two such systems. The lemma follows. ∎

We remark $c_{\beta_{i}}=1$ where $\beta_{i}$ ($i=1,\cdots,m$) are the classes
corresponding to each of the irreducible components of the divisor
$\pi^{-1}(\partial P)$. If $X$ is Fano, then $c_{\beta}=0$, for
$\beta\neq\beta_{i}$. But this may not be the case if $X$ is not Fano.

We now will use our perturbed moduli space to define a structure of filtered
$A_{\infty}$ algebra on the de Rham cohomology
$H(L(u);\Lambda_{0}^{{\mathbb{R}}})\cong(H(L(u),{\mathbb{R}}))\otimes\Lambda_{0}$.
We will write it $\mathfrak{m}^{can}$.

We take a $T^{n}$ equivariant Riemannian metric on $L(u)$. We observe that a
differential form $\rho$ on $L(u)$ is harmonic if and only if $\rho$ is
$T^{n}$ equivariant. So we identify $H(L(u),{\mathbb{R}})$ with the set of
$T^{n}$ equivariant forms from now on.

We consider the evaluation map

$ev=(ev_{1},\cdots,ev_{k},ev_{0}):\mathcal{M}^{\text{\rm
main}}_{k+1}(L(u),\beta)^{\mathfrak{s}_{\beta}}\to L(u)^{k+1}.$

Let $\rho_{1},\cdots,\rho_{k}$ be $T^{n}$ equivariant differential forms on
$L(u)$. We define

$\mathfrak{m}^{can}_{k,\beta}(\rho_{1},\cdots,\rho_{k})=(ev_{0})_{!}(ev_{1},\cdots,ev_{k})^{*}(\rho_{1}\wedge\cdots\wedge\rho_{k}).$
(11.6)

We remark that integration along fiber $(ev_{0})_{!}$ is well defined and
gives a smooth form, since $ev_{0}$ is a submersion. (It is a consequence of
$T^{n}$ equivariance.) More precisely, we apply Definition 16.7 in section 16
as follows. We put $\mathcal{M}=\mathcal{M}^{\text{\rm
main}}_{k+1}(L(u),\beta)$, $L_{s}=L^{k}$, $L_{t}=L$.
$ev_{s}=(ev_{1},\cdots,ev_{k}):\mathcal{M}\to L_{s}$,
$ev_{t}=ev_{0}:\mathcal{M}\to L_{t}$. Thus we are in the situation we
formulate at the beginning of section 16. Then using Lemma 16.9 and Remark
16.8 (1), we put

$\mathfrak{m}^{can}_{k,\beta}(\rho_{1},\cdots,\rho_{k})=(\mathcal{M};ev_{s},ev_{t},\mathfrak{s}_{\beta})_{*}(\rho_{1}\times\cdots\times\rho_{k}).$
(11.7)

We remark that the right hand side of (11.7) is again $T^{n}$ equivariant
since $\mathfrak{s}_{\beta}$ etc. are $T^{n}$ equivariant.

Lemma 11.2 (4) implies

$\displaystyle\partial\mathcal{M}^{\text{\rm main}}_{k+1}(L(u),\beta)$
$\displaystyle=\bigcup_{k_{1}+k_{2}=k+1}\bigcup_{\beta_{1}+\beta_{2}=\beta}\bigcup_{l=1}^{k_{2}}\mathcal{M}^{\text{\rm
main}}_{k_{1}+1}(L(u),\beta_{1}){}_{ev_{0}}\times_{ev_{l}}\mathcal{M}^{\text{\rm
main}}_{k_{2}+1}(L(u),\beta_{2}).$

Therefore using Lemmata 16.9, 16.10, we have the following formula.

