1 Introduction
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Non-Compact Symplectic Toric Manifolds

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Non-Compact Symplectic Toric ManifoldsThis paper is a contribution to the Special Issue on Poisson Geometry in Mathematics and Physics. The full collection is available at http://www.emis.de/journals/SIGMA/Poisson2014.html

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Yael KARSHON  and Eugene LERMAN 

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Y. Karshon and E. Lerman

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 Department of Mathematics, University of Toronto,
 40 St. George Street, Toronto, Ontario, Canada M5S 2E4 \EmailDkarshon@math.toronto.edu

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 Department of Mathematics, The University of Illinois at Urbana-Champaign,
 1409 W. Green Street, Urbana, IL 61801, USA \EmailDlerman@math.uiuc.edu

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Received August 15, 2014, in final form July 10, 2015; Published online July 22, 2015

\Abstract

A key result in equivariant symplectic geometry is Delzant’s classification of compact connected symplectic toric manifolds. The moment map induces an embedding of the quotient of the manifold by the torus action into the dual of the Lie algebra of the torus; its image is a unimodular (“Delzant”) polytope; this gives a bijection between unimodular polytopes and isomorphism classes of compact connected symplectic toric manifolds. In this paper we extend Delzant’s classification to non-compact symplectic toric manifolds. For a non-compact symplectic toric manifold the image of the moment map need not be convex and the induced map on the quotient need not be an embedding. Moreover, even when the map on the quotient is an embedding, its image no longer determines the symplectic toric manifold; a degree two characteristic class on the quotient makes an appearance. Nevertheless, the quotient is a manifold with corners, and the induced map from the quotient to the dual of the Lie algebra is what we call a unimodular local embedding. We classify non-compact symplectic toric manifolds in terms of manifolds with corners equipped with degree two cohomology classes and unimodular local embeddings into the dual of the Lie algebra of the corresponding torus. The main new ingredient is the construction of a symplectic toric manifold from such data. The proof passes through an equivalence of categories between symplectic toric manifolds and symplectic toric bundles over a fixed unimodular local embedding. This equivalence also gives a geometric interpretation of the degree two cohomology class.

\Keywords

Delzant theorem; symplectic toric manifold; Hamiltonian torus action; completely integrable systems

\Classification

53D20; 53035; 14M25; 37J35

1 Introduction

In the late 1980s, Delzant classified compact connected symplectic toric manifolds [7] by showing that the map

is a bijection onto the set of unimodular (also referred to as “smooth” or “Delzant”) polytopes. This beautiful work has been widely influential. The goal of this paper is to extend Delzant’s classification theorem to non-compact manifolds.

Delzant’s classification is built upon convexity and connectedness theorems of Atiyah and Guillemin–Sternberg [1, 10]. Compactness plays a crucial role in the proof of these theorems. Indeed, for a non-compact symplectic toric manifold the moment map image need not be convex and the fibers of the moment map need not be connected. And even when the fibers of the moment map are connected the moment map image need not uniquely determine the corresponding symplectic toric manifold. Thus, the passage to noncompact symplectic toric manifolds requires a different approach. As a first step we make the following observation (the proof is given in Appendix B):

Proposition 1.1.

Let be a symplectic toric -manifold. Then the quotient is naturally a manifold with corners and the induced map

is a unimodular local embedding. See Definitions A.16 and 2.5.)

Definition 1.2.

Given a symplectic toric -manifold and a -quotient map , we refer to the map that is defined by as the orbital moment map.

See Remarks 1.4 and 1.5 for the origin of the notion of orbital moment map and its relation to developing map in affine geometry. The fact that the quotient  is a manifold with corners is closely related to the fact that for a completely integrable system with elliptic singularities the space of tori is a manifold with corners [3, 31].

Our classification result can be stated as follows.

Theorem 1.3.

