Reduced phase space and toric variety coordinatizations of Delzant spaces

Under consideration for publication in Math. Proc. Camb. Phil. Soc. 147 Reduced phase space and toric variety coordinatizations of Delzant spaces By JOHANNES J. DUISTERMAAT AND ALVARO PELAYO Mathematisch Instituut, Universiteit Utrecht P.O. Box 80 010, 3508 TA Utrecht, The Netherlands. e-mail: J.J.Duistermaat@uu.nl and University of California Berkeley, Mathematics Department, 970 Evans Hall # 3840 Berkeley, CA 94720-3840, USA. e-mail: apelayo@math.berkeley.edu (Received 6 May 2008; revised 11 August 2008) Abstract In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained. 1. Introduction Let (M, σ) be a smooth compact and connected symplectic manifold of dimension 2n and let T be a torus which acts effectively on (M, σ) by means of symplectomorphisms. If the action of T on (M, σ) is moreover Hamiltonian, then dim T n, and the image of the momentum mapping µ T : M t is a convex polytope in the dual space t of t, where t denotes the Lie algebra of T. In the maximal case when dim T = n, (M, σ) is called a Delzant space. Delzant [3, (*) on p. 323] proved that in this case the polytope is very special, a socalled Delzant polytope, of which we recall the definition in Section 2. Furthermore Delzant [3, Th. 2.1] proved that two Delzant spaces are T -equivariantly symplectomorphic if and only if their momentum mappings have the same image up to a translation by an element Research stimulated by KNAW professorship Research partially funded by an NSF postdoctoral fellowship

148 Johannes J. Duistermaat and Alvaro Pelayo of t. Thirdly Delzant [3, pp. 328, 329] proved that for every Delzant polytope there exists a Delzant space such that µ T (M) =. This Delzant space is obtained as the reduced phase space for a linear Hamiltonian action of a torus N on a symplectic vector space E, at a value λ N of the momentum mapping of the Hamiltonian N-action, where E, N and λ N are determined by the Delzant polytope. Finally Delzant [3, Sec. 5] observed that the Delzant polytope gives rise to a fan (= éventail in French), and that the Delzant space with Delzant polytope is T - equivariantly diffeomorphic to the toric variety M toric defined by the fan. Here M toric is a complex n-dimensional complex analytic manifold, and the action of the real torus T on M toric has an extension to a complex analytic action on M toric of the complexification T C of T. In our description in Section 5 of the toric variety M toric we do not use fans. The information, for each vertex v of, which codimension one faces of contain v, already suffices to define M toric. In this note we show that the construction of the Delzant space M as a reduced phase space leads, for every vertex v of the Delzant polytope, to a natural coordinatization ϕ v of a T -invariant open cell M v in M, where M v contains the unique fixed point m v in M of the T -action such that µ T (m v ) = v. We give an explicit construction of the inverse ψ v of ϕ v, which is a maximal diffeomorphism in the sense of Remark 3 9. The construction of ψ v originated in an attempt to extend the equivariant symplectic ball embeddings from (B 2n r, σ 0 ) (C n, σ 0 ) into the Delzant space (M, σ) in Pelayo [11] by maximal equivariant symplectomorphisms from open neighborhoods of the origin in C n into the Delzant space (M, σ). If v and w are two different vertices, then the coordinate transformation ϕ w ϕ 1 v is given by the explicit formulas (4.3), (4.4). This system of coordinates gives a new construction of the symplectic manifold with torus action from the Delzant polytope. After we wrote this paper V. Guillemin informed us that he had also considered the idea of this construction. Let Σ be the set of all strata of the orbit type stratification of M for the T -action. Then the domain of definition M v of ϕ v is equal to the union of all S Σ such that the fixed point m v belongs to the closure of S in M, see Corollary 5 6. The strata S Σ are also the orbits in the toric variety M toric M for the action of the complexification T C of the real torus T, and the domain of definition M v of ϕ v is equal to the domain of definition of a natural complex analytic T C -equivariant coordinatization Φ v of a T C -invariant open cell. The diffeomorphism Φ v ϕ 1 v, which sends the reduced phase space coordinates to the toric variety coordinates, maps U v := ϕ v (M v ) diffeomorphically onto a complex vector space, and is given by the explicit formulas (5.10). In the toric variety coordinates the complex structure is the standard one and the coordinate transformations are relatively simple Laurent monomial transformations, whereas the symplectic form is generally given by quite complicated algebraic functions. On the other hand, in the reduced phase space coordinates the symplectic form is the standard one, but the coordinate transformations, and also the complex structure, have a more complicated appearance. While these completely explicit coordinate formulas are the main novelty of the paper, we also use them to reprove many of the known results, leading to an efficient and hopefully attractive exposition of the subject. Let F denote the set of all d codimension one faces of and, for every vertex v of, let F v denote the set of all f F such that v f. Note that #(F v ) = n for every vertex v of. For any sets A and B, let A B denote the set of all A-valued functions on B. If A is a field and the set B is finite, then A B is a #(B)-dimensional vector space over

