Non-Compact Symplectic Toric Manifolds

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.

1. Introduction, notation, conventions 1 2. Background on symplectic toric manifolds 6 3. Uniqueness 10 4. Collapsing a symplectic toric bundle to a symplectic toric manifold 13 5. Characteristic classes of symplectic toric manifolds 20 6. Equivalence of categories between symplectic toric bundles and symplectic toric manifolds 24 Appendix A. Manifolds with corners 26 References 27 1. Introduction, notation, conventions By a theorem of Delzant [De], a compact connected symplectic toric manifold is determined up to isomorphism by its moment map image, which is a unimodular polytope. Moreover, every unimodular polytope is the moment map image of a compact connected symplectic toric manifold. It was noted without proof in [K] that, with the techniques of Condevaux-Dazord-Molino [CDM], Delzant's proof generalizes 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. Lerman, Meinrenken, Tolman, and Woodward developed a different approach, and this generalization of Delzant's theorem can be easily extracted from their results; see [LMTW,Thm. 4.3(a,b)] and [LT,section 7]. Their method actually leads to a more general classification result: it allows one to classify arbitrary symplectic toric manifolds This research is partially supported by the Natural Sciences and Engineering Research Council of Canada and by the National Science Foundation.
The contents of the paper. To state precisely the main result of the paper, we first need to make a few definitions and list some auxiliary results. We begin with the definition of a unimodular cone: 1.1. Definition. Let g * denote the dual of the Lie algebra of a torus G. A unimodular cone in g * is a subset C of g * of the form where ǫ is a point in g * and {v 1 , . . . , v k } is a basis of the integral lattice of a subtorus K of G.
We proceed with the definition of a unimodular local embedding.
1.4. Definition. Let W be a manifold with corners and g * the dual of the Lie algebra of a torus G. A smooth map ψ : W → g * is a unimodular local embedding if for each point x in W there exists a unimodular cone C ⊂ g * and open sets T ⊂ W and U ⊂ g * such that ψ(T ) = C ∩ U and such that ψ| T : T → C ∩ U is a diffeomorphism of manifolds with corners.
As a first step we will prove, in section 2, that the orbital moment map is a unimodular local embedding: 1.5. Proposition. Let (M, ω, µ : M → g * ) be a symplectic toric G manifold. Then the orbit space M/G is a manifold with corners (cf. A.5), and the orbital moment map µ : M/G → g * is a unimodular local embedding.
1.6. Remark. It is easy to construct examples where the orbital moment map is not an embedding. Consider, for instance, a 2-dimensional torus G. Removing the origin from the dual of its Lie algebra g * gives us a space homotopy equivalent to a circle. Thus the fibers of the universal covering map p : W → g * {0} have countably many points. The pullback p * (T * G) of the principal G-bundle µ : T * G → g * along p is a symplectic G-toric manifold with orbit space W and orbital moment map p, which is certainly not an embedding. Similarly, let S 2 be the unit sphere in R 3 with the standard area form, and equip S 2 × S 2 with the standard toric action of (S 1 ) 2 with moment map µ((x 1 , x 2 , x 3 ), (y 1 , y 2 , y 3 )) = (x 3 , y 3 ). Its image is the square I 2 = [−1, 1] × [−1, 1]. Remove the origin, and let p : W → I 2 {0} be the universal covering. Then the fiber product W × I 2 {0} (S 2 × S 2 ) is a symplectic toric manifold; it is a Z-fold covering of (S 2 × S 2 ) (the equator × the equator). Like in the previous example, the orbital moment map is not an embedding. Unlike the previous example, here the toric action is not free.
We now introduce the category of symplectic toric manifolds over a unimodular local embedding: 1.7. Definition. We define the category STM(W ψ → g * ) of symplectic toric G manifolds over a unimodular local embedding ψ : W → g * of a manifold with corners W into the dual of the Lie algebra of a torus G.
1.8. Remark. We will informally write π : M → W or simply M for an object of STM(W ψ → g * ) and ϕ : M → M ′ for a morphism between two objects. If W is a subset of g * and ψ : W → g * is the inclusion map, we will use the term symplectic toric G manifold over W (instead of "over ψ : W → g * ") and we will write it as (M, ω, µ : M → W ) (omitting π).
We are now in position to state the first main result of this paper.
1.9. Theorem. Let ψ : W → g * be a unimodular local embedding of a manifold with corners W into the dual of the Lie algebra of a torus G. Then there exists a bijection between the set of isomorphism classes of objects of the category STM(W ψ → g * ) of symplectic toric G manifolds over ψ : W → g * and H 2 (W ; Z G × R), the degree 2 cohomology of W with coefficients in the abelian group Z G × R, where Z G denotes the integral lattice of the torus G.
1.10. Remark. Let G be a 3-dimensional torus. Then g * {0} is homotopy equivalent to a 2-sphere, hence it has the integers as the second integral cohomology: H 2 (g * {0}; Z) = Z. By Theorem 1.9 above, there is exactly one equivalence class of symplectic toric manifolds over g * {0} for each element of H 2 (g * {0}; Z G × R) ∼ = Z 3 × R.
More generally, given an n dimensional torus G ∼ = (S 1 ) n , let U be an open subset of the dual of its Lie algebra g * , and let A = H 2 (U ; Z). Note that every finitely generated abelian group can occur as such an A for sufficiently large n. The inclusion map U → g * is a unimodular embedding. Theorem 1.9 associates one equivalence class of symplectic toric manifolds to each element of A ⊗ (Z n × R).
The bijection between isomorphism classes in STM(W ψ → g * ) and elements of H 2 (W ; Z G × R) that is obtained in Theorem 1.9 depends on the choice of an object in STM(W ψ → g * ). To avoid such a choice, in Section 5, we explicitly associate to a symplectic toric G manifold (M, ω, µ) over W a class c 1 (M ) in H 2 (W ; Z G ), the Chern class, that encodes the "twistedness" of M as a G manifold over W , and a class c hor (M ) in H 2 (W ; R), the horizontal class, that encodes the horizontal part of the symplectic form ω. These classes determine the symplectic toric G manifold up to isomorphism. Identifying W with M/G, we obtain the following variant of Theorem 1.9: 1.11. Theorem.
(2) Let (M 1 , ω 1 , µ 1 : M 1 → g * ) and (M 2 , ω 2 , µ 2 : M 2 → g * ) be symplectic toric G manifolds. Let µ 1 : M 1 /G → g * and µ 2 : M 2 /G → g * be the maps induced from the moment maps. Then M 1 and M 2 are isomorphic as symplectic toric G manifolds if and only if there exists a diffeomorphism F : Once we know that the category STM(W ψ → g * ) is non-empty, the proof of Theorem 1.9 is identical to the argument of Boucetta and Molino [BM]. Similar arguments appear in [DD, D, Z]. Showing that the category is non-empty is not entirely trivial when W has infinitely many facets.
To show that the category is non-empty, we introduce a new category, STB(W ψ → g * ), of symplectic toric G bundles over the unimodular local embedding W ψ → g * , and a functor Because STB is non-empty, STM is non-empty. The construction of the functor collapse is the most technical part of this paper.
If W ψ → g * is the inclusion map of a unimodular ("Delzant") polytope, and if the symplectic toric G bundle (M, ω, µ : M → W ) is the restriction to W of a symplectic toric G bundle ( M ,ω,μ : M → W ) over a neighborhood W of W in g * , then the collapse of M is essentially the same thing as the symplectic cut of M with respect to the polytope W . We note that for compact symplectic toric G manifolds, the idea to obtain their classification by expressing these manifolds as the symplectic cuts of symplectic toric G manifolds with free G actions is due to Eckhard Meinrenken; see chapter 7, section 5, of [Mei]. Also see Remark 6.2.
Finally, the functor (1.12) turns out to be an equivalence of categories: 1.13. Theorem. Let W be a manifold with corners and ψ : W → g * a unimodular local embedding. Then the functor (1.12) is an equivalence of categories.
We proceed as follows.
In Section 2, we recall the local normal form for neighborhoods of torus orbits in symplectic toric manifolds. We use it to prove that (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 Section 3, we recall the argument due to Boucetta and Molino [BM] (in the form independently rediscovered by Lerman, Tolman and Woodward, see Section 7 of [LT]): Given a symplectic toric G manifold ((M, ω, µ : M → g * ), π : M → W ) over ψ : W → g * , we introduce its sheaf of automorphisms A, show that the equivalence classes of symplectic toric G manifolds over ψ : W → g * that are locally isomorphic to M are in bijective correspondence with theČech cohomology classes in H 1 (W ; A), and recall that H 1 (W ; A) is isomorphic to H 2 (W ; Z G × R). We also recall how to deduce the classical Delzant theorem and its generalization to the case that the moment map is proper as a map to a convex subset of g * -Proposition 3.9.