Let be the dual of the Lie algebra of a torus and a unimodular local embedding of a manifold with corners. Then

  1. There exists a symplectic toric -manifold with -quotient map and orbital moment map .

  2. The set of isomorphism classes of symplectic toric -manifolds with -quotient map and orbital moment map is in bijective correspondence with the set of cohomology classes

    where denotes the integral lattice of the torus .

The main difficulty in proving Theorem 1.3 lies in establishing part (1). Results similar to part (2) hold in a somewhat greater generality for completely integrable systems with elliptic singularities (under a mild properness assumption) [3, 31, 32]: once one knows that there is one completely integrable system with the space of tori , the space of isomorphism classes of all such systems is classified by the second cohomology of  with coefficients in an appropriate sheaf (q.v. op. cit.). The existence part for completely integrable systems, called the realization problem by Zung [32], is much more difficult. For instance in [32] the realization problem is only addressed for 2-dimensional spaces of tori. The solution to the realization problem announced in [3] and a similar solution in [31] is difficult to apply in practice. The solution, is, roughly, as follows. Given an integral affine manifold with corners one shows first that there is an open cover of  such that over each  the realization problem has a solution . Then, if there exists a collection of isomorphisms satisfying the appropriate cocycle condition, the realization problem has a solution for . Compare this with Theorem 1.3(1) which asserts that the realization problem for completely integrable torus actions always has a solution. We believe that the realization problem for completely integrable systems with elliptic singularities also always has a solution. We will address this elsewhere.

Our proof of part (1) of Theorem 1.3 proceeds as follows. We define symplectic toric -bundles over : these are symplectic principal -bundles over manifolds with corners with orbital moment map . They form a category, which we denote by . This category is always non-empty: it contains the pullback . We then construct a functor

(1.1)

from the category of symplectic toric -bundles over to the category of symplectic toric -manifolds over . The functor c trades corners for fixed points; it is a version of a symplectic cut [16]. It follows, since is non-empty, that there always exist symplectic toric -manifolds over a given unimodular local embedding of a manifold with corners .

More is true. We show that the functor c is an equivalence of categories. Hence, it induces a bijection, , between the isomorphism classes of objects of our categories:

The geometric meaning of the cohomology classes in that classify symplectic toric -manifolds over now becomes clear: the elements of classify principal -bundles, and the elements of keep track of the “horizontal part” of the symplectic forms on these bundles.

We note that, for compact symplectic toric -manifolds, the idea to obtain their classification by expressing these manifolds as the symplectic cuts of symplectic toric -manifolds with free  actions is due to Eckhard Meinrenken (see [23, Chapter 7, Section 5]).

The paper is organized as follows. In Section 2, after introducing our notation and conventions, we construct the functor (1.1). In Section 3 we show that any two symplectic toric  bundles over the same unimodular local embedding are locally isomorphic (Lemma 3.1). In Section 4 we prove that the functor c in (1.1) is an equivalence of categories. In Section 5, we give the classification of symplectic toric -bundles over a fixed unimodular local embedding in terms of two characteristic classes, the Chern class , which is in and encodes the “twistedness” of the bundle, and the horizontal class , which is in and encodes the “horizontal part” of the symplectic form on the bundle. We show that the map

is a bijection. Since the map is a bijection, the composite

is a bijection as well. This classifies (isomorphism classes of) symplectic toric -manifolds over .

Finally, in Section 6 we discuss those symplectic toric manifolds that are determined by their moment map images. In Proposition 6.5 we use Theorem 1.3 to derive Delzant’s classification theorem and its generalization in the case of symplectic toric -manifolds that are not necessarily compact but whose moment maps are proper as maps to convex subsets of . (In fact, already in [15] it was noted that, with the techniques of Condevaux–Dazord–Molino [4], Delzant’s proof should generalize to non-compact manifolds if the moment map is proper as a map to a convex open subset of the dual of the Lie algebra.) In Theorem 6.7, which was obtained in collaboration with Chris Woodward, we characterize those symplectic toric manifolds that are symplectic quotients of the standard  by a subtorus of the standard torus . In Example 6.9 we construct a symplectic toric manifold that cannot be obtained by such a reduction.

