Delzant s Garden. A one-hour tour to symplectic toric geometry
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1 Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem Delzant s construction Examples Post-Delzant 1 Different faces of toric geometry 1.1 Toric varieties in algebraic geometry Toric variety = normal variety with (Zariski) open, dense, (C ) n orbit. Fan construction: strong convex rational polyhedral cone semigroup algebra affine toric fan=family of S.C.P.C glue along the intersection=toric variety 1.2 Toric varieties in complex geometry Projective toric variety = smooth projective variety with open dense (C ) n orbit. Monomial embedding construction: Suppose is Delzant, Z n = {α(1),, α(d + 1)}, c is any parameter. let Φ c = [c 1z α(1) : : c d+1 z α(d+1) ] : (C ) n CP d. Toric variety M is the Zariski-closure of the image of Φ c in CPd. 1.3 Toric manifolds in symplectic geometry Toric manifold = (M 2n, ω) equipped with an effective Hamiltonian T n action. Delzant s construction: Delzant, d number of facets, N d n T d subtorus determined by. Toric manifold M is the symplectic quotient of C d by linear N action. 1
2 1.4 Why toric? They provide a remarkably fertile testing ground for general theories (Fulton) It is related to many different subjects in mathematics: algebraic geometry commutative algebra complex geometry toric geometry string theory mirror symmetry symplectic geometry combinatorics integrable systems 2 Symplectic toric manifolds Definition 1 A compact connected symplectic manifold (M 2n, ω) is toric if it admits an effective 1 Hamiltonian T n action. Examples: (1) The sphere S 2 with the standard T 1 action the rotation. (2) The complex projective space CP 2 = C 3 / with the T 2 = T 3 /T 1 action. By the Atiyah-Guillemin-Sternberg convexity theorem, the image of the moment map is always a convex polytope in R n. The moment polytopes of the previous examples are S 2 CP 2 1 A G-action is effective if every e g G moves at least one p M. One can show that if the T n action is not effective, then the action reduces to a T n 1 action. On the other hand, if a G-action is effective, then dim M 2 dim G. 2
3 Definition 2 A polytope R n is called 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. An algebraic description of Delzant s polytope: Let v i, 1 i d, be the primitive outward-pointing normal vectors to the facets of, then = {x (R n ) x, v i λ i, i = 1,, d}. Theorem 1 For any symplectic toric manifold (M, ω), its moment polytope is Delzant. Proof: Let x M G, then p = µ(x) is a vertex. We have seen from last talk (proof of convexity theorem) that the moment polytope near p is {p + s i w i s i 0} where w 1,, w n are the weights of the linearized T n -action on T x M. Thus satisfies the conditions (1) and (2). Suppose does not satisfies the condition (3), then the matrix W consisting of w i s is not invertible as Z-matrix. We take τ W 1 (Z n ) such that τ Z n. (If W is not invertible, we can take τ be any non-integer vector in the kernel of W ). Now we have w i, τ Z for all i, i.e. exp(τ) acts trivially on a neighborhood of x, but exp(τ) is not the identity element. Contradiction. So satisfies (3). 3 Delzant s theorem Theorem 2 (Delzant, 1990) Moment polytope determines the toric manifold M: symplectic toric manifolds Delzant polytopes T n -equivariant symplectomorphisms translations The proof is divided into several steps: Step 1: M is toric = µ(m) is Delzant. (Done) Step 2: is Delzant construct compact connected symplectic manifold M. Step 3: Check M is toric and µ(m ) =. Step 4: M 1 M 2 = µ(m 1 ) µ(m 2 ). Step 5: µ(m 1 ) = µ(m 2 ) = M 1 M 2. 3
4 More details in Step 2: Step 2.a: The (d n)-torus N. Let be Delzant with d facets, = {x (R n ) x, v i λ i, i = 1,, d}. Let e 1,, e d be the standard basis of R n. 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 map π : T d T n. Let N = Ker(π). It is a (d n)-subtorus of T d. We have the following exact sequences 0 N i T d π T n 0, 0 n i R d π R n 0, and 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 iθn ) (z 1,, z n ) = (e iθ 1 z 1,, e iθn z n ). The action is Hamiltonian with moment map φ : C d (R d ), φ(z 1,, z n ) = 1 2 ( z 1 2,, z n 2 ) + c. Thus the induced N-action is Hamiltonian with moment map i φ : C d n. We choose c = λ = (λ 1,, λ n ). Step 2.c: The zero level set Z = (i φ) 1 (0) is compact. Let = π ( ). According to π (x), e i = x, v i together with our choice of c, one can show Im(π ) Im(φ) =. Thus Z = (i φ) 1 (0) = φ 1 (Ker(i )) = φ 1 (Im(π )) = φ 1 ( ) is compact. ( compact and φ proper.) Step 2.d: N acts freely on Z, thus M is compact symplectic. 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 }. By Delzant s conditions, one can show that the restriction map of π, π : (T d ) z T n, is injective. (Proof: If π([t]) = π([s]), then π(t) π(s) = i I z (t i s i )v i Z n, so t i s i Z, and [t] = [s].) This implies N z = N (T d ) z = i(n) (T d ) z = Ker(π) (T d ) z = Ker(π (T d ) z ) = {1}. 4
