LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

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1 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9 we showed ker(dµ m ) = (T m (G m)) ωm. Proposition 1.1. Let (M, ω, T k, µ) be a compact connected Hamiltonian T k -space, then for any m M, then orbit T k m is an isotropic submanifold of M. Proof. The moment map µ is T k -invariant, so on the orbit T k m, µ takes a constant value ξ t. It follows that the differential dµ m : T m M T ξ t t maps the subspace T m (T k m) to 0. In other words, T m (T k m) ker(dµ m ) = (T m (T k m)) ωm. So T k m is an isotropic submanifold of M. Effective torus actions. Definition 1.2. An action of a Lie group G on a smooth manifold M is called effective (or faithful) if each group element g e moves at least one point m M, i.e. G m = {e}. m M (Equivalently, if the group homomorphism τ : G Diff(M) is injective.) Remark. If a group action τ of G on M is not effective, then ker(τ) is a normal subgroup of G, and the action τ induces a smooth action of G/ker(τ) on M which is effective. A remarkable fact on effective T k -action is 1

2 2 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Theorem 1.3. Suppose T k acts on M effectively. Then the set of points where the action is free, M = {m M G m = {e}}, is an open and dense subset in M. For a proof, c.f. Guillemin-Ginzburg-Karshon, Moment Maps, Cobordisms, and Hamiltonian Group Actions, appendix B, corollary B.48. An important consequence is Corollary 1.4. Let (M, ω, T k, µ) be a compact connected Hamiltonian T k -space, If the T k -action is effective, then dim M 2k. Proof. Pick any point m in M where the T k -action is free, i.e. (T k ) m = {e}. Then the orbit T k m is diffeomorphic to T k /(T k ) m = T k, and thus has dimension k. But we have just seen that T k m is an isotropic submanifold of M. So Symplectic Toric manifolds. k = dim(t k m) 1 dim M. 2 Definition 1.5. A compact connected symplectic manifold (M, ω) of dimension 2n is called a symplectic toric manifold is it is equipped with an effective Hamiltonian T n -action. Example. C n admits an effective Hamiltonian T n action, and thus is a symplectic toric manifold. (t 1,, t n ) (z 1,, z n ) = (t 1 z 1,, z n t n ), Example. CP n admits an effective Hamiltonian T n action, (t 1,, t n ) [z 0 : z 1 : : z n ] = [z 0 : t 1 z 1 : : t n z n ] and thus is a symplectic toric manifold. The image of the moment map is the simplex in R n with n + 1 vertices 1e 2 i and (0,, 0), where e i = (0,, 1,, 0). Example. The products of toric manifolds is still toric. Remark. A symplectic toric manifold is a special complete integrable system because for any X, Y t, {µ X, µ Y }(m) = ω m (X M (m), Y M (m)) = 0.

3 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS 3 Delzant polytopes. According to the Atiyah-Guillemin-Sternberg convexity theorem, the image of the moment map is always a convex polytope in R n. The moment polytope of CP 1, CP 2 and CP 1 CP 1 are S 2 CP 2 CP 1 CP 1 Definition 1.6. A polytope R n is called a Delzant polytope if (1) (simplicity) there are n edges meeting at every vertex p. (2) (rationality) the edges meeting at p are of the form p + tu i, with u i Z n. (3) (smoothness) at each p, u 1,, u n form a Z-basis of Z n. Obviously the previous examples are Delzant polytopes. Delzant polytopes More examples of (0,1) (1,1) (0,0) (4,0) (0,0,1) (0,1,0) (0,0,0) (1,0,0) The following polytopes are not Delzant: (0,2) (2,2) (0,0) (3,0) (0,0,1) (0,1,0) (1,1,0) (0,0,0) (1,0,0) Remark. Suppose a Delzant polytope has d faces. Let v i, 1 i d, be the primitive outward-pointing normal vectors to the faces of, then can be described via a set of inequalities x, v i λ i, i = 1,, d Moment polytopes are Delzant. Now we are ready to prove Theorem 1.7. For any symplectic toric manifold (M, ω), its moment polytope is a Delzant polytope.