The paper has two appendices. Appendix A contains background on manifolds with corners. In Appendix B, we recall the local normal form for neighborhoods of torus orbits in symplectic toric manifolds, and we use it to prove the following facts, which are known but maybe hard to find in the literature:

  1. orbit spaces of symplectic toric manifolds are manifolds with corners;

  2. orbital moment maps of symplectic toric manifolds are unimodular local embeddings; and

  3. any two symplectic toric manifolds over the same unimodular local embedding are locally isomorphic.

In the remainder of this section, following referees’ suggestions, we describe some relations of our work to existing literature on integral affine structures and Lagrangian fibrations.

Remark 1.4 (orbital moment maps).

An equivariant moment map for an action of a Lie group on a symplectic manifold descends to a continuous map between orbit spaces. This map was introduced by Montaldi [26] under the name of orbit momentum map and was used to study stability and persistence of relative equilibria in Hamiltonian systems. An analogue of this map in contact geometry was used by Lerman to classify contact toric manifolds [17].

The content of Proposition 1.1 is that for a symplectic toric manifold the orbit space is not just a topological space. It has a natural structure of a manifold with corners and that the induced orbital moment map is .

Symplectic toric manifolds, in addition to being examples of symplectic manifolds with Hamiltonian torus actions, are also a particularly nice class of completely integrable systems with elliptic singularities. Viewed this way is a developing map for an integral affine structure on the manifold with corners (see also Remark 1.5 below).

Remark 1.5 (integral affine structures).

An integral affine structure on a manifold with corners is usually defined in terms of an atlas of coordinate charts with integral affine transition maps; see, for example, [31]. It is not hard to see that such an atlas on a manifold  defines a Lagrangian subbundle of the cotangent bundle with two properties:

  1. the fiber is a lattice;

  2. if lies in a stratum of of codimension then there is a local frame of defined near ( so that the first 1-forms annihilate the vectors tangent to the stratum.

Conversely, any such Lagrangian subbundle defines on an atlas of coordinate charts with integral affine transition maps.

In general the bundle may have no global frame. And even if it does have a global frame the one forms (which are necessarily closed) need not be exact. But if there is a global exact frame of , then we have a smooth map . Such a map  is a developing map for the integral affine structure on .

Observe that a unimodular local embedding defines an integral affine structure on as follows. Since is a local embedding, the cotangent bundle is the pullback by of the cotangent bundle . Consequently the standard Lagrangian lattice pulls back to a Lagrangian subbundle of . A choice of a basis of of the integral lattice defines a map . It is given by

The map is a developing map for .

Symplectic toric manifolds and proper Lagrangian fibrations

Let be a symplectic toric -manifold with a  quotient map . Restricting to the interior of  (as a manifold with corners), we get a completely integrable system in the sense that was studied by Duistermaat [8], namely, a proper Lagrangian fibration with connected fibers. These were revisited and generalized by Dazord and Delzant [6]. For a detailed exposition see [20].

Remark 1.6 (the integral affine structure and the monodromy).

As Duistermaat explains, a proper Lagrangian fibration with connected fibers defines an integral affine structure on the base . Each covector determines a vector field along by the equation , and the Lagrangian lattice sub-bundle is

Duistermaat’s monodromy measures the non-triviality of the Lagrangian lattice sub-bundle . When it is trivial, the bundle of tori becomes a trivial bundle with fiber, say, , and become trivial bundles with fibers and , and becomes a principal bundle. In this case, an orbital moment map is also a developing map for the integral affine structure. Having a moment map in this context exactly means that the integral affine structure on  is globally developable.

Remark 1.7 (the characteristic classes).

Let be a proper Lagrangian fibration with connected fibers. The fibers of the bundle of tori act freely and transitively on the fibers of . Moreover, every point in has a neighborhood over which and are isomorphic; this is Duistermaat’s formulation of the Arnold–Liouville theorem on the local existence of action angle variables. Globally, such fibrations are classified by the first cohomology group

of the sheaf of Lagrangian sections of .

