September 2011 at Varna

Transcription

1 September 2011 at Varna University of California at Riverside, and Tsinghua University In collaboration with Daniele Grandini and Brian Rolle

2 overview Classical deformation of complex structures (Kodaira-Spencer-Kuranishi) Extended deformation (Kontsevich-Barrannikov, Merkulov, Zhou), (algebraic) Generalized Complex Structures, (Hitchin, Gaultieri) (geometric) (generalized extended) Issues: Cohomology stability Counter Examples (cohomology jumping e.g. CP 2, Hopf) Motivation: Kodaira-Thurston surface. Infinitesimal vs Integrability Application to hyper-symplectic nilmanifolds

3 Classical deformation of complex structures Complex structure. J : T T, J J = identity. differentials = T (1,0) T (0,1). tangents = T 1,0 T 0,1. Integrability: T 1,0 : Nijenhuis tensor. T (0,1) differential ideal. A deformation is a change of the splitting of tangents, (and differentials), determined by a graph of a map: Γ : T 0,1 T 1,0. T 1,0 Γ = {l + Γl : l T 1,0 }, T 0,1 Γ = {l + Γl : l T 0,1 }. Integrability: closure of space of sections. Γ [Γ, Γ] = 0. Linearize: if Γ(t), Γ(0) = 0, and dγ dt (0) = Γ 1, Γ 1 = 0. Γ 1 H 1 (T 1,0 ).

4 Extended Deformation, Differential Gerstenhaber algebra G = (T (1,0) T (0,1) ). G p,q = T p,0 M T (0,q), G k = p+q=k G p,q. k = deg DGA(M, J) := (G = G k,, [, ], ). construction of and [, ]. ( as in Γ [Γ, Γ] = 0) (G,, ), (a b) = (a) b + ( 1) deg a a b. (G, [, ], ), [a, b] = [a, b] + ( 1) deg a+1 [a, b]. Distributive: [a, b c] = [a, b] c + ( 1) (deg a+1) deg b b [a, c]. The cohomology of, H (M, J), a Gerstenhaber algebra. The structure of the DGA controls the deformation theory.

5 Geometric realization in infinitesimal deformation Classical Γ 1 H 1 (T 1,0 ), Γ 1 : T 0,1 T 1,0. Extended: Γ 1 p,q H p ( q T 1,0 ). Generalized: Γ 1 H 0 (T 2,0 ) H 1 (T 1,0 ) H 2 (O). Integrability:from Γ 1 to Γ: Γ [Γ, Γ] = 0. Γ 1 = Λ 2 T 1,0, Λ : T (1,0) T 1,0, Λ : T (0,1) T 0,1. If Λ 2 T 1,0, then by type consideration Λ = 0, [Λ, Λ] = 0. Holomorphic Poisson structures. Known: if Λ is non-degenerate over every point, Λ comes from a symplectic structure.

6 From classical to generalized complex structures (M, J) a complex space. J : T T, ( J 0 0 J ) ( T T ) ( T T ). +i eigen-space: L = T 1,0 T (0,1). X ijx, α + ij α. DGA(J) = ( L,, [, ], ), H k = p+q=k H q (M, p T 1,0 ). (N, ω) a symplectic space: ω : T T, non-degenerate. ω = ω. ( ) ( ) ( ) 0 ω 1 T T ω 0 T T. L = {X iω(x ) : X T }. DGA(ω) = ( L,, [, ], ω ). When L = T, ω = d. H k = HdeR k. Then there is the bracket.

7 An identification Complex structure L L = T T. A pairing: X, α = α(x ). Deformation L Γ L Γ = T T. L = L. on L [, ] on L = L. L Γ = L Γ. Γ on L Γ. L L Γ = T T, L Γ = L [, ] on L Γ = L Γ, Γ on L. Γ : L L Γ = + [Γ, ]. DGA(J Γ ) = DGA( L,, [, ], + [Γ, ]).

8 Issue Given a generalized (i.e. degree-2) deformation Γ(t) of a classical complex structure, deformation of DGAs, DGA(J Γ(t) ) = DGA( L,, [, ], + [Γ(t), ]). deformation of cohomology as GA: H (J Γ(t) ), with, [, ]. Problem Given a complex structure J with Γ(t) p+q=2 C (T (0, q) p T 1,0 ) and Γ(t) [Γ(t), Γ(t)] = 0, when will the GA structure of H (J Γ(t) ) be invariant along the deformation family determined by Γ(t)? Γ 1 p+q=2 H q ( p T 1,0 ) = H 0 (M, T 2,0 ) H 1 (M, T 1,0 ) H 2 (M, O). Holomorphic Poisson, classical, B-field.

9 Example. Hopf: U(1) SU(2). Λ H 0 (anti-canonical bundle). Z + ω C (L) = T 1,0 T (0,1), Λ (Z + ω) = 0: Z + ω + [Λ, Z] + [Λ, ω] = 0. ω = 0, [Λ, Z] = 0, Z + [Λ, ω] = 0. Λ = X 0 X, ω = a(ln z 2 ) + ψ = aσ + ψ. Z + ω = a([x, ln z 2 ] + σ) + bx 0 + cx + Λ ψ. H 1 ( Λ ) = [X, ln z 2 ] + σ, X 0, X. H 1 () = H 1 (M, O) H 0 (M, T 1,0 ) = σ u(1) su(2), Jumping cohomology.