4 4 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Proof. Let m M be a fixed point of the Hamiltonian torus action, then p = µ(m) is a vertex of the moment polytope. We have seen from the proof of the Atiyah- Guillemin-Sternberg convexity theorem that the moment polytope near p is n {p + s i w i s i 0} i=1 where w 1,, w n are the weights of the linearized isotropic action of the torus on T m M. Thus satisfies the conditions (1) and (2). Suppose does not satisfy the condition (3). Let W be the Z-matrix whose row vectors are the vectors w i s. Then W is not invertible as a Z-matrix. We take a vector τ Z n such that W τ Z n. (If W is not invertible, we can take τ be any non-integer vector in the kernel of W. If W is invertible as an R-matrix but not invertible as a Z-matrix, then W 1 can not map all Z-vectors to Z vectors ). So we have w i, τ Z for all i. Recall that in a neighborhood of m, there exists coordinate system (z 1,, z n ) so that the action of T n is given by exp(x) (z 1,, z n ) = (e 2πi w 1,X z 1,, e 2πi wn,x z n ). So exp(τ) acts trivially on a neighborhood of m, but exp(τ) is not the identity element in T n. This contradicts with the fact that in a dense open subset of M the action is free. So satisfies (3). 2. Delzant s theorem Statement of main theorem. The main result is the following classification for symplectic toric manifold, which says that symplectic toric manifolds are characterized by their moment polytopes: Theorem 2.1 (Delzant, 1990). There is a one-to-one correspondence between symplectic toric manifolds (up to T n equivariant symplectomorphisms) and Delzant polytopes. More precisely, (1) The moment polytope of a toric manifold is a Delzant polytope. (2) Every Delzant polytope is the moment polytope of a symplectic toric manifold. (3) Two toric manifolds with the same moment polytope are equivariantly symplectomorphic. The proof is divided into several steps: Step 1: M toric = µ(m) Delzant. (Done as theorem 1.7.) Step 2: Delzant construct compact connected symplectic manifold M.

5 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS 5 Step 3: Check M is toric and µ(m ) =. Step 4: (M 1 ) = (M 2 ) M 1 M 2. Construction of M from Delzant polytope. Now let be a Delzant polytope in R n. Suppose has d facets, then by the algebraic description one can find primitive outward pointing vectors v 1,, v d so that = {x (R n ) x, v i λ i, i = 1,, d}. By translation we may assume 0, and thus λ i 0 for all i. We shall construct M as the symplectic quotient of R d by a Hamiltonian action of a torus N of dimension d n. Step 2.a The (d n)-torus N. Let e 1,, e d be the standard basis of R d. Define linear map π : R d R n, e i v i Then since is Delzant, π is onto and maps Z d onto Z n. So we get an induced surjective Lie group homomorphism π : T d = R d /Z d T n = R n /Z n. Let N = ker(π). It is a (d n)-subtorus of T d. Note that from the exact sequence of Lie group homomorphisms 0 N i T d π T n 0 one gets an exact sequence of Lie algebras 0 n i R d π R n 0, and thus an exact sequence of dual Lie algebras 0 (R n ) π (R d ) i n 0. Step 2.b The Hamiltonian N-action on C d. The standard T d -action on C d is given by (e iθ 1,, e iθ d ) (z 1,, z d ) = (e iθ 1 z 1,, e iθ d z d ). The action is Hamiltonian with moment map φ : C d (R d ), φ(z 1,, z d ) = 1 2 ( z 1 2,, z d 2 ) + c. We choose c = λ = (λ 1,, λ d ). Since N is a sub-torus of T d, the induced N-action on C d is Hamiltonian with moment map ι φ : C d n. Step 2.c The zero level set Z = (ι φ) 1 (0) is compact. Let = π ( ). Then is compact. We claim

