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Lagrangian surgery and
Rigid analytic family of Floer homologies
Kenji Fukaya
A part of this talk is based on joint work withYong‐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 waythen homological Mirror symmetry conjecture will be proved.
2
( , )X ω a sympectic manifold
L X⊂ (relatively spin) Lagrangian submanifold
0 0: ( ; ) ( ; ), 0,1, 2,kkm H L H L k⊗Λ → Λ = …
F‐Oh‐Ohta‐Ono(AMS/IP 46)
10
0
( ) ( ; ) ( , , ) 0kk
L b H L m b b∞
=
⎧ ⎫= ∈ Λ =⎨ ⎬⎩ ⎭
∑ …M
(Filtered) A infinity structure
Maurer‐Cartan Scheme
( )i iLb ∈M 1 1 2 2 0, ), , ); )(( (HF L b L b Λ
Floer homology3
L X⊂Suppose Maslov index of is 0.
0 , 0, limii i i iii
a T aλ λ λ→∞
⎧ ⎫Λ = ∈ ≥ = ∞⎨ ⎬
⎩ ⎭∑ C
1 1 2 2 0, ), , ); )(( (HF L b L b Λ is Z graded.
4
10[ ]T −Λ = Λ
Homological Mirror symmetry conjecture (Kontsevitch)
( , )X ω ( , )X J∨Mirror complex manifold.
L X⊂ ( )E L X ∨→ Object of derived categoryof coherent sheaves.
5
1 2( , )HF L L 1 2Ext( ( ), ( ))E L E L≅
Guess (related to (a version of) Stronminger‐Yau‐Zaslow conjecture)
( ( ), )( ) ( , ( ( ), ); )L u bE L HF L L u b= Λ
( ( ))u B
X L u∨
∈
=∪M ( )L u : a family of Lagrangian submanifoldparametrized by
Mirror Object = family of Floer homologies
6
u B∈
7
L
B
Xi
ππ is a locally trivial fiber bundle
1( ) ( )L u uπ −=
( ) : ( )L ui L u X→
is a Lagrangian embedding.
1( ( ); )uT B H L u→ R is an isomorphism.Assume
( )L u X⊂Suppose Maslov index of is 0.
B has flat affine structure.
:s B → L is a section.
Theorem 1 (to be written up, a part is in arXiv:0908.0148)
8
1( ) ( ( )) ( ( );2 1 )u B
L u H L u π∈
= −∪M L M Z
has a structure of rigid analytic space.
(1)
(2) L′ is a another Lagrangian submanifold (relatively spin, Maslov = 0).
( )b L′ ′∈M
( , ) (( , ), ( ( ), ); )u b HF L b L u b′ ′ Λ
If
then
defines an object of derived category of coherent sheaves on ( )M L
9
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).
10
10
0( ) ( ; ) ( , , ) 0k
kL b H L m b b
∞
=
⎧ ⎫= ∈ Λ =⎨ ⎬⎩ ⎭
∑ …M
22
0 1. ,rank0
( ) ( , , ) ( ( ); ), ( ) ( ( ))lu k u u l H
k
P x m x x H L u P x P x∞
==
= ∈ Λ =∑ ……
i ix x=∑ e ie is a basis of 1( ( ); )H L u R
, 11
( ) ( , , )il lu i u m
iP x T P y yλ
∞
=
=∑ …
exp( )i iy x=
' 1 1( ' , , ' ) ( , , )u m u mP y y P y y=… …
1 11( , , )i i m mu u u uu u
i i i my T y Q T y T y′ ′′− −−′ = + …
,1 1 1 11 , 1
1( , , ) ( , , )i jm m m mu u u uu u u u
i m i j mj
Q T y T y T Q T y T yλ∞
′ ′′ ′− −− −
=
= ∑… …
1 1, 1 1 1( , , ) , , , , ,l
i u m m mP y y y y y y− −⎡ ⎤∈ ⎣ ⎦… … …R
,u ui i′ are affine coordinate of ,u u′
Theorem 1 is good enough to construct homological Mirror functor for torus.
11
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.
12
L
B
Xi
ππ is a locally trivial fiber bundle over
1( ) ( )L u uπ −=
( ) : ( )L ui L u X→
is a Lagrangian embedding.
1( ( ); )ouT B H L u→ R is an isomorphism.Assume
oB B−
oB
is a finitely many point
dim 4X =R1 2( ) ( )L u u Tπ −= =
ou B B∈ − 1( ) ( )L u uπ −=
ou B∈
is immersed with one self intersection point which is transversal.
2S
Theorem 2 (to be written up)
13
1( ) ( ( )) ( ( );2 1 )u B
L u H L u π∈
= −∪M L M Z
has a structure of rigid analytic space.
