can be made arbitrarily large , and in this limit the limit cycles approach a homoclinic orbit connecting the stable and unstable manifold of one of the plane wave fixed points ( Fig .
LaTeXMLRef a ) .
Some of these infinite period MAWs have also been referred to as “ homoclinic ” holes , and have been studied extensively recently LaTeXMLCite ; they are qualitatively different from the well-known Nozaki-Bekki holes LaTeXMLCite .
Even if the coefficients LaTeXMLMath and LaTeXMLMath are fixed , MAWs are not uniquely determined .
Counting arguments , similar to those developed in LaTeXMLCite , yield that in general we may expect a two-parameter family of solutions .
Let us first perform the counting for the homoclinic orbits .
As shown in LaTeXMLCite , these orbits connect the one-dimensional unstable manifold of a fixed point with its two-dimensional stable manifold .
In general , one needs to satisfy one condition to make such a connection , in other words , such a homoclinic orbit is of codimension one .
Since the coherent structure Ansatz ( LaTeXMLRef ) has two freely adjustable parameters ( LaTeXMLMath and LaTeXMLMath ) , we therefore expect a one parameter family of homoclinic orbits .
The situation for the limit cycles of the ODEs is even simpler .
Limit cycles are of codimension zero in parameter space , and so we expect a two parameter family of limit cycles .
In other words , if we have found a limit cycle for certain values of LaTeXMLMath and LaTeXMLMath , then we expect this limit cycle to persist for nearby values of the parameters LaTeXMLMath and LaTeXMLMath .
Obviously , we can parameterize this family of limit cycle coherent structures by LaTeXMLMath and LaTeXMLMath , but this is not very insightful .
Instead we will use the following two quantities that are more directly accessible in studies of the CGLE : the spatial period LaTeXMLMath of the MAWs , and their average phase gradient LaTeXMLMath .
Note that for homoclinic holes , LaTeXMLMath simply goes to infinity ; thus homoclinic orbits and limit cycles are members of a single family .
The multiplicity of the MAWs can also be obtained by considering the instability of the plane wave solutions from which the MAWs emerge LaTeXMLCite ( see section LaTeXMLRef below ) .
The plane waves form a one-parameter ( LaTeXMLMath ) family and undergo the well-known Eckhaus instability when the coefficients LaTeXMLMath are increased beyond certain critical values which depend on LaTeXMLMath .
In the unstable regime , a plane wave with wavenumber LaTeXMLMath is unstable to a whole band of perturbations with wavenumbers LaTeXMLMath LaTeXMLCite .
For finite systems of size LaTeXMLMath , this instability thus only appears when LaTeXMLMath .
Therefore for each LaTeXMLMath there is a unique one-parameter ( LaTeXMLMath ) family of perturbations that can render the plane wave unstable and at each of the corresponding bifurcations a new MAW solution emerges .
Hence also by this line of reasoning MAWs form a two-parameter family .
The general counting arguments given in the previous section do not provide information on the range of existence of MAWs as a function of the coefficients LaTeXMLMath and LaTeXMLMath and the parameters LaTeXMLMath and LaTeXMLMath .
Here we will focus our analysis on the LaTeXMLMath case since this is most relevant for the transition to defect chaos LaTeXMLCite ; the LaTeXMLMath case will be treated elsewhere LaTeXMLCite .
All bifurcation computations have been performed with the aid of the software package AUTO94 LaTeXMLCite .
AUTO94 can trace MAW solutions through parameter space , and when it detects bifurcations it can follow the newly emerging branches .
AUTO94 discretizes the ODEs ( LaTeXMLRef ) on a periodic domain of length LaTeXMLMath , and LaTeXMLMath will play the role of the period LaTeXMLMath of the MAWs .
Control of the average phase gradient LaTeXMLMath is implemented via the integral constraint LaTeXMLMath .
Since periodic boundary conditions result in translational invariance , we introduce an additional “ pinning ” condition LaTeXMLMath in order to obtain unique solutions .
Under these conditions , the continuation procedure works as follows .
First of all , LaTeXMLMath and LaTeXMLMath are set to fixed values , and throughout this paper we will set LaTeXMLMath .
Starting from a known solution such as a plane wave or a coherent structure obtained by other means , AUTO94 is set up to trace the MAWs along trajectories in LaTeXMLMath , LaTeXMLMath space , while calculating the parameters LaTeXMLMath and LaTeXMLMath of these MAWs .
The results of our bifurcation analysis are summarized in Fig .
LaTeXMLRef .
When LaTeXMLMath or LaTeXMLMath is increased , the uniformly oscillating state of the CGLE ( LaTeXMLMath ) becomes unstable via a Hopf bifurcation , from which stationary MAWs emerge ( section LaTeXMLRef ) .
These stationary , left-right symmetric solutions undergo a drift pitchfork bifurcation , which leads to left and right traveling MAWs ( section LaTeXMLRef , see also Fig .
LaTeXMLRef b ) ; as discussed later , these are the solutions relevant for the dynamics in the phase chaotic regime .
Following these branches of traveling MAWs , we encounter a saddle-node bifurcation where an “ upper ” and “ lower ” branch of MAWs merge ( section LaTeXMLRef , see also Fig .
LaTeXMLRef ) ; this bifurcation limits the range of existence of MAWs and is closely related to the formation of defects .
The upper branch MAWs can be continued back to negative values of LaTeXMLMath , where they terminate in a solution consisting of a periodic array of shocks and stationary Nozaki-Bekki holes LaTeXMLCite .
Upper branch MAWs with LaTeXMLMath have been studied under the name homoclinic holes LaTeXMLCite .
It should be noted that , without loss of generality , we focus here on solutions with LaTeXMLMath , for which the main peak of the phase gradient profile is positive ( see Fig .
LaTeXMLRef ) .
Solutions with LaTeXMLMath can be obtained from right moving MAWs by applying the mapping LaTeXMLMath .
Since the average phase gradient LaTeXMLMath is conserved across bifurcations , we start the continuation procedure from the uniformly oscillating solution LaTeXMLMath that has LaTeXMLMath .
On an infinite domain this uniformly oscillating solution becomes unstable via the so-called Benjamin-Feir instability when LaTeXMLMath LaTeXMLCite .
In a finite domain of size LaTeXMLMath , the onset of this instability is shifted to higher values of the product LaTeXMLMath LaTeXMLCite ; this finite size effect is relevant for our studies since the spatial period LaTeXMLMath is fixed in the continuation procedure .
In the ODEs ( LaTeXMLRef ) , the fixed point LaTeXMLMath corresponds to the homogeneously oscillating solution .
For given values of the period LaTeXMLMath , this fixed point undergoes a Hopf bifurcation at values of LaTeXMLMath and LaTeXMLMath where in the CGLE ( LaTeXMLRef ) the mode with wavenumber LaTeXMLMath becomes unstable LaTeXMLCite .
This Hopf bifurcation was analytically shown to be supercritical for sufficiently small LaTeXMLMath and large LaTeXMLMath in earlier studies LaTeXMLCite ; our numerical results are consistent with this .
For finite LaTeXMLMath , the solution bifurcating from the fixed point is a limit cycle which approaches a homoclinic orbit in the limit LaTeXMLMath .
The solutions of the CGLE that correspond to these orbits are stationary , reflection symmetric MAWs ; an example of these is shown in Fig .
LaTeXMLRef a .
When the CGLE coefficients LaTeXMLMath and/or LaTeXMLMath are increased further , the stationary MAW undergoes a drift pitchfork bifurcation LaTeXMLCite from which two new branches of asymmetric ( LaTeXMLMath ) MAWs emerge ( see Fig .
LaTeXMLRef b ) ; one of these moves to the left , one to the right .
The locations of both the Hopf and the drift pitchfork bifurcation approach the Benjamin-Feir-Newell curve for large LaTeXMLMath ( Fig .
LaTeXMLRef a ) , while for smaller LaTeXMLMath the drift pitchfork occurs for increasingly larger coefficients LaTeXMLMath and LaTeXMLMath .
However , only when these coefficient lie in the range shown as the shaded area in Fig .
LaTeXMLRef b , the pitchfork bifurcation can occur .
Otherwise , only stationary MAWs are found .
For increasing LaTeXMLMath and LaTeXMLMath these MAWs become pulse-like and finally approach the solitonic solutions of the nonlinear Schrödinger equation LaTeXMLCite ( Fig .
LaTeXMLRef c , d ) .
For the case LaTeXMLMath LaTeXMLCite , the initial plane wave already breaks the reflection symmetry , the initial MAW has nonzero velocity and the drift pitchfork bifurcation is replaced by its typical unfolding LaTeXMLCite .
Along the branch of right traveling MAWs that we described above , the maximum of the phase gradient grows with increasing LaTeXMLMath and LaTeXMLMath until a saddle-node ( SN ) bifurcation is reached , where these MAWs merge with another branch of MAW-like solutions .
To distinguish these branches we refer to them as the “ lower ” and the “ upper ” branch ; for examples see Figs .
LaTeXMLRef , LaTeXMLRef .
The lower branch MAWs are the key to understand more of the phenomenology of phase chaos .
The upper branch MAWs can , similarly to the lower branch MAWs , be parameterized by LaTeXMLMath and LaTeXMLMath , but for the same parameters , they present more pronounced modulations ( see Fig .
LaTeXMLRef ) .
The most important aspect of the saddle-node bifurcation is that it limits the range of existence of MAWs , since we will show that this limit is responsible for the transition from phase to defect chaos .
Fixing LaTeXMLMath , the locations of these bifurcations form a two-dimensional manifold in the three dimensional space spanned by LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
In Fig .
LaTeXMLRef a the saddle-node curves are shown in the LaTeXMLMath coefficient plane for a number of fixed periods LaTeXMLMath ; for larger LaTeXMLMath , the values of LaTeXMLMath where the bifurcation takes place decrease .
In Fig .
LaTeXMLRef b the saddle-node curves for a number of fixed values of LaTeXMLMath are shown in the LaTeXMLMath plane ; for larger LaTeXMLMath ( LaTeXMLMath ) , the saddle-node occurs for smaller values of LaTeXMLMath and LaTeXMLMath ( LaTeXMLMath ) LaTeXMLCite .
Once the coefficients LaTeXMLMath and LaTeXMLMath are fixed , we define LaTeXMLMath as the period for which the saddle-node bifurcation occurs .
Note that there is also a range of coefficients LaTeXMLMath and LaTeXMLMath ( between the LaTeXMLMath and LaTeXMLMath curve where the saddle-node bifurcation does not occur .
In this section we will show that many basic aspects of the phenomenology of the CGLE can be understood from a typical bifurcation diagram of MAWs such as shown in Fig .
LaTeXMLRef .
We have chosen fixed coefficients LaTeXMLMath and LaTeXMLMath and varied the spatial period LaTeXMLMath of MAWs that exist at these coefficients .
Three families of solutions are represented : the homogeneous oscillation , the lower branch ( LB ) and the upper branch ( UB ) MAWs .
The shaded area schematically indicates the near-MAW structures observed in phase chaotic states such as shown in Fig .
LaTeXMLRef ( f , i , j ) .
The arrows in Fig .
LaTeXMLRef represent the dynamical evolution of perturbed MAWs , and their direction can be obtained by performing a linear stability analysis .
Linear stability - As discussed in section LaTeXMLRef , the homogeneous solution is stable against short wavelength perturbations ( arrow 1 ) , and turns unstable via the Hopf bifurcation that also generates the lower branch MAWs ( arrows 2 ) .
As discussed in LaTeXMLCite , upper branch MAWs have at least one unstable eigenvalue , and the dynamical evolution of perturbations is directed away from upper branch MAWs ( arrows 3,4 ) .
The linear stability of lower branch MAWs will be discussed in more detail in section LaTeXMLRef .
It turns out that perturbations of lower branch MAWs can evolve in many ways , but in almost all cases the ensuing dynamics remains close to the lower MAW branch ( shaded area in Fig .
LaTeXMLRef ) .
The only exception we have found to this rule is when a MAW is pushed beyond the saddle-node bifurcation ( arrow 5 ) .
Nonlinear evolution - Here we want to go beyond the linear analysis and study the nonlinear evolution of MAWs along the arrows of Fig .
LaTeXMLRef .
The examples ( at different choices of the coefficients ) of the dynamics shown below are not exhaustive , but should serve to illustrate typical behavior which appears to be very robust .
arrow 2 - When the uniform oscillation becomes linearly unstable perturbations grow .
To the left of the saddle-node , perturbations evolve to dynamics dominated by lower branch MAWs ( Fig .
LaTeXMLRef ) .
For small system sizes , stable MAWs may occur ( Fig .
LaTeXMLRef a , b ) , while for larger systems periodic sequences of MAWs are unstable with respect to the so-called interaction or splitting instabilities LaTeXMLCite that will be discussed in section LaTeXMLRef .
Hence a perturbed unstable homogeneous state typically does not converge to a train of coherent MAWs , but instead evolves to phase chaos ( Fig .
LaTeXMLRef c ) .
In the context of the bifurcation diagram , note that the disordered structures observed in the phase chaotic evolution are quite similar to lower branch MAWs .
The shaded area in Fig .
LaTeXMLRef represents this “ near-MAW ” behavior .
arrows 3,4 - Upper branch MAWs are always unstable due to the positive eigenvalue associated with the saddle-node bifurcation .
The resulting incoherent dynamics has been studied quite extensively in the context of hole-defect dynamics LaTeXMLCite .
( i ) When a perturbation has pushed an upper branch MAW towards the “ lower ” part of the bifurcation diagram , the structure decays towards lower branch MAWs ( arrow 3 ) .
An example of a space time plot for the decay towards a lower branch MAW is shown in Fig .
LaTeXMLRef a .
( ii ) When the perturbation pushes the MAW towards the “ upper ” side of the diagram , the phase gradient peak that characterizes MAWs grows without bound , and at the same time the minimum of LaTeXMLMath approaches zero : a defect is formed ( arrow 4 ) .
The dynamics after such a defect has formed depends on the values of the coefficients LaTeXMLMath and LaTeXMLMath .
Two different examples are shown in Fig .
LaTeXMLRef b , c .
For more details see section LaTeXMLRef .
arrow 5 - So far we have encountered two scenarios : if the phase gradient peak of a structure is “ larger ” than that of an upper branch MAW , then it will grow out to form defects .
If it is “ smaller ” , it will decay back in the direction of the lower branch MAWs .
The latter process frequently occurs in phase chaos , preventing the formation of defects , while the former process needs to be initiated by appropriate initial conditions .
However , when the upper and lower branches approach each other and disappear in a saddle-node bifurcation , there are no structures left to prevent arbitrary small perturbations to grow out to defects .
This dynamical process , which is represented by arrow 5 in Fig .
LaTeXMLRef , is the core of our argument : defect formation takes place beyond the saddle-node bifurcation .
An alternative approach to describe the creation of defects from phase chaotic states is via blow-ups in so-called phase-equations LaTeXMLCite .
Phase equations are based on the observation that close to the onset of phase chaos ( near the Benjamin-Feir-Newell curve ) the amplitude is “ slaved ” to the phase dynamics .
In this situation a phase equation can be obtained by a gradient expansion LaTeXMLCite .
The expansion including all parity-symmetric terms up to fourth order LaTeXMLCite reads LaTeXMLEquation where LaTeXMLMath .
The lowest order description of phase chaos is obtained when the parameters LaTeXMLMath and LaTeXMLMath are set equal to zero ; the resulting equation is known as the Kuramoto-Sivashinsky equation LaTeXMLCite .
The phase equations with higher order terms included have been studied via direct integration by Sakaguchi LaTeXMLCite .
For the full Eq .
( LaTeXMLRef ) , Sakaguchi observed finite time divergences of the phase gradient for coefficients close to the transition from phase to defect chaos in the CGLE .
He attributed such divergences to the occurrence of defects in the CGLE .
No blow-up of the phase gradient is observed for Eq .
( LaTeXMLRef ) without the last term , or for the simple Kuramoto-Sivashinsky equation .
Recently , Abel et al .
LaTeXMLCite quantified the increasing discrepancies between the phase equations of different orders and the full dynamics in the CGLE with increasing distance from the Benjamin-Feir-Newell curve and identified the relative importance of the various terms in Eq .
( LaTeXMLRef ) .
Since the essential ingredient of our theory is the occurrence of a saddle-node bifurcation , we have investigated the bifurcation scenario for various truncations of the phase equations ( LaTeXMLRef ) .
In the context of phase dynamics , our Ansatz ( LaTeXMLRef ) becomes of the form LaTeXMLEquation .
We have studied MAW-like structures occurring in the phase equations by employing the same methodology as for the CGLE ; the average phase gradient value LaTeXMLMath is fixed to LaTeXMLMath and LaTeXMLMath parameterizes the spatial period of the MAW .
In Fig .
LaTeXMLRef we compare bifurcation diagrams and MAW profiles for different expansions at the parameters LaTeXMLMath .
For all phase equations considered here the coherent structures are again born in a Hopf and undergo a drift pitchfork bifurcation , beyond which the maximal phase gradients increase .
This leads to increasing discrepancies between different approximations .
In particular , the coherent structures for Eq .
( LaTeXMLRef ) exhibit saddle-node bifurcations at parameter values not far from those for corresponding MAWs in the CGLE ; nevertheless the MAWs of Eq .
( LaTeXMLRef ) deviate substantially from the CGLE MAWs for the upper branch of MAWs .
The Kuramoto-Sivashinsky equation , and Eq .
( LaTeXMLRef ) without the last term , do not exhibit a saddle-node bifurcation .
Since these latter two models do not experience blow-up , we can safely conclude that these observations confirm our picture , and that the saddle-node bifurcations of coherent structures play the same crucial role in both the full CGLE and its phase equations .
In this section we will study the dynamical evolution of the CGLE near the transitions from phase to defect chaos .
The transition between these two states can either be hysteretic or continuous : in the former case , the transition is referred to as LaTeXMLMath , in the latter as LaTeXMLMath LaTeXMLCite .
How are defects generated from phase chaos ?
Let us start to consider a small system in which a stable lower branch MAWs has been created .
When we fix the coefficients LaTeXMLMath and LaTeXMLMath and steadily increase the size of the system , and hence the period LaTeXMLMath of the MAW , we find that as soon as we push LaTeXMLMath beyond LaTeXMLMath , the MAW structure blows up to form defects .
An example of this is shown in Fig .
LaTeXMLRef a .
In a similar fashion , defects are created when the system size LaTeXMLMath is fixed , and either LaTeXMLMath or LaTeXMLMath are increased until LaTeXMLMath ( Fig .
LaTeXMLRef c , d ) .
How is this related to phase chaos ?
As shown in Fig .
LaTeXMLRef b , typical phase chaotic states show much more incoherent dynamics , containing many MAW like structures but of much smaller period .
Our central conjecture is therefore that the transitions from phase to defect chaos are triggered by the occurrence of near-MAW structures in a phase chaotic state with LaTeXMLMath LaTeXMLCite and periods larger than LaTeXMLMath , the spatial period of the critical nucleus for defect creation .
To test this conjecture , we have numerically investigated the distribution of inter-peak spacings LaTeXMLMath of the phase gradient profile ( see Fig .
LaTeXMLRef e , f ) .
In section LaTeXMLRef we discuss the definition of LaTeXMLMath and the details of our numerical analysis .
In particular , we have examined in the LaTeXMLMath plane 17 different “ cuts ” across the LaTeXMLMath and LaTeXMLMath transition lines .
In section LaTeXMLRef the results of our numerics along a cut through the LaTeXMLMath transition line are presented , while section LaTeXMLRef is devoted to the LaTeXMLMath transition .
We will show that the presence of inter-peak spacings LaTeXMLMath larger than LaTeXMLMath accurately predicts the transition from phase to defect chaos ( Fig .
LaTeXMLRef ) .
In the last section LaTeXMLRef we will show that a reasonable , parameter-free estimate of the numerically observed transitions can be obtained via a linear stability analysis of the MAWs .
To verify our main conjecture , we have to characterize the MAW structures occurring in the phase-chaotic regime .
In general this is a complicated task , since the phase gradient profile of a typical phase chaotic state ( see Fig .
LaTeXMLRef e , f and LaTeXMLRef ) consists of many peaks of different size , spacing and shape ; a priori it is unclear how to compare these to MAW profiles .
However , a close inspection of the defect forming process reveals that while closely spaced phase gradient peaks evolve in a quite erratic way , well spaced peaks appear to have a more regular dynamics and frequently their overall shape resembles that of MAWs ( see Fig .
LaTeXMLRef ) .
These large period near-MAWs modify their shape quite slowly with respect to the other structures present in the chaotic field , and propagate over a disordered background .
Therefore we study the distribution of inter-peak distances LaTeXMLMath , keeping in mind that the tail of this distribution is relevant for defect generation .
The phase gradient profile of a coherent MAW ( see Figs .
LaTeXMLRef a and Fig .
LaTeXMLRef a ) shows a secondary maximum .
To obtain the correct period LaTeXMLMath of a near-MAW , such small extrema should be neglected when the inter-peak spacing LaTeXMLMath is measured .
We introduce a cutoff for the size of the phase gradient peak equal to the size of the secondary extremum of the MAW with the largest LaTeXMLMath .
As an additional result of this cutoff , small fluctuations are not considered as MAW peaks .
It should be noted that the tail of the distribution of LaTeXMLMath is rather insensitive to the precise value of this cutoff .
In order to estimate the probability density LaTeXMLMath , for every time interval LaTeXMLMath , the inter-peak periods LaTeXMLMath of the spatial profile of the phase gradient are determined .
In addition , for every snapshot the largest value LaTeXMLMath of the inter-peak spacing LaTeXMLMath is stored separately , and this leads to the distribution LaTeXMLMath .
From the spatial profile of LaTeXMLMath the distribution LaTeXMLMath and the minimal amplitude value LaTeXMLMath can be derived .
This latter quantity is used to detect defects : when LaTeXMLMath falls below a value of 0.1 , we take this as an indication of a defect .
Extensive simulations have been made possible thanks to an innovative time-splitting code which ensures precision and stability comparable with pseudo-spectral codes , but is noticeably faster LaTeXMLCite .
The spatial resolution LaTeXMLMath has been set to LaTeXMLMath and the integration time step to LaTeXMLMath .
Simulations have been carried out for integration times ranging from LaTeXMLMath to LaTeXMLMath and for a typical system size LaTeXMLMath ; occasionally , runs have been performed with LaTeXMLMath and LaTeXMLMath .
Typically , our runs start from random initial conditions of the type LaTeXMLMath ( where LaTeXMLMath and LaTeXMLMath ) with LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are random numbers uniformly distributed in LaTeXMLMath and LaTeXMLMath .
This initial condition ( LaTeXMLRef ) leads to a smooth phase and the formation of defects due to initial discontinuities is avoided .
In sections LaTeXMLRef and LaTeXMLRef we will consider in detail two particular cuts in the LaTeXMLMath coefficient space , one across the LaTeXMLMath and one across the LaTeXMLMath curve .
In particular , we will analyze the behavior of the probability densities LaTeXMLMath , LaTeXMLMath and LaTeXMLMath for both transitions .
In this section we concentrate on the LaTeXMLMath transition that is observed when the value of LaTeXMLMath is fixed at 3.0 and LaTeXMLMath is varied .
Transition to defect chaos - Starting from random initial conditions we have integrated the dynamics of the CGLE for long durations .
For a fixed system size LaTeXMLMath we observe that , as a function of the total integration time , the value of LaTeXMLMath for which defects are formed appears to decrease .
Similar behavior occurs when the system size LaTeXMLMath is increased for fixed integration times .
For example , for an integration time of LaTeXMLMath and LaTeXMLMath we find for system size LaTeXMLMath , LaTeXMLMath and LaTeXMLMath critical values LaTeXMLMath and LaTeXMLMath , respectively .
For a size LaTeXMLMath and integration times LaTeXMLMath a critical value of LaTeXMLMath is also found .
Note that even the lowest value of LaTeXMLMath for the numerically measured transition obtained here is far above the lower bound LaTeXMLMath which is the value of LaTeXMLMath where the size of the critical nucleus for defect formation diverges ( LaTeXMLMath ) .
Below , we will give an estimate of the critical value LaTeXMLMath for which the defect density should vanish in the thermodynamic limit by extrapolating finite time and finite size data .
Distribution of LaTeXMLMath - Let us now consider the distribution of LaTeXMLMath ’ s for various coefficients LaTeXMLMath near the LaTeXMLMath transition .
It is clear from the data reported in Fig .
LaTeXMLRef that the shape of these distributions is quite insensitive to the presence or absence of defects .
This can be partly explained by the fact that just above the LaTeXMLMath transition defects arise in the system as rare isolated events occurring during the spatio-temporal evolution , as shown in Fig .
LaTeXMLRef c. This is fully consistent with earlier observations that the LaTeXMLMath transition is continuous LaTeXMLCite .
We focus on the tail of the probability density LaTeXMLMath , since this gives information on the probability to observe defects .
Our numerical results suggest an exponential decay , i.e. , LaTeXMLMath with LaTeXMLMath for sufficiently large LaTeXMLMath .
Similarly to the apparent transition value LaTeXMLMath , the values associated to extremal events LaTeXMLMath and LaTeXMLMath depend on integration times and system sizes .
By assuming that LaTeXMLMath remains finite ( but likely exponentially small ) for large LaTeXMLMath , we can expect that for long enough times , rare events associated with large values LaTeXMLMath will occur , and hence , defects can form after possibly very long transients .
Crossover behavior - A good order parameter to identify the occurrence of the transition starting from the defect chaos phase near the LaTeXMLMath transition is the defect density LaTeXMLMath which measures the number of defects occurring per space and time unity .
In the defect chaos regime LaTeXMLMath , while it vanishes at the LaTeXMLMath -transition .
Now we can relate this order parameter to the tail of the distribution of LaTeXMLMath .
Our conjecture states that defects should arise when LaTeXMLMath , therefore the defect density LaTeXMLMath should be related to the probability to have structures of period LaTeXMLMath , i.e .
, LaTeXMLEquation where LaTeXMLMath has been used .
If we now assume that the distribution LaTeXMLMath does not vary significantly across the transition ( as is evident from Fig .
LaTeXMLRef ) , then the change in the probability to have LaTeXMLMath is dominated by the changes in LaTeXMLMath with LaTeXMLMath .
A reasonable fit of our bifurcation data for LaTeXMLMath ( see Fig .
LaTeXMLRef ) in the interval LaTeXMLMath is LaTeXMLEquation where LaTeXMLMath .
Combining this result with the Ansatz ( LaTeXMLRef ) , we immediately obtain the following expression for the defect density : LaTeXMLEquation .
A similar expression was proposed in LaTeXMLCite for the defect density near the LaTeXMLMath transition .
In order to verify if the expression ( LaTeXMLRef ) is reasonable also for our choice of the parameters , we have estimated the probability LaTeXMLCite LaTeXMLEquation to observe an amplitude less than LaTeXMLMath .
This quantity gives a more precise characterization of the LaTeXMLMath -transition than LaTeXMLMath , because it measures not only the extreme events corresponding to true defects , but also the tendency of the system to generate structures characterized by small LaTeXMLMath .
We estimated the quantity ( LaTeXMLRef ) for several LaTeXMLMath values and for various LaTeXMLMath parameter values in the defect chaos regime .
Reporting LaTeXMLMath as a function of LaTeXMLMath a reasonable linear scaling is observed in the range LaTeXMLMath , for LaTeXMLMath , with the choice LaTeXMLMath .
The value LaTeXMLMath where the defect density should asymptotically vanish is much smaller than LaTeXMLMath obtained via direct numerical simulations but still bigger than LaTeXMLMath where LaTeXMLMath .
We can now easily estimate the integration time needed to observe a tiny shift of the apparent value LaTeXMLMath towards the corresponding asymptotic value LaTeXMLMath .
Limiting our analysis to system size LaTeXMLMath , a typical time-scale to observe a defect at LaTeXMLMath is LaTeXMLMath .
At this value of LaTeXMLMath , LaTeXMLMath , while for LaTeXMLMath , LaTeXMLMath .
Invoking the exponential decay of LaTeXMLMath , one immediately finds that the time scale to observe a defect at LaTeXMLMath is of order LaTeXMLMath , which is completely outside the reach of present day computers .
In order to characterize the LaTeXMLMath transition from phase to defect chaos in more detail LaTeXMLMath has been fixed , while the coefficient LaTeXMLMath is varied .
The LaTeXMLMath transition is hysteretic LaTeXMLCite : to the left of LaTeXMLMath one may have phase or defect chaos depending on the initial conditions .
Beyond the LaTeXMLMath phase chaos breaks down and defects occur spontaneously for any initial condition .
In order to study the dynamics across this transition we therefore initialized the simulations with initial conditions ( LaTeXMLRef ) , ( LaTeXMLRef ) or used relaxed phase chaos configurations corresponding to values of LaTeXMLMath far below the LaTeXMLMath line .
The probability densities LaTeXMLMath and LaTeXMLMath are shown in Fig .
LaTeXMLRef .
For LaTeXMLMath all distributions collapse on a unique curve , but as soon as defects arise the distributions change substantially .
Whenever a defect is generated , hole-defect dynamics takes place ( see Fig .
LaTeXMLRef b ) .
As a result phase chaos is replaced by defect chaos .
The noticeable modification of the distributions thus reflects the fact that the LaTeXMLMath transition is discontinuous .
Also the probability density for LaTeXMLMath changes abruptly across the LaTeXMLMath transition .
When approaching the transition to defect chaos from the Benjamin-Feir-Newell curve , three parameter regions , corresponding to different dynamical regimes , can be distinguished ( Fig .
LaTeXMLRef ) .
The first encountered region corresponds to infinite values of LaTeXMLMath : here we expect no defects to occur , irrespectively of system size and integration time .
The phase chaos is the asymptotic regime in this first region .
Then , when LaTeXMLMath and/or LaTeXMLMath are increased , a crossover regime is reached where extreme events ( large inter-peak spacings ) may lead to defect formation .
Here phase chaos can persist as a long lived transient , but eventually we expect it to break down .
Then , when LaTeXMLMath and/or LaTeXMLMath are even further increased , we experience a dramatic drop in transient times , and defect chaos sets in quite rapidly .
We understand this drop to occur when typical values of LaTeXMLMath ( and not rare extreme events ) become larger than the corresponding LaTeXMLMath values .
An approximate prediction for the location of the apparent phase to defect chaos transition ( numerically obtained from the defect density ) can be achieved in terms of a simple linear stability analysis of the MAWs ( Figs .
LaTeXMLRef and LaTeXMLRef ) .
A key element in our framework is the “ typical large value ” of LaTeXMLMath as a function of coefficients LaTeXMLMath and LaTeXMLMath ; below we will identify two linear instabilities that act to either increase or decrease LaTeXMLMath , and their balance sets a scale for typical LaTeXMLMath that will predict the location of the transition from phase to defect chaos rather well .
Due to translational and phase symmetries both MAW branches have neutral modes , i.e. , Goldstone modes .
The eigenvalue associated with the saddle-node bifurcation is positive for MAWs of the upper branch and negative for the lower branch .
In what follows the lower branch MAWs are considered exclusively .
Splitting - The spatial structure of a MAW of large period consists , roughly , of a homogeneous plane wave part and a local peak part .
For the parameter regime we consider here , fully extended plane waves are linearly unstable , and so we may expect that the MAW spectrum will be dominated by this instability for sufficiently large values of LaTeXMLMath .
Our linear stability analysis indeed shows that for appropriate parameters ( LaTeXMLMath ) and small enough LaTeXMLMath , all eigenvalues LaTeXMLMath , but when we increase LaTeXMLMath , MAWs become linearly unstable ( LaTeXMLMath , Fig .
LaTeXMLRef ) .
The shape of the unstable eigenmodes ( Fig .
LaTeXMLRef b ) suggests that this instability leads to the growth of a new peak in the homogeneous part of the MAW , and this is indeed the behavior observed in numerical simulations of the perturbed MAW ( Fig .
LaTeXMLRef c , d ) .
As a result two ( or more ) short MAWs with smaller LaTeXMLMath will appear .
We interpret this process as the splitting of a MAW in two or more smaller MAWs and we call the eigenmodes associated to such instability “ splitting modes ” .
Clearly , this instability tends to reduce the peak-to-peak distances LaTeXMLMath and prevents MAWs to cross the SN boundary ; in the phase chaotic regime this instability tends to inhibit defect generation .
Interaction - By using a Bloch Ansatz LaTeXMLCite , we extended the stability analysis to systems with LaTeXMLMath identical pulses ( LaTeXMLMath ) .
For LaTeXMLMath , an additional instability may appear LaTeXMLCite ( see Fig .
LaTeXMLRef ) .
Eigenvalues LaTeXMLMath are found mainly for small LaTeXMLMath ( typically LaTeXMLMath ) .
The shape of the eigenmodes , i.e. , an alternating sequence of positive and negative translational Goldstone modes ( Fig .
LaTeXMLRef b ) , suggests that the instability is due to the interaction between adjacent MAWs .
This interaction shifts adjacent peaks into opposite directions , thereby creating occasional larger values of LaTeXMLMath ( Fig .
LaTeXMLRef c , d ) .
In phase chaos this process leads to an increase of the spacing LaTeXMLMath between some peaks , thus enhancing the generation of defects .
Competition of Instabilities - Both the splitting and interaction mechanisms are similar to instabilities observed in the Kuramoto-Sivashinsky equation LaTeXMLCite .
We believe that phase chaos is governed by the competition of these two mechanisms that tend to increase or decrease the inter-peak spacings LaTeXMLMath .
Almost independent of the coefficients the splitting instability dominates for MAWs with LaTeXMLMath .
This can explain why large inter-peak spacings LaTeXMLMath become rare as reported in Figs .
