# Generalized Thomas–Yau Uniqueness Theorems

Yohsuke Imagi

###### Abstract

We generalize Thomas–Yau’s uniqueness theorem [17, Theorem 4.3] in two ways.
We prove a stronger statement for special Lagranigans and include minimal
Lagrangians in Kähler–Einstein manifolds or more generally $J$-minimal
Lagrangians introduced by Lotay and Pacini [13, 14]. In every case the heart
of the proof is to make certain Hamiltonian perturbations. For this we use the
method by Imagi, Joyce and Oliveira dos Santos [8, Theorem 4.7].

##  1 Introduction

In this paper we improve and generalize Thomas–Yau’s uniqueness theorem [17,
Theorem 4.3]. We state first our result for special Lagrangians. For the more
complete statement see Corollary 6.3 (ii).

######  Theorem 1.1.

Let $X$ be a Kähler manifold equipped with a holomorphic volume form. Let the
cohomology Fukaya category $H{\mathbin{\cal F}}(X)$ have two isomorphic
objects supported near two closed irreducibly-immersed special Lagrangians
$L_{1},L_{2}$ respectively. Then $L_{1}=L_{2}\subseteq X.$

######  Remark 1.2.

We do not suppose that $L_{1},L_{2}$ have even after shift the same grading.
This is a consequence of the theorem. Also we do not suppose that either
$L_{1}$ or $L_{2}$ itself underlies an object of $H{\mathbin{\cal F}}(X)$ nor
do we suppose even that it is cleanly immersed as Fukaya [3] does. But we do
suppose that we can perturb $L_{1},L_{2}$ both to generically immersed
Lagrangians which underlie objects of $H{\mathbin{\cal F}}(X).$ This is how we
deal with badly immersed Lagrangians in the Floer-theory context in this
paper.

We state next our generalization to minimal Lagrangians in Kähler–Einstein
manifolds or more generally to $J$-minimal Lagrangians introduced by Lotay and
Pacini [13, 14]. At the moment we state the result only in the Kähler–Einstein
case. For the more complete statement see Corollary 6.3 (i).

######  Theorem 1.3.

Let $X$ be a ${\mathbin{\mathbb{Z}}}_{k}$-graded Kähler–Einstein manifold of
complex dimension $n\leqslant\nobreak k;$ and $L_{1},L_{2}\subseteq X$ two
closed irreducibly-immersed ${\mathbin{\mathbb{Z}}}_{k}$-graded minimal
Lagrangians with the same grading on $L_{1}\cap L_{2}.$ Let
${\mathbf{b}}_{1},{\mathbf{b}}_{2}\in H{\mathbin{\cal F}}(X)$ be two objects
supported respectively near $L_{1},L_{2}$ and such that:

either ${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ or $k\geqslant 2n-1$ and
${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ up to shift; and (1.1) $\displaystyle
HF^{i}({\mathbf{b}}_{1},{\mathbf{b}}_{1})\cong
HF^{i}({\mathbf{b}}_{2},{\mathbf{b}}_{2})\neq 0\text{ for }i=0,n.$ (1.2)

Then $L_{1}=L_{2}\subseteq X.$

######  Remark 1.4.

The more precise meaning of $L_{1},L_{2}$ having the same grading on
$L_{1}\cap L_{2}$ is given in Definition 2.1 below. We make a more geometric
definition of gradings than Seidel [16] does.

In the circumstances of Theorem 1.3, if $c_{1}(X)>0$ there may be many
automorphisms of $X$ and many zero objects of $H{\mathbin{\cal F}}(X)$
especially in the toric case as Fukaya, Oh, Ohta and Ono [5] study. So we can
hardly expect a general uniqueness statement to hold. But for instance, to the
Clifford torus

$T^{n}:=\\{[z_{0},\cdots,z_{n}]\in{\mathbin{\mathbb{C}}}P^{n}:|z_{0}|=\cdots=|z_{n}|\\}$
(1.3)

we can apply Theorem 1.3 with the latter alternative of (1.1). We can in fact
take $k=2n+2$ as we show in Example 3.5 below. Also according to Fukaya, Oh,
Ohta and Ono [5] there are $(n+1)$ objects ${\mathbf{b}}\in H{\mathbin{\cal
F}}(X)$ supported on $T^{n}$ with $HF^{*}({\mathbf{b}},{\mathbf{b}})\cong
H^{*}(T^{n}).$ So (1.2) holds and we have:

######  Corollary 1.5.

Let $L\subseteq{\mathbin{\mathbb{C}}}P^{n}$ another closed irreducibly-
immersed ${\mathbin{\mathbb{Z}}}_{2n+2}$-graded minimal Lagrangian such that
$L,T^{n}$ have the same grading on $L\cap T^{n}.$ Let $H{\mathbin{\cal F}}(X)$
have two isomorphic objects, one supported near $L$ and the other exactly on
$T^{n}$ as above. Then $L=T^{n}.$

On the other hand, if $c_{1}(X)\leqslant\nobreak 0$ the nonzero condition
(1.2) holds automatically in certain circumstances. For instance, if either
$L_{1}$ or $L_{2}$ is embedded then (1.2) holds automatically by Fukaya, Oh,
Ohta and Ono’s theorem [4, Theorem E]. If $c_{1}(X)=0$ there is a stronger
version of it which we use for Theorem 1.1.

If $c_{1}(X)<0$ Theorem 1.2 extends to $J$-minimal Lagrangians. They are
locally unique, so the result is interesting only in the global context.

For the proofs we follow in outline that by Thomas and Yau [17, Theorem 4.3].
In every case the heart of the proof is to make certain Hamiltonian
perturbations of the two Lagrangians. The Floer-theory codition implies that
the two perturbed Lagrangians have at least one intersection point of index
$0$ or $n$ at which we do by analysis a sort of unique continuation.

But for the Hamiltonian perturbation process we use the method by Imagi, Joyce
and Oliveira dos Santos [8, Theorem 4.7] because we can unfortunately not
justify the original Morse-theory method [17, Theorem 4.3]. The modified
method works only in the real analytic category. This causes no problem in the
Kähler–Einstein case because in that case everything is automatically analytic
by elliptic regularity. On the other hand, in Theorem 1.1 and in the
$J$-minimal version of Theorem 1.3 we work in the $C^{\infty}$ category and
the real analyticity condition is achieved by another Hamiltonian-perturbation
process.

We begin in §2 with our geometric treatment of graded Lagrangians. In §3 we
recall the relevant facts about Maslov forms in Lotay–Pacini’s sense. In §4 we
give a minimum account of Floer theory which we use in this paper. In §5 we
carry out the key step of making Hamiltonian perturbations. In §6 we state and
prove all our results. They are essentially at the chain level because §5 is
independent of Floer theory.

Finally we remark that Thomas–Yau’s uniqueness theorem is just a bit of their
whole proposal [17] which is now improved by Joyce [10].

Acknowledgements. The author would like to thank Mohammed Abouzaid and Kenji
Fukaya for helpful conversations. This work was supported financially by the
National Natural Science Foundation of China (11950410501).

##  2 Geometric Gradings

We make a geometric definition of graded Lagrangians:

######  Definition 2.1.

Let $(X,\omega)$ a symplectic manifold, $J$ an $\omega$-compatible almost-
complex structure on $X$ and $K_{X}$ the canonical bundle over $(X,J).$ Let
$k$ be either a positive integer or infinity, write
${\mathbin{\mathbb{Z}}}_{k}:={\mathbin{\mathbb{Z}}}/k{\mathbin{\mathbb{Z}}}$
for $k$ finite and write
${\mathbin{\mathbb{Z}}}_{\infty}:={\mathbin{\mathbb{Z}}}.$ By a
${\mathbin{\mathbb{Z}}}_{k}$-grading of $(X,\omega,J)$ we mean for $k<\infty$
the pair of a complex line bundle $K_{X}^{2/k}$ and a bundle isomorphism
$(K_{X}^{2/k})^{k}\cong K_{X}^{2},$ and for $k=\infty$ a nowhere vanishing
section $\Psi$ of $K_{X}^{2}.$ Suppose given such a
${\mathbin{\mathbb{Z}}}_{k}$-grading of $X$ and a Hermitian metric $g$ on
$(X,J).$ Then for every $x\in X$ and for every Lagrangian plane $\ell\subseteq
T_{x}X$ there is a $g$-orthogonal decomposition $T_{x}X=\ell\oplus J\ell$ and
accordingly a canonical $g$-unit section of $K_{X}^{2}$ which we denote by
$\Omega_{\ell}^{2};$ that is, if $e_{1},\cdots,e_{n}\in\ell$ are a
$g$-orthonormal basis then

$\Omega_{\ell}^{2}:=\pm e^{1}\wedge\cdots\wedge e^{n}\wedge
Je^{1}\wedge\cdots\wedge Je^{n}.$

There is also a canonical $k$-fold cover $\operatorname{Lag}^{k}TX$ of the
Lagrangian Grassmannian $\operatorname{Lag}TX$ defined for $k<\infty$ by

$\operatorname{Lag}^{k}TX:=\\{(\ell,\alpha)\in\operatorname{Lag}TX\times_{X}K_{X}^{2/k}:\Omega_{\ell}^{2}=\alpha^{k}\\}$

and for $k=\infty$ by
$\operatorname{Lag}^{\infty}TX:=\\{(\ell,\phi)\in\operatorname{Lag}TX\times_{X}{\mathbin{\mathbb{R}}}:\Omega_{\ell}^{2}=e^{i\phi}\Psi\\}.$
By a ${\mathbin{\mathbb{Z}}}_{k}$-grading of an immersed Lagrangian
$L\subseteq(X,\omega)$ we mean a lift to $\operatorname{Lag}^{k}TX$ of the
tangent-space map $L\to\operatorname{Lag}TX,$ that is, for $k<\infty$ a
section $\alpha:L\to K_{X}^{2/k}$ with $\alpha^{k}=\Omega_{L}^{2}$ and for
$k=\infty$ a section $\phi:L\to S^{1}$ with $e^{i\phi}\Psi=\Omega_{L}^{2}.$ It
is unique up to ${\mathbin{\mathbb{Z}}}_{k}$-shifts, where the shift $[1]$ is
defined for $k<\infty$ by $(L,\alpha)[1]:=(L,e^{2\pi i/k}\alpha)$ and for
$k=\infty$ by $(L,\phi)[1]:=(L,\phi+2\pi).$

######  Remark 2.2.

The advantage of this definition is that given two Lagrangians $L_{i}$
$(i=1,2)$ graded by $\alpha_{i}$ or $\phi_{i}$ as above, we can compare the
two gradings $\alpha_{i}$ or $\phi_{i}.$ In particular, it makes sense to say
that they are equal or not.

