LECTURE11:SYMPLECTICTORICMANIFOLDSCONTENTS1Symplectictoricmanifolds1.42.Delzant's theorem83.Symplecticcut1.SYMPLECTIC TORICMANIFOLDSTOrbit of torus actions.Recall that in lecture 9 we showed ker(dμm) = (Tm(G .m))mProposition 1.1.Let (M,w, Th,μ) be a compact connected Hamiltonian T-space,then for any m E M, then orbit Tk.m is an isotropic submanifold of M.Proof. The moment map μ is Tk-invariant, so on the orbit Tk .m, μ takes a constantvalue et*.It follows that thedifferential dμm:TmM-→Tt* t*maps thesubspaceTm(T.m)to0.Inotherwords,Tm(Tk. m) C ker(dμm) = (Tm(Tk . m)"mSoTk.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 calledeffective (or faithful)if each group element g+e moves at least one point m EM,i.e.N Gm = [e].mEM(Equivalently, if the group homomorphism T:G→Diff(M) is injective.)Remark. If a group action T of G on M is not effective, then ker(+) is a normalsubgroup of G, and the action induces a smooth action of G/ker() on M whichis effective.A remarkable fact on effective Tk-action is1
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
2LECTURE11:SYMPLECTICTORICMANIFOLDSTheorem 1.3. Suppose Tk acts on M effectively.Then the set of points where theaction is free,M=(mEMIGm=[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 isCorollary 1.4. Let (M,w, Tk, μ) be a compact connected Hamiltonian Tk-space, IftheTk-action is effective, then dimM ≥2k.Proof. Pick any point m in M where the Tk-action is free, i.e. (T*)m = [e]. Thenthe orbit Tk.m is diffeomorphic to T*/(T)m = Tk, and thus has dimension k. Butwe have just seen that Tk.m is an isotropic submanifold of M. So1k = dim(Tk . m) ≤dimM.口 Symplectic Toric manifolds.Definition 1.5. A compact connected symplectic manifold (M,w) of dimension 2nis called a symplectic toric manifold is it is equipped with an effective HamiltonianT"-action.Erample. Cn admits an effective Hamiltonian Tn action,(t1,**,tn).(z1,..,zn)= (tiz1,*.,zntn),and thus is a symplectictoricmanifoldErample. Cpn admits an effective Hamiltonian Tn action(ti,*.,tn) [2z0 : z1 :.*.: zn] = [20 : tiz1 :*-.: tnzn]and thus is a symplectic toric manifold. The image of the moment map is thesimplex in Rn with n +1 vertices e and (0,...,0), where e,=(0,...,1,...,0).Erample.Theproducts of toricmanifolds isstilltoricRemark.A symplectic toric manifold is a special complete integrable system becausefor any X,Y et,(μ,μ(m) =Wm(Xm(m), Ym(m)) = 0
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.
3LECTURE11:SYMPLECTICTORICMANIFOLDS Delzant polytopes.According to the Atiyah-Guillemin-Sternberg convexity theorem, the image ofthe moment map is always a convex polytope in IRn. The moment polytope of CplCp2 and Cpl × Cpl areS2CP2CPl×CPDefinition 1.6. A polytope E Rn is called a Delzant polytope if(simplicity) there are n edges meeting at every vertex p.112(rationality)the edges meeting at p are of the form p+tui, with u, e Zn(3) (smoothness) at each p, ui, ... , un form a Z-basis of Zn.Obviously the previous examples areDelzant polytopes.More examples ofDelzantpolytopes(0,1)(1,1)(0,0,1)(0,X0)(0,0)(4,0)(0,0,0)(1,0,0)The following polytopes are not Delzant:(0,2)(2.2)(0,0,1)0X0)(1,1.0)(0,0)(3,0)(0,0,0)(1,0,0)Remark. Suppose a Delzant polytope has d faces. Let vi, 1 ≤i< d, be the primitiveoutward-pointing normal vectors to the faces of , then can be described via aset of inequalities(c,vi)≤Ai，i=l,..,d Moment polytopes are Delzant.NowwearereadytoproveTheorem 1.7. For any symplectic toric manifold (M,w), its moment polytope isa Delzant polytope
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
4LECTURE11:SYMPLECTICTORICMANIFOLDSProof. Let m e M be a fixed point of the Hamiltonian torus action, then p =μ(m)is a vertex of themoment polytope.Wehaveseen fromtheproof of theAtiyah.Guillemin-Sternberg convexity theorem that themoment polytopenear p is(p+ siwi / si≥ 0]i=1wherewi,...,wn are the weights of the linearized isotropic action of the torus onTmM. Thus satisfies the conditions (1) and (2).Suppose does not satisfy the condition (3). Let W be the Z-matrix whoserow vectors are the vectors w,'s. Then W is not invertible as a Z-matrix. We takea vector T g Zn such that W e Zn. (If W is not invertible, we can take - be anynon-integer vector in the kernel of W. If W is invertible as an R-matrix but notinvertible as a Z-matrix,then W-1 can not map all Z-vectors to Z vectors ).So wehave(wi,T)EZfor all i.Recall that in a neighborhood of m, there exists coordinate system (zi,.., zn)so that the action of Tn is given byexp(X) (21, ** ,zn) = (e2mi(w,X)21, e2元i(wm,X)zm)So exp(+) acts trivially on a neighborhood of m, but exp(+) is not the identityelement in Tn. This contradicts with the fact that in a dense open subset of M the口action isfree.Sosatisfies (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 poly-topes:Theorem 2.1 (Delzant, 1990). There is a one-to-one correspondence between sym-plectic toric manifolds (up to Tn equivariant symplectomorphisms)and Delzant poly-topes. 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 sym-plectomorphic.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
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∆.
5LECTURE11:SYMPLECTICTORICMANIFOLDSStep 3: Check M is toric and μ(M) = △Step 4: △(M) =△(M2) M1 ~ M2 Construction of M from Delzant polytope △Now let be aDelzant polytope in Rn.Supposehas d facets, then by thealgebraic description one can find primitive outward pointing vectors Ui,.,Ud sothat△={re (R")* I <r, ) ≤入, i=l, .-.,d).By translation we may assume 0 e, and thus A,≥ 0 for all i.We shall constructM as the symplectic quotient of Rd by a Hamiltonian action of a torus N ofdimension d - n.Step 2.aThe (d-n)-torus N.Let ei,... , ed be the standard basis of Rd. Define linear mapT: Rd→R",e→viThen since is Delzant, π is onto and maps Zd onto Zn. So we get an inducedsurjective Lie group homomorphismπ: Td- Rd/Zd→ T"- Rn/ZnLet N = ker(π). It is a (d - n)-subtorus of Td.Note that from the exact sequence of Lie group homomorphismsONTd"T0one gets an exact sequence of Lie algebras0nRdRn0,and thus an exact sequence of dual Lie algebras0 → (Rn)* (Rd)*=n*→0.Step2.bThe Hamiltonian N-action on Cd.The standard Td-action on Cd is given by(eii, ... ,ea) (z1,*., zd) = (eiz1, ,eidzd)The action is Hamiltonian with moment mapΦ: Cd → (Rd)*, Φ(z1,*-, zd) =-(1212, .. zal) + c.We choose c=入=(1,...,>a).Since N is a sub-torus of Td, the induced N-action on Cd is Hamiltonian withmoment map t* o Φ : Cd → n*.Step 2.cThezero level set Z = (*o @)-1(O) is compactLet △'= π*(). Then △ is compact. We claim
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