$\sum_{\beta_{1}+\beta_{2}=\beta}\sum_{k_{1}+k_{2}=k+1}\sum_{l=1}^{k_{1}}(-1)^{*}\mathfrak{m}^{can}_{k_{1},\beta_{1}}(\rho_{1},\cdots,\mathfrak{m}^{can}_{k_{2},\beta_{2}}(\rho_{l},\cdots),\cdots,\rho_{k}).$
(11.8)

Here $*=\sum_{i=1}^{l}(\deg\rho_{i}+1)$. (See the end of section 16 for sign.)
We now put

$\mathfrak{m}_{k}^{can}(\rho_{1},\cdots,\rho_{k})=\sum_{\beta}T^{\beta\cap\omega/(2\pi)}\mathfrak{m}_{k,\beta}^{can}(\rho_{1},\cdots,\rho_{k}).$
(11.9)

We extend (11.9) to $\rho\in H(L(u);\Lambda^{{\mathbb{R}}}_{0})$ such that it
is $\Lambda_{0}$ multilinear. Then (11.8) implies that it defines a structure
of a filtered $A_{\infty}$ structure on $H(L(u);\Lambda_{0}^{{\mathbb{R}}})$
in the sense of section 3.

We also remark that our filtered $A_{\infty}$ algebra is unital and the
constant $0$ form $1\in H^{0}(L;{\mathbb{R}})$ is a unit. This is a
consequence of Lemma 11.2 (3).

We next calculate our filtered $A_{\infty}$ structure in the case when
$\rho_{i}$ are degree $1$ forms.

######  Lemma 11.8.

For $\mathfrak{x}\in H^{1}(L(u),\Lambda_{0})$ and $\beta\in\pi_{2}(X,L)$ with
$\mu(\beta)=2$, we have

$\mathfrak{m}_{k,\beta}^{can}(\mathfrak{x},\cdots,\mathfrak{x})=\frac{c_{\beta}}{k!}(\partial\beta\cap\mathfrak{x})^{k}\cdot
PD([L(u)]).$

Here $PD([L(u)])$ is the Poincaré dual to the fundamental class. In other
words it is the $n$ form with $\int_{L(u)}PD([L(u)])=1$.

###### Proof.

It suffices to consider the case $\mathfrak{x}=\rho\in
H^{1}(L(u);{\mathbb{R}})$, and show

$\int_{L(u)}\mathfrak{m}^{can}_{k,\beta}(\rho,\cdots,\rho)=\frac{c_{\beta}}{k!}(\partial\beta\cap\mathfrak{x})^{k}.$
(11.10)

Let

$C_{k}=\\{(t_{1},\cdots,t_{k})\mid 0\leq t_{1}\leq\cdots\leq t_{k}\leq 1\\}.$
(11.11)

We define an iterated blow-up, denoted by $\widehat{C}_{k}$, of $C_{k}$ in the
following way. Let $S=\partial D$ be the boundary of the unit disc
$D=D^{2}\subset{\mathbb{C}}$ and $\beta_{D}\in H_{2}({\mathbb{C}},S)$ be the
homology class of the unit disc. We consider the moduli space
$\mathcal{M}_{k+1}({\mathbb{C}},S;\beta_{D})$ and the evaluation map
$\vec{ev}=(ev_{0},\cdots,ev_{k}):\mathcal{M}_{k+1}({\mathbb{C}},S;\beta_{D})\to(S^{1})^{k+1}.$
We fix a point $p_{0}\in S\subset{\mathbb{C}}$ and put

$\widehat{C}_{k}:=ev_{0}^{-1}(p_{0})\subset\mathcal{M}_{k+1}({\mathbb{C}},S;\beta_{D}).$

We make the identification $S^{1}\setminus\\{p_{0}\\}\cong(0,1)$. Then
$\vec{ev}$ induces a diffeomorphism

$\widehat{C}_{k}\cap\mathcal{M}_{k+1}^{\rm
reg}({\mathbb{C}},S;\beta_{D})\to\text{\rm Int }C_{k}$

given by

$[w,z_{0},\cdots,z_{k}]\mapsto(w(z_{1})-w(z_{0}),\cdots,w(z_{k})-w(z_{0}))$

where

$\text{\rm Int }C_{k}=\\{(t_{1},\cdots,t_{k})\mid
0<t_{1}<\cdots<t_{k}<1\\}\subset C_{k}.$

In this sense $\widehat{C}_{k}$ is regarded as an iterated blow up of $C_{k}$
along the diagonal (that is the set of points where $t_{i}=t_{i+1}$ for some
$i$). We identify $\partial D=S\cong{\mathbb{R}}/{\mathbb{Z}}\cong S^{1}$. We
have