We make another definition in the circumstances of Definition 2.1:

######  Definition 2.3.

For $i=1,2$ let $L_{i}\subseteq(X,\omega)$ be a Lagrangian with a
${\mathbin{\mathbb{Z}}}_{k}$-grading $\alpha_{i}:L\to K_{X}^{2/k},$
$k<\infty.$ Suppose $L_{1},L_{2}$ mutually transverse at a point $x\in
L_{1}\cap L_{2}.$ We define the Maslov index
$\mu_{L_{1},L_{2}}(x)\in{\mathbin{\mathbb{Z}}}_{k}.$ Define $e^{i\tau}\in
S^{1}$ by $\alpha_{2}=e^{i\tau}\alpha_{1}\in K_{X}^{2/k}$ over $x.$ Take an
isomorphism $(T_{x}X,J|_{x},g|_{x})\cong{\mathbin{\mathbb{C}}}^{n}$ which maps
$T_{x}L_{1}$ to
${\mathbin{\mathbb{R}}}^{n}\subseteq{\mathbin{\mathbb{C}}}^{n}$ and
$T_{x}L_{2}$ to
$\\{(e^{i\theta_{1}}x_{1},\cdots,e^{i\theta_{n}}x_{n})\in{\mathbin{\mathbb{C}}}^{n}:x_{1},\cdots,x_{n}\in{\mathbin{\mathbb{R}}}\\}.$
for some $\theta_{1},\cdots,\theta_{n}\in(0,\pi).$ These
$\theta_{1},\cdots,\theta_{n}$ are unique up to order and we have
$\Omega_{L_{2}}^{2}=\pm
e^{-2i(\theta_{1}+\cdots+\theta_{n})}\Omega_{L_{1}}^{2}.$ So we can define

$\mu_{L_{1},L_{2}}(x):=\frac{1}{2\pi}(2\theta_{1}+\cdots+2\theta_{n}+k\tau)\in{\mathbin{\mathbb{Z}}}\text{
modulo }k{\mathbin{\mathbb{Z}}}.$ (2.1)

If $k=\infty$ and if for $i=1,2$ the $L_{i}$ is
${\mathbin{\mathbb{Z}}}$-graded by $\phi_{i}:L_{i}\to{\mathbin{\mathbb{R}}}$
then

$\mu_{L_{1},L_{2}}(x):=\frac{1}{2\pi}(2\theta_{1}+\cdots+2\theta_{n}-\phi_{1}(x)+\phi_{2}(x))\in{\mathbin{\mathbb{Z}}}.$
(2.2)

We recall now the notion of special Lagrangians:

######  Definition 2.4.

Let $(X,\omega)$ a symplectic manifold, $J$ an $\omega$-compatible almost-
complex structure on $X$ and $\Psi$ a ${\mathbin{\mathbb{Z}}}$-grading of
$(X,\omega,J).$ We say that an immersed Lagrangian in $(X,\omega,J,\Psi)$ is
special if it is ${\mathbin{\mathbb{Z}}}$-graded by a constant.

######  Remark 2.5.

In the symplectic context including this definition and §4 we do not need $J$
to be integrable. But in the more geometric context we suppose $J$ integrable
and Harvey–Lawson’s theorem applies as in Theorem 3.2.

We prove a lemma which we use in Corollary 4.7:

######  Lemma 2.6.

Let $(X,\omega,J,\Psi)$ be such as in Definition 2.4 and $L_{1},L_{2}\subseteq
X$ two mutually-transverse special Lagrangian submanifolds which have not even
after shifts the same grading. Then there exists $i\in{\mathbin{\mathbb{Z}}}$
such that the Maslov indices of $L_{1},L_{2}$ all belong to $[i,i+n-1].$ This
will still hold under Lagrangian perturbations of $L_{1},L_{2}.$

###### Proof.

Let $\phi_{1},\phi_{2}\in{\mathbin{\mathbb{R}}}$ be gradings of $L_{1},L_{2}.$
Then $\phi-\phi^{\prime}\notin{\mathbin{\mathbb{Z}}}$ and this condition is
preserved under Lagrangian perturbations of $L_{1},L_{2}.$ Let $x\in L_{1}\cap
L_{2}$ and let $\theta_{1},\cdots,\theta_{n}\in(0,\pi)$ be as in (2.1). Then
by (2.2) we have $\mu_{L_{1},L_{2}}(x)\in[i,i+n-1]$ for some
$i\in{\mathbin{\mathbb{Z}}}.$ This $i$ is independent of $x$ which completes
the proof. ∎

We prove another lemma which we use in §6:

######  Lemma 2.7.

Let $(X,\omega)$ be a symplectic manifold of real dimension $2n,$
${\mathbin{\mathbb{Z}}}_{k}$-graded with $k\geqslant n,$ equipped with a
compatible almost complex-structure $J$ and equipped with a Hermitian metric
$g.$ Let $L_{1},L_{2}\subseteq X$ be two closed immersed
${\mathbin{\mathbb{Z}}}_{k}$-graded Lagrangians with the same grading on
$L_{1}\cap L_{2}.$ Denote by $S\subseteq L_{1}\cap L_{2}$ the set of points at
which $L_{1},L_{2}$ have at least one common tangent plane. Then for every
open neighbourhood $U\subseteq X$ of $S$ there exists $\epsilon>0$ such that
every Hamiltonian $\epsilon$-perturbation of $L_{2}$ that intersects $L_{1}$
generically $($that is, only at transverse double points$)$ has no
intersection point with $L_{1}\cap U$ of index $0$ or $n$ modulo
$k{\mathbin{\mathbb{Z}}}.$

###### Proof.

If this fails there are an open neighbourhood $U\subseteq X$ of $S,$ a
sequence of Hamiltonian perturbations $L_{2}^{i}$ of $L_{2}$ all transverse to
$L_{1}$ and tending smoothly to $L_{2},$ and a sequence of intersection points
$x^{i}\in L_{1}\cap L_{2}^{i}\setminus U$ of index $0$ or $n.$ By hypothesis
$L_{1}\cap L_{2}^{i}\setminus U$ is compact, so there is a subsequence of
$x^{i}$ tending to some point $x\in L_{1}\cap L_{2}\setminus U.$ Suppose now
that $k<\infty;$ the other case $k=\infty$ may be treated in the same manner.
Give $L_{2}^{i}$ a grading $\alpha^{i}$ by the obvious homotopy, and write
$\alpha^{i}=\exp(\sqrt{-1}\tau^{i})\alpha$ over $x^{i}.$ By hypothesis
$L_{1},L_{2}$ have the same grading at $x^{i}$ so we may suppose that for each
$i$ we have $|\tau^{i}|<2\pi/k$ and the sequence $\tau^{i}$ tends to $0$ as
$i\to\infty.$ Denote by $\theta_{1}^{i},\cdots,\theta_{n}^{i}\in(0,\pi)$ the
$n$ angles between the two tangent planes at $x^{i}.$ Then
$2\theta_{1}^{i}+\cdots+2\theta_{n}^{i}+k\tau^{i}=0\text{ or }n\pi\text{
modulo }2k\pi{\mathbin{\mathbb{Z}}}.$ In fact, since $k\geqslant n$ it follows
that this holds without modulo $2k\pi{\mathbin{\mathbb{Z}}}.$ So the limits of
the $n$ angles sum up to $0$ or $n\pi;$ that is, at the limit point $x$ there
is a common tangent plane to $L_{1},L_{2}.$ But this contradicts the
definition of $S.$ ∎

##  3 Maslov Forms

Following Lotay and Pacini [14] we make:

######  Definition 3.1.

Let $(X,\omega,J)$ be a Kähler manifold of complex dimension $n$ and $K_{X}$
the canonical bundle over $(X,J).$ Let $L\subseteq(X,\omega)$ be a Lagrangian
submanifold, and $g$ a Hermitian metric on $(X,J)$ which need not be the
Kähler metric of $(X,\omega,J)$ nor even Kähler. As in Definition 2.1 take the
canonical $g$-unit section $\Omega_{L}^{2}$ of $K_{X}^{2}$ over $L.$ Denote by
$\nabla$ the Chern connection on $(X,J,g)$ and by $A$ the real $1$-form on
$K_{X}^{2}$ over $L$ defined by
$\nabla\Omega_{L}^{2}=2iA\otimes\Omega_{L}^{2}.$ We call $A$ the Maslov form
on $L$ relative to $g$ and say that $L$ is $g$-Maslov-zero if $A\equiv 0.$

Here it is not essential that $J$ is integrable. Many results by Lotay and
Pacini [13, 14] hold with $J$ an $\omega$-compactible almost-complex structure
and with $\nabla$ a connection such that $\nabla J=\nabla g=0.$ But it is
convenient for us to suppose $J$ integrable. We can then take the Chern
connection; in §5 we can use the underlying real analytic structure of $(X,J)$
which will be convenient for us to state our results; and there are as in
Theorem 3.7 nice deformation-theory results for special Lagrangians and
$J$-minimal Lagrangians which we use in Corollary 6.2.