Next, we introduce the category STB(W ψ → g * ) of symplectic toric G bundles over a unimodular local embedding. It is non-empty.
In Section 4, we construct the functor (1.12) from the category of symplectic toric G bundles over a unimodular local embedding to the category of symplectic toric G manifolds over the same unimodular local embedding. Because the category STB(W ψ → g * ) is non-empty, it follows that the category STM(W ψ → g * ) is non-empty as well. Together with the results of the previous sections, this gives Theorem 1.9 In Section 5, we define the classes c 1 (M ) and c hor (M ), for M ∈ STM(W ψ → g * ), and we prove Theorem 1.11. The purpose of this approach is to clarify the geometric meaning of the Z G and R parts of the classifying cohomology class.
In Section 6, we prove Theorem 1.13.
The reader familiar with Delzant's paper may wonder which of the the symplectic toric manifolds that we classify can be obtained as symplectic quotients of some standard C N by an action of a subtorus of the standard torus T N . The following theorem and its proof are the result of our discussion with Chris Woodward. We thank Chris for bringing up the question and helping us prove the answer.
1.14. Theorem. A symplectic toric G manifold (M, ω, µ : M → g * ) is isomorphic to a symplectic quotient of C N by a subtorus of the standard torus T N if and only if its orbital moment map µ : M/G → g * is an embedding and its image is a closed polyhedral subset of g * with at most N (≥ dim G) facets and at least one vertex.
Proof. We first argue that the conditions of the theorem are necessary: if a symplectic toric manifold (M, ω, µ) is a symplectic quotient of C N by a subtorus K ֒→ T N , then the orbital moment mapμ is an embedding and the moment map image µ(M ) is a unimodular polyhedral subset of g * with at least one vertex.
Recall that for the standard action of T N on C N the map ψ : C N → R N , ψ(z 1 , . . . , z N ) = (|z 1 | 2 , . . . |z N | 2 ), is a moment map (with an appropriate identification of the dual of the Lie algebra of T N with R N ). The image of ψ is the positive orthant R N + := {t ∈ R N | t i ≥ 0, ∀i}. Since ψ : C N → R N + is also a T N -orbit map, the orbital moment mapψ : C N /T N = R N + → R N is simply an inclusion. The inclusion K ֒→ T N induces an inclusion i T : k ֒→ Lie(T N ) of Lie algebras and, dually, the projection i T : Lie(R N ) * = R N → k * . Then ϕ := i T • ψ : C N → k * is a moment map for the action of K on C N . Suppose ν ∈ k * is a point such that the action of K on its preimage ϕ −1 (ν) is free. Then ν is necessarily a regular value of ϕ. Moreover, it is well known that the symplectic quotient M := ϕ −1 (ν)/K is a symplectic toric G manifold for G = T N /K (cf. [De]). Recall that the moment map µ : M → g * has the following description. Since g * is canonically isomorphic to the annihilator k • of k in Lie(T N ) * = R N and since (i T ) −1 (ν) = λ + k • for some λ ∈ (i T ) −1 (ν), we have an identification of g * with the affine plane (i T ) −1 (ν). With this identification, the restriction Since all the fibers of ψ are T N -orbits, the fibers of µ are T N /K = G-orbits. Sinceψ is an open map to its image, so isμ. Hence the orbital moment mapμ : , it has at most N facets. Now we argue that the image has to have at least one vertex. We give a symplectic (as opposed to a convex geometry) argument.
The function f (z) = |z j | 2 : C N → R is proper, non-negative, T N -invariant, and its vector field generates a circle action. Hence under the standard argument f | ϕ −1 (ν) descends to a proper non-negative G-invariant periodic Hamiltonianf on M . Sincef is non-negative and proper, it achieves a minimum somewhere on M . Sincef is a periodic and G-invariant, the set of points wheref achieves its minimum is a G-invariant symplectic submanifold N of M . Sincef is proper, N is compact. Since the action of G on N is Hamiltonian, it has to have a fixed point. The image of this fixed point under the moment map µ is a vertex of the polyhedral set µ(M ). Now suppose that the orbital moment map µ : M/G → g * of a symplectic toric G-manifold (M, ω, µ : M → g * ) is an embedding, that its image A = µ(M/G) is an intersection of N ≥ dim G many closed half-spaces, and that A has a vertex * . Let H 1 , . . . , H n denote the half-spaces whose supporting hyperplanes form the facets meeting at * . Since A is unimodular, n = dim G and the intersection H 1 ∩ . . . ∩ H n is isomorphic (as a unimodular cone) to R n + . Consequently there is an isomorphism G → T n so that the image of a corresponding moment map ψ : C n → g * is H 1 ∩ . . . ∩ H n . Let H n+1 , . . . , H N denote the remaining half-spaces so that A = H 1 ∩ . . . ∩ H N . We now successively apply the symplectic cut construction to C n using the half-spaces H n+1 , . . . , H N [L1]. This amounts to taking a symplectic quotient of C n × C N −n by an action of (S 1 ) N −n . Since A is unimodular, the quotient M is smooth [LMTW]. Thus we obtain a symplectic G-toric manifold (M, ω, µ) with moment map image µ(M ) = A. It is not hard to see that the orbital moment map µ : M/G → g * is an embedding.

Background on symplectic toric manifolds
We begin by recalling the symplectic slice representation.
2.1. Definition. Let a compact Lie group G act on a symplectic manifold (M, ω) with a moment map µ : M → g * . The symplectic slice representation at a point x of M is the linear symplectic action of the stabilizer G x on the symplectic vector space V 2.2. We recall that in a Hamiltonian torus action all the orbits are isotropic. We note that in Definition 2.1, if the orbit G · x is isotropic, then the symplectic slice is simply V The following theorem is a consequence of the equivariant constant rank embedding theorem of Marle [Mar] (a simpler and more limited version can be found in [LS]). The proof in the case where the group is a torus was given earlier by Guillemin and Sternberg [GS2].
2.3. Theorem. Let a compact Lie group G act on symplectic manifolds (M, ω) and (M ′ , ω ′ ) with moment maps µ : M → g * and µ ′ : M ′ → g * . Fix a point x of M and a point x ′ of M ′ . Suppose that x and x ′ have the same stabilizer, their symplectic slice representations are linearly symplectically isomorphic, and they have the same moment map value. Then there exists an equivariant symplectomorphism from an invariant neighborhood of x in M to an invariant neighborhood of x ′ in M ′ that respects the moment maps and that sends x to x ′ .
We use Theorem 2.3 only when G is a torus and only in the proofs of Lemmas 2.5 and 2.14.
Recall the local normal form for neighborhoods of orbits in a symplectic toric G manifold: 2.5. Lemma. Let (M, ω, µ : M → g * ) be a symplectic toric G manifold and x ∈ M a point. Let (1) There exists an isomorphism τ K : K → T k such that the symplectic slice representation at x is isomorphic to the action of K on C k obtained from the composition of τ K with the standard action of T k on C k . (2) Let τ : G → T ℓ × T k be an isomorphism of Lie groups such that τ (a) = (1, τ K (a)) for all a ∈ K. Then there exists a G-invariant open neighborhood U of x in M and a τ -equivariant open symplectic embedding j : Here T ℓ acts on T * T ℓ by the lift of the left multiplication and T k acts on C k by [t 1 , . . . , t k ] · (z 1 , . . . , z k ) = (e 2π where φ((q 1 , . . . , q ℓ , p 1 , . . . p ℓ ), (z 1 , . . . , z k )) = ( (p 1 , . . . , p ℓ ) , |z j | 2 e * j ) , e * 1 , . . . , e * k is the canonical basis of the weight lattice (Z k ) * , and τ * : (R * ) ℓ × (R * ) k → g * is the isomorphism on duals of Lie algebras that is induced by τ .
Part (1) of Lemma 2.5 follows from the facts that every linear symplectic action of a compact group preserves some compatible Hermitian structure and that every k dimensional abelian subgroup of U (k) is conjugate to the subgroup of diagonal matrices. Part (2) follows from Theorem 2.3.