$\mathcal{M}^{\text{\rm
main}}_{k+1}(L(u),\beta)^{\mathfrak{s}}\cong\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)^{\mathfrak{s}}\times\widehat{C}_{k}.$ (11.12)

In fact, Corollary 11.5 implies $\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)^{\mathfrak{s}}$ consists of finitely many free $T^{n}$
orbits (with multiplicity $\in{\mathbb{Q}}$) and $\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)^{\mathfrak{s}}=\mathcal{M}^{\text{\rm
main,reg}}_{1}(L(u),\beta)^{\mathfrak{s}}$. By Lemma 11.2 (3) we have a map
$\mathcal{M}^{\text{\rm
main,reg}}_{k+1}(L(u),\beta)^{\mathfrak{s}}\to\mathcal{M}^{\text{\rm
main,reg}}_{1}(L(u),\beta)^{\mathfrak{s}}$. It is easy to see that the fiber
can be identified with $\widehat{C}_{k}$.

Under this identification, the evaluation map $\vec{ev}$ is induced by

$ev_{i}(\mathfrak{u};t_{1},\cdots,t_{k})=[t_{i}\partial\beta]\cdot
ev(\mathfrak{u})$ (11.13)

for $(\mathfrak{u};t_{1},\cdots,t_{k})\in\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)\times\text{\rm Int }C_{k}\subset\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)\times\widehat{C}_{k}$.

Here $\partial\beta\in H_{1}(L(u);{\mathbb{Z}})$ is identified to an element
of the universal cover $\widetilde{L}(u)\cong{\mathbb{R}}^{n}$ of $L(u)$ and
$[t_{i}\partial\beta]\in L(u)$ acts as a multiplication on the torus.
$ev(\mathfrak{u})$ is defined by the evaluation map $ev:\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)\to L(u)$. We also have :

$ev_{0}(\mathfrak{u};t_{1},\cdots,t_{k})=ev(\mathfrak{u}).$ (11.14)

We remark that $ev:\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i})\to L(u)$
is a diffeomorphism (See Theorem 11.1 (3)). Now we have

$\int_{L(u)}\mathfrak{m}^{dR}_{k,\beta}(\rho,\cdots,\rho)=c_{\beta}\text{\rm
Vol}(C_{k})\left(\int_{\partial\beta}\rho\right)^{k}=\frac{c_{\beta}}{k!}(\partial\beta\cap\mathfrak{x})^{k}.$

The proof of Lemma 11.8 is now complete. ∎

######  Remark 11.9.

We can prove that our filtered $A_{\infty}$ algebra
$(H(L(u);\Lambda_{0}^{{\mathbb{R}}}),\mathfrak{m}_{*}^{can})$ is homotopy
equivalent to the one in [FOOO3] Theorem A (= [FOOO2] Theorem A). The proof is
a straight forward generalization of section 7.5 [FOOO3] (= section 33 of
[FOOO2]) and is omitted here. In fact we do not need to use this fact to prove
Theorem 1.5 if we use de Rham version in all the steps of the proof of Theorem
1.5 without involving the singular homology version.

######  Remark 11.10.

We constructed our filtered $A_{\infty}$ structure directly on de Rham
cohomology group $H(L(u);\Lambda_{0}^{{\mathbb{R}}})$. The above construction
uses the fact that wedge product of harmonic forms are again harmonic. This is
a special feature of our situation where our Lagrangian submanifold $L$ is a
torus. (In other words we use the fact that rational homotopy type of $L$ is
formal.)

Alternatively we can construct filtered $A_{\infty}$ structure on the de Rham
complex
$\Omega(L(u))\widehat{\otimes}_{{\mathbb{R}}}\Lambda_{0}^{{\mathbb{R}}}$ and
reduce it to the de Rham cohomology by homological algebra. Namely we consider
smooth forms $\rho_{i}$ which are not necessarily harmonic and use (11.6) and
(11.9) to define $\mathfrak{m}_{k}^{dR}(\rho_{1},\cdots,\rho_{k})$. (The proof
of $A_{\infty}$ formula is the same.) Using the formality of $T^{n}$, we can
show that the canonical model of
$(\Omega(L(u))\widehat{\otimes}_{{\mathbb{R}}}\Lambda_{0}^{{\mathbb{R}}},\mathfrak{m}^{dR}_{*})$
is the same as $(H(L(u);\Lambda_{0}^{{\mathbb{R}}}),\mathfrak{m}^{can})$. We
omit its proof since we do not use it.