LaTeXMLRef , LaTeXMLRef .
We suggest a connection between the interchanging dominance of these two different instabilities and the sudden change of LaTeXMLMath ( near LaTeXMLMath ) or the transient times before defect occurrence ( near LaTeXMLMath ) .
We calculated the linear stability spectra for a variety of coefficients and periods LaTeXMLMath close to LaTeXMLMath .
From these we obtain a curve in coefficient space ( Fig .
LaTeXMLRef ) where the real parts of interaction and splitting eigenvalues are equal .
For larger LaTeXMLMath or LaTeXMLMath , interaction becomes stronger , and we expect larger LaTeXMLMath ’ s and defect formation , while for smaller LaTeXMLMath and LaTeXMLMath , splitting dominates , LaTeXMLMath ’ s are decreased and defect formation becomes rare .
As shown in Fig .
LaTeXMLRef , the curve where the two instabilities are equally strong near the saddle-node bifurcation gives a rather good estimate of where the apparent transition from phase to defect chaos occurs .
Notice that in this “ balance of instabilities ” picture , there is no tunable parameter : once we have calculated LaTeXMLMath and the instabilities of the MAWs for a range of coefficients , a precise prediction for the “ transition ” from phase to defect chaos can be given .
In this section we report some open questions related to defect formation , together with some final remarks and a brief outlook .
Further Refinements - In order to accurately test our results , we have measured for each of the 17 cuts and for several values of the coefficients across the LaTeXMLMath - or LaTeXMLMath -lines the amplitude distribution LaTeXMLMath and the phase gradient peak-to-peak spacing distribution LaTeXMLMath .
We conjectured that defects occur if and only if LaTeXMLMath .
Indeed , we observe that in 11 out of 17 points such conjecture is fulfilled .
On the remaining 6 points the theoretical conjecture leads to an estimation of the transition lines within a maximal error bar of LaTeXMLMath % .
The points determined following the conjecture are indicated as empty circles in Fig .
LaTeXMLRef .
The small deviations may have different reasons , that we summarize below : ( i ) If fluctuations occurring during the phase chaotic dynamics are only moderate , such as happens near the LaTeXMLMath transition line or for small system sizes , more complex coherent structures can survive for a short time .
Here we analyzed only the shortest coherent structures characterized by a single hump .
We believe that this is sufficient to understand the main aspects of the dynamics of large systems .
However , longer combined MAWs with more than one hump emerge from periodic MAWs via period doubling bifurcations .
The existence of the long combined MAWs is limited by saddle-node bifurcations analogously to single MAWs , but these bifurcations occur at slightly bigger values of the parameters LaTeXMLMath and LaTeXMLMath .
Therefore the appearance of these more complicated structures can delay defect formation even if one inter-peak spacing within the structure is bigger than LaTeXMLMath of the single MAW .
( ii ) Near the LaTeXMLMath line the dynamical fluctuations in the phase chaotic regime are stronger than in the proximity of the LaTeXMLMath line .
In this case and for sufficiently high values of the parameter LaTeXMLMath we observed situations where not only the structure with the longest inter-peak spacing but also the neighboring structures were involved in the defect formation .
( iii ) The assumption to consider MAWs with LaTeXMLMath is only an approximation .
If the average phase gradient locally ( on scales LaTeXMLMath ) deviates from LaTeXMLMath then the saddle-node bifurcation slightly shifts towards smaller coefficients LaTeXMLCite .
As far as the ( numerically ) improved LaTeXMLMath and LaTeXMLMath lines are concerned , we observe that both these lines lie to the left of the ones determined in earlier numerical studies LaTeXMLCite .
This is due to the fact that our simulations are of longer duration then those performed previously .
This confirms the expectation that such transition lines will shift towards the Benjamin-Feir-Newell curve for increasing systemsize and integration times LaTeXMLCite .
Moreover , some authors claim that indeed in the thermodynamic limit LaTeXMLMath and LaTeXMLMath will coincide with the Benjamin-Feir-Newell curve and the phase chaos regime will disappear LaTeXMLCite .
On the basis of our simulations we can not exclude such a possibility for higher space dimensions , but based on the results presented in this paper we conjecture that the saddle-node line for LaTeXMLMath provides a lower boundary for the transition from phase to defect chaos in the one-dimensional CGLE .
Final Remarks - We have presented a systematic study of modulated amplitude waves ( MAWs ) in the complex Ginzburg-Landau equation ( CGLE ) .
These periodic coherent structures originate from supercritical bifurcations from the homogeneous oscillation of the CGLE due to the Benjamin-Feir instability .
The range of existence of MAWs is bounded by saddle-node bifurcations occurring for values of LaTeXMLMath and LaTeXMLMath that depend on the period LaTeXMLMath of the MAWs .
Approaching the transition from phase to defect chaos , near-MAWs with large LaTeXMLMath occur in phase chaos , and defects are generated when the period of these near-MAWs becomes larger than the spatial period LaTeXMLMath of the critical nucleus .
This scenario is valid for both the LaTeXMLMath and LaTeXMLMath transition .
The divergence of LaTeXMLMath for coefficients in the phase-chaos regime led us to conjecture that there is a lower bound for the transition from phase to defect chaos .
Considerations of the linear stability properties of MAWs in light of their tendency to increase or decrease the typical period LaTeXMLMath in phase chaos , has led us to a fit-free estimate of the apparent transition from phase to defect chaos that fits the numerical data well .
Altogether , our study leaves little space for doubt that the transition from phase chaos to defect chaos in the CGLE is governed by coherent structures and their bifurcations .
From a general viewpoint , our analysis shows that there is no collective behavior that drives the transition .
Instead , strictly local fluctuations drive local structures beyond their saddle-node bifurcation and create defects .
Outlook - We want to stress here that the extension of the analysis to MAWs with nonzero average phase gradients LaTeXMLCite , will be of considerable interest for experimentalists , because in some recent experiments concerning Rayleigh-Bénard or Marangoni convection in quasi-one-dimensional geometries , supercritical Eckhaus instabilities of plane wave trains and the corresponding emergence of stable saturated MAWs have been observed LaTeXMLCite .
These states are analogous to what happens for the 1d CGLE when phase chaotic solutions with LaTeXMLMath are considered LaTeXMLCite .
The relevance of MAWs for two-dimensional structures is suggested by recent experimental evidence of MAWs observed in connection with superspiral and spiral breakup occurring in a Belousov-Zhabotinsky reaction LaTeXMLCite .
Moreover , in the phase chaotic regime of the 2d CGLE the correspondence between long inter-peak spacings ( here diameter of cells ) and the strength of the local modulation has already been noticed numerically LaTeXMLCite .
Additional mechanisms present in 2d remain to be explored .
Thereby it might turn out that phase chaos exists in the thermodynamic limit in 1d only but not in 2d as previously conjectured LaTeXMLCite .
It is a pleasure to acknowledge discussions with H. Chaté , M. Howard and L. Kramer .
AT and MB are grateful to ISI Torino for providing a pleasant working environment during the Workshop on “ Complexity and Chaos ” in October 1999 .
AT would also thank Caterina , Daniel , Katharina and Sara for providing him with a faithful representation of a chaotic evolution .
MGZ is supported from a post-doctoral grant of the MEC ( Spain ) and FOMEC-UBA ( Argentina ) .
MvH acknowledges financial support from the EU under contract ERBFMBICT 972554 .
We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions .
Following the automorphism case , we introduce two notions for endomorphisms of type III factors : modular endomorphisms and Connes-Takesaki modules .
Several applications to compact groups of automorphisms and subfactors of type III factors are given from the viewpoint of ergodic theory .
The canonical extension of an automorphism of a type III factor gives rise to a natural isomorphism from the automorphism group of the type III factor into that of the crossed product by the modular automorphism group .
Although the notion itself had been known to specialists before , ( e.g .
LaTeXMLCite ) , the first systematic analysis of the canonical extension was proposed and accomplished by U. Haagerup and E. Størmer .
In LaTeXMLCite , LaTeXMLCite , they introduced several classes of automorphisms of von Neumann algebras and investigated their structure using the canonical extension .
One of the purposes of the present notes is to generalize the notion of the canonical extension to an endomorphism whose image has a finite index LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Our analysis is based on techniques of the common continuous decomposition of inclusions and endomorphisms of type III factors , mainly developed by Ph .
Loi LaTeXMLCite , T. Hamachi-H. Kosaki LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , R. Longo LaTeXMLCite , LaTeXMLCite , and H. Kosaki-R. Longo LaTeXMLCite .
In LaTeXMLCite , J. E. Roberts introduced an action of the dual object of a compact group on a von Neumann algebra , now called a Roberts action , which is a functor from the category of the finite dimensional unitary representations to that of the endomorphisms of the von Neumann algebra .
As several invariants of discrete group actions on type III factors can be captured by the canonical extension LaTeXMLCite , LaTeXMLCite , it is natural to expect that the canonical extension of endomorphisms plays an important role in the analysis of compact group actions through the dual Roberts actions .
Indeed , we obtain several new results on minimal actions of compact groups on type III factors .
Study of minimal actions of compact groups was initiated by S. Popa and A. Wassermann using subfactor techniques developed in LaTeXMLCite .
Interested readers are recommended to consult LaTeXMLCite for related topics .
We introduce two new notions for endomorphisms corresponding to extended modular automorphisms and the Connes-Takesaki module in the automorphism case LaTeXMLCite : modular endomorphisms and the Connes-Takesaki module of endomorphisms .
As in the case of extended modular automorphisms , a modular endomorphism carries a unitary group valued cocycle of the flow of weights , and it gives rise to a Roberts action of the dual object of the essential range of the cocycle , the notion developed by G. Mackey and R. Zimmer LaTeXMLCite , LaTeXMLCite .
The dual actions of Roberts actions consisting of modular endomorphisms provide a large variety of new examples of minimal actions of compact groups on type III LaTeXMLMath factors .
Also , we give a strong constraint for the possible types of an algebra-fixed point algebra pair for a minimal action of a compact semisimple Lie group using these two notions .
In LaTeXMLCite , Zimmer introduced the notion of extensions of ergodic transformation groups with relatively discrete spectrum , which is a relative version of the notion of ergodic transformations with pure point spectrum .
It turns out that a non-commutative generalization of this notion is closely related to modular endomorphisms ( Theorem 5 .
11 ) , which may be expected because both objects are described in terms of cocycles of ergodic transformation groups with values in compact groups .
In particular , we characterize a subfactor , not necessarily of a finite index , coming from an ergodic extension of the flow of weights in terms of a group-subgroup subfactor ( c.f .
LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite ) .
Our original purpose for introducing the canonical extension of endomorphisms is to settle a problem left unsolved in LaTeXMLCite .
When a type III factor and its subfactor with a finite index have the common flow of weights , their “ type II principal graph ” makes sense via the common crossed product decomposition LaTeXMLCite , LaTeXMLCite .
In LaTeXMLCite , we characterized the situation where the type II principal graph does not coincide with the principal graph of the original subfactor for the case of III LaTeXMLMath , LaTeXMLMath factors .
Namely , the two graphs do not coincide if and only if some power of the canonical endomorphism for the inclusion contains a non-trivial modular automorphism .
It would be tempting to conjecture that the same statement should be true in the type III LaTeXMLMath case if the modular automorphism is replaced with an extended modular automorphism .
However , it turns out that the right counterpart for our purpose is a modular endomorphism ( Theorem 3.7 ) .
One can observe a similar phenomenon in the recent work of T. Masuda LaTeXMLCite ( c.f .
LaTeXMLCite ) .
Final part of this work was finished while the author stayed at MSRI , and he would like to thank them for their hospitality .
He also would like to thank T. Hamachi and H. Kosaki for stimulating discussions .
Some of the main results in this paper were announced in LaTeXMLCite .
First , we briefly summarize notation used in this paper .
Our basic references are LaTeXMLCite and LaTeXMLCite for Tomita-Takesaki theory , LaTeXMLCite for index theory of type III factors , LaTeXMLCite for sector theory and infinite index inclusions , and LaTeXMLCite for cocycles of ergodic transformation groups .
Undefined terms and notations used in this paper should be found in these references .
We always assume that von Neumann algebras have separable preduals , Hilbert spaces are separable , and locally compact groups are second countable .
Automorphisms and endomorphisms of von Neumann algebras are always assumed to be LaTeXMLMath -preserving and unital .
For a von Neumann algebra LaTeXMLMath , we denote by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath the sets of automorphisms , inner automorphisms , and endomorphisms respectively .
We denote by LaTeXMLMath and LaTeXMLMath the sets of unitary equivalence classes of automorphisms and endomorphisms respectively .
A member in LaTeXMLMath is called a sector .
For LaTeXMLMath , the intertwiner space from LaTeXMLMath to LaTeXMLMath is defined by LaTeXMLEquation .
When LaTeXMLMath is a factor and LaTeXMLMath is irreducible , that is , LaTeXMLMath , LaTeXMLMath is a Hilbert space with the inner product LaTeXMLEquation .
Every action of a topological group on a von Neumann algebra LaTeXMLMath is assumed to be continuous with respect to the LaTeXMLMath -topology in LaTeXMLMath .
For a group action LaTeXMLMath on LaTeXMLMath , LaTeXMLMath denotes the fixed point subalgebra of LaTeXMLMath under the action LaTeXMLMath .
When LaTeXMLMath acts on a Hilbert space LaTeXMLMath , we denote by LaTeXMLMath the representation of LaTeXMLMath on LaTeXMLMath defined by LaTeXMLEquation .
We denote by LaTeXMLMath and LaTeXMLMath the left and right regular representations of LaTeXMLMath respectively .
For a weight LaTeXMLMath on LaTeXMLMath or an operator valued weight from LaTeXMLMath to a subalgebra , we denote by LaTeXMLMath and LaTeXMLMath the domain of LaTeXMLMath and the left ideal corresponding to LaTeXMLMath as usual .
Let LaTeXMLMath be a faithful normal semifinite weight on LaTeXMLMath .
We denote by LaTeXMLMath the crossed product of LaTeXMLMath by the modular automorphism group LaTeXMLMath , which is the von Neumann algebra generated by LaTeXMLMath and the implementing one-parameter unitary group LaTeXMLMath , where LaTeXMLMath .
We often omit LaTeXMLMath when there is no possibility of confusion .
We denote by LaTeXMLMath and LaTeXMLMath the dual action of LaTeXMLMath and the natural trace constructed from the dual weight of LaTeXMLMath on LaTeXMLMath and the generator of LaTeXMLMath .
Then , the triple LaTeXMLMath does not depend on the choice of LaTeXMLMath under the following identification LaTeXMLEquation where LaTeXMLMath is another faithful normal weight on LaTeXMLMath and LaTeXMLMath is the Connes cocycle derivative .
As for the relationship between LaTeXMLMath and LaTeXMLMath , we have the following LaTeXMLCite : Let LaTeXMLMath be a von Neumann algebra and LaTeXMLMath be a faithful normal semifinite weight on LaTeXMLMath .
Then , for every automorphism LaTeXMLMath , there exists a unique automorphism LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation Moreover , the map LaTeXMLMath is a homomorphism from LaTeXMLMath into LaTeXMLMath .
LaTeXMLMath is called the canonical extension of LaTeXMLMath .
The goal of this section is to introduce the canonical extension of endomorphisms with finite statistical dimensions generalizing the automorphism case .
Let LaTeXMLMath be an infinite factor .
We denote by LaTeXMLMath the set of endomorphisms of LaTeXMLMath with finite statistical dimensions .
For LaTeXMLMath , we denote by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , the minimal conditional expectation from LaTeXMLMath onto the image of LaTeXMLMath LaTeXMLCite , the standard left inverse of LaTeXMLMath , and the statistical dimension of LaTeXMLMath , respectively : that is , LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath is the minimum index of LaTeXMLMath .
We denote by LaTeXMLMath the unitary equivalence classes of the endomorphisms in LaTeXMLMath .
LaTeXMLMath has three natural operations , forming a direct sum , composing two endomorphisms ( regarded as a product ) , and taking conjugation .
The statistical dimensions are additive , multiplicative , and invariant under these three operations on LaTeXMLMath respectively LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Let LaTeXMLMath be an inclusion of type III factors with a finite index .
Then , we regard LaTeXMLMath as a subalgebra of LaTeXMLMath through the identification LaTeXMLMath , where LaTeXMLMath is the minimal expectation from LaTeXMLMath onto LaTeXMLMath and LaTeXMLMath is a faithful normal semifinite weight on LaTeXMLMath .
We denote by LaTeXMLMath the extension of LaTeXMLMath to LaTeXMLMath leaving LaTeXMLMath fixed , which is a normal conditional expectation from LaTeXMLMath onto LaTeXMLMath LaTeXMLCite .
Even when LaTeXMLMath has an infinite index , if LaTeXMLMath is a unique normal conditional expectation from LaTeXMLMath onto LaTeXMLMath , we use the same convention .
Before introducing the canonical extension , we need some preparation .
Let LaTeXMLMath be an infinite factor .
A pair LaTeXMLMath consisting of a faithful normal semifinite weight LaTeXMLMath on LaTeXMLMath and LaTeXMLMath is said to be an invariant pair if the following hold : LaTeXMLEquation .
LaTeXMLEquation Note that the above conditions are equivalent to LaTeXMLMath thanks to LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be a type LaTeXMLMath factor .
Then , the following hold : If LaTeXMLMath is a faithful normal semifinite weight on LaTeXMLMath and LaTeXMLMath , then LaTeXMLMath is a faithful normal semifinite weight on LaTeXMLMath .
If LaTeXMLMath is an invariant pair , then LaTeXMLMath commutes with LaTeXMLMath for LaTeXMLMath .
Let LaTeXMLMath be a dominant weight ( see LaTeXMLCite for the definition ) .
Then , for every sector in LaTeXMLMath , there exists a representative LaTeXMLMath such that LaTeXMLMath is an invariant pair .
Let LaTeXMLMath be a dominant weight on LaTeXMLMath and LaTeXMLMath .
If LaTeXMLMath is an invariant pair , then there exist irreducible LaTeXMLMath and isometries LaTeXMLMath LaTeXMLMath such that each LaTeXMLMath belongs to the centralizer LaTeXMLMath and LaTeXMLEquation .
Moreover , each LaTeXMLMath is an invariant pair .
Let LaTeXMLMath be a dominant weight and LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath are invariant pairs , then LaTeXMLMath is in the centralizer LaTeXMLMath .
( 1 ) .
Since LaTeXMLMath is a faithful normal semifinite weight on LaTeXMLMath , so is LaTeXMLMath on LaTeXMLMath .
( 2 ) .
Let LaTeXMLMath and LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath , which is a faithful normal semifinite weight on LaTeXMLMath by assumption .
Then , we have LaTeXMLEquation .
LaTeXMLEquation Regarding LaTeXMLMath as an isomorphism from LaTeXMLMath to LaTeXMLMath , we get LaTeXMLEquation ( 3 ) .
This is LaTeXMLCite .
( 4 ) .
Let LaTeXMLMath and LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath .
Then , we have common continuous decomposition LaTeXMLCite , LaTeXMLCite LaTeXMLEquation such that LaTeXMLEquation .
Therefore , each non-zero projection LaTeXMLMath is equivalent to 1 in LaTeXMLMath ( more strongly in LaTeXMLMath ) , and there exist isometries LaTeXMLMath , LaTeXMLMath such that each LaTeXMLMath is a minimal projection in LaTeXMLMath and LaTeXMLMath forms a partition of unity .
Setting LaTeXMLMath , we get the first assertion .
Since the minimal expectation for LaTeXMLMath is given by LaTeXMLMath we have LaTeXMLEquation thanks to the local index formula for the minimal conditional expectation LaTeXMLCite and the fact that LaTeXMLMath is an isomorphism from LaTeXMLMath to LaTeXMLMath .
This implies LaTeXMLEquation .
Thus , LaTeXMLEquation .
Since LaTeXMLMath is in the centralizer of LaTeXMLMath , this implies that LaTeXMLMath is an invariant pair .
( 5 ) .
Thanks to ( 4 ) , the general case is reduced to the case where LaTeXMLMath and LaTeXMLMath are irreducible , and we make this assumption .
If LaTeXMLMath , we have nothing to prove , and so we assume LaTeXMLMath for some unitary LaTeXMLMath .
Since LaTeXMLMath commutes with LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is proportional to LaTeXMLMath , and there exists a scalar LaTeXMLMath such that LaTeXMLMath .
However , the KMS condition and LaTeXMLMath imply LaTeXMLMath , which shows LaTeXMLMath .
∎ Now we introduce the canonical extension of an endomorphism .
Let LaTeXMLMath be a type LaTeXMLMath factor and LaTeXMLMath be a faithful normal semifinite weight on LaTeXMLMath .
Then , for every LaTeXMLMath , there exists a unique endomorphism LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation Under the identification LaTeXMLMath , LaTeXMLMath does not depend on the choice of the weight LaTeXMLMath .
Let LaTeXMLMath be a dominant weight .
First we assume that LaTeXMLMath is an invariant pair , and LaTeXMLMath and LaTeXMLMath are as in the proof of Lemma 2.3 , ( 4 ) .
Thanks to Lemma 2.3 , ( 2 ) , we may consider LaTeXMLMath an isomorphism from LaTeXMLMath onto LaTeXMLMath intertwining LaTeXMLMath and LaTeXMLMath .
Therefore , LaTeXMLMath extends to an isomorphism LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath sending LaTeXMLMath to LaTeXMLMath .
Since LaTeXMLMath satisfies LaTeXMLMath , we may regard LaTeXMLMath as a subalgebra of LaTeXMLMath identifying LaTeXMLMath with LaTeXMLMath .
We show that LaTeXMLMath satisfies the required property .
Indeed , LaTeXMLEquation .
Regarding LaTeXMLMath as an isomorphism from LaTeXMLMath onto LaTeXMLMath again , we have LaTeXMLEquation and so LaTeXMLEquation .
Now , we treat the general case .
Thanks to Lemma 2.3 , ( 3 ) , it suffices to show the statement for LaTeXMLMath , where LaTeXMLMath is a unitary in LaTeXMLMath , and LaTeXMLMath and LaTeXMLMath are as before .
Indeed , LaTeXMLEquation .
Therefore , LaTeXMLMath has the desired property .
∎ The same formula of the Connes cocycle as above also appears in LaTeXMLCite .
We call the above LaTeXMLMath the canonical extension of LaTeXMLMath .
When , LaTeXMLMath is an automorphism , the left inverse LaTeXMLMath is nothing but the inverse of LaTeXMLMath .
Therefore , our definition of the canonical extension generalizes that in the automorphism case .
Let LaTeXMLMath be properly infinite von Neumann algebras .
Then , the standard representation of LaTeXMLMath is automatically standard for LaTeXMLMath and the product of two modular conjugation LaTeXMLMath makes sense .
The canonical endomorphism LaTeXMLMath for the inclusion LaTeXMLMath is defined by LaTeXMLMath LaTeXMLCite .
In general , modular conjugations depend on the choice of natural cones .
However , a different choice of modular conjugations amounts to only a perturbation of LaTeXMLMath by an inner automorphism of LaTeXMLMath .
Therefore , we call any endomorphism in LaTeXMLMath of the form LaTeXMLMath with a unitary LaTeXMLMath the canonical endomorphism as well .
Let LaTeXMLMath be a type LaTeXMLMath factor and LaTeXMLMath .
Then , the following hold : LaTeXMLMath .
LaTeXMLMath .
Let LaTeXMLMath be a subfactor of LaTeXMLMath with a finite index and LaTeXMLMath be the canonical endomorphism for LaTeXMLMath .
Then , LaTeXMLMath is the canonical endomorphism for LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath .
In the proof of Theorem 2.4 , we have already shown that for LaTeXMLMath and a unitary LaTeXMLMath , we have LaTeXMLEquation .
Thus , in order to prove the statements we may replace LaTeXMLMath and LaTeXMLMath with unitary equivalent endomorphisms .
Thanks to Lemma 2.3 , ( 3 ) , we may and do assume that LaTeXMLMath and LaTeXMLMath are invariant pairs , where LaTeXMLMath is a dominant weight .
Therefore , we have LaTeXMLEquation .
In this situation , ( 2 ) , ( 4 ) , and ( 5 ) are obvious and ( 1 ) follows from Lemma 2.3 , ( 5 ) .
To prove ( 3 ) , we consider the common continuous decomposition of LaTeXMLMath .
We may and do assume that LaTeXMLMath is of the form LaTeXMLMath , where LaTeXMLMath is a dominant weight on LaTeXMLMath and LaTeXMLMath is the minimal expectation from LaTeXMLMath to LaTeXMLMath .
Then , we have the common continuous decomposition LaTeXMLEquation .
Let LaTeXMLMath be the trace whose dual weight is LaTeXMLMath .
Then , thanks to LaTeXMLCite , the canonical endomorphism of LaTeXMLMath scales LaTeXMLMath by LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the modular conjugations of LaTeXMLMath and LaTeXMLMath , and LaTeXMLMath be the corresponding canonical endomorphism of LaTeXMLMath .
To define the canonical endomorphism of LaTeXMLMath and LaTeXMLMath , we utilize the modular objects of these algebras naturally coming from the modular objects of LaTeXMLMath and LaTeXMLMath through the crossed products as computed in LaTeXMLCite .
Let LaTeXMLMath and LaTeXMLMath be such canonical endomorphisms of LaTeXMLMath and LaTeXMLMath respectively .
Then , LaTeXMLMath is an invariant pair LaTeXMLCite .
On the other hand , applying LaTeXMLCite twice and using LaTeXMLCite for the computation of LaTeXMLMath and LaTeXMLMath , we know that LaTeXMLMath is an extension of LaTeXMLMath leaving LaTeXMLMath invariant .
Therefore , LaTeXMLMath coincides with the canonical extension of LaTeXMLMath .
∎ Let LaTeXMLMath be a type III factor and LaTeXMLMath be an automorphism of LaTeXMLMath .
In LaTeXMLCite , it was shown that LaTeXMLMath is inner if and only if LaTeXMLMath is the composition of an inner automorphism with an extended modular automorphism .
We adopt this property as the definition of an endomorphism counterpart of an extended modular automorphism .
Let LaTeXMLMath be a type III factor and LaTeXMLMath .
We say that LaTeXMLMath is a modular endomorphism if LaTeXMLMath is an inner endomorphism , that is , there exist isometries LaTeXMLMath with mutually orthogonal ranges and LaTeXMLMath such that LaTeXMLEquation .
The system of isometry LaTeXMLMath satisfying the above condition is called an implementing system for LaTeXMLMath .
We denote by LaTeXMLMath the set of modular endomorphisms of LaTeXMLMath .
Thanks to Proposition 2.5 , ( 1 ) , an endomorphism unitary equivalent to a modular endomorphism is again a modular endomorphism , and so it makes sense to call a sector modular if its representatives are modular endomorphisms .
We denote by LaTeXMLMath the set of sectors of modular endomorphisms .
It is also easy to see from Proposition 2.5 , ( 1 ) , ( 2 ) that LaTeXMLMath is closed under forming a direct sum and a product of finitely many elements .
What is not clear for the moment is whether LaTeXMLMath is closed under conjugation and irreducible decomposition , which will be shown later .
The number LaTeXMLMath of the isometries of the implementing system in Definition 3.1 is nothing but LaTeXMLMath , which also will be shown later .
Let LaTeXMLMath be as above and LaTeXMLMath be the center of LaTeXMLMath .
Then , the restriction of LaTeXMLMath to LaTeXMLMath is an ergodic action of the real number group LaTeXMLMath .
Thus , by Mackey ’ s point realization theorem LaTeXMLCite , there exist a standard Borel space LaTeXMLMath , a probability measure LaTeXMLMath on LaTeXMLMath , and an ergodic flow LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
( For simplicity , we often omit LaTeXMLMath and denotes LaTeXMLMath just by LaTeXMLMath ) .
LaTeXMLMath is called the smooth flow of weights LaTeXMLCite .
Let LaTeXMLMath be a compact group .
A Borel map LaTeXMLMath is said to be a cocycle of LaTeXMLMath if for fixed LaTeXMLMath , the cocycle relation LaTeXMLEquation holds for almost all LaTeXMLMath .
Two cocycles agree on the outside of a null set are identified as usual .
We denote by LaTeXMLMath the set of all LaTeXMLMath -valued cocycles of LaTeXMLMath .
Two cocycles LaTeXMLMath and LaTeXMLMath are said to be equivalent ( or cohomologous ) if there exists a Borel map LaTeXMLMath such that for fixed LaTeXMLMath LaTeXMLEquation holds for almost all LaTeXMLMath .
A cocycle equivalent to a constant function LaTeXMLMath is said to be a coboundary .
We denote by LaTeXMLMath the set of equivalence classes of LaTeXMLMath -valued cocycles of LaTeXMLMath .
For the significance of cocycles from the viewpoint of Mackey ’ s notion of virtual groups , readers are refered to LaTeXMLCite and LaTeXMLCite .
As Connes and Takesaki showed that the group of extended modular automorphisms divided by the inner automorphism group is isomorphic to LaTeXMLMath LaTeXMLCite , we show that LaTeXMLMath is “ isomorphic ” to the union LaTeXMLMath of the unitary group LaTeXMLMath -valued cohomology classes .
Let LaTeXMLMath and LaTeXMLMath be an implementing system for LaTeXMLMath .
Since LaTeXMLMath is globally preserved by LaTeXMLMath , LaTeXMLMath belongs to LaTeXMLMath .
LaTeXMLMath is regarded as a LaTeXMLMath -valued Borel function on LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be as above .
Then , LaTeXMLMath is a LaTeXMLMath -valued cocycle of LaTeXMLMath , whose cohomology class depends only on the sector of LaTeXMLMath .
Let LaTeXMLMath be a cocycle equivalent to LaTeXMLMath .
Then , there exists an implementing system LaTeXMLMath for LaTeXMLMath such that LaTeXMLMath .
( 1 ) .
The cocycle relation of LaTeXMLMath follows from LaTeXMLEquation .
Let LaTeXMLMath be another implementing system for LaTeXMLMath , and set LaTeXMLMath .
Since LaTeXMLMath is a matrix valued function that is unitary , we have LaTeXMLMath .
Setting LaTeXMLMath , we get LaTeXMLEquation which shows that LaTeXMLMath and LaTeXMLMath are equivalent .
Let LaTeXMLMath be a unitary of LaTeXMLMath and LaTeXMLMath .
Then , LaTeXMLMath is an implementing system for LaTeXMLMath .
Since LaTeXMLMath is fixed by LaTeXMLMath , LaTeXMLMath gives the same cohomology class as LaTeXMLMath .
( 2 ) .
We take a LaTeXMLMath -valued function LaTeXMLMath satisfying LaTeXMLEquation .
Then , LaTeXMLMath has a desired property .
∎ We introduce a map LaTeXMLEquation sending LaTeXMLMath to LaTeXMLMath in the above lemma .
As described in LaTeXMLCite , LaTeXMLMath has natural three operations in analogous to the unitary representation theory of compact groups : direct sum LaTeXMLMath , tensor product LaTeXMLMath and complex conjugate LaTeXMLMath .
LaTeXMLMath inherits these operations and we use the same notation for the cohomology classes as well .
Let LaTeXMLMath be as above and LaTeXMLMath .
Then , LaTeXMLMath is a bijection .
We have LaTeXMLMath We have LaTeXMLMath LaTeXMLMath is the conjugate sector of LaTeXMLMath if and only if LaTeXMLMath .
In particular , LaTeXMLMath is closed under conjugation .
LaTeXMLMath is grade preserving in the sense that LaTeXMLMath if and only if LaTeXMLMath .
( 1 ) .
First we show that LaTeXMLMath is injective .
Let LaTeXMLMath satisfying LaTeXMLMath .
Then , thanks to Lemma 3.2 , ( 2 ) , there exist implementing systems LaTeXMLMath and LaTeXMLMath for LaTeXMLMath and LaTeXMLMath respectively such that they give the same cocycle .
We set LaTeXMLMath , which is a unitary in LaTeXMLMath satisfying LaTeXMLMath .
This shows that LaTeXMLMath is injective .
Next we show that LaTeXMLMath is surjective .
Let LaTeXMLMath be a dominant weight on LaTeXMLMath and LaTeXMLMath be the continuous decomposition .
We take the trace LaTeXMLMath on LaTeXMLMath whose dual weight is LaTeXMLMath , and take the implementing one-parameter unitary group LaTeXMLMath for LaTeXMLMath .
We assume that LaTeXMLMath acts on a Hilbert space LaTeXMLMath .
Then , the Takesaki duality theorem LaTeXMLCite implies that there exists an isomorphism LaTeXMLMath from LaTeXMLMath to LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation where LaTeXMLMath is the left regular representation of LaTeXMLMath and LaTeXMLMath is the multiplication operator of LaTeXMLMath .
Moreover , LaTeXMLMath is identified with LaTeXMLMath under LaTeXMLMath , or more precisely we have LaTeXMLMath .
Using LaTeXMLMath , we identify the flow LaTeXMLMath with LaTeXMLEquation .
Let LaTeXMLMath be a given cocycle , where LaTeXMLMath is understood as the point realization of LaTeXMLMath .
We take a system of isometries LaTeXMLMath in LaTeXMLMath satisfying LaTeXMLMath and set LaTeXMLEquation .
Then , LaTeXMLMath is a unitary satisfying LaTeXMLEquation which means that LaTeXMLMath is a LaTeXMLMath -cocycle .
Since LaTeXMLMath is stable LaTeXMLCite , every LaTeXMLMath -cocycle is a coboundary and there exists a unitary LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath .
Then , LaTeXMLMath is an isometry in LaTeXMLMath satisfying LaTeXMLMath and LaTeXMLEquation .
