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(dµm) = (Tm(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 dµm : TmM → Tξt ∗ ' t ∗ maps the subspace Tm(T k · m) to 0. In other words, Tm(T k · m) ⊂ ker(dµm) = (Tm(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 6= e moves at least one point m ∈ M, i.e. \ m∈M Gm = {e}. (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 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, Mf = {m ∈ M | Gm = {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 k = dim(T k · m) ≤ 1 2 dim M. ¶ Symplectic Toric manifolds. 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, (t1, · · · , tn) · (z1, · · · , zn) = (t1z1, · · · , zntn), and thus is a symplectic toric manifold. Example. CPn admits an effective Hamiltonian T n action, (t1, · · · , tn) · [z0 : z1 : · · · : zn] = [z0 : t1z1 : · · · : tnzn] and thus is a symplectic toric manifold. The image of the moment map is the simplex in R n with n + 1 vertices 1 2 ei and (0, · · · , 0), where ei = (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, µY }(m) = ωm(XM(m), YM(m)) = 0.
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 CP1 , CP2 and CP1 × CP1 are S 2 ❅ ❅ ❅ ❅ ❅ ❅ CP2 CP1 × CP1 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 + tui , with ui ∈ Z n . (3) (smoothness) at each p, u1, · · · , un form a Z-basis of Z n . Obviously the previous examples are Delzant polytopes. More examples of Delzant polytopes ◗◗◗◗◗◗◗◗◗ (0,0) (4,0) (0,1) (1,1) ❅ ❅ ❅ ❅ ❅ ❅ (0,0,0) (1,0,0) (0,0,1) (0,1,0) The following polytopes are not Delzant: ❆ ❆ ❆ ❆ ❆ ❆ (0,0) (3,0) (0,2) (2,2) ❅ ❅ ❅ ❅ ❅ ❅ ❍❍❍❍❍❍❍❍ (0,0,0) (1,0,0) (0,0,1) (0,1,0) (1,1,0) Remark. Suppose a Delzant polytope has d faces. Let vi , 1 ≤ i ≤ d, be the primitive outward-pointing normal vectors to the faces of ∆, then ∆ can be described via a set of inequalities hx, vii ≤ λi , 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 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 AtiyahGuillemin-Sternberg convexity theorem that the moment polytope near p is {p + Xn i=1 siwi | si ≥ 0} where w1, · · · , wn are the weights of the linearized isotropic action of the torus on TmM. 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 wi ’s. Then W is not invertible as a Z-matrix. We take a vector τ 6∈ 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 hwi , τ i ∈ Z for all i. Recall that in a neighborhood of m, there exists coordinate system (z1, · · · , zn) so that the action of T n is given by exp(X) · (z1, · · · , zn) = (e 2πihw1,Xi z1, · · · , e2πihwn,Xi zn). 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∆.
LECTURE 11: SYMPLECTIC TORIC MANIFOLDS 5 Step 3: Check M∆ is toric and µ(M∆) = ∆. Step 4: ∆(M1) = ∆(M2) ⇐⇒ M1 ' M2. ¶ 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 v1, · · · , vd so that ∆ = {x ∈ (R n ) ∗ | hx, vii ≤ λi , i = 1, · · · , d}. By translation we may assume 0 ∈ ∆, and thus λi ≥ 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. I Step 2.a The (d − n)-torus N. Let e1, · · · , ed be the standard basis of R d . Define linear map π : R d → R n , ei 7→ vi 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. I Step 2.b The Hamiltonian N-action on C d . The standard T d -action on C d is given by (e iθ1 , · · · , eiθd ) · (z1, · · · , zd) = (e iθ1 z1, · · · , eiθd zd). The action is Hamiltonian with moment map φ : C d −→ (R d ) ∗ , φ(z1, · · · , zd) = − 1 2 (|z1| 2 , · · · , |zd| 2 ) + c. We choose c = λ = (λ1, · · · , λd). 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 ∗ . I Step 2.c The zero level set Z = (ι ∗ ◦ φ) −1 (0) is compact. Let ∆0 = π ∗ (∆). Then ∆ is compact. We claim
6 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Claim I: Im(π ∗ ) ∩ Im(φ) = ∆0 . According to this claim, Z = (i ∗ ◦ φ) −1 (0) = φ −1 (ker(i ∗ )) = φ −1 (Im(π ∗ )) = φ −1 (∆0 ). So Z is compact since the map φ proper. Proof of claim I:. Obviously ∆0 = π ∗ (∆) ⊂ Im(π ∗ ). By the definition φ, Im(φ) consists of those points y with hy, eii ≤ λi . For any x ∈ ∆, hπ ∗ (x), eii = hx, vii ≤ λi . So π ∗ (∆) ⊂ Im(φ), and thus ∆0 ⊂ Im(π ∗ ) ∩ Im(φ). Conversely suppose y = π ∗ (z) = φ(w). Then hz, vii = hπ ∗ (z), eii ≤ λi . In other words, z ∈ ∆. It follows Im(π ∗ ) ∩ Im(φ) ⊂ ∆0 . I Step 2.d N acts freely on Z. For z ∈ Z d , let Iz = {i | zi = 0}. Then (T d )z = {t ∈ T d | ti = 1 for i 6∈ Iz}. Claim II: The restriction map of π, π : (T d )z → T n , is injective. This implies Nz = N ∩ (T d )z = i(N) ∩ (T d )z = ker(π) ∩ (T d )z = ker(π|(Td)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) = ∆0 . So one can find x ∈ ∆ so that φ(z) = π ∗ (x). Thus i ∈ Iz ⇔ zi = 0 ⇔ hφ(z), eii = λi ⇔ hπ ∗ (x), eii = λi ⇔ hx, vii = λi . In other words, x is a point in the intersection of facets whose normal vectors are vi . As a consequence, we see that the set of vectors {vi | i ∈ Iz} are linearly independent. Now let [t], [s] ∈ (T d )z. If π([t]) = π([s]), then π(t) − π(s) = X i∈Iz (ti − si)vi ∈ Z n . It follows that ti − si ∈ Z for i ∈ Iz. So [t] = [s].
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 K¨ahler, with more works one can prove that M∆ is actually a K¨ahler 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: I 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. dim(T d )z = n. So by claim II, the map 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 . I 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. I Step 3.c The moment polytope of M∆ is ∆. µ(M∆) = µ ◦ pr(Z) = ej ∗ ◦ φ ◦ j(Z) = ej ∗ ◦ φ((i ∗ ◦ φ) −1 (0)) = ej ∗ (ker(i ∗ )) = ej ∗ (π ∗ (∆)) = (π ◦ ej) ∗ (∆) = ∆
8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS ¶ Equivariantly symplectomorphic toric manifolds. I Step 4.a If (M1, ω1, T n , µ1) and (M2, ω2, T n , µ2) are two equivariantly diffeomorphic toric manifolds, then µ1(M1) and µ2(M2) differ by a translation. In fact, if we let Φ : M1 → M2 be the equivariant diffeomorphism. Then Φ∗µ2 is also a moment map for the T n -action on M1, because dhΦ ∗µ2, Xi = Φ∗ dhµ2, Xi = Φ∗ ιXM2 ω2 = ιXM1 ω1. So there exists a constant ξ ∈ t ∗ so that Φ∗µ2 = µ1 + ξ. It follows that µ2(M2) = Φ∗µ2(M1) = µ1(M1) + ξ. Now suppose (M1, ω1, T n , µ1) and (M2, ω2, T n , µ2) are two symplectic toric manifolds with µ1(M1) = µ2(M2). We would like to prove that there exists an equivariant diffeomorphism that sends M1 to M2. I Step 4.b If µ1(M1) = µ2(M2), one can construct by induction a diffeomorphism that intertwines the torus actions and moment maps. I Step 4.c Show that the cohomology class of ω is determined by the moment polytope. I Step 4.d Apply Moser’s trick. For more details, c.f. Kai Cieliebak’s notes, P. 40-45. 3. Symplectic cut Student presentation