Coordinatizations of Delzant spaces 149 A. One of the technical points in this paper is the efficient organization of proofs and formulas made possible by viewing the Delzant space as a reduction of the vector space C F, and letting, for each vertex v, the coordinatizations ϕ v and Φ v take their values in C Fv. This leads to a natural projection ρ v : C F C Fv obtained by the restriction of functions on F to F v F. For each vertex v the complex vector space C Fv is isomorphic to C n, but the isomorphism depends on an enumeration of F v, the introduction of which would lead to an unnecessary complication of the combinatorics. Similarly our torus T is isomorphic to R n /Z n, but the isomorphism depends on the choice of a Z-basis of the integral lattice t Z in the Lie algebra t of T. As for each vertex v a different Z-basis of t Z appears, we also avoid such a choice, keeping T in its abstract form. We hope and trust that this will not lead to confusion with our main references Delzant [3], Audin [2] and Guillemin [8] about Delzant spaces, where C F, each C Fv, and T is denoted as C d, C n, and R n /Z n, respectively. The organization of this manuscript is as follows. In Section 2 we review the definition of the reduced phase Delzant space, and introduce the notations which will be convenient for our purposes. In Section 3 we define the reduced phase space coordinatizations. In Section 4 we give explicit formulas for the coordinate transformations and describe the reduced phase space Delzant space as obtained by gluing together bounded open subsets of n-dimensional complex vector spaces with these coordinate transformations as the gluing maps. In Section 5 we review the definition of the toric variety defined by the Delzant polytope, prove that the natural mapping from the reduced phase space to the toric variety is a diffeomorphism, and compare the coordinatizations of Section 3 with the natural coordinatizations of the toric variety. In Section 6 we discuss the de Rham cohomology classes of Kähler forms on the toric manifold, which actually are equal to the de Rham cohomology classes of the symplectic forms of the model Delzant spaces. In Section 7 we present these computations for the two simplest classes of examples, the complex projective spaces and the Hirzebruch surfaces. 2. The reduced phase space Let T be an n-dimensional torus, a compact, connected, commutative n-dimensional real Lie group, with Lie algebra t. It follows that the exponential mapping exp : t T is a surjective homomorphism from the additive Lie group t onto T. Furthermore, t Z := ker(exp) is a discrete subgroup of (t, +) such that the exponential mapping induces an isomorphism from t/t Z onto T, which we also denote by exp. Note that t Z is defined in terms of the group T rather than only the Lie algebra t, but the notation t Z has the advantage over the more precise notation T Z that it reminds us of the fact it is a subgroup of the additive group t. Because t/t Z is compact, t Z has a Z-basis which at the same time is an R-basis of t, and each Z-basis of t Z is an R-basis of t. Using coordinates with respect to an ordered Z-basis of t Z, we obtain a linear isomorphism from t onto R n which maps t Z onto Z n, and therefore induces an isomorphism from T onto R n /Z n. For this reason, t Z is called the integral lattice in t. However, because we do not have a preferred Z-basis of t Z, we do not write T = R n /Z n. Let be an n-dimensional convex polytope in t. We denote by F and V the set of all codimension one faces and vertices of, respectively. Note that, as a face is defined as the set of points of the closed convex set on which a given linear functional attains its minimum, see Rockafellar [12, p.162], every face of is compact. For every v V, we

150 Johannes J. Duistermaat and Alvaro Pelayo write F v = {f F v f}. is called a Delzant polytope if it has the following properties, see Guillemin [8, p. 8]. i) For each f F there is an X f t Z and λ f R such that the hyperplane which contains f is equal to the set of all ξ t such that X f, ξ + λ f = 0, and is contained in the set of all ξ t such that X f, ξ + λ f 0. The vector X f and constant λ f are made unique by requiring that they are not an integral multiple of another such vector and constant, respectively. ii) For every v V, the X f with f F v form a Z-basis of the integral lattice t Z in t. It follows that = {ξ t X f, ξ + λ f 0 for every f F }. (2.1) Also, #(F v ) = n for every v V, which already makes the polytope quite special. In the sequel we assume that is a given Delzant polytope in t. For any z C F and f F we write z(f) = z f, which we view as the coordinate of the vector z with the index f. Let π be the real linear map from R F to t defined by π(t) := f F t f X f, t R F. (2.2) Because, for any vertex v, the X f with f F v form a Z-basis of t Z which is also an R-basis of t, we have π(z F ) = t Z and π(r F ) = t. It follows that π induces a surjective homomorphism of Lie groups π from the torus R F /Z F = (R/Z) F onto t/t Z, and we have the corresponding surjective homomorphism exp π from R F /Z F onto T. Write n := ker π, a linear subspace of R F, and N = ker(exp π ), a compact commutative subgroup of the torus R F /Z F. Actually, N is connected, see Lemma 3 1 below, and therefore isomorphic to n/n Z, where n Z := n Z F is the integral lattice in n of the torus N. 1 On the complex vector space C F of all complex-valued functions on F we have the action of the torus R F /Z F, where t R F /Z F maps z C F to the element t z C F defined by (t z) f = e 2π i t f z f, f F. The infinitesimal action of Y R F = Lie(R F /Z F ) is given by (Y z) f = 2π i Y f z f, which is a Hamiltonian vector field defined by the function z Y, µ(z) = f F Y f z f 2 /2 = f F Y f (x f 2 + y f 2 )/2, (2.3) and with respect to the symplectic form σ := (i /4π) f F dz f dz f = (1/2π) f F dx f dy f, (2.4) if z f = x f + i y f, with x f, y f R. Here the factor 1/2π is introduced in order to avoid an integral lattice (2π Z) F instead of our Z F. 1 We did not find a proof of the connectedness of N in [3], [2], or [8].

Coordinatizations of Delzant spaces 151 Because the right hand side of (2.3) depends linearly on Y, we can view µ(z) as an element of (R F ) R F, with the coordinates µ(z) f = z f 2 /2 = (x f 2 + y f 2 )/2, f F. (2.5) In other words, the action of R F /Z F on C F is Hamiltonian, with respect to the symplectic form σ and with momentum mapping µ : C F (Lie(R F /Z F )) given by (2.3), or equivalently (2.5). It follows that the subtorus N of R F /Z F acts on C F in a Hamiltonian fashion, with momentum mapping µ N := ι n µ : C F n, (2.6) where ι n : n R F denotes the identity viewed as a linear mapping from n R F to R F, and its transposed ι n : (R F ) n is the map which assigns to each linear form on R F its restriction to n. Write λ N = ι n(λ), where λ denotes the element of (R F ) R F with the coordinates λ f, f F. It follows from Guillemin [8, Th. 1.6 and Th. 1.4] that λ N is a regular value of µ N, hence the level set Z := µ N 1 ({λ N }) of µ N for the level λ N is a smooth submanifold of C F, and that the action of N on Z is proper and free. As a consequence the N-orbit space M = M := Z/N is a smooth 2n-dimensional manifold such that the projection p : Z M exhibits Z as a principal N-bundle over M. Moreover, there is a unique symplectic form σ M on M such that p σ M = ι Z σ, where ι Z is the identity viewed as a smooth mapping from Z to C F. Remark 2 1 Guillemin [8] used the momentum mapping µ N λ N instead of µ N, such that the reduction is taken at the zero level of his momentum mapping. We follow Audin [2, Ch. VI, Sec. 3.1] in that we use the momentum mapping µ N for the N-action, which does not depend on λ, and do the reduction at the level λ N. The symplectic manifold (M, σ M ) is the Marsden-Weinstein reduction of the symplectic manifold (C F, σ) for the Hamiltonian N-action at the level λ N of the momentum mapping, as defined in Abraham and Marsden [1, Sec. 4.3]. On the N-orbit space M, we still have the action of the torus (R F /Z F )/N T, with momentum mapping µ T : M t determined by π µ T p = (µ λ) Z. (2.7) The torus T acts effectively on M and µ T (M) =, see Guillemin [8, Th. 1.7]. Actually, all these properties of the reduction will also follow in a simple way from our description in Section 3 of Z in term of the coordinates z f, f F. The symplectic manifold M together with this Hamiltonian T -action is called the Delzant space defined by, see Guillemin, [8, p. 13]. This proves the existence part [3, pp. 328, 329] of Delzant s theory. 3. The reduced phase space coordinatizations. For any v V, let ι v := ρ v : (R Fv ) (R F ) denote the transposed of the restriction projection ρ v : R F R Fv. If in the usual way we identify (R Fv ) and (R F ) with R Fv and R F, respectively, then ι v : R Fv R F is the embedding defined by ι v (x) f = x f if f F v and ι v (x) f = 0 if f F, f / F v. Because ι v maps Z Fv into Z F and