For LaTeXMLMath we set , LaTeXMLEquation .
LaTeXMLEquation and LaTeXMLMath .
Then , LaTeXMLMath is an endomorphism of LaTeXMLMath commuting with LaTeXMLMath , LaTeXMLMath is a left inverse of LaTeXMLMath satisfying LaTeXMLMath , and LaTeXMLMath is a LaTeXMLMath -preserving conditional expectation from LaTeXMLMath onto the image of LaTeXMLMath .
We introduce LaTeXMLMath that is an extension of LaTeXMLMath to LaTeXMLMath leaving LaTeXMLMath invariant .
Indeed , such LaTeXMLMath exists because the following hold : LaTeXMLEquation .
LaTeXMLEquation In a similar way , we define a left inverse LaTeXMLMath of LaTeXMLMath and a conditional expectation LaTeXMLMath from LaTeXMLMath onto the image of LaTeXMLMath by the relations LaTeXMLEquation and LaTeXMLMath .
Note that LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath are extensions of LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath respectively .
We claim that LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath is an invariant pair .
Indeed , it suffices to show LaTeXMLMath , and the other claims can be shown easily .
We define LaTeXMLMath by LaTeXMLEquation .
Then , for LaTeXMLMath and LaTeXMLMath , we have LaTeXMLEquation which shows LaTeXMLMath is a one-element Pimsner-Popa basis and LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
To show that LaTeXMLMath is minimal , it suffices to show LaTeXMLMath for all LaTeXMLMath LaTeXMLCite .
Thanks to the Connes-Takesaki relative commutant theorem LaTeXMLCite , for LaTeXMLMath we have LaTeXMLEquation and LaTeXMLEquation .
Thus , LaTeXMLEquation which shows the claims .
Since LaTeXMLMath is an invariant pair , LaTeXMLMath is an extension of LaTeXMLMath leaving LaTeXMLMath invariant .
Therefore , we have LaTeXMLEquation which shows that LaTeXMLMath is a modular endomorphism carrying LaTeXMLMath .
( 2 ) and ( 3 ) are easy , and ( 5 ) follows from the above proof .
( 4 ) .
Assume that LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be implementing systems for LaTeXMLMath and LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation We define two isometries LaTeXMLMath and LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath and LaTeXMLMath belong to LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
Thus , LaTeXMLMath LaTeXMLCite .
Since LaTeXMLMath is a bijection , this also shows that LaTeXMLMath implies LaTeXMLMath and LaTeXMLMath .
∎ The following is a useful criterion for an irreducible endomorphism to be modular .
Let LaTeXMLMath be a type LaTeXMLMath factor .
Let LaTeXMLMath be an irreducible endomorphism .
If there exists a non-zero element in LaTeXMLMath , LaTeXMLMath is a modular endomorphism .
Every irreducible component of a modular endomorphism is again a modular endomorphism .
In particular LaTeXMLMath is closed under irreducible decomposition .
( 1 ) .
First , we show that LaTeXMLMath contains at least one isometry if LaTeXMLMath .
Let LaTeXMLMath be the set of partial isometries in LaTeXMLMath .
Using the polar decomposition , we know that LaTeXMLMath is not empty .
We introduce an order into LaTeXMLMath as follows : For two element LaTeXMLMath , we say that LaTeXMLMath dominates LaTeXMLMath if the following holds LaTeXMLEquation .
Note that when this is the case , LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath hold .
We show that LaTeXMLMath is an inductively ordered set with this order .
Let LaTeXMLMath be a totally ordered subset of LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are increasing nets of projections in LaTeXMLMath and LaTeXMLMath , they converge to projections in LaTeXMLMath and in LaTeXMLMath respectively in strong topology .
We claim that the net LaTeXMLMath converges in strong LaTeXMLMath topology .
Let LaTeXMLMath with LaTeXMLMath dominating LaTeXMLMath , and LaTeXMLMath be a vector in the Hilbert space that LaTeXMLMath acts on .
Then , LaTeXMLEquation .
LaTeXMLEquation Therefore , the strong LaTeXMLMath limit exists , and it gives a majorant of LaTeXMLMath , which shows that LaTeXMLMath is inductively ordered .
Now , we apply Zorn ’ s lemma to LaTeXMLMath and take a maximal element LaTeXMLMath .
We show that LaTeXMLMath is an isometry .
Suppose LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath acts on LaTeXMLMath ergodically , there exists some LaTeXMLMath such that LaTeXMLMath .
Therefore , LaTeXMLMath dominates LaTeXMLMath , which is contradiction .
Thus , LaTeXMLMath is an isometry .
Let LaTeXMLMath be a maximal set of isometries in LaTeXMLMath with mutually orthogonal ranges ( such a set exists thanks to Zorn ’ s lemma again .
) , and let LaTeXMLMath .
If LaTeXMLMath , we are done , and so we assume LaTeXMLMath .
Let LaTeXMLMath be the subset of elements in LaTeXMLMath with range projections orthogonal to LaTeXMLMath .
We claim that there exists a non-zero element in LaTeXMLMath .
Indeed , since LaTeXMLMath , there exists some LaTeXMLMath such that LaTeXMLMath , which shows that there exists non-zero LaTeXMLMath with LaTeXMLMath .
Using the polar decomposition , we get the claim .
We introduce an order into LaTeXMLMath as before and take a maximal element LaTeXMLMath .
Since LaTeXMLMath is a maximal set , LaTeXMLMath is not an isometry , and we set LaTeXMLMath .
We claim LaTeXMLMath .
Indeed , by maximality of LaTeXMLMath , LaTeXMLMath holds for LaTeXMLMath , and LaTeXMLMath for LaTeXMLMath as well for all LaTeXMLMath because LaTeXMLMath is globally invariant under LaTeXMLMath .
Thus , we get LaTeXMLEquation .
Since LaTeXMLMath , this implies LaTeXMLMath .
Let LaTeXMLMath .
Then , LaTeXMLMath have mutually orthogonal ranges such that LaTeXMLEquation .
Thus , we get LaTeXMLEquation .
Since there exists a conditional expectation from LaTeXMLMath to LaTeXMLMath satisfying the Pimsner-Popa inequality LaTeXMLCite , LaTeXMLCite , LaTeXMLMath is a finite set , and we identify LaTeXMLMath with LaTeXMLMath .
We take LaTeXMLMath satisfying LaTeXMLMath , and set LaTeXMLMath to be the Borel subset corresponding LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then , LaTeXMLMath and LaTeXMLMath satisfy LaTeXMLEquation .
LaTeXMLEquation LaTeXMLEquation .
LaTeXMLEquation We set LaTeXMLMath , which is regarded as an LaTeXMLMath by LaTeXMLMath matrix-valued function on LaTeXMLMath .
However LaTeXMLMath satisfies LaTeXMLEquation which is contradiction .
Thus , we conclude LaTeXMLMath , and LaTeXMLEquation ( 2 ) .
Let LaTeXMLMath be a modular endomorphism with an implementing system LaTeXMLMath for LaTeXMLMath , and LaTeXMLMath be an irreducible component of LaTeXMLMath .
We take a non-zero element LaTeXMLMath .
Thanks to Proposition 2.5 , ( 1 ) , LaTeXMLMath belongs to LaTeXMLMath , LaTeXMLMath .
Since LaTeXMLMath , there exists some LaTeXMLMath such that LaTeXMLMath is non-zero .
Thus , LaTeXMLMath is a modular endomorphism thanks to ( 1 ) .
∎ Though it is shown in the above that LaTeXMLMath is closed under irreducible decomposition , we have not given a criterion for a modular endomorphism to be irreducible , or that of how to decompose it when reducible , in terms of ergodic theory yet .
We discuss this issue now .
Let LaTeXMLMath be a standard Borel space , LaTeXMLMath be a probability measure on LaTeXMLMath , and LaTeXMLMath be a locally compact group ergodically acting on LaTeXMLMath as a non-singular transformation group .
For a compact group LaTeXMLMath , we define a LaTeXMLMath -valued cocycle LaTeXMLMath as in the case of the flow .
We denote by LaTeXMLMath the closed subgroup generated by the image of LaTeXMLMath .
We collect necessary results on cocycles from R. Zimmer ’ s fundamental paper LaTeXMLCite in the next theorem in order to introduce a few notions for cocycles .
Though Zimmer treated only the measure preserving case in LaTeXMLCite , his argument works for the non-singular case with a formal modification ( see also LaTeXMLCite ) : Let the notations be as above , and LaTeXMLMath .
Then , the following hold : There exists a closed subgroup LaTeXMLMath such that LaTeXMLMath is equivalent to an LaTeXMLMath -valued cocycle LaTeXMLMath , and LaTeXMLMath is never equivalent to LaTeXMLMath with LaTeXMLMath a proper subgroup of LaTeXMLMath .
LaTeXMLMath is uniquely determined up to conjugacy .
We call such LaTeXMLMath a minimal cocycle and call LaTeXMLMath the minimal subgroup of LaTeXMLMath .
Let LaTeXMLMath be a compact group containing LaTeXMLMath .
Then , if LaTeXMLMath is a minimal cocycle as a LaTeXMLMath -valued cocycle , it is the case as a LaTeXMLMath -valued cocycle .
Let LaTeXMLMath be a minimal LaTeXMLMath -valued cocycle such that LaTeXMLMath , and LaTeXMLMath be a continuous homomorphism from LaTeXMLMath to a compact group LaTeXMLMath .
Then , LaTeXMLMath is a minimal cocycle .
Let LaTeXMLMath be a minimal LaTeXMLMath -valued cocycle with LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath be finite dimensional unitary representations of LaTeXMLMath .
If LaTeXMLMath is a Borel map from LaTeXMLMath to LaTeXMLMath such that for every fixed LaTeXMLMath , LaTeXMLEquation holds for almost all LaTeXMLMath , then LaTeXMLMath is constant almost everywhere with the value in LaTeXMLMath .
We say that a closed subgroup LaTeXMLMath is irreducible if the defining representation of LaTeXMLMath on LaTeXMLMath is irreducible .
Let LaTeXMLMath be a type LaTeXMLMath factor and LaTeXMLMath be a modular endomorphism with LaTeXMLMath .
Then , LaTeXMLMath is irreducible if and only if the minimal subgroup of LaTeXMLMath is irreducible .
Thanks to theorem 3.3 , it suffices to show that the minimal subgroup of LaTeXMLMath is irreducible if and only if LaTeXMLMath is never equivalent to a direct sum of two cocycles .
Assume that the minimal subgroup of LaTeXMLMath is irreducible .
If LaTeXMLMath were equivalent to LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , with LaTeXMLMath , LaTeXMLMath could be considered as a LaTeXMLMath -valued cocycle , and the minimal subgroup of LaTeXMLMath would be conjugate to a subgroup of LaTeXMLMath in LaTeXMLMath thanks to Theorem 3.5 , ( 2 ) .
This is contradiction and LaTeXMLMath is never equivalent to a direct sum of two cocycles .
The converse also follows from Theorem 3.5 , ( 2 ) .
∎ We end this section with giving an answer to the problem about the type II and type III principal graphs described in Introduction .
Let LaTeXMLMath be an inclusion of type III factors of a finite index with the minimal expectation LaTeXMLMath .
We say that LaTeXMLMath and LaTeXMLMath has the common flow of weights if LaTeXMLMath .
When this is the case , we have common central decomposition LaTeXMLEquation and we could define the type II principal graph for that of LaTeXMLMath for LaTeXMLMath in a conull set .
However , it would be cumbersome to treat “ measurable field of subfactors ” ( though it should not be too hard to do so , c.f .
LaTeXMLCite . )
Instead , we formulate the problem using a global term .
Let LaTeXMLEquation be the tower for LaTeXMLMath .
Then , using the minimal conditional expectation in each step , we have the “ tower for the core inclusion ” LaTeXMLCite : LaTeXMLEquation .
We still denote by LaTeXMLMath the natural extension of LaTeXMLMath to LaTeXMLMath leaving the Jones projections fixed .
Then , we have LaTeXMLEquation .
We say that graph change occurs if LaTeXMLMath does not coincides with LaTeXMLMath for some LaTeXMLMath .
Let LaTeXMLMath be the minimal expectation from LaTeXMLMath onto LaTeXMLMath and LaTeXMLMath be the natural extension of LaTeXMLMath to LaTeXMLMath , which is a conditional expectation from LaTeXMLMath onto LaTeXMLMath .
Then , using LaTeXMLMath and LaTeXMLMath , we can show that LaTeXMLMath implies LaTeXMLMath .
In a similar way using downward basic construction and mirroring LaTeXMLMath , LaTeXMLMath , we can also show that graph change occurs for LaTeXMLMath if and only if it occurs for LaTeXMLMath .
A modular endomorphism is said to be non-trivial if it is irreducible and not equivalent to identity .
Let LaTeXMLMath be an inclusion of type LaTeXMLMath factors with a finite index and with the common flow of weights .
Then , graph change occurs for LaTeXMLMath if and only if a non-trivial modular endomorphism appears in some power of the canonical endomorphism LaTeXMLMath for LaTeXMLMath .
Assume that graph change occurs for LaTeXMLMath .
Taking sufficiently large LaTeXMLMath , we may assume LaTeXMLMath .
Let LaTeXMLMath be the canonical endomorphism for LaTeXMLMath .
Then , thanks to Proposition 2.5 , this is equivalent to LaTeXMLEquation .
Therefore , either of the following two holds : ( i ) LaTeXMLMath contains two irreducibles LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath .
( ii ) LaTeXMLMath contains an irreducible LaTeXMLMath such that LaTeXMLMath .
Indeed , let LaTeXMLMath be a partition of unity consisting of minimal projections in LaTeXMLMath .
If LaTeXMLEquation there exist some LaTeXMLMath and LaTeXMLMath such that LaTeXMLEquation .
If the irreducible components corresponding to LaTeXMLMath and LaTeXMLMath are different , ( i ) occurs .
If they are the same , ( ii ) occurs .
In the case ( i ) , the Frobenius reciprocity implies LaTeXMLMath ( though LaTeXMLMath is not a factor , the argument in LaTeXMLCite works ) .
Thanks to Lemma 3.3 , this shows that LaTeXMLMath , which is contained in LaTeXMLMath , contains a modular endomorphism .
Since LaTeXMLMath does not contain identity , it is indeed a non-trivial one .
In the case ( ii ) , we take LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath be isometries satisfying the usual relation LaTeXMLCite LaTeXMLEquation .
Then , LaTeXMLMath .
Let LaTeXMLMath be an irreducible sector contained in LaTeXMLMath that is not equivalent to identity , and LaTeXMLMath be a basis of LaTeXMLMath .
If LaTeXMLMath is not zero for some LaTeXMLMath , we are done thanks to Lemma 3.3 .
Thus , we assume LaTeXMLMath for every LaTeXMLMath not equivalent to identity .
This would imply that LaTeXMLMath .
However , since LaTeXMLMath commutes with LaTeXMLMath , we would get LaTeXMLEquation which is contradiction .
Thus , LaTeXMLMath contains a non-trivial modular endomorphism , and so does LaTeXMLMath .
The converse can be shown easily .
∎ For a type LaTeXMLMath , LaTeXMLMath factor , the flow is a transitive and every LaTeXMLMath -valued cocycle is equivalent to a direct sum of LaTeXMLMath -valued cocycles that come from homomorphisms from the stabilizer subgroup of a point LaTeXMLCite , LaTeXMLMath for LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
This means that every modular endomorphism is decomposed into usual modular automorphisms .
Therefore , the above theorem generalizes the previous one obtained in LaTeXMLCite .
There exist plenty of examples in the type LaTeXMLMath case , where higher dimensional non-trivial modular endomorphisms appear in the canonical endomorphisms ( see Section 5 and Proposition A.5 ) .
For a type III factor LaTeXMLMath , we denote by LaTeXMLMath the set of automorphisms of LaTeXMLMath that commute with the restriction of LaTeXMLMath to LaTeXMLMath .
For an automorphism LaTeXMLMath of LaTeXMLMath , the restriction of LaTeXMLMath to LaTeXMLMath belongs to LaTeXMLMath , which is called the Connes-Takesaki module of LaTeXMLMath .
This correspondence gives a homomorphism LaTeXMLEquation which is called the fundamental homomorphism .
( Though the original definition of Connes and Takesaki looks different from ours LaTeXMLCite , this is an equivalent description of it . )
The purpose of this section is to introduce the Connes-Takesaki module for endomorphisms .
Let LaTeXMLMath be a type III factor and LaTeXMLMath .
We say that LaTeXMLMath has a Connes-Takesaki module if LaTeXMLMath has the common flow of weights .
When LaTeXMLMath has a Connes-Takesaki module , we denote by LaTeXMLMath the restriction of LaTeXMLMath to LaTeXMLMath .
LaTeXMLMath is called the Connes-Takesaki module of LaTeXMLMath .
We denote by LaTeXMLMath ( respectively LaTeXMLMath ) the set of endomorphisms in LaTeXMLMath with Connes-Takesaki modules ( respectively trivial Connes-Takesaki modules ) .
Note that LaTeXMLMath depends only on the sector of LaTeXMLMath .
We denote by LaTeXMLMath and LaTeXMLMath the corresponding subsets of LaTeXMLMath .
Let LaTeXMLMath be a type LaTeXMLMath factor .
Then , If LaTeXMLMath have Connes-Takesaki modules , then so does LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath , LaTeXMLMath satisfying LaTeXMLMath .
Then , LaTeXMLMath has a Connes-Takesaki module if and only if each LaTeXMLMath does and LaTeXMLMath does not depend on LaTeXMLMath .
When it is the case , LaTeXMLMath .
LaTeXMLMath has a Connes-Takesaki module if and only if the conjugate LaTeXMLMath does .
When it is the case , LaTeXMLMath .
Every modular endomorphism has a trivial Connes-Takesaki module .
Let LaTeXMLMath be a subfactor with a finite index .
If LaTeXMLMath and LaTeXMLMath have the common flow of weights , then the canonical endomorphism LaTeXMLMath for LaTeXMLMath has a trivial Connes-Takesaki module .
( 1 ) .
This is trivial .
( 2 ) .
We take isometries LaTeXMLMath satisfying LaTeXMLEquation .
Assume LaTeXMLMath .
Thanks to Lemma 2.4 , ( 1 ) , we have LaTeXMLEquation which shows that LaTeXMLMath for every LaTeXMLMath and LaTeXMLMath does not depend on LaTeXMLMath .
Next we assume that LaTeXMLMath for every LaTeXMLMath and LaTeXMLMath does not depend on LaTeXMLMath , which is denoted by LaTeXMLMath .
Then , LaTeXMLEquation which shows LaTeXMLMath .
( 3 ) .
Since LaTeXMLMath is the dual inclusion of LaTeXMLMath , the first part is obvious .
The second part follows from ( 1 ) and ( 2 ) because LaTeXMLMath contains identity .
( 4 ) .
This follows from the definition of modular endomorphisms .
( 5 ) .
This follows from Proposition 2.5 , ( 3 ) .
∎ Let LaTeXMLMath be an inclusion of type LaTeXMLMath , LaTeXMLMath factors with a finite index , and LaTeXMLMath be the minimal expectation from LaTeXMLMath onto LaTeXMLMath .
We say that LaTeXMLMath has common discrete decomposition if there exist an inclusion of type LaTeXMLMath factors LaTeXMLMath with a finite index , a minimal expectation LaTeXMLMath from LaTeXMLMath to LaTeXMLMath preserving the trace , and a trace scaling automorphism LaTeXMLMath of LaTeXMLMath globally preserving LaTeXMLMath such that LaTeXMLEquation .
Note that in LaTeXMLCite , common discrete decomposition with respect to any faithful normal expectation is discussed , while here we consider the minimal expectation only .
The following lemma is standard and we omit the proof .
Let LaTeXMLMath be an inclusion of type LaTeXMLMath , LaTeXMLMath factors with a finite index , and LaTeXMLMath be the minimal expectation from LaTeXMLMath to LaTeXMLMath .
Then , the following are equivalent : LaTeXMLMath has common discrete decomposition .
LaTeXMLMath has the common flow of weights .
For some ( and any ) generalized trace LaTeXMLMath on LaTeXMLMath ( see LaTeXMLCite for the definition , it is also called LaTeXMLMath -trace in LaTeXMLCite ) , LaTeXMLMath is a generalized trace on LaTeXMLMath .
Let LaTeXMLMath be a type LaTeXMLMath , LaTeXMLMath , factor , and LaTeXMLMath .
Then , If LaTeXMLMath is irreducible , LaTeXMLMath has a Connes-Takesaki module .
LaTeXMLMath has common discrete decomposition if and only if LaTeXMLMath has a Connes-Takesaki module , which is further equivalent to that every irreducible component of LaTeXMLMath has the same Connes-Takesaki module .
( 1 ) .
This follows from the condition ( 3 ) of Lemma 4.3 .
( 2 ) .
This follows from Proposition 4.2 , ( 2 ) and Lemma 4.3 .
∎ Let LaTeXMLMath be a type LaTeXMLMath factor , LaTeXMLMath , and LaTeXMLMath be a generalized trace on LaTeXMLMath .
We set LaTeXMLMath .
Then , the modular automorphism group LaTeXMLMath has period LaTeXMLMath and we consider LaTeXMLMath an action of LaTeXMLMath .
We denote by LaTeXMLMath the crossed product LaTeXMLMath , which is a type LaTeXMLMath factor generated by a copy of LaTeXMLMath and the implementing unitary representation LaTeXMLMath of LaTeXMLMath .
If LaTeXMLMath is another weight on LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath is a LaTeXMLMath -cocycle as a LaTeXMLMath action , and we identify LaTeXMLMath with LaTeXMLMath .
We denote by LaTeXMLMath and LaTeXMLMath the natural trace on LaTeXMLMath and the dual action of LaTeXMLMath respectively .
In analogous to the canonical extension , we can construct a functor from LaTeXMLMath to LaTeXMLMath .
Note that thanks to Proposition 4.2 , LaTeXMLMath is a full subcategory of LaTeXMLMath .
Let LaTeXMLMath be a type LaTeXMLMath factor and LaTeXMLMath be a generalized trace .
Then , for every LaTeXMLMath , there exists a unique endomorphism LaTeXMLMath satisfying LaTeXMLEquation .
LaTeXMLEquation Moreover , for LaTeXMLMath , the following hold : LaTeXMLMath .
LaTeXMLMath .
Let LaTeXMLMath be a subfactor of LaTeXMLMath with a finite index and LaTeXMLMath be the canonical endomorphism for LaTeXMLMath .
If LaTeXMLMath has common discrete decomposition , then , LaTeXMLMath is the canonical endomorphism for LaTeXMLMath .
LaTeXMLMath .
LaTeXMLMath .
Since the center of LaTeXMLMath is generated by LaTeXMLMath , we have LaTeXMLMath by assumption , which implies LaTeXMLEquation .
Therefore , there exists a unitary LaTeXMLMath such that LaTeXMLMath thanks to LaTeXMLCite .
This means that LaTeXMLMath is an invariant pair .
The rest of the proof is the same as those of Theorem 2.4 and Proposition 2.5 if the dominant weight is replaced with the generalized trace .
∎ We end this section with a remark on the type LaTeXMLMath case .
Let LaTeXMLMath be a type LaTeXMLMath factor and LaTeXMLMath be a faithful normal semifinite trace on LaTeXMLMath .
Then , the canonical extension and the Connes-Takesaki modules make sense for LaTeXMLMath as well .
We introduce a scalar valued module LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation .
Note that every inclusion of type LaTeXMLMath factors with a finite index comes from the tensor product of a common type I factor and an inclusion of LaTeXMLMath factors .
Therefore , the restriction of LaTeXMLMath to the image of LaTeXMLMath is a semifinite trace and the above definition makes sense .
The center of LaTeXMLMath is generated by LaTeXMLMath , and we have LaTeXMLEquation .
This means that LaTeXMLMath has a Connes-Takesaki module if and only if LaTeXMLMath is trace preserving .
When this is the case , we have LaTeXMLEquation .
Therefore , Proposition 4.2 , ( 2 ) implies that LaTeXMLMath is an extremal inclusion ( see LaTeXMLCite for the definition ) if and only if each irreducible component of LaTeXMLMath has the same LaTeXMLMath .
In this section , we investigate the structure of minimal actions of compact groups on type III factors applying our machinery to the corresponding Roberts actions .
Modular endomorphisms and the Connes-Takesaki modules for endomorphisms provide new invariants for minimal actions of compact groups through Roberts actions .
Let LaTeXMLMath be a compact group and LaTeXMLMath be the tensor category of finite dimensional unitary representations of LaTeXMLMath .
Roughly speaking , a Roberts action of the dual of LaTeXMLMath on a von Neumann algebra LaTeXMLMath is a functor from a ( sufficiently big ) subcategory LaTeXMLMath of LaTeXMLMath , ( called a ring in LaTeXMLCite ) , to LaTeXMLMath , where the set of arrows from LaTeXMLMath to LaTeXMLMath is LaTeXMLMath .
For the precise definition of the Roberts action and the crossed product by it , we refer to LaTeXMLCite and LaTeXMLCite ( see also LaTeXMLCite ) .
For each equivalence class of irreducible representations of LaTeXMLMath , we choose a representative LaTeXMLMath , and denote by LaTeXMLMath the collection of them .
We always assume that LaTeXMLMath contains every member of LaTeXMLMath and its complex conjugate representation .
We often omit to specify LaTeXMLMath in a Roberts action , and say “ a Roberts action of LaTeXMLMath ” if there is no possibility of confusion ( although it is a bit sloppy terminology ) .
For LaTeXMLMath , we denote by LaTeXMLMath the set of LaTeXMLMath -homomorphisms from LaTeXMLMath to LaTeXMLMath , that is , the set of arrows from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be a factor and LaTeXMLMath be an action of LaTeXMLMath on LaTeXMLMath .
LaTeXMLMath is said to be minimal if LaTeXMLMath is faithful and the relative commutant LaTeXMLMath of the fixed point subalgebra LaTeXMLMath is trivial LaTeXMLCite .
Note that when LaTeXMLMath is minimal in our sense , the crossed product LaTeXMLMath is automatically a factor isomorphic to LaTeXMLMath .
Indeed , to show this , we may assume that LaTeXMLMath is infinite by taking tensor product with a type I factor if necessary .
Then , a similar argument as in LaTeXMLCite implies that for each irreducible representation LaTeXMLMath of LaTeXMLMath , there exists a Hilbert space LaTeXMLMath in LaTeXMLMath that is globally invariant under LaTeXMLMath such that the restriction of LaTeXMLMath to LaTeXMLMath is equivalent to LaTeXMLMath .
Therefore , LaTeXMLCite implies the claim .
More strongly , LaTeXMLCite shows that LaTeXMLMath is the crossed product LaTeXMLMath by some Roberts action LaTeXMLMath , and that LaTeXMLMath is its dual action LaTeXMLMath , where LaTeXMLMath .
The relationship between LaTeXMLMath and LaTeXMLMath is as follows : Let LaTeXMLMath be an orthonormal basis for LaTeXMLMath , where LaTeXMLMath is the dimension of LaTeXMLMath .
Then , LaTeXMLMath is given by LaTeXMLEquation .
On the other hand , LaTeXMLMath is recovered from LaTeXMLMath via LaTeXMLEquation .
Note that LaTeXMLMath is uniquely determined up to equivalence defined in LaTeXMLCite .
We will use these notations throughout this section whenever a minimal action LaTeXMLMath of a compact group LaTeXMLMath on a factor LaTeXMLMath is given .
Let LaTeXMLMath be an infinite factor , and LaTeXMLMath be a Roberts action of LaTeXMLMath on LaTeXMLMath .
Then , the dual action of LaTeXMLMath is minimal if and only if LaTeXMLMath is irreducible for every irreducible representation LaTeXMLMath .
We set LaTeXMLMath , LaTeXMLMath , and LaTeXMLEquation where LaTeXMLMath is the Haar measure of LaTeXMLMath .
Let LaTeXMLMath be as above .
For LaTeXMLMath , we define LaTeXMLMath by LaTeXMLEquation .
Then , LaTeXMLMath has the following formal expansion ( see LaTeXMLCite ) : LaTeXMLEquation .
More precisely , the above summation converges in the GNS Hilbert space topology with respect to an LaTeXMLMath -invariant normal state , and LaTeXMLMath completely determines LaTeXMLMath .
Using this expansion , we can see that LaTeXMLMath if and only if LaTeXMLMath .
Therefore , LaTeXMLMath is minimal if and only if LaTeXMLMath does not contain identity for every non-trivial irreducible representation LaTeXMLMath .
However , the latter is equivalent to the statement that LaTeXMLMath is irreducible for very irreducible LaTeXMLMath thanks to the Frobenius reciprocity for the usual compact group representations and that of sectors LaTeXMLCite .
∎ Let LaTeXMLMath be an LaTeXMLMath -cocycle .
We denote by LaTeXMLMath the perturbation LaTeXMLMath of LaTeXMLMath by the cocycle LaTeXMLMath .
LaTeXMLMath is called stable if every LaTeXMLMath -cocycle is a coboundary .
The following statements are probably well-known among specialists , and actually some of them already exist in the literature .
However , since we can not find complete proofs in the literature , we provide them here for convenience of readers .
Let LaTeXMLMath be a minimal action of a compact group LaTeXMLMath on a factor LaTeXMLMath .
We denote by LaTeXMLMath the unique normal conditional expectation from LaTeXMLMath onto LaTeXMLMath obtained by taking average over LaTeXMLMath .
Then , the following hold : Every cocycle perturbation of LaTeXMLMath is again minimal .
If LaTeXMLMath is of type LaTeXMLMath , so is LaTeXMLMath .
LaTeXMLMath is stable in this case LaTeXMLCite .
If LaTeXMLMath is of type LaTeXMLMath , so is LaTeXMLMath .
LaTeXMLMath is stable in this case .
If LaTeXMLMath is of type LaTeXMLMath , LaTeXMLMath is of type LaTeXMLMath .
LaTeXMLMath is stable in this case ( more generally , LaTeXMLMath is stable whenever LaTeXMLMath is of type LaTeXMLMath ) .
LaTeXMLMath is of type LaTeXMLMath , LaTeXMLMath and LaTeXMLMath is of type LaTeXMLMath , if and only if the center of LaTeXMLMath contains the modular automorphism group LaTeXMLMath , for some faithful normal semifinite weight LaTeXMLMath on LaTeXMLMath ( which is actually a trace if this is the case ) .
LaTeXMLMath is not stable in this case .
( 1 ) .
First we claim that the second dual action LaTeXMLMath on LaTeXMLMath is minimal whenever LaTeXMLMath is so .
To show the claim , we may assume that LaTeXMLMath is infinite as usual .
Thus , we have a Hilbert space LaTeXMLMath in LaTeXMLMath for each LaTeXMLMath as before .
This implies that there exists a Hilbert space LaTeXMLMath in LaTeXMLMath such that the restriction of LaTeXMLMath to LaTeXMLMath is equivalent to the regular representation .
Therefore , LaTeXMLMath is conjugate to LaTeXMLMath , which shows the claim .
Let LaTeXMLMath be a cocycle perturbation of LaTeXMLMath .
Since LaTeXMLMath is an irreducible inclusion and there exists a projection LaTeXMLMath such that LaTeXMLEquation which shows that LaTeXMLMath is minimal .
( 2 ) , ( 3 ) .
The first part is easy .
Stability in these two cases ( and the case ( 4 ) as well ) follows from ( 1 ) and Connes ’ 2 by 2 matrix trick LaTeXMLCite .
( 4 ) .
We show that if LaTeXMLMath is of type II , LaTeXMLMath is either of type II or of type LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be a faithful normal semifinite trace on LaTeXMLMath .
Then , the centralizer LaTeXMLMath is an intermediate subfactor between LaTeXMLMath and LaTeXMLMath , and in particular LaTeXMLMath is a factor .
Therefore , the Connes spectrum LaTeXMLMath coincides with the Arveson spectrum LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
Thus , LaTeXMLMath is either of type II or type LaTeXMLMath , LaTeXMLMath depending on the period of the modular automorphism group LaTeXMLMath .
( 5 ) .
Note that part of the proof has been already done in the above .
Assume that LaTeXMLMath contains LaTeXMLMath , for some faithful normal semifinite weight LaTeXMLMath on LaTeXMLMath .
Then , the centralizer LaTeXMLMath is an intermediate subfactor of LaTeXMLMath .
Let LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath , which is a normal conditional expectation from LaTeXMLMath to LaTeXMLMath .
Since the restriction of LaTeXMLMath to LaTeXMLMath is nothing but LaTeXMLMath and it is semifinite , LaTeXMLMath is of type II , and so is LaTeXMLMath as well because LaTeXMLMath is trivial .
This finishes the proof of the first statement .
Now , we show that LaTeXMLMath is not stable in the above situation with type III LaTeXMLMath .
First we assume that LaTeXMLMath is of type LaTeXMLMath .
Let LaTeXMLMath be a non-zero finite projection , and LaTeXMLMath be an isometry satisfying LaTeXMLMath .
We set LaTeXMLMath , which is an LaTeXMLMath -cocycle .
It is easy to show that LaTeXMLMath is finite and LaTeXMLMath is not a coboundary .
When LaTeXMLMath is of type LaTeXMLMath , a similar construction identifying LaTeXMLMath with a corner of LaTeXMLMath works .
∎ In the above situation , assume that LaTeXMLMath is of type LaTeXMLMath with a faithful normal trace LaTeXMLMath .
Then , the type of LaTeXMLMath ( or the period of LaTeXMLMath ) is completely determined by LaTeXMLMath for the Roberts action LaTeXMLMath satisfying LaTeXMLMath , where LaTeXMLMath is the scalar Connes-Takesaki module introduced in Remark 4.6 .
