47

数理解析研究所講究録
第2004巻 2016年 47-56

HOLOMORPHIC CURVETECHNIQUE IN SYMPLECTICGEOMETRYKAORU 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 type ofthe 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, which can be captured as critical points of a certain
functional on the loop space (the least action principle). Conley and Zehnder usedfinite dimensional approximation of the functional, hence reduced the problem in a
finite dimensional setting. A formal computation leads that the gradient flow linesof the functional can 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 ofMorse‐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 in this proceedings. In this note, we would like to present a glimpseof the method of holomorphic curves without going into details following the lecturein the workshop.
1Gromov called this theorem as squeezing theorem. However, it states. somehow, the ball cannotbesqueezed to a symplectic cyclinder of smaller width.

theorem.

So it is now often called non‐squeezing

48

2. A LITTLE PRELIMINARY

In this section, we collect several notion, definitions in symplectic geometry. \mathrm{A}
symplectic structure on a manifold X is to equip X with a closed non‐degenerate 2‐
form $\omega$ (symplectic form). Namely, d $\omega$=0 and $\omega$ induces the following isomorphismof vector bundles:v\in TX\mapsto i(v) $\omega$\in T^{*}X.
Thus $\omega$ also induces the one‐to‐one correspondence between vector fields and 1‐forms.
The most basic example of symplectic manifolds is the symplectic vector space. Let( x_{1} yl,
,

.

.

.

,

x_{n}, y_{n} ) be linear coordiinates on \mathbb{R}^{2n}$\omega$_{0} is a symplectic form on \mathbb{R}^{2n}.

.

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

Clearly

Darbouxs theorem guarantees that any symplectic

manifold is locally diffeomorphic to a symplectic vector space.
any point on (X, $\omega$) , we can take local coordinates
written as $\omega$_{0}.

(x_{1} , yl,

.

.

.

,

In other words, atx_{n}, y_{n} ), in which $\omega$ is

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.
For a smooth function h on X , we define the Hamiltonian vector field X_{h} associatedwith h by i(X_{h}) $\omega$=dh.

On the symplectic vector space, the Hamiltonian vector

field associated with h is given by

X_{h}=\displayst le\sum_{i=1}^{n}(\frac{\parti lh}{\parti ly_{i}\frac{\parti l}{\parti lx_{i}-\frac{\parti lh}{\parti lx_{i}\frac{\parti l}{\parti ly_{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 H be a smooth function on [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 is called a Hamiltonian 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 describedaso‐calledbygenerating function. Then the fixed points are critical points of thegenerating 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
.

,

.

,

.

.

,

(T_{p}S)^{\perp_{ $\omega$}}=\{v\in T_{p}X| $\omega$(v, w)=0, \forall w\in T_{p}S\}.

49

S is called coisotropic, if (TS)^{1_{ $\omega$}}\subset TS

manifold,

wehave \dim S\leq(resp.

\geq ).

For an isotropic\displaystyle \frac{1}{2}\dim X.

(resp. coisotropic) sub‐A particularly important class ofsubmanifolds is that of Lagrangian submanifolds, which are isotropic and coisotropic.
Typical examples of Lagrangian submanifolds are the graph of closed 1‐forms in the

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 of a symplectic diffeomorphism of (X, $\omega$) considered as asubmanifold in (X\times X, - $\omega$\oplus $\omega$).

symplectic transformations (linear isomorphism on a
symplectic vector space preserving the symplectic structure) contains the unitarygroup 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 of the tangent bundle
Since the group of linearwith J^{2}=-id ) such that g_{J}(, ) := $\omega$ (, Je) is a Riemannian metric on X (almost

complex structure compatible with $\omega$ ). Moreover, the space of almost complex struc‐tures compatible with $\omega$ is contractible. 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 ,to the fact that J ‐holomorphic functionsof holomorphic curves.