10 Motivation. Kodaira-Thurston in Crelle, 2006 Γ 1 K H even H odd = p,q H q (M, p T 1,0 ). H, V horizontal and vertical holomorphic vector fields. ω, ρ the dual (1,0)-forms. Γ 1 = t 0 + t 1 ω ρ + t 2 ρ V + t 3 ω H + t 4 H V + t 5 ω ρ H V s 0 ρ + s 1 ω + s 2 V + s 3 ρ H V + s 4 ω H V + s 5 ω ρ H. Γ = Γ 1 s 0t4 1 t 2 T s 0s 3 1 t 2 ρ H. Solve for MC: s 0 = 0 or s 3 = 0 and t 4 = 0. Variation of wedge product on cohomology spaces. Weak Frobenius structure on Kuranishi space

11 When s 0 = 0 Γ = t 0 + t 1 ω ρ + t 2 ρ V + t 3 ω H + t 4 H V + t 5 ω ρ H V +s 1 ω + s 2 V + s 3 ρ H V + s 4 ω H V + s 5 ω ρ H. t 1 t 2 t 3 t 4 s 1 s 2 t 4 1 t 2 t t t t 2 s 4 0 t s 4 t s 4 0 t s s 1 t 2 s 4 0 s t 2 t 4 s 2 1 t 2 s 4 0 s 4 0 s 3 1 t 2 s 4 0 s 4 s 4

12 Conditions of stability is non-trivial. Necessary conditions: Suppose Φ(t) : L L, induces a DGA-homomorphism for each t: Φ(t) : ( L, [, ],, Γ(t) ) ( L, [, ],, ). For any A, B in L, Φ Γ(t) = Φ, Φ(A B) := Φ(A) Φ(B), [Φ(A), Φ(B)] = Φ([A, B]). Infinitesimal: Γ(t) = tγ 1 + O(2), Φ = I + tφ + O(2), φ : L L = T 1,0 T (0,1) ; (φa) φ(a) = [Γ 1, A]; [φa, B] + [A, φb] = φ[a, B]; φ(a B) = (φa) B + A (φb).

13 From infinitesimal to Integrable Γ 1 and φ are compatible if Γ 1 2 (T 1,0 T (0,1) ), and Γ 1 = 0, φ : L L = T 1,0 T (0,1) (φa) φ(a) = [Γ 1, A]; [φa, B] + [A, φb] = φ[a, B]; φ(a B) = (φa) B + A (φb). Theorem Compatible pairs are integrable. Γ(t) = ( 1) n 1 1 n! tn φ n 1 Γ 1, Φ(t) = n=1 n=0 1 n! tn φ n.

14 Holomophic Poisson Γ 1 H 0 (M, T 2,0 ) H 1 (M, T 1,0 ) H 2 (M, O). Given Λ, Λ = 0, [Λ, Λ] = 0, how to find φ? ( ) ( ) ( I II T 1,0 T 1,0 φ = : III IV T (0,1) T (0,1) ) A type checking observation: Only component II is relevant. Theorem Given Λ holomorphic Poisson, if φ : T (0,1) T 1,0 is compatible with Γ 1, then Φ(t) = 1 + tφ yields an isomorphism. φ φ = 0. Type-(1,1) bivector. Problem now: For Kodaira-Thurston and similar objects with given Λ, find compatible pairs.

15 Kodaira-Thurston as a nilmanifold M = (R H 3 )/, H 3, the 3-dimensional Heisenberg group. [e 1, e 2 ] = e 3. de 3 = e 1 e 2. e 3, e 4 h = e 1, e 2, e 3, e 4 e 1, e 2 Kodaira: elliptic fibration over elliptic curve. Je 1 = e 2, Je 3 = e 4. Thurston: symplectic non-kählerian. (ω invariant symplectic forms) Normizu: The inclusion DGA(h, ω) DGA(M, ω) induces isomorphism of DeRham cohomology spaces (as vector spaces). A general result. Fino, Console, Pedersen, Grantcharov, Poon, Rollenske: The inclusion DGA(h, J) DGA(M, J) induces isomorphism of Dolbeault cohomology spaces. A rather general statement holds for nilmanifolds with invariant complex structures.

16 A hypersymplectic space T (g, ω) a real nilpotent Lie algebra, a symplectic structure. V the underlying vector space of g. a symplectic rep γ : g sp(v, ω). torsion-free flat connection on G. Make a semi-direct product T := g γ V. [(x, 0), (y, 0)] = ([x, y], 0), [(x, 0), (0, v)] = (0, γ(x)v). J(x, y) = ( y, x). Ω 1 ((x, u), (y, v)) := ω(x, v) ω(u, y). Ω 2 ((x, u), (y, v)) := ω(x, y) ω(u, v). Ω 3 ((x, u), (y, v)) := ω(x, y) + ω(u, v). Ω 1, Ω 2, Ω 3 are algebraically related. Hypersymplectic. T is so simple that the DGA are given by algebras.

17 Solution Ω c = Ω 1 + iω 2 is type (2,0). Ω 3 is type (1,1). Ω c : T 1,0 T (0,1). Λ = Ω 1 c : T (0,1) T 1,0. Ω 3 T (1,0) T (0,1), Ω 3 : T 1,0 T (0,1). Ω 1 3 : T (0,1) T 1,0. First attempt: φ = λω 1 3? More type (1,1)-form Ω 4. Second attempt: φ = λω3 1 + µω 1 4, with additional metric structure on g and γ. For some pseudo-kählerian space (Kodaira-Thurston), Ω 4 exists so that Λ and φ are compatible. (Andranda s list) There are examples (Kodaira-Thurston) for which DGA(J tλ ) is invariant along tλ. t = 0, complex. t 0, symplectic.