152 Johannes J. Duistermaat and Alvaro Pelayo ι v (R Fv ) Z F = ι v (Z Fv ), it induces an embedding of the n-dimensional torus R Fv /Z Fv into R F /Z F, which we also denote by ι v. Lemma 3 1. With these notations, R F, Z F, and R F /Z F are the direct sum of n and ι v (R Fv ), n Z n and ι v (Z Fv ), and N and ι v (R Fv /Z Fv ), respectively. It follows that N is connected, a torus, with integral lattice equal to n Z F. It also follows that π ι v is an isomorphism from the torus R Fv /Z Fv onto the torus T. Proof. Let t R F. Because the X f, f F v, form an R-basis of t, there exists a unique t v R Fv, such that π(t) = f F v (t v ) f X f = π(ι v (t v )), that is, t ι v (t v ) n. Moreover, because the X f, f F v, also form a Z-basis of t Z, we have that t v Z Fv, and therefore t ι v (t v ) n Z F, if t Z F. Lemma 3 2. We have z Z if and only if µ(z) λ π (t ). More explicitly, if and only if there exists a ξ t such that z f 2 /2 λ f = X f, ξ for every f F. (3.1) When z Z, the ξ in ( 3.1) is uniquely determined. Furthermore, Z = (µ λ) 1 (π ( )), (µ λ)(z) = π ( ), and Z is a compact subset of C F. Proof. The kernel of ι n is equal to the space of all linear forms on R F which vanish on n := ker π, and therefore ker ι n is equal to the image of π : t (R F ). Because π is surjective, π is injective, which proves the uniqueness of ξ. It follows from (3.1) that X f, ξ + λ f 0 for every f F, and therefore ξ in view of (2.1). Conversely, if ξ, then there exists for every f F a complex number z f such that z f 2 /2 = X f, ξ +λ f, which means that z Z and (µ λ)(z) = π (ξ). The set π ( ) is compact because is compact and π is continuous. Because the mapping µ λ is proper, it follows that Z = (µ λ) 1 (π ( )) is compact. Let v V. The X f, f F v, form an R-basis of t, and therefore there exists for each z C Fv a unique ξ = µ v (z) t such that (3.1) holds for every f F v. That is, the mapping µ v : C Fv t is defined by the equations z f 2 /2 λ f = X f, µ v (z), z C Fv, f F v. (3.2) In other words, µ v is defined by the formula ρ v π µ v = ρ v (µ λ) ι v, (3.3) where ρ v denotes the restriction projection from R F onto R Fv. Lemma 3 3. If we let T act on C Fv via R Fv /Z Fv by means of (t, z) (π ι v ) 1 (t) z, then µ v : C Fv t is a momentum mapping for this Hamiltonian action of T on C Fv, with µ v (0) = v. Here the symplectic form on C Fv is equal to σ := (i /4π) dz f dz f = (1/2π) dx f dy f, (3.4) f F v f F v that is, ( 2.4) with F replaced by F v.

Coordinatizations of Delzant spaces 153 Let ρ v denote the restriction projection from C F onto C Fv, and let U v be the interior of the subset ρ v (Z) of C Fv. Write v := \ f. (3.5) f F \F v Then ρ v (Z) = µ v 1 ( ), µ v (ρ v (Z)) =, U v = µ v 1 ( v ), and µ v (U v ) = v. In particular ρ v (Z) is a compact subset of C Fv, and U v is a bounded and connected open neighborhood of 0 in C Fv. Proof. The first statement follows from (3.3), the fact that ρ v µ ι v is a momentum mapping for the standard R Fv /Z Fv action on C Fv, and the fact that a momentum mapping for a Hamiltonian action plus a constant is a momentum mapping for the same Hamiltonian action. It follows in view of (3.2) that X f, µ v (0) +λ f = 0 for every f F v, hence µ v (0) = v in view of i) in the definition of a Delzant polytope, and the fact that {v} is the intersection of all the f F v. It follows from (3.2), Lemma 3 2, that z Z if and only if z f 2 /2 = X f, µ v (ρ v (z)) + λ f for every f F, (3.6) where we note that these equations are satisfied by definition for the f F v. Therefore, if z Z, then (3.6) and (2.1) imply that µ v (ρ(z)). Conversely, if ξ, then it follows from Lemma 3 2 that there exists z Z such that π (ξ) = µ(z) λ, of which the restriction to F v yields ξ = µ v (ρ(z)). If ξ v, z v C Fv, µ v (z v ) = ξ, then X f, µ v (z v ) + λ f > 0 for every f F \ F v, which will remain valid if we replace z v by z v in a sufficiently small neighborhood of z v in C Fv. It follows that we can find z C F such that ρ v ( z) = z v and (3.6) holds with z replaced by z. That is, z Z, and we have proved that z v U v. Let conversely z U v C Fv. We have in view of (3.2) that z f 2 /2 = X f, µ v (z) µ v (0) = X f, µ v (z) v for every f F v. Therefore µ v (z) v is multiplied by c 2 if we replace z by c z, c > 0. Because z is in the interior of ρ v (Z), we have c z v ρ v (Z), hence µ v (c z) for c > 1, c sufficiently close to 1. On the other hand, if ξ belongs to a face of which is not adjacent to v, then v + τ (ξ v) / for any τ > 1. It follows that µ v (z) does not belong to any f F \ F v, that is, µ v (z) v. The equation (3.6) can be written in the form z f = r f (µ v (ρ v (z))), where, for each f F, the function r f : R 0 is defined by r f (ξ) := (2( X f, ξ + λ f )) 1/2, f F, ξ. (3.7) We now view the equations (3.6) for z Z as equations for the coordinates z f, f F \ F v, with the z f, f F v as parameters, where the latter constitute the vector z v = ρ v (z). If z v U v, then for each f F \ F v the coordinate z f lies on the circle about the origin with strictly positive radius r f (µ v (z v ). Lemma 3 1 implies that the homomorphism which assigns to each element of N its projection to R F \Fv /Z F \Fv is an isomorphism, and the latter torus is the group of the coordinatewise rotations of the z f, f F \ F v. On the other hand, if we let Z v := ρ 1 v (U v ) Z, which is an open subset of Z C F, the differential of µ N in (2.6) evaluated at any z = (z f ) f F Z v is surjective; indeed, write z f = x f + i y f for every f F and suppose by contradiction that µ N is not