arei.e., Jodf=df oj. In contrast

rare, thereareat least

locally plenty

Non‐integrability of J , measured by Nijenhuis tensor, gives

restrictions for J ‐holomorphic submanifolds. In the case of real two‐dimension (com‐plex one‐dimension), Nijenhuis tensor vanishes, hence no restriction. In particular,call (the image of) a holomorphic map from the Riemann sphere a holomorphic
sphere. Moreover, the deformation theory of holomorphic maps from closed Riemannsurfaces (resp. compact Riemann surface with Lagrangian boundary condition) is
controlled by two‐step elliptic complex (elliptic operator). The compactness of themoduli space is also estabilished in works starting with [7].
we3. HOLOMORPHIC CURVES
In this section, we discuss a couple of results in [7] and try to give rough ideas of

proofs.

Non‐squeezing theorem. Non‐squeezing theorem is a manifestation of sym‐
plectic rigidity. We briefly present its statement and implication followed by a flavorof the proof. Define the ball of radius R and a cylinder of width R by
3.1.

B^{2n}(R)=\displaystyle \{(x_{i}, y_{i})\in \mathb {R}^{2n}|\sum_{i=1}^{n}(x_{i}^{2}+y_{i}^{2})<R\},Z^{2n}(R)=\{(x_{i}, y_{i})\in \mathbb{R}^{2n}|x_{1}^{2}+y_{1}^{2}<R\}=B^{2}(R)\times \mathbb{R}^{2n-2}50Theorem.(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'.
.

If we replace the condition that $\psi$ preserves the symplectic structureby that $\psi$ preserves the volume form, or if we replace Z^{2n}(R') by another kind ofcylinder 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.
Remark.

,( 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
Theorem.

.

\mathbb{R}^{2n} in C^{0} ‐topology, $\psi$ is a symplectic diffeomorphism.

A very rough sketch of the proof of non‐squeezing theorem goes as follows.
wetake a sufficiently large L such that, after translation in 3‐rd,

.

.

.

,

Firstly,

2n ‐th coordi‐$\psi$(B^{2n}(R)) is contained in B^{2}(R)\mathrm{x}((L/4,3L/4))^{2n-2} Embed B^{2}(R') toS^{2}(A) the sphere of area A= $\pi$ R^{2}\prime Then B^{2n}(R) is symplectically embedded inS^{2}(A)\times \mathbb{R}^{2n-2}/L\mathbb{Z}^{2n-2} Denote by $\omega$ the product symplectic structure. Let J_{0} benates,.

.

,.

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

J_{0}\displaystyle\frac{\partial}{\partialx_{i} =\frac{\partial}{\partialy_{i} , J_{0}\displaystyle \frac{\partial}{\partial y_{i} =-\frac{\partial}{\partial x_{i} Pick.

analmost complex structure J compatible with $\omega$ which is an extension of $\psi$_{*}(J_{0}) andcoincides with J_{0} outside S^{2}(A)\times[L/4, 3L/4]^{2n-2}
as asubset of \mathbb{R}^{2n-2}/L\mathbb{Z}^{2n-2}

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

.

.

.

(Here we regard [L/4,3L/4]^{2n-2}

Then we can find J‐holomorphic sphere S^{2}(A)\times\{p\}Using the deformation theory and compactness theorem

for holomorphic curves, we can show that there is a family of holomorphic spheres inthe 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 ofB^{2n}(R)area.

The monotoniciy formula for the area of holomorphic curves, the symplecticof S which is A is at least $\pi$ R^{2}

small,

,

we

,

.

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

Since $\epsilon$>0 can be arbitraryobtain R\leq R'.

3.2. Non‐existence of exact Lagrangian submanifold in (\mathbb{R}^{2n}, $\omega$_{0}).

A symplec‐tic manifold (X, $\omega$) is called exact, if $\omega$ is an exact 2‐form, i.e., $\omega$=d $\lambda$ for some 1‐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 of convex domains in (\mathbb{R}^{2n}, $\omega$_{0}) are also exact symplectic manifolds. \mathrm{A}Lagrangian submanifold L in (X, $\omega$=d $\lambda$) is called exact, if the restriction of $\lambda$ is anexact 1‐form on L Note that the Lagrangian condition implies that the restrictionof $\lambda$ to L is a closed 1‐form. The following result is also due to Gromov [7].
.