The short exact sequence of sheaves

gives an exact sequence

Noting that Lagrangian sections of are the same as closed one-forms, and identifying the  of their sheaf with , we get an exact sequence

(1.2)

The second of these maps is Duistermaat’s Chern class. When the monodromy is trivial, Duistermaat’s Chern class is the Chern class of as a principle bundle. If is the -quotient map of a symplectic toric -manifold, then Duistermaat’s Chern class for coincides with ours under the identification .

If the monodromy and Chern class both vanish, Duistermaat defines a class in , which is often called the Lagrangian class; it is the cohomology class of the pullback of by a global smooth section. If is the quotient map of a symplectic toric -manifold, and if additionally the Chern class vanishes, then Duistermaat’s Lagrangian class for coincides with our horizontal class under the identification .

Remark 1.8.

If is the -quotient map for a symplectic toric -manifold and , our characteristic class gives a splitting

that is consistent with (1.2). Moreover, our construction provides a geometric meaning to the Lagrangian class in .

In this case

  • every element of gives rise to a symplectic toric -manifold, and

  • distinct elements of represent non-isomorphic symplectic toric -manifolds.

Both of these facts are not necessarily true in the more general situation that is addressed by Duistermaat and Dazord–Delzant.

2 A functor from symplectic toric bundles
to symplectic toric manifolds

The purpose of this section is to construct a functor

from the category of symplectic toric -bundles to the category of symplectic toric -manifolds over a given unimodular local embedding of a manifold with corners . Once this functor is constructed, we deduce Theorem 1.3(1) almost immediately. In Section 4 we prove that c is an equivalence of categories. We start by establishing our notation and recording a few necessary definitions.

Notation and conventions. A torus is a compact connected abelian Lie group. A torus of dimension is isomorphic, as a Lie group, to and to . We denote the Lie algebra of a torus by , the dual of the Lie algebra, , by , and the integral lattice, , by . The weight lattice of is the lattice dual to ; we denote it by . When a torus acts on a manifold , we denote the action of an element by and the vector field induced by a Lie algebra element by ; by definition

We write the canonical pairing between  and  as . Our sign convention for a moment map for a Hamiltonian action of a torus  on a symplectic manifold  is that it satisfies

(2.1)

For us a symplectic toric -manifold is a triple where is a manifold, is a symplectic form and is a moment map for an effective Hamiltonian action of a torus with .

Definition 2.1.

A unimodular cone in the dual of the Lie algebra of a torus is a subset  of  of the form

where is a point in and is a basis of the integral lattice of a subtorus of . We record the dependence of the cone on the data and by writing

Remark 2.2.

The set is a unimodular cone defined by the empty basis of the integral lattice  of the trivial subtorus  of .

Remark 2.3.

A unimodular cone is a manifold with corners. Moreover, it is a manifold with faces (q.v. Definition A.10).

For a unimodular cone the facets are the sets

The vector in the formula above is the inward pointing primitive normal to the facet . (Recall that a vector in the lattice is primitive if for any the equation for implies that .)

Lemma 2.4.

The primitive inward pointing normal to a facet of a unimodular cone is uniquely determined by any open neighborhood of a point  of  in .

Proof.

The affine hyperplane spanned by is uniquely determined by the intersection . Up to sign, such a hyperplane has a unique primitive normal. The sign of the normal is determined by requiring that at the point the normal points into . ∎

Definition 2.5 (unimodular local embedding (u.l.e.)).

Let be a manifold with corners and  the dual of the Lie algebra of a torus. A smooth map is a unimodular local embedding (a u.l.e.) if for each point  in  there exists an open neighborhood of the point and a unimodular cone such that is contained in and is an open embedding. That is, is open in and is a diffeomorphism.

Remark 2.6.