154 Johannes J. Duistermaat and Alvaro Pelayo surjective. Then there exists X n, X 0, such that f F X f (x f x f + y f y f ) = 0 for every z, z C F. Because z Z v, z f 0 for every f F \ F v. By taking f F \ F v, and z f = x f + i y f = 0 for every f f, as well as z f arbitrary, we conclude that X f = 0. Because f is arbitrary, X ι v (R Fv ). On the other hand by Lemma 3 1, n ι v (R Fv ) = {0}, so X = 0, a contradiction. These facts lead us to the following conclusions, where in order to make the presentation self-contained, we do not assume that Z is a smooth submanifold of C F of codimension equal to the dimension of N. Proposition 3 4. Let v be a vertex of. The open subset Z v := ρ 1 v (U v ) Z of Z is a connected smooth submanifold of C F of real dimension 2n+(d n), where d = #(F ) and d n = dim N. The action of the torus N on Z v is free, and the projection ρ v : Z v U v exhibits Z v as a principal N-bundle over U v. It follows that we have a reduced phase space M v := Z v /N, which is a connected smooth symplectic 2n-dimensional manifold, which carries an effective Hamiltonian T -action with momentum mapping as in ( 2.7), with Z replaced by Z v. There is a unique global section s v : U v Z v of ρ v : Z v U v such that s v (z) f R >0 for every z U v and f F \ F v. Actually, s v (z) f = r f (µ v (z)) when z U v and f F \ F v, and therefore the section s v is smooth. If p v : Z v M v = Z v /N denotes the canonical projection, then ψ v := p v s v is a T -equivariant symplectomorphism from U v onto M v, where T acts on U v via R Fv /Z Fv, as in Lemma 3 3. Remark 3 5 When z belongs to the closure ρ v (Z) = U v of U v in C Fv, see Lemma 3 3, we can define s v (z) C F by s v (z) f = z f when f F v and s v (z) f = r f (µ v (z)) when f F \ F v. This defines a continuous extension s v : U v C F of the mapping s v : U v Z. Therefore s v (U v ) Z, and ψ v := p s v : U v M is a continuous extension of the diffeomorphism ψ v : U v M v. The continuous mapping ψ v : U v M is surjective, but the restriction of it to the boundary U v := U v \ U v of U v in C Fv is not injective. If z v U v, then the set G of all f F \ F v such that s v (z v ) f = 0, or equivalently µ T (ψ v (z v )) f, is not empty. The fiber of ψ v over ψ v (z v ) is equal to the set of all t v z v, where the t v R Fv /Z Fv are of the form t v f = g G (v) f g t g, f F v, where t g R/Z. It follows that each fiber is an orbit of some subtorus of R Fv /Z Fv on C Fv. acting Recall the definition (3.5) of the open subset v of the Delzant polytope. Because the union over all vertices v of the v is equal to, we have the following corollary. Corollary 3 6. The sets Z v, v V, form a covering of Z. As a consequence, we recover the results mentioned in Section 2 that Z is a smooth submanifold of C F of real dimension n + d, the action of the torus N on Z is free, and we have a reduced phase space M := Z/N, which is a compact and connected smooth 2n-dimensional symplectic manifold, which carries an effective Hamiltonian T -action with momentum mapping µ T : M T as in ( 2.7). Since Z is the level set of µ N for the level λ N, it follows that λ N is a regular value of µ N.

Coordinatizations of Delzant spaces 155 Moreover, the sets M v, v V, form an open covering of M and the ϕ v := (ψ v ) 1 : M v U v form an atlas of T -equivariant symplectic coordinatizations of the Hamiltonian T -space M. For each v V, we have M v = µ T 1 ( v ), and µ T Mv = µ v ϕ v. For a characterization of M v in terms of the orbit type stratification in M for the T - action, see Corollary 5 6, which also implies that M v is an open cell in M. Corollary 3 7. For every f F the set µ T 1 (f) is a real codimension two smooth compact connected smooth symplectic submanifold of M. For each v V, the set M v is dense in M, and the diffeomorphism ψ v : U v M v is maximal among all diffeomorphisms from open subsets of C Fv onto open subsets of M. Proof. If f F, then for each v V we have that µ v 1 (f) = {z U v z f = 0} (3.8) if v f, that is, f v. This follows from (3.2) and i) in the description of in the beginning of Section 2. On the other hand, µ v 1 (f) = if f / v. Because µ T 1 (f) M v = ψ v (µ v 1 (f)), and the M v, v V, form an open covering of M, this proves the first statement. The second statement follows from the first one, because the complement of M v in M is equal to the union of the sets µ T 1 (f ) with f F \ F v. Remark 3 8 It follows from the proof of Corollary 3 7, that µ 1 T (f) is a connected component of the fixed point set in M of the of the circle subgroup exp(r X f ) of T. Actually, µ 1 T (f) is a Delzant space for the action of the (n 1)-dimensional torus T/ exp(r X f ), with Delzant polytope P (t/(r X f )) such that the image of P in t under the embedding (t/(r X f )) t is equal to a translate of f. In a similar way, if g is a k-dimensional face of, then µ T 1 (g) is a 2k-dimensional Delzant space for the quotient of T by the subtorus of T which acts trivially on µ T 1 (g). Remark 3 9 Let ι : T R n /Z n be an isomorphism of tori, which allows us to let t T act on C n via R n /Z n by means of (t z) j = e 2π i ι(t)j z j, 1 j n. Let U be a connected T -invariant open neighborhood of 0 in C n, provided with the symplectic form (2.4) with F replaced by {1,..., n}. Let ψ : U M be a T -equivariant symplectomorphism from U onto an open subset ψ(u) of M. Because 0 is the unique fixed point for the T -action in U, and the fixed points for the T -action in M are the preimages under µ T of the vertices of, there is a unique v V such that µ T (ψ(0)) = v. Let I v : C Fv C n denote the complex linear extension of the tangent map of the torus isomorphism ι (π ι v ). In terms of the notation of Lemma 3 3 and Proposition 3 4, we have that U I v (U v ) and ψ v = ψ I v on I 1 v (U), which leads to an identification of ψ with the restriction of ψ v to the connected open subset I 1 v (U) of U v, via the isomorphism I 1 v. The ψ s, with U equal to a ball in C n centered at the origin, are the equivariant symplectic ball embeddings in Pelayo [11], and the second statement in Corollary 3 7 shows

156 Johannes J. Duistermaat and Alvaro Pelayo that the diffeomorphisms ψ v are the maximal extensions of these equivariant symplectic ball embeddings. 4. The coordinate transformations Recall the description in Lemma 3 3, for every vertex v, of the open subset U v = µ 1 v ( v ) of C Fv. Let v, w V. Then U v, w := ϕ v (M v M w ) = U v ψ v 1 ψ w (U w ) = {z v U v (z v ) f 0 for every f F v \ F w }. (4.1) In this section we will give an explicit formula for the coordinate transformations ϕ w ϕ v 1 = ψ w 1 ψ v : U v, w U w, v, which then leads to a description of the Delzant space M as obtained by gluing together the subsets U v with the coordinate transformations as the gluing maps. Let f F. Because the X g, g F w, form a Z-basis of t Z, and X f t Z, there exist unique integers (w) g f, g F w, such that X f = g F w (w) g f X g. (4.2) Note that if f F w, then (w) g f = 1 when g = f and (w)g f = 0 otherwise. For the following lemma recall that r g is defined by expression (3.7). Lemma 4 1. Let v, w V, z v U v, w. Then z w := ϕ w ϕ 1 v (z v ) U w C Fw is given by zg w = (zf v ) (w) g f / zf v (w) g f (4.3) f F v if g F w F v, and if g F w \ F v. z w g f F v\f w = (zf v ) (w) g f rg (µ v (z v ))/ f F v f F v\f w zf v (w) g f (4.4) Proof. The element z w U w is determined by the condition that s w (z w ) belongs to the N-orbit of s v (z v ). That is, for some t R F such that s w (z w ) f = e i t f s v (z v ) f for every f F t f X f = 0. (4.5) f F It follows from (4.5), (4.2) and the linear independence of the X g, g F w, that t n if and only if t g = (w) g f t f for every g F w. (4.6) f F \F w Note that µ v (z v ) = µ T (m) = µ w (z w ), where m = ψ v (z v ) = ψ w (z w ). It follows from the definition of the sections s v and s w, see Proposition 3 4, that

Coordinatizations of Delzant spaces 157 i) s v (z v ) f = zf v and s w(z w ) f = zf w if f F v F w, ii) s v (z v ) f = zf v and s w(z w ) f = r f (µ w (z w )) = r f (µ v (z v )) if f F v \ F w, iii) s v (z v ) f = r f (µ v (z v )) and s w (z w ) f = zf w if f F w \ F v, and iv) s v (z v ) f = r f (µ v (z v )) = r f (µ w (z w )) = s w (z w ) f if f F \ (F v F w ). It follows from ii) and iv) that t f = arg zf v and t f = 0 modulo 2π if f F v \ F w and f F \ (F v F w ), respectively. Then (4.6) implies that, modulo 2π, t g = (w) g f arg zv f for every g F w. f F v\f w It now follows from i) and iii) that if g F w, then zg w = s w (z w ) g = e i tg s v (z v ) g is equal to e i tg zg v = (zf v ) (w) g f / zf v (w) g f f F v if g F v, and equal to e i tg zg v = (zf v ) (w) g f z v g / f F v f F v\f w f F v\f w zf v (w) g f if g / F v, respectively. Here we have used that if g F w, then (w) g f = 1 if f = g and (w) g f = 0 if f F w, f g. Because zg v = r g (µ v (z v )) if g / F v, see (3.6) and (3.7), this completes the proof of the lemma. Remark 4 2 Note that z v U v, w means that z v U v and zf v 0 if f F v \ F w. Furthermore, z v U v implies that if g / F v, then µ v (z v ) / g, and therefore r g is smooth on a neighborhood of µ v (z v ). Finally, note that if g F w and f F v F w, then (w) g f {0, 1}, and therefore each of the factors in the right hand sides of (4.3) and (4.4) is smooth on U v, w. Remark 4 3 In (4.3) and (4.4) only the integers (w) g f appear with f F v and g F w. Let (w v) denote the matrix (w) g f, where f F v and g F w. Then (w v) is invertible, with inverse equal to the integral matrix (v w). These integral matrices also satisfy the cocycle condition that (w v) (v u) = (w u), if u, v, w V. These properties follow from the fact that (4.2) shows that (w v) is the matrix which maps the Z-basis X g, g F w, onto the Z-basis X f, f F v, of t Z. It is no surprise that these base changes enter in the formulas which relate the models in the vector spaces C Fv for the different choices of v V. Corollary 4 4. Let, for each v V, the mapping µ v : C Fv t be defined by ( 3.2), which is a momentum mapping for a Hamiltonian T -action via R Fv /Z Fv on the symplectic vector space C Fv as in Lemma 3 3. Define U v := µ v 1 ( v ). If also w V, define U v, w as the right hand side of ( 4.1), and, if z v U v, w, define ϕ w, v (z v ) := z w, where z w C Fw is given by ( 4.3) and ( 4.4). Then ϕ w, v is a T -equivariant symplectomorphism from U v, w onto U w, v such that µ w = µ v ϕ v, w on U w, v. The ϕ w, v satisfy the cocycle condition ϕ w, v ϕ v, u = ϕ w, u where the left hand side is defined. Glueing together the Hamiltonian T -spaces U v, v V, with the momentum maps µ v, by means of the gluing maps ϕ w, v, v, w V, we obtain a compact

158 Johannes J. Duistermaat and Alvaro Pelayo connected smooth symplectic manifold M with an effective Hamiltonian T -action with a common momentum map µ : M T such that µ( M) =. In other words, M is a Delzant space for the Delzant polytope. The Delzant space M is obviously isomorphic to the Delzant space M = µ 1 ({λ})/n introduced in Section 2, and actually the isomorphism is used in the proof that M is a Delzant space for the Delzant polytope. The only purpose of Corollary 4 4 is to exhibit the Delzant space as obtained from gluing together the U v, v V, by means of the gluing maps ϕ v, w, v, w V. 5. The toric variety Let T := {z C z = 1} denote the unit circle in the complex plane. The mapping t u where u f = e 2π i t f for every f F is an isomorphism from the torus R F /Z F onto T F, where T F acts on C F by means of coordinatewise multiplication and R F /Z F acted on C F via the isomorphism from R F /Z F onto T F. The complexification T C of the compact Lie group T is the multiplicative group C of all nonzero complex numbers, and the complexification of T F is equal to T F C := (T C) F = (C ) F, which also acts on C F by means of coordinatewise multiplication. The complexification N C of N is the subgroup exp(n C ) of UC F, where n C := n i n C F denotes the complexification of n, viewed as a complex linear subspace, a complex Lie subalgebra, of the Lie algebra C F of T F C. In view of (4.6), we have, for every v V, that N C is equal to the set of all t T F C such that t g = f F \F v This implies that N C is a closed subgroup of T F C is a reductive complex algebraic group. If we define (v) t g f f, g Fv. (5.1) \Fv isomorphic to TF C, and therefore N C C F v := {z C F z f 0 for every f F \ F v }, (5.2) then it follows from (5.1) that the action of N C on C F v is free and proper. It follows that the action of N C on C F V = C F v (5.3) is free. Lemma 5 1. The action of N C on C F V v V is proper. Proof. As the referee observed, this does not follow immediately from the properness of the N C -action on each of the C F v s, and because we did not find a proof in the literature, we present one here. Also, G. Schwarz observed that in view of Luna s slice theorem it is sufficient to prove that the N C -orbits are closed subsets of C F V but we could not find a proof for the closedness of the orbits which is much simpler than the proof of the properness of the action. Finally, the statement of the lemma is implicitly contained in the statement in Audin [2, bottom of p. 155] that U Σ X Σ is a principal K C -bundle. For each subset E of F, let C(E) = f E R 0 X f denote the polyhedral cone in t spanned by the vectors X f, f E. Let v, w V, X C(F v ) C(F w ), and ξ. That

is, Coordinatizations of Delzant spaces 159 X = c f X f = d g X g, f F v g F w with c f, d g R 0, and X f, ξ + λ f 0 for every f F. Because X f, v + λ f = 0 for every f F v, it follows that X, ξ v 0, with equality if and only if c f = 0 or ξ f for every f F v. Similarly X, ξ w 0, with equality if and only if d g = 0 or ξ g for every g F w. For ξ = (1/2) (v + w) we have f F v F w if ξ f F v or ξ g F w, hence c f = 0 for every f F v \ F w and d g = 0 for every g F w \ F v. We therefore have proved that the collection of simplicial cones C(F v ), v V, has the fan property of Demazure [4, Déf. 1 in 4] that C(F v ) C(F w ) = C(F v F w ) for every v, w V. The argument below that the fan property implies the properness of the N C -action on C F V is inspired by the proof of Danilov [5, bottom of p. 133] that a toric variety defined by a fan is separated. What we have to prove is that if x C F V is close to x0 C F V, t N C, and y = t x C F V is close to y 0 C F V, then t remains in a compact subset of N C, that is, t f remains bounded and bounded away from 0 for every f F. It follows from (5.3) that we have x 0 C F v and y 0 C F w for some v, w V. Then (5.2) implies that for every f F \ (F v F w ) both x f and y f remain bounded and bounded away from zero, hence t f = y f /x f remains bounded and bounded away from zero. The fan property implies that there exists a linear form ξ on t such that X f, ξ > 0 for every f F v \F w, X f, ξ = 0 for every f F v F w, and X g, ξ < 0 for every g F w \F v. We can arrange that X f, ξ Z for every f F v, which implies that X f, ξ Z for every f F because the X f, f F v form a Z-basis of F. For each f F v \ F w it follows from (5.1) that t f = g F w\f v t (v) f g g h F \(F v F w) t (v)f h h, where the second factor remains bounded and bounded away from zero. Using (4.2) we therefore obtain f = g ϕ, (5.4) f F v\f w t Xf, ξ g F w\f v t Xg, ξ where the factor ϕ remains bounded and bounded away from zero. It follows from y 0 C F w that y f remains bounded away from zero for every f F v \ F w, and because x f remains bounded, t f = y f /x f remains bounded away from zero. On the other hand it follows from x 0 F v that, for each g F w \ F v, x g remains bounded away from zero, and because y g remains bounded, it follows that t g = y g /x g remains bounded. Because X g, ξ < 0 for every g F w \ F v, it follows that the right hand side in (5.4) remains bounded, and therefore the left hand side as well. Because X f, ξ > 0 for every f F v \ F w, and each t f, f F v \ F w, remains bounded away from zero, it follows that t f remains bounded for every f F v \ F w. This in turn implies that, for each f F v \ F w, x f = y f /t f remains bounded away from zero, because y f does so. Therefore x 0 C F w, and because also y 0 C F w, it follows that the t remain in a compact subset of N C because the N C -action on C F w is proper.

160 Johannes J. Duistermaat and Alvaro Pelayo Therefore the N C -orbit space M toric := C F V /N C (5.5) has the unique structure of a Hausdorff complex analytic manifold of complex dimension n such that the canonical projection from C F V onto M toric exhibits C F V as a principal N C- bundle over M toric. On M toric we still have the complex analytic action of the complex Lie group group T F C /N C, which is isomorphic to the complexification T C of our real torus T induced by the projection π. The complex analytic manifold M toric together with the complex analytic action of T C on it is the toric variety defined by the polytope in the title of this section. If v V and z C F v, then it follows from (5.1) that there is a unique t N C such that t f = z f for every f F \ F v, or in other words, z = t ζ, where ζ C F is such that ζ f = 1 for every f F \ F v. Let S v : C Fv C F v be defined by S v (z v ) f = zf v when f F and S v (z v ) = 1 when f F \ F v, as in Audin [2, p. 159]. If P v : C F v C F v /N C denotes the canonical projection from C F v onto the open subset Mv toric := C F v /N C of M toric, then Ψ v := P v S v is a complex analytic diffeomorphism from C Fv onto M toric v. It is T C - equivariant if we let T C act on C Fv via T Fv C as in Lemma 3 3. We use the diffeomorphism Φ v := Ψ 1 v from Mv toric onto C Fv as a coordinatization of the open subset Mv toric of M toric. If v, w V, then Uv, toric w := Φ v (Mv toric Mw toric ) = C Fv Ψ 1 v Ψ w (C Fw ) = {z v C Fv (z v ) f 0 for every f F v \ F w }. (5.6) Moreover, with a similar argument as for Lemma 4 1, actually much simpler, we have that for every z v Uv, toric w the element z w := Φ w Φ 1 v (z v ) C Fw is given by zg w = (zf v ) (w) g f, g Fw, (5.7) f F v where we define (z v f )0 = 1 when z v f = 0, which can happen when f F v F w. In this way the coordinate transformation Φ w Φ v 1 is a Laurent monomial mapping, much simpler than the coordinate transformation (4.3), (4.4). It follows that the toric variety M toric can be alternatively described as obtained by gluing the n-dimensional complex vector spaces C Fv, v V, together, with the maps (5.7) as the gluing maps. This is the kind of toric varieties as introduced by Demazure [4, Sec. 4]. For later use we mention the following observation of Danilov [5, Th. 9.1], which is also of interest in itself. Lemma 5 2. M toric is simply connected. Proof. Let w V. It follows from (5.6), for all v V, that the complement of Mw toric in M toric is equal to the union of finitely closed complex analytic submanifolds of complex codimension one, whereas Mw toric is contractible because it is diffeomorphic to the complex vector space C Fw. Because complex codimension one is real codimension two, any loop in M toric with base point in M toric w the complement of M toric w can be contracted within M toric w can be slightly deformed to such a loop which avoids in M toric, that is, which is contained in Mw toric, after which it to the base point in M toric w. Recall the definition in Section 2 of the reduced phase space M = Z/N.

Coordinatizations of Delzant spaces 161 Theorem 5 3. The identity mapping from Z into C F V, followed by the canonical projection P from C F V to M toric = C F V /N C, induces a T -equivariant diffeomorphism ϖ from M = Z/N onto M toric. It follows that each N C -orbit in C F V intersects Z in an N-orbit in Z. Proof. Because N is a closed Lie subgroup of N C, we have that the mapping P : Z C F V /N C induces a mapping ϖ : Z/N C F V /N C, which moreover is smooth. If v V, z Z v, then it follows from (5.1) that the t f, f F \ F v, of an element t N C can take arbitrary values, and therefore the z f, f F \ F v can be moved arbitrarily by means of infinitesimal N C -actions. Because Z is defined by prescribing the z f, f F \ F v, as a smooth function of the z f, f F v, and the Z v, v V, form an open covering of Z, this shows that at each point of Z the N C -orbit is transversal to Z, which implies that ϖ is a submersion. It follows that ϖ(m) is an open subset of M toric. Because M is compact and ϖ is continuous, ϖ(m) is compact, and therefore a closed subset of M toric. Because M toric is connected, the conclusion is that ϖ(m) = M toric, that is, ϖ is surjective. Because ϖ is a surjective submersion, dim R M = 2n = dim R M toric, and M is connected, we conclude that ϖ is a covering map. Because M toric is simply connected, see Lemma 5 2, we conclude that ϖ is injective, that is, ϖ is a diffeomorphism. Remark 5 4 Theorem 5 3 is the last statement in Delzant [3], with no further details of the proof. Audin [2, Prop. 3.1.1] gave a proof using gradient flows, whereas the injectivity has been proved in [8, Sec. A1.2] using the principle that the gradient of a strictly convex function defines an injective mapping. Note that in the definition of the toric variety M toric, the real numbers λ f, f F, did not enter, whereas these numbers certainly enter in the definition of M, the symplectic form on M, and the diffeomorphism ϖ : M M toric. Therefore the symplectic form σλ toric := (ϖ 1 ) (σ) on M toric will depend on the choice of λ R F. On the symplectic manifold (M toric, σλ toric ), the action of the maximal compact subgroup T of T C is Hamiltonian, with momentum mapping equal to µ toric λ := µ ϖ 1 : M toric t, (5.8) where µ toric λ (M toric ) =, where we note that in (2.1) depends on λ. In the following lemma we compare the reduced phase space coordinatizations with the toric variety coordinatizations. Lemma 5 5. Let v V. Then M toric v = ϖ(m v ), and θ v := Ψ v 1 ϖ ψ v (5.9) is a T Fv -equivariant diffeomorphism from U v onto C Fv. For each z v U v, the element ζ v := θ v (z v ) is given in terms of z v by ζf v = zf v r f (µ v (z v )) (v) f f, f F v, (5.10) f F \F v where the functions r f : R 0 are given by ( 3.7). We have µ v (z v ) = µ T (ψ v (z v )) = µ toric λ (Ψ v (ζ v )), (5.11)

162 Johannes J. Duistermaat and Alvaro Pelayo and z v = θ 1 v (ζ v ) is given in terms of ζ v by zf v = ζf v r f (ξ) (v) f f, f F v, (5.12) f F \F v coordtranstoric where ξ is the element of equal to the right hand side of ( 5.11). Proof. It follows from Lemma 3 3 and the paragraph preceding Proposition 3 4 that if z v ρ v (Z), then z v U v if and only if zf v 0 for every f F \ F v. That is, the set Z v in Proposition 3 4 is equal to Z C F v. It therefore follows from Theorem 5 3 that each N C -orbit in the N C -invariant subset C F v of C F V intersects the N-invariant subset Z v of Z in an N-orbit in Z v, that is, M toric v = P v (C F v ) = ϖ(p v (Z v )) = ϖ(m v ). If z v U v, then Proposition 3 4 implies that s v (z v ) f = z v f for every f F v and s v (z v ) f = r f (µ v (z v )), f F \ F v. If we define t T F C by tf = rf (µv(z v )) 1, f F \ Fv, t f = f F \F v r f (µ v (z v )) (v) f f, f F v, then (t s v (z v )) t = 1 for every t F \ F v and, for every f F v, ζf v := (t s v) f is equal to the right hand side of (5.10). That is, t s v (z v ) = S v (ζ v ), see the definition of S v in the paragraph preceding (5.6). On the other hand, it follows from (5.1) that t N C, and therefore Ψ v (ζ v ) = P v (t s v (z v )) = P v (s v (z v )) = ϖ p v (s v (z v )) = ϖ ψ v (z v ), that is, ζ v = Ψ v 1 ϖ ψ v (z v ). Corollary 5 6. Let s be the relative interior of a face of. Then µ 1 T (s) is equal to a stratum S of the orbit type stratification in M of the T -action, and also equal to the preimage under ϖ : M M toric of a T C -orbit in M toric. If s = {v} for a vertex v, then µ 1 T (s) = {m v } for the unique fixed point m v in M for the T -action such that µ T (m v ) = v. The mapping s µ 1 T (s) is a bijection from the set Σ of all relative interiors of faces of onto the set Σ of all strata of the orbit type stratificiation in M for the action of T. If s, s Σ then s is contained in the closure of s in if and only if µ 1 T (s) is contained in the closure of µ 1 T (s ) in M. The domain of definition M v of ϕ v in M is equal to the union of the S Σ such that m v belongs to the closure of S in M. The domain of definition Mv toric = ϖ(m v ) of Φ v is equal to the union of the corresponding strata of the T -action in M toric, each of which is a single T C -orbit in M toric. M v and Mv toric are open cells in M and M toric, respectively. Proof. There exists a vertex v of such that v belongs to the closure of s in t, which implies that s is disjoint from all f F \ F v. Let F v, s denote the set of all f F v such that s f, where F v, s = if and only if s is the interior of. For any subset G of F v, let C Fv G denote the set of all z such that z CFv f = 0 if f G and z f 0 if f F v \ G. It follows from µ v = µ T ψ v and (3.8) that ψ 1 v (µ T 1 (s)) is equal to U v C Fv G with