Theorem.

(non‐existence of exact Lagrangian submanifolds)

embedded Lagrangian submanifold in (\mathbb{R}^{2n}, $\omega$_{0}).

Let L be a closed

Then L is not exact.

51

In particular, this theorem implies that a closed embedded Lagrangian submanifold(\mathbb{R}^{2n}, $\omega$_{0}) has non‐zero first Betti number. For example, the sphere S^{n} cannot beinembedded in (\mathbb{R}^{2n}, $\omega$_{0}) as a Lagrangian submanifold for n>1.

It contrast to the factthat S^{3} can be embedded in \mathb {C}^{3} as a totally real submanifold [1], [8] page 193.

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_{$\gam a$}$\lambda$\neq O. It suffices to find a non‐constantHere isaholomorphic disc u in (\mathbb{R}^{2n}, J_{0}) with boundary on L,

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

sincewhere 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\Vert a\in \mathbb{C}^{n} with

a

1 and

p_{0}\in L.

Consider thepair of u:(D^{2}, \partial D^{2})\rightarrow(\mathbb{C}^{n}, L) and s\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\displaystyle\frac{\partialu}{\partial\overline{z}:=\frac{1}{2}(\frac{\partialu}{\partialx}+J_{0}\frac{\partialu}{\partialy})=s\cdot\mathrm{a}.

(1)When s=0, u must be the constant map to p_{0}map u with\displaystyle \int_{D^{2} $\omega$_{0}=0.

This is because a holomorphic

must be a constant map. When s is sufficiently large, there

solutions u for (1). The reason is the following. If u is a solution of (1), u is

are noharmonic, i.e., \triangle u=0.

\displayte\frac{ptialu}{\partil\overin{z} is also harmonic. By the

Hence

mean

value theorem forharmonic functions and Stokes formula, we find thatLet D=\displaystyle \max

by 2D

,

\displaystyle\frac{\partialu}{\partial\overline{z}(0)=\frac{1}{\int_{D^{2}dx y}\int_{D^{2}\frac{\partialu}{\partial\overline{z}dx y=\frac{-\sqrt{-1}{\int_{D^{2}dx y}\int_{\partialD^{2}
p\Vert|p\in L\}

.

udz.

norm of the right hand side is bounded
Hence, if s>2D the equation (1) has noThen thewhile the left hand side is s\cdot \mathrm{a}.

,solution.
The energy of a map u is defined bycomputation, we find that

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

By a simple

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

Since the first term on the right hand side vanishes and the second term is boundedby 4D^{2} from the above, E(u) is uniformly bounded. Now the compactness argument(Gromovs compactness, removal of singularities, etc.) yields the following. Whenthe energy is uniformly bounded and s_{n} converges to s_{\infty} the sequence u_{n} of solutions,for (1) with s=s_{n} converges to a solution u_{\infty} of (1) with s=s_{\infty} away from a finite

Rescaling u_{n} suitably around these finitely many points, thesequence converges to a non‐constant holomorphic map from either a Riemannsphere or a disc with Lagrangian boundary condition. In our setting, there are no
number of points.
new

52

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 nobubble 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 toasolution at s=2D , which is a contradiction. Therefore a non‐constant holomorphic
disc v : (D^{2}, \partial D^{2})\rightarrow(\mathbb{C}^{n}, L) must appear as a bubble between s=0 and s=4D^{2}.

Remark.