Indeed , as in LaTeXMLCite , we can obtain the action of the modular automorphism group on LaTeXMLMath as follows : LaTeXMLEquation .
It is possible to show that the canonical extension gives continuous homomorphism from LaTeXMLMath to LaTeXMLMath in the LaTeXMLMath -topologies in the same spirit of the Cones-Takesaki ’ s proof of continuity of the fundamental homomorphism LaTeXMLCite .
However , we just mention here that when LaTeXMLMath is an action of a locally compact group LaTeXMLMath on LaTeXMLMath with a faithful normal invariant state , then it is particularly easy to show that LaTeXMLMath is continuous .
On the other hand , when LaTeXMLMath is a Roberts action of LaTeXMLMath on a factor LaTeXMLMath , Proposition 2.5 shows that LaTeXMLMath extends to LaTeXMLMath on LaTeXMLMath via the canonical extension .
Let LaTeXMLMath be an action of a compact group LaTeXMLMath on a factor LaTeXMLMath , and LaTeXMLMath be the normal conditional expectation from LaTeXMLMath onto LaTeXMLMath given by the average of LaTeXMLMath over LaTeXMLMath .
We regard LaTeXMLMath as a subalgebra of LaTeXMLMath identifying LaTeXMLMath with LaTeXMLMath , where LaTeXMLMath is a faithful normal semifinite weight on LaTeXMLMath .
Then , The fixed point subalgebra of LaTeXMLMath under LaTeXMLMath coincides with LaTeXMLMath .
When LaTeXMLMath is a minimal action that is the dual action of a Roberts action LaTeXMLMath of LaTeXMLMath on LaTeXMLMath .
Then , LaTeXMLMath is naturally isomorphic to LaTeXMLMath .
( 1 ) .
We set LaTeXMLMath .
Let LaTeXMLMath be the normal conditional expectation from LaTeXMLMath onto the fixed point subalgebra of LaTeXMLMath under LaTeXMLMath obtained by average of LaTeXMLMath over LaTeXMLMath .
Since LaTeXMLMath leaves LaTeXMLMath fixed for all LaTeXMLMath and LaTeXMLMath , the image of LaTeXMLMath coincides with LaTeXMLMath .
( 2 ) .
Let LaTeXMLMath be a dominant weight on LaTeXMLMath , and LaTeXMLMath .
For each LaTeXMLMath , LaTeXMLMath , we may and do assume that LaTeXMLMath is an invariant pair thanks to Lemma 2.3 , ( 3 ) .
Then , the modular automorphism group LaTeXMLMath acts on LaTeXMLMath trivially LaTeXMLCite and LaTeXMLMath leaves LaTeXMLMath invariant .
Therefore , LaTeXMLMath is given by LaTeXMLEquation .
Thus , ( 1 ) and LaTeXMLCite imply that LaTeXMLMath and LaTeXMLMath is the dual action of LaTeXMLMath .
∎ The purpose of this subsection is to show how a system of modular endomorphisms gives rise to a Roberts action , and to see the structure of its dual action .
There are two options to do so , ( of course they are essentially the same ) , and we start with the one using the minimal subgroup of a cocycle .
Let LaTeXMLMath be a type III factor and LaTeXMLMath be a countable family of modular endomorphisms of LaTeXMLMath .
By adding new endomorphisms to LaTeXMLMath if necessary , we may and do assume the following conditions : LaTeXMLMath is closed under taking a product of any two members in LaTeXMLMath .
For every pair LaTeXMLMath there exists LaTeXMLMath such that LaTeXMLMath .
If LaTeXMLMath is contained in LaTeXMLMath , there exists an endomorphism in LaTeXMLMath that is equivalent to LaTeXMLMath .
Every LaTeXMLMath has conjugate ( up to equivalence ) in LaTeXMLMath .
For each LaTeXMLMath we choose LaTeXMLMath satisfying LaTeXMLMath , and set LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the unitary group .
Thanks to Theorem 3.5 , we may and do assume that LaTeXMLMath is already a minimal cocycle .
We denote by LaTeXMLMath the minimal subgroup of LaTeXMLMath .
Let LaTeXMLMath be the representation of LaTeXMLMath on LaTeXMLMath that is the projection onto the LaTeXMLMath -component .
Then , we have LaTeXMLMath .
We denote by LaTeXMLMath the collection of LaTeXMLMath , LaTeXMLMath .
Now , we construct a Roberts action LaTeXMLMath of LaTeXMLMath on LaTeXMLMath sending LaTeXMLMath to LaTeXMLMath .
Thanks to Lemma 3.2 , ( 2 ) , we can choose an implementing system LaTeXMLMath for LaTeXMLMath giving LaTeXMLMath .
Let LaTeXMLMath be the linear span of LaTeXMLMath , which is a Hilbert space in LaTeXMLMath .
For LaTeXMLMath , we denote by LaTeXMLMath the linear span of elements of the form LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , which is naturally identified with LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is the fixed point subset of LaTeXMLMath under LaTeXMLMath , Theorem 3.5 , ( 4 ) implies that the above identification carries LaTeXMLMath onto LaTeXMLMath .
Therefore , we define a functor LaTeXMLMath setting LaTeXMLMath , LaTeXMLMath to be the corresponding element in LaTeXMLMath .
Note that since LaTeXMLMath maps every irreducible object to an irreducible object , Lemma 5.1 implies that the resulting crossed product gives irreducible inclusion of factors LaTeXMLMath , and the dual action LaTeXMLMath is minimal .
Another way to construct LaTeXMLMath from LaTeXMLMath is to apply the Doplicher-Roberts duality theorem LaTeXMLCite to LaTeXMLMath ( see also LaTeXMLCite ) .
For this purpose , we have to specify the permutation symmetry LaTeXMLMath for every pair LaTeXMLMath .
Indeed , we set LaTeXMLEquation .
Note that LaTeXMLMath belongs to LaTeXMLMath and it does not depend on the choice of implementing systems .
It is a routine work to show that LaTeXMLMath is a permutation symmetry for LaTeXMLMath , and that the resulting crossed product by LaTeXMLMath and the dual action constructed in LaTeXMLCite are the same as LaTeXMLMath and LaTeXMLMath above .
In general , LaTeXMLMath does not uniquely determine either the compact group LaTeXMLMath or the Roberts action LaTeXMLMath LaTeXMLCite .
( When LaTeXMLMath is abelian , the obstruction is nothing but the usual LaTeXMLMath obstruction for uniqueness of liftings of the LaTeXMLMath -kernel with a trivial 3 cocycle LaTeXMLCite , LaTeXMLCite . )
Whenever we deal with Roberts actions consisting of modular endomorphisms in the rest of this paper , we always assume that we make the above choice of LaTeXMLMath on the arrows ( or equivalently , the above choice of the permutation symmetries ) .
Let the notations be as above .
Then , LaTeXMLMath is generated by LaTeXMLMath and the relative commutant LaTeXMLMath , and LaTeXMLMath coincides with the center LaTeXMLMath of LaTeXMLMath .
The smooth flow of weights LaTeXMLMath of LaTeXMLMath is given by the skew product LaTeXMLMath , that is : LaTeXMLMath , LaTeXMLMath , and the flow is given by LaTeXMLEquation .
The factor map from LaTeXMLMath onto LaTeXMLMath corresponding to the inclusion LaTeXMLMath is the projection onto the first component .
The Connes-Takesaki module LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is given by LaTeXMLEquation .
Before starting the proof , we simplify the notation a little .
We may and do assume LaTeXMLMath thanks to the assumption ( iii ) .
For LaTeXMLMath , LaTeXMLMath , we simply denote LaTeXMLMath and LaTeXMLMath .
( 1 ) We take LaTeXMLMath and an orthonormal basis LaTeXMLMath as before , where LaTeXMLMath is transformed by LaTeXMLMath in the same way as the canonical basis of LaTeXMLMath .
Let LaTeXMLMath be the unique normal conditional expectation from LaTeXMLMath onto LaTeXMLMath , and LaTeXMLMath be the natural extension of LaTeXMLMath to LaTeXMLMath that is a normal conditional expectation from LaTeXMLMath onto LaTeXMLMath .
As in the proof of Lemma 5.1 , for LaTeXMLMath we set LaTeXMLEquation .
Then , LaTeXMLMath is in LaTeXMLMath if and only if LaTeXMLMath for every LaTeXMLMath and every LaTeXMLMath .
This is further equivalent to that there exists LaTeXMLMath such that LaTeXMLEquation for every LaTeXMLMath and every LaTeXMLMath .
Note that thanks to Proposition 4.2 , ( 4 ) and Lemma 5.4 , LaTeXMLMath commutes with every element in LaTeXMLMath .
Therefore , in the same way as in LaTeXMLCite , we can conclude that LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath .
Since LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath , it is also generated by LaTeXMLMath and LaTeXMLMath as well .
Thus , to prove LaTeXMLMath , it suffices to show that LaTeXMLMath generate a commutative von Neumann algebra .
Let LaTeXMLMath be the von Neumann algebra generated by LaTeXMLMath .
We show that there exists a LaTeXMLMath isomorphism from LaTeXMLMath onto LaTeXMLMath sending LaTeXMLMath to LaTeXMLMath .
For this , it suffices to check that both LaTeXMLMath and LaTeXMLMath have the same algebraic relations because we have LaTeXMLMath , where LaTeXMLEquation .
First , we show LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be isometries defined by LaTeXMLEquation .
LaTeXMLEquation Then , the definition of the crossed product by a Roberts action implies that we also have LaTeXMLEquation .
LaTeXMLEquation Thus , we get LaTeXMLEquation .
For the same reason , we have LaTeXMLEquation which implies LaTeXMLEquation .
Thus , LaTeXMLMath is abelian .
For the same reason again , there exist LaTeXMLMath such that LaTeXMLEquation .
LaTeXMLEquation which implies LaTeXMLEquation .
Note that thanks to the Peter-Weyl theorem , LaTeXMLMath is obtained by LaTeXMLEquation .
Thus , we have LaTeXMLEquation .
This shows that LaTeXMLEquation holds as well thanks to the Peter-Weyl theorem again .
Therefore , there exists a desired isomorphism from LaTeXMLMath onto LaTeXMLMath .
Let LaTeXMLMath be a faithful normal state of LaTeXMLMath whose restriction is given by the measure LaTeXMLMath , and we set LaTeXMLMath .
Then , for LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
This shows that LaTeXMLMath .
LaTeXMLMath acts on LaTeXMLMath as LaTeXMLEquation which shows that the flow is given by the skew product .
( 2 ) Since LaTeXMLMath acts on LaTeXMLMath trivially , we have LaTeXMLEquation .
This means that the Connes-Takesaki module of LaTeXMLMath is given as in the statement .
∎ Let LaTeXMLMath be a type III factor , LaTeXMLMath be a compact group , and LaTeXMLMath be a minimal cocycle with LaTeXMLMath .
For each LaTeXMLMath , we choose LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath be the subset of LaTeXMLMath generated by LaTeXMLMath and satisfying the above conditions ( i ) - ( iv ) .
We denote by LaTeXMLMath the above crossed product LaTeXMLMath and call it the skew product of LaTeXMLMath and LaTeXMLMath by the cocycle LaTeXMLMath .
We call LaTeXMLMath the dual action of the skew product .
Note that LaTeXMLMath depends only on the class LaTeXMLMath .
Our next task is to show the “ converse ” of the above theorem , and we need some preparation .
Let LaTeXMLMath be an action of a locally compact group LaTeXMLMath on a von Neumann algebra LaTeXMLMath , and LaTeXMLMath be the standard Hilbert space of LaTeXMLMath .
Let LaTeXMLMath be the modular conjugation of LaTeXMLMath acting on LaTeXMLMath as given by LaTeXMLCite .
Then , we have LaTeXMLEquation .
Let LaTeXMLMath be the modular conjugation of LaTeXMLMath on LaTeXMLMath .
Then , LaTeXMLMath is given by LaTeXMLEquation where LaTeXMLMath is the modular function of LaTeXMLMath and LaTeXMLMath is the canonical implementation of LaTeXMLMath on LaTeXMLMath .
Direct computation yields LaTeXMLEquation .
LaTeXMLEquation Thus , we get LaTeXMLEquation ∎ Let LaTeXMLMath be an irreducible inclusion of factors with a normal conditional expectation LaTeXMLMath .
We assume that LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath .
Then , the restriction of LaTeXMLMath to LaTeXMLMath is a conditional expectation onto LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath satisfies the trace property LaTeXMLMath .
Indeed , for LaTeXMLMath in the domain of LaTeXMLMath , we have LaTeXMLEquation which shows LaTeXMLMath .
Let LaTeXMLMath be a type LaTeXMLMath factor and LaTeXMLMath be a von Neumann algebra whose center includes LaTeXMLMath .
We assume that there exist an ergodic LaTeXMLMath -action LaTeXMLMath on LaTeXMLMath extending the flow of LaTeXMLMath , and a faithful normal conditional expectation LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath such that LaTeXMLMath , LaTeXMLMath and LaTeXMLMath for LaTeXMLMath .
Then , there exists a unique irreducible inclusion LaTeXMLMath of type LaTeXMLMath factors with a normal conditional expectation LaTeXMLMath such that LaTeXMLMath and LaTeXMLEquation .
First , we show uniqueness of LaTeXMLMath .
Let LaTeXMLMath be a von Neumann algebra including LaTeXMLMath with a unique conditional expectation LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath .
We assume that LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath .
Then , the linear span of LaTeXMLMath is dense in LaTeXMLMath by assumption .
For a faithful normal state LaTeXMLMath on LaTeXMLMath and LaTeXMLMath , LaTeXMLMath , we have LaTeXMLEquation .
This means that the structure of LaTeXMLMath is uniquely determined by LaTeXMLMath and LaTeXMLEquation .
On the other hand , LaTeXMLMath completely determine the original inclusion via the Takesaki duality theorem .
Therefore , such LaTeXMLMath as in the statement is unique if it exists .
We set LaTeXMLMath as in LaTeXMLCite , LaTeXMLCite .
More precisely , let LaTeXMLMath be the standard Hilbert space of LaTeXMLMath .
We introduce an inner product into the algebraic tensor product LaTeXMLMath by LaTeXMLEquation .
Let LaTeXMLMath be the completion of LaTeXMLMath , and LaTeXMLMath be the natural map from LaTeXMLMath to LaTeXMLMath .
LaTeXMLMath and LaTeXMLMath naturally act on the first and the second tensor components of LaTeXMLMath , and we denote by LaTeXMLMath the von Neumann algebra generated by LaTeXMLMath and LaTeXMLMath in LaTeXMLMath .
In terms of direct integral , LaTeXMLMath can be expressed as follows ( though we do not use it in the proof ) : Let LaTeXMLEquation .
LaTeXMLEquation be disintegration of LaTeXMLMath and LaTeXMLMath over LaTeXMLMath .
Then LaTeXMLMath is given by LaTeXMLEquation .
Thanks to the Connes-Takesaki relative commutant theorem and Lemma 5.7 , we have LaTeXMLMath .
We show LaTeXMLMath using this and applying LaTeXMLCite ( or alternatively , using the above direct integral expression ) .
To do so , we need to separate LaTeXMLMath and LaTeXMLMath by a type I von Neumann algebra .
Indeed , let LaTeXMLMath be the commutant of LaTeXMLMath in LaTeXMLMath .
Then , LaTeXMLMath naturally acts on LaTeXMLMath and we have LaTeXMLMath and LaTeXMLMath , where the commutant LaTeXMLMath is considered in LaTeXMLMath .
Applying LaTeXMLCite twice , we get LaTeXMLEquation .
We introduce a 1-parameter automorphism group LaTeXMLMath on LaTeXMLMath extending both LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath be the canonical implementation of LaTeXMLMath on LaTeXMLMath .
We set LaTeXMLEquation .
Then , LaTeXMLMath extends to a 1-parameter unitary group on LaTeXMLMath satisfying LaTeXMLEquation .
We define LaTeXMLMath to be the restriction of LaTeXMLMath to LaTeXMLMath .
Next , we show that there exists a faithful normal conditional expectation LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath satisfying LaTeXMLEquation .
Let LaTeXMLMath be a faithful normal state on LaTeXMLMath , and LaTeXMLMath be the cyclic and separating vector of LaTeXMLMath giving LaTeXMLMath .
We set LaTeXMLMath , and claim that LaTeXMLMath is cyclic and separating vector for LaTeXMLMath .
Indeed , LaTeXMLMath is clearly a cyclic vector .
To show that it is separating , we introduce a conjugation LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath is the modular conjugation of LaTeXMLMath .
It is easy to show that LaTeXMLMath extends to a conjugation such that LaTeXMLMath .
Since LaTeXMLMath is cyclic for LaTeXMLMath , LaTeXMLMath is a separating for LaTeXMLMath .
We denote by LaTeXMLMath the state of LaTeXMLMath defined by LaTeXMLMath , LaTeXMLMath .
Then , it is a routine work to show that LaTeXMLMath is the modular conjugation for LaTeXMLMath , and the modular automorphism group LaTeXMLMath is given by LaTeXMLEquation .
Thanks to Takesaki ’ s theorem LaTeXMLCite , there exists a LaTeXMLMath -preserving normal conditional expectation LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath .
Since LaTeXMLMath for LaTeXMLMath , LaTeXMLMath , LaTeXMLMath satisfies LaTeXMLMath .
We set LaTeXMLMath and LaTeXMLMath to be the restriction of LaTeXMLMath to LaTeXMLMath .
Since LaTeXMLMath satisfies LaTeXMLMath , LaTeXMLMath is a faithful normal conditional expectation from LaTeXMLMath onto LaTeXMLMath .
Therefore , LaTeXMLMath is of type III LaTeXMLCite .
Let LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath , where LaTeXMLMath is a faithful normal semifinite weight on LaTeXMLMath .
Then , thanks to the Landstad theorem LaTeXMLCite , LaTeXMLMath and LaTeXMLMath is the dual action of LaTeXMLMath .
We claim that LaTeXMLMath , and in particular LaTeXMLMath is a factor .
Indeed , we have LaTeXMLEquation .
Since LaTeXMLMath acts on LaTeXMLMath ergodically , we get LaTeXMLMath .
Instead of showing that LaTeXMLMath comes from the modular automorphism group directly , we show it for the dual action of LaTeXMLMath .
We define LaTeXMLMath , which is a faithful normal semifinite trace on LaTeXMLMath satisfying LaTeXMLMath .
Indeed , since LaTeXMLMath is a trace , LaTeXMLMath acts on LaTeXMLMath trivially .
On the other hand , since LaTeXMLMath is a faithful normal conditional expectation from LaTeXMLMath onto LaTeXMLMath and LaTeXMLMath , the restriction of LaTeXMLMath to LaTeXMLMath is the same as the restriction of LaTeXMLMath to LaTeXMLMath LaTeXMLCite , which is trivial as we saw before .
Thus , LaTeXMLMath is a trace with the scaling property LaTeXMLEquation .
This means that the dual action of LaTeXMLMath is the modular automorphism group for the dual weight LaTeXMLMath .
Thus , identifying LaTeXMLMath with LaTeXMLMath using the Takesaki duality theorem , we get LaTeXMLEquation .
LaTeXMLEquation Let LaTeXMLMath be the generator of LaTeXMLMath .
Then , we have LaTeXMLMath , which implies LaTeXMLMath .
Thus , we get LaTeXMLMath , which implies LaTeXMLMath and LaTeXMLMath .
∎ Let LaTeXMLMath be an action of a compact group LaTeXMLMath on a type LaTeXMLMath factor LaTeXMLMath .
If the kernel of LaTeXMLMath is trivial , then LaTeXMLMath is minimal .
The smooth flow of weight of the fixed point algebra LaTeXMLMath is the factor flow of LaTeXMLMath by LaTeXMLMath .
There exists a minimal cocycle LaTeXMLMath with LaTeXMLMath , unique up to equivalence , such that LaTeXMLMath is the skew product LaTeXMLMath , and LaTeXMLMath is the dual action of the skew product .
We use the same notation as in Lemma 5.4 .
Lemma A.1 implies that the smooth flow of weights of LaTeXMLMath is of the form LaTeXMLMath with the flow and LaTeXMLMath given by LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is a minimal cocycle with LaTeXMLMath .
According to the above splitting of LaTeXMLMath , we set LaTeXMLMath and LaTeXMLMath that is the fixed point subalgebra of LaTeXMLMath under LaTeXMLMath .
Note that the factor flow LaTeXMLMath and the class of LaTeXMLMath are uniquely determined by LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath is an equivariant copy of LaTeXMLMath , the Landstad ’ s type theorem for coactions LaTeXMLCite implies that LaTeXMLMath is the crossed product LaTeXMLMath by a coaction of LaTeXMLMath , where we use the fact that LaTeXMLMath is the fixed point subalgebra of LaTeXMLMath under LaTeXMLMath .
Moreover , since LaTeXMLMath commutes with LaTeXMLMath , the coaction is trivial and we actually have LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath are identified with LaTeXMLMath and LaTeXMLMath in the right-hand side respectively .
This implies LaTeXMLMath , LaTeXMLMath and LaTeXMLMath .
Since LaTeXMLMath acts on LaTeXMLMath ergodically , LaTeXMLMath is a factor .
If LaTeXMLMath were semifinite , LaTeXMLMath would be a coboundary , which would imply that LaTeXMLMath is trivial and LaTeXMLMath .
This is contradiction because LaTeXMLMath is of type III .
Therefore , LaTeXMLMath is a type III factor .
Now , Theorem 5.5 and Lemma 5.8 show that LaTeXMLMath is the skew product LaTeXMLMath .
Since the Galois group for LaTeXMLMath coincides with LaTeXMLMath LaTeXMLCite , LaTeXMLCite and every member LaTeXMLMath in the Galois group is specified by LaTeXMLMath , we conclude that LaTeXMLMath is the dual action of the skew product .
∎ If LaTeXMLMath is an AFD factor , then for every compact subgroup LaTeXMLMath of LaTeXMLMath , there exists a minimal action LaTeXMLMath of LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath .
LaTeXMLMath is unique up to conjugacy : namely if LaTeXMLMath also satisfies the condition , then there exists an automorphism LaTeXMLMath with a trivial Connes-Takesaki module such that LaTeXMLEquation .
This is an easy consequence of Theorem 5.5 , Theorem 5.9 , and the fact that AFD type III factors are determined by their flows LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
∎ As for the existence of a homomorphic lifting in the above corollary , a more general statement is known : there always exists a homomorphic lifting LaTeXMLMath from LaTeXMLMath to LaTeXMLMath for every AFD type III factor LaTeXMLMath LaTeXMLCite .
Now we characterize ( not necessarily finite index ) subfactors whose canonical endomorphism is decomposed into modular endomorphisms .
Let LaTeXMLMath be an irreducible inclusion of factors with a unique normal conditional expectation from LaTeXMLMath onto LaTeXMLMath .
We denote by LaTeXMLMath and LaTeXMLMath the basic extension and the dual operator valued weight from LaTeXMLMath to LaTeXMLMath respectively LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
We say that the inclusion LaTeXMLMath is discrete if the restriction of LaTeXMLMath on the relative commutant LaTeXMLMath is semifinite .
We say that a discrete inclusion is unimodular if LaTeXMLMath is invariant under LaTeXMLMath , where LaTeXMLMath is the modular conjugation of LaTeXMLMath ( see LaTeXMLCite for conditions equivalent to these . )
Let LaTeXMLMath be an inclusion of type LaTeXMLMath factors with a unique normal conditional expectation LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath .
Then , the following conditions are equivalent : LaTeXMLMath is a unimodular inclusion and the restriction of the canonical endomorphism LaTeXMLMath to LaTeXMLMath is decomposed into modular endomorphisms .
LaTeXMLMath is a discrete inclusion and LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath is an extension of the flow LaTeXMLMath with relatively discrete spectrum in the sense of Definition A.2 , and LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath .
There exists a compact group LaTeXMLMath , a faithful ergodic action LaTeXMLMath of LaTeXMLMath on a von Neumann algebra LaTeXMLMath , a minimal action LaTeXMLMath of LaTeXMLMath on a factor LaTeXMLMath such that the kernel of LaTeXMLMath is trivial and LaTeXMLEquation .
In the case ( 4 ) , LaTeXMLMath is determined by the cocycle LaTeXMLMath corresponding to the system of modular endomorphisms appearing in the decomposition of LaTeXMLMath , LaTeXMLMath is the skew product LaTeXMLMath , and LaTeXMLMath is the dual action of the skew product .
LaTeXMLMath and LaTeXMLMath are uniquely determined from LaTeXMLMath by Theorem A.4 .
When one ( and all ) of the above conditions holds , LaTeXMLMath is AFD if and only if LaTeXMLMath is AFD .
( 1 ) LaTeXMLMath ( 2 ) .
We assume that ( 1 ) holds and the irreducible decomposition of LaTeXMLMath is given by LaTeXMLEquation .
Thanks to LaTeXMLCite , the multiplicity LaTeXMLMath of LaTeXMLMath is always finite .
We set LaTeXMLEquation .
Then , the dimension of LaTeXMLMath is LaTeXMLMath and LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath LaTeXMLCite .
Let LaTeXMLMath be an orthonormal basis of LaTeXMLMath .
We choose a dominant weight LaTeXMLMath on LaTeXMLMath and assume that LaTeXMLMath is an invariant pair for every LaTeXMLMath as before .
Then , since LaTeXMLMath is unimodular , LaTeXMLCite imply that LaTeXMLMath is in the centralizer of LaTeXMLMath .
By assumption , there exists an implementing system LaTeXMLMath for LaTeXMLMath .
As LaTeXMLMath is in the centralizer of LaTeXMLMath , it commutes with LaTeXMLMath , and so LaTeXMLEquation .
Since LaTeXMLMath is generated LaTeXMLMath and LaTeXMLMath , it is also generated by LaTeXMLMath and LaTeXMLMath .
( 2 ) LaTeXMLMath ( 3 ) .
We assume ( 2 ) , and set LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath .
Then , LaTeXMLMath is a normal conditional expectation with the trace property LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be the basic extension of LaTeXMLMath .
As before , we regard LaTeXMLMath as a subalgebra of LaTeXMLMath identifying LaTeXMLMath with LaTeXMLMath .
Then , LaTeXMLMath is the basic extension of LaTeXMLMath with respect to LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
On the other hand , we consider the inclusion LaTeXMLMath acting on the standard Hilbert space LaTeXMLMath of LaTeXMLMath .
Then , the basic extension of this inclusion is LaTeXMLMath .
Let LaTeXMLMath be the canonical implementation of LaTeXMLMath , and LaTeXMLMath be the modular conjugation of LaTeXMLMath .
We use the same notation as in the proof of Lemma 5.8 , and identify LaTeXMLMath with LaTeXMLMath .
Note that LaTeXMLMath naturally acts on LaTeXMLMath through the action on the second tensor component .
In the same way as before , we get the following : LaTeXMLEquation where everything takes place in LaTeXMLMath .
Thus , LaTeXMLMath and LaTeXMLEquation .
LaTeXMLEquation where LaTeXMLMath is the fixed point subalgebra of LaTeXMLMath under the relative modular automorphism group LaTeXMLMath LaTeXMLCite .
Since LaTeXMLMath is discrete , LaTeXMLCite implies that LaTeXMLMath is a direct sum of matrix algebras and the restriction of LaTeXMLMath to LaTeXMLMath is still semifinite .
Therefore , if LaTeXMLMath is the center valued trace LaTeXMLMath of LaTeXMLMath , LaTeXMLMath is an extension of the flow LaTeXMLMath with relatively discrete spectrum .
Indeed , let LaTeXMLMath be a faithful normal state on LaTeXMLMath .
Then using the spatial derivative , we can show that the modular automorphism group of LaTeXMLMath is implemented by the modular operator LaTeXMLMath of LaTeXMLMath on LaTeXMLMath LaTeXMLCite .
However , since LaTeXMLMath is a trace and LaTeXMLMath is trivial , LaTeXMLMath is a trace as well .
This means that LaTeXMLMath holds with some density LaTeXMLMath affiliated with LaTeXMLMath .
Let LaTeXMLMath be the Jones projection for LaTeXMLMath , and LaTeXMLMath .
Then , LaTeXMLMath is considered as a field of scalar multiples of rank one projections in the central decomposition of LaTeXMLMath .
Since LaTeXMLEquation we actually have LaTeXMLMath .
( 3 ) LaTeXMLMath ( 4 ) .
We assume ( 3 ) .
Thanks to Theorem A.4 , there exists a compact group LaTeXMLMath , a minimal cocycle LaTeXMLMath with LaTeXMLMath , and a faithful ergodic action LaTeXMLMath of LaTeXMLMath on a von Neumann algebra LaTeXMLMath such that LaTeXMLEquation ( see Theorem A.4 for the notations ) .
Let LaTeXMLMath be the skew product LaTeXMLMath and LaTeXMLMath be the dual action .
We set LaTeXMLMath .
Then , thanks to ergodicity of LaTeXMLMath , LaTeXMLMath is an irreducible inclusion of factors .
Let LaTeXMLMath be the unique LaTeXMLMath -invariant trace LaTeXMLCite .
Then , the restriction of LaTeXMLMath to LaTeXMLMath gives a normal conditional expectation from LaTeXMLMath onto LaTeXMLMath .
Thanks to Lemma 5.8 , to prove the statement it suffices to show that LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath , and LaTeXMLEquation .
Lemma 5.4 and Theorem 5.9 show that LaTeXMLMath is actually given by LaTeXMLEquation where LaTeXMLMath is the left regular representation .
Thus , we have LaTeXMLEquation which shows LaTeXMLMath .
Note that LaTeXMLMath acts on LaTeXMLMath trivially and it acts on LaTeXMLMath as in Theorem 5.9 .
Now we assume that LaTeXMLMath acts on LaTeXMLMath , where LaTeXMLMath is the standard Hilbert space of LaTeXMLMath and LaTeXMLMath .
Let LaTeXMLMath and LaTeXMLMath be the canonical implementations of LaTeXMLMath on LaTeXMLMath and LaTeXMLMath on LaTeXMLMath respectively .
Then , LaTeXMLMath is given by LaTeXMLEquation .
We introduce a unitary operator LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation which satisfies LaTeXMLMath Let LaTeXMLMath be the restriction of LaTeXMLMath on LaTeXMLMath , which is an automorphism of LaTeXMLMath .
Then , the above relation implies LaTeXMLEquation .
LaTeXMLEquation For LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath we have LaTeXMLEquation .
Thus , we get LaTeXMLEquation ( 4 ) LaTeXMLMath ( 1 ) .
We assume ( 4 ) and set LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be the unique LaTeXMLMath -invariant trace on LaTeXMLMath , LaTeXMLMath be the GNS Hilbert space of LaTeXMLMath , and LaTeXMLMath be the natural map from LaTeXMLMath into LaTeXMLMath .
We denote by LaTeXMLMath the canonical implementation of LaTeXMLMath on LaTeXMLMath .
For each LaTeXMLMath , we denote by LaTeXMLMath the spectral subspace corresponding to LaTeXMLMath , which is the image of the map LaTeXMLEquation .
It is known that the multiplicity of LaTeXMLMath is finite LaTeXMLCite and there exists an orthonormal basis LaTeXMLMath , LaTeXMLMath , LaTeXMLMath with respect to the inner product of LaTeXMLMath such that LaTeXMLEquation .
Since LaTeXMLMath is given by the average of LaTeXMLMath over LaTeXMLMath , the Peter-Weyl theorem implies LaTeXMLEquation .
Note that LaTeXMLMath form an orthonormal basis of LaTeXMLMath .
Though expansion of LaTeXMLMath with this basis does not converge in weak topology in general , we can still show that LaTeXMLMath generate LaTeXMLMath using the same argument as in LaTeXMLCite .
The point of the argument is that when a von Neumann subalgebra is globally invariant under the modular automorphism group of a state , the corresponding subspace in the GNS subspace determines the subalgebra .
We use this principle in the rest of the proof without mentioning it .
Let LaTeXMLMath be the unique normal conditional expectation from LaTeXMLMath onto LaTeXMLMath , LaTeXMLMath be a faithful normal state on LaTeXMLMath , and LaTeXMLMath .
We denote by LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath the GNS Hilbert space of LaTeXMLMath , the natural map from LaTeXMLMath into LaTeXMLMath , and the canonical implementation of LaTeXMLMath on LaTeXMLMath respectively .
We regard the subspace of LaTeXMLMath -invariant vectors in LaTeXMLMath as the GNS Hilbert space LaTeXMLMath of the restriction of LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be the Roberts action of LaTeXMLMath on LaTeXMLMath whose dual action is LaTeXMLMath , and LaTeXMLMath , LaTeXMLMath be an orthonormal basis of LaTeXMLMath as before .
Thanks to Theorem 5.9 , LaTeXMLMath is a modular endomorphism for all LaTeXMLMath .
Note that LaTeXMLMath is globally invariant under the modular automorphism group LaTeXMLMath LaTeXMLCite .
Let LaTeXMLEquation .
Thanks to LaTeXMLCite , if LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath , LaTeXMLMath is a discrete inclusion .
Now we set LaTeXMLEquation .
Then , LaTeXMLMath are isometries with mutually orthogonal ranges .
The definition of the crossed product by a Roberts action implies LaTeXMLEquation .
Therefore , direct computation using the orthonormal basis LaTeXMLMath yields LaTeXMLEquation .
Thus , LaTeXMLMath is generated by LaTeXMLMath and LaTeXMLMath , and so LaTeXMLMath is a discrete inclusion .