In Definition 2.5, the cone is not uniquely determined by the point ; for instance it can have facets that do not pass through . For example, let , let be the inclusion map of the positive quadrant, and let . If the neighborhood of meets the non-negative axis, then the cone must be the positive quadrant too. Otherwise, the natural choice for is the closed upper half plane, but for suitable choices of the cone can also be the intersection of the closed upper half plane with a half plane of the form for and or of the form for and .

Remark 2.7.

Proposition 1.1 shows that the orbital moment map of a symplectic toric manifold is a unimodular local embedding.

Example 2.8.

It is easy to construct examples where the orbital moment map is not an embedding. Consider, for instance, a 2-dimensional torus . Removing the origin from the dual of its Lie algebra gives us a space that is homotopy equivalent to a circle. Thus the fibers of the universal covering map have countably many points. The pullback along of the principal -bundle is a symplectic toric -manifold with orbit space  and orbital moment map , which is certainly not an embedding.

Similarly, let be the unit sphere in with the standard area form, and equip  with the standard toric action of with moment map . Its image is the square . Remove the origin, and let be the universal covering. Then the fiber product is a symplectic toric manifold; it is a -fold covering of . As in the previous example, the orbital moment map is not an embedding. Unlike the previous example, the torus action here is not free.

Definition 2.9.

Let be a manifold with corners and a unimodular local embedding. A symplectic toric manifold over is a symplectic toric -manifold , equipped with a quotient map for the action of on (q.v. Definition A.16), such that

Remark 2.10.

Since the moment map together with the symplectic form encodes the action of the group on and since the quotient map together with encode , we may regard a symplectic toric -manifold over as a triple .

We now fix a u.l.e.

of a manifold with corners into the dual of the Lie algebra of a torus , and proceed to define the category of symplectic toric -manifolds over and the category of symplectic toric -bundles over .

Definition 2.11 (the category of symplectic toric -manifolds over ).

We define an object of the category to be a symplectic toric -manifold over . A morphism from to is a -equivariant symplectomorphism such that .

Notation 2.12.

Informally, we may sometimes write for an object of and for a morphism between two such objects. Also, we may write as shorthand for .

Definition 2.13 (the category of symplectic toric -bundles over ).

An object of the category is a principal -bundle over a manifold with corners (cf. Definition A.17) together with a -invariant symplectic form so that is a moment map for the action of  on . We call the triple a symplectic toric -bundle over the u.l.e. , or a symplectic toric -bundle over  for short. A morphism  from to is a -equivariant symplectomorphism  with .

Remark 2.14.

The categories and are groupoids, that is, all of their morphisms are invertible.

If is a u.l.e., is a symplectic toric manifold over and is open, then the restriction is also a u.l.e., and

is a symplectic toric -manifold over . The restriction map extends to a functor

Given an open subset of we get the restriction . The three restriction functors are compatible:

In other words, the assignment

is a (strict) presheaf of groupoids.

A reader familiar with stacks will have little trouble checking that the presheaf satisfies the descent condition with respect to any open cover of  and that thus is a stack on the site of open subsets of  with the cover topology. The stack  is not a geometric stack.

Similarly, a symplectic toric bundle over a manifold with corners  restricts to a symplectic toric bundle over an open subset of . These restrictions define a presheaf of groupoids . A reader familiar with stacks can check that is also a stack; see also Lemma 4.7 below.

Remark 2.15.

If is a u.l.e. and is a manifold without corners (i.e., a manifold) then

If is an arbitrary manifold with corners, then its interior (q.v. Definition A.3) is a manifold, and so

The functor

Next we outline the construction of the functor from the category of symplectic toric -bundles to the category of symplectic toric -manifolds over a u.l.e. .

Step 1: characteristic subtori. We show that attaches to each point a subtorus  of  together with a choice of a basis of its integral lattice .

A basis of the integral lattice of a torus defines a linear symplectic representation , which we may regard as a symplectic toric -manifold (here is the associated moment map with ). Thus for each point we also have a symplectic toric -manifold .