As Lotay and Pacini [13, Theorem 2.4] prove, if $g$ is Kähler the $g$-Maslov-
zero condition and the $J$-minimal condition are equivalent. If also $L$ is
$g$-Lagrangian—that is, $g(Jv,v^{\prime})=0$ for any $v,v^{\prime}\in TL$—then
$L$ is $J$-minimal if and only if $L$ is $g$-minimal in the ordinary sense.

We work with $g$-Maslov-zero Lagrangians rather than with $J$-minimal
Lagrangians because for special Lagrangians we can take $g$ to be merely
conformally Kähler as we recall now. Let $(X,\omega,J)$ a Kähler manifold and
$\Psi$ a $J$-holomorphic volume form on $X.$ Joyce [9] and other authors call
$(X,\omega,J,\Psi)$ an almost Calabi–Yau manifold but this does not mean that
$J$ is merely an almost complex structure; $J$ is integrable. It means
$\omega$ not necessarily Ricci-flat. Denote by $g$ the conformally Kähler
metric on $(X,J)$ associated with $\psi^{2}\omega$ where
$\psi:X\to{\mathbin{\mathbb{R}}}^{+}$ is defined by

$\psi^{2n}\omega^{n}/n!=(-1)^{n(n-1)/2}(i/2)^{n}\Psi\wedge\overline{\Psi}.$
(3.1)

Then for every Lagrangian submanifold $L\subseteq X$ the canonical $g$-unit
section $\Omega_{L}^{2}$ is of the form $\alpha\Psi^{2}|_{L}$ where
$\alpha:L\to S^{1}$ is a smooth function. This $L$ has zero Maslov $1$-form if
and only if $\alpha$ is constant, that is, if and only if $L$ is special. In
these circumstances Harvey–Lawson’s theorem [7] may be stated as follows:

######  Theorem 3.2.

If $L$ is special then $L$ is orientable and calibrated by $\sqrt{\alpha}\Psi$
in Harvey–Lawson’s sense where the choice of $\sqrt{\alpha}$ corresponds to
the orientation of $L.$ And $L$ is area-minimizing with respect to $g$ and
$J$-minimal in $(X,J,g).$

Lotay and Pacini [14, Proposition 4.5] prove a formula for the Maslov $1$-form
$A$ and the $J$-mean-curvature vector $H_{J}$:

$-JA=(H_{J}+T_{J})\lrcorner g$ (3.2)

where $T_{J}$ comes from the torsion of the Chern connection $\nabla.$ If $L$
is special then by definition we have $A=0$ and $H_{J}$ is the ordinary mean
curvature which is also zero. So by (3.2) we have $T_{J}=0$ too. Conversely,
if $L$ is merely $J$-minimal then $T_{J}$ may be nonzero. This justifies us
working with $g$-Maslov-zero Lagrangians rather than $J$-minimal Lagrangians.

We compute now the relative first Chern class $2c_{1}(X,L)\in
H^{2}(X,L;{\mathbin{\mathbb{Z}}})$ for $g$-Maslov-zero Lagrangians $L\subseteq
X.$ We recall therefore:

######  Lemma 3.3.

Let $(X,\omega)$ be a symplectic manifold, $J$ an $\omega$-compatible complex
structure, $K_{X}$ the canonical bundle over $(X,J)$ and
$L\subseteq(X,\omega)$ a Lagrangian submanifold. Then the relative de-Rham
class $2c_{1}(X,L)\in H^{2}(X,L;{\mathbin{\mathbb{R}}}),$ which is integral,
may be represented by the curvature $2$-form of any connection on $K_{X}^{2}$
that is flat over $L.$

So by Definition 3.1 we have:

######  Corollary 3.4.

Let $(X,\omega,J)$ be a Kähler manifold, $g$ a Hermitian metric on $(X,J),$
and $L\subseteq(X,\omega)$ a $g$-Maslov-zero Lagrangian submanifold. Then
$2c_{1}(X,L)$ may be represented by the curvature $(1,1)$-form of $K_{X}^{2}$
relative to the Chern connection on $(X,J,g).$

######  Example 3.5.

Let $X={\mathbin{\mathbb{C}}}P^{n},$ $\omega$ the Fubini–Study form and $g$
the Fubini–Study metric. This is Kähler–Einstein and the curvature
$(1,1)$-form of $K_{X}$ relative to $g$ is $-(n+1)\omega.$ The Clifford torus
$T^{n}\subseteq{\mathbin{\mathbb{C}}}P^{n}$ defined by (1.3) is a minimal
Lagrangian. It is $J$-minimal and $g$-Maslov-zero so we can apply to it
Corollary 3.4.

We prove that ${\mathbin{\mathbb{C}}}P^{n}$ and $T^{n}$ may both be
${\mathbin{\mathbb{Z}}}_{2n+2}$-graded. Since
$H_{1}({\mathbin{\mathbb{C}}}P^{n},{\mathbin{\mathbb{Z}}})=0$ it follows
according to Seidel [16, Lemma 2.6] that ${\mathbin{\mathbb{C}}}P^{n}$ may be
${\mathbin{\mathbb{Z}}}_{2n+2}$-graded because
$2c_{1}({\mathbin{\mathbb{C}}}P^{n})=-2(n+1)[\omega].$ Also $T^{n}$ may be
${\mathbin{\mathbb{Z}}}_{k}$-graded if and only if
$2c_{1}({\mathbin{\mathbb{C}}}P^{n},T^{n})$ is divisible by $k.$ By Corollary
3.4 we have $c_{1}({\mathbin{\mathbb{C}}}P^{n},T^{n})=-(n+1)[\omega]\in
H^{2}({\mathbin{\mathbb{C}}}P^{n},T^{n};{\mathbin{\mathbb{R}}}).$ Computation
shows that $H_{2}({\mathbin{\mathbb{C}}}P^{n},T^{n};{\mathbin{\mathbb{Z}}})$
is generated by the image of
$H_{2}({\mathbin{\mathbb{C}}}P^{n},{\mathbin{\mathbb{Z}}})$ and the $n$ discs
$D_{a}\subseteq{\mathbin{\mathbb{C}}}P^{n},$ $a\in\\{1,\cdots,n\\},$ defined
by $|z_{a}|\leqslant\nobreak 1$ and $z_{b}=1$ for every $b\neq a.$ We have

$\omega|_{D_{a}}=\frac{ni}{2\pi}\frac{dz_{a}\wedge
d\bar{z}_{a}}{(n+|z_{a}|^{2})^{2}}\text{ and }\int_{D_{a}}\omega=1.$

So $[\omega]\in
H^{2}({\mathbin{\mathbb{C}}}P^{n},T^{n};{\mathbin{\mathbb{R}}})$ is integral,
q.e.d.

We turn now to the deformation theory for $g$-Maslov-zero Lagrangians. We
recall Lotay–Pacini’s lemma [13, Lemma 4.1] which we use in §5:

######  Lemma 3.6.

Let $(X,\omega,J)$ be a Kähler manifold and $g$ a Hermitian metric on $(X,J).$
Let $L\subseteq(X,\omega,J)$ be a closed immersed $g$-Maslov-zero Lagrangian
and $NL$ the $g$-normal bundle to $L\subseteq X.$ For every sufficiently small
$v\in C^{\infty}(NL)$ denote by $Av$ the $g$-Maslov $1$-form on the graph of
$v$ embedded in $X$ by the $g$-exponential map, and denote by $A^{\prime}$ the
linearization of $A$ at $0\in C^{\infty}(NL).$ Then for every $v\in
C^{\infty}(NL)$ we have

$A^{\prime}v=dd^{*}\hat{v}+\sum_{i=1}^{n}g(T_{\nabla}(v,e_{i}),e_{i})+v\lrcorner
F_{\nabla}$ (3.3)

where $\hat{v}:=-Jv\lrcorner g,$ $\\{e_{1},\cdots,e_{n}\\}$ is a
$g$-orthonormal frame on $TL,$ $\nabla$ the Chern connection of $(X,J,g),$
$T_{\nabla}$ its torsion tensor and $F_{\nabla}$ its curvature $(1,1)$-form
for $K_{X}.$

###### Proof.

This is a version of Lotay–Pacini’s lemma [13, Lemma 4.1]. They suppose
$(X,J,g)$ almost Kähler but again this is not essential; the result applies to
every almost Hermitian manifold $(X,J,g)$ with connection $\nabla$ such that
$\nabla J=\nabla g=0.$ Our formula (3.3) is simpler than Lotay–Pacini’s in two
respects. One is that their $J$-volume function
$\rho_{J}:L\to{\mathbin{\mathbb{R}}}^{+}$ is identically equal to $1$ and the
other is that their projection operator $\pi_{L}$ is $g$-orthogonal. These
both hold because $L$ is $g$-Lagrangian, which completes the proof. ∎

We remark that we can compute the torsion term in the conformally Kähler case.
If $g$ is a conformally Kähler metric on $(X,J)$ associated with
$\psi^{2}\omega$ then $\nabla-{\partial}\log\psi^{2}\otimes\mathop{\rm
id}\nolimits$ is the Levi-Civita connection for $\omega.$ Hence by computation
we see that for any vector fields $u,v$ on $X$ we have

$T_{\nabla}(u,v)=d\log\psi\llcorner(u\otimes v+Ju\otimes Jv-v\otimes
u-Jv\otimes Ju).$

So the torsion term on (3.3) is equal to $(n-1)d\log\psi$ and

$A^{\prime}v=d(\psi^{n-1}d^{*}\psi^{1-n}\hat{v})+v\lrcorner F_{\nabla}.$ (3.4)

This has application to deformation theory; that is, by (3.4) we can apply
Lotay–Pacini’s result [13, Proposition 4.5] with $\psi^{n-1}$ in place of
$\rho_{J}.$ So Lotay–Pacini’s uniqueness and persistence theorem [13, Theorem
5.2] extends to $g$-Maslov-zero Lagrangians with $g$ conformally Kähler. But
the corresponding preturbations are just Lagrangian and not necessarily
Hamiltonian as we want in the Floer-theory context.