Let LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath , which is a conditional expectation from LaTeXMLMath onto LaTeXMLMath .
Then , we have LaTeXMLEquation .
This shows that the inclusion is unimodular LaTeXMLCite .
When LaTeXMLMath is AFD , LaTeXMLMath is AFD and LaTeXMLMath is AFD thanks to the proof of Lemma 5.8 and Theorem A.4 .
Therefore , LaTeXMLMath is AFD .
Since there exists a conditional expectation from LaTeXMLMath onto LaTeXMLMath , the converse also holds LaTeXMLCite .
∎ In the above , if LaTeXMLMath is commutative , LaTeXMLMath in ( 4 ) is commutative , and there exists a unique closed subgroup LaTeXMLMath such that LaTeXMLMath where LaTeXMLMath action is given by the right translation .
Thus , LaTeXMLEquation .
This characterizes an inclusion coming from a ( commutative ) ergodic extension of the flow of weights .
For related topics , see LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite , LaTeXMLCite .
Under the condition that the kernel of LaTeXMLMath is trivial , ( and LaTeXMLMath does not contain non-trivial normal subgroups , which is regarded as a trivial constraint ) , LaTeXMLMath are unique up to conjugacy .
However , if we drop this condition , there are essentially different description of LaTeXMLMath in terms of a group-subgroup subfactor LaTeXMLCite , LaTeXMLCite .
Another easy but interesting case is when LaTeXMLMath is a matrix algebra , that is , LaTeXMLMath is a Wassermann ’ s type subfactor .
In this case , LaTeXMLMath and LaTeXMLMath have the common flow and graph change occurs ( c.f .
LaTeXMLCite ) .
In this subsection , we show how the Connes-Takesaki modules of endomorphisms give constraints to minimal actions of compact groups , in particular semisimple compact Lie groups .
Let LaTeXMLMath be a compact group and LaTeXMLMath be a Roberts action of LaTeXMLMath on a factor LaTeXMLMath such that LaTeXMLMath is irreducible and has a Connes-Takesaki module for every LaTeXMLMath .
We denote LaTeXMLMath for simplicity .
Then , thanks to Proposition 4.2 , LaTeXMLMath form a group , which is denoted by LaTeXMLMath .
We treat LaTeXMLMath as a discrete group , and we denote by LaTeXMLMath its dual group .
Let LaTeXMLMath be a compact group and LaTeXMLMath be a Roberts action of LaTeXMLMath on a factor LaTeXMLMath such that LaTeXMLMath is irreducible for every LaTeXMLMath .
Then , If LaTeXMLMath is connected , LaTeXMLMath has a Connes-Takesaki module for every LaTeXMLMath .
If LaTeXMLMath has a Connes-Takesaki module for every LaTeXMLMath , then there exists a continuous isomorphism LaTeXMLMath from LaTeXMLMath into the center LaTeXMLMath of LaTeXMLMath satisfying LaTeXMLEquation ( 1 ) .
Suppose that there exists LaTeXMLMath such that LaTeXMLMath does not have a Connes-Takesaki module .
Let LaTeXMLMath , and we consider the Loi-Hamachi-Kosaki decomposition of LaTeXMLMath LaTeXMLCite , LaTeXMLCite , LaTeXMLCite as follows : LaTeXMLEquation where LaTeXMLMath and LaTeXMLMath are determined by the conditions LaTeXMLEquation .
LaTeXMLEquation Since LaTeXMLMath does not have a Connes-Takesaki module , either LaTeXMLMath or LaTeXMLMath occurs .
We may assume LaTeXMLMath by considering LaTeXMLMath if necessary .
Then , LaTeXMLMath contains the the canonical endomorphism LaTeXMLMath for LaTeXMLMath .
Note that the dual inclusion LaTeXMLMath satisfies the condition of Theorem 5.11 and LaTeXMLMath is a finite to one extension of the flow LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
Thus , Theorem 5.11 implies that LaTeXMLMath generates a Roberts action of the dual LaTeXMLMath of a finite group LaTeXMLMath , which is a subset of LaTeXMLMath .
We regard LaTeXMLMath as a subset of LaTeXMLMath .
We define a closed normal subgroup LaTeXMLMath by LaTeXMLEquation .
Since LaTeXMLMath is a finite set of irreducible representations closed under complex conjugate and irreducible decomposition of products of any members in LaTeXMLMath , the quotient group LaTeXMLMath is finite LaTeXMLCite .
However , this is contradiction because LaTeXMLMath is connected .
Therefore , LaTeXMLMath has a Connes-Takesaki module for every LaTeXMLMath .
( 2 ) .
For LaTeXMLMath , we consider a family of unitary operators LaTeXMLEquation .
Then , thanks to Theorem 4.2 and the Tannaka duality theorem LaTeXMLCite , there exists a unique element LaTeXMLMath such that LaTeXMLMath holds .
Since LaTeXMLMath is a scalar for every irreducible LaTeXMLMath , LaTeXMLMath is in the center LaTeXMLMath .
LaTeXMLMath is clearly a homomorphism from LaTeXMLMath into LaTeXMLMath , which is injective as the set of irreducible representations separates points of LaTeXMLMath .
LaTeXMLMath is continuous because the topology of LaTeXMLMath is the same as that induced by the weak topology of the image of LaTeXMLMath by the direct sum representation of all members in LaTeXMLMath .
∎ Let LaTeXMLMath be a minimal action of a compact connected semisimple Lie group LaTeXMLMath on a type LaTeXMLMath factor LaTeXMLMath .
Then , If LaTeXMLMath is of type LaTeXMLMath , so is LaTeXMLMath .
If LaTeXMLMath is of type LaTeXMLMath , LaTeXMLMath , there exists a positive integer LaTeXMLMath such that LaTeXMLMath is of type LaTeXMLMath .
Let LaTeXMLMath be the Roberts action of LaTeXMLMath on LaTeXMLMath whose dual action is LaTeXMLMath , then LaTeXMLMath .
If LaTeXMLMath is of type LaTeXMLMath , so is LaTeXMLMath .
( The flow of LaTeXMLMath and LaTeXMLMath could be very much different in this case . )
Since the center of a semisimple Lie group is a finite group , Proposition 5.2 , ( 2 ) implies that LaTeXMLMath is of type III .
As before , we regard LaTeXMLMath as a subalgebra of LaTeXMLMath using a unique conditional expectation LaTeXMLMath .
Let LaTeXMLMath be the Roberts action of LaTeXMLMath on LaTeXMLMath whose dual action is LaTeXMLMath .
We first assume that LaTeXMLMath is of type LaTeXMLMath , LaTeXMLMath .
Since every non-trivial modular endomorphism of LaTeXMLMath is a composition of an inner automorphism and a modular automorphism in this case , no non-trivial modular endomorphisms appear in LaTeXMLMath because a connected semisimple Lie group has no non-trivial 1 dimensional representation .
Thus , the same computation as in the proof of Theorem 5.5 implies LaTeXMLEquation .
Thanks to Theorem 5.13 , LaTeXMLMath has a Connes-Takesaki module for every LaTeXMLMath , which implies LaTeXMLEquation .
This shows that if LaTeXMLMath is of type LaTeXMLMath , so is LaTeXMLMath .
When LaTeXMLMath is of type LaTeXMLMath , LaTeXMLMath , we have LaTeXMLMath .
Since LaTeXMLMath is a finite group , LaTeXMLMath is finite and its dual group LaTeXMLMath is a finite subgroup of LaTeXMLMath .
Thus , LaTeXMLMath is a finite cyclic group and so is LaTeXMLMath .
Now we assume that LaTeXMLMath is of type LaTeXMLMath .
Then , in the same way as above we can show LaTeXMLEquation .
Since LaTeXMLMath is a finite group , LaTeXMLMath is a non-transitive ergodic flow and so is LaTeXMLMath .
Therefore , LaTeXMLMath is of type LaTeXMLMath .
∎ We give a simple example of a minimal action of a compact group with non-trivial LaTeXMLMath .
Let LaTeXMLMath be a positive number with LaTeXMLMath , and set LaTeXMLMath .
We define a state of the 3 by 3 matrix algebra LaTeXMLMath by LaTeXMLEquation where LaTeXMLMath means a diagonal matrix .
We set LaTeXMLEquation which is a Powers factor of type LaTeXMLMath .
For LaTeXMLMath , we define LaTeXMLEquation .
Then , LaTeXMLMath is a minimal action of LaTeXMLMath .
It is easy to show LaTeXMLMath , which implies that LaTeXMLMath is of type LaTeXMLMath .
Note that the restriction of LaTeXMLMath to LaTeXMLMath induces a minimal action of LaTeXMLMath , whose center is trivial .
Thus , Corollary 5.14 implies that LaTeXMLMath and LaTeXMLMath has the common flow and LaTeXMLMath is of type LaTeXMLMath .
Corollary 5.14 again implies that the endomorphism in the Roberts action corresponding to the spin LaTeXMLMath representation has a non-trivial Connes-Takesaki module .
In this appendix , we collect some results from ergodic theory , ( or its simple non-commutative generalization ) , used in the main body of the present notes .
We provide proofs here because we could not find appropriate references , though some ( or all ) of them are probably well-known to specialists .
For a locally compact group LaTeXMLMath , we say that LaTeXMLMath is a LaTeXMLMath -space if LaTeXMLMath is a standard Borel space , LaTeXMLMath is a probability measure on LaTeXMLMath , and LaTeXMLMath acts on LaTeXMLMath as a non-singular Borel transformation group .
We say that a LaTeXMLMath -action is faithful if the corresponding action on LaTeXMLMath is faithful .
We say that two LaTeXMLMath -spaces LaTeXMLMath and LaTeXMLMath are isomorphic if the corresponding LaTeXMLMath -actions on the von Neumann algebras ( or equivalently , measure algebras ) LaTeXMLMath and LaTeXMLMath are conjugate .
For LaTeXMLMath and a Borel subset LaTeXMLMath , we use the notation LaTeXMLMath .
Let LaTeXMLMath be a locally compact group and LaTeXMLMath be an ergodic LaTeXMLMath -space .
We assume that a compact group LaTeXMLMath faithfully acts on LaTeXMLMath commuting with LaTeXMLMath .
We use the notation with LaTeXMLMath acting on LaTeXMLMath from right and LaTeXMLMath acting on LaTeXMLMath from left .
Let LaTeXMLMath be the factor space of LaTeXMLMath by LaTeXMLMath .
Then , there exists a minimal LaTeXMLMath -valued cocycle LaTeXMLMath with LaTeXMLMath such that LaTeXMLMath is isomorphic to the skew product LaTeXMLMath , that is , under this isomorphism , the factor map from LaTeXMLMath onto LaTeXMLMath corresponds to the projection onto the first component , and the LaTeXMLMath -action corresponds to the following action : LaTeXMLEquation .
LaTeXMLEquation LaTeXMLMath is unique up to equivalence .
Considering the Gelfand spectrum of an appropriate weakly dense separable LaTeXMLMath -subalgebra of LaTeXMLMath , we may and do assume that LaTeXMLMath is a compact metric space , on which LaTeXMLMath acts continuously , and that LaTeXMLMath is invariant under LaTeXMLMath .
Since LaTeXMLMath is compact , every LaTeXMLMath -orbit is closed and there exists a Borel subset LaTeXMLMath that meets each LaTeXMLMath -orbit exactly once ( see , for example , LaTeXMLCite ) .
Let LaTeXMLMath be the set of all closed subgroups of LaTeXMLMath .
Then , LaTeXMLMath has a Polish space structure such that LaTeXMLMath continuously acts on LaTeXMLMath by LaTeXMLMath , LaTeXMLMath , LaTeXMLMath LaTeXMLCite .
We denote by LaTeXMLMath the class of LaTeXMLMath in the quotient space of LaTeXMLMath .
For each LaTeXMLMath , let LaTeXMLMath be the stabilizer subgroup of LaTeXMLMath .
Then , thanks to LaTeXMLCite , the map LaTeXMLEquation is a Borel map satisfying LaTeXMLMath for LaTeXMLMath .
Thanks to the ergodicity of the LaTeXMLMath -action , there exist LaTeXMLMath and LaTeXMLMath -invariant Borel null set LaTeXMLMath such that for every LaTeXMLMath , we have LaTeXMLMath .
We set LaTeXMLMath .
Let LaTeXMLMath be the normalizer subgroup of LaTeXMLMath in LaTeXMLMath .
Choosing a Borel cross section from LaTeXMLMath to LaTeXMLMath , we have a Borel map LaTeXMLMath such that LaTeXMLMath .
We set LaTeXMLEquation .
Then , LaTeXMLMath is a Borel subset of LaTeXMLMath that meets each LaTeXMLMath -orbit exactly once and LaTeXMLMath for all LaTeXMLMath .
Choosing a Borel cross section LaTeXMLMath , we set LaTeXMLEquation .
Thanks to LaTeXMLCite , LaTeXMLMath is a Borel isomorphism from LaTeXMLMath onto LaTeXMLMath .
Since LaTeXMLMath is LaTeXMLMath -invariant , LaTeXMLMath is of the form LaTeXMLMath , where LaTeXMLMath is the ( normalized ) Haar measure of LaTeXMLMath .
We introduce a LaTeXMLMath -action on LaTeXMLMath from the action on LaTeXMLMath through LaTeXMLMath .
When we regard LaTeXMLMath as the factor space of LaTeXMLMath by the LaTeXMLMath -action , we denote the action of LaTeXMLMath on LaTeXMLMath by LaTeXMLMath , LaTeXMLMath .
Then , for each LaTeXMLMath and LaTeXMLMath , there exists a unique LaTeXMLMath such that LaTeXMLMath .
Using a Borel cross section from LaTeXMLMath to LaTeXMLMath again , we can take LaTeXMLMath to be a Borel map from LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath be the quotient map .
Commutativity of the LaTeXMLMath -action and LaTeXMLMath -action implies that for each fixed LaTeXMLMath , LaTeXMLMath takes its values in LaTeXMLMath almost everywhere .
Moreover , LaTeXMLMath is an LaTeXMLMath -valued cocycle and LaTeXMLEquation .
This implies that for every subset LaTeXMLMath , LaTeXMLMath is LaTeXMLMath -invariant .
Thus , thanks to the ergodicity of the LaTeXMLMath -action , we conclude that whenever LaTeXMLMath is an open subset of LaTeXMLMath , LaTeXMLEquation which implies LaTeXMLMath , that is , LaTeXMLMath is a normal subgroup of LaTeXMLMath .
Since the LaTeXMLMath -action is faithful , we get LaTeXMLMath .
Uniqueness of LaTeXMLMath up to equivalence is obvious .
∎ In LaTeXMLCite , Zimmer introduced the notion of ergodic extensions with relatively discrete spectrum .
The following is a slight generalization of this notion to the non-commutative case .
Let LaTeXMLMath be locally compact group and LaTeXMLMath be an ergodic LaTeXMLMath -space .
Let LaTeXMLMath be a von Neumann algebra with an ergodic LaTeXMLMath -action LaTeXMLMath such that the center of LaTeXMLMath equivariantly includes LaTeXMLMath , that is LaTeXMLEquation .
Let LaTeXMLMath be the canonical implementation of LaTeXMLMath on the standard Hilbert space for LaTeXMLMath .
We say that LaTeXMLMath is an extension of LaTeXMLMath with relatively discrete spectrum if the von Neumann algebra LaTeXMLMath is decomposed into a direct sum of type I factors , and for each minimal projection LaTeXMLMath , LaTeXMLMath is bounded , where LaTeXMLMath is the ( non-normalized ) center valued trace on LaTeXMLMath , ( in other words , the restriction of LaTeXMLMath to LaTeXMLMath is semifinite ) .
Note that the above definition coincides with Zimmer ’ s one LaTeXMLCite when LaTeXMLMath is commutative ( although the two definitions appear different ) .
Let LaTeXMLMath be a locally compact group , LaTeXMLMath be a compact group , LaTeXMLMath be a LaTeXMLMath -space , and LaTeXMLMath be a cocycle .
If LaTeXMLMath is an action of LaTeXMLMath on a von Neumann algebra LaTeXMLMath standardly acting on a Hilbert space LaTeXMLMath , then there exists an action LaTeXMLMath of LaTeXMLMath on LaTeXMLMath such that for all LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLEquation holds for almost every LaTeXMLMath .
Let LaTeXMLMath be the canonical implementation of LaTeXMLMath on LaTeXMLMath .
We introduce a unitary LaTeXMLMath , LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation .
Then , LaTeXMLMath is a representation of LaTeXMLMath , which is a Borel map from LaTeXMLMath to the unitary group of LaTeXMLMath with respect to the Borel structure coming from the weak ( and strong ) topology .
Since any unitary Borel representation of a locally compact group is continuous , ( or more strongly , any Borel homomorphism between Polish groups is continuous LaTeXMLCite ) , so is LaTeXMLMath .
LaTeXMLMath is defined by the restriction of LaTeXMLMath to LaTeXMLMath .
∎ The proof of ( 1 ) of the following theorem is a straightforward generalization of Zimmer ’ s argument in LaTeXMLCite except for the proof of uniqueness , which does not exist there .
( Zimmer ’ s comment about uniqueness in LaTeXMLCite does not seem to be relevant . )
Let LaTeXMLMath be a locally compact group , LaTeXMLMath be an ergodic LaTeXMLMath -space , and LaTeXMLMath be an action of LaTeXMLMath on a von Neumann algebra LaTeXMLMath .
We assume that LaTeXMLMath is an extension of LaTeXMLMath with relatively discrete spectrum .
Then , There exist a compact group LaTeXMLMath , a faithful ergodic action LaTeXMLMath of LaTeXMLMath on a von Neumann algebra LaTeXMLMath , and a minimal cocycle LaTeXMLMath with LaTeXMLMath such that LaTeXMLEquation that is , there exists an isomorphism from LaTeXMLMath onto LaTeXMLMath that intertwines LaTeXMLMath and LaTeXMLMath , and is identity on LaTeXMLMath .
LaTeXMLMath is always an injective von Neumann algebra , and there exists a unique faithful normal conditional expectation LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath satisfying LaTeXMLMath , LaTeXMLMath and LaTeXMLMath , LaTeXMLMath .
LaTeXMLMath satisfying the condition of ( 1 ) are unique in the following sense : if LaTeXMLMath satisfy the same condition , then there exist a group isomorphism LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath and an isomorphism LaTeXMLMath from LaTeXMLMath onto LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath are equivalent and LaTeXMLMath , LaTeXMLMath .
( 1 ) .
By assumption , we have a direct sum decomposition LaTeXMLEquation where LaTeXMLMath is a type I factor .
We take a system of matrix units LaTeXMLMath of LaTeXMLMath with LaTeXMLMath either a natural number or possibly infinite such that LaTeXMLMath form a partition of unity consisting of minimal projections of LaTeXMLMath .
Since LaTeXMLMath , LaTeXMLMath normalizes LaTeXMLMath , it does LaTeXMLMath as well , and we set LaTeXMLMath to be the restriction of LaTeXMLMath to LaTeXMLMath .
Then , we have LaTeXMLMath and in particular , LaTeXMLEquation .
Thus , by ergodicity LaTeXMLMath is a constant , say LaTeXMLMath .
In what follows , when we claim that “ some statement involving LaTeXMLMath and LaTeXMLMath holds for almost all LaTeXMLMath ” , we always mean that for each fixed LaTeXMLMath , there exists a conull set LaTeXMLMath such that the statement holds for all LaTeXMLMath .
The annoying fact that LaTeXMLMath varies according to LaTeXMLMath is usually taken care of by Mackey ’ s argument in LaTeXMLCite .
We consider the disintegration of LaTeXMLMath over LaTeXMLMath LaTeXMLEquation .
Then , LaTeXMLMath means that the dimension of LaTeXMLMath is LaTeXMLMath for almost all LaTeXMLMath .
We take LaTeXMLMath , LaTeXMLMath such that for almost all LaTeXMLMath LaTeXMLMath is an orthonormal basis of LaTeXMLMath .
Since LaTeXMLMath is a unitary satisfying LaTeXMLMath for LaTeXMLMath , for LaTeXMLMath and LaTeXMLMath we have , on one hand , LaTeXMLEquation and on the other hand , LaTeXMLEquation .
This implies , LaTeXMLEquation for almost all LaTeXMLMath .
Thus , LaTeXMLEquation is an orthonormal basis of LaTeXMLMath for almost all LaTeXMLMath .
We set LaTeXMLEquation .
Then , LaTeXMLMath is an element of LaTeXMLMath .
We define a compact group LaTeXMLMath and a cocycle LaTeXMLMath by LaTeXMLEquation .
LaTeXMLEquation Changing the system LaTeXMLMath if necessary , we may and do assume that LaTeXMLMath is minimal .
We denote by LaTeXMLMath the minimal subgroup of LaTeXMLMath .
Note that LaTeXMLMath is minimal for all LaTeXMLMath and that if LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are not equivalent ( otherwise , LaTeXMLMath would be bigger ) .
Let LaTeXMLMath and LaTeXMLMath be Hilbert spaces with basis LaTeXMLMath and LaTeXMLMath respectively .
We set LaTeXMLEquation .
Since LaTeXMLMath is an orthonormal basis of LaTeXMLMath for almost all LaTeXMLMath , we identify LaTeXMLMath with LaTeXMLMath , and LaTeXMLMath with LaTeXMLMath .
There exists a natural embedding of LaTeXMLMath in the group LaTeXMLMath of all unitaries LaTeXMLMath as those unitaries acting “ only on LaTeXMLMath , LaTeXMLMath ” .
Through this embedding , we also regard LaTeXMLMath as a subgroup of LaTeXMLMath .
Then , we have LaTeXMLEquation for almost all LaTeXMLMath .
Let LaTeXMLEquation be the disintegration of LaTeXMLMath over LaTeXMLMath .
Then , the map LaTeXMLMath is measurable with respect to the Effros Borel structure of the set of von Neumann algebras on LaTeXMLMath LaTeXMLCite , LaTeXMLCite .
For LaTeXMLMath , ( A.1 ) implies LaTeXMLEquation for almost all LaTeXMLMath .
Since LaTeXMLMath has countable generators ( we assume that Hilbert spaces are separable throughout this paper ) , this implies LaTeXMLEquation for almost all LaTeXMLMath .
Using the Effros Borel structure in the same way as in the proof of LaTeXMLCite and passing to an equivalent cocycle if necessary ( and changing LaTeXMLMath and LaTeXMLMath on a null set ) , we may and do assume that there exists a von Neumann algebra LaTeXMLMath such that LaTeXMLMath and LaTeXMLMath normalizes LaTeXMLMath for all LaTeXMLMath .
Since LaTeXMLMath is a minimal cocycle with LaTeXMLMath , LaTeXMLMath normalizes LaTeXMLMath , and so we define LaTeXMLMath to be the restriction of LaTeXMLMath , LaTeXMLMath to LaTeXMLMath .
Then , ( A.2 ) implies LaTeXMLMath .
Since LaTeXMLMath is ergodic , so is LaTeXMLMath .
( 2 ) .
Thanks to LaTeXMLCite , LaTeXMLMath is injective and so is LaTeXMLMath .
Let LaTeXMLMath be a normal conditional expectation from LaTeXMLMath onto LaTeXMLMath .
Then , since LaTeXMLMath is the standard Hilbert space of LaTeXMLMath , there exists LaTeXMLMath whose vector state is LaTeXMLMath , where LaTeXMLMath means the state of LaTeXMLMath corresponding to the measure LaTeXMLMath .
Thus , for all LaTeXMLMath and LaTeXMLMath we have LaTeXMLEquation .
We introduce a measurable field of normal states LaTeXMLMath of LaTeXMLMath by LaTeXMLEquation .
Then , ( A.4 ) implies LaTeXMLMath for almost all LaTeXMLMath .
Moreover , if LaTeXMLMath commutes with LaTeXMLMath , we get LaTeXMLEquation for almost all LaTeXMLMath .
Let LaTeXMLMath be the set of all normal states of LaTeXMLMath , which is a Polish space in the norm topology .
Since LaTeXMLMath is continuous in LaTeXMLMath -topology , LaTeXMLMath acts on LaTeXMLMath as a continuous transformation group .
Applying the same argument as above to LaTeXMLMath and passing to an equivalent cocycle ( and changing LaTeXMLMath and LaTeXMLMath on a null set ) , we may assume that there exists LaTeXMLMath such that LaTeXMLMath for all LaTeXMLMath and LaTeXMLMath holds .
Thus , LaTeXMLMath is a LaTeXMLMath -invariant state , which is unique because of ergodicity of LaTeXMLMath .
Thus , LaTeXMLMath is uniquely determined .
LaTeXMLMath has the trace property as LaTeXMLMath is a trace thanks to LaTeXMLCite .
( 3 ) .
We assume that LaTeXMLMath are also given by LaTeXMLMath , LaTeXMLMath .
Let LaTeXMLMath be the unique LaTeXMLMath -invariant trace , LaTeXMLMath be the GNS Hilbert space of LaTeXMLMath , and LaTeXMLMath be the canonical implementation of LaTeXMLMath , LaTeXMLMath .
Then , we have LaTeXMLMath and LaTeXMLMath is given by LaTeXMLEquation .
Since LaTeXMLMath is a compact group , we have the irreducible decomposition LaTeXMLEquation where LaTeXMLMath is the multiplicity space , which may be zero .
Since the LaTeXMLMath -action on LaTeXMLMath is ergodic and LaTeXMLMath is a minimal cocycle with LaTeXMLMath we have LaTeXMLEquation .
Thus , we identify LaTeXMLMath with the subset of LaTeXMLMath consisting of the irreducibles appearing in LaTeXMLMath , and identify LaTeXMLMath with LaTeXMLMath when LaTeXMLMath is identified with LaTeXMLMath .
In what follows , we abuse notation and use only LaTeXMLMath , instead of LaTeXMLMath , if there is no possibility of confusion .
We set LaTeXMLEquation and regard it as a subgroup of the unitary group LaTeXMLMath as before .
Since LaTeXMLMath is faithful , we identify LaTeXMLMath with LaTeXMLMath .
Let LaTeXMLMath be the measurable field of unitaries describing the identity map on LaTeXMLMath with respect to the two distinct splitting LaTeXMLMath and LaTeXMLMath .
LaTeXMLMath is decomposed as LaTeXMLEquation where LaTeXMLMath is a measurable field of unitaries .
Then , we have LaTeXMLEquation .
LaTeXMLEquation for almost all LaTeXMLMath .
Thanks to uniqueness of the minimal subgroup up to conjugacy , there exists a unitary LaTeXMLMath from LaTeXMLMath to LaTeXMLMath such that if we set LaTeXMLEquation then LaTeXMLMath .
We denote by LaTeXMLMath the restriction of LaTeXMLMath to LaTeXMLMath .
Let LaTeXMLMath , and LaTeXMLEquation .
LaTeXMLEquation Then , we have LaTeXMLEquation .
LaTeXMLEquation for almost all LaTeXMLMath .
Since LaTeXMLMath and LaTeXMLMath are mutually equivalent minimal LaTeXMLMath -valued cocycles with LaTeXMLMath , LaTeXMLCite implies that there exist LaTeXMLMath normalizing LaTeXMLMath and a Borel map LaTeXMLMath such that LaTeXMLMath for almost all LaTeXMLMath .
Let LaTeXMLMath , then we get LaTeXMLEquation .
LaTeXMLEquation for almost all LaTeXMLMath .
Let LaTeXMLMath be the restriction of LaTeXMLMath to LaTeXMLMath .
Then LaTeXMLMath and LaTeXMLMath have the desired property .
∎ The following is an easy generalization of LaTeXMLCite to the non-singular case , which asserts that there are plenty of non-trivial modular endomorphisms for any type LaTeXMLMath factors .
Let LaTeXMLMath be an ergodic non-transitive LaTeXMLMath -space .
Then , for every compact group LaTeXMLMath , there exists a minimal cocycle LaTeXMLMath with LaTeXMLMath .
As in LaTeXMLCite , it suffices to show the statement for a LaTeXMLMath -action instead of the LaTeXMLMath -action thanks to Ambrose-Kakutani ’ s theorem LaTeXMLCite .
Let LaTeXMLMath be a given non-singular and non-transitive ergodic transformation on LaTeXMLMath .
We show that there exists a minimal cocycle LaTeXMLMath with LaTeXMLMath .
We take a non-transitive ergodic transformation LaTeXMLMath on LaTeXMLMath such that LaTeXMLMath is LaTeXMLMath -invariant .
Thanks to LaTeXMLCite , there exists a minimal cocycle LaTeXMLMath such that LaTeXMLMath .
Let LaTeXMLMath be the LaTeXMLMath -action on LaTeXMLMath given by LaTeXMLMath .
We define a cocycle LaTeXMLMath on LaTeXMLMath by LaTeXMLEquation which is a minimal cocycle with LaTeXMLMath .
Since LaTeXMLMath is orbit equivalent to LaTeXMLMath LaTeXMLCite , we get the result .
∎
For a stratified symplectic space , a suitable concept of stratified Kähler polarization encapsulates Kähler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kähler space which establishes an intimate relationship between nilpotent orbits , singular reduction , invariant theory , reductive dual pairs , Jordan triple systems , symmetric domains , and pre-homogeneous spaces : The closure of a holomorphic nilpotent orbit or , equivalently , the closure of the stratum of the associated pre-homogeneous space of parabolic type carries a ( positive ) normal Kähler structure .
In the world of singular Poisson geometry , the closures of principal holomorphic nilpotent orbits , positive definite hermitian JTS ’ s , and certain pre-homogeneous spaces appear as different incarnations of the same structure .
The closure of the principal holomorphic nilpotent orbit arises from a semisimple holomorphic orbit by contraction .
Symplectic reduction carries a positive Kähler manifold to a positive normal Kähler space in such a way that the sheaf of germs of polarized functions coincides with the ordinary sheaf of germs of holomorphic functions .
Symplectic reduction establishes a close relationship between singular reduced spaces and nilpotent orbits of the dual groups .
Projectivization of holomorphic nilpotent orbits yields exotic ( positive ) stratified Kähler structures on complex projective spaces and on certain complex projective varieties including complex projective quadrics .
The space of ( in general twisted ) representations of the fundamental group of a closed surface in a compact Lie group or , equivalently , a moduli space of central Yang-Mills connections on a principal bundle over a surface , inherits a ( positive ) normal ( stratified ) Kähler structure .
Physical examples are provided by certain reduced spaces arising from angular momentum zero .
Given a quantizable system with constraints , the issue arises whether reduction after quantization coincides with quantization after reduction .
Reduction after quantization makes sense provided the unreduced phase space is a quantizable smooth symplectic manifold in such a way that the symmetries can be quantized as well ; it may then be studied within the usual framework of geometric quantization .
Up to now , the available methods have been insufficient to attack the problem of quantization of reduced observables , though , once the reduced phase space is no longer a smooth manifold ; we will refer to this situation as the singular case .
The singular case is the rule rather than the exception .
For example , simple classical mechanical systems and the solution spaces of classical field theories involve singularities ; see e. g. LaTeXMLCite and the references there .
In the presence of singularities , the naive restriction of the quantization problem to a smooth open dense part , the “ top stratum ” , may lead to a loss of information and in fact to inconsistent results ; see what is said below .
Trying to overcome these difficulties on the classical level , we were led to isolate a certain class of “ Kähler spaces with singularities ” , to which the present paper is devoted .
Examples thereof arise from symplectic reduction , applied to Kähler manifolds .
Other examples arise from certain moduli spaces , of central Yang-Mills connections over a closed surface as well as ( equivalently ) of suitable representations of the fundamental group of a closed surface in a compact Lie group .
Still other examples arise from taking the closure of a holomorphic nilpotent orbit in a real semisimple ( or reductive ) Lie algebra of hermitian type ( see what is said below ) .
A typical example of the kind of Kähler space with singularities we will study is the complex plane with its ordinary complex analytic structure , but with a Poisson algebra of continuous functions which has the origin as a singularity so that geometrically this plane should then be viewed as a half-cone rather than the ordinary plane ; the cone point , i. e. the singularity , can not be detected by the complex analytic structure .
Albeit looking like a toy example , it illustrates the general phenomenon that , under such circumstances , playing off against each other the complex and real ( semi-analytic ) structures yields geometric insight .
Actually , this “ exotic ” plane is the closure in LaTeXMLMath of what we call a holomorphic nilpotent orbit : A real semisimple Lie algebra LaTeXMLMath with Cartan decomposition LaTeXMLMath together with an element LaTeXMLMath in the center of LaTeXMLMath , referred to as an LaTeXMLMath - element , such that the restriction of LaTeXMLMath to LaTeXMLMath yields a complex structure on the latter is said to be of hermitian type LaTeXMLCite ; the classification of semisimple Lie algebras of hermitian type plainly parallels E. Cartan ’ s classification of semisimple hermitian symmetric spaces .
We will say that an orbit LaTeXMLMath in such a Lie algebra LaTeXMLMath is holomorphic provided the projection from LaTeXMLMath to LaTeXMLMath is a diffeomorphism onto its image in such a way that the resulting complex structure LaTeXMLMath on LaTeXMLMath combines with its ( Kostant-Kirillov-Souriau ) symplectic structure to a positive Kähler structure .
Here LaTeXMLMath is identified with its dual by means of an appropriate positive multiple of the Killing form , made precise later in the paper .
The holomorphicity of an orbit does not depend on the choice of LaTeXMLMath -element or , equivalently , on the Cartan decomposition ; however , the induced complex structure on the orbit does depend on that choice ; see ( 3.7.5 ) and ( 3.7.6 ) below .
The name “ holomorphic ” is intended to hint at the fact that the holomorphic discrete series representations arise from holomorphic quantization on integral semisimple holomorphic orbits LaTeXMLMath ( the requisite complex structure being different from LaTeXMLMath , though , see what is said below ) .