Using the continuous family of perturbations, this construction can be
generalized to the case of arbitrary relatively spin Lagrangian submanifold in
a symplectic manifold. See [Fu5].

###### Proof of Proposition 4.3.

Proposition 4.3 immediately follows from Corollary 11.5, Lemma 11.8 and Lemma
11.9 : We just take the sum

$\displaystyle\sum_{k=0}^{\infty}\mathfrak{m}^{can}_{k}(b,\cdots,b)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{\beta\in\pi_{2}(X,L(u))}T^{\omega\cap\beta/2\pi}\mathfrak{m}^{can}_{k,\beta}(b,\cdots,b)$
(11.15) $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\sum_{\beta}T^{\omega\cap\beta/2\pi}\mathfrak{m}^{can}_{k,\beta}(b,\cdots,b)$
$\displaystyle=$
$\displaystyle\sum_{\beta}\sum_{k=0}^{\infty}\frac{c_{\beta}}{k!}(\partial\beta\cap
b)^{k}T^{\beta\cap\omega/2\pi}\cdot PD([L(u)]).$

Note, by the degree reason, we need to take sum over $\beta$ with
$\mu(\beta)=2$.

Since $b$ is assumed to lie in $H^{1}(L(u),\Lambda_{+})$ not just in
$H^{1}(L(u),\Lambda_{0})$, the series appearing as the scalar factor in
(11.15) converges in non-Archimedean topology of $\Lambda_{0}$ and so the sum
$\sum_{k=0}^{\infty}\mathfrak{m}^{can}_{k}(b,\cdots,b)$ is a multiple of
$PD([L(u)])$. Hence $b\in\widehat{\mathcal{M}}_{\text{\rm weak}}(L(u))$ by
definition (4.1). We remark that the gauge equivalence relation in Chapter 4
[FOOO3] is trivial on $H^{1}(L(u);\Lambda_{0})$ and so
$H^{1}(L(u);\Lambda_{+})\hookrightarrow{\mathcal{M}}_{\text{\rm weak}}(L(u))$.
We omit the proof of this fact since we do not use it in this paper. ∎

###### Proof of Theorem 4.5.

Suppose that there is no nontrivial holomorphic sphere whose Maslov index is
nonpositive. Then Theorem 11.1 (5) implies that if $\mu(\beta)\leq 2$,
$\beta\neq\beta_{i}$, $\beta\neq 0$ then $\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)$ is empty. Therefore again by dimension counting as in
Corollary 11.5, we obtain

$\sum_{k=0}^{\infty}\mathfrak{m}_{k}^{dR,can}(x,\cdots,x)=\sum_{i=1}^{m}\sum_{k=0}^{\infty}T^{\omega\cap\beta_{i}/2\pi}\mathfrak{m}_{k,\beta_{i}}^{dR,can}(x,\cdots,x)$

for $x\in H^{1}(L(u),\Lambda_{+})$. On the other hand, we obtain

$\displaystyle\mathfrak{PO}(x;u)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\sum_{k=0}^{\infty}\frac{1}{k!}(\partial\beta_{i}\cap
x)^{k}T^{\ell_{i}(u)}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\sum_{k=0}^{\infty}\frac{1}{k!}\langle
v_{i},x\rangle^{k}T^{\ell_{i}(u)}=\sum_{i=1}^{m}e^{\langle
v_{i},x\rangle}T^{\ell_{i}(u)}$

from (6.8), (11.15) and the definition of $\mathfrak{PO}$. Writing
$x=\sum_{i=1}^{n}x_{i}\text{\bf e}_{i}$ and recalling $y_{i}=e^{x_{i}}$, we
obtain $e^{\langle v_{i},x\rangle}=y_{1}^{v_{i,1}}\cdots y_{n}^{v_{i,n}}$ and
hence the proof of Theorem 4.5. ∎