Step 2: a topological version of the functor c. The collection of the subtori defines for each principal -bundle an equivalence relation in a functorial manner. We show that

  1. Each quotient is a topological -space with orbit space and the action of on is free over the interior . (Here, stands for “topological cut”.)

  2. For every map of principal -bundles over we naturally get a -equivariant homeomorphism .

  3. These data define a functor

  4. Moreover, is a map of presheaves of groupoids. In particular, for every open subset  of ,

Step 3: the actual construction of c. We show that for every point there is an open neighborhood so that for every symplectic toric -bundle the symplectic quotient

is a symplectic toric -manifold over .

As in Step 2 the mapping (i.e., the restriction to  followed by ) from symplectic toric -bundles over to symplectic toric manifolds over extends to a functor. In particular for every map of symplectic toric -bundles over we have a map of symplectic toric -manifolds over .

At the same time, for each symplectic toric -bundle we construct a collection

of equivariant homeomorphisms that have the following two compatibility properties:

  1. For a fixed bundle and any two points , the map

    is a map of symplectic toric -manifolds over .

  2. For a point and a map of symplectic toric bundles over the diagram

    (2.2)

    commutes.

The first property tells us that the family of homeomorphisms defines on the structure of a symplectic toric -manifold over . We denote this manifold, which is an object of , by . The second property tells us that defines a map of symplectic toric -manifolds over . This gives rise to the desired functor c.

We now proceed to fill in the details of the construction.

Details of Step 1.

We start by proving

Lemma 2.16.

Given a unimodular local embedding and a point there exists a unique subtorus of and a unique basis of its integral lattice such that the following holds. There exists an open neighborhood of in so that

is an open embedding of manifolds with corners.

Proof.

By definition of a u.l.e., there exists an open neighborhood of and a unimodular cone such that is contained in and is an open embedding of manifolds with corners. Since is an open embedding it maps the interior of  to an open subset of the interior of . We may assume that  is a neighborhood with faces. Then the stratum  of  containing  lies in exactly  facets of , where  is the codimension of . For each  the image  is an open subset of a unique facet  of  and  is an open neighborhood of in . By Lemma 2.4 the pair uniquely determines the primitive inward pointing normal  of the facet  of . Since is a basis of an integral lattice of a subtorus of , its subset is also a basis of an integral lattice of a possibly smaller subtorus of . We set , . We note that

To obtain the neighborhood we delete from the manifold with faces  all the faces that do not contain . ∎

Remark 2.17.

The basis and the corresponding torus do not depend on our choice of the cone : by construction is the primitive normal to the affine hyperplane spanned by that points into . In fact the only way we use the existence of the unimodular cone is to insure that the set of normals to the facets of forms a basis of an integral lattice of a subtorus of the torus .

Similarly, the basis does not depend on the choice of either.

Remark 2.18.

For each stratum of the function is locally constant, hence constant. Consequently the subtorus  depends only on the stratum of containing the point  and not on the point  itself. Similarly the basis depends only on the stratum of  containing .

Remark 2.19.

For we can read off the group from the face structure of and the set . Namely

We also note that the subset

of forms a basis of the integral lattice of .

Lemma 2.20.

A manifold with corners  that admits a u.l.e.  is a manifold with faces q.v. [13] and Definition A.10 below. In particular, for any symplectic toric -manifold , the quotient is a manifold with faces.

Proof.

The map sends a neighborhood of a point in a codimension 1 stratum of to a relatively open subset of an affine hyperplane whose normal lies the integral lattice of . Consequently sends all of to and is a local diffeomorphism. The lemma follows from this observation. ∎

Remark 2.21.

It follows from the proof of Lemma 2.20 that the map attaches to every (connected) codimesion 1 stratum  of  a primitive vector (namely, the corresponding primitive inward normal). The function is the analogue of the characteristic function of Davis and Januszkiewicz [5] and of the characteristic bundle of Yoshida [31].

Recall that any symplectic representation of a torus is complex hence has well-defined weights. These weights do not depend on a choice of an invariant complex structure compatible with the symplectic form since the space of such structures is path connected.