Combining the argument above with (figure‐eight trick, Gromov derived
following result on Lagrangian intersection. The condition for tameness of a
symplectic manifold gurantees good control of holomorphic curves at the end of thesymplectic manifold. For example, (\mathbb{R}^{2n}, $\omega$_{0}) cotangent bundles equipped with thestandard symplectic structure, Liouville domains are tame symplectic manifolds.
the,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 THEORYIn this section, we explain Floers idea very briefly. Let L, L be closed embedded
Lagrangian submanifolds in a closed 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 theresulting cohomology is invariant under Hamiltonian deformations of L'.
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$\alpha$_{$\gam a$}($\xi$):=\displaystyle\int_{0}^{1}$\omega$($\xi$(t),\dot{$\gam a$}(t) dt,
where \dot{$\gam a$} is the velocity vector of $\gamma$ and $\xi$ is a tangent vector of \mathcal{P}(L, L') at $\gamma$ , in otherwords, $\xi$ is a section of $\gamma$^{*}TX such that $\xi$(0) (resp. $\xi$(1) ) tangents to L (resp.

We can see that $\alpha$ is an exact 1‐formasL

follows. Fix $\gamma$_{0}\in \mathcal{P}(L, L') and consider[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$\gamma$ close to $\gamma$_{0}Then there is a map w :

.

,

,

,

.

,

find such w with the image close to $\gamma$_{0} Such w is unique up to homotopy respecting
the boundary condition above. Then we set.

\displaystyle \mathcal{A}^{loc}( $\gam a$):=\int_{[0,1]\times[0,1]}w^{*} $\omega$.

53

computation shows that d\mathcal{A}^{loc}= $\alpha$ on a neighborhood of $\gamma$_{0} Thus $\alpha$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
L ).
A direct

has

a

.

local

,

\mathcal{P}\underline{(}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 theofpull‐back$\eta$ 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 somethingelse). 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. (Itdepends 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 index in,,.

finite dimensional case.

Using the Riemannian metric g_{J}by,define an inner product on the tangent

we canspace T_{ $\gamma$}\mathcal{P}(L, L')\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(gradientvector field of \mathcal{A} is given by\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathcal{A}( $\gamma$)=-J\dot{ $\gamma$}.
We can interpret gradient flow trajectories as solutions 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'\displaystyle \frac{\partial u}{\partial $\tau$}( $\tau$, t)+J(u)\frac{\partial u}{\partial t}( $\tau$, t)=0.

(2)

Note that the equation (2) may not have a solution for a given $\gamma$ such that$\gamma$(t)$\gamma$.

.

A compactness argument for holomorphic curves implies that for a solution(2),p^{\pm}\in L\cap L'ofu(0, t)=In other words, there may not exist \mathrm{a} (gradient flow trajectory passing throughE(u)the energy.

is finite if and

if

only

\displaystyle \lim_{ $\tau$\rightar ow\pm\inf ty}\underline{u(} $\tau$,t)=p^{\pm}

for

u

some

We call such u a Floer trajectory. We denote by M(p^{-},p^{+}) the space ofsolutions of (2) such that \displaystyle \lim_{ $\tau$\rightar ow\pm\infty}u( $\tau$, t)=p^{\pm}.

Since (2) is invariant under the shift\overline{\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
in $\tau$ ‐direction, \mathbb{R} acts on.

We write its quotient space by \mathcal{M}(p^{-},p^{+})=:.

2\mathrm{I}\mathrm{n} order to define $\delta$ it is necessary to make sense of the cardinality of \mathcal{M}^{\dim=0} using perturbation,of J (some cases) and/or abstract pertrubation. In general situation, p\in L\cap L' should be replacedby the inverse images in the covering space, just as in Novikov theory for closed 1‐forms on a finite
dimensional case. We must also take Novikov completion in order to define CF^{\cdot}(L', L) in general.
,

54

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 of 0 ‐dimensional components of\mathcal{M}(p^{-},p^{+})\mathcal{M}(p^{-}, p^{+}) is not canonically oriented. In the case that L(or more generally (L, L') is a relatively
then
are
in
orientedaconsistent way [5]. In order to show thatspin pair),\mathcal{M}(p^{-},p^{+})$\delta$ 0 $\delta$=0 we study the ends of 1‐dimensional components of \mathcal{M}(p^{-},p^{+}) There are
and L