We are now ready to prove that the quotient of a symplectic toric manifold is a manifold with corners: Proof of Proposition 1.5. By lemma 2.5 we may assume that G = T ℓ × T k , that M = T * T ℓ × C k with the action as above, and that the moment map is the map µ : M → (R * ) ℓ × (R * ) k given by µ((q 1 , . . . , q ℓ , p 1 , . . . p ℓ ), (z 1 , . . . , z k )) = ( (p 1 , . . . , p ℓ ) , where, as before, e * 1 , . . . , e * k is the canonical basis of the weight lattice (Z k ) * . Then Hence, µ(M ) is a unimodular cone; in particular, it is a manifold with corners. We now argue that µ : M → µ(M ) is a quotient map in the category of manifolds with corners; cf. A.5. Clearly, the fibers of µ are precisely the G-orbits. So it remains to show that for any manifold with corners N and any G-invariant smooth map f : M → N there exists a unique smooth mapf : Clearly, there exists a unique mapf with the above property, and our task is to show thatf is actually smooth. Without loss of generality we assume that N = R. The smoothness off then follows from a special case of a theorem of Schwarz [Sch]. The key point is that since the functions |z 1 | 2 , . . . , |z k | 2 generate the ring of T k invariant polynomials on C k , for any smooth T k -invariant function h on C k there is a smooth functionh on R k with h(z 1 , . . . , z k ) =h(|z 1 | 2 , . . . |z k | 2 ).
We now spell out some elementary facts about unimodular cones and unimodular local embeddings. 2.7. Let G be a torus with Lie algebra g and integral lattice Z G . Let ǫ be an element of g * and let (v 1 , . . . , v k ) be a sequence of vectors in Z G that form a basis of the integral lattice Z K of a subtorus K of G. We denote the corresponding unimodular cone by C (v 1 ,...,v k ;ǫ) , that is, Every subset {i 1 , . . . , i s } of the indices {1, . . . , k} uniquely determines a face F = F i 1 ,...,is of C: ..,is is the unique face of C whose relative interior contains η. Note also that the cone C itself is a face of codimension zero, its interiorC is the subtorus K C is trivial, and the map (2.9) assigns the empty set to every point ofC.
2.10. Lemma. Let G be a torus, let v 1 , . . . , v k be a basis for the integral lattice of a subtorus K, and let ǫ be an element of g * . Let C = C (v 1 ,...,v k ;ǫ) . Then (1) The subtorus K is uniquely determined by any neighborhood of ǫ in the cone C.
(2) The basis {v 1 , . . . , v k } is determined by any neighborhood of ǫ in C.
(3) The element ǫ is not uniquely determined by C, but its image under the natural projection i * K : g * → k * is uniquely determined by any neighborhood of ǫ in C. Lemma 2.10 is used in the proofs of Lemmas 2.11 and 2.14.
Proof of Lemma 2.10. If U is a neighborhood of ǫ in C then C = R + · U . So it is enough to show that C determines the subtorus K, the basis {v 1 , . . . , v k }, and the image of ǫ under the projection g * → k * .
The facets of C span affine hyperspaces that are parallel to ker v i ; this determines the vectors v i up to scalar. The fact that v i are primitive elements of the lattice Z G further determines these vectors up to sign. The signs are determined by the fact that the functions η → η, v i are bounded from below on C. The affine spans of the facets of C are given by the equations η, v i = ǫ, v i ; this fact determines the constants ǫ, v i . These constants, in turn, determine the image of ǫ under the projection map g * → k * : this image is The following two lemmas will be used to define the collapse functor in Section 4: 2.11. Lemma. Let ψ : W → g * be a unimodular local embedding. For every point w in W there exist a unique subtorus K w of G and a unique basis v w 1 , . . . , v w k of the integral lattice of K w such that the following is true.
Let ǫ = ψ(w), and let C w = C (v w 1 ,...,v w k ;ǫ) . For every sufficiently small neighborhood U w of w in W , the image ψ(U w ) is contained in C w , and the restriction ψ| Uw : U w → C w is an open embedding. Moreover, if w and w ′ belong to the same face of W , then Proof of Lemma 2.11. Let w be a point in W . By the definition of unimodular local embedding, there exists a unimodular cone Uniqueness of v w 1 , . . . , v w k is a consequence of Lemma 2.10. The faces of W are the equivalence classes under the minimal equivalence relation for which, if U is an open subset of W and ψ| U : U → C is an open embedding into a unimodular cone, then, for every face F of C with relative interiorF , all the points in the same connected component of ψ −1 (F ) are equivalent to each other. But for all the points w in ψ −1 (F ), the subtorus K w is equal to K F and the basis v w 1 , . . . , v w k is equal to the basis {v , . . . , v ψ(w) s }, in the notation of (2.9). This implies the last part of the theorem.
2.12. Lemma. Let ψ : W → g * be a unimodular local embedding. Let ǫ and ǫ ′ be elements of g is an open embedding, there exist points in U ∩ U ′ , arbitrarily close to w ′ , that are sent by ψ to the relative interior of the facet of C corresponding to v i . Let w be such a point. Because ψ| U ∩U ′ : U ∩ U ′ → C is an open embedding, w must belong to the relative interior of a facet of U ∩ U ′ . Let S be the connected component of the relative interior of the facet of U ∩ U ′ containing the point w.
is an open subset of one of the facets of C ′ , say, the facet corresponding to the vector v ′ j . Necessarily, v i = v ′ j , being the unique primitive lattice vector that is normal to ψ(S) and points into ψ To finish the section, we prove that symplectic toric G manifolds over the same unimodular local embedding are locally isomorphic.
Proof. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G manifold over a unimodular local embedding ψ : W → g * . Let w be a point of W and x a point in π −1 (w). Every invariant Let K be the stabilizer of x in G, and let k be the dimension of K. Lemma 2.5 implies that the symplectic slice representation at x is linearly symplectically isomorphic to the action of K on represent the weights of the K action on C k . Let v 1 , . . . , v k be the dual basis, in Z K , and let ǫ = µ(x). The equation (2.6) for the moment map implies that every neighborhood of x contains a smaller invariant neighborhood of x whose moment map image is a neighborhood of ǫ in the cone C (v 1 ,...,v k ;ǫ) . By this and Lemma 2.10, the symplectic slice representation is determined up to linear symplectic isomorphism by the image of any invariant neighborhood of G · x which is sufficiently small. This image is exactly ψ(U w ), where U w is a sufficiently small neighborhood of w in W . So the germ of ψ at w determines the symplectic slice representation up to linear symplectic isomorphism. Clearly, this germ also determines the moment map value, µ(x), which is equal to ψ(w). The result then follows from Theorem 2.3.

Uniqueness
We begin by defining the sheaf of automorphisms of a symplectic toric G manifold: 3.1. Definition. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G manifold over a unimodular local embedding ψ : W → g * . Its sheaf of automorphisms is the sheaf A on W that associates to every open set U ⊂ W the group of automorphisms of the symplectic toric G manifold π −1 (U ) over U → g * .

3.2.
Remark. The definition of the sheaf A depends on the choice of a symplectic toric G manifold over W . Two such choices result in sheaves that are isomorphic over sufficiently small open sets, by Lemma 2.14.
). The result of the gluing is a symplectic toric G manifold M f over ψ : W → g * . Different choices of cocycles representing c give rise to isomorphic manifolds.
Conversely, suppose that ((M ′ , ω ′ , µ ′ : M ′ → g * ), π ′ : M ′ → W ) is another symplectic toric manifold over ψ : W → g * . By Lemma 2.14 there exists an open covering The cohomology class of this cocycle is independent of the choices of the isomorphisms h i .
3.4. Lemma. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G manifold over a unimodular local embedding ψ : W → g, and let A be its sheaf of automorphisms (cf. (3.1)). Then A is a sheaf of abelian groups, and Proof. The argument is analogous to those of [DD], [BM], and [LT,Proposition 7.3]. We claim that there exists a short exact sequence of sheaves of groups for an open subset U ⊂ W , to the time-one flow of the Hamiltonian vector field of the function f • π : π −1 (U ) → R. We refer the reader to the proof of [LT,Prop. 7.3] for the details of the proof that (3.5) is an exact sequence of sheaves. We only note here that the most difficult part is to show that Λ is onto as a map of sheaves. We record this fact below as Lemma 3.6.
Because the sheaves Z G × R and C ∞ W are abelian and the sequence (3.5) is exact, the sheaf A is abelian.
Because the sheaf C ∞ W admits partitions of unity, its cohomology groups of degree ≥ 1 are zero. The required isomorphism them follows from the long exact sequence that corresponds to the short exact sequence (3.5).