Coordinatizations of Delzant spaces 163 G = F v, s. The diffeomorphism θ v maps this set onto the set C Fv G with G = F v, s. Because the sets of the form C Fv G with G F v are the strata of the orbit type stratification of the T Fv -action on C Fv, and also equal to the (T C ) Fv -orbits in C Fv, the first statement of the corollary follows. The second statement follows from µ 1 v ({v}) = {0} and the fact that 0 is the unique fixed point of the T Fv -action in U v. If s Σ and v V, then M v belongs to the closure of µ 1 T (s) if and only if s is not contained in any f F \F v. This proves the characterization of the domain of definition M v := Z v /N = µ 1 T ( v ) of ϕ v. The last statement follows from the fact that Φ v is a diffeomorphism from Mv toric onto the vector space C Fv, and ϖ is a diffeomorphism from M v onto Mv toric. The stratification of M toric by T C -orbits is one of the main tools in the survey of Danilov [5] on the geometry of toric varieties. Remark 5 7 If v, w V, then ϕ w ϕ v 1 = ψ w 1 ψ v = θ w 1 (Ψ w 1 Ψ v ) θ v = θ w 1 (Φ w Φ v 1 ) θ v. Using the formula (5.7) for Φ w Φ v 1, this can be used in order to obtain the formulas (4.3), (4.4) as a consequence of (5.10). In the proof, it is used that ξ := µ v (z v ) = µ w (z w ), z v f = r f (ξ) if f F v \ F w, and if f F \ F v and g F w. f F v (v) f f (w) g f = (w)g f In the following corollary we describe the symplectic form σλ toric M toric in the toric variety coordinates. on the toric variety Corollary 5 8. For each v V, the symplectic form (Φ v 1 ) (σ toric λ ) on C Fv is equal to (θ v 1 ) (σ v ), where σ v is the standard symplectic form on C Fv given by ( 3.4). Because r f (µ v (z v )) 2 is an inhomogeneous linear function of the quantities zf v 2, it follows from (5.10) that the equations which determine the zf v 2 in terms of the quantities ζf v 2 are n polyomial equations for the n unknowns zv f 2, f F v, where the coefficients of the polyomials are inhomogeneous linear functions of the ζf v, f F v. In this sense the zf v 2, f F v, are algebraic functions of the ζf v 2, f F v, and substituting these in (5.10) we obtain that the diffeomorphism θ 1 v from C Fv onto U v is an algebraic mapping. If is a simplex, when M toric is the n-dimensional complex projective space, we have an explicit formula for θ 1 v, see Subsection 7 1. However, already in the case that is a planar quadrangle, when M toric is a complex two-dimensional Hirzebruch surface, we do not have an explicit formula for θ 1 v. See Subsection 7 2. Summarizing, we can say that in the toric variety coordinates the complex structure is the standard one and the coordinate transformations are the relatively simple Laurent monomial transformations (5.7). However, in the toric variety coordinates the λ-dependent symplectic form in general is given by quite complicated algebraic functions. On the other hand, in the reduced phase space coordinates the symplectic form is the standard one, but the coordinate transformations (4.3), (4.4) are more complicated. Also

164 Johannes J. Duistermaat and Alvaro Pelayo the complex structure in the reduced phase space coordinates, which depends on λ, is given by more complicated formulas. Remark 5 9 It is a challenge to compare the formula in Corollary 5 8 for the symplectic form in toric variety coordinates with Guillemin s formula in [8, Th. 3.5 on p. 141] and [9, (1.3)]. Note that in the latter the pullback by means of the momentum mapping appears of a function on the interior of, where in general we do not have a really explicit formula for the momentum mapping in toric variety coordinates. 6. Cohomology classes of Kähler forms on toric varieties For the construction of the toric variety by gluing the C Fv, v V together by means of the gluing maps (5.7), one only needs the integral F w F v -matrices (w v) g f := (w)g f Z, g F w, f F v as the data. The Laurent monomial coordinate transformations Uv, toric w Uw, toric v : z v z w are diffeomorphisms if and only if the integral matrices (w v) are invertible, and the gluing defines an equivalence relation if and only if the matrices satisfy the cocycle condition (u w) = (u v) (v w) for all u, v, w V. If this holds, then the same gluing procedure allows to glue the R Fv, Z Fv, T F v, and (C ) Fv together to an n-dimensional vector space t, an integral lattice t Z in t, an n-dimensional torus T, and the complexification T C of T, respectively, where T is the unique maximal compact subgroup of T C. Here the exponentation t e 2π i t on each coordinate defines the isomorphism t/t Z T. For each v V and f Fv the standard Z-basis vector e f Z Fv is mapped to an element X f t, where the X f, f F v, form a Z-basis of t Z. The manifold M toric obtained by means of the gluing process is Hausdorff = separated in the algebraic geometric terminology, if and only if the vectors X f t, f F, have the fan property. The toric variety M toric is compact if and only if the fan is complete, which means that t is equal to the union of all the cones C(F v ), v V. In the above construction, F and V are just abstract finite sets, and in particular do not yet have the interpretation of being the set of faces and vertices, respectively, of a Delzant polytope in t. We now will construct, for each element λ R F satisfying suitable conditions, a Delzant polytope = λ, such that F and V can be identified as the set of faces and vertices of λ, respectively. For a given λ R F, there is a unique solution ξ = v λ t of the linear equations X f, ξ + λ f = 0 for all f F v. On the other hand we have the subset = λ of t defined by (2.1). The condition of having a complete fan is equivalent to the existence of a choice of λ f s such that λ is a convex polytope in t with the v λ, v V as its vertices. This means that X f, v λ + λ f > 0 for each f F \ F v. These λ s form a convex open cone Λ in R F. Identifying v λ t with v, we obtain for each λ Λ the symplectic form σλ toric := (ϖ 1 λ ) σ on M toric, where ϖ λ = ϖ is the diffeomorphism from the Delzant space M = M λ = Z λ /N onto M toric. Here Z = Z λ denotes the level set of the momentum mapping µ N at the level ι n(λ). Let [σλ toric ] Hde 2 Rham(M toric ) denote the de Rham cohomology class of σλ toric. For each f F the linear form ɛ f : C F C : z z f induces a surjective homomorphism of tori N T, hence an isomorphism N/ ker(ɛ f N ) T, and therefore N/ ker(ɛ f N ) is a circle group. It follows from Duistermaat and Heckman [6, (2.10)] that, for each f F, [σ toric λ ] λ f = ι de Rham (c f ). (6.1)