Lagrangian surgery and Rigid analytic family of Floer homologies Kenji Fukaya A part of this talk is based on joint work with Yong Geun Oh, Kaoru Ono, Hiroshi Ohta 1
Why Family of Floer cohomology? It is expected that if family Floer cohomology is built in an ideal way then homological Mirror symmetry conjecture will be proved. 2
( X, ω) L X a sympectic manifold (relatively spin) Lagrangian submanifold F Oh Ohta Ono (AMS/IP 46) m : H( L; Λ ) H( L; Λ ), k = 0,1,2, k M k 0 0 1 ( L) = b H ( L; Λ 0) mk ( b,, b) = 0 k = 0 (Filtered) A infinity structure Maurer Cartan Scheme b M( L ) HF(( L1, b1), ( L2, b2); Λ0) i i Floer homology 3
L X Suppose Maslov index of is 0. λ i Λ 0 = at i ai C, λi 0, limλi = i i Λ=Λ [ T 1 ] 0 HF(( L, b), ( L, b ); Λ ) is Z graded. 1 1 2 2 0 4
Homological Mirror symmetry conjecture (Kontsevitch) ( X, ω ) ( X, J) Mirror complex manifold. L X EL ( ) X Object of derived category of coherent sheaves. HF( L1, L2) Ext( EL ( 1), EL ( 2)) 5
Guess (related to (a version of) Stronminger Yau Zaslow conjecture) X = M ( L( u)) Lu ( ) u B : a family of Lagrangian submanifold parametrized by u B EL ( ) = HFL (,( Lu ( ), b); Λ) ( Lu ( ), b) Mirror Object = family of Floer homologies 6
L i X π is a locally trivial fiber bundle π Lu 1 ( ) = π ( u) B s: B L is a section. i : ( ) Lu ( ) L u X is a Lagrangian embedding. Assume TB H 1 ( Lu ( ); R) u B is an isomorphism. has flat affine structure. Lu ( ) X Suppose Maslov index of is 0. 7
Theorem 1 (to be written up, a part is in arxiv:0908.0148) (1) = ML M Z 1 ( ) ( Lu ( )) H( Lu ( );2π 1 ) u B has a structure of rigid analytic space. (2) If L M is a another Lagrangian submanifold (relatively spin, Maslov = 0). b ( L ) then ( ub, ) HF(( L, b ),( Lu ( ), b); Λ) defines an object of derived category of coherent sheaves on ML ( ) 8
Kontsevich Soibelman proposed to use Rigid analytic geometry to study homological Mirror symmetry around 2000. Various operators etc. appears in Floer theory and Gromov Witten theory is one over (universal) Novikov ring = a kind of formal power series ring, and its convergence is not know. An idea to use rigid analytic geometry is first to construct everything in the lebel of formal power series (Novikov ring) and prove a version of Mirror symmetry (formal power series version) and use GAGA of rigid analytic geometry to prove convergence later (in the complex side). 9
M 1 ( L) = b H ( L; Λ 0) mk ( b,, b) = 0 k = 0 2 l u( ) = k(,, ) ( ( ); Λ 0), u( ) = ( u( )) l= 1.,rank H k = 0 P x m x x H L u P x P x 2 x= x i e e i i is a basis of H 1 ( L( u); R) y i = exp( x ) i l i l u = λ i, u 1 m i= 1 P ( x) T P ( y,, y ) R l 1 1 Piu, ( y1,, ym) y1,, ym, y1,, y m P ( y',, y' ) = P ( y,, y ) u' 1 m u 1 m u u u, u i i are affine coordinate of, y = T y + Q T y T y i i u1 u1 m m (,, ) u u u u i i i 1 m λ i 1 m = i, j 1 m j= 1 Q T y T y T Q T y T y 1 1, ( u u u,, m u m i j ) ( u1 u1 u,, m u m ) 10
Theorem 1 is good enough to construct homological Mirror functor for torus. To go beyond the case of torus we need to include singular fiber. The main result of this talk says that we can do it in the case of simplest singular fiber. 11
L i X π is a locally trivial fiber bundle over o B π u B o Lu 1 ( ) = π ( u) B i : ( ) Lu ( ) L u X is a Lagrangian embedding. Assume o TB H 1 ( Lu ( ); R) u is an isomorphism. dimr X = 4 B B o u B B is a finitely many point o Lu ( ) = ( u) = T Lu 1 2 π 1 ( ) = π ( u) S is immersed with one self intersection point which is transversal. 2 12