5 More details in Step 3: Step 3.a: Hamiltonian T n -action on M. Suppose z is a point such that ψ(z) = π (p) for a vertex p. Then by counting dimension, we see that π : (T d ) z T n is bijective. Thus 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, the T n action commutes with the N-action, thus we have an induced Hamiltonian T n action of T n on M, whose moment map µ satisfies µ p = j φ j, where p 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 T d, and thus T n, acts freely on the open dense subset φ 1 (π (int )) Z. Step 3.c: The moment polytope of M is. µ(m ) = µ p(z) = j φ j(z) = j φ((i φ) 1 (0)) = j (Ker(i )) = j (π ( )) = (π j) ( ) =. Step 4: This is obvious: different moment maps for the same Hamiltonian T n -action differ by a constant in the dual Lie algebra, thus the corresponding moment polytopes differ by a translation. Little details in Step 5: Construct by induction a diffeomorphism that intertwines the torus actions and moment maps. Show that the cohomology class of ω is determined by the moment polytope. Apply Moser s trick. For more details, c.f. Kai Cieliebak s notes, P Corollary 1 Symplectic toric manifolds M inherit Kähler structure from C N. (So symplectic toric manifolds are always Kähler toric manifolds!) Next time: Victor Guillemin will talk about this Kähler metric on M. 5
6 4 Some examples S 2 = CP 1 is the only 2-dimensional symplectic toric manifold. The standard n-simplex is the only n-dimensional Delzant polytope with n + 1 vertices = CP n is the only 2n dimensional symplectic toric manifold with n + 1 fixed points. All the 2-dimensional Delzant polytopes with 4 vertices are trapezoids with vertices (0, 0), (0, 1), (l, 1), (l + n, 0), where n is nonnegative integer and l > 0: The corresponding symplectic toric manifolds are called Hirzebruch surfaces. They are the only 4-dimensional symplectic toric manifolds with four fixed points. Cutting the Delzant polytope at a vertex blowing up the symplectic toric manifold at the corresponding fixed point. For example, the first Hirzebruch surface is blowing up of CP 2 : 5 Post-Delzant stories 5.1 Other classification theorems In 1997, E. Lerman and S. Tolman classified symplectic toric orbifolds (constructed by removing the smoothness from the Delzant conditions) by simple rational polytopes AND positive integers attached to each facets. In 2002, E. Lerman classified contact toric manifolds ((M 2n+1, ξ) admitting effective T n action which preserves the contact structure ξ) by rational polyhedral cones. In 2003, Y. Karshon and S. Tolman classified complexity one spaces (dim M/2 dim T n = 1). 6
7 5.2 Quotients of C d There are many other symplectic manifolds that can be realized as quotient of the complex space C d. Examples includes the Grassmannian, the coadjoint orbits of U(n), and more generally the quiver varieties, plus their symplectic cuts. They are all spherical varieties the non-abelian analogue of toric varieties. Definition 3 Let G be a connected reductive group, B G be a Borel subgroup. An irreducible normal G-variety X is spherical if B has an dense orbit in X. Conjecture 1 (Guillemin) Spherical varieties can be realized as symplectic quotients of C d. 6 References Symplectic toric geometry: Victor Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian T n -Spaces Ana Cannas da Silva, Symplectic toric manifolds M. Aubin, The Topology Of Torus Action On Symplectic Manifolds Kai Cieliebak, Symlectic geometry, part B Toric varieties in algebraic geometry: W. Fulton, Introduction to Toric Varieties Monomial embedding: I. Gelfand, M. Kapranov and A. Zelevinsky, Discriminants, resultants and multidimensional determinants 7
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