On the other hand, we can make Hamiltonian perturbations of special
Lagrangians and $J$-minimal Lagangians. The result for special Lagrangians
goes back to McLean’s theorem [15] and that for $J$-minimal Lagrangians is
proved by Lotay–Pacini [13, Theorem 5.6].

######  Theorem 3.7.

(i) Let $(X,\omega,J)$ be a Kähler manifold and $\Psi$ a $J$-holomorphic
volume form on $X.$ Then under perturbations of $(J,\Psi),$ as
$\omega$-compatible complex structures and holomorphic volume forms relative
to them, every closed immersed special Lagrangian persists as Hamiltonian
perturbations.
(ii) Let $(X,\omega,J)$ be a Kähler manifold such that $-\omega$ is the Ricci
$(1,1)$-form of another Kähler metric $g$ on $(X,J).$ Then under perturbations
of $(J,g),$ as $\omega$-compatible complex structures and Kähler metrics
relative to them whose Ricci $(1,1)$-forms are all equal to $-\omega,$ every
closed immersed $J$-minimal Lagrangian persists as Hamiltonian perturbations.

##  4 $HF^{*}$ Nonvanishing Theorems

Following Fukaya, Oh, Ohta and Ono [4] given a unital commutative ring $R$ we
define the Novikov ring

$\Lambda=\left\\{\sum_{i=0}^{\infty}a_{i}T^{\lambda_{i}}:a_{i}\in
R[e,e^{-1}],\lambda_{i}\in{\mathbin{\mathbb{R}}}\text{ for each }i\text{ and
}\lim_{i\to\infty}\lambda_{i}=\infty\right\\}$ (4.1)

where $T,e$ are two formal variables. They are relevant respectively to the
areas and Maslov numbers of pseudoholomorphic curves. We denote by
$\Lambda^{0}\subseteq\Lambda$ the subring defined by the same formula (4.1) as
$\Lambda$ but with $\lambda_{i}\geqslant 0$ in place of
$\lambda_{i}\in{\mathbin{\mathbb{R}}}.$ We denote by
$\Lambda^{+}\subseteq\Lambda^{0}$ the ideal defined by (4.1) with with
$\lambda_{i}>0$ in place of $\lambda_{i}\in{\mathbin{\mathbb{R}}}.$ We write
$\approx$ for $=$ modulo $\Lambda^{+}.$

Fukaya categories are defined over $\Lambda.$ If the ambient symplectic
manifold is spherically nonnegative we can take $R={\mathbin{\mathbb{Z}}}$ as
Fukaya, Oh, Ohta and Ono [6] do. But in general we have to do the
pseudoholomorphic-curve counts over ${\mathbin{\mathbb{Q}}}$ and we shall
therefore suppose that $R$ contains ${\mathbin{\mathbb{Q}}}.$ Otherwise, in
our applications we need no condition on $R;$ that is, we work under the
following:

######  Hypothesis 4.1.

$(X,\omega)$ is a ${\mathbin{\mathbb{Z}}}_{k}$-graded symplectic manifold of
real dimension $2n,$ either compact or $\omega$-convex at infinity. $R$ is a
unital commutative ring such that if $X$ is spherically nonnegative then
${\mathbin{\mathbb{Q}}}\subseteq R.$ If $2\neq 0$ in $R$ then $X$ is given a
background bundle for relative spin structures on its Lagrangians, and the
Lagrangians concerned are all given relative spin structures.

Under this hypothesis there is a Fukaya category ${\mathbin{\cal F}}(X)$ and
its cohomology category $H{\mathbin{\cal F}}(X).$ In this paper, following
Akaho and Joyce [1] we suppose that every object of ${\mathbin{\cal F}}(X)$
consists of a closed generically-immersed Lagrangian $L\subseteq X$ and its
additional structures. We can include also a suitable class of local systems
on $L$ and the results in this paper will hold for them but we shall omit this
in our formal treatment. Objects of $H{\mathbin{\cal F}}(X)$ contain bounding
cochains as we recall below.

We recall first the homologically perturbed version of Floer complexes. Given
a closed manifold $\,\widehat{\\!L}$ and a generic Lagrangian immersion
$\,\widehat{\\!L}\to L\subseteq X$ we write $L\cap
L:=\,\widehat{\\!L}\times_{X}\,\widehat{\\!L}.$ This is the disjoint union of
the diagonal $\,\widehat{\\!L}$ and self-intersection pairs. It depends upon
the choice of $\,\widehat{\\!L}\to X$ but is unique up to automorphisms of
$\,\widehat{\\!L}.$ We write $C_{L}^{*}:=H^{*}(L\cap L,R)[-\mu_{L,L}]$ where
the diagonal has no degree shift and every self-intersection pair has degree
equal to its Maslov index. If $L$ is embedded, this $C_{L}^{*}$ is
${\mathbin{\mathbb{Z}}}$-graded and supported in degrees $0,\cdots,n.$ If $L$
is ${\mathbin{\mathbb{Z}}}_{k}$-graded, so is $C_{L}^{*}.$ If also $L$ has on
its every self-intersection pair two gradings nearly equal, then $C_{L}^{*}$
is supported in degrees $0,\cdots,n$ modulo $k{\mathbin{\mathbb{Z}}}.$

We use the following version of $A_{\infty}$ algebra structure constructions.
There are on $C_{L}^{0}$ a unit and the intersection pairing so we can speak
of unitality and cyclic symmetry as Fukaya [2] does.

######  Theorem 4.2.

There is on $C_{L}^{*}\otimes\Lambda^{0}$ a cohomologically-unital cyclic
curved $A_{\infty}$ algebra structure $(\mathfrak{m}^{i})_{i=0}^{\infty}$ with
$\mathfrak{m}^{1}\approx 0$ and $\mathfrak{m}^{2}\approx\pm\wedge$ where
$\wedge$ is the cup-product map. If also $L$ is embedded, we can make this
strictly unital.

######  Remark 4.3.

The embedded case is proved by Fukaya [2]. This will extend to the immersed
case if we give up having the strict unit and satisfy ourselves with a
cohomological unit. To have the strict unit we ought to perturb in a certain
manner the relevant pseudoholomorphic-curve moduli spaces. But in the immersed
case there would be those of constant maps to self-intersection points. If we
perturbed these, the cyclic symmetry would fail.

The cyclic symmetry is used in Lemma 4.4 below. But also we give it another
proof in which we use the open-closed map.

By a bounding cochain on
$(C_{L}^{*}\otimes\Lambda^{0};\mathfrak{m}^{0},\mathfrak{m}^{1},\mathfrak{m}^{2},\cdots)$
we mean an element ${\mathbf{b}}\in C^{1}_{L}\otimes\Lambda^{+}$ with
$\sum_{i=0}^{\infty}\mathfrak{m}^{i}({\mathbf{b}},\cdots,{\mathbf{b}})=0.$
Given such a ${\mathbf{b}}$ there is a natural way of giving
$C_{L}^{*}\otimes\Lambda^{0}$ a cohomologically-unital cyclic ordinary
$A_{\infty}$ algebra structure
$(\mathfrak{m}^{i}_{\mathbf{b}})_{i=0}^{\infty}$ with
$\mathfrak{m}_{\mathbf{b}}^{1}\approx 0$ and
$\mathfrak{m}_{\mathbf{b}}^{2}\approx\pm\wedge.$

We denote by $1\in C_{L}^{0}$ the unit, which is the cohomological unit in
Theorem 4.2. We denote by $*1\in C_{L}^{n}$ the volume form supported on the
diagonal, which is if $2=0$ in $R$ the Poincaré dual to a point. We denote
also symbolically by $(x,y):=\pm\int_{L\cap L}x\wedge y$ the intersection
pairing on $C_{L}^{*}.$ The sign does not matter to us. We prove now:

######  Lemma 4.4.

For every bounding cochain ${\mathbf{b}}\in C_{L}^{1}\otimes\Lambda^{+}$ we
have:

  * (i)

if $L$ is embedded, we have
$*1\notin\operatorname{im}\mathfrak{m}^{1}_{\mathbf{b}};$ and

  * (ii)

if $\mathfrak{m}^{1}_{\mathbf{b}}*1=0$ then $[*1]\neq 0$ in
$H^{n}(C_{L}^{*}\otimes\Lambda^{0},\mathfrak{m}_{\mathbf{b}}^{1}).$

###### Proof.

Suppose contrary to (i) that $*1=\mathfrak{m}^{1}_{\mathbf{b}}x$ for some
$x\in C^{*}_{L}\otimes\Lambda.$ Then using the strict unit we find

$0=(\mathfrak{m}^{1}_{\mathbf{b}}1,x)=\pm(1,\mathfrak{m}^{1}_{\mathbf{b}}x)=\pm(1,*1)\approx\pm\int_{L}\mathfrak{m}^{2}_{\mathbf{b}}(1,*1)\approx\pm\int_{L}*1=1.$
(4.2)

This is impossible which proves (i).