For later reference, we record one fact that was used in the above proof: 3.6. Lemma. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric manifold over a unimodular local embedding ψ : W → g * . Let g : M → M be an isomorphism. Then, for every point w of W , there exist a neighborhood U w in W and a smooth function f : U w → R such that the restriction of the isomorphism g to the preimage π −1 (U w ) is the time one flow of the Hamiltonian vector field of the function f • π : π −1 (U w ) → R.
Proof. We refer the reader to [LT,Prop. 7.3] for the proof. We note that this proof relies on a non-trivial lemma of Gerald Schwartz, which appeared in [HS,Theorem 3.1]: if M is a manifold with an action of a torus G, and if h : M → M is an equivariant diffeomorphism that carries every orbit to itself, then there exists a smooth invariant function ϕ : We now state the "uniqueness" part of Theorem 1.9: 3.7. Proposition. Let ψ : W → g * be a unimodular local embedding. Fix a symplectic toric G manifold M over W . Then there exists a bijection where π 0 STM(W ψ → g * ) denotes the set of isomorphism classes of objects of the category Proof. This result follows immediately from Lemmas 3.3 and 3.4.
3.8. Remark. It is the artifact of our construction that the bijection above depends on the choice of the manifold M .
We now recall how to obtain Delzant's classification theorem and its generalization to the case that the manifold is not necessarily compact but the moment map is proper as a map to a convex subset of g * : 3.9. Proposition. Let (M, ω, µ) be a connected symplectic toric G manifold. Suppose that there exists a convex subset T of g * that contains the image µ(M ) and such that the map µ : M → T is proper. Then the image µ(M ) determines the triple (M, ω, µ) up to isomorphism: if (M ′ , ω ′ , µ ′ ) is another connected symplectic toric G manifold, and if there exists a convex subset T ′ of g * that contains µ ′ (M ′ ) and such that the map µ ′ : Proof. The assumptions of the proposition imply that the image ∆ := µ(M ) is a convex subset of g * , the level sets of µ are connected, and the map µ is open as a map to ∆. See e.g. [LMTW] or [BK]. It follows that the orbital moment mapμ : M/G → g * also has the convex image ∆, connected level sets, and the mapμ is open as a map to ∆. From this and Proposition 1.5, we deduce that the orbital moment mapμ : M/G → g * is a diffeomorphism of M/G with ∆. The same argument applies to M ′ . Thus, M and M ′ may be viewed as elements of the category STM(∆ inclusion −−−−−→ g * ) .
It remains to prove that, for any unimodular local embedding ψ : W → g * , the category STM(W ψ → g * ) of symplectic G-toric manifolds over ψ is non-empty. Our strategy is to define a new category STB(W ψ → g * ), of symplectic toric G bundles over ψ, and to construct a functor (3.10) collapse : It will follow from the definition of the category STB(W ψ → g * ) that it is non-empty for any ψ.
Consequently, STM(W ψ → g * ) is non-empty as well. In fact, it suffices for our purposes to construct the map (3.10) on objects only.
3.11. Definition. We define the category STB(W ψ → g * ) of symplectic toric G bundles over a unimodular local embedding ψ : W → g * . An object of this category is a principal G bundle π : P → W equipped with a symplectic form ω and a moment map µ : P → g * such that µ = ψ • π (cf. A.7). The morphisms in this category are G-equivariant symplectomorphisms that commute with the maps to W .
3.12. Notation. Let ψ : W → g * be a unimodular local embedding. We write an object of the category STB(W ψ → g * ) as ((P, ω, µ : P → g * ), π : P → W ). If W is a subset of g * and ψ : W → g * is the inclusion map, we call this object a symplectic toric G bundle over W (instead of "over ψ : W → g * ") and we write it as (P, ω, µ : M → W ) (omitting π). We often use the symbol M rather than the symbol P for the total space of the bundle.
3.13. Remark. The standard lifted action of the torus G on its cotangent bundle T * G makes T * G into a symplectic toric G bundle over the identity map id : g * → g * . If ψ : W → g * is a unimodular local embedding, the pullback of T * G → g * by ψ gives a symplectic toric G bundle over ψ : W → g * .
3.14. Example. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G manifold over a unimodular local embedding ψ : W → g * . LetM be the interior of M and letW be the interior of W , cf. (A.2). Then ((M , ω, µ :M → g * ), π :M →W ) is a symplectic toric G bundle over ψ :W → g * . Indeed, the slice theorem for compact group actions implies that the quotientM →W admits local sections, so it is a principal G bundle.

Collapsing a symplectic toric bundle to a symplectic toric manifold
The purpose of this section is to define the collapse functor from symplectic toric G bundles to symplectic toric G manifolds. We begin by defining the collapse of a symplectic toric bundle as a topological space. 4.1. Notation. If ∼ is an equivalence relation on a topological space X and π : X → X/∼ is the quotient map, we write [x] for π(x) for any x ∈ X.
4.2. Definition. Let W ψ → g * be a unimodular local embedding. For each face F of W , let K F denote the subtorus of G that corresponds to F as described in Lemma 2.11. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G bundle over W ψ → g * . Let ∼ be the smallest equivalence relation on M such that, for every face F of W , we have m ′ ∼ m whenever π(m) ∈ F and m ′ ∈ K F · m. The collapse of M , as a topological space, is the quotient for two morphisms of symplectic toric G bundles. In other words, 4.2 defines a functor from the category of symplectic toric G bundles over ψ : W → g * to topological G spaces (in fact, topological G manifolds) over ψ : W → g * . In the rest of this section we construct on the topological space collapse(M ) the structure of a symplectic toric G manifold, so that collapse becomes a functor Thus, given a symplectic toric G bundle π : M → W over W ψ → g * , we would like to construct a symplectic toric G manifold over W ψ → g * whose underlying topological space is collapse(M ). An important special case is when M is obtained from a symplectic toric G bundle M over an open subset of g * by taking the preimage in M of a unimodular cone in g * . In this case, the collapse of M is essentially the same thing as the symplectic cut of M with respect to this cone, as described in [LMTW]. Thus, in preparation for the construction of the collapse functor, we first recall how to apply the symplectic cut construction, with respect to a unimodular cone, to a symplectic toric G bundle over an open subset of g * . 4.7. Notation. Let G be a torus. Let ǫ be an element of g * and v 1 , . . . , v k an ordered basis of the integral lattice Z K of a subtorus K of G. Denote by 2π dz j ∧ dz j , and with the K-action generated by the moment map z → i * Denote by −C k (v 1 ,...,v k ;ǫ) the space C k with the same K action as before, with the opposite symplectic form −ω k C , and with the moment map − i * K ǫ + |z j | 2 v * j .
4.8. Let G be a torus, let U be an open subset of g * , and let ( M , ω, µ : M → U ) be a symplectic toric G bundle over U . Let ǫ be an element of g * , and let v 1 , . . . , v k be an ordered basis of the integral lattice Z K of a subtorus K of G. Let Note that M := µ −1 (C) is a symplectic toric G bundle over U := U ∩ C, so we can construct the topological quotient M (v 1 ,...,v k ;ǫ) = collapse(M ) = µ −1 (C)/∼ as described in 4.2. This quotient is equipped with a G action and with a map to C. We have an open dense embedding j :M = µ −1 (C) → M (v 1 ,...,v k ;ǫ) that intertwines the G actions and the maps to C.
Consider the product M × (−C k (v 1 ,...,v k ;ǫ) ), that is, M × C k , with the symplectic form ω ⊕ (−ω C k ) and with the K action generated by the moment map Because the K action on M is free, the K action on M × C k is also free. So 0 is a regular value of the moment map Φ, the level set Φ −1 (0) is a smooth submanifold of M × C k , and the quotient is a manifold. Moreover, there exists a unique two-form on this quotient whose pullback to Φ −1 (0) is equal to the pullback of ω ⊕ (−ω C k ) through the inclusion map Φ −1 (0) → M × C k . These assertions immediately follow from the symplectic reduction procedure for M × C k , a.k.a the symplectic cut procedure for M with respect to the cone C (v 1 ,...,v k ;ǫ) . Moreover, the G action on the first factor of M × C k induces a symplectic toric G action on ( from M to Φ −1 (0) descends to a homeomorphism Finally, the restriction of the map (4.9) toM = µ −1 (C) is a moment map preserving equivariant symplectomorphism ofM with an open dense subset of ( M × C k )// 0 K.

4.10.
Remark. The K action on C k (v 1 ,...,v k ;ǫ) depends on the ordering of the basis v 1 , . . . , v k . Reordering the basis changes (M × C k )// 0 K by an isomorphism. On the other hand, (M × C k )// 0 K does not really depend on ǫ; it depends only on the projection of ǫ to the dual of the Lie algebra of K, or, equivalently, it depends only on the numbers ǫ, v 1 , . . . , ǫ, v k .
In the above construction, the zero level set Φ −1 (0) is entirely contained in the manifold with corners M ×C k = µ −1 (C)×C k ⊂ M ×C k . We now reformulate this same construction while starting directly with a symplectic toric G bundle M over an open subset of the cone C (v 1 ,...,v k ;ǫ) without passing through an ambient smooth manifold M . One possible source of confusion is that, whereas Φ −1 (0) is a smooth submanifold of M × C k and is contained in the manifold with corners M × C k , it is not a submanifold with (empty) corners of M × C k in the sense of, say, [Mat]. Nevertheless, the manifold structure on Φ −1 (0) is induced from its inclusion into M × C k : a function on Φ −1 (0) is smooth if and only if it is equal to the restriction to Φ −1 (0) of a smooth function on M × C k ; cf. A.3. 4.11. Lemma. Let G be a torus, let ǫ be an element of g * , and let (v 1 , . . . , v k ) be an ordered basis of the integral lattice Z K of a subtorus K of G. Let U be an open subset of the unimodular cone C (v 1 ,...,v k ;ǫ) . Let (M, ω, µ : M → U ) be a symplectic toric G bundle over U . Consider the manifold with corners the projection map on the duals of the Lie algebras. Note that the zero level set, ..,v k ;ǫ) ). Then the zero level set Φ −1 (0) is a manifold (without corners), and the quotient Φ −1 (0)/K is a manifold (without corners). (The word "is" is defined in A.3 and A.5.) 4.12. Notation. In the setup of Lemma 4.11, we denote (4.13) (M × C k )// 0 K := Φ −1 (0)/K.
Proof of Lemma 4.11. To show that Φ −1 (0) is a manifold, it is enough to show that for every point in M there exists a neighborhood Ω and a manifold structure on Φ −1 (0) ∩ (Ω × C k ) with which a real valued function is smooth if and only if it is equal to the restriction to Φ −1 (0) ∩ (Ω × C k ) of a smooth function on Ω × C k . Once we know that Φ −1 (0) is a manifold, the fact that Φ −1 (0)/K is a manifold follows from the general fact that the quotient of a smooth manifold by a free action of a compact Lie group is a manifold. Fix a point m ∈ M . By the definition of symplectic toric G bundle there exist a neighborhood V of µ(m) in U and, setting Ω := µ −1 (V ), a diffeomorphism ψ : Ω → G × V that carries the moment map µ : Ω → V to the projection map G × V → V . The map ψ × Id is then a diffeomorphism from Ω × C k to G × V × C k , and it carries Φ −1 (0) ∩ (Ω × C k ) to (4.14) { So it is enough to show that the subset (4.14) of G × V × C k is a manifold (in the sense of A.3). Let L be a complementary torus to K in G and let i * L : g * → l * be the projection map on the dual of Lie algebras. Possibly shrinking V , we may assume that the affine isomorphism u → (i * K (u − ǫ), i * L (u − ǫ)) from g * to k * × l * carries V to a product V K × V L where V K is an open subset of the cone {η ∈ k * | η, v j > 0 for all 1 ≤ j ≤ k} and where V L is an open neighborhood of the origin in l * . Let O = {z ∈ C k | |z j | 2 v * j ∈ V K }. The smooth map (g, u, z) → (g, i * L (u − ǫ), z) from G × V × C k to G × l * × C k carries the subset (4.14) to the open subset G × V L × O. Its restriction to (4.14) has a smooth inverse, namely, the map (g, u L , z) → (g, ǫ + |z j | 2 v * j + u L , z) where, by slight abuse of notation, the element u L of l * is identified with the element of g * whose image in l * is u L and whose image in k * is 0. This map carries the manifold structure of G × V L × O to a manifold structure on (4.14) with the required properties. 4.15. Lemma. Let U be an open subset of the unimodular cone C (v 1 ,...,v k ;ǫ) . Let M be a symplectic toric G bundle over U . The map M → (M × C k ) given by descends to a homeomorphism which is G equivariant and which respects the maps to U .
Proof. This result is essentially the same as the analogous result for symplectic cutting. It follows from the following facts, which are not difficult to check: the map (4.17) is well defined, continuous, one-to-one and onto, G equivariant, respects the maps to U , and proper.
Starting with a symplectic toric G bundle M over an open subset of a unimodular cone C v 1 ,...,v k ;ǫ , Lemmas 4.11 and 4.15 combine to give a manifold structure on the topological space M (v 1 ,...,v k ;ǫ) = M/∼ such that the homeomorphism j fromM to an open dense subset of M (v 1 ,...,v k ;ǫ) (namely, to the preimage ofC (v 1 ,...,v k ;ǫ) ) is a diffeomorphism. The next step is to show that the G invariant symplectic structure (j −1 ) * ω on the open dense subset of M (v 1 ,...,v k ;ǫ) extends smoothly to a (necessarily unique and G invariant) symplectic structure on M (v 1 ,...,v k ;ǫ) . We do this by locally identifying M (v 1 ,...,v k ;ǫ) with a symplectic cut, in the usual sense, of an extension of M to a symplectic toric G bundle that does not have boundary and corners. The following lemma guarantees that such extensions exist. 4.18. Lemma. Let C be a unimodular cone in g * and U an open subset of C. Let (M, ω, µ : M → g * ) be a symplectic toric G bundle over U . Let w be a point in U . Then there exist a neighborhood Ω of w in g * such that C ∩ Ω is contained in U , a symplectic toric G bundle over Ω, ( M ,ω,μ : M → Ω), and an isomorphism of symplectic toric G bundles over C ∩ Ω betweenμ −1 (C ∩ Ω) and µ −1 (C ∩ Ω).
Proof. By the definition of G bundle, there exists a neighborhood V of w in U and an equivariant diffeomorphism from π −1 (V ) onto G × V that carries the moment map µ to the projection map G × V → V . Without loss of generality we may assume that w is the origin in g * and that G is the standard torus T n = R n /Z n . Write the coordinates on G = T n as t 1 , . . . , t n , with t j ∈ R/Z, and the coordinates on g * = R n as x 1 , . . . , x n . Then, on a neighborhood of µ −1 (w) in M , we get coordinates t 1 , . . . , t n , x 1 , . . . , x n , taking values in (R/Z) n × V . Because the torus action is generated by the vector fields ∂/∂t j and the moment map is (t 1 , . . . , t n , x 1 , . . . , x n ) → (x 1 , . . . , x n ), the symplectic form must have the form dx j ∧ dt j + a ij dx i ∧ dx j . Because the coefficients a ij are invariant smooth functions (cf. A.7), there exist smooth functions a ij on V such that a ij (t 1 , . . . , t n , x 1 , . . . , x n ) = a ij (x 1 , . . . , x n ).
By the Poincare lemma (see A.8), if Ω is a sufficiently small ball around the origin in g * (= R n ), there exists a one-form b j dx j on V ∩ Ω such that a ij dx i ∧ dx j = d b j dx j . Letb j = b j (x 1 , . . . , x n ) be a smooth function on Ω whose restriction to Ω ∩ V is equal to b j .
The result of the lemma then holds with the manifold M = G × Ω and the symplectic form ω = dx j ∧ dt j + d b j (x 1 , . . . , x n )dx j . 4.19. Lemma. Let (M, ω, µ : M → U ) be a symplectic toric G bundle over an open subset of a unimodular cone C (v 1 ,...,v k ;ǫ) . Consider the set (M × C k )// 0 K given in (4.13) with its manifold structure given in Lemma 4.11. Let j 0 :M → (M × C k )// 0 K be the composition of the open dense embedding j :M ֒→ M (v 1 ,...,v k ;ǫ) (cf. (4.5), (4.6)) with the homeomorphism (4.17). Then there exists a unique symplectic form on (M × C k )// 0 K whose pullback through j 0 is equal to the given symplectic form onM . Moreover, the manifold (M × C k )// 0 K, with this symplectic form and with the G action and moment map induced from M , is a symplectic toric G manifold.
Proof. Because j 0 is an open dense embedding, if such a symplectic form exists, then it is unique. For the same reason, two such symplectic forms that are defined on open subsets must coincide on the intersection of these subsets. Thus, it is enough to find such a symplectic form locally. Locally, we obtain such a symplectic form by applying the symplectic cut procedure 4.8 to the "smooth extension" obtained in Lemma 4.18. 4.20. Remark. Let C ⊂ g * be a unimodular cone, and suppose that U is contained in the interior C of C. Then the embedding j 0 of Lemma 4.19 is an isomorphism of Hamiltonian G manifolds We next observe that the collapse construction is functorial: where the vertical arrows are the homeomorphisms described in Lemma 4.15. Moreover,φ and collapse(ϕ) are equivariant open embeddings, and collapse(ϕ) is an equivariant symplectomorphism of (M 1 × C k )// 0 K with an open invariant subset of (M 2 × C k )// 0 K that respects the moment maps.
Proof. Since ϕ intertwines the G moment maps µ 1 and µ 2 , it intertwines the K moment maps For the last assertion, it is enough to notice that there exists an open dense subset of (M 1 × C k )// 0 K (namely, j 0 (M 1 )) on which collapse(ϕ) is an equivariant symplectomorphism that respects the moment maps (cf. Lemma 4.19).
We now generalize Remark 4.20 to an important technical result.
where f is the homeomorphism in (4.17), f ′ is its analogue for L, κ([m]) = [m] for all m, and ν is an isomorphism of symplectic toric G manifolds.
Proof of Lemma 4.23. We recall that ( where, if j ∈ {j 1 , . . . , j r }, then we set z j = µ(m) − ǫ, v j . This map is smooth, because µ(m) − ǫ, v j is strictly positive on M for all j ∈ {j 1 , . . . , j r }. We take ν : Φ −1 L (0)/L → Φ −1 K (0)/K to be the function induced from (4.25). To see that ν is a diffeomorphism, we note that it has a smooth inverse, namely, the function Φ −1 K (0)/K → Φ −1 L (0)/L that is induced from the projection map (z 1 , . . . , z k )) → (m, (z j 1 , . . . , z jr )). (4.26) To show that ν respects the G actions, the symplectic forms, and the moment maps, it is enough to show this on an open dense set. This we get from Lemma 4.19.
We are now ready to define the functor 4.27. We first define this functor on objects. Let ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G bundle over a unimodular local embedding ψ : W → g * . As a topological space, collapse(M ) = M/∼ is the space defined in (4.2). It is a Hausdorff second countable topological space equipped with a G action and with a quotient map to W . By Definition 1.4 of a unimodular local embedding, for every point w in W there exists a neighborhood U w and a unimodular cone C w = C (v 1 ,...,v k ;ǫ) such that π(U w ) is contained in C w and such that π| Uw : U w → C w is an open embedding. For every such U w , the inclusion map of M | Uw := π −1 (U w ) into M descends to an open embedding of (M | Uw ) (v 1 ,...,v k ;ǫ) into collapse(M ). We equip the image of this embedding with the structure of a symplectic toric manifold that is induced from the homeomorphism 4.29. Finally, we define the collapse functor on arrows. Fix a unimodular local embedding ψ : W → g * , and let π 1 : M 1 → W and π 2 : M 2 → W be two objects in STB(W ψ → g * ), that is, two principal G bundles over the manifold with corners W with G invariant symplectic forms so that ψ•π i : M i → g * are the corresponding moment maps. If ϕ : M 1 → M 2 is a map of principal bundles over W , then ϕ descends to a continuous G equivariant map collapse(ϕ) : It remains to check that collapse(ϕ) is smooth and symplectic. Since smoothness is a local property, it follows from the functoriality of cuts by unimodular cones (Lemma 4.22). Finally, the fact that collapse ϕ is symplectic follows from 4.28.
This concludes our construction of the functor collapse : Consequently, as we explained above, since the category STB(W ψ → g * ) is non-empty for any unimodular local embedding ψ : W → g * (we can always pull back T * G → g * by ψ; cf. Remark 3.13), the category of symplectic toric manifolds STM(W ψ → g * ) is non-empty as well. Together with Proposition 3.7, this completes the proof of our main result, Theorem 1.9. 4.30. Remark (The functor collapse is local). Given the unimodular local embedding ψ : W → g * , the restriction of ψ to an open subset U of W is also a unimodular local embedding. Furthermore, the category of symplectic toric bundles STB(W ψ → g * ) and the category of symplectic toric manifolds That is, we have restriction functors commutes.
To verify that this diagram commutes for objects, let M be a symplectic toric G bundle over W ψ → g * . The fact that (collapse M )| U is equal to collapse(M | U ) as symplectic toric G manifolds follows from 4.21. It means that (collapse M )| U and collapse(M | U ) are equal as sets, and, moreover, that the two symplectic toric G manifold structures on this set, obtained by first collapsing and then restricting or first restricting and then collapsing, are equal. We may then use the notation collapse M | U without ambiguity.
To verify that the diagram commutes for arrows, let ϕ : M 1 → M 2 be an isomorphism of symplectic toric G bundles over W ψ → g * . Then each of (collapse ϕ)| U and collapse(ϕ| U ) is an isomorphism of symplectic toric G manifolds from collapse M 1 | U to collapse M 2 | U . To show that these isomorphisms are equal, it is enough to show that they are equal as maps of sets, and this follows easily from the definitions. 4.31. The locality of collapse can be rephrased as follows: the assignments are presheaves of categories over W , and collapse is a map of presheaves. It is not hard to check that the two presheaves are, in fact, stacks and the functor collapse is, in fact, an isomorphism of stacks (cf. section 6). Moreover, we expect to be able to construct "categorified differential characters", in the spirit of Hopkins-Singer [HoS], that would "classify" the category STB(W ψ → g * ). The π 0 of the category of characters will be the cohomology H 2 (W ; Z G × R). In other words, we believe that Theorem 1.9 can be categorified, to use a fashionable turn of phrase. We leave the details to a subsequent paper.

Characteristic classes of symplectic toric manifolds
By Theorem 1.9, symplectic toric G manifolds over a unimodular local embedding W ψ → g * are in bijection with elements of H 2 (W ; R × Z G ). The actual bijection that is given in the proof depends on the initial choice of one such manifold.
In this section we define the invariants directly rather than as measurements of the global difference between two locally isomorphic manifolds. We associate to a symplectic toric G manifold M = ((M, ω, µ : M → g * ), π : M → W ) over a unimodular local embedding W ψ → g * two cohomology classes, the Chern class, which encodes the "twistedness" of M as a G manifold over W , and the horizontal class, which encodes the "horizontal part" of the symplectic form ω. These classes determine the symplectic toric G manifold up to isomorphism. They are analogous to classes that were defined in [BM, DD, D, Z]. The purpose of this approach is to clarify the geometric meaning of the Z G and R parts of the classifying cohomology class.
Let ψ : W → g * be a unimodular local embedding. Restriction to the interiorW (cf. Example 3.14) defines a functor An examination of the proofs of Lemmas 3.3 and 3.4 shows that the isomorphisms obtained in these lemmas are consistent with restrictions to open subsets. Since the inclusion mapW → W induces an isomorphism in cohomology, we deduce that the functor (5.3) induces a bijection on equivalences classes. Thus, it is enough to define the Chern class (5.1) and the horizontal class (5.2) for elements of STM(W ψ → g * ).
Recall that, becauseW has no boundary, STM(W ψ → g * ) = STB(W ψ → g * ). We proceed to define the Chern class and horizontal class for arbitrary symplectic toric bundles (not just over manifolds without boundary or corners such asW ).
Recall that, for a torus G, principal G bundles over a fixed manifold or manifold with corners B are classified by their Chern classes, which are elements of H 2 (B; Z G ). We briefly recall how this works. Fix a principal G bundle π : P → B. Let U be an open covering of B. For each U ∈ U , let ϕ U : π −1 (U ) → G×U be a trivialization. For each U and V in U , let h V U : U ∩V → g be a logarithm of the transition map, i.e., the automorphism ϕ is aČech 2-cocycle with coefficients in Z G . Its cohomology class is the Chern class of the principal G bundle π : P → B. We define the Chern class of a symplectic toric G bundle over W ψ → g * to be its Chern class in the usual sense, as a principal G bundle over W .
We now turn to the real-valued part of the characteristic class.
5.4. Lemma. Let W be a manifold with corners and ψ : W → g * a unimodular local embedding.
Recall that, for a principal G bundle π : P → B, a form α on P is basic (i.e., there exists a form β on B such that α = π * β) iff the form is horizontal (i.e., ι(ξ P )α = 0 for all ξ ∈ G) and G-invariant.
Proof of Lemma 5.4. The two-form ω − d µ, Θ is G-invariant because so are ω, µ, and Θ. It is horizontal because, for every ξ ∈ g, Thus, it is basic. Let Θ ′ be another connection one-form on π : M → W . Then there exists a one-form γ on W such that Θ ′ = Θ + π * γ. Let β ′ be the two-form on W that is determined by Θ ′ as in part (1) of the lemma. Then Thus, β ′ is equal to β − d(something), which is cohomologous to β. 5.5. Remark. We recall that dΘ itself descends to a g-valued two-form on W , which is, by definition, the curvature of the connection. By the Chern-Weil recipe, the image of the first Chern class c 1 (π : M → W ) under the mapȞ 2 (W, Z G ) → H 2 (W ; g) is the cohomology class of the curvature. 5.6. Remark. The connection determines a splitting of each tangent space of M into a vertical subspace and a horizontal subspace. The two-form ω splits, accordingly, into a two-form ω V V on the vertical space, a two-form ω HH on the horizontal space, and a pairing ω V H between the vertical and horizontal spaces. Because the level sets of a toral moment map are isotropic, the vertical component ω V V is zero. The component ω V H is determined by the moment map equation. Thus, the only freedom in choosing ω is in the horizontal component ω HH . The form β determines this horizontal component.

5.7.
Remark. In the "minimal coupling" construction for a symplectic fiber bundle, a connection with Hamiltonian holonomy determines a closed two-form on the total space that extends the fiberwise symplectic forms, and one can add to it the pullback of a two-form from the base to obtain a symplectic form on the total space. See [St] or [GLS,chap. 1]. Our construction can be viewed as a form of "un-coupling": we start with a symplectic form on the total space, subtract a closed twoform that is determined by a connection, and obtain a two-form on the base. One difference is that our two-form on the total space has a zero vertical-vertical component and a prescribed verticalhorizontal component, whereas the two-form for "minimal coupling" has a prescribed verticalvertical component and zero vertical-horizontal component. is the class in H 2 (W ; R) that is represented by a two-form β on W , whereπ * β =ω − d μ, Θ for a connection Θ on π :M → W . This class is well defined by Lemma 5.4. 5.9. Lemma. Let W be a manifold with corners, and let ψ : W → g * be a unimodular local embedding. Let π : M → W be a principal G bundle. Let µ = ψ • π : M → g * . Let ω 0 and ω 1 be G-invariant symplectic forms on M for which µ is a moment map. Suppose that where β is an exact two-form on W . Then there exists an equivariant diffeomorphism ψ 1 : M → M that respects π and such that ψ * 1 ω 1 = ω 0 . Proof. As usual in Moser's method, we seek a family of equivariant diffeomorphisms ψ t : M → M such that ψ * t ω * t = ω. Differentiating, where α is a one-form on W with dα = β and where dψt dt = X t • ψ t . We achieve this by setting X t to be the vector field that satisfies The vector field X t is invariant, because so are ω t and π * α.
We claim that π * X t = 0. Because µ = ψ • π and ψ is a local embedding, it is enough to show that µ * X t = 0, that is, that ι(X t )dµ ξ = 0 for every ξ ∈ g. And, indeed, ι(X t )dµ ξ = −ω(ξM , X t ) = ι(ξM )π * α = 0: the first equality follows from the definition of the moment map, the second follows from the definition of X t , and the third follows from π being G-invariant.
Because π * X t = 0, and because π : M → W is proper and its level sets are manifolds without boundary or corners, we can integrate the vector field X t . This yields an isotopy ψ t with the required properties. 5.10. Lemma. Let M 1 = ((M 1 , ω 1 , µ 1 : M 1 → g * ), π 1 : M 1 → W ) and M 2 = ((M 2 , ω 2 , µ 2 : M 2 → g * ), π 2 : M 2 → W ) be symplectic toric G bundles over a unimodular local embedding ψ : W → g * . Suppose that their Chern classes and horizontal classes are equal. Then they are isomorphic in Proof. Because c 1 (M 1 ) = c 1 (M 2 ), M 1 and M 2 are isomorphic as principal G bundles over W . Without loss of generality we may assume that they are equal as principal G bundles over W . Because c hor (M 1 ) = c hor (M 2 ), Lemma 5.9 implies that they are isomorphic as symplectic toric G bundles over W .
We now define the characteristic classes for symplectic toric G manifolds. 5.11. Definition. Let M = ((M, ω, µ : M → g * ), π : M → W ) be a symplectic toric G manifold over a unimodular local embedding W ψ → g * . LetM = M |W . The Chern class c 1 (M ) is the class in H 2 (W ; Z G ) whose pullback toW is the Chern class of the principal G bundleM →W . The horizontal class c hor (M ) is the class in H 2 (W ; R) whose pullback toW is the horizontal class of the symplectic toric G bundleM →W . 5.12. Remark. Suppose that W is a manifold (without boundary or corners) and ψ : W → g * is a local embedding ("unimodular" is now automatic). Then symplectic toric G bundles over W ψ → g * are the same thing as symplectic toric G manifolds over W ψ → g * . In this case, the classes c 1 (M ) and c hor (M ) that are associated to a symplectic toric G bundle M over W ψ → g * are the same as the classes that are associated to M as a symplectic toric G manifold. Since |W : H 2 (W ; Z G ) → H 2 (W ; Z G ) and |W : H 2 (W ; R) → H 2 (W ; R) are isomorphisms, we conclude that c 1 (M ) = c 1 (M ) and c hor (M ) = c hor (M ).
We conclude this section by re-stating the classification of symplectic toric G manifolds in terms of their characteristic classes. 5.14. Proposition. Let W be a manifold with corners and ψ : W → g * a unimodular local em- Proof. By Lemma 5.10, M 1 |W is isomorphic to M 2 |W . As explained after Equation (5.3), this implies that M 1 is isomorphic to M 2 .
We will use the following lemma twice: to prove part (1) of Theorem 1.11 and to prove Proposition 6.1. 5.15. Lemma. Let W be a manifold with corners and ψ : W → g * a unimodular local embedding. Let A ∈ H 2 (W ; Z G ) and let B ∈ H 2 (W ; R). Then there exists a symplectic toric G bundleM with c 1 (M ) = A and c hor (M ) = B.
Proof. Letπ :M → W be a principal G bundle over W with Chern class A. Letμ = ψ •π :M → g * , let Θ be a connection one-form onM → W , let β be a two-form on W with cohomology class B, and letω = d µ, Θ + π * β.
The connection determines a decomposition T pM ∼ = g ⊕ g * for every p ∈M . With respect to this decomposition, and with respect to a choice of basis in g and the dual basis in g * , the two-formω is represented by a matrix of the form 0 −I I β .
Thus, the two-formω is non-degenerate.
We now prove Theorem 1.11. (1)  Proof of part (2) of Theorem 1.11. Let W = M 2 /G and let ψ : W → g * be induced from the moment map µ 2 : M 2 → g * . By Proposition 1.5, W is a manifold with corners and ψ is a unimodular local embedding. Take π 2 : M 2 → W to be the quotient map, and take π 1 : M 1 → W to be the composition of the quotient map M 1 → M 1 /G with the diffeomorphism F : M 1 /G → M 2 /G. Then M 1 and M 2 become elements of STM(W ψ → g * ), and c 1 (M 1 ) = c 1 (M 2 ) and c hor (M 1 ) = c hor (M 2 ) as cohomology classes on W . By Proposition 5.14, M 1 is isomorphic to M 2 .

Equivalence of categories between symplectic toric bundles and symplectic toric manifolds
Let W be a manifold with corners and ψ : W → g * a unimodular local embedding. The purpose of this section is to prove that the functor collapse : STB(W ψ → g * ) → STM(W ψ → g * ) is an equivalence of categories.
The first step is to show that the functor collapse is essentially surjective: 6.2. Remark. Our proof that collapse is essentially surjective uses the classifications of symplectic toric manifolds and symplectic toric bundles. If one proves essential surjectivity directly, the classification of symplectic toric manifolds can then be deduced from the classification of symplectic toric bundles. For compact toric manifolds, this was essentially done by Eckhard Meinrenken (chapter 7, section 5 of [Mei]). (He obtained M as a symplectic cut of symplectic toric manifold M over a neighborhood of W . The preimage of W in M is then a symplectic toric bundle over W whose collapse is M .) The next step is to prove that the functor collapse : STB → STM is fully faithful. For this, we give an analogue of Lemma 3.6 for symplectic toric bundles: 6.3. Lemma. LetM = ((M ,ω,μ :M → g * ),π :M → W ) be a symplectic toric bundle over a unimodular local embedding ψ : W → g * .