Lemma 2.22.

Let , , be two symplectic representations of a torus with the same set of weights. Then there exists a symplectic linear isomorphism of representations .

Proof.

Choose -invariant compatible complex structures on  and . As complex representations, each of  and  decomposes into one-dimensional complex representations. Because the weights are the same, it is enough to consider the case that  and  are the same complex vector space and its complex dimension is one. In this case, because  and are both compatible with the complex structure, one must be a positive multiple of the other: for some scalar . We may then take . ∎

Lemmas 2.16 and 2.22 imply that to any point of a manifold with corners a u.l.e.  unambiguously attaches a symplectic toric -manifold : the weights of the representation is the basis of the weight lattice dual to the basis . If is another symplectic representation of with the same set of weights as  then the symplectic toric -manifolds and are linearly isomorphic as symplectic toric manifolds. ∎

Details of Step 2.

Given a principal -bundle we define to be the smallest equivalence relation on such that whenever and , lie on the same  orbit. We give the set the quotient topology. Since the action of  on the fiber of  above commutes with the action of , the topological space

is naturally a -space. For the same reason descends to a quotient map . Since for the points in the interior of  the groups  are trivial, the action of  on  is free.

If is a map of principal -bundles over , then it maps fibers to fibers and -orbits to orbits thereby inducing . Explicitly is given by

Here, as before denotes the equivalence class of and denotes the corresponding class in .

It is easy to check that the map

is a functor that commutes with restrictions to open subsets of . ∎

Details of Step 3.

We start by extending the symplectic reduction theorem of Marsden–Weinstein and Meyer [22, 24] to manifolds with corners.

Theorem 2.23.

Suppose is a symplectic manifold with corners with a proper Hamiltonian action of a Lie group and an associated equivariant moment map . Suppose further:

  1. For any point the stabilizer of is trivial;

  2. there is an extension of to a manifold containing as a domain q.v. Definition A.8) with .

Then is a manifold without corners and the quotient

is naturally a symplectic manifold.

Remark 2.24.

The main issue in proving the theorem is in showing that is actually a manifold and that it has the right dimension. In other words the issue is transversality for manifolds with corners. To be more specific if is a manifold with corners, is a smooth function and is a regular value of , then it is not true in general that is a manifold, with or without corners. Take, for example,

and . Then is a regular value of but

which is clearly not a manifold, with or without boundary.

The standard approach to transversality for manifolds with corners [28] is to impose an additional requirement that the kernel of the differential of is transverse to the strata of . However, in the situation we care about we have tangency instead. Moreover, it is easy to write down an example of a smooth function so that the graph of is tangent to the plane but the set

is not a manifold. This is why we make an awkward assumption on the level set in Theorem 2.23. On the other hand, this assumption is easy to check in practice.

Before proving the theorem we first prove

Lemma 2.25.

Let be a smooth function on a manifold with corners . Suppose is a manifold without corners containing  as a domain, and is an extension of  with

If is a regular value of that is, if for all the map is onto, then  is naturally a smooth manifold of dimension in the sense of Definition A.14.

Proof.

Since is a regular value of , and since , the value is also regular for . Consequently, is naturally a manifold of dimension . Since , we conclude that  is naturally a manifold. ∎

Remark 2.26.

Note that the assumptions of the lemma force to be tangent to the strata of : otherwise cannot hold.

Proof of Theorem 2.23.

Once we know that is actually a manifold, the classical arguments of Marsden–Weinstein [22] and of Meyer [24] apply to show that is basic and that its kernel is precisely the directions of the orbits. Consequently the restriction descends to a closed nondegenerate 2-form on the manifold .

By Lemma 2.25 it is enough to show that 0 is a regular value of . This will follow from our assumption that the action on is free. Again, the argument is standard. Indeed, let . To show that the differential is surjective, we need to show that the annihilator of its image is zero. Let be in the annihilator of this image: