HOLOMORPHIC CURVE TECHNIQUE IN SYMPLECTIC GEOMETRY (Intelligence of Low-dimensional Topology)

HOLOMORPHIC CURVE

_TECHNIQUE

IN SYMPLECTIC GEOMETRY

KAORU ONO RIMS, KYOTO UNIVERSITY

1. INTRODUCTION

In the middle of 1980' \mathrm{s}, Gromov introduced

pseudo‐holomorphic

curves in sym‐

plectic

manifolds and derived many

significant

results

[7].

For

example,

(1) (non‐)

squeezing

theoreml, (2)

non‐existence of embedded exact

_Lagrangian

submanifolds

in a

symplectic

vector_space,

₍₃₎

_description

of closed

_symplectic

four‐manifolds con‐

taining

a

symplectically

embedded

sphere

of

_{non‐negative}

self intersection number

(McDuff

developed

the

_theory

after Gromov’s seminal

_{work), (4)}

_homotopy

_typeof the

_{symplectic diffeomorphism}

_group of

_{(S^{2}, $\omega$)\times(S^{2}, $\omega$)}

and the fact that the fun‐ damental _group of that of

_{(S^{2}, $\omega$)\mathrm{x}(S^{2}, c $\omega$)}

, c>1, contains an element of infinite order. Around the same

time,

Conley

and Zehnder

proved

Arnold’s

_conjecture

for

fixed

_points

of Hamiltonian

_{diffeomoprhisms}

on tori

[2].

Fixed

points

of a Hamil‐ tonian

_{diffeomorphism correspond}

to

_1‐periodic

solutions of the

_{corresponding}

time‐

dependent

Hamiltonian_system, whichcan be

captured

as critical

points

ofacertain

functional on the

loop

_space

(the

least action

principle).

Conley

and Zehnder used

finite dimensional

_{approximation}

of the

_functional,

hence reduced the

_problem

in a

finite dimensional

_setting.

A formal

_computation

leads that the

_gradient

flow lines

of the functionalcan be

thought

of solutions of

Cauchy‐Riemann

_equation

perturbed

by

Hamiltonian term.

_Shortly

after these

_{works, combining}

the

_holomorphic

curve

technique

and the variational

_approach,

Floer initiated

_{\displaystyle \frac{\infty}{2}}

dimensional”

_analog

of Morse‐Novikov

_theory,

which is

_nowadays

called Floer

_theory

_[3].

The

_original

mo‐ tivation is Arnold’s

_conjecture

for fixed

_points

of Hamiltonian

_{diffeomorphisms.}

It

related to a

_question

on

Lagrangian

intersection

closely.

Since

then,

Floer

theory

has been

_developed

in various direction

_{including Heegaard}

Floer

_{theory presented}

in

_Tange’s

article inthis

_proceedings.

Inthis_note, wewould like topresent a

glimpse

ofthe method of

_holomorphic

curveswithout

going

into details

following

the lecture in the

_workshop.

1Gromovcalled this theoremassqueezingtheorem. However,itstates. _somehow,the ballcannot

be _“squeezed” to a_{symplectic cyclinder}of smaller width. So it is nowoften called non‐squeezing

2. A LITTLE PRELIMINARY

In this

_section,

we collect several

notion,

definitions in

symplectic

_geometry. \mathrm{A}

symplectic

structure on amanifold X is to

_equip

X with aclosed

non‐degenerate

2‐

form $\omega$

(symplectic form).

Namely,

d $\omega$=0 and $\omega$ induces the

following isomorphism

ofvector bundles:

v\in TX\mapsto i(v) $\omega$\in T^{*}X.

Thus $\omega$alsoinduces the one‐to‐one

correspondence

betweenvectorfieldsand 1‐forms. The mostbasic

_example

of

_symplectic

manifolds is the

_symplectic

_{vector space.} Let

(

x_{1},yl,. .._,x_{n}, y_{n}

)

be linear coordiinates on \mathbb{R}^{2n}. Write

$\omega$_{0}=\displaystyle \sum_{i=1}^{n}dx_{i}\wedge dy_{i}

.

Clearly

$\omega$_{0} is a

symplectic

form on \mathbb{R}^{2n}. Darboux’s theorem

guarantees

that any

symplectic

manifold is

_{locally diffeomorphic}

to a

symplectic

_{vector space. In} other

words,

at any

point

on

(X, $\omega$)

_, we can take local coordinates

(

_{x_{1},yl,}. ..

,x_{n}, y_{n}

),

in which $\omega$ is written as _{$\omega$_{0}.}

Other

_examples

include oriented surfaces

_equipped

with area

form,

_cotangent

bun‐ dle

_(with

a standard

symplectic

form

(dp\wedge dq'')

_, Kähler

manifolds,

in

particular,

complex projective

spaces.

Forasmooth functionhonX,wedefine the Hamiltonianvectorfield

X_{h}

associated

with h

_by

_{i(X_{h}) $\omega$=dh}

. On the

symplectic

vector space, the Hamiltonian vector

field associated with h is

_given

_by

X_{h}=\displaystyle \sum_{i=1}^{n}(\frac{\partial h}{\partial y_{i}}\frac{\partial}{\partial x_{i}}-\frac{\partial h}{\partial x_{i}}\frac{\partial}{\partial y_{i}})

.

By

Cartan’s

_formula,

_{X_{h}}

satisfies

_{\mathcal{L}_{X_{h}} $\omega$=0.}

A

_{diffeomorphism $\psi$}

of X is called a

symplectic

diffeomorphism

(symplectomor‐

phism),

if

_{$\psi$^{*} $\omega$= $\omega$}

. There is a

specific

class of

symplectic

diffeomorphism

called

Hamiltonian

_{diffeomorphisms.}

Let Hbe asmooth functionon

[0, 1]\times X

,

(with

com‐

pact

support).

We set

_{h_{t}=H(t, )}

.

Integrating

\{X_{h_{t}}\}

, we obtain an

isotopy

$\varphi$_{H}^{t}

with

_{$\varphi$_{H}^{0}=id_{X}}

. A

diffeomorphism $\psi$

of X iscalled aHamiltonian

diffeomorphism,

if

there exists H such that

_{$\psi$=$\varphi$_{H}^{1}}

. A

diffeomorphism

on an m‐dimensional manifold

is

_{locally expressed by}

m functions of m‐variables. A Hamiltonian

diffeomorphism

$\psi$

on a 2n‐dimensional

symplectic

manifold

_is,

in a _sense, described

by

a function on 2n‐variables.

(For

example,

if

$\psi$

is close to the

identity,

it is

locally

described

by

a so‐called

_generating

function.

Then the fixed

_points

are critical

_points

of the

generating

function.)

A submanifold S in

_{(X, $\omega$)}

is called

_symplectic,

if the restriction of $\omega$ to S is a

symplectic

form on S. S is called

_isotropic,

if the restriction of $\omega$ to S vanishes

everywhere

on S, in other

words,

TS\subset(TS)^{\perp_{ $\omega$}}

Here

S is called

_coisotropic,

if

_{(TS)^{1_{ $\omega$}}\subset TS}

. For an

isotropic

(resp. coisotropic)

sub‐

manifold,

we have

\dim S\leq

(resp.

\geq

)

\displaystyle \frac{1}{2}\dim X

. A

particularly

important

class of

submanifoldsisthat of

_Lagrangian

_{submanifolds,}

which are

isotropic

and

coisotropic.

Typical examples

of

_Lagrangian

submanifolds arethe

graph

of closed 1‐forms inthe

cotangent

bundle of a smooth manifold M_, the conormal bundle of a submanifold

of M in the

_cotangent

bundle of M, the real part of

non‐singular algebraic

variety

definied over \mathbb{R}, the

graph

ofa

symplectic

diffeomorphism

of

(X, $\omega$)

considered as a

submanifold in

_{(X\times X, - $\omega$\oplus $\omega$)}

.

Since the group of linear

symplectic

transformations

(linear

isomorphism

on a

symplectic

vector _space

_preserving

the

_symplectic

_structure)

contains the

_unitary

group as a maximal _compact

subgroup,

the _{structure group} of the _tangent bun‐

dle of a

symplectic

manifold can be reduced to the

_unitary

group. In other

words,

there exists an almost

complex

strcutre J

_(an

_endomrophism

ofthe _tangent bundle with J^{2}=-id

₎

suchthat

_{g_{J}(, )}

:= $\omega$

(, Je)

is a Riemannian metric on X

(almost

complex

structure

_compatible

with $\omega$

).

Moreover,

the_spaceof almost

complex

struc‐ tures

_compatible

with $\omega$ iscontractible. A_map

f

between almost

complex

manifolds

( $\Sigma$, j)

and

_{(X, J)}

is called J

_{‐holomorphic}

_(or

_simply

_{holomorphic),}

if the differen‐

tial

_df

is

_complex

linear with _respect to

_j

and J,

i.e.,

Jodf=df

oj.

In contrast to the fact that J

_{‐holomorphic}

functions are _rare, there are at least

_{locally plenty}

of

_holomorphic

curves.

Non‐integrability

of J_, measured

by

Nijenhuis

_tensor,

_gives

restrictionsfor J

_{‐holomorphic}

submanifolds. Inthecaseof real two‐dimension

(com‐

plex

one‐dimension),

Nijenhuis

tensor

_vanishes,

hence no restriction. In

particular,

we call

(the

image

of)

a

holomorphic

_map from the Riemann

sphere

a

holomorphic

sphere.

Moreover,

the deformation

_theory

of

_holomorphic

_mapsfrom closed Riemann surfaces

_(resp.

compact Riemann surface with

_{Lagrangian boundary}

_condition)

is

controlled

_by

_two‐step

_{elliptic complex}

_{(elliptic operator).}

The _compactness of the

moduli space is also estabilished in works

_starting

with

[7].

3. HOLOMORPHIC CURVES

In this

_section,

we discuss a

couple

of results in

[7]

and

_try

to

_give

_rough

ideas of

proofs.

3.1.

_{Non‐squeezing}

theorem.

_{Non‐squeezing}

theorem is a manifestation of_sym‐

plectic rigidity.

We

_briefly

presentits statement and

_implication

followed

_by

aflavor

of the

_proof.

Define the ball of radius R and a

cylinder

ofwidth R

by

B^{2n}(R)=\displaystyle \{(x_{i}, y_{i})\in \mathbb{R}^{2n}|\sum_{i=1}^{n}(x_{i}^{2}+y_{i}^{2})<R\},

Theorem.(non‐squeezing theorem)

Let

_$\psi$

:

B^{2n}(R)\rightarrow Z^{2n}(R')

be a

symplectic

embedding,

i.e.,

an

embedding

such that

$\psi$^{*}$\omega$_{0}|_{Z^{2n}(R')}=$\omega$_{0}|_{B^{2n}(R)}

. Then

R\leq R'.

Remark. If we

replace

the condition that

$\psi$

_preserves the

symplectic

structure

by

that

_$\psi$

preserves the volume

form,

or if we

replace

Z^{2n}(R')

by

another kind of

cylinder

defined

_by

_{x_{1}^{2}+x_{2}^{2}<R}

, the conclusion does not hold. As a

corollary

of

non‐squeezing theorem,

the

_following

result holds.

Theorem.

₍

C^{0}

‐rigidity

of

_symplectic

_{diffeomorphisms)}

Let

_{$\psi$_{n}}

be a _sequence of

symplectic diffeomorphisms

of

_{(\mathbb{R}^{2n}, $\omega$_{0})}

. If

$\psi$_{n}

converges to a

diffeomorphism‘ $\psi$

of

\mathbb{R}^{2n} in C^{0}

_‐topology,

_$\psi$

is a

symplectic diffeomorphism.

Avery

rough

sketch of the

proof

of

non‐squeezing

theoremgoes asfollows.

Firstly,

we take a

sufficiently large

L such

that,

after translation in

3‐rd,

. . . _,2n‐th coordi‐

nates,

$\psi$(B^{2n}(R))

is contained in

_{B^{2}(R)\mathrm{x}((L/4,3L/4))^{2n-2}}

. Embed

B^{2}(R')

to

S^{2}(A)

, the

sphere

of area

A= $\pi$ R^{2}\prime

. Then

B^{2n}(R)

is

symplectically

embedded in

S^{2}(A)\times \mathbb{R}^{2n-2}/L\mathbb{Z}^{2n-2}

. Denote

by

$\omega$ the

product symplectic

structure. Let

J_{0}

be

the standard

_complex

structure on \mathbb{R}^{2n} such that

J_{0}\displaystyle \frac{\partial}{\partial x_{i}}=\frac{\partial}{\partial y_{i}}, J_{0}\displaystyle \frac{\partial}{\partial y_{i}}=-\frac{\partial}{\partial x_{i}}

. Pickan

almost

_complex

structure J

_compatible

with $\omega$ which is an extension of

$\psi$_{*}(J_{0})

and

coincides with

_{J_{0}}

outside

_{S^{2}(A)\times[L/4, 3L/4]^{2n-2}}

.

(Here

we

regard

[L/4,3L/4]^{2n-2}

as a subset of

\mathbb{R}^{2n-2}/L\mathbb{Z}^{2n-2}

. Then we can find J

‐holomorphic sphere

S^{2}(A)\times\{p\}

with

_{p\not\in[L/4, 3L/4]^{2n-2}}

.

Using

the deformation

theory

and compactness theorem

for

_holomorphic

_curves, we canshowthatthere is a

family

of

holomorphic spheres

in

the same

homology

class_{sweep the} whole _space

S^{2}(A)\times \mathbb{R}^{2n-2}/L\mathbb{Z}^{2n-2}

. In

particu‐

lar,

there is a

holomorphic sphere

S

passing

through

$\psi$(O)

, where O is the

origin

of

B^{2n}(R)

. The

monotoniciy

formula for the areaof

holomorphic

curves, the

symplectic

area of S, which is A, is at least

$\pi$ R^{2}

. Hence

$\pi$ R^{2}\leq A

. Since $\epsilon$>0 canbe

arbitrary

small,

we obtain

R\leq R'.

3.2. Non‐existence ofexact

_Lagrangian

submanifold in

_{(\mathbb{R}^{2n}, $\omega$_{0})}

. A

symplec‐

ticmanifold

_{(X, $\omega$)}

iscalledexact, if $\omega$isanexact

_2‐form,

_i.e.,

$\omega$=d $\lambda$ forsome1‐form

$\lambda$. A

typical example

is

(\mathbb{R}^{2n}, $\omega$_{0})

, where

$\omega$_{0}=d(\displaystyle \sum_{i=1}^{n}x_{i}dy_{i})

, for

example.

Cotan‐

gent bundles with the standard

_symplectic

_structure, Liouville

_domains,

which are

generalization

ofconvexdomains in

(\mathbb{R}^{2n}, $\omega$_{0})

are also exact

_symplectic

manifolds. \mathrm{A}

Lagrangian

submanifold L in

_{(X, $\omega$=d $\lambda$)}

iscalled exact, if the restriction of $\lambda$ is an exact 1‐form on L. Note that the

Lagrangian

condition

implies

that the restriction

of $\lambda$ to L is aclosed 1‐form. The

following

result is also due to Gromov

[7].

Theorem.

_{(non‐existence}

ofexact

_Lagrangian

_{submanifolds)}

Let L be a closed

In

_particular,

this theorem

_implies

thataclosed embedded

Lagrangian

submanifold

in

_{(\mathbb{R}^{2n}, $\omega$_{0})}

has non‐zero first Betti number. For

example,

the

sphere

S^{n} cannot be

embedded in

_{(\mathbb{R}^{2n}, $\omega$_{0})}

as a

Lagrangian

submanifoldfor n>1. It contrast tothefact

that

S^{3}

canbe embedded in \mathbb{C}^{3} as a

totally

real submanifold

[1], [8]

_{page 193.} Here is a

rough

sketch of the

proof.

In order to show that

_{$\lambda$|_{L}}

is not _exact, we will find a

loop

_{$\gamma$ in L} such that

\displaystyle \int_{ $\gamma$} $\lambda$\neq

O. It suffices to find a non‐constant

holomorphic

disc u in

(\mathbb{R}^{2n}, J_{0})

with

boundary

on L, since

\displaystyle \int_{\partial D^{2}} $\lambda$=\int_{D^{2}}u^{*}$\omega$_{0}>0,

where the first

_equality

is due to Stokes’ formula and the second

_inequality

follows from

_{compatibility}

of

_{J_{0}}

and $\omega$_{0} as well as

non‐constancy

of u. The main

body

of

the

_proof

is to find such a

holomorphic

disc.

Pick and fix a \in \mathbb{C}^{n} with

\Vert

a 1 and

p_{0}\in L

. Consider the

pair

of u :

(D^{2}, \partial D^{2})\rightarrow(\mathbb{C}^{n}, L)

ands\in \mathbb{R}

_satisfying

_{u(1)=p_{0},}

_{u_{*}[D^{2}, \partial D^{2}]=0}

in

_{H_{2}(\mathbb{C}^{n}, L;\mathbb{Z})}

and

(1)

\displaystyle \frac{\partial u}{\partial\overline{z}}:=\frac{1}{2}(\frac{\partial u}{\partial x}+J_{0}\frac{\partial u}{\partial y})=s\cdot \mathrm{a}.

When

_s=0,

u must be the constant map to p_{0}. This is because a

holomorphic

map u with

\displaystyle \int_{D^{2}}$\omega$_{0}=0

must be a _{constant map.} When s is

sufficiently large,

there

are no solutions u for

(1).

The reason is the

following.

If u is asolution of

(1),

u is

harmonic, i.e.,

\triangle u=0. Hence

\displaystyle \frac{\partial u}{\partial\overline{z}}

is also harmonic.

By

the mean value theorem for

harmonic functions and Stokes’

_formula,

wefind that

\displaystyle \frac{\partial u}{\partial\overline{z}}(0)=\frac{1}{\int_{D^{2}}dxdy}\int_{D^{2}}\frac{\partial u}{\partial\overline{z}}dxdy=\frac{-\sqrt{-1}}{\int_{D^{2}}dxdy}\int_{\partial D^{2}}

udz.

Let D=\displaystyle \max

_{p\Vert|p\in L\}}

. Then the norm of the

right

hand side is bounded

by

2D, while the left hand side is s\cdot \mathrm{a}.

Hence,

if s>2D

, the

equation

(1)

has no

solution.

The energy of a _map u is defined

by

E(u)=\displaystyle \frac{1}{2}\int_{D^{2}}\Vert du\Vert^{2}dxdy

.

By

a

simple

computation,

we find that

E(u)=\displaystyle \int_{D^{2}}u^{*}$\omega$_{0}+\int_{D^{2}}\Vert\frac{\partial u}{\partial\overline{z}}\Vert^{2}dxdy.

Since the firsttermon the

right

hand side vanishes and the second term isbounded

by

4D^{2} from the

_above,

_E(u)

is

_uniformly

bounded. Now the _compactness

_argument

(Gromov’s

compactness, removal of

_{singularities,}

_etc.)

_yields

the

_following.

When theenergy is

uniformly

bounded and s_{n}converges to s_{\infty}, thesequenceu_{n}of solutions

for

₍₁₎

with _{s=s_{n} converges}to asolution _{u_{\infty}} of

(1)

with _{s=s_{\infty} away} from afinite

number of

_points.

_Rescaling

u_{n}

suitably

around these

finitely

many

points,

the new_{sequence converges to} anon‐constant

holomorphic

_map from either a Riemann

compact

holomorphic

curves in \mathbb{C}^{n}_, hence the

only

possibility

is a

holomorphic

disc.

By

the deformation

_theory

and

_{bubbling‐off}

_argument of

_holomorphic

_{curves, if}no

bubble appear

during

0\leq s\leq 2D, the space of solutions

(s, u)

of

(1)

is a one‐

dimensional manifold and the constant solution to p_{0} at s=0 is deformed to a

solution at s=2D, which is acontradiction. Therefore a non‐constant

holomorphic

disc v :

(D^{2}, \partial D^{2})\rightarrow(\mathbb{C}^{n}, L)

must _appear as abubble between s=0 and s=4D^{2}.

Remark.

_Combining

the_argumentabove with

_{(figure‐eight trick”,}

Gromov derived the

_following

result on

Lagrangian

intersection. The condition for tameness of a

symplectic

manifold_gurantees

_good

control of

_holomorphic

curves at theend of the

symplectic

manifold. For

_example,

_{(\mathbb{R}^{2n}, $\omega$_{0})}

,

cotangent

bundles

equipped

with the

standard

_symplectic

_structure, Liouville domains are tame

_symplectic

manifolds. Theorem.

_(persistence

of

_Lagrangian

_{intersection)}

Let

_{(X, $\omega$)}

be an exact sym‐

plectic

manifold. Let L be a closed embedded exact

_Lagrangian

submanifold and

_$\psi$

a Hamiltonian

diffeomorphism

of

(X, $\omega$)

. Then we have

L\cap $\psi$(L)\neq\emptyset.

4. NAIVE IDEA OF LAGRANGIAN FLOER THEORY

In this

_section,

we

explain

Floer’s idea _very

briefly.

Let

L,

L be closed embedded

Lagrangian

submanifolds in aclosed

symplectic

manifold

(X, $\omega$)

such that L and L'

intersects

_{transversally.}

In

_good

_situations,

we can define a

complex

(CF^{\cdot}(L, L), $\delta$)

,

where

_{CF^{\cdot}(L', L)}

is

_{generated by}

intersection

_points

of L and L, such that the

resulting cohomology

is invariant under Hamiltonian deformations ofL'. Let us consider the_space of

paths

from L to L‘:

\mathcal{P}(L, L)=\{ $\gamma$ : [0, 1]\rightarrow X| $\gamma$(0)\in L, $\gamma$(1)\in L\}.

Define a “1‐form” $\alpha$ on

\mathcal{P}(L, L)

by

$\alpha$_{ $\gamma$}( $\xi$):=\displaystyle \int_{0}^{1} $\omega$( $\xi$(t),\dot{ $\gamma$}(t))dt,

where

_{\dot{ $\gamma$}}

is the

_velocity

vectorof _$\gamma$ and

_$\xi$

is a

_tangent

vectorof

_{\mathcal{P}(L, L')}

at _$\gamma$, in other

words,

$\xi$

is asection of

$\gamma$^{*}TX

such that

$\xi$(0)

(resp. $\xi$(1) )

_tangents

to L

_(resp.

L

We can see that $\alpha$ is an “exact 1‐form” as follows. Fix

$\gamma$_{0}\in \mathcal{P}(L, L')

and consider $\gamma$ close to $\gamma$_{0}. Then there is a map w :

[0

,1

]

\mathrm{x}[0, 1]\rightarrow X

such that

w(0, t)=$\gamma$_{0}(t)

,

w(1, t)= $\gamma$(t)

,

w(s, 0)\in L

and

w(s, 1)\in L

. For $\gamma$

sufficiently

close to $\gamma$_{0}_, we can

find such wwith the

image

closeto_{$\gamma$_{0}}. Suchw is

unique

up to

homotopy

respecting

the

_boundary

condition above. Then we set

A direct

_computation

shows that

d\mathcal{A}^{loc}= $\alpha$

on a

neighborhood

of $\gamma$_{0}. Thus $\alpha$

has a local

primitive

function around any $\gamma$_{0}_, hence $\alpha$ is \mathrm{a} (closed 1‐form” The

function

\mathcal{A}^{loc}

may not be extended to a

globally

well‐defined function on

\mathcal{P}\underline{(}L,

L

).

However,

we have a well‐defined function \mathcal{A} on a suitalble

_covering

_space

\mathcal{P}(L, L')

of

_{\mathcal{P}(L,}

L

_(For

any closed 1‐form $\eta$, there exists a

covering

space on which the

pull‐back

of _$\eta$ becomes

_exact.)

We mimick Morse

_complex

_(or

Novikov

_complex

of

a closed

1‐form),

although

we cannot follow the construction in finite dimension

by

the

_following

reasons.

(1)

the

gradient

flow lines

(see below)

_{may not exist}

_passing

through

a

_{given point}

(no

gradient

flow), (2)

the Hessian of the function has

_infinitely

many

positive

and

negative

subspaces

(Morse

index must be

replaced

by something

else).

For each intersection

_point

_{p\in L\cap L}

, we take a

path

$\Lambda$(t)

in the space of

Lagrangian

subspaces

joining

T_{p}L

to

_{T_{p}L}

‘ _and

can

assign

Maslov‐ViterUo index.

(It

depends

on the

path

$\Lambda$(t)

.

Sometimes,

we can take these

paths

at each intersection

points

in a coherent

way.)

We use Maslov‐Viterbo index in

place

of Morse indexin

finite dimensionalcase.

Using

the Riemannian _{metric g_{J}}, we can define an inner

product

on the tangent space

T_{ $\gamma$}\mathcal{P}(L, L')

by

\displaystyle \{$\xi$_{1}, $\xi$_{2}\}:=\int_{0}^{1}g_{J}($\xi$_{1}(t), $\xi$_{2}(t))dt.

With _respect to this inner

_product,

a formal

computation

yields

that the

‘(gradient

vector field”’ of\mathcal{A} is

_given

_by

\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathcal{A}( $\gamma$)=-J\dot{ $\gamma$}.

Wecan

interpret

gradient

flow

_trajectories

assolutions of the

Cauchy‐Riemann

_equa‐ tion for

_{u:\mathbb{R}\times[0, 1]\rightarrow X}

with

_{u(\mathbb{R}\times\{0\})\subset L}

and

_{u(\mathbb{R}\times\{1\})\subset L'}

(2)

\displaystyle \frac{\partial u}{\partial $\tau$}( $\tau$, t)+J(u)\frac{\partial u}{\partial t}( $\tau$, t)=0.

Note that the

_equation

₍₂₎

may not have asolution for a

given

_$\gamma$ such that

u(0, t)=

$\gamma$(t)

. Inother

words,

theremaynotexist\mathrm{a}

(‘gradient

flow

trajectory”’

passing

through

$\gamma$. A compactness argument for

holomorphic

curves

implies

that for a solution u

of

_(2),

the _energy

_E(u)

is finite if and

_only

if

_{\displaystyle \lim_{ $\tau$\rightarrow\pm\inf ty}\underline{u(} $\tau$,}

t

₎

_=p^{\pm}

for some

p^{\pm}\in L\cap L'

. We call suchu aFloer

trajectory.

We denote

by

M(p^{-},p^{+})

thespace of

solutions of

₍₂₎

such that

_{\displaystyle \lim_{ $\tau$\rightarrow\pm\infty}u( $\tau$, t)=p^{\pm}}

. Since

(2)

is invariant under theshift

in $\tau$

‐direction,

\mathbb{R} acts on

\overline{\mathcal{M}}(p^{-},p^{+})

. We write its

quotient

space

by

\mathcal{M}(p^{-},p^{+})=

\overline{\mathcal{M}}(p^{-},p^{+})/\mathbb{R}

. Floer

coboundary

operator $\delta$ :

CF^{\cdot}(L, L)\rightarrow CF^{+1}(L', L)

is

defined2

2\mathrm{I}\mathrm{n}ordertodefine $\delta$,it isnecessary tomakesenseof thecardinalityof\mathcal{M}^{\dim=0} using perturbation

of J

_{(some cases) and/or}

abstract_{pertrubation.} In_{generalsituation,p\in L\cap L'} should be_replaced bythe inverse_imagesinthe_{coveringspace,just}asinNovikovtheoryfor closed 1‐formson afinite

by

counting

number of Floer

_{trajectories joining}

_{p^{-} and}

_p^{+}

$\delta$ p^{-}=\displaystyle \sum\#_{2}\mathcal{M}^{\dim=0}(p^{-},p^{+})p^{+}.

Here

_{\#_{2}\mathcal{M}^{\dim=0}(p^{-},p^{+})}

is the

_cardinality

modulo 2 of0‐dimensional _components of

\mathcal{M}(p^{-},p^{+})

. In

general,

\mathcal{M}(p^{-}, p^{+})

is not

canonically

oriented. In the case that L

and L‘

are

equipped

with

_spin

structures

_(or

more

generally

(L, L')

is a

relatively

spin

pair),

then

_{\mathcal{M}(p^{-},p^{+})}

areoriented in aconsistent way

[5].

Inordertoshow that

$\delta$ 0 $\delta$=0, we

study

the ends of 1‐dimensional components of

\mathcal{M}(p^{-},p^{+})

. There are

the

_following

_{possibilities}

of ends. The first

_type

is

_splitting

intoseveral Floer

_trajec‐

tories. This is similar to limit behavor of

_gradient

_trajectories

in finite dimensional Morse

_theory.

The second

_type

is

_{bubbling‐off}

of

_holomorphic

discs at the

_boundary

of

_{\mathbb{R}\times[0}

,1

]

. There may also

happen

bubbling‐off

of

holomorphic spheres.

However

it occurs in real codimension

2,

while the second _type occurs in real codimension 1. Hence the last _type canbe excluded

by

pertubation

of J

(under

_certain

_assumption

ofX and

L,

L

₎

or abstract

_perturbation

_technique.

_We can see the

bubbling‐off

of

holomorphic

discs inthe

_{following simple}

local

_example.

Let L=\mathbb{R} and L' the unit

circle around the

_origin

in \mathbb{C}. Then consider Floer

trajectories

from -1 to itself. In

this case,

\mathcal{M}(-1, -1)

is an interval

(-1,1)

. The

boundary

point

1

corresponds

to

splitting

into two Floer

_trajectories

_(upper

hemidisc as Floer

_trajectory

from -1 to 1 and lower hemidisc as Floer

_trajectory

from 1 to -1

):

The

_boundary

_point

-1

corresponds

to

_{bubbling‐off}

ofa

holomorphic disc,

i.e.,

aconstantFloer

_trajectory

at -1with the unitdisc as a

holomorphic

disc bubble. In

fact,

we can seethat

$\delta$ 0 $\delta$\neq 0

in this case.

Ifwe can exclude the

bubbling‐off

of

holomorphic discs,

we can see $\delta$\circ $\delta$=0 and

obtain Floer cochain

_complex

_{(CF\cdot(L', L), $\delta$)}

. The

resulting cohomology

denoted

by

HF^{\cdot}(L', L)

is called Floer

_cohomology

of

_L,

L . Wecanalso show that

HF^{\cdot}(L, L)\cong

HF^{\cdot}( $\psi$(L), L)

for aHamiltonian

diffeomorphism $\psi$

such that L and

$\psi$(L')

intersects

transversally.

If we can _compute Floer

cohomology,

we can

give

a lower bound of

the number ofintersection

_points

_provided

_they

intersect

_{transversally.}

Floer

_[3]

realized these lines of ideas under the condition that

_{$\pi$_{2}(X, L)=0}

and

L'= $\psi$(L)

forsome Hamiltonian

diffeomorphism $\psi$

. Wecan also

study

Floer

theory

for Hamiltonian

_{diffeomorphisms.}

_(By

_taking

the

_graph

ofa Hamiltoniandiffeomor‐

phism,

it can bealso _put in

Lagrangian

intersection

setting.)

In

[4],

Floer succeeded

the construction for a Hamiltonian

diffeomorphism

on monotone

symplectic

mani‐

folds. Here

_monotonicity

is that the first Chern class is

_{positively proportional}

to

the de Rham

_cohomology

class

_represented

_by

the

_symplectic

form.

_Yong‐Geun

Oh considered an

analogous

situation in

Lagrangian

intersection

_setting

and

_performed

Since the_righthand side of the definition of $\delta$_maybeaninfinitesum. Soweallow the infinitesum as longasthe value of the function\mathcal{A} _{grows to +\infty.}

the construction of Floer

_cohomology

of

_{(L, L')}

under the condition that

_L,

L' are monotone

_Lagrangian

submanifolds3

with minimal Maslov number >2.

Later,

he

obtained the construction for monotoen L and

_{L'= $\psi$(L)}

with minimal Maslov number 2.

Aswementioned

before,

we _{may not}have $\delta$ 0 $\delta$=0. Suchanobstruction is caused

by

bubbling‐off

of

_holomorphic

discs. In order to understand

_{obstructions,}

we have to

_study

all

_holomorphic

discsin a

systematic

_way. Thisisdone

by Fukaya, Oh,

Ohta

and the author

_[5].

We formulate it in terms of

_{filtered A_{\infty} ‐algebra}

associated with

Lagrangian

submanifolds.

_(We

cannot

_explain

_terminology

here and would like to inviteinterested readersto

_[5].)

If the Maurer‐Cartan

_equation

admits weaksolutions for L and L' with the same

potential

value,

then we can

rectify

$\delta$ to a differential

of

_{CF^{\cdot}(L, L)}

anddefine Floer

_cohomology

_depends

on

(weak)

solutions of Maurer‐

Cartan

_equations.

The

_{resulting cohomology}

is also invariant under Hamiltonian deformations of L' in a suitable sense.

Another way to deform Floer

coboundary

operator

is bulk

_{deformations. Namely,}

we candeformall constructions such as filtered

A_{\infty}

‐algebras,

filtered

A_{\infty}‐bimodules,

etc.

_using

an

cycle

in X. These

machinery

may be considered as too abstract.

However,

happily enough,

we can see the

efficiency

of all these

machinery

in

appli‐

cation to concrete

_examples

such as

Lagrangian

torus fibers in

_compact

Kählertoric

manifolds,

see, e.g., a_survey article

[6].

REFERENCES

[1]

P.Ahern and W. Rudin, Totallyreal_{embeddings of}S^{3} in\mathrm{C}^{3}, Proc.Amer. Math. Soc.

94

_(1985),

460‐462

[2]

C. C. _Conleyand E. _Zehnder, The_BirkhoffLewts_{fixed point}theorem anda_conjecture

by V. I. _Arnold, Invent. Math.

_73(1983),

33‐49.

[3] A. _Floer, Morse _{theoryfor Lagrangian intersections,} J. Differential Geom.

_28(1988),

513‐547.

[4]

A. Floer, Holomorphic spheres and _{symplectic fixed points,} Comm. Math. _Phys.

120(1989),

576‐611.

[5]

K. _Fukaya, Y.‐G. _Oh, H. Ohta and K. _{Ono, Lagrangian} intersection Floer _theory, anomaly and_obstruction, Part Iand_II,

_AMS/IP

StudiesinAdvanced_{Math., 46‐1,2,} Amer. Math. Soc. and InternationalPress, 2009.

[6]

K. _Fukaya,Y.‐G. _Oh, H. Ohtaand K. _{Ono, Lagrangian}Floer_theory on compacttoric

manifolds, survey,InSurveysinDifferentialGeometryVol.XVII, InternationalPress,

2012, 229‐298. 231‐268.

[7]

M. _{Gromov, Pseudoholomorphic} curves in symplectic manifolds, Invent. Math. 82

(1985),

307‐347.

[S]

M. _Gromov,Partial Differentail_{Relations, Springer Verlag,} 1986.

3ALagrangiansubmanifold L is_monotone, if the Maslov class of L is_{positivelyproportional}to

Research Institute for Mathematical Sciences

Kyoto University

Kyoto

606‐8502

Japan

Email address:

_{[email protected]‐u.ac.jp}