Theorem 2 (to be written up) (1) = ML M Z 1 ( ) ( Lu ( )) H( Lu ( );2π 1 ) u B has a structure of rigid analytic space. (2) If L M b ( L ) is a another Lagrangian submanifold (relatively spin, Maslov = 0). then ( ub, ) HF(( L, b ),( Lu ( ), b); Λ) defines an object of derived category of coherent sheaves on ML ( ) 13
Application Construction of homological Mirror functor for K3 surface. Note: Homological Mirrror symmetry was proved for quartic surface by P. Seidel. 14
Method of proof: Lagrangian surgery and behavior of Floer homologies via surgery. (F,Oh,Ohta,Ono; Chapter 10 of Lagrangian Floer thoery book.) Floer theory of Immersed Lagrangian submanifold. (M. Akaho D. Joyce, arxiv:0803.0717) 15
Lagrangian surgery X L1 L2 Two Lagragian submanifolds which intersect at one point ± 1 ± 2 transversaly. L = L # L is the connected sum, embedded in X L 2 L + L 1 L (Lalonde Sikovar, Polterovich) 16
Theorem 3 (F Oh Ohta Ono, will be in the revised version of Chapter 10) Assueme Maslov index of L i are zero. (1) M M M (2) If b ( L) ( L± ) = ( L1) ( L2) There exist long exact sequences i M b ( L ) i M HF(( L, b ),( L, b )) HF(( L, b ),( L,( b, b )) HF(( L, b ),( L, b )) and 2 2 1 2 1 1 HF(( L, b ),( L, b )) HF(( L, b ),( L,( b, b )) HF(( L, b ),( L, b )) 1 1 + 1 2 2 2 Note: Proved before by P. Seidel in case L 1 or L 2 is a sphere and exact case (= the case of Dehn twist). 17
Idea of the proof Becomes holomorphic 2 gon Holomorphic triangle n 2 S Becomes parametrized family of holomorphic 2 gons 18
Floer theory of Immersed Lagrangian submanifold. (M. Akaho D. Joyce, arxiv:0803.0717) Special case: L(0) is immersed 2 sphere with one self intersection point which is transversal. M( L (0)) =Λ Λ 0 0 HF(( L b ),( L(0), b)) is parametrized by M b= ( x, x ) ( L(0)) =Λ Λ 1 2 0 0 19
pq = T x x M k k pq β ω k1 k2, # (( β,, );, ) ( x1, x2) 1 2 1 2 ( β, k, k ) k, k 0,1,2, 1 2 1 2= L q 2 r L(0) 1 p 20
M (( β, k, k ); p, q) 1 2 L p q β r r r r k = 1, k = 3 1 2 21
Resolve singularity by surgery. There are 2 parameter family of smooth Lagrangian obtained. T 2 22
Vanishing cycle is realized by Holomorphic disc = B Wall crossing line 23
Becomes holomorphic 2 gon 0 S = 2points = Holomorphic triangle n 2 S Becomes parametrized family of holomorphic 2 gons 24
β ω T y 1 p One disc q β k1 k2 T ω β ω x x T x 1 2 1 β ω T ( y ± y y ) 1 1 2 Two discs 25
(2) Bifurcation of the moduli space of holomorphic strip around `type one singular fiber (F Oh Ohta Ono 2000 version of [FOOO]) (3) (1) (4) Moduli of holomorphic strips (1) (2) (3) (4) (1) 26
Vanishing cycle is realized by Holomorphic disc = B Wall crossing line 27
p q = T x x M k k p q β ω k1 k2, # (( β,, );, ) ( x1, x2) 1 2 1 2 ( β, k, k ) 1 2 = ± β ω k1 1 1 k2, ( ) # ((,, );, ) 1 y pq T y y y y M β k k pq 2 1 1 ( y, ) 1 2 1 2 ( β, k, k ) 1 2 = ± β ω k1 k2, ( ) # ((,, );, ) 1 y pq T y yy y M β k k pq 2 1 ( y, ) 1 1 2 1 2 ( β, k, k ) 1 2 28
β ω T y 1 p One disc q β k1 k2 T ω β ω x x T x 1 2 1 β ω T ( y ± y y ) 1 1 2 Two discs 29
y = y ± y y 1 1 1 2 This coordinate change is the same as one appearing in the work by Gross Siebert. x = y 1 1 1 1 x2 = y ± y y x 2 2 = 1 1 1 x = y ± y y 1 1 1 2 1 y y = y ± y y 1 1 1 1 2 y =± y y 1 1 1 2 This is the monodromy by the Dehn twist. (= singularity of affine structure). 30