Indeed , for any real positive LaTeXMLMath , the LaTeXMLMath -orbit in LaTeXMLMath generated by LaTeXMLMath is semisimple and holomorphic ; it realizes the smooth manifold underlying the corresponding symmetric domain as a symplectic manifold .
In Section 3 we will prove that a simple Lie algebra of hermitian type and split rank LaTeXMLMath has exactly LaTeXMLMath holomorphic nilpotent orbits LaTeXMLMath which are linearly ordered in such a way that LaTeXMLMath and that the projection from LaTeXMLMath to the constituent LaTeXMLMath in the Cartan decomposition is a homeomorphism .
This entails the fact that every orbit in the closure LaTeXMLMath of a holomorphic nilpotent orbit is itself holomorphic and that the projection from LaTeXMLMath to LaTeXMLMath is a homeomorphism onto its image which , via the complex structure on LaTeXMLMath induced by the LaTeXMLMath -element LaTeXMLMath ( which is part of the hermitian type structure ) , turns LaTeXMLMath into a complex affine variety .
The “ exotic ” plane mentioned earlier is a very special case thereof .
In general , LaTeXMLMath may be seen as arising from the corresponding semisimple holomorphic orbits by contraction ; this contraction does not change the complex analytic structure .
A comment about the complex structures is perhaps in order : The complex structure LaTeXMLMath on a semisimple holomorphic orbit LaTeXMLMath arises from the projection to LaTeXMLMath as does the complex structure on LaTeXMLMath , whence complex analytically these spaces are just a copy of LaTeXMLMath .
However , for a semisimple orbit , this is not the hermitian symmetric space complex structure .
Our Kähler spaces with singularities are stratified symplectic spaces with additional structure .
Reduced spaces stratified into smooth symplectic manifolds ( which arise as symmetry types ) occur already in LaTeXMLCite ( where smooth infinite dimensional pieces are not excluded ) .
The notion of stratified symplectic structure on a space LaTeXMLMath which we will use is that of LaTeXMLCite ; it is a stratification into smooth finite dimensional manifolds ( in a sense stronger than that of LaTeXMLCite ) together with a Poisson algebra LaTeXMLMath of continuous functions on LaTeXMLMath which , on each stratum , restricts to a symplectic Poisson algebra of smooth functions , not necessarily consisting of all smooth functions on the stratum .
We deliberately write LaTeXMLMath even though this algebra need not be an algebra of ordinary smooth functions ; the algebra LaTeXMLMath is then referred to as a smooth structure on LaTeXMLMath .
A stratified symplectic space is a stratified space LaTeXMLMath together with a stratified symplectic structure LaTeXMLMath , referred to henceforth as the stratified symplectic Poisson algebra ( of the stratified symplectic space ) .
In LaTeXMLCite it is shown that the reduced space for a hamiltonian action of a compact Lie group on a smooth symplectic manifold inherits a stratified symplectic structure , the requisite stratified symplectic Poisson algebra being that given in LaTeXMLCite .
Stratified symplectic structures have been constructed on various moduli spaces LaTeXMLCite including moduli spaces of central Yang-Mills connections for bundles on a surface and spaces of ( in general twisted ) representations of a surface group in a compact Lie group ( where “ twisted ” means suitable representations of the universal central extension ) , and the stratified symplectic structure on such a space yields a description of its symplectic singularity behaviour ; see LaTeXMLCite for overviews .
In the present paper we will explore stratified symplectic spaces arising from holomorphic nilpotent orbits and from reduction of Kähler manifolds ; in physics , examples come from standard hamiltonian systems in mechanics , in particular from angular momentum .
The relationship between angular momentum zero and certain nilpotent orbits in the real symplectic Lie algebras has been observed already in LaTeXMLCite .
In fact , the nilpotent orbits described explicitly in that paper are precisely the holomorphic ones but this was not observed there .
The Poisson structure of a stratified symplectic space encapsulates the mutual positions of the symplectic structures on the strata .
The new key notion which we will introduce in this paper is that of stratified polarization , which likewise encapsulates the mutual positions of polarizations on the strata .
We will show that , for certain nilpotent orbits LaTeXMLMath of a semisimple Lie algebra of hermitian type including the holomorphic ones ( already mentioned above ) , the closure LaTeXMLMath carries what we will call a complex analytic stratified Kähler polarization .
Furthermore , for a ( positive ) Kähler manifold with a hamiltonian action of a compact Lie group LaTeXMLMath which preserves the Kähler polarization and extends to an action of the complexification of LaTeXMLMath , the reduced space inherits a complex analytic stratified Kähler polarization .
In general , we refer to a space with a complex analytic stratified Kähler polarization as a complex analytic stratified Kähler space .
We also introduce the somewhat more general notion of stratified Kähler polarization ( not necessarily complex analytic ) ; on each stratum , this kind of polarization boils down to an ordinary Kähler polarization , and a space equipped with a stratified Kähler polarization will be referred to as a stratified Kähler space .
While a Kähler polarization on an ordinary smooth symplectic manifold is well known to be equivalent to a complex structure for which the symplectic structure is a Kähler form , we do not know whether a stratified Kähler polarization ( in our sense ) on a stratified symplectic space necessarily determines a corresponding complex analytic structure .
The stratification of a complex analytic stratified Kähler space is in general finer than the ordinary complex analytic stratification and the stratified symplectic structure then exhibits singularities which are complex analytically spurious ; an illustration is the complex plane with a cone point mentioned above .
More striking examples are certain exotic complex projective spaces .
By an exotic complex projective space we mean complex projective space LaTeXMLMath with its ordinary complex structure , but with a non-standard stratified symplectic structure LaTeXMLMath which is compatible with the complex structure where “ non-standard ” means that there are at least two strata .
For example , complex projective 3-space has an exotic structure whose singular locus is a Kummer surface , and the stratified Poisson structure detects the Kummer surface and its 16 singular points ; see 4.3 below .
Complex projective 3-space has yet another exotic structure , having as its singular locus a complex quadric ; see Section 10 below .
Such exotic spaces even arise from classical constrained systems in mechanics : The reduced space LaTeXMLMath ( say ) of LaTeXMLMath harmonic oscillators in some LaTeXMLMath with total angular momentum zero and constant energy is an exotic complex projective space of complex dimension LaTeXMLMath in such a way that , for LaTeXMLMath , the reduced spaces LaTeXMLMath ( say ) of LaTeXMLMath harmonic oscillators in some LaTeXMLMath with total angular momentum zero and ( same ) constant energy embed naturally into LaTeXMLMath and constitute an ascending chain LaTeXMLMath of ( compact ) complex analytic stratified Kähler spaces which , as complex analytic spaces , are projective determinantal varieties , and such that the LaTeXMLMath are the strata of the decomposition of any LaTeXMLMath ( LaTeXMLMath ) .
These examples are particular cases of systematic classes of exotic stratified Kähler structures on complex projective spaces which will be given in the paper .
The question whether there exist exotic structures on complex projective spaces is also prompted by the following observation : For a compact Kähler manifold LaTeXMLMath which is complex analytically a projective variety ( i. e. Hodge manifold ) , with reference to the standard structure on complex projective space ( i. e. Fubini-Study metric ) the Kodaira embedding will not in general be symplectic , but complex projective space might still carry an exotic structure which , via the Kodaira embedding , restricts to the Kähler structure on LaTeXMLMath .
More generally , given a Hodge manifold LaTeXMLMath together with an appropriate group of symmetries and momentum mapping , reduction carries it to a complex analytic stratified Kähler space LaTeXMLMath which is as well a projective variety , and the question arises whether in such a situation the requisite complex projective space carries an exotic structure which , via the Kodaira embedding , restricts to the complex analytic stratified Kähler structure on LaTeXMLMath .
Our approach in particular demonstrates that this situation actually occurs in “ mathematical nature ” .
The particular case LaTeXMLMath ( LaTeXMLMath , LaTeXMLMath ) arising from angular momentum is a good illustration .
We close the introduction with a guide through the paper .
In Section 1 we reproduce some material from the relationship between Poisson algebras and Lie-Rinehart algebras .
In Section 2 we introduce stratified polarized spaces .
The aim of Section 3 is to show that the closure of any holomorphic nilpotent orbit carries a normal complex analytic stratified Kähler structure ; see Theorem 3.2.1 for details .
We will , furthermore , give a complete classification of all holomorphic nilpotent orbits , and we will explicitly describe the real ( semi-algebraic ) and the complex analytic structure of the closure of any such holomorphic nilpotent orbit .
We will also classify what we call pseudoholomorphic nilpotent orbits ; a pseudoholomorphic nilpotent orbit LaTeXMLMath is one which under the projection to the constituent LaTeXMLMath of the Cartan decomposition is still mapped diffeomorphically onto its image ( but the resulting Kähler structure is not constrained to be positive ) .
Along the way , we obtain several results on holomorphic nilpotent orbits that look interesting in their own right : For a classical Lie algebra LaTeXMLMath , let LaTeXMLMath be its standard representation , and consider the ordinary associative algebra LaTeXMLMath of endomorphisms thereof .
We will show that , in the standard cases ( LaTeXMLMath ) , an element LaTeXMLMath of LaTeXMLMath generates a pseudoholomorphic nilpotent orbit if and only if LaTeXMLMath is zero in LaTeXMLMath and that , likewise , an element LaTeXMLMath of LaTeXMLMath generates a pseudoholomorphic nilpotent orbit if and only if LaTeXMLMath is zero in LaTeXMLMath .
The Lie algebra LaTeXMLMath sits of course inside LaTeXMLMath and an element LaTeXMLMath satisfying LaTeXMLMath or LaTeXMLMath is plainly nilpotent ( i. e. LaTeXMLMath is a nilpotent endomorphism of LaTeXMLMath ) but the requirement that LaTeXMLMath satisfy LaTeXMLMath or LaTeXMLMath has no direct meaning within LaTeXMLMath itself .
Another interesting side result is the observation that , when LaTeXMLMath is simple , a pseudoholomorphic nilpotent orbit LaTeXMLMath is actually holomorphic ( or antiholomorphic , which means that LaTeXMLMath is holomorphic ) if and only if the projection from the closure LaTeXMLMath to LaTeXMLMath is a homeomorphism onto its image in LaTeXMLMath ; see Theorem 3.2.1 .
Our results are conclusive , except that we do not give explicit defining equations for the complex analytic structures of the closures of the rank 1 holomorphic nilpotent orbits in the two exceptional cases .
In LaTeXMLCite , given a Lie algebra of hermitian type LaTeXMLMath , a nilpotent orbit LaTeXMLMath in LaTeXMLMath is defined to be holomorphic provided it corresponds , under the Kostant-Sekiguchi correspondence LaTeXMLCite , to an orbit in the holomorphic constituent LaTeXMLMath of the complexification LaTeXMLMath , and it is then proved , via the Kronheimer flow LaTeXMLCite , that the restriction of the projection to LaTeXMLMath of such an orbit is a diffeomorphism onto its image , that is to say , that such an orbit is ( pseudo ) holomorphic in our sense .
The approach in the present paper is somewhat different , though : We define the property of holomorphicity of an orbit in terms of the projection to LaTeXMLMath —this definition is more elementary than that involving the Kostant-Sekiguchi correspondence—and we classify all holomorphic nilpotent orbits ( in our sense ) by means of a geometric description which bypasses the Kostant-Sekiguchi correspondence ; in particular , our approach does not involve the Kronheimer flow at all and is therefore likely to allow for generalization over other ( formally real ) fields .
It also enables us to show that the projection from a holomorphic nilpotent orbit LaTeXMLMath to LaTeXMLMath extends to a homeomorphism from the closure LaTeXMLMath onto its image in LaTeXMLMath .
We do not know whether this property of holomorphic nilpotent orbits can formally be deduced from properties of the Kronheimer flow since passing to the closure amounts to changing the topological type of the corresponding bundle on LaTeXMLMath .
This property of holomorphic nilpotent orbits plainly implies the fact that , for a holomorphic nilpotent orbit , the Kostant-Sekiguchi correspondence is compatible with passing to the closure—this is actually true for any nilpotent orbit in a real semisimple Lie algebra , cf .
LaTeXMLCite —but is considerably stronger than just the compatibility of the Kostant-Sekiguchi correspondence with respect to the closure .
In Section 4 we show that reduction carries an ordinary ( positive ) Kähler polarization to a ( positive ) complex analytic stratified Kähler polarization .
As an application , we prove that the moduli spaces mentioned earlier inherit normal ( complex analytic stratified ) Kähler space structures .
Classically , such a structure was obtained only in the non-singular case , where these spaces are smooth Kähler manifolds .
Our approach bypasses the usual geometric invariant theory construction ( which does not give the requisite stratified symplectic structure ) and yields an analytic construction of these moduli spaces .
A particular case is the space LaTeXMLMath of representations of the fundamental group of a Riemann surface of genus two in LaTeXMLMath ; in view of results of Narasimhan-Ramanan LaTeXMLCite , this space , realized as the moduli space of semistable holomorphic vector bundles of rank two , degree zero , and trivial determinant , is complex analytically a copy of LaTeXMLMath , and the subspace of classes of semistable holomorphic vector bundles which are not stable is a Kummer surface .
We will show that , indeed , LaTeXMLMath carries a normal ( complex analytic stratified ) Kähler structure having the Kummer surface as its singular locus .
This is the exotic structure on LaTeXMLMath mentioned earlier , and the Kummer surface , being viewed as a Kodaira embedding , inherits its ( now singular ) normal Kähler structure from the embedding .
In Section 5 we describe the holomorphic nilpotent orbits ( for the classical cases ) as reduced spaces for suitable momentum mappings ; this involves versions of the First and Second Main Theorem of Invariant Theory LaTeXMLCite ; the Second Main Theorem of Invariant Theory will actually be a consequence of our approach to invariant theory .
Our tools are the original observation of Kempf-Ness LaTeXMLCite and an extension of the basic construction in LaTeXMLCite ; in that reference , this construction is given only for the case of the classical groups over the complex numbers .
We extend this construction to the appropriate real forms of the classical groups .
The Kempf-Ness observation is then exploited to identify a holomorphic nilpotent orbit as a symplectic quotient with the corresponding complex categorical quotient ( in the sense of geometric invariant theory ) .
The construction in Section 5 has a consequence for singular reduction which is interesting in its own right : Let LaTeXMLMath be a compact Lie group which may be written as the direct product of copies of classical groups of the kind LaTeXMLMath , LaTeXMLMath , LaTeXMLMath ; for each of these , let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath be the standard unitary representation , LaTeXMLMath being viewed as a subgroup of LaTeXMLMath in the usual way .
Pick natural numbers LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , consider the representations LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , LaTeXMLMath and , finally , take the sum LaTeXMLMath of them .
This is a unitary LaTeXMLMath -representation .
Here LaTeXMLMath is the conjugate representation of LaTeXMLMath .
The representations LaTeXMLMath ( being real ) and LaTeXMLMath ( being quaternionic ) are actually self-conjugate whence there is no need to take conjugate representations for the LaTeXMLMath ’ s , nor for the LaTeXMLMath ’ s .
Let LaTeXMLMath be the unique LaTeXMLMath -momentum mapping of LaTeXMLMath having the value zero at the origin , and let LaTeXMLMath be the real reductive Lie algebra which arises , accordingly , as a sum copies of the LaTeXMLMath ’ s , LaTeXMLMath ’ s , LaTeXMLMath ’ s .
Corollary 5.4.4 below says that , under these circumstances , the LaTeXMLMath -reduced space LaTeXMLMath is as a normal Kähler space isomorphic to the closure of a holomorphic nilpotent orbit in LaTeXMLMath .
Furthermore , every holomorphic nilpotent orbit in LaTeXMLMath arises in this way .
It is interesting to point out , cf .
LaTeXMLCite , that the groups LaTeXMLMath and the direct product LaTeXMLMath of the corresponding copies of LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , constitute a real reductive dual pair in LaTeXMLMath , the symplectic structure on LaTeXMLMath being determined by its unitary one .
More generally , let LaTeXMLMath be a Lie group which may be written as the direct product of copies of classical groups of the kind LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , and let LaTeXMLMath be an LaTeXMLMath -representation which is a sum of standard representations .
In view of LaTeXMLCite , the groups LaTeXMLMath and LaTeXMLMath then still constitute a reductive dual pair in LaTeXMLMath , the symplectic structure on LaTeXMLMath being the obvious one on LaTeXMLMath determined by the data ( see Section 5 below for details ) .
Let LaTeXMLMath be the unique LaTeXMLMath -momentum mapping of LaTeXMLMath having the value zero at the origin .
Corollary 5.5.4 below says that the LaTeXMLMath -reduced space LaTeXMLMath ( which we define to be that of closed LaTeXMLMath -orbits in LaTeXMLMath ) is as a stratified space isomorphic to a space LaTeXMLMath which is a union of pseudoholomorphic nilpotent orbits in a Lie algebra LaTeXMLMath of the above kind in such a way that the map from LaTeXMLMath to LaTeXMLMath is a Poisson map .
Furthermore , every pseudoholomorphic nilpotent orbit of LaTeXMLMath arises in this way ( as a stratum of a reduced space of the kind LaTeXMLMath ) .
In Section 6 we give a construction which yields the holomorphic nilpotent orbits of LaTeXMLMath by singular reduction with respect to the non-compact group LaTeXMLMath ; for arbitrary LaTeXMLMath , it does not seem possible to obtain these orbits by singular reduction with respect to a compact group , though .
In Section 7 we relate holomorphic nilpotent orbits with positive definite hermitian J ( ordan ) T ( riple ) S ( ystem ) s , and in Section 8 we give some relevant information for the exceptional cases in terms of JTS ’ s .
To explain what we achieve at that stage , let LaTeXMLMath be a real reductive Lie algebra of hermitian type , with Cartan decomposition LaTeXMLMath and complexification LaTeXMLMath , and let LaTeXMLMath and LaTeXMLMath denote the adjoint group and maximal compact subgroup with LaTeXMLMath , respectively .
We will establish the following facts , cf .
Theorem 7.3 for the classical cases and Theorems 8.4.1 and 8.4.2 for the exceptional ones : Under the homeomorphism ( induced by the projection from LaTeXMLMath to LaTeXMLMath ) from the closure LaTeXMLMath of the principal holomorphic nilpotent orbit LaTeXMLMath onto the complex vector space LaTeXMLMath , the LaTeXMLMath -orbit stratification of LaTeXMLMath passes to the stratification of LaTeXMLMath by Jordan rank , with reference to the JTS-structure on LaTeXMLMath , and the latter stratification , in turn , coincides with the LaTeXMLMath -orbit stratification of LaTeXMLMath .
In this fashion , the induced stratified symplectic Poisson structure on LaTeXMLMath turns the latter into a normal complex analytic stratified Kähler space , and the Poisson structure detects the stratification by Jordan rank .
This reduces the study of holomorphic nilpotent orbits in LaTeXMLMath to that of the hermitian Jordan triple system LaTeXMLMath : All geometric information of the holomorphic nilpotent orbits is encoded in that JTS , even the stratified symplectic Poisson structure since the latter is determined by the Lie bracket in LaTeXMLMath , which can be reconstructed from the JTS ; this parallels the well known fact that all geometric information of a symmetric domain is encoded in its JTS , cf .
e. g. LaTeXMLCite .
Our results entail in particular that the pre-homogeneous spaces studied and classified in LaTeXMLCite —these are precisely those which arise from positive definite hermitian JTS ’ s—carry more structure ; indeed , such a space necessarily underlies the closure of the principal holomorphic nilpotent orbit ( viewed as a complex analytic stratified Kähler space ) of a semisimple Lie algebra of hermitian type .
As a byproduct , we obtain the Bernstein-Sato polynomials for the regular pre-homogeneous spaces under discussion ; see Theorems 7.1 and 8.4.1 for details .
For example , for LaTeXMLMath , this polynomial is Freudenthal ’ s generalized LaTeXMLMath -determinant over a complex split octonion algebra LaTeXMLCite or , equivalently , Jacobson ’ s generic norm .
In Section 9 we show how the closure of the principal holomorphic nilpotent orbit arises from the corresponding semisimple holomorphic orbits by contraction .
In Section 10 we projectivize the holomorphic nilpotent orbits by means of stratified symplectic reduction with respect to the circle group .
This gives systematic classes of examples of exotic projective spaces .
In particular , the principal holomorphic nilpotent orbit in LaTeXMLMath is a copy of LaTeXMLMath having as singular locus in the sense of stratified Kähler spaces an affine quadric , and projectivization thereof yields the exotic projective space LaTeXMLMath having as singular locus a projective quadric .
Thus the projective quadric inherits a Kähler , in fact Hodge structure whose symplectic constituent does not arise ( via the embedding ) from the Fubini-Study metric on projective space , and the underlying real space arises from the non-zero non-principal holomorphic nilpotent orbit in LaTeXMLMath by projectivization .
This is a particular example of a Kodaira embedding of the kind mentioned earlier , where the symplectic structure results from a proper stratified symplectic Poisson algebra on projective space .
For the exceptional Lie algebra LaTeXMLMath , we obtain an exotic Kähler structure on LaTeXMLMath having as singular locus the projective cubic defined by the generic norm over the octonions mentioned earlier , and this cubic hypersurface , in turn , inherits a complex analytic stratified Kähler structure with two strata whose underlying real structure arises from the closure of the rank 2 holomorphic nilpotent orbit in LaTeXMLMath by projectivization .
Actually , this cubic has played a major role in the development of the exceptional Lie groups ; a version of it was studied by E. Cartan .
In the standard cases ( LaTeXMLMath ) , via the principle of reduction in stages , the projectivized holomorphic nilpotent orbits may also be obtained by ordinary symplectic reduction , as explained in Theorem 10.1 ; somewhat amazingly , this does not seem to be possible for LaTeXMLMath for general LaTeXMLMath , nor for LaTeXMLMath or LaTeXMLMath ; in these cases , we obtain the projectivized holomorphic nilpotent orbits only by proper stratified symplectic reduction , with respect to the circle group .
Finally , in Section 11 we compare our notion of stratified Kähler space with that of Kähler space with singularities introduced by Grauert LaTeXMLCite and with subsequent refinements thereof LaTeXMLCite , referred to in LaTeXMLCite as stratified Kählerien spaces .
There is a substantial difference between complex analytic stratified Kähler spaces in our sense and stratified Kählerien spaces , though .
Our emphasis is on the Poisson structure , and this is crucial for issues related with quantization ; stratified Kählerien spaces do not involve Poisson structures at all .
Thus the notion of stratified Kähler space establishes an intimate relationship between nilpotent orbits , singular reduction , invariant theory , reductive dual pairs , Jordan triple systems , symmetric domains , and pre-homogeneous spaces ; in particular , in the world of singular Poisson geometry , the closures of principal holomorphic nilpotent orbits , positive definite hermitian JTS ’ s , and certain pre-homogeneous spaces appear as different incarnations of the same structure and , in the classical cases , that structure arises , via invariant theory and singular reduction , from a non-singular one phrased in terms of dual pairs .
This Poisson geometry is , perhaps , already lurking behind Lie ’ s theory of function groups , cf .
the notion of dual pair of Poisson mappings in LaTeXMLCite .
Somewhat amazingly , Jordan algebras ( and generalizations thereof : JTS ’ s ) —which have been invented in search for new models of quantum mechanics ( cf .
e. g. what is said in LaTeXMLCite ) —are now seen to illuminate certain classically reduced phase spaces ( e. g. that of finitely many harmonic oscillators with total angular momentum zero ) .
The length of the paper is to some extent explained by the need for various different descriptions of holomorphic nilpotent orbits : To obtain a complete list thereof , the relative root systems suffice ; to see that the list is complete , though , we need the classification of nilpotent orbits in the literature in terms of classical matrix realizations , matrix realizations are needed as well to explain the relationship with singular reduction and , at times , structural insight is more easily obtained from matrix realizations than from root systems .
To describe the link with JTS ’ s—the exceptional Lie algebras are actually constructed from JTS ’ s—we reproduce a description of the requisite JTS ’ s taylored to our purposes .
We conclude with a somewhat vague comment about normality , a notion which so far does not seem to have played a major role in symplectic or Poisson geometry .
The results of the present paper indicate that , perhaps , normal Kähler spaces constitute the correct category for doing Kähler reduction .
While smoothness is not in general preserved , in the circumstances studied in our paper , normality is preserved under reduction .
What remains to be done is to develop a general reduction procedure for ( in general singular ) normal Kähler spaces .
In a follow-up paper LaTeXMLCite , we have extended the ordinary holomorphic quantization scheme to stratified Kähler spaces .
Using this extension , we have established the fact that , indeed , in a suitable sense , in the framework of Kähler quantization , reduction after quantization is equivalent to quantization after reduction , observables included .
In particular , in that paper , explicit examples are given which justify the claim made earlier that , in the presence of singularities , restricting quantization to a smooth open dense part ( the top stratum ) leads in general to a loss of information and in fact to inconsistent results .
In another follow-up paper LaTeXMLCite , we have shown that the geometry of certain Severi varieties may be elucidated in terms of holomorphic nilpotent orbits .
I am much indebted to A. Weinstein and T. Ratiu for discussions , and for their encouragement to carry out the research program the present paper is part of .
I am much indebted as well to M. Duflo for having introduced me into Satake ’ s book LaTeXMLCite and for having suggested a possible relationship between certain nilpotent orbits and Jordan algebras .
Thanks are also due to R. Buchweitz , R. Cushman , J. Hilgert , F. Hirzebruch , F. Lescure , O. Loos , L. Manivel , J. Marsden , E. Neher , J. Sekiguchi , P. Slodowy , T. Springer , J. Stasheff , M. Vergne , and R. Weissauer for support at various stages of the project .
I owe a special debt to Satake ’ s book LaTeXMLCite ; without this book , the paper would not have been materialized in its present form .
Table of contents 1 .
Poisson algebras and Lie-Rinehart algebras 2 .
Stratified polarized spaces 3 .
The closure of a holomorphic nilpotent orbit 4 .
Reduction and stratified Kähler spaces 5 .
Associated representations and singular reduction 6 .
Associated representations for the remaining classical case 7 .
Hermitian Jordan triple systems and pre-homogeneous spaces 8 .
The exceptional cases 9 .
Contraction of semisimple holomorphic orbits 10 .
Projectivization and exotic projective varieties 11 .
Comparison with other notions of Kähler space with singularities References 1 .
Poisson algebras and Lie-Rinehart algebras Let LaTeXMLMath be a commutative ring with 1 taken as ground ring which , for the moment , may be arbitrary .
For a commutative LaTeXMLMath -algebra LaTeXMLMath , we denote by LaTeXMLMath the LaTeXMLMath -Lie algebra of derivations of LaTeXMLMath , with its standard Lie algebra structure .
Recall that a Lie-Rinehart algebra LaTeXMLMath consists of a commutative LaTeXMLMath -algebra LaTeXMLMath and an LaTeXMLMath -Lie algebra LaTeXMLMath together with an LaTeXMLMath -module structure on LaTeXMLMath and an LaTeXMLMath -module structure LaTeXMLMath on LaTeXMLMath ( that is , LaTeXMLMath acts on LaTeXMLMath by derivations ) , and these are required to satisfy two compatibility conditions modeled on the properties of the pair LaTeXMLMath , LaTeXMLMath being the ring of smooth functions LaTeXMLMath on a smooth manifold LaTeXMLMath and LaTeXMLMath the Lie algebra LaTeXMLMath of smooth vector fields on LaTeXMLMath .
Explicitly , these compatibility conditions read LaTeXMLMath and LaTeXMLMath , where LaTeXMLMath and LaTeXMLMath .
Given an arbitrary commutative algebra LaTeXMLMath over LaTeXMLMath , another example of a Lie-Rinehart algebra is the pair LaTeXMLMath , with the obvious action of LaTeXMLMath on LaTeXMLMath and obvious LaTeXMLMath -module structure on LaTeXMLMath .
There is an obvious notion of morphism of Lie-Rinehart algebras and , with this notion of morphism , Lie-Rinehart algebras constitute a category .
Given a Lie-Rinehart algebra LaTeXMLMath we shall occasionally refer to LaTeXMLMath as an LaTeXMLMath - Lie algebra .
More details may be found in Rinehart LaTeXMLCite and in our papers LaTeXMLCite and LaTeXMLCite .
Let LaTeXMLMath be a Poisson algebra , and let LaTeXMLMath be the LaTeXMLMath -module of formal differentials of LaTeXMLMath .
For LaTeXMLMath , the assignment LaTeXMLMath yields an LaTeXMLMath -valued 2-form LaTeXMLMath on LaTeXMLMath , the Poisson 2- form for LaTeXMLMath .
Its adjoint LaTeXMLEquation is a morphism of LaTeXMLMath -modules , and the formula LaTeXMLEquation yields a Lie bracket LaTeXMLMath on LaTeXMLMath , viewed as an LaTeXMLMath -module .
The LaTeXMLMath -module structure on LaTeXMLMath , the bracket LaTeXMLMath , and the morphism LaTeXMLMath of LaTeXMLMath -modules endow the pair LaTeXMLMath with a Lie-Rinehart algebra structure in such a way that LaTeXMLMath is a morphism of Lie-Rinehart algebras .
See LaTeXMLCite ( 3.8 ) for details .
We write LaTeXMLMath and , for LaTeXMLMath , we write LaTeXMLMath .
We now take as ground ring that of the reals LaTeXMLMath or that of the complex numbers LaTeXMLMath ; we shall occasionally use the neutral notation LaTeXMLMath for either of them .
We shall consider spaces LaTeXMLMath with an algebra of continuous LaTeXMLMath -valued functions , deliberately denoted by LaTeXMLMath , for example ordinary smooth manifolds and ordinary smooth functions ; such an algebra LaTeXMLMath will then be referred to as a smooth structure on LaTeXMLMath and will be viewed as part of the structure , and the pair LaTeXMLMath will be referred to as a smooth space .
A space may support distinct smooth structures , though .
Given a space LaTeXMLMath with a smooth structure LaTeXMLMath , we shall write LaTeXMLMath for the space of formal differentials with those differentials divided out that are zero at each point , cf .
LaTeXMLCite .
More precisely : Let LaTeXMLMath , and view LaTeXMLMath as a LaTeXMLMath -module by means of the evaluation map which assigns to a function LaTeXMLMath its value LaTeXMLMath at LaTeXMLMath ; we shall occasionally denote this LaTeXMLMath -module by LaTeXMLMath .
A formal differential is zero at the point LaTeXMLMath of LaTeXMLMath provided it passes to zero in LaTeXMLMath .
For example , when LaTeXMLMath is an ordinary smooth manifold and LaTeXMLMath an ordinary smooth function , in local coordinates LaTeXMLMath , a formal differential of the kind LaTeXMLMath is non-zero but is zero at each point .
For a general smooth space LaTeXMLMath , the induced LaTeXMLMath -module morphism LaTeXMLEquation is clearly an isomorphism .
At a point LaTeXMLMath of LaTeXMLMath , the object LaTeXMLMath amounts to the ordinary space of differentials , in the following sense : Denote by LaTeXMLMath the ideal of functions in LaTeXMLMath that vanish at the point LaTeXMLMath .
The LaTeXMLMath -linear map which assigns to a function LaTeXMLMath in LaTeXMLMath its formal differential induces isomorphisms LaTeXMLEquation of LaTeXMLMath -vector spaces , and each of these may be taken as the space of differentials at LaTeXMLMath ; over the reals , this space , in turn , may be identified with the cotangent space LaTeXMLMath .
When LaTeXMLMath is an ordinary smooth manifold , LaTeXMLMath amounts to the space of smooth sections of the cotangent bundle .
For a general smooth space LaTeXMLMath over the reals , when LaTeXMLMath is endowed with a Poisson structure , the formula ( 1.2 ) yields a Lie-bracket LaTeXMLMath on the LaTeXMLMath -module LaTeXMLMath and the 2-form LaTeXMLMath is still defined on LaTeXMLMath ; its adjoint then yields an LaTeXMLMath -linear map LaTeXMLEquation .
The LaTeXMLMath -module structure on LaTeXMLMath , the bracket LaTeXMLMath , and the morphism LaTeXMLMath of LaTeXMLMath -modules endow the pair LaTeXMLMath with a Lie-Rinehart algebra structure in such a way that LaTeXMLMath is a morphism of Lie-Rinehart algebras .
Indeed the obvious projection map from LaTeXMLMath to LaTeXMLMath is compatible with the structure .
The non-triviality of the kernel of this projection map does not cause any problem .
∎ We shall write LaTeXMLMath as LaTeXMLMath .
When LaTeXMLMath is an ordinary smooth manifold , the range LaTeXMLMath of the adjoint map LaTeXMLMath from LaTeXMLMath to LaTeXMLMath boils down to the space LaTeXMLMath of smooth vector fields on LaTeXMLMath .
In this case , the Poisson structure on LaTeXMLMath is symplectic , that is , arises from a ( uniquely determined ) symplectic structure on LaTeXMLMath , if and only if LaTeXMLMath , which may now be written as a morphism of smooth vector bundles from the cotangent bundle to the tangent bundle , is an isomorphism .
Thus the 2-form LaTeXMLMath , which is defined for every Poisson algebra , generalizes the symplectic form of a symplectic manifold ; see Section 3 of LaTeXMLCite for details .
2 .
Stratified polarized spaces The aim of this section is to develop a satisfactory notion of Kähler polarization for a stratified symplectic space .
Let LaTeXMLMath be a symplectic manifold , with symplectic Poisson algebra LaTeXMLMath , and write LaTeXMLMath for its Poisson 2-form .
Since LaTeXMLMath is symplectic , the complexification LaTeXMLEquation of the adjoint LaTeXMLMath is an isomorphism of LaTeXMLMath -Lie algebras .
Hence a complex polarization ( i. e. integrable Lagrangian distribution ) LaTeXMLMath for LaTeXMLMath corresponds to a certain LaTeXMLMath -Lie subalgebra LaTeXMLMath of LaTeXMLMath which is just the pre-image under ( 2.1 ) of the space LaTeXMLMath of sections of LaTeXMLMath .
The polarized complex functions determined by LaTeXMLMath are smooth complex valued functions LaTeXMLMath defined on open sets of LaTeXMLMath satisfying LaTeXMLMath for every smooth vector field LaTeXMLMath in LaTeXMLMath ; when LaTeXMLMath is a Kähler manifold and LaTeXMLMath the holomorphic polarization , these are ordinary holomorphic functions .
In general , non-zero polarized functions exist at most locally .
Following LaTeXMLCite , we shall refer to a continuous function LaTeXMLMath , defined on an open subset of a space LaTeXMLMath , as a function in LaTeXMLMath .
For general LaTeXMLMath and LaTeXMLMath , given a function LaTeXMLMath in LaTeXMLMath , for a suitable compactly supported smooth bump function LaTeXMLMath on LaTeXMLMath ( the notion of bump function will be made precise below ) , the differential LaTeXMLMath is defined everywhere on LaTeXMLMath .
Notice that we can extend LaTeXMLMath to a smooth function LaTeXMLMath on LaTeXMLMath in such a way that LaTeXMLMath , so the differential LaTeXMLMath really lies in the LaTeXMLMath -module of differentials , for the algebra LaTeXMLMath of smooth functions on the whole space LaTeXMLMath .
Then LaTeXMLMath is generated by these differentials .
A little thought reveals that , for every such LaTeXMLMath and LaTeXMLMath , with appropriate bump functions LaTeXMLMath and LaTeXMLMath so that the differentials LaTeXMLMath and LaTeXMLMath are defined everywhere on LaTeXMLMath , the Lie bracket ( 1.2 ) induces a bracket on LaTeXMLMath .
This bracket is given by the expression LaTeXMLEquation notice that the third term LaTeXMLMath ( which has initially to occur , cf .
( 1.2 ) ) is zero since LaTeXMLMath and LaTeXMLMath are polarized .
Let LaTeXMLMath be a stratified space , and let LaTeXMLMath be a smooth structure on LaTeXMLMath which , on each stratum , restricts to an algebra of ordinary smooth functions ; henceforth all smooth spaces coming into play in the paper will be of this kind .
Given such a space LaTeXMLMath and an open subset LaTeXMLMath thereof , with its induced smooth structure , and a smooth function LaTeXMLMath on LaTeXMLMath , we refer to LaTeXMLMath as a smooth function in LaTeXMLMath .
When LaTeXMLMath is a stratified symplectic space , we will occasionally refer to the stratification as the symplectic stratification .
Let LaTeXMLMath be a stratified symplectic space , with complex Poisson algebra LaTeXMLMath , let LaTeXMLMath be the corresponding LaTeXMLMath -Lie algebra , and let LaTeXMLMath be an arbitrary stratum .
Then the restriction map from LaTeXMLMath to LaTeXMLMath is Poisson and hence induces a morphism LaTeXMLEquation of Lie-Rinehart algebras .
Given a LaTeXMLMath -Lie subalgebra LaTeXMLMath of LaTeXMLMath , let LaTeXMLMath be the LaTeXMLMath -submodule of LaTeXMLMath generated by the image of LaTeXMLMath under ( 2.2 ) ; it inherits the structure of a LaTeXMLMath -Lie subalgebra of LaTeXMLMath .
A LaTeXMLMath -Lie subalgebra LaTeXMLMath of LaTeXMLMath will be said to be a stratified ( complex ) polarization for LaTeXMLMath if , for every stratum LaTeXMLMath , under the isomorphism ( 2.1 ) of LaTeXMLMath -Lie algebras ( for the smooth symplectic manifold LaTeXMLMath ) , the LaTeXMLMath -Lie subalgebra LaTeXMLMath of LaTeXMLMath is identified with the space of sections of a complex polarization in the usual sense .
Given a stratified polarization LaTeXMLMath for LaTeXMLMath , a smooth complex function LaTeXMLMath in LaTeXMLMath , that is , a function which is defined on an open subset LaTeXMLMath of LaTeXMLMath and belongs to the induced smooth structure LaTeXMLMath will be said to be polarized provided it satisfies LaTeXMLEquation .
Here and henceforth “ smooth ” refers to the structure algebras of functions LaTeXMLMath on LaTeXMLMath and LaTeXMLMath on open subsets LaTeXMLMath thereof ; thus a “ smooth function ” in our sense is not necessarily an ordinary smooth function .
In general polarized functions will exist at most locally , and we can talk about polarized functions in LaTeXMLMath or the sheaf of germs of polarized functions .
Infinitesimally , the notion of stratified polarization comes down to this : For LaTeXMLMath , the Poisson 2-form LaTeXMLMath passes to a 2-form LaTeXMLMath on the cotangent space LaTeXMLMath .
Given a stratified polarization , when LaTeXMLMath lies in the stratum LaTeXMLMath , the restriction map from LaTeXMLEquation to LaTeXMLMath induces a surjective LaTeXMLMath -linear map LaTeXMLEquation and LaTeXMLMath is a Lagrangian subspace of LaTeXMLMath with respect to the complexified symplectic structure LaTeXMLMath on LaTeXMLMath .
When LaTeXMLMath is a smooth symplectic manifold , viewed as a stratified symplectic space with a single stratum , this notion of polarization manifestly boils down to the ordinary one .
For a general stratified symplectic space , a stratified polarization encapsulates the mutual positions of the polarizations on the strata .
Given a stratified symplectic space LaTeXMLMath , a stratified polarization LaTeXMLMath for LaTeXMLMath will be said to be a ( positive ) stratified Kähler polarization if , for every stratum LaTeXMLMath , the image of LaTeXMLMath under ( 2.1 ) is the space of sections of a ( positive ) Kähler polarization for LaTeXMLMath in the usual sense .
A stratified symplectic space with a stratified Kähler polarization will be said to be a stratified Kähler space .
Since a Kähler polarization on a symplectic manifold determines a Kähler structure thereupon it is obvious that each stratum of a stratified Kähler space inherits a Kähler structure .
Recall that a symplectic structure LaTeXMLMath is said to be compatible with a complex structure LaTeXMLMath provided LaTeXMLMath is a symplectic operator so that LaTeXMLMath defines a not necessarily positive Kähler structure .
Given a real symplectic structure LaTeXMLMath on a complex domain LaTeXMLMath of complex dimension LaTeXMLMath , let LaTeXMLMath be the Poisson structure determined by LaTeXMLMath .
In terms of holomorphic coordinates LaTeXMLMath , LaTeXMLMath is compatible with the complex structure LaTeXMLMath if and only if LaTeXMLMath for LaTeXMLMath .
Under these circumstances , the Kähler polarization LaTeXMLMath ( in our sense ) is the free LaTeXMLMath -submodule of LaTeXMLMath generated by the differentials LaTeXMLMath .
For a smooth function LaTeXMLMath in LaTeXMLMath , the Cauchy-Riemann equations then amount to LaTeXMLEquation .
The differentials LaTeXMLMath freely generate LaTeXMLMath as a LaTeXMLMath -module , and LaTeXMLMath is , likewise , identifies the free LaTeXMLMath -module generated by the vector fields LaTeXMLMath .
The symplectic structure LaTeXMLMath on LaTeXMLMath is compatible with the complex structure LaTeXMLMath if and only if the isomorphism ( 2.1 ) of LaTeXMLMath -Lie algebras from LaTeXMLMath onto LaTeXMLMath identifies the LaTeXMLMath -linear span of the LaTeXMLMath with the LaTeXMLMath -linear span of the LaTeXMLMath ( so that this span contains the hamiltonian vector fields LaTeXMLMath of the LaTeXMLMath ’ s ( LaTeXMLMath ) and therefore coincides with the span of these hamiltonian vector fields ) and , likewise , the LaTeXMLMath -linear span of the LaTeXMLMath with the LaTeXMLMath -linear span of the LaTeXMLMath .
The latter property , in turn , is equivalent to the vanishing of LaTeXMLMath for LaTeXMLMath .
See for example the discussion in LaTeXMLCite ( 5.4 p. 92 ) .
Under these circumstances , the Kähler polarization ( in our sense ) LaTeXMLMath is generated by the differentials LaTeXMLMath , and the polarized smooth complex functions LaTeXMLMath in LaTeXMLMath or , equivalently , the germs of polarized smooth complex functions are ( classes of ) functions LaTeXMLMath in LaTeXMLMath which satisfy LaTeXMLMath for LaTeXMLMath , where LaTeXMLMath refers to the action of LaTeXMLMath on LaTeXMLMath reproduced in Section 1 above .
By construction , for every smooth complex function LaTeXMLMath in LaTeXMLMath , given LaTeXMLMath , we have LaTeXMLMath .
Hence the polarized smooth complex functions LaTeXMLMath in LaTeXMLMath are those functions which satisfy the equations LaTeXMLMath for LaTeXMLMath .
However , for every coordinate function LaTeXMLMath and every smooth complex valued function LaTeXMLMath in LaTeXMLMath , LaTeXMLEquation .
Since LaTeXMLMath is symplectic , the matrix LaTeXMLMath is invertible whence the Cauchy-Riemann equations amount to the vanishing of LaTeXMLMath for LaTeXMLMath .
∎ Under the circumstances of Proposition 2.3 , the holomorphic functions in LaTeXMLMath coincide with the smooth complex valued functions in LaTeXMLMath which are polarized in our sense .
Let LaTeXMLMath be a stratified Kähler space ; we will say that a complex analytic structure on LaTeXMLMath is compatible with the stratified Kähler structure if ( i ) the stratification of LaTeXMLMath ( as a stratified symplectic space ) is a refinement of the complex analytic stratification , if ( ii ) every holomorphic function in LaTeXMLMath is smooth in LaTeXMLMath and if , ( iii ) the sheaf of germs of holomorphic functions in LaTeXMLMath is contained in its sheaf of germs of polarized functions .
Given a stratified Kähler space LaTeXMLMath with a compatible complex analytic structure , for any stratum , endowed with the complex structure coming from the induced Kähler polarization on that stratum , the restriction map from LaTeXMLMath to that stratum is plainly complex analytic .
Given a stratified Kähler space LaTeXMLMath together with a normal compatible complex analytic structure , if the stratification of LaTeXMLMath ( as a stratified symplectic space ) has an open and dense stratum , the sheaf of germs of holomorphic functions coincides with that of polarized functions .
The open and dense stratum is necessarily contained in the top complex analytic stratum .
A continuous function in LaTeXMLMath which is holomorphic in the strata , in particular in the open and dense stratum , is necessarily itself holomorphic , since on a normal space Riemann ’ s extension theorem holds .
See e. g. Whitney ’ s book LaTeXMLCite or Grauert-Remmert LaTeXMLCite for details .
Consequently a polarized smooth complex function in LaTeXMLMath is necessarily holomorphic .
∎ We do not know whether for a general stratified Kähler space having a compatible complex analytic structure the sheaf of germs of holomorphic functions actually coincides with that of polarized functions , nor do we know whether a stratified Kähler polarization on a stratified symplectic space LaTeXMLMath necessarily determines a compatible complex analytic structure on LaTeXMLMath .
Borrowing some terminology from the language of sheaves , we will say that a smooth space LaTeXMLMath , with smooth structure LaTeXMLMath , is fine provided for an arbitrary locally finite open covering LaTeXMLMath of LaTeXMLMath , there is a partition of unity LaTeXMLMath subordinate to LaTeXMLMath ( i. e. LaTeXMLMath ) with LaTeXMLMath for every LaTeXMLMath ; such a function LaTeXMLMath will then be referred to as a bump function .
The notion of fine space will help us avoid having to talk about the sheaf of germs of smooth functions .
Let LaTeXMLMath be a fine stratified symplectic space , endowed with a complex analytic structure .
Suppose that the symplectic stratification is a refinement of the complex analytic one and that germs of holomorphic functions belong to the stratified symplectic structure , that is , are smooth functions in LaTeXMLMath .
Then the LaTeXMLMath -submodule LaTeXMLMath of LaTeXMLMath generated by differentials of the kind LaTeXMLMath where LaTeXMLMath is holomorphic in LaTeXMLMath and LaTeXMLMath a bump function is a stratified Kähler polarization for LaTeXMLMath , necessarily compatible with the complex analytic structure , if and only if , for every pair LaTeXMLMath of holomorphic functions in LaTeXMLMath , the Poisson bracket LaTeXMLMath vanishes .
Given differentials LaTeXMLMath and LaTeXMLMath where LaTeXMLMath and LaTeXMLMath are bump functions and LaTeXMLMath and LaTeXMLMath holomorphic functions in LaTeXMLMath , LaTeXMLEquation .
Suppose that LaTeXMLMath vanishes for any pair of holomorphic functions LaTeXMLMath and LaTeXMLMath in LaTeXMLMath .
The identity LaTeXMLMath implies that the LaTeXMLMath -submodule LaTeXMLMath of LaTeXMLMath is then closed under the Lie bracket .
Let LaTeXMLMath be a stratum of LaTeXMLMath .
We must show that the LaTeXMLMath -submodule LaTeXMLMath of LaTeXMLMath generated by the image of LaTeXMLMath under ( 2.2 ) passes under ( 2.1 ) ( where LaTeXMLMath plays the role of LaTeXMLMath in ( 2.1 ) ) to the space of sections of an ordinary Kähler polarization on LaTeXMLMath .
Since the problem is local , we may suppose that LaTeXMLMath is the zero locus of a finite set LaTeXMLMath of holomorphic functions on a polydisc LaTeXMLMath in some LaTeXMLMath , with coordinates LaTeXMLMath .
Restricted to the stratum LaTeXMLMath , the holomorphic functions LaTeXMLMath will not be independent but , since LaTeXMLMath is a smooth complex submanifold of LaTeXMLMath , the LaTeXMLMath -submodule LaTeXMLMath of LaTeXMLMath generated by the differentials LaTeXMLMath coincides with the LaTeXMLMath -submodule LaTeXMLMath of LaTeXMLMath generated by the holomorphic differentials in LaTeXMLMath .
More precisely , suitable holomorphic coordinate functions LaTeXMLMath on LaTeXMLMath may be written as holomorphic functions in the LaTeXMLMath , whence the LaTeXMLMath -submodule LaTeXMLMath of LaTeXMLMath generated by the differentials LaTeXMLMath coincides with that generated by the differentials LaTeXMLMath .
By virtue of Proposition 2.3 , the latter generate the Kähler polarization in our sense .
Conversely , when LaTeXMLMath is a stratified Kähler polarization for LaTeXMLMath , given holomorphic functions LaTeXMLMath and LaTeXMLMath in LaTeXMLMath , in view of Proposition 2.3 , the bracket LaTeXMLMath is zero on every stratum , that is , LaTeXMLMath is zero .
∎ Under the circumstances of Theorem 2.5 , the stratified Kähler polarization LaTeXMLMath is determined by the complex analytic structure ; if , on the other hand , LaTeXMLMath determines the complex analytic structure in the sense that the sheaf of germs of polarized functions actually coincides with that of holomorphic functions , LaTeXMLMath will be said to be a complex analytic stratified Kähler polarization , and the corresponding space will be referred to as a complex analytic stratified Kähler space .
A complex analytic stratified Kähler space which , as a complex analytic space , is normal will be said to be a normal Kähler space .
Normal Kähler spaces constitute a particularly nice class of stratified Kähler spaces .
All the examples of stratified Kähler spaces known to us are normal Kähler spaces .
The following is now a mere observation ; yet we spell it out since it looks exceedingly attractive .
A function LaTeXMLMath in a complex analytic stratified Kähler space LaTeXMLMath is holomorphic if and only if LaTeXMLMath for every holomorphic function LaTeXMLMath in LaTeXMLMath .
Moreover , near any point LaTeXMLMath of LaTeXMLMath , the requisite complex analytic stratified Kähler polarization LaTeXMLMath has a finite set of LaTeXMLMath -module generators LaTeXMLMath where LaTeXMLMath are holomorphic functions in LaTeXMLMath defined near LaTeXMLMath , and a function LaTeXMLMath in LaTeXMLMath is holomorphic ( near LaTeXMLMath ) if and only if LaTeXMLEquation .
Under such circumstances , the equations ( 2.6.1 ) may be viewed as Cauchy-Riemann equations for the complex analytic stratified Kähler space LaTeXMLMath .
Thus the Cauchy-Riemann equations make sense even though LaTeXMLMath is not necessarily smooth .
Henceforth every stratified symplectic space will be assumed to be fine .
When LaTeXMLMath is a smooth symplectic manifold , viewed as a ( fine ) stratified symplectic space with a single stratum , Theorem 2.5 comes down to the usual characterization of a Kähler structure on LaTeXMLMath in terms of a Kähler polarization .
However , to attack the question whether a stratified Kähler polarization on a general stratified symplectic space LaTeXMLMath necessarily determines a compatible complex analytic structure on LaTeXMLMath , one would have to develop a theory of obstructions to realizing germs of stratified Kähler spaces as complex analytic germs in a compatible fashion .
In the smooth ( i. e. manifold ) case , there is no such realization problem since manifolds are locally modeled on affine spaces , and complex structures on affine spaces are handled by standard linear algebra .
In this section we will show that the closure of a holomorphic nilpotent orbit inherits a normal complex analytic stratified Kähler structure .
Moreover , we will give a complete classification of holomorphic nilpotent orbits , and we will explicitly describe the real and complex structures of the closure of any holomorphic nilpotent orbit .
3.1 .
Reductive Lie algebras of hermitian type .
Following LaTeXMLCite ( p. 54 ) , we define a ( semisimple ) Lie algebra of hermitian type to be a pair LaTeXMLMath which consists of a real semisimple Lie algebra LaTeXMLMath with a Cartan decomposition LaTeXMLMath and a central element LaTeXMLMath of LaTeXMLMath , referred to as an LaTeXMLMath - element , such that LaTeXMLMath is a ( necessarily LaTeXMLMath -invariant ) complex structure on LaTeXMLMath .
Sometimes we will then say that LaTeXMLMath is of hermitian type , without explicit reference to an LaTeXMLMath -element .
Slightly more generally , a reductive Lie algebra of hermitian type is a reductive Lie algebra LaTeXMLMath together with an element LaTeXMLMath whose constituent LaTeXMLMath ( say ) in the semisimple part LaTeXMLMath of LaTeXMLMath is an LaTeXMLMath -element for LaTeXMLMath LaTeXMLCite ( p. 92 ) .
Given two semisimple ( or reductive ) Lie algebras LaTeXMLMath and LaTeXMLMath of hermitian type , a morphism LaTeXMLMath of Lie algebras is said to be an LaTeXMLMath - homomorphism provided LaTeXMLMath .
For a real semisimple Lie algebra LaTeXMLMath , with Cartan decomposition LaTeXMLMath , we write LaTeXMLMath for its adjoint group and LaTeXMLMath for the ( compact ) connected subgroup of LaTeXMLMath with LaTeXMLMath ; the requirement that LaTeXMLMath be of hermitian type is equivalent to LaTeXMLMath being a ( non-compact ) hermitian symmetric space .
Given LaTeXMLMath , the space of LaTeXMLMath -elements or , equivalently , the manifold of corresponding Cartan decompositions , may be identified with a disjoint union LaTeXMLMath of two copies of the homogeneous space LaTeXMLMath in an obvious fashion , LaTeXMLMath and LaTeXMLMath being the connected components containing LaTeXMLMath and LaTeXMLMath , respectively .
For example , for LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are the two connected components of a standard hyperboloid .
Given a real semisimple Lie algebra LaTeXMLMath of hermitian type , the underlying Lie algebra LaTeXMLMath decomposes as LaTeXMLEquation where LaTeXMLMath is the maximal compact semisimple ideal and where LaTeXMLMath are non-compact and simple .
For a non-compact simple Lie algebra with Cartan decomposition LaTeXMLMath , the LaTeXMLMath -action on LaTeXMLMath coming from the adjoint representation of LaTeXMLMath is faithful and irreducible whence the center of LaTeXMLMath is then at most one-dimensional ; indeed LaTeXMLMath has an LaTeXMLMath -element turning it into a Lie algebra of hermitian type if and only if the center of LaTeXMLMath has dimension one .
In view of E. Cartan ’ s infinitesimal classification of irreducible hermitian symmetric spaces , the classical Lie algebras LaTeXMLMath ( LaTeXMLMath , where LaTeXMLMath ) , LaTeXMLMath ( LaTeXMLMath ) , LaTeXMLMath ( LaTeXMLMath ) , LaTeXMLMath ( LaTeXMLMath ) , LaTeXMLMath ( LaTeXMLMath ) together with the real forms LaTeXMLMath and LaTeXMLMath of the exceptional Lie algebras LaTeXMLMath and LaTeXMLMath , respectively , constitute a complete list of simple Lie algebras of hermitian type .
We refer to LaTeXMLMath , LaTeXMLMath and LaTeXMLMath as the standard cases .
3.2 .
Holomorphic and pseudoholomorphic orbits .
Let LaTeXMLMath be a semisimple Lie algebra of hermitian type .
Then , cf .
LaTeXMLCite , LaTeXMLMath and LaTeXMLMath , that is , the Cartan decomposition LaTeXMLMath is encapsulated in the choice of LaTeXMLMath -element LaTeXMLMath .
As usual , let LaTeXMLMath be the decomposition of the complexification LaTeXMLMath of LaTeXMLMath into LaTeXMLMath - and LaTeXMLMath -eigenspaces of LaTeXMLMath , respectively .
Thus LaTeXMLMath is the span of LaTeXMLMath ( LaTeXMLMath ) and LaTeXMLMath that of LaTeXMLMath ( LaTeXMLMath ) where LaTeXMLMath runs through LaTeXMLMath .
The association LaTeXMLMath is an isomorphism of complex vector spaces from LaTeXMLMath , endowed with the complex structure LaTeXMLMath , onto LaTeXMLMath .
We will refer to an adjoint orbit LaTeXMLMath having the property that the projection map from LaTeXMLMath to LaTeXMLMath , restricted to LaTeXMLMath , is a diffeomorphism onto its image , as a pseudoholomorphic orbit .
In Lemma 3.3.6 below we shall show that a pseudoholomorphic nilpotent orbit maps ( diffeomorphically ) onto a smooth complex submanifold of LaTeXMLMath , and in ( 3.4 ) below we shall show that a non-nilpotent pseudoholomorphic orbit necessarily maps diffeomorphically onto LaTeXMLMath .
Thus a pseudoholomorphic orbit LaTeXMLMath inherits a complex structure from the complex structure LaTeXMLMath on LaTeXMLMath , and this complex structure , combined with the Kostant-Kirillov-Souriau form on LaTeXMLMath , viewed as a coadjoint orbit by means of ( a positive multiple of ) the Killing form , turns LaTeXMLMath into a ( not necessarily positive ) Kähler manifold .
Indeed , a choice of ( complex ) basis LaTeXMLMath of LaTeXMLMath yields holomorphic coordinate functions on LaTeXMLMath which , together with their complex conjugates LaTeXMLMath , may be used as coordinates for the smooth complex functions on LaTeXMLMath ; since , by assumption , the composite of the injection of LaTeXMLMath into LaTeXMLMath ( identified with LaTeXMLMath via a suitable multiple of the Killing form , see below ) , combined with the projection to LaTeXMLMath , is a diffeomorphism onto its image , these functions yield coordinate functions on LaTeXMLMath which , with an abuse of notation , we still write as LaTeXMLMath and LaTeXMLMath ; we shall see below that every smooth complex function on LaTeXMLMath may be written as a function in these variables .
Since the constituents LaTeXMLMath and LaTeXMLMath are abelian subalgebras of LaTeXMLMath , cf .
LaTeXMLCite ( p. 313 ) , LaTeXMLCite ( p. 55 ) , on LaTeXMLMath the Poisson-Lie brackets LaTeXMLMath vanish since the Poisson bracket on LaTeXMLMath is induced from the Lie bracket on LaTeXMLMath .
Furthermore , since LaTeXMLMath , the Poisson-Lie brackets LaTeXMLMath are given by the restriction to LaTeXMLMath of the resulting complex functions on LaTeXMLMath determined by the Lie bracket in LaTeXMLMath .
In view of Proposition 2.3 , the complex structure and the symplectic structure corresponding to the Poisson structure thus combine to a ( not necessarily positive ) Kähler structure on LaTeXMLMath ; this will be made precise in Theorem 3.7.1 below .
For nilpotent orbits , we will make this somewhat roundabout reasoning more perspicuous in Theorem 3.3.11 via the classification given in Theorem 3.3.3 below .
We now choose a positive multiple of the Killing form .
We will say that a pseudoholomorphic orbit LaTeXMLMath is holomorphic provided the resulting Kähler structure on LaTeXMLMath is positive .
Any positive multiple of the Killing form will do ; passing from one choice to another one amounts to rescaling the resulting hermitian form .
The notion of holomorphic orbit relies on the choice of LaTeXMLMath -element LaTeXMLMath ( while the notion of pseudoholomorphic orbit does not ) ; given a holomorphic orbit LaTeXMLMath , when LaTeXMLMath is replaced by LaTeXMLMath , LaTeXMLMath ( and not LaTeXMLMath ) will be holomorphic with respect to LaTeXMLMath .
Sometimes we will refer to an orbit which is holomorphic with respect to LaTeXMLMath as an antiholomorphic orbit .
The name “ holomorphic ” is intended to hint at the fact that the holomorphic discrete series representations arise from holomorphic quantization on integral semisimple holomorphic orbits but , beware , the requisite complex structure is not the one arising from projection to the symmetric constituent of the Cartan decomposition .
In particular , we can talk about holomorphic nilpotent orbits .
A different definition of holomorphic nilpotent orbit may be found in LaTeXMLCite which , as we will show in Remark 3.3.13 , is equivalent to ours .
Henceforth the closure is always understood to be taken in the ordinary ( not Zariski ) topology .
Let LaTeXMLMath be a semisimple Lie algebra of hermitian type .
With reference to the decomposition ( 3.1.1 ) , any ( pseudo ) holomorphic nilpotent orbit LaTeXMLMath in LaTeXMLMath may be written as LaTeXMLMath where , for LaTeXMLMath , each LaTeXMLMath is a ( pseudo ) holomorphic nilpotent orbit in LaTeXMLMath , with reference to the component LaTeXMLMath of LaTeXMLMath in LaTeXMLMath .
Consequently , any orbit LaTeXMLMath in the closure of LaTeXMLMath may be written as LaTeXMLMath where , for LaTeXMLMath , each LaTeXMLMath is in the closure of LaTeXMLMath .
This reduces the study of general ( pseudo ) holomorphic nilpotent orbits ( and of their closures ) to that of ( pseudo ) holomorphic nilpotent orbits in simple hermitian Lie algebras .
We now spell out the main result of this section .
Given a semisimple Lie algebra LaTeXMLMath of hermitian type , for any holomorphic nilpotent orbit LaTeXMLMath , the diffeomorphism from LaTeXMLMath onto its image in LaTeXMLMath extends to a homeomorphism from the closure LaTeXMLMath onto its image in LaTeXMLMath , this homeomorphism turns LaTeXMLMath into a complex affine variety in such a way that , LaTeXMLMath being suitably identified with its dual so that LaTeXMLMath is identified with the closure of a coadjoint orbit , the following hold : The induced complex analytic structure on LaTeXMLMath combines with the resulting Poisson structure to a normal complex analytic stratified Kähler structure , and the closure LaTeXMLMath is a union of finitely many holomorphic nilpotent orbits .
Furthermore , when LaTeXMLMath is simple , a pseudoholomorphic nilpotent orbit LaTeXMLMath having the property that the diffeomorphism from LaTeXMLMath onto its image in LaTeXMLMath extends to a homeomorphism from the closure LaTeXMLMath onto its image in LaTeXMLMath is necessarily holomorphic or antiholomorphic .
Theorem 3.2.1 will be a consequence of Theorem 3.3.3 below save that the complex analytic stratified Kähler structure will be explained in Subsection 3.7 below .
In the classical cases , the real and complex analytic structures of the closure LaTeXMLMath of any holomorphic nilpotent orbit LaTeXMLMath will be elucidated in Theorems 3.5.4 , 3.5.5 , and 3.6.2 below .
The normality will be explained in Theorems 3.5.5 , 3.6.3 , 5.3.3 , 8.4.1 , 8.4.2 .
3.2.2 .
The case of LaTeXMLMath .
This Lie algebra is ( well known to be ) of hermitian type , in the following fashion : The nilpotent elements LaTeXMLMath and LaTeXMLMath together with the semisimple one LaTeXMLMath span LaTeXMLMath , subject to the relations LaTeXMLEquation and LaTeXMLMath is an LaTeXMLMath -element .
The Cartan decomposition determined by the choice of LaTeXMLMath has the form LaTeXMLMath in such a way that LaTeXMLMath spans LaTeXMLMath and LaTeXMLMath and LaTeXMLMath span LaTeXMLMath .
The induced complex structure LaTeXMLMath on LaTeXMLMath is given by LaTeXMLEquation and LaTeXMLMath is a basis of LaTeXMLMath , viewed as a complex vector space .
Thus LaTeXMLMath is of hermitian type .
More generally , for any LaTeXMLMath , the product LaTeXMLMath of LaTeXMLMath copies of LaTeXMLMath is of hermitian type .
For intelligibility we reconcile the ( well known ) classification of orbits in LaTeXMLMath with the notions of holomorphic and antiholomorphic orbit .
The nilcone in LaTeXMLMath consists of all LaTeXMLMath with LaTeXMLMath and decomposes into the disjoint union of the orbits LaTeXMLMath and LaTeXMLMath ( say ) of LaTeXMLMath and LaTeXMLMath , respectively , together with the orbit which consists merely of the origin .
For LaTeXMLMath , the two-sheeted hyperboloid consisting of all LaTeXMLMath with LaTeXMLMath decomposes into the two elliptic LaTeXMLMath -orbits of LaTeXMLMath and LaTeXMLMath which we denote by LaTeXMLMath and LaTeXMLMath , respectively .
We will now show that the orbits LaTeXMLMath and LaTeXMLMath are holomorphic and antiholomorphic , respectively ( LaTeXMLMath ) .
To this end , let LaTeXMLMath , LaTeXMLMath , LaTeXMLMath .
For LaTeXMLMath , LaTeXMLMath and LaTeXMLMath are the LaTeXMLMath -orbits of LaTeXMLMath and LaTeXMLMath , respectively and , when LaTeXMLMath denote the coordinate functions in this basis , LaTeXMLMath .
Thus , for LaTeXMLMath , LaTeXMLEquation .
LaTeXMLEquation The Lie algebra LaTeXMLMath may be identified with its dual via any positive multiple of the Killing form LaTeXMLMath ( LaTeXMLMath ) ; the positivity is necessary in order for the isomorphism from LaTeXMLMath to LaTeXMLMath to preserve the natural orientations given by the complex structures .
To obtain simple formulas , we will now identify LaTeXMLMath and LaTeXMLMath with their duals by means of the adjoint LaTeXMLEquation of the trace pairing ( which is one fourth of the Killing form ) , that is , for LaTeXMLMath , LaTeXMLMath .
Then LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , or , equivalently , when LaTeXMLMath are viewed as linear functions on LaTeXMLMath , LaTeXMLEquation .
The Poisson structure on LaTeXMLMath induced via LaTeXMLMath is determined by ( 3.2.2.1 ) and is therefore given by the formulas LaTeXMLEquation .
The orthogonal projection from LaTeXMLMath to the plane LaTeXMLMath , restricted to LaTeXMLMath or to LaTeXMLMath ( LaTeXMLMath ) , is plainly a diffeomorphism onto its image in LaTeXMLMath which is onto LaTeXMLMath for LaTeXMLMath ; for LaTeXMLMath , this diffeomorphism extends to a homeomorphism from LaTeXMLMath as well as from LaTeXMLMath onto LaTeXMLMath .
For LaTeXMLMath , we now parametrize the orbits LaTeXMLMath by the plane LaTeXMLMath .
For LaTeXMLMath , we take LaTeXMLMath and LaTeXMLMath as coordinate functions for the plane LaTeXMLMath while , for LaTeXMLMath , the requisite ( non-standard ) smooth structure LaTeXMLMath on LaTeXMLMath is the algebra of smooth functions in the variables LaTeXMLMath , subject to the relation LaTeXMLMath .
The Poisson structures induced on LaTeXMLMath from the Poisson structures on the orbits via the parametrizations are now given by LaTeXMLEquation when LaTeXMLMath and by ( 3.2.2.2 ) , subject to the relation LaTeXMLMath , when LaTeXMLMath .
In the latter case , the Poisson structure extends the symplectic Poisson structure on LaTeXMLMath ( as well as that on LaTeXMLMath ) to one on LaTeXMLMath ( or LaTeXMLMath ) ( cf .
LaTeXMLCite ) .
The induced complex structure LaTeXMLMath on LaTeXMLMath is standard , with holomorphic coordinate LaTeXMLMath .
Consequently , when LaTeXMLMath is used to parametrize LaTeXMLMath , the resulting hermitian form may be written as LaTeXMLMath and , when LaTeXMLMath is used to parametrize LaTeXMLMath , this form may accordingly be written as LaTeXMLMath .
Thus , the orbits LaTeXMLMath are holomorphic and the orbits LaTeXMLMath are antiholomorphic , whatever LaTeXMLMath , and the Poisson and complex analytic structures turn LaTeXMLMath into a normal positive Kähler space and , likewise , LaTeXMLMath into a normal “ negative ” Kähler space .
Moreover , for LaTeXMLMath , the Kähler manifold LaTeXMLMath contracts into the normal Kähler space LaTeXMLMath and , likewise , LaTeXMLMath contracts into LaTeXMLMath .
Hence , LaTeXMLMath being viewed as a deformation parameter , the singular normal Kähler structure on LaTeXMLMath deforms into an ordinary Kähler structure ; under this deformation , the non-standard Poisson structure “ resolves ” into an ordinary smooth symplectic Poisson structure while the complex analytic structure is not deformed .
In Section 9 we will recall the notion of contraction and show how this relationship between LaTeXMLMath and the LaTeXMLMath ’ s ( LaTeXMLMath ) extends to arbitrary holomorphic nilpotent orbits .
For LaTeXMLMath , the complex structure LaTeXMLMath ( say ) on LaTeXMLMath and LaTeXMLMath induced from the complex structure LaTeXMLMath on LaTeXMLMath via the projection to LaTeXMLMath is not the hermitian symmetric space structure .
In order to see this , we note that , for LaTeXMLMath , the stereographic projection LaTeXMLEquation identifies each LaTeXMLMath with the generalized Poincaré disc LaTeXMLMath in the LaTeXMLMath -plane , generalized in the sense that , in the literature , for LaTeXMLMath , this is the familiar Poincaré disc ; see e. g. ( 3.7.5 ) in LaTeXMLCite .
A straightforward calculation gives LaTeXMLMath where LaTeXMLMath whence the induced symplectic structure LaTeXMLMath on LaTeXMLMath may be written as LaTeXMLMath .
This is the Kähler form of the hermitian metric LaTeXMLMath on LaTeXMLMath with holomorphic coordinate LaTeXMLMath to which the hermitian symmetric space hermitian form on LaTeXMLMath passes under the stereographic projection .
Thus the latter is a symplectomorphism but it is not compatible with the complex structures when LaTeXMLMath is endowed with the complex structure LaTeXMLMath since then the coordinate change LaTeXMLMath is not holomorphic .
Thus , when endowed with the Kähler structure determined by LaTeXMLMath , LaTeXMLMath is only symplectically a model for the Siegel space ( the space of complex structures on the plane compatible with its standard symplectic structure ) .
Remark .
The hermitian symmetric space Kähler metric on LaTeXMLMath ( equivalently : on LaTeXMLMath , with the metric LaTeXMLMath ) has constant negative curvature equal to LaTeXMLMath while the hermitian form LaTeXMLMath has strictly negative curvature which is constant only along LaTeXMLMath -orbits , LaTeXMLMath being the maximal compact subgroup of LaTeXMLMath determined by the choice of LaTeXMLMath .
Indeed , a calculation shows that , for LaTeXMLMath , LaTeXMLMath has curvature equal to LaTeXMLMath ; notice that LaTeXMLMath when LaTeXMLMath .
This shows again that , for LaTeXMLMath , LaTeXMLMath does not yield the hermitian symmetric space structure on LaTeXMLMath .
Moreover , this formula entails that , in particular , LaTeXMLMath ( as well as LaTeXMLMath ) is flat ( has zero curvature ) .
3.3 .
Classification of pseudoholomorphic nilpotent orbits .
Let LaTeXMLMath be a real Lie algebra .
As usual , we will refer to a triple LaTeXMLMath of elements of LaTeXMLMath as an LaTeXMLMath - triple in LaTeXMLMath provided LaTeXMLMath .
An LaTeXMLMath -triple LaTeXMLMath plainly determines and is determined by a homomorphism LaTeXMLMath of real Lie algebras via LaTeXMLMath .
Suppose in addition that LaTeXMLMath is semisimple , with Cartan involution LaTeXMLMath and Cartan decomposition LaTeXMLMath .
We will say that an LaTeXMLMath -triple LaTeXMLMath in LaTeXMLMath is invariant provided LaTeXMLMath and LaTeXMLMath .
An LaTeXMLMath -triple LaTeXMLMath in LaTeXMLMath is plainly invariant if and only if the corresponding Lie algebra homomorphism LaTeXMLMath is compatible with the Cartan involutions .
Let LaTeXMLMath be a simple Lie algebra of Hermitian type , with Cartan decomposition LaTeXMLMath .
We will say that an invariant LaTeXMLMath -triple is holomorphic or an LaTeXMLMath -triple provided the Lie algebra homomorphism from LaTeXMLMath to LaTeXMLMath it determines is an LaTeXMLMath -homomorphism .
An invariant LaTeXMLMath -triple LaTeXMLMath is plainly an LaTeXMLMath -triple if and only if LaTeXMLEquation .
We will deduce the classification of holomorphic nilpotent orbits in LaTeXMLMath from the relative root system .
We will systematically denote the split rank , also called real rank ( the dimension of a maximal abelian subalgebra of LaTeXMLMath ) by LaTeXMLMath .
The following result is a version of an observation spelled out as Remark 3 on p. 62 of LaTeXMLCite and we therefore label it as a Proposition .
This observation , in turn , is a consequence of the existence of LaTeXMLMath strongly orthogonal ( absolute ) roots relative to the complexification of a compact Cartan subalgebra ( referred to as well as compactly embedded Cartan subalgebra in the literature ) , cf .
Lemma II.4.3 on p. 60 of LaTeXMLCite .
Given a simple Lie algebra LaTeXMLMath of Hermitian type and split rank LaTeXMLMath , with Cartan decomposition LaTeXMLMath , a choice of maximal abelian subalgebra LaTeXMLMath of LaTeXMLMath determines a system LaTeXMLMath , … , LaTeXMLMath of LaTeXMLMath -triples which combine to an LaTeXMLMath -embedding of LaTeXMLMath into LaTeXMLMath in such a way that the following hold .
1 ) The elements LaTeXMLMath constitute a basis of LaTeXMLMath .
2 ) With the notation LaTeXMLMath for the basis of LaTeXMLMath dual to LaTeXMLMath , the relative root system LaTeXMLMath determined by LaTeXMLMath is given by LaTeXMLEquation 3 ) The LaTeXMLMath -element LaTeXMLMath of LaTeXMLMath equals LaTeXMLMath if and only if LaTeXMLMath is of type LaTeXMLMath .
4 ) The LaTeXMLMath -embedding of LaTeXMLMath into LaTeXMLMath extends to an LaTeXMLMath -embedding of LaTeXMLMath into LaTeXMLMath , LaTeXMLMath being endowed with the appropriate LaTeXMLMath -element , and this embedding identifies the two LaTeXMLMath -elements if and only if LaTeXMLMath is of type LaTeXMLMath .
We will say that a simple Lie algebra of hermitian type is regular provided its relative root system is of type LaTeXMLMath , and we will say that it is non-regular otherwise .
This notion of regularity fits with that of regularity of pre-homogeneous spaces to be examined in Section 7 below .
For the corresponding symmetric domain , regularity amounts to the property of being a tube domain .
To adjust the notation on p. 110 of LaTeXMLCite to ours , recall that in LaTeXMLCite ( p. 92 and p. 109 ) certain root vectors LaTeXMLMath ( LaTeXMLMath ) are given satisfying the equations LaTeXMLEquation .
Then the triples LaTeXMLMath given by LaTeXMLEquation ( cf .
III.1.6 .
on p. 91 and III.2.9 on p. 97 of LaTeXMLCite ) will have the properties asserted in 1 ) and 2 ) .
The statement 3 ) is a consequence of Corollary III.1.6 on p. 94 , combined with Remark 1 in III.8 on p. 150 of LaTeXMLCite .
To justify statement 4 ) we observe that , in the regular case , appropriate root vectors for the relative root system LaTeXMLMath span a copy of LaTeXMLMath in LaTeXMLMath .
In the non-regular case , the relative root system LaTeXMLMath contains the subsystem LaTeXMLMath and , likewise , appropriate root vectors for the latter span a copy of LaTeXMLMath in LaTeXMLMath .
∎ Remark .
For intelligibility , we list the various types of the relative root systems that occur here , cf .
LaTeXMLCite p. 115 : LaTeXMLMath : LaTeXMLMath ; LaTeXMLMath , LaTeXMLMath : LaTeXMLMath ; LaTeXMLMath : LaTeXMLMath LaTeXMLMath ; LaTeXMLMath : LaTeXMLMath ; LaTeXMLMath : LaTeXMLMath ; LaTeXMLMath : LaTeXMLMath ; LaTeXMLMath : LaTeXMLMath .
Given a simple Lie algebra LaTeXMLMath , with adjoint group LaTeXMLMath , for LaTeXMLMath , let LaTeXMLMath and let LaTeXMLMath ; this is plainly a nilpotent orbit in LaTeXMLMath .
We will refer to LaTeXMLMath as the type of LaTeXMLMath , to LaTeXMLMath as its rank , and to LaTeXMLMath as its signature .
For later reference , we will write LaTeXMLMath ( LaTeXMLMath ) .
For a simple Lie algebra LaTeXMLMath of hermitian type and split rank LaTeXMLMath , the LaTeXMLMath nilpotent orbits LaTeXMLMath LaTeXMLMath constitute a complete list of the pseudoholomorphic nilpotent orbits .
Given such an orbit LaTeXMLMath , an orbit LaTeXMLMath distinct from LaTeXMLMath is in the closure of LaTeXMLMath if and only if LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
In particular , the orbits LaTeXMLMath LaTeXMLMath and LaTeXMLMath LaTeXMLMath are precisely the holomorphic and antiholomorphic ones , respectively .
For a nilpotent orbit LaTeXMLMath of a simple Lie algebra LaTeXMLMath of hermitian type , the property of being holomorphic , antiholomorphic , or pseudoholomorphic does not depend on the choice of LaTeXMLMath -element LaTeXMLMath .
More precisely : Once an LaTeXMLMath -element LaTeXMLMath has been chosen , when LaTeXMLMath is holomorphic or antiholomorphic with respect to LaTeXMLMath , it will be so with respect to any LaTeXMLMath in the space LaTeXMLMath of LaTeXMLMath -elements ; when LaTeXMLMath is pseudoholomorphic with respect to LaTeXMLMath , it will be so with respect to any LaTeXMLMath in the space LaTeXMLMath .
The proof requires some preparation : Following the procedure in III.4 of LaTeXMLCite ( p. 110 ) , we choose the basis LaTeXMLMath of LaTeXMLMath where LaTeXMLMath for LaTeXMLMath and where LaTeXMLMath in the regular case and LaTeXMLMath in the non-regular case .
Let LaTeXMLMath be the resulting decomposition into positive and negative relative roots , so that LaTeXMLEquation .
For LaTeXMLMath , let LaTeXMLMath be its root space and , for any LaTeXMLMath , write LaTeXMLMath .
Thus LaTeXMLMath decomposes as LaTeXMLMath where LaTeXMLMath is the centralizer of LaTeXMLMath in LaTeXMLMath , LaTeXMLMath being the centralizer of LaTeXMLMath in LaTeXMLMath .
With the notation LaTeXMLMath , the resulting Iwasawa decomposition has the form LaTeXMLMath ; accordingly , we write LaTeXMLMath for the corresponding global Iwasawa decomposition—in particular , this fixes the nilpotent group LaTeXMLMath —and , for later reference , we write LaTeXMLMath .
For LaTeXMLMath , let LaTeXMLEquation this yields the LaTeXMLMath -triple LaTeXMLMath and we write LaTeXMLMath for the resulting LaTeXMLMath -homomorphism .
Here and below the ordering of the LaTeXMLMath -triples LaTeXMLMath ( LaTeXMLMath ) does not matter .
For LaTeXMLMath , the parabolic subalgebra LaTeXMLMath of LaTeXMLMath ( written as LaTeXMLMath in LaTeXMLCite III.2 ( p. 95 ) ) which is determined by LaTeXMLMath or , what amounts to the same , by LaTeXMLMath , has Levi decomposition LaTeXMLMath where LaTeXMLMath is the ( reductive ) centralizer of LaTeXMLMath in LaTeXMLMath , and LaTeXMLEquation .
Let LaTeXMLMath , LaTeXMLMath , for LaTeXMLMath , let LaTeXMLEquation and , more generally , given LaTeXMLMath , let LaTeXMLEquation .
We note that LaTeXMLMath and LaTeXMLMath correspond to LaTeXMLMath and LaTeXMLMath in III.2 of LaTeXMLCite ( p. 95 ff .
) , respectively .
The following is well known ( and easy to prove ) .
As a vector space , the nilpotent Lie algebra LaTeXMLMath decomposes as LaTeXMLEquation and the decomposition yields a graded Lie algebra ( “ graded ” being understood in the obvious naive sense ) whence , in particular , LaTeXMLMath is abelian .
Furthermore , in the regular case , LaTeXMLMath is zero while , in the non-regular case , LaTeXMLMath which , as a vector space , is the direct sum LaTeXMLMath , may be written as a central Lie algebra extension LaTeXMLMath with abelian quotient , i. e. as a Heisenberg algebra .
More generally , for LaTeXMLMath , as a vector space , LaTeXMLMath decomposes as LaTeXMLMath .
∎ To spell out the next lemma we recall that the linear LaTeXMLMath -action on LaTeXMLMath , viewed as a complex vector space via the complex structure LaTeXMLMath , extends canonically to a linear LaTeXMLMath -action on this vector space .
Each nilpotent orbit LaTeXMLMath ( where LaTeXMLMath ) is pseudoholomorphic , that is , the projection from LaTeXMLMath to LaTeXMLMath is a diffeomorphism onto its image in LaTeXMLMath , and the image of LaTeXMLMath in LaTeXMLMath is the LaTeXMLMath -orbit of LaTeXMLMath .
Write LaTeXMLMath and LaTeXMLMath for the span of LaTeXMLMath and LaTeXMLMath , respectively .
We claim that LaTeXMLMath .
Indeed , LaTeXMLMath is the union of the orbits LaTeXMLMath ( LaTeXMLMath ) .
By Lemma 3.3.5 , LaTeXMLMath ; hence LaTeXMLMath since LaTeXMLMath acts trivially on LaTeXMLMath .
Furthermore , LaTeXMLMath is closed under the LaTeXMLMath -action .
Thus , by virtue of the Iwasawa decomposition LaTeXMLMath of LaTeXMLMath , LaTeXMLMath as asserted .
Consequently the orbit LaTeXMLMath decomposes into the LaTeXMLMath -orbits of the elements LaTeXMLEquation where LaTeXMLMath with LaTeXMLMath ( LaTeXMLMath ) .
In view of the action of the relative Weyl group , which contains a copy of the symmetric group on LaTeXMLMath letters , we normalize LaTeXMLMath by requiring LaTeXMLMath and LaTeXMLMath .
Likewise the orbit LaTeXMLMath decomposes into the LaTeXMLMath -orbits of the elements LaTeXMLEquation where LaTeXMLMath with LaTeXMLMath ( LaTeXMLMath ) .
The image in LaTeXMLMath of LaTeXMLMath under the orthogonal projection to LaTeXMLMath is the element LaTeXMLMath .
Using the ( known ) structure of the stabilizer of any nilpotent element in a semisimple Lie algebra , cf .
e. g. Theorem 1.7 in LaTeXMLCite , we conclude that the stabilizer LaTeXMLMath in LaTeXMLMath of LaTeXMLMath coincides with the stabilizer LaTeXMLMath in LaTeXMLMath of LaTeXMLMath .
More precisely : The nilpotent element LaTeXMLMath belongs to the invariant LaTeXMLMath -triple LaTeXMLMath where LaTeXMLEquation and LaTeXMLMath ; we note that LaTeXMLMath is an LaTeXMLMath -triple if and only if LaTeXMLMath and if LaTeXMLMath .
The stabilizer LaTeXMLMath of LaTeXMLMath in the adjoint group LaTeXMLMath of LaTeXMLMath is contained in the Jacobson-Morozow parabolic subgroup LaTeXMLMath ( say ) associated with LaTeXMLMath ( where LaTeXMLMath is the unipotent radical and LaTeXMLMath the Levi subgroup ) , and the reductive constituent of LaTeXMLMath is contained in the ( reductive ) stabilizer LaTeXMLMath of the entire triple ( where LaTeXMLMath refers to the copy of LaTeXMLMath in LaTeXMLMath generated by LaTeXMLMath ) whence LaTeXMLMath is contained in LaTeXMLMath , in fact , equals LaTeXMLMath ; the notation in LaTeXMLCite is LaTeXMLMath for our LaTeXMLMath .
Noting that , with reference to the Cartan involution LaTeXMLMath on LaTeXMLMath , LaTeXMLMath , we must show that the inclusion LaTeXMLMath is the identity .
In order to justify this assertion , we note first that , when LaTeXMLMath and LaTeXMLMath , we have LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath ; this implies that LaTeXMLMath ( since LaTeXMLMath commutes with the adjoint action of LaTeXMLMath on LaTeXMLMath ) where LaTeXMLMath refers to the Levi subgroup of the parabolic subgroup LaTeXMLMath associated with the LaTeXMLMath -triple LaTeXMLMath .
Thus LaTeXMLMath now coincides with the stabilizer in LaTeXMLMath of LaTeXMLMath .
The latter , in turn , equals the stabilizer LaTeXMLMath of LaTeXMLMath in LaTeXMLMath ; since the factor LaTeXMLMath is irrelevant , we conclude that LaTeXMLMath is equal to LaTeXMLMath in this special case .
Next we consider the case where LaTeXMLMath and LaTeXMLMath .
Noting that LaTeXMLMath and LaTeXMLMath are conjugate to LaTeXMLMath and LaTeXMLMath , respectively , in the complexification LaTeXMLMath , taking suitable conjugates of stabilizers in the complexification of LaTeXMLMath , we conclude that LaTeXMLMath coincides with LaTeXMLMath .
Finally , we may reduce the general case to the one just established by noting that LaTeXMLMath and LaTeXMLMath arise from LaTeXMLMath and LaTeXMLMath respectively as the result of the adjoint action with a suitable element from the constituent LaTeXMLMath in the Iwasawa-decomposition LaTeXMLMath .
Consequently LaTeXMLMath coincides with LaTeXMLMath in the general case .
We conclude that the projection from LaTeXMLMath onto its image in LaTeXMLMath is a diffeomorphism .
Furthermore , with the above normalization , the orbits LaTeXMLMath ( LaTeXMLMath and LaTeXMLMath ) are mutually disjoint , and their union coincides with the LaTeXMLMath -orbit of LaTeXMLMath in LaTeXMLMath as may be see from writing out the LaTeXMLMath -action on LaTeXMLMath in terms of the polar decomposition LaTeXMLMath of LaTeXMLMath .
In particular , given LaTeXMLMath and LaTeXMLMath , for some LaTeXMLMath , we have LaTeXMLEquation .
Hence , for LaTeXMLMath with LaTeXMLMath , the projection from LaTeXMLMath onto its image in LaTeXMLMath is a diffeomorphism onto the LaTeXMLMath -orbit of LaTeXMLMath in LaTeXMLMath .
In particular , LaTeXMLMath is pseudoholomorphic as asserted .
∎ Remark .
Let LaTeXMLMath be the ( reductive ) centralizer of LaTeXMLMath in the adjoint group LaTeXMLMath and let LaTeXMLMath .
The union of the orbits LaTeXMLMath ( LaTeXMLMath ) equals LaTeXMLMath , and the projection to LaTeXMLMath is induced by the canonical surjection from LaTeXMLMath to LaTeXMLMath given by the assignment to LaTeXMLMath of LaTeXMLMath where LaTeXMLMath refers to the image of LaTeXMLMath under the projection to LaTeXMLMath .
From this observation , one may as well deduce that the projection from LaTeXMLMath onto its image in LaTeXMLMath is a diffeomorphism onto the LaTeXMLMath -orbit of LaTeXMLMath in LaTeXMLMath .
For example , for LaTeXMLMath , the real rank LaTeXMLMath equals LaTeXMLMath , LaTeXMLMath amounts to the space LaTeXMLMath of real symmetric LaTeXMLMath -matrices , LaTeXMLMath to that of complex symmetric LaTeXMLMath -matrices LaTeXMLMath , associating LaTeXMLMath to LaTeXMLMath amounts to associating to a real symmetric LaTeXMLMath -matrix the same matrix , viewed as a complex symmetric LaTeXMLMath -matrix , and the canonical surjection from LaTeXMLMath to LaTeXMLMath amounts to the familiar map from LaTeXMLMath to LaTeXMLMath .
The restriction of this map to the LaTeXMLMath -span of any LaTeXMLMath -orbit of non-degenerate real symmetric LaTeXMLMath -matrices is a diffeomorphism onto the space of non-degenerate complex symmetric LaTeXMLMath -matrices .
Lemma 3.3.6 says that the orbits LaTeXMLMath are pseudoholomorphic .
It remains to justify the claim as to the closures of the orbits LaTeXMLMath and to explain how the holomorphic and antiholomorphic orbits are singled out .
To this end we observe that every nilpotent orbit of the semisimple Lie algebra of hermitian type LaTeXMLMath is pseudoholomorphic ; these orbits are generated by the LaTeXMLMath ’ s where LaTeXMLMath ( LaTeXMLMath ) .
The classification of holomorphic orbits of LaTeXMLMath reproduced in ( 3.2.2 ) above entails at once that the orbits involving only LaTeXMLMath are the holomorphic ones and that , likewise , the orbits involving only LaTeXMLMath are the antiholomorphic ones .
Furthermore , when LaTeXMLMath is the LaTeXMLMath -orbit of LaTeXMLMath , the orbits in the closure of LaTeXMLMath are precisely the orbits generated by those LaTeXMLMath -tuples which arise from LaTeXMLMath by replacing some LaTeXMLMath with zero .
However , within LaTeXMLMath , the relative Weyl group LaTeXMLMath permutes the LaTeXMLMath copies of LaTeXMLMath and hence in particular the entries of the LaTeXMLMath whence , within LaTeXMLMath , what remains invariant under the relative Weyl group , are the rank LaTeXMLMath and the signature LaTeXMLMath .
Thus an orbit LaTeXMLMath distinct from LaTeXMLMath is in the closure of LaTeXMLMath if and only if LaTeXMLMath for some LaTeXMLMath and LaTeXMLMath with LaTeXMLMath .
Furthermore , since the complex structure LaTeXMLMath on LaTeXMLMath is LaTeXMLMath -invariant , the orbits LaTeXMLMath ( LaTeXMLMath ) and LaTeXMLMath ( LaTeXMLMath ) are precisely the holomorphic and antiholomorphic ones , respectively .
Finally we establish the fact that the orbits LaTeXMLMath LaTeXMLMath exhaust the pseudoholomorphic nilpotent orbits .
Inspection of the possible nilpotent orbits in LaTeXMLMath shows that , given a nilpotent orbit which is different from one of the kind LaTeXMLMath , the projection to the constituent LaTeXMLMath of the Cartan decomposition of LaTeXMLMath is no longer injective .
For the classical cases we postpone the details to the proofs of Theorems 3.5.3 and 3.6.2 below .
We now settle the two exceptional cases LaTeXMLMath and LaTeXMLMath by inspection .
As before , LaTeXMLMath denotes the adjoint group of the corresponding simple Lie algebra LaTeXMLMath under discussion .
The Lie algebra LaTeXMLMath has 12 non-zero nilpotent orbits .
The orbits numbered ( 1 ) to ( 5 ) in Table X of LaTeXMLCite ( p. 512 ) are precisely the pseudoholomorphic ones ; with the appropriate choice of LaTeXMLMath -element , the orbits ( 1 ) and ( 3 ) coincide with the orbits LaTeXMLMath and LaTeXMLMath , respectively , the orbits ( 2 ) and ( 4 ) are antiholomorphic and amount to the orbits LaTeXMLMath and LaTeXMLMath , while the orbit ( 5 ) is our pseudoholomorphic nilpotent orbit LaTeXMLMath which is neither holomorphic nor antiholomorphic .
There are no other pseudoholomorphic nilpotent orbits in LaTeXMLMath .
Indeed , in this case , LaTeXMLMath , and column 4 ) of Table X of LaTeXMLCite gives the complex dimensions of the stabilizers of the corresponding LaTeXMLMath -orbits in LaTeXMLMath where LaTeXMLMath is the maximal compact subgroup of the adjoint group LaTeXMLMath .
Since LaTeXMLMath , inspection of the dimensions of the stabilizers shows that , under the projection from LaTeXMLMath to LaTeXMLMath , none of the orbits ( 6 ) – ( 12 ) can map diffeomorphically onto its image in LaTeXMLMath .
The Lie algebra LaTeXMLMath has 22 non-zero nilpotent orbits .
The orbits numbered ( 1 ) to ( 9 ) in Table XIII of LaTeXMLCite ( p. 516 ) are precisely the pseudoholomorphic ones ; with the appropriate choice of LaTeXMLMath -element , the orbits ( 1 ) , ( 3 ) , and ( 6 ) coincide with , respectively , the orbits LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , the orbits ( 2 ) , ( 4 ) , and ( 7 ) are antiholomorphic and amount to , respectively , the orbits LaTeXMLMath , LaTeXMLMath , and LaTeXMLMath , while the remaining orbits ( 5 ) , ( 8 ) , ( 9 ) amount to our pseudoholomorphic nilpotent orbits respectively LaTeXMLMath , LaTeXMLMath , LaTeXMLMath , which are neither holomorphic nor antiholomorphic .
There are no other pseudoholomorphic nilpotent orbits in LaTeXMLMath .
Indeed , in this case , LaTeXMLMath , and the complex dimensions of the stabilizers of the corresponding LaTeXMLMath -orbits in LaTeXMLMath may be found in column 4 ) of Table X of LaTeXMLCite where LaTeXMLMath is the maximal compact subgroup of the adjoint group LaTeXMLMath ; the group LaTeXMLMath is locally isomorphic to LaTeXMLMath ( LaTeXMLMath being a compact form of LaTeXMLMath ) and LaTeXMLMath .
Inspection of the dimensions of the stabilizers shows that , under the projection from LaTeXMLMath to LaTeXMLMath , none of the orbits ( 10 ) – ( 22 ) can map diffeomorphically onto its image in LaTeXMLMath .
∎ Let LaTeXMLMath and LaTeXMLMath be two LaTeXMLMath -elements , and use the notation LaTeXMLMath , LaTeXMLMath , etc .
with reference to LaTeXMLMath .
Then , for LaTeXMLMath , LaTeXMLMath provided LaTeXMLMath and LaTeXMLMath are in the same component of the space of LaTeXMLMath -elements , and LaTeXMLMath otherwise .
∎ Remark 3.3.7 .
In the standard cases , the holomorphicity of the orbits LaTeXMLMath ( LaTeXMLMath ) , in particular , the assertion in the above proof of Lemma 3.3.6 involving the stabilizers , will also be a consequence of Theorem 5.3.3 below .
The projection from the closure LaTeXMLMath of a pseudoholomorphic nilpotent orbit LaTeXMLMath to the constituent LaTeXMLMath of the Cartan decomposition is injective if and only if LaTeXMLMath is holomorphic or antiholomorphic .
Indeed , a pseudoholomorphic nilpotent orbit LaTeXMLMath is one of the kind LaTeXMLMath .
If LaTeXMLMath or LaTeXMLMath , for each LaTeXMLMath , there is only a single rank LaTeXMLMath nilpotent orbit in the closure LaTeXMLMath .
If LaTeXMLMath and LaTeXMLMath , the projection from the closure LaTeXMLMath to LaTeXMLMath is no longer injective since the closure contains at least two orbits of the same rank but having different signature ; the projection to LaTeXMLMath identifies the two .
∎ An orbit which belongs to the closure of a ( pseudo ) holomorphic nilpotent orbit is necessarily ( pseudo ) holomorphic , and the closure of a ( pseudo- ) holomorphic nilpotent orbit is a union of finitely many ( pseudo ) holomorphic nilpotent orbits .
This is an almost immediate consequence of Theorem 3.3.3 .
∎ This follows readily from Theorem 3.3.3 .
We leave the details to the reader .
∎ Let LaTeXMLMath be a simple Lie algebra of hermitian type and real rank LaTeXMLMath ; for LaTeXMLMath , we will henceforth denote the holomorphic nilpotent orbit LaTeXMLMath by LaTeXMLMath .
Theorem 3.3.3 entails that the holomorphic nilpotent orbits LaTeXMLMath are linearly ordered in such a way that LaTeXMLEquation .
Borrowing and extending terminology from LaTeXMLCite , where the regular ( complex ) nilpotent orbit is said to be principal , we will refer to the top orbit LaTeXMLMath as the principal holomorphic nilpotent orbit .
The principal holomorphic nilpotent ( in our sense ) may be viewed as the regular holomorphic nilpotent orbit ; it is in particular unique , and its closure contains every holomorphic nilpotent orbit .
For the principal holomorphic nilpotent orbit LaTeXMLMath , the composite of the projection from the closure LaTeXMLMath to LaTeXMLMath with the complex linear isomorphism from LaTeXMLMath to LaTeXMLMath is a homeomorphism LaTeXMLMath .
Under this homeomorphism , the LaTeXMLMath -orbit stratification of LaTeXMLMath passes to the LaTeXMLMath -orbit stratification of LaTeXMLMath .
Thus , for LaTeXMLMath , restricted to LaTeXMLMath , this homeomorphism is a LaTeXMLMath -equivariant diffeomorphism from LaTeXMLMath onto the LaTeXMLMath -orbit in LaTeXMLMath of LaTeXMLMath .
This is a consequence of Lemma 3.3.6 , except perhaps the fact that the indicated map is onto LaTeXMLMath ; the latter , in turn , is implied by the observation that LaTeXMLMath is the disjoint union of its LaTeXMLMath -orbits .
∎ Remark 3.3.12 .
In LaTeXMLCite ( p. 185 ) a nilpotent orbit LaTeXMLMath of LaTeXMLMath is said to be of convex type provided it is contained in a proper generating invariant cone , and a complete classification of orbits of convex type is given .
The orbits of convex type in this sense are precisely the holomorphic and antiholomorphic ones .
Remark 3.3.13 .
It is known ( Theorem 1.9 in LaTeXMLCite ) that , for a general real semisimple Lie algebra LaTeXMLMath , the assignment to LaTeXMLMath of LaTeXMLMath where LaTeXMLMath runs through invariant LaTeXMLMath -triples in LaTeXMLMath yields a bijective correspondence LaTeXMLMath , referred to in the literature as Kostant-Sekiguchi correspondence , between nilpotent ( LaTeXMLMath - ) orbits in LaTeXMLMath and nilpotent LaTeXMLMath -orbits in LaTeXMLMath ( i. e. LaTeXMLMath -orbits in LaTeXMLMath which arise as intersection of a nilpotent orbit in LaTeXMLMath with LaTeXMLMath ) .
In particular , for a semisimple Lie algebra LaTeXMLMath of hermitian type , a little thought reveals that the holomorphic nilpotent orbits LaTeXMLMath in LaTeXMLMath are precisely those which have the property that the orbit LaTeXMLMath lies in LaTeXMLMath .
In LaTeXMLCite , a nilpotent orbit LaTeXMLMath of LaTeXMLMath is defined to be holomorphic provided LaTeXMLMath lies in LaTeXMLMath ; it is then observed that , for a nilpotent orbit LaTeXMLMath having the property that LaTeXMLMath lies in LaTeXMLMath , the projection from LaTeXMLMath to LaTeXMLMath is a diffeomorphism onto its image .
The argument in LaTeXMLCite relies on the fact , established in LaTeXMLCite that , for an arbitrary nilpotent orbit LaTeXMLMath , the Kostant-Sekiguchi correspondence may be realized by a diffeomorphism from LaTeXMLMath to LaTeXMLMath ; this diffeomorphism involves a certain flow constructed by Kronheimer in LaTeXMLCite and is referred to in the literature as Kronheimer-Vergne diffeomorphism .
Our approach avoids the detour via this diffeomorphism .
Remark 3.3.14 .
The classification of nilpotent orbits in the real exceptional Lie algebras given in LaTeXMLCite ( coming into play in the proof of Theorem 3.3.3 ) relies on the Kostant-Sekiguchi correspondence .
We now indicate briefly how , in the two exceptional cases , the holomorphic nilpotent orbits can be classified without reference to the Kostant-Sekiguchi correspondence : Over the complex numbers , the classification of nilpotent orbits is in terms of “ regular ” subalgebras of LaTeXMLMath of minimal rank containing a representative of the nilpotent orbit being classified , the requisite subalgebra being written as LaTeXMLMath in LaTeXMLCite and LaTeXMLMath in LaTeXMLCite .
These subalgebras actually play a role similar to that of the indecomposable types in LaTeXMLCite for the classical cases which we exploited for the proofs of Theorems 3.5.3 and 3.6.2 .
As before , LaTeXMLMath refers to the adjoinnt group of the Lie algebra LaTeXMLMath under discussion .
LaTeXMLMath : The orbits LaTeXMLMath and LaTeXMLMath complexify to the orbits in LaTeXMLMath given in terms of the subalgebras LaTeXMLMath and/or LaTeXMLMath written as LaTeXMLMath and LaTeXMLMath , respectively , on p. 152 of LaTeXMLCite and in Table 1 on p. 447 of LaTeXMLCite .
Inspection of these tables shows that no other nilpotent orbit in LaTeXMLMath can be holomorphic : given any such orbit , it complexifies to an orbit involving a regular subalgebra of minimal rank which does not arise by complexification from a reductive Lie algebra of hermitian type .
Hence LaTeXMLMath has no holomorphic nilpotent orbits other than LaTeXMLMath .
A slight extension of this reasoning shows that the orbits LaTeXMLMath where LaTeXMLMath and LaTeXMLMath constitute a complete list of pseudoholomorphic nilpotent orbits