(1) Let f : W → R be a smooth function. Then the time one Hamiltonian flow of f :π :M → R is well defined (and it is an isomorphism ofM ).
(2) Let f 1 , f 2 : W → R be smooth functions. Suppose that the time one Hamiltonian flow of f 1 •π is equal to that of f 2 •π. Then, on every connected component of W , the difference f 1 − f 2 : W → R coincides with the pullback of an integral affine function to this component.
where , is the pairing between g * and g.) (3) Let g :M →M be an isomorphism. Then every point w of W has a neighborhood U w in W and a smooth function f : U w → R such that the restriction of the isomorphism g to the preimageπ −1 (U w ) is the time one flow of the Hamiltonian vector field of the function f •π :π −1 (U w ) → R.
Proof. Let ξ f be the Hamiltonian vector field of f •π: Because f •π is, locally, a collective function (which means that it is of the form g • µ where g is smooth on g * ), its Hamiltonian vector field is tangent to the G orbits. This implies that the time one flow of this vector field is well defined, proving the first part of the lemma. For w ∈ W , the restriction toπ −1 (w) of the Hamiltonian vector field f •π is equal to the vector field ξM induced from a Lie algebra element ξ. The Lie algebra element ξ is the image of df •π under the natural isomorphism (T π(w) g * ) * ∼ = g. The time one vector field of ξM | π −1 (w) is the identity map iff ξ ∈ Z G . These facts imply the second part of the lemma.
To prove the third part of the lemma, we follow an argument from the proof of Proposition 7.3 of [LT]. Let g :M →M be an isomorphism. Let ϕ : W → G be the smooth function such that g(m) = ϕ(m) · m for all m ∈ M . (Existence and smoothness of ϕ follows fromπ :M → W being a principal G bundle. In the analogous lemma for symplectic toric manifolds, Lemma 3.6, one uses Lemmas of Gerald Schwarz to deduce that ϕ exists and is smooth.) Let w ∈ W . Let U w be a neighborhood of w in W and let h : U w → g be such that h • π is a logarithm of ϕ, i.e., ϕ(m) = exp(h(π(m))) for all m ∈ U z . Then g|π−1 (Uw) :M | Uw →M | Uw is the time one flow of the vector field ξ onM | Uw that is given by ξ| m = h(π(m))| M | m . Denote this flow g t . Thus, g = g 1 .
So the one-form α is closed. Because dα = 0 and the pullback of α toπ −1 (w) is zero, after possibly shrinking U w we may assume that α is exact, and, moreover, a primitive function for α is constant on fibers ofπ. Thus, there exists a smooth function f : U w → R such that d(f •π) = α. The time one Hamiltonian flow of f •π is then g.
Proof. LetM 1 andM 2 be symplectic toric G bundles over W . Let M 1 = collapse(M 1 ) and M 2 = collapse(M 2 ). We need to prove that the functor collapse induces a bijection between Hom(M 1 ,M 2 ) and Hom(M 1 , M 2 ). We have already shown that the functor collapse induces a bijection on equivalence classes. Thus, it is enough to consider the case thatM 1 =M 2 =, say,M .
Let M = collapse(M ). We need to prove that the functor collapse induces a bijection from Aut(M ) to Aut(M ). This follows from Lemmas 3.6 and 6.3 and from the following fact.
Let f : W → g * be a smooth function. Letĝ :M →M be the time one map of f •π. Let g : M → M be the time one map of f • π. Then collapse(ĝ) = g.

Appendix A. Manifolds with corners
In this appendix we collect relevant facts about manifolds with corners. We also refer the reader to [DH].
A.1. Let V be a subset of R n . A map ϕ : V → R m is smooth if for every point p of V there exist an open subset Ω in R n containing p and a smooth map from Ω to R m whose restriction to Ω ∩ V coincides with ϕ| Ω∩V . A map ϕ from V to a subset of R m is smooth if it is smooth as a map to R m . A map ϕ from a subset of R n to a subset of R m is a diffeomorphism if it is a bijection and both it and its inverse are smooth.
We denote by R + the set of non-negative real numbers. A sector is the set R k + × R n−k where n is a non-negative integer and k is an integer between 0 and n. Let X be a Hausdorff second countable topological space. A chart on an open subset U of X is a homeomorphism ϕ from U to an open subset V of a sector. Charts ϕ : U → V and ϕ ′ : U ′ → V ′ are compatible if ϕ ′ • ϕ −1 is a diffeomorphism from ϕ(U ∩ U ′ ) to ϕ ′ (U ∩ U ′ ). An atlas on X is a set of pairwise compatible charts whose domains cover X. Two atlases are equivalent if their union is an atlas. A manifold with corners is a Hausdorff second countable topological space equipped with an equivalence class of atlases.
A.2. Let X be a manifold with corners. The dimension of X is n if the charts take values in sectors in R n . A point x of X has index k if there exists a chart ϕ from a neighborhood of x to R k + × R n−k such that ϕ(x) = 0; the index of a point is well defined. The k-boundary, X (k) , of X is the set of points of index ≥ k. The boundary of X is the 1-boundary, ∂X := X (1) . The interior of X is the complement of the boundary:X := X ∂X. The connected components of the sets X (k) X (k+1) form a stratification of X. If the dimension of X is n, we call the closures of strata of dimension less than n faces, and we call the closures of strata of dimension n − 1 facets.
A.3. Definition. A subset Y of a manifold with corners X is a smooth manifold if there exists a manifold structure on Y that has the following property: for every function f : Y → R, the function f is smooth on Y if and only if for every p ∈ Y there exists a neighborhood Ω of p in X and a smooth function Ω → R whose restriction to Ω ∩ Y coincides with f | Ω∩Y . If such a manifold structure exists then it is unique.
A.4. Remark. Definition A.3 is adapted to the particular application that we need in this paper. In this definition, the condition on Y does not imply that Y is a submanifold with corners of X in the sense of, say, [Mat]. It is more convenient here to use the language of differential structures (cf. [Si1,Si]): the condition is that the differential structure on Y that is induced from its inclusion into X makes Y into a smooth manifold.
A.5. Definition. A group action of a Lie group G on a manifold with corners X is a homomorphism ρ from G to the group of diffeomorphisms of X such that the map G × X → X given by (a, x) → ρ(a)(x) is smooth. Given such an action, we say that a smooth map π from X to another manifold with corners W is a quotient map if for every G-invariant smooth map f : X → Y there exists a unique smooth map f : W → Y such that f = f • π. We say that the geometric quotient X/G is a manifold, or a manifold with corners, if there exists a manifold, or a manifold with corners, structure on X/G such that the orbit map X → X/G is a quotient map; if such a structure exists then it is unique.
A.6. Let W be a manifold with corners and G a Lie group. A principal G bundle over W is a manifold with corners X, equipped with a G-action and a map π : X → W , such that for every w in W there exists a neighborhood V ⊂ W and a diffeomorphism of π −1 (V ) with G × V that carries the map π to the projection map G × V → V and that carries the G action on π −1 (V ) to action of G on G × V by left multiplication on the first factor.
A.7. The tangent space T x X of a manifold with corners X at a point x ∈ X is the space of derivations at x of germs at x of smooth functions defined near x. Thus, the tangent space is a vector space even if the point x is in the boundary of X. A differential form on X is a multilinear form on each tangent space that is carried through the charts to differential forms whose coefficients are smooth functions on open subsets of sectors. Differential forms pull back through smooth maps. A differential two-form is non-degenerate if it is non-degenerate on each tangent space. It is closed if it is carried through the charts to closed forms on open subsets of sectors. A symplectic form is a non-degenerate closed two-form. An action of a Lie group G gives rise to ("kinematic") vector fields ξ X for Lie algebra elements ξ ∈ g. A moment map for a G-action on a symplectic manifold with corners (X, ω) is a map µ : X → g * that satisfies d µ, ξ = −ι(ξ X )ω for all ξ ∈ g.
A.8. On a manifold with corners M , the de Rham cohomology is (well defined and) invariant under homotopy. Consequently, the Poincaré lemma holds: every closed form is locally exact. The proofs are exactly as for ordinary manifolds. A key step is that if β is a k-form on [0, 1] × M then i * 1 β − i * 0 β = π * dβ + dπ * β where i t : M → M × [0, 1] is i t (m) = (m, t) and where π * is fiber integration.