We give (ii) two proofs. Firstly, if $*1=\mathfrak{m}^{1}_{\mathbf{b}}x$ for
some $x\in C^{*}_{L}\otimes\Lambda$ then as in (4.2) we have

$0\approx\pm\int_{L}\mathfrak{m}_{\mathbf{b}}^{2}(1,*1)$ (4.3)

but $\mathfrak{m}_{\mathbf{b}}^{2}(1,*1)$ need not be $*1.$ However $1$ is a
cohomological unit and $*1$ an $\mathfrak{m}_{\mathbf{b}}^{1}$-cocycle, so

$\mathfrak{m}_{\mathbf{b}}^{2}(1,*1)=*1+\mathfrak{m}_{\mathbf{b}}^{1}y\approx*1\approx*1.$
(4.4)

Now by (4.3) and (4.4) we have $0\approx\pm\int_{L\cap L}*1\approx\pm 1\in
R\setminus\\{0\\}.$ This is impossible and so completes the first proof.

In the other proof of (ii) we use neither the cyclic symmetry nor the
unitality but the open-closed map constructed by Fukaya, Oh, Ohta and Ono [4,
§3.8]. This extends to generically immersed Lagrangians. We use in particular
the map
$\mathfrak{p}_{\mathbf{b}}:H^{n}(C_{L}^{*}\otimes\Lambda^{0},\mathfrak{m}_{\mathbf{b}}^{1})\to
H^{2n}(X,R)\otimes\Lambda^{0}.$ Denote by $*_{X}1$ the volume form on $X$ and
to distinguish it from that on $L$ write $*_{L}1:=*1\in C^{n}_{L}.$ Then by
the extended version of Fukaya, Oh, Ohta and Ono’s lemma [4, Lemma 6.4.2] we
have $\mathfrak{p}_{\mathbf{b}}[*_{L}1]\approx[*_{X}1].$ So $[*_{L}1]\neq 0.$
∎

We prove now our version of Fukaya, Oh, Ohta and Ono’s theorem [4, Theorem E].
We recall therefore that the $A_{\infty}$ structure of Theorem 4.2 is gapped;
that is, if we take on $X$ an $\omega$-compatible almost-complex structure $J$
we can write $\mathfrak{m}_{\mathbf{b}}^{1}$ as the sum of
$\mathfrak{m}_{{\mathbf{b}},\beta}^{1}$ of degree $1-[\beta]\cdot 2c_{1}(X,L)$
where $\beta\in H_{2}(X,L;{\mathbin{\mathbb{Z}}})$ is a genus-zero stable
$J$-holomorphic curve class. Also, given a bounding cochain ${\mathbf{b}}\in
C_{L}^{1}\otimes\Lambda^{+}$ we define
$HF^{*}({\mathbf{b}},{\mathbf{b}}):=H^{*}(C_{L}^{*}\otimes\Lambda,\mathfrak{m}_{\mathbf{b}}^{1})\otimes\Lambda.$

######  Theorem 4.5.

Let Hypothesis 4.1 hold and let $L\subseteq X$ be a closed generically-
immersed Lagrangian. Suppose also that:

  * (i)

$L$ is embedded and there is on $X$ an $\omega$-compatible almost-complex
structure $J$ such that for every $J$-holomorphic curve class $\beta\in
H_{2}(X,L;{\mathbin{\mathbb{Z}}})$ we have $\beta\cdot
2c_{1}(X,L)\leqslant\nobreak 0;$ or

  * (ii)

$k\geqslant n+2$ and $L$ has on its every self-intersection pair the two
gradings nearly equal.

Then for $i=0,n$ and for every object ${\mathbf{b}}\in H{\mathbin{\cal F}}(X)$
supported on $L$ we have $HF^{i}({\mathbf{b}},{\mathbf{b}})\neq 0.$

###### Proof.

The case (i) is due to Fukaya, Oh, Ohta and Ono [4, Theorem E] but we recall
the proof. Since $L$ is embedded it follows that $C_{L}^{*}$ is
${\mathbin{\mathbb{Z}}}$-graded and supported in degrees $0,\cdots,n.$ Also,
since $\beta\cdot 2c_{1}(X,L)\leqslant\nobreak 0$ it follows that
$\mathfrak{m}_{\mathbf{b}}^{1}$ has degree $\geqslant 1.$ So under
$\mathfrak{m}_{\mathbf{b}}^{1}$ nothing goes to $1$ and $*1$ goes to zero.
Since the cohomological unit is by definition an
$\mathfrak{m}_{\mathbf{b}}^{1}$-cycle it follows now that $[1]\neq 0$ in
$H^{0}(C_{L}^{*}\otimes\Lambda^{0},\mathfrak{m}_{\mathbf{b}}^{1})$ and
accordingly in $HF^{0}({\mathbf{b}},{\mathbf{b}}).$ On the other hand, we have
proved that $*1$ is also an $\mathfrak{m}_{\mathbf{b}}^{1}$-cycle. So by Lemma
4.4 (i) we have $[*1]\neq 0$ in
$H^{n}(C_{L}^{*}\otimes\Lambda^{0},\mathfrak{m}_{\mathbf{b}}^{1})$ and
accordingly in $HF^{n}({\mathbf{b}},{\mathbf{b}}).$

In the case (ii) we have $C_{L}^{*}$ merely
${\mathbin{\mathbb{Z}}}_{k}$-graded. But $k\geqslant n+2$ so
$C_{L}^{-1}=C_{L}^{n+1}=0.$ Now $d_{\mathbf{b}}$ has degree $1$ modulo
$k{\mathbin{\mathbb{Z}}}$ so again under $\mathfrak{m}_{\mathbf{b}}^{1}$
nothing goes to $1$ and $*1$ goes to zero. So $[1]\neq 0$ in
$HF^{0}({\mathbf{b}},{\mathbf{b}})$ and now by Lemma 4.4 (ii) we have
$[*1]\neq 0$ in $HF^{n}({\mathbf{b}},{\mathbf{b}}).$ This completes the proof.
∎

We apply Theorem 4.5 to $g$-Maslov-zero Lagrangians. We say that a closed
immersed Lagrangian $L\subseteq X$ nearly underlies ${\mathbin{\cal F}}(X)$ if
there is an arbitrarily-small Hamiltonian deformation of $L$ which underlies
an object of $H{\mathbin{\cal F}}(X).$ Such an object is said to be supported
near $L$.

######  Corollary 4.6.

Let Hypothesis 4.1 hold with $X$ given an $\omega$-compatible almost-complex
structure $J.$ Let $g$ be another Hermitian metric on $(X,J)$ and
$L\subseteq(X,\omega,J,g)$ a close ${\mathbin{\mathbb{Z}}}_{k}$-graded
immersed $g$-Maslov-zero Lagrangian. Suppose that either: (i) $L$ is embedded;
or (ii) $k\geqslant n+2$ and $L$ has on its every self-intersection pair the
two gradings nearly equal. Then for $i=0,n$ and for every object
${\mathbf{b}}\in H{\mathbin{\cal F}}(X)$ supported near $L$ we have
$HF^{i}({\mathbf{b}},{\mathbf{b}})\neq 0.$

###### Proof.

In the case (i) it follows from Corollary 3.4 that the condition (i) of
Theorem 4.5 holds. This concerns only integral homology and cohomology classes
and so is preserved under perturbations of $L.$ Taking one of them which
underlies ${\mathbf{b}}$ we see from Theorem 4.5 that
$HF^{i}({\mathbf{b}},{\mathbf{b}})\neq 0$ as we want. Also in the case (ii)
the condition (ii) of Theorem 4.5 is preserved under perturbations of $L$ so
the conclusion follows in the same way. ∎

For special Lagrangians we can say more:

######  Corollary 4.7.

Let Hypothesis 4.1 hold with $X$ given an $\omega$-compatible almost-complex
structure $J$ and a $J$-holomorphic volume form. Then we have:

  * (i)

Then for $i=0,n$ and for every object ${\mathbf{b}}\in H{\mathbin{\cal F}}(X)$
supported near a closed immersed special Lagrangian, we have
$HF^{i}({\mathbf{b}},{\mathbf{b}})\neq 0.$

  * (ii)

Let $L_{1},L_{2}\subseteq X$ be two closed immersed special Lagrangians and
let $H{\mathbin{\cal F}}(X)$ have two isomorphic objects
${\mathbf{b}}_{1},{\mathbf{b}}_{2}$ supported respectively near $L_{1},L_{2}.$
Then $L_{1},L_{2}$ have up to shift the same grading.

###### Proof.

The part (i) follows immediately from Corollary 4.6 with $k=\infty.$ If (ii)
fails then by Lemma 2.6 we have $HF^{*}({\mathbf{b}}_{1},{\mathbf{b}}_{2})$
supported in degrees $i,\cdots,i+n-1$ for some $i\in{\mathbin{\mathbb{Z}}}.$
So either $HF^{0}({\mathbf{b}}_{1},{\mathbf{b}}_{2})=0$ or
$HF^{n}({\mathbf{b}}_{1},{\mathbf{b}}_{2})=0.$ But
${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ so either
$HF^{0}({\mathbf{b}}_{1},{\mathbf{b}}_{1})=0$ or
$HF^{n}({\mathbf{b}}_{1},{\mathbf{b}}_{1})=0.$ This however contradicts (i). ∎

##  5 Hamiltonian Perturbation Theorem

In this section we prove the following theorem. We work in the real analytic
category and for our applications in §6 we can take the underlying real
analytic structure of a complex manifold $(X,J).$ In fact any other real
analytic structure will do if it makes everything analytic but the statement
will then be too long.