###### Proof of Theorem 4.6.

Let $\beta\in\pi_{2}(X)$, $\mu(\beta)=2$ and $\mathcal{M}_{\text{\rm
weak}}(\beta)\neq\emptyset$. Theorem 11.1 (5) implies that

$\partial\beta=\sum
k_{i}\partial\beta_{i},\quad\beta=\sum_{i}k_{i}\beta_{i}+\sum_{j}\alpha_{j}.$

Hence

$\sum_{k}T^{\beta\cap\omega/2\pi}\mathfrak{m}_{k,\beta}^{can}(b,\cdots,b)$

becomes one of the terms of the right hand side of (4.7). We remark that class
$\beta$ with $\mu(\beta)\geq 4$ does not contribute to
$\mathfrak{m}_{k}^{can}(b,\cdots,b)$ by the degree reason.

When all the vertices of $P$ lie in ${\mathbb{Q}}^{n}$, then the symplectic
volume of all $\alpha_{j}$ are in $2\pi{\mathbb{Q}}$. Moreover
$\omega\cap\beta_{i}\in 2\pi{\mathbb{Q}}$. Therefore the exponents
$\beta\cap\omega/2\pi$ are rational.

The proof of Theorem 4.6 is complete. ∎

######  Remark 11.11.

We remark that in Lemma 11.2 we constructed a system of multisections only for
$\mathcal{M}_{k+1}^{\text{main}}(L(u),\beta)$ with $\beta\cap\omega<E$. So we
obtain only an $A_{n,K}$ structure instead of a filtered $A_{\infty}$
structure. Here $(n,K)=(n(E),K(E))$ depends on $E$ and
$\lim_{E\to\infty}(n(E),K(E))=(\infty,\infty)$. It induces an $A_{n,K}$
structure $\mathfrak{m}^{(E)}$ on $H(L;\Lambda_{0})$. (See subsection 7.2.7
[FOOO3] = subsection 30.7 [FOOO2].) In the same way as section 7.2 [FOOO3] (=
section 30 [FOOO2]), we can find $(n^{\prime}(E),K^{\prime}(E))$ such that
$(n^{\prime}(E),K^{\prime}(n))\to(\infty,\infty)$ as $E\to\infty$ and the
following holds.

If $E_{1}<E_{2}$ then $A_{n(E_{1}),K(E_{1})}$ structure
$\mathfrak{m}^{(E_{1})}$ is $(n^{\prime}(E),K^{\prime}(E))$-homotopy
equivalent to $\mathfrak{m}^{(E_{2})}$.

This implies that we can extend $\mathfrak{m}^{(E_{1})}$ (regarded as
$A_{n^{\prime}(E_{1}),K^{\prime}(E_{1})}$ structure) to a filtered
$A_{\infty}$ structure by Theorem 7.2.72 [FOOO3] (= Theorem 30.72 [FOOO2]).
(We also remark that for all the applications in this paper we can use
filtered $A_{n,K}$ structure for sufficiently large $n,K$, in place of
filtered $A_{\infty}$ structure.)

Moreover we can use Lemma 11.7 to show the following. If $\mathfrak{x}_{i}\in
H^{1}(L(u);{\mathbb{Q}})$ then
$\mathfrak{m}_{k,\beta}^{(E)}(\mathfrak{x}_{1},\cdots,\mathfrak{x}_{k})$ is
independent of $E$. So in particular it coincides to one of filtered
$A_{\infty}$ structure we define as above.

In other words, since the number $c_{\beta}$ is independent of the choice of
the system of $T^{n}$ invariant multisections it follows that the potential
function in Theorem 4.6 is independent of it. However we do not know how to
calculate it.

######  Remark 11.12.

We used de Rham cohomology to go around the problem of transversality among
chains in the classical cup product. One drawback of this approach is that we
lose control of the rational homotopy type. Namely we do not prove here that
the filtered $A_{\infty}$ algebra (partially) calculated above is homotopy
equivalent to the one in Theorem A [FOOO3] (= Theorem A [FOOO2]) over
${\mathbb{Q}}$. (Note all the operations we obtain is defined over
${\mathbb{Q}}$, however.) We however believe that they are indeed homotopy
equivalent over ${\mathbb{Q}}$. There may be several possible ways to prove
this statement, one of which is to use the rational de Rham forms used by
Sullivan.

Moreover since the number $c_{\beta}$ is independent of the choices we made,
the structure of filtered $A_{\infty}$ algebra on
$H(L(u),\Lambda^{{\mathbb{Q}}})$ is well defined, that is independent of the
choices involved. The ${\mathbb{Q}}$-structure is actually interesting in our
situation. See for example Proposition 7.13. However homotopy equivalence of
the ${\mathbb{Q}}$-version of Lemma 11.9 is not used in the statement of
Proposition 7.13 or in its proof.

##  12\. Non-unitary flat connection on $L(u)$

In this section we explain how we can include (not necessarily unitary) flat
bundles on Lagrangian submanifolds in Lagrangian Floer theory following [Fu3],
[Cho].

######  Remark 12.1.

We need to use flat complex line bundle for our purpose by the following
reason. In [FOOO3] we assumed that our bounding cochain $b$ is an element of
$H(L;\Lambda_{+})$ since we want the series

$\mathfrak{m}_{1}^{b}(x)=\sum_{k,\ell}\mathfrak{m}_{k+\ell+1}(b^{\otimes
k},x,b^{\otimes\ell})$

to converge. There we used convergence with respect to the non-Archimedean
norm. For the case of Lagrangian fibers in toric manifold, the above series
converges for $b\in H^{1}(L;\Lambda_{0})$. The convergence is the usual
(classical Archimedean) topology on ${\mathbb{C}}$ on each coefficient of
$T^{\lambda}$.

This is not an accident and was expected to happen in general. (See Conjecture
3.6.46 [FOOO3] = Conjecture 11.46 [FOOO2].) However for this convergence to
occur, we need to choose the perturbations on $\mathcal{M}_{k+1}^{\text{\rm
main}}(L,\beta)$ so that it is consistent with
$\mathcal{M}_{k^{\prime}+1}^{\text{\rm main}}(L,\beta)$ ($k^{\prime}\neq k$)
via the forgetful map. We can make this choice for the current toric situation
by Lemma 11.2 (3). In a more general situation, we need to regard
$\mathcal{M}_{1}^{\text{\rm main}}(L,\beta)$ as a chain in the free loop
space. (See [Fu4].)

On the other hand, if we use the complex structure other than the standard
one, we do not know whether Lemma 11.2 (3) holds or not. So in the proof of
independence of Floer cohomology under the various choices made, there is a
trouble to use a bounding cochain $b$ lying in $H^{1}(L;\Lambda_{0})$. The
idea, which is originally due to Cho [Cho], is to change the leading order
term of $\mathfrak{x}$ by twisting the construction using _non-unitary_ flat
bundles on $L$.

Let $X$ be a symplectic manifold and $L$ be its relatively spin Lagrangian
submanifold. Let $\rho:H_{1}(L;{\mathbb{Z}})\to{\mathbb{C}}\setminus\\{0\\}$
be a representation and $\mathfrak{L}_{\rho}$ be the flat ${\mathbb{C}}$
bundle induced by $\rho$.

We replace the formula (11.9) by

$\mathfrak{m}_{k}^{\rho,can}=\sum_{\beta\in
H_{2}(M,L)}\rho(\partial\beta)\,\,\mathfrak{m}^{can}_{k,\beta}\otimes
T^{\omega(\beta)/2\pi}.$

(Compare this with (4.3) in section 4.)

######  Proposition 12.2.

$(H(L(u);\Lambda_{0}^{{\mathbb{R}}}),\mathfrak{m}_{k}^{\rho,can})$ is a
filtered $A_{\infty}$ algebra.