6 6 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Claim I: Im(π ) Im(φ) =. According to this claim, Z = (i φ) 1 (0) = φ 1 (ker(i )) = φ 1 (Im(π )) = φ 1 ( ). So Z is compact since the map φ proper. Proof of claim I:. Obviously = π ( ) Im(π ). By the definition φ, Im(φ) consists of those points y with y, e i λ i. For any x, π (x), e i = x, v i λ i. So π ( ) Im(φ), and thus Im(π ) Im(φ). Conversely suppose y = π (z) = φ(w). Then z, v i = π (z), e i λ i. In other words, z. It follows Im(π ) Im(φ). Step 2.d N acts freely on Z. For z Z d, let I z = {i z i = 0}. Then (T d ) z = {t T d t i = 1 for i I z }. Claim II: The restriction map of π, π : (T d ) z T n, is injective. This implies N z = N (T d ) z = i(n) (T d ) z = ker(π) (T d ) z = ker(π (T d ) z ) = {1}. So the action is free. Proof of claim II. Suppose z Z = (ι φ) 1 (0), i.e. φ(z) ker(ι ) = Im(π ). Then φ(z) Im(π ) Im(φ) =. So one can find x so that φ(z) = π (x). Thus i I z z i = 0 φ(z), e i = λ i π (x), e i = λ i x, v i = λ i. In other words, x is a point in the intersection of facets whose normal vectors are v i. As a consequence, we see that the set of vectors {v i i I z } are linearly independent. Now let [t], [s] (T d ) z. If π([t]) = π([s]), then π(t) π(s) = i I z (t i s i )v i Z n. It follows that t i s i Z for i I z. So [t] = [s].

7 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS 7 In conclusion, we see that M = C d /N = Z/N is a compact symplectic manifold of dimension 2d 2(d n) = 2n.. Remark. Since M is constructed via C d which is Kähler, with more works one can prove that M is actually a Kähler manifold. The moment polytope of M is. We need to show that M admits a Hamiltonian T n action which is effective. The action is actually very natural: Step 3.a Hamiltonian T n -action on M. Suppose z is a point such that φ(z) = π (x) for a vertex x of. Then from the proof of claim II we see that dim(t d ) z equals the number of facets of that meets at p, i.e. So by claim II, the map dim(t d ) z = n. π : (T d ) z T n is bijective. By identifying T n with (T d ) z we get an embedding j : T n T d with π j = Id. So T n acts on C d in a Hamiltonian way, with moment map j φ. Moreover, this T n - action commutes with the N-action we constructed above. Thus by the reduction by stages arguments (presented by Chao en last time), we get an induced Hamiltonian T n -action on M, whose moment map µ satisfies µ pr = j φ j, where pr is the projection from Z to M, and j is the inclusion from Z to C d. Step 3.b The above T n -action is effective. Since the T d action is effective, this induced T n -action is also effective. Step 3.c The moment polytope of M is. µ(m ) = µ pr(z) = j φ j(z) = j φ((i φ) 1 (0)) = j (ker(i )) = j (π ( )) = (π j) ( ) =.

8 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Equivariantly symplectomorphic toric manifolds. Step 4.a If (M 1, ω 1, T n, µ 1 ) and (M 2, ω 2, T n, µ 2 ) are two equivariantly diffeomorphic toric manifolds, then µ 1 (M 1 ) and µ 2 (M 2 ) differ by a translation. In fact, if we let Φ : M 1 M 2 be the equivariant diffeomorphism. Then Φ µ 2 is also a moment map for the T n -action on M 1, because d Φ µ 2, X = Φ d µ 2, X = Φ ι XM2 ω 2 = ι XM1 ω 1. So there exists a constant ξ t so that Φ µ 2 = µ 1 + ξ. It follows that µ 2 (M 2 ) = Φ µ 2 (M 1 ) = µ 1 (M 1 ) + ξ. Now suppose (M 1, ω 1, T n, µ 1 ) and (M 2, ω 2, T n, µ 2 ) are two symplectic toric manifolds with µ 1 (M 1 ) = µ 2 (M 2 ). We would like to prove that there exists an equivariant diffeomorphism that sends M 1 to M 2. Step 4.b If µ 1 (M 1 ) = µ 2 (M 2 ), one can construct by induction a diffeomorphism that intertwines the torus actions and moment maps. Step 4.c Show that the cohomology class of ω is determined by the moment polytope. Step 4.d Apply Moser s trick. For more details, c.f. Kai Cieliebak s notes, P Student presentation 3. Symplectic cut