In general,.

areequipped with spin structures,.

the following possibilities of ends. The first type is splitting into several 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 boundaryof \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 can be excluded by pertubation of J (under certain assumptionof X and L, L ) or abstract perturbation technique. We can see the bubbling‐off ofholomorphic discs in the following simple local example. Let L=\mathbb{R} and L' the unit
circle around the origin in \mathb {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 tosplitting 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 of a holomorphic disc, i.e., a constant Floer trajectory at-1 with the unit disc as a holomorphic disc bubble. In fact, we can see that $\delta$ 0 $\delta$\neq 0.

.

in this case.

If we can exclude the bubbling‐off of holomorphic discs, we can see $\delta$\circ $\delta$=0 andobtain Floer cochain complex (CF\cdot(L', L), $\delta$)

The resulting cohomology denoted by.

HF^{\cdot}(L', L) is called Floer cohomology of L, L.

We can also show that HF^{\cdot}(L, L)\cong

HF^{\cdot}( $\psi$(L), L) for a Hamiltonian 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 of intersection points provided they intersect transversally.

[3] realized these lines of ideas under the condition that $\pi$_{2}(X, L)=0 and
L'= $\psi$(L) for some Hamiltonian diffeomorphism $\psi$ We can also study Floer theory
for Hamiltonian diffeomorphisms. (By taking the graph of a Hamiltonian diffeomor‐
phism, it can be also put in Lagrangian intersection setting.) In [4], Floer succeeded
Floer

.

the construction for a Hamiltonian diffeomorphism on monotone symplectic mani‐folds.

Here monotonicity is that the first Chern class is positively proportional tothe de Rham cohomology class represented by the symplectic form.

Yong‐Geun Oh

considered an analogous situation in Lagrangian intersection setting and performed
Since the right hand side of the definition of $\delta$ may be an infinite sum. So we allow the infinite sum
aslong as the value of the function \mathcal{A} grows to +\infty.

55

the construction of Floer cohomology of (L, L') under the condition that L, L' aremonotone

Lagrangian submanifolds3 with minimal Maslov number >2

obtained the construction for monotoen L and.

Later, he

L'= $\psi$(L) with minimal Maslov

number 2.

As we mentioned before, we may not have $\delta$ 0 $\delta$=0 Such an obstruction is causedby bubbling‐off of holomorphic discs. In order to understand obstructions, we haveto study all holomorphic discs in a systematic way. This is done 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.

invite interested readers to [5].) If the Maurer‐Cartan equation admits weak solutions

for L and L' with the same potential value, then we can rectify $\delta$ to a differentialof CF^{\cdot}(L, L) and define 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.
we canNamely,deform all constructions such as filtered A_{\infty} ‐algebras, filtered A_{\infty}‐bimodules,

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 Kähler toricetc.

.

manifolds, see, e.g., a survey article [6].
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[1]
[2]
[3]

P. Ahern and W. Rudin,

Totally real embeddings of S^{3} in \mathrm{C}^{3},

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94 (1985), 460‐462

C. C. Conley and E. Zehnder, The Birkhoff Lewts fixed point theorem and a conjecture

by V. I. Arnold, Invent. Math. 73(1983), 33‐49.
A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom.

28(1988),513‐547.

[4]

A.

[5]

K.

Floer, Holomorphic spheres and symplectic fixed points,

Comm.

Math.

Phys.

120(1989), 576‐611.

[7]

Fukaya, Y.‐G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory,
anomaly and obstruction, Part I and II, AMS/IP Studies in Advanced Math., 46‐1,2,Amer. Math. Soc. and International Press, 2009.
K. Fukaya, Y.‐G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric
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3A Lagrangian submanifold L is monotone, if the Maslov class of L is positively proportional to

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Research Institute for Mathematical Sciences

Kyoto University
Kyoto 606‐8502
Japan
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