######  Theorem 5.1.

(i) Let $(X,\omega,J)$ be a real analytic Kähler manifold of complex dimension
$n$ and $g$ a real analytic Hermitian metric on $(X,J).$ Let
$L_{1},L_{2}\subseteq X$ be two distinct closed irreducibly-immersed
$g$-Maslov-zero Lagrangians. Then there exists a neighbourhood $U$ of
$S\subseteq X$ and arbitrarily-small Hamiltonian deformations
$L_{1}^{\prime},L_{2}^{\prime}$ of $L_{1},L_{2}$ respectively which intersect
generically each other with no intersection point in $U$ of index $0$ or $n.$
This will still hold if $L_{1},L_{2}$ are $g$-Maslov-zero only near $L_{1}\cap
L_{2}.$ If also Hypothesis 4.1 holds and $L_{1},L_{2}$ nearly underlie
$H{\mathbin{\cal F}}(X)$ we can make $L_{1}^{\prime},L_{2}^{\prime}$ underlie
$H{\mathbin{\cal F}}(X).$
(ii) The assertion will hold even if $\omega,g$ are not analytic but if there
are $\omega$-compatible complex structures $J_{t},$ $J_{t}$-Hermitian metrics
$g_{t}$ and $g_{t}$-Maslov-zero Hamiltonian perturbations $L_{1t},L_{2t}$ of
$L_{1},L_{2}$ all parametrized smoothly by $t\in{\mathbin{\mathbb{R}}}$ with
$(J_{0},g_{0},L_{10},L_{20})=(J,g,L_{1},L_{2})$ and such that for $t\neq 0$
the $\omega,g_{t}$ are $J_{t}$-analytic.

######  Remark 5.2.

(i) The theorem is of local nature so in fact we need the $g$-Maslov-zero
condition and the real analyticity condition only near $L_{1}\cap L_{2}.$
(ii) The index in $U$ may be defined without grading
$L_{1}^{\prime},L_{2}^{\prime};$ that is, near $S$ we write $L_{2}^{\prime}$
over $L_{1}^{\prime}$ as the graph of an exact $1$-form $df$ and we take the
Morse indices of $f.$ This definition agrees with that in §2 if $L_{1},L_{2}$
are graded with the same grading on $L_{1}\cap L_{2}$ and if
$L_{1}^{\prime},L_{2}^{\prime}$ are graded by the obvious homotopies.

There are a few key estimates which we use for the proof of Theorem 5.1.
Firstly, according to Łojasiewicz [12] for every function
$f:{\mathbin{\mathbb{R}}}^{n}\to{\mathbin{\mathbb{R}}}$ with $f(0)=0$ and
analytic near $0\in{\mathbin{\mathbb{R}}}^{n}$ there exist two constants $c>0$
and $p\in(0,1)$ such that near $0\in{\mathbin{\mathbb{R}}}^{n}$ we have
$|f|^{p}\leqslant\nobreak c|df|.$ This implies readily that we have:

######  Lemma 5.3.

In the circumstances of Theorem 5.1 (i) take a Weinstein neighbourhood of
$L_{2}\subseteq(X,\omega)$ and write $L_{1}$ near $S$ as the graph over
$L_{2}$ of a closed $1$-form $u.$ Then near $S$ we have $|u|\ll|\nabla u|\ll
1;$ that is, for every $\epsilon>0$ there exists a neighbourhood of
$S\subseteq L_{2}$ on which $|u|\leqslant\nobreak\epsilon|\nabla
u|\leqslant\nobreak\epsilon^{2}.$ Here $|\bullet|,\nabla$ are both computed
with respect to the induced metric on $L_{2}.$

We show next that the Maslov $1$-form operator may be well approximated by
$dd^{*}$:

######  Lemma 5.4.

Let $(X,\omega,J)$ be a Kähler manifold and $g$ a Hermitian metric on $(X,J).$
Let $L\subseteq X$ be a closed immersed $g$-Maslov-zero Lagrangian, and $NL$
the $g$-normal bundle to $L\subseteq X.$ Then there exists a constant $c>0$
such that for every sufficiently $C^{1}$-small $v\in C^{\infty}(NL)$ whose
graph $($embedded in $X$ by the $g$-exponential map$)$ is $g$-Maslov-zero, we
have at every point of $L$

$|dd^{*}\hat{v}|\leqslant\nobreak c|v|_{C^{1}}(1+|v|_{C^{2}})$ (5.1)

where $\hat{v}$ is the same as in Lemma 3.6, $|v|_{C^{1}}:=|v|+|\nabla v|$ and
$|v|_{C^{2}}:=|v|+|\nabla v|+|\nabla^{2}v|;$ the latter two are computed
pointwise on $L$ with respect to $g.$

###### Proof.

Let $v\in C^{\infty}(NL)$ be sufficiently $C^{1}$-small and $Av\in
C^{\infty}(T^{*}L)$ the Maslov $1$-form of the graph of $v.$ The tangent plane
to the graph of $v$ involves the first derivative $\nabla v$ so the canonical
section of $K_{X}^{2}$ over the graph of $v$ is of the form $B(v,\nabla v)$
where $B:NL\oplus(T^{*}L\otimes NL)\to NL$ is a smooth function. Then $Av$
involves the differentiation of $B(v,\nabla v)$ so

$Av=A_{1}(v,\nabla v)+A_{2}(v,\nabla v)\cdot\nabla^{2}v$ (5.2)

where $A_{1}:NL\oplus(T^{*}L\otimes NL)\to NL$ and
$A_{2}:NL\oplus(T^{*}L\otimes NL)\to NL\otimes TL\otimes TL$ are some smooth
functions. Denote by $A^{\prime}$ the linearization of $A$ at $0\in
C^{\infty}(NL).$ Then by (5.2) there is a $v$-independent constant $c_{1}>0$
such that

$|(A-A^{\prime})v|\leqslant\nobreak c_{1}[|v|^{2}+|\nabla
v|^{2}+|\nabla^{2}v|(|v|+|\nabla v|)]\leqslant\nobreak
c_{1}|v|_{C^{1}}|v|_{C^{2}}$ (5.3)

where the same constant $c_{1}$ will do for both the inequalities. Now by
Lemma 3.6 $A^{\prime}$ is, up to lower-order terms, equal to $dd^{*}\hat{v};$
that is, there is a $v$-independent constant $c_{2}>0$ such that

$|A^{\prime}v-dd^{*}\hat{v}|\leqslant\nobreak c_{2}|v|_{C^{1}}.$

So by (5.3) we have
$|(A-A^{\prime})v|\leqslant\nobreak\max\\{c_{1},c_{2}\\}|v|_{C^{1}}(1+|v|_{C^{2}}).$
This implies indeed that if $Av=0$ then (5.1) holds. ∎

We give a corollary to the lemma above:

######  Corollary 5.5.

In the circumstances of Lemma 5.4 take a Weinstein neighbourhood of
$L\subseteq X.$ Then there exists a constant $c>0$ such that for every
sufficiently $C^{1}$-small $1$-form $u\in C^{\infty}(T^{*}L)$ whose graph is
also minimal, we have at every point of $L$

$|dd^{*}u|\leqslant\nobreak c|u|_{C^{1}}(1+|u|_{C^{2}}).$ (5.4)

###### Proof.

The two embeddings, one given by the exponential map and the other given by
the Weinstein neighbourhood, are mutually related by a diffeomorphism $F$
between neighbourhoods of the zero sections of $T^{*}L,NL$ respectively which
induces the identity on the zero section $L,$ and over every point of $L,$ a
diffeomorphism of the two fibres. So there is a smooth function
$F_{1}:T^{*}L\to NL\otimes TL$ such that for every sufficiently $C^{1}$-small
$u\in C^{\infty}(T^{*}L)$ we have $Fu=F_{1}u\cdot u.$ In particular there is a
$u$-independent constant $c>0$ such that

$|Fu|_{C^{1}}\leqslant\nobreak|u|_{C^{1}}\text{ and
}|Fu|_{C^{2}}\leqslant\nobreak c|u|_{C^{2}}.$ (5.5)

The function $Gu:=-JFu\lrcorner g$ is also of the same form; that is, there is
a smooth function $G_{1}:T^{*}L\to T^{*}L\otimes TL$ such that $Gu=G_{1}u\cdot
u.$ This implies in turn that we have, making $c$ large enough,

$|dd^{*}Gu|\leqslant\nobreak c|dd^{*}u|+c|u|_{C^{1}}(1+|u|_{C^{2}})$ (5.6)

Now by (5.1) we have, making $c$ large enough,

$|dd^{*}Gu|\leqslant\nobreak c|Fu|_{C^{1}}(1+|Fu|_{C^{2}}).$

So by (5.6) we have (5.4) with $c$ large enough. ∎

Now we prove Theorem 5.1 in three steps:

###### Step 1: proof of the first part of (i).

In this case we take $L_{1}^{\prime}=L_{1}$ and perturb only $L_{2}.$ We take
a Weinstein neighbourhood of $L_{2}\subseteq(X,\omega)$ and define
$L_{2}^{\prime}$ as the graph of a certain exact $1$-form $df$ on $L_{2}$ with
$f:L_{2}\to{\mathbin{\mathbb{R}}}$ taken as follows. Since $L_{1},L_{2}$ are
both connected and analytic with $L_{1}\neq L_{2}$ as subsets of $X$ it
follows that $L_{2}\setminus L_{1}$ is dense in $L_{2}.$ So there is a Morse
function $f:L_{2}\to{\mathbin{\mathbb{R}}}$ with $df\neq 0$ on $L_{1}\cap
L_{2}.$ Since $S\subseteq L_{1}\cap L_{2}$ it follows then that there is a
constant $c>0$ such that we have near $S$

$|f|_{C^{2}}<c|df|.$ (5.7)

This condition is preserved under rescalings and smooth perturbations of $f$
so we can make $L_{2}^{\prime}$ arbitrarily $C^{\infty}$ close to $L_{2}$ and
transverse to $L_{1}.$

Near $S$ take a local component of $L_{1}$ if need be (in the strictly
immersed case) and write it as the graph over $L_{2}$ of some closed $1$-form
$u.$ This is possible by the definition of $S$ and in fact we have also
$u=\nabla u=0$ on $S.$ So by Lemma 5.3 we have near $S$

$|u|\ll|\nabla u|\ll 1.$ (5.8)

On the other hand, by Corollary 5.5 with $L_{2}$ in place of $L,$ there is a
constant $c^{\prime}>0,$ which depends upon $|u|_{C^{2}},$ such that we have
near $S$

$|dd^{*}u|\leqslant\nobreak c^{\prime}(|u|+|\nabla u|)\leqslant\nobreak
2c^{\prime}|\nabla u|\ll 1$ (5.9)

where the second inequality follows from (5.8). Applying to $d^{*}u$ the
Łojasiewicz estimate, we find a constant $\epsilon>0$ such that we have near
$S$

$|d^{*}u|\leqslant\nobreak|dd^{*}u|^{1+\epsilon}\leqslant\nobreak(2c^{\prime}|\nabla
u|)^{1+\epsilon}\ll|\nabla u|$ (5.10)

where the second inequality follows from (5.9).

We show now that we have on $L_{1}\cap L_{2}^{\prime}$ near $S$

$|d^{*}(u-df)|\leqslant\nobreak n^{-1/2}|\nabla(u-df)|.$ (5.11)

On $L_{1}\cap L_{2}^{\prime}$ near $S$ we have $u=df$ and so by (5.7)

$|\nabla(u-df)|\geqslant|\nabla u|-|\nabla df|\geqslant|\nabla
u|-c|df|=|\nabla u|-c|u|\geqslant\frac{1}{2}|\nabla u|$ (5.12)

where the last inequality follows from (5.8). On the other hand by (5.10) we
have on $L_{1}\cap L_{2}^{\prime}$ near $S$

$|d^{*}(u-df)|\leqslant\nobreak|d^{*}u|+c|df|\leqslant\nobreak\frac{1}{4}n^{-1/2}|\nabla
u|+c|u|\leqslant\nobreak\frac{1}{2}n^{-1/2}|\nabla u|.$ (5.13)

where the last inequality follows again from (5.8). By (5.12) and (5.13) we
have (5.11) as we want.

Finally we take a point $x\in L_{1}\cap L_{2}^{\prime}$ near $S$ and compute
its index, which is the index of the nondegenerate symmetric bilinear form
$\nabla(u-df)$ on $T_{x}L_{2}.$ Diagonalize it by an orthonormal basis of
$T_{x}L_{2}$ and denote by $\lambda_{1},\cdots,\lambda_{n}$ the diagonal
entries. Then by (5.11) we have

$\textstyle|\lambda_{1}+\cdots+\lambda_{n}|\leqslant\nobreak
n^{-1/2}(\lambda_{1}^{2}+\cdots+\lambda_{n}^{2})^{1/2}\leqslant\nobreak\max\\{|\lambda_{1}|,\cdots,|\lambda_{n}|\\}.$

But $\lambda_{1},\cdots,\lambda_{n}$ are all nonzero so they have not all the
same sign; that is, the index is neither $0$ nor $n$ as we want. ∎

###### Step 2: proof of the latter part of (i).

We make first a generic Hamiltonian perturbation of $L_{2}$ which underlies
$H{\mathbin{\cal F}}(X)$ and we define $L_{2}^{\prime}$ as a further
Hamiltonian perturbation of it. As in the step 1 we take a Weinstein
neighbourhood of $L_{2}$ and define $L_{2}^{\prime}$ as the graph of some
exact $1$-form $dh_{2}$ over $L_{2}$ with $h_{2}\neq 0$ on $S.$ Also as in the
step 1 write $L_{1}$ near $S$ as the graph of some real analytic closed
$1$-form $u$ on $L_{2}$ near $S.$ Take a generic Hamiltonian perturbation
$L_{1}^{\prime}$ of $L_{1}$ which underlies $H{\mathbin{\cal F}}(X)$ and is
close enough to $L_{1}$ to be written near $S$ as the graph of $u+dh_{1}$ over
$L_{2}$ with

$|h_{1}|_{C^{2}}\ll|h_{2}|_{C^{2}}.$ (5.14)

Since $dh_{2}\neq 0$ on $S$ it follows that there is a constant $c>0$ such
that near $S$ we have $|h_{2}|_{C^{2}}<\frac{c}{4}|dh_{2}|.$ And then putting
$f:=h_{2}-h_{1}$ we have

$|h_{2}|_{C^{2}}<\frac{c}{4}(|df|+|dh_{1}|)\leqslant\nobreak\frac{c}{4}|df|+\frac{1}{2}|h_{2}|_{C^{2}}$
(5.15)

where the last inequality follows from (5.14). The whole estimate (5.15)
implies $|h_{2}|_{C^{2}}<\frac{c}{2}|df|.$ Hence using again (5.14) we find

$|f|_{C^{2}}\leqslant\nobreak|h_{2}|_{C^{2}}+|h_{1}|_{C^{2}}<\frac{c}{2}|df|+|h_{2}|_{C^{2}}<c|df|;$

that is, the estimate (5.7) holds as in the step 1. Following the subsequent
estimates we see also that (5.11) holds now on $L_{1}^{\prime}\cap
L_{2}^{\prime}$ near $S.$ This implies again that
$L_{1}^{\prime},L_{2}^{\prime}$ have near $S$ no intersection point of index
$0$ or $n.$ ∎

###### Step 3: proof of Theorem 5.1 (ii).

As in the step 2 we make Hamiltonian perturbations
$L_{1}^{\prime},L_{2}^{\prime}$ of $L_{1},L_{2}$ both underlying
$H{\mathbin{\cal F}}(X).$ The $L_{2}^{\prime}$ is the graph of $dh_{2}$ over
$L_{2}$ and the $L_{1}^{\prime}$ near $S$ is the graph of $u+dh_{1}$ over
$L_{2}$ near $S.$ The difference $f:=h_{2}-h_{1}$ satisfies (5.7) as in the
step 1. We extend these to smooth families. By Theorem 3.7 there are
$g_{t}$-Maslov-zero Hamiltonian perturbations $L_{1t},L_{2t}$ of
$L_{1},L_{2}.$ Extend the Weinstein neighbourhood of $L_{2}$ to those of
$L_{2t}$ and write $L_{2}^{\prime}$ as the graph of some $dh_{2t}$ over
$L_{2t}.$ Denote by $S_{t}\subseteq L_{1t}\cap L_{2t}$ the set of intersection
points at which $L_{1t},L_{2t}$ have at least one common tangent plane. Write
$L_{1t}$ near $S_{t}$ as the graph of a real analytic closed $1$-form $u_{t}$
over $L_{2t}$ near $S_{t}$ and write $L_{1}^{\prime}$ near $S_{t}$ as the
graph of $u_{t}+dh_{1t}$ over $L_{2t}$ near $S_{t}.$ Then with $t$ small
enough the estimate (5.7) holds for $f_{t}:=h_{2t}-h_{1t}$ in place of $f$ and
we can follow the subsequent estimates in the step 1. We can do this in a
$t$-independent neighbourhood $U$ of $S\subseteq X$ because $S_{t}$ tends to
$S.$ So $L_{1}^{\prime},L_{2}^{\prime}$ have in $U$ no intersection point of
index $0$ or $n.$ ∎

##  6 Conclusions

From Lemma 2.7 and Theorem 5.1 we deduce:

######  Theorem 6.1.

Let $(X,\omega,J)$ be a real analytic Kähler manifold of complex dimension
$n,$ ${\mathbin{\mathbb{Z}}}_{k}$-graded with $k\geqslant n$ and given another
real analytic Hermitian metric $g.$ Let $L_{1},L_{2}\subseteq X$ two distinct
closed irreducibly-immersed $g$-Maslov-zero Lagrangians with the same grading
on $L_{1}\cap L_{2}.$ Then there exist arbitrarily-small Hamiltonian
deformations $L_{1}^{\prime},L_{2}^{\prime}$ of $L_{1},L_{2}$ which intersect
generically each other with no intersection point of index $0$ or $n.$ If also
Hypothesis 4.1 holds and $L_{1},L_{2}$ nearly underlie $H{\mathbin{\cal
F}}(X)$ we can make $L_{1}^{\prime},L_{2}^{\prime}$ underlying
$H{\mathbin{\cal F}}(X).$

###### Proof.

Let $U$ be as in Theorem 5.1 and corresponding to this $U$ let $\epsilon$ be
as in Lemma 2.7. Then corresponding to this $\epsilon$ let $L_{i}^{\prime}$
$(i=1,2)$ be such $\epsilon$-perturbations of $L_{i}$ as in Theorem 5.1.
Applying Lemma 2.7 to the intersection points outside $U$ and Theorem 5.1 to
those in $U$ we see that $L_{1}^{\prime},L_{2}^{\prime}$ are such as we want.
∎

We make now the $C^{\infty}$ version of Theorem 6.1:

######  Theorem 6.2.

Let $(X,\omega,J)$ be a Kähler manifold of complex dimension $n$ which is
either: ${\mathbin{\mathbb{Z}}}$-graded by a $J$-holomorphic volume form
$\Psi$ and given the conformally Kähler metric $g$ as in $\S\ref{sec2}$ by
using $\eqref{ps};$ or ${\mathbin{\mathbb{Z}}}_{k}$-graded with $k\geqslant n$
and given another Kähler metric $g$ whose the Ricci $(1,1)$-form is $-\omega.$
Let $L_{1},L_{2}\subseteq(X,\omega,J,g)$ be two distinct closed irreducibly-
immersed $g$-Maslov-zero Lagrangians with the same grading on $L_{1}\cap
L_{2}.$ Suppose that one of the following three conditions holds: (i) $X$ is
compact; (ii) $X$ is $\omega$-convex, $\omega$ is exact and $J$ is Stein; or
(iii) either $L_{1}$ or $L_{2}$ is generically immersed. Then the same
conclusion holds as in Theorem 6.1.