(1)
(2) L′ is a another Lagrangian submanifold (relatively spin, Maslov = 0).
( )b L′ ′∈M
( , ) (( , ), ( ( ), ); )u b HF L b L u b′ ′ Λ
If
then
defines an object of derived category of coherent sheaves on ( )M L
14
Application
Construction of homological Mirror functor for K3 surface.
Note: Homological Mirrror symmetry was proved for quartic surface by P. Seidel.
15
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)
1 2X L L⊃ ∪ Two Lagragian submanifolds which intersect at one point transversaly.
1 2#L L L± ±= is the connected sum, embedded in X
1L
2LL+
L−
Lagrangian surgery
16(Lalonde‐Sikovar, Polterovich)
Theorem 3 (F‐Oh‐Ohta‐Ono, will be in the revised version of Chapter 10)
There exist long exact sequences
Note: Proved before by P. Seidel in case or is a sphere and exact case(= the case of Dehn twist).
2L1L
2 2 1 2 1 1(( , ), ( , )) (( , ), ( , ( , )) (( , ), ( , ))HF L b L b HF L b L b b HF L b L b−′ ′ ′ ′ ′ ′→ → → →
17
iLAssueme Maslov index of are zero.
1 2( ) ( ) ( )L L L± = ×M M M(1)
(2) ( )i ib L∈M ( )b L′ ′∈MIf
1 1 1 2 2 2(( , ), ( , )) (( , ), ( , ( , )) (( , ), ( , ))HF L b L b HF L b L b b HF L b L b+′ ′ ′ ′ ′ ′→ → → →
and
Holomorphic triangle
Becomes holomorphic 2 gon
Becomes parametrizedfamily of holomorphic 2 gons
2nS −
Idea of the proof
18
19
Floer theory of Immersed Lagrangian submanifold.
(M. Akaho – D. Joyce, arXiv:0803.0717)
Special case: (0)L is immersed 2‐sphere with one self intersection point which is transversal.
(( ), ( (0), ))HF L b L b′ ′
0 0( (0))L = Λ ⊕ΛM
is parametrized by
1 2 0 0( , ) ( (0))b x x L= ∈ = Λ ⊕ΛM
20
(0)LL′
p
q
1 2
1 2
1 21 2
( , ) 1 2 1 2( , , )
, 0,1,2,
, # (( , , ); , )k kx x
k kk k
p q T x x M k k p qβ ω
β
β∩
=
∂ = ∑…
1
2 r
21
1 2(( , , ); , )M k k p qβ
β
1 21, 3k k= =
p qL′
r r r r
22
Resolve singularity by surgery.
There are 2 parameter family of smooth Lagrangianobtained.
2T
23
BVanishing cycle is realized by Holomorphic disc
Wall crossing line
=
Holomorphic triangle
Becomes holomorphic 2 gon
Becomes parametrizedfamily of holomorphic 2 gons
2nS −
=24
0 2 pointsS =
25
1 21 2 1k kT x x T xβ ω β ω∩ ∩
1T yβ ω∩
1 1 2( )T y y yβ ω∩ ±
One disc
Two discs
p
q
26
(1)
(2)
(3)
(4)
(1) (2) (3) (4) (1)
Moduli of holomorphic strips
Bifurcation of the moduli space of holomorphic striparound `type one’ singular fiber(F‐Oh‐Ohta‐Ono 2000 version of [FOOO])
27
BVanishing cycle is realized by Holomorphic disc
Wall crossing line
=
28
1 2
1 2
1 2
( , ) 1 2 1 2( , , )
, # (( , , ); , )k kx x
k kp q T x x M k k p qβ ω
β
β∩∂ = ∑
1 2
1 2 1 1
1 2
1 1( , ) 1 2 1 2
( , , )
, ( ) # (( , , ); , )k ky y
k k
p q T y y y y M k k p qβ ω
β
β∩ − −∂ = ±∑
1 2
1 2 1
1 2
( , ) 1 1 2 1 2( , , )
, ( ) # (( , , ); , )k ky y
k k
p q T y y y y M k k p qβ ω
β
β−∩∂ = ±∑
29
1 21 2 1k kT x x T xβ ω β ω∩ ∩
1T yβ ω∩
1 1 2( )T y y yβ ω∩ ±
One disc
Two discs
p
q
30
1 1
1 11 1
2 2
x y
x y y y− −
=
= ±1
1 1 1 21
2
x y y y
x y −
′ ′= ±
′=
1 1 1 2y y y y′ ′= ±
11 1 1 2y y y y−′′ = ±
11 1 2y y y−′′= ±
This is the monodromy by the Dehn twist.(= singularity of affine structure).
This coordinate change is the same as one appearing in the work by Gross‐Siebert.