###### Proof.

We reduce the problem to the real analytic case by a further perturbation
process. In the case (i) we use the fact that for every compact Kähler
manifold $(X,\omega,J)$ there exists on $(X,J)$ a smooth family
$(\omega_{t})_{t\in{\mathbin{\mathbb{R}}}}$ of Kähler forms with
$\omega_{0}=\omega$ and $\omega_{t},$ $t\neq 0,$ analytic. This seems well
known but for clarity we give it a proof. Denote by $g_{\omega}$ the Kähler
metric of $(X,\omega,J)$ and take on $(X,J)$ a smooth family of Riemannian
metrics $g_{t}^{\prime}$ with $g_{0}^{\prime}=g_{\omega}$ and
$g_{t}^{\prime},$ $t\neq 0,$ analytic. Then
$g_{t}:={\frac{1}{2}}(g_{t}^{\prime}+J^{*}g_{t}^{\prime})$ defines on $(X,J)$
a smooth family of Hermitian metrics with $g_{0}=g_{\omega}$ and $g_{t},$
$t\neq 0,$ analytic. Denote by ${\partial}_{t}^{*},\bar{\partial}_{t}^{*}$ the
respective formal $g_{t}$-adjoints of ${\partial},\bar{\partial}$ and
following Kodaira and Spencer [11, §6] introduce the smooth family of elliptic
operators

$E_{t}:={\partial}\bar{\partial}\bar{\partial}_{t}^{*}{\partial}_{t}^{*}+\bar{\partial}_{t}^{*}{\partial}_{t}^{*}{\partial}\bar{\partial}+\bar{\partial}_{t}^{*}{\partial}{\partial}_{t}^{*}\bar{\partial}+{\partial}_{t}^{*}\bar{\partial}\bar{\partial}_{t}^{*}{\partial}+{\partial}_{t}^{*}{\partial}+\bar{\partial}_{t}^{*}\bar{\partial}.$

Since $g_{0}$ is Kähler it follows readily that $E_{0}$ is apart from the last
two terms just the squared Laplacian. The last two terms are so added that the
kernel of $E_{t}$ consists of closed forms. Denote by $\omega_{t}^{\prime}$
the projection of $\omega$ onto the kernel of $E_{t}$ in the space of
$(1,1)$-forms. Then by Kodaira–Spencer’s result [11, Proposition 8] the $\ker
E_{t}$ has dimension independent of $t.$ So by another result of them [11,
Theorem 5] the $\omega_{t}^{\prime}$ is smooth with respect to $t.$ Putting
$\omega_{t}:={\frac{1}{2}}(\omega_{t}+\overline{\omega}_{t})$ we see that
$\omega_{t}$ is such as we want.

In the case (ii) we write $\omega=dJdp$ for some
$p:X\to{\mathbin{\mathbb{R}}}$ and perturb $p$ to real analytic functions. In
the case (iii) we take a Stein open set in $(X,J)$ on which $\omega$ is exact
and which contains the generically immersed Lagrangian. This is possible
because every its self-intersection point is a transverse double point. So as
a plurisubharmonic function near it we can take the product of squared
distances from the two components of the generically immersed Lagrangian.

In every case there are Kähler perturbations $\omega_{t}$ of $\omega$ and by
Moser’s theorem diffeomorphisms $\Phi_{t}:X\to X$ with
$\omega=\Phi_{t}^{*}\omega_{t}.$ Then $\omega$ is analytic with respect to
$J_{t}:=\Phi_{t}^{*}J$ and for $i=1,2$ the Lagrangian
$\Phi_{t}^{*}L_{i}\subseteq(X,\omega)$ is nearly-zero Maslov-form relative to
$g_{t}:=\Phi_{t}^{*}g.$ So by Theorem 3.7 there is a $g_{t}$-Maslov-zero
Hamiltonian perturbation $L_{it}$ of $\Phi_{t}^{*}L_{i}$ to which we can apply
Theorem 5.1 (ii). Combining it with Lemma 2.7 as in the proof of Theorem 6.1
we see that Theorem 6.2 holds. ∎

Finally using the results of §4 we get:

######  Corollary 6.3.

(i) Let $(X,\omega,J,g,L_{1},L_{2})$ be such as in Theorem 6.1 or Theorem 6.2
except that $L_{1},L_{2}\subseteq X$ may be equal. Let Hypothesis 4.1 hold and
let ${\mathbf{b}}_{1},{\mathbf{b}}_{2}\in H{\mathbin{\cal F}}(X)$ be two
objects supported respectively near $L_{1},L_{2}$ such that:

either ${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ or $k\geqslant 2n-1$ and
${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ up to shift. (6.1)

Suppose also that

$HF^{i}({\mathbf{b}}_{1},{\mathbf{b}}_{1})\cong
HF^{i}({\mathbf{b}}_{2},{\mathbf{b}}_{2})\neq 0\text{ for }i=0,n.$ (6.2)

Or more generally, suppose as in Corollary 4.6 that either:
$c_{1}(X)\leqslant\nobreak 0$ and one of $L_{1},L_{2}$ is embedded; or
$k\geqslant n+2$ and one of $L_{1},L_{2}$ has on its every self-intersection
pair the two gradings nearly equal. Then $L_{1}=L_{2}\subseteq X.$
(ii) Let $(X,\omega,J)$ be a Kähler manifold equipped with a holomorphic
volume form; and $L_{1},L_{2}\subseteq X$ two closed irreducibly-immersed
special Lagrangians. Suppose that either $\omega$ is $J$-analytic or one of
(i)–(iii) of Theorem 6.2 holds. Let Hypothesis 4.1 hold and let
$H{\mathbin{\cal F}}(X)$ have two isomorphic objects supported respectively
near $L_{1},L_{2}.$ Then $L_{1}=L_{2}\subseteq X.$

###### Proof.

Suppose contrary to the assertion that $L_{1}\neq L_{2}.$ Then perturbing
$L_{1},L_{2}$ as in Theorem 6.1 or Theorem 6.2 we see that
$HF^{*}({\mathbf{b}}_{1},{\mathbf{b}}_{2})$ is supported in degrees
$1,\cdots,n-1$ modulo $k{\mathbin{\mathbb{Z}}}.$ But by hypothesis we have
$k\geqslant n$ so
$HF^{0}({\mathbf{b}}_{1},{\mathbf{b}}_{2})=HF^{n}({\mathbf{b}}_{1},{\mathbf{b}}_{2})=0.$
If also ${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ then
$HF^{0}({\mathbf{b}}_{1},{\mathbf{b}}_{1})=HF^{n}({\mathbf{b}}_{1},{\mathbf{b}}_{1})=0$
which however contradicts (6.2) or Corollary 4.6. This is the first case of
(6.1).

In the second case we have $k\geqslant 2n-1$ but
${\mathbf{b}}_{1}\cong{\mathbf{b}}_{2}$ only up to shift. Then from Theorem
6.1 we see only that $HF^{*}({\mathbf{b}}_{1},{\mathbf{b}}_{1})$ is supported
in degrees $i,\cdots,i+n-2$ for some $i\in{\mathbin{\mathbb{Z}}}.$ But again
we have $HF^{0}({\mathbf{b}}_{1},{\mathbf{b}}_{1})\neq 0$ and
$HF^{n}({\mathbf{b}}_{1},{\mathbf{b}}_{1})\neq 0$ so after translating
$[i,i+n-2]$ by $k{\mathbin{\mathbb{Z}}}$ we may suppose $0\in[i,i+n-2]$ and
$n\in[i+j,i+j+n-2]$ for some $j\in k{\mathbin{\mathbb{Z}}}.$ Then
$i\in[2-n,0]\cap[2-j,n-j]$ so $0\geqslant 2-j$ and $2-n\leqslant\nobreak n-j;$
that is, $j\in[2,2n-2].$ So $0<j<2n-1\leqslant\nobreak k$ which however
contradicts $j\in k{\mathbin{\mathbb{Z}}}.$ This proves Corollary 6.3 (i).

For Corollary 6.3 (ii) we use Corollary 4.7 (ii) to see that $L_{1},L_{2}$
have up to shifts the same grading. We can then apply Corollary 6.3 (i) with
$k=\infty$ which completes the proof. ∎

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