###### Proof.

Suppose that $[f]\in\mathcal{M}_{k+1}^{\text{\rm main}}(L,\beta)$ is a fiber
product of $[f_{1}]\in\mathcal{M}_{\ell+1}^{\text{\rm main}}(L,\beta_{1})$ and
$[f_{2}]\in\mathcal{M}_{k-\ell}^{\text{\rm main}}(L,\beta_{2})$. Namely
$\beta_{1}+\beta_{2}=\beta$ and $ev_{0}(f_{2})=ev_{i}(f_{1})$ for some $i$.
Then it is easy to see that

$\rho(\partial\beta)=\rho(\partial\beta_{1})\rho(\partial\beta_{2}).$ (12.1)

(12.1) and (11.8) imply the filtered $A_{\infty}$ relation. ∎

The unitality can also be proved in the same way. The well definedness (that
is, independence of various choices up to homotopy equivalence) can also be
proved in the same way.

######  Remark 12.3.

We have obtained our twisted filtered $A_{\infty}$ structure on the
(untwisted) cohomology group $H^{*}(L;\Lambda_{0})$. This is because the flat
bundle $Hom(\mathfrak{L}_{\rho},\mathfrak{L}_{\rho})$ is trivial. In more
general situation where we consider a flat bundle $\mathfrak{L}$ of higher
rank, we obtain a filtered $A_{\infty}$ structure on cohomology group with
local coefficients with values in $Hom(\mathfrak{L},\mathfrak{L})$.

The filtered $A_{\infty}$ structure $\mathfrak{m}_{k}^{\rho,can}$ is different
from $\mathfrak{m}_{k}^{can}$ in general as we can see from the expression of
the potential function given in Lemma 4.9.

In the rest of this section, we explain the way how the Floer cohomology
detects the Lagrangian intersection. Namely we sketch the proof of Theorem
3.11 in our case and its generalization to the case where we include the
nonunital flat connection $\rho$.

Let $\psi_{t}:X\to X$ be a Hamiltonian isotopy with $\psi_{0}=$ identity. We
put $\psi_{1}=\psi$. We consider the pair

$L^{(0)}=L(u),\quad L^{(1)}=\psi(L(u))$

such that $L^{(1)}$ is transversal to $L^{(0)}$. We then take a one-parameter
family $\\{J_{t}\\}_{t\in[0,1]}$ of compatible almost complex structures such
that $J_{0}=J$ which is the standard complex structure of $X$ and
$J_{1}=\psi_{*}(J)$.

Let $p,q\in L^{(0)}\cap L^{(1)}$. We consider the homotopy class of a maps

$\varphi:{\mathbb{R}}\times[0,1]\to X$ (12.2)

such that

  1. (1)

$\lim_{\tau\to-\infty}\varphi(\tau,t)=p$,
$\lim_{\tau\to+\infty}\varphi(\tau,t)=q$.

  2. (2)

$\varphi(\tau,0)\in L^{(0)}$, $\varphi(\tau,1)\in L^{(1)}$.

We denote by $\pi_{2}(L^{(1)},L^{(0)};p,q)$ the set of all such homotopy
classes. There are obvious maps

$\displaystyle\pi_{2}(L^{(1)},L^{(0)};p,r)\times\pi_{2}(L^{(1)},L^{(0)};r,q)\to\pi_{2}(L^{(1)},L^{(0)};p,q),$
(12.3)
$\displaystyle\pi_{2}(X;L^{(1)})\times\pi_{2}(L^{(1)},L^{(0)};p,q)\to\pi_{2}(L^{(1)},L^{(0)};p,q),$
$\displaystyle\pi_{2}(L^{(1)},L^{(0)};p,q)\times\pi_{2}(X;L^{(0)})\to\pi_{2}(L^{(1)},L^{(0)};p,q).$

We denote them by $\\#$.

######  Definition 12.4.

We consider the moduli space of maps (12.2) satisfying (1), (2) above, in
homotopy class $B\in\pi_{2}(L^{(1)},L^{(0)};p,q)$, and satisfies the following
equation :

$\frac{\partial\varphi}{\partial\tau}+J_{t}\left(\frac{\partial\varphi}{\partial
t}\right)=0.$ (12.4)

We denote it by $\widehat{\mathcal{M}}^{\text{reg}}(L^{(1)},L^{(0)};p,q;B).$
We take its stable map compactification and denote it by
$\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B).$ We divide this space by the
${\mathbb{R}}$ action induced by the translation of $\tau$ direction to obtain
${\mathcal{M}}(L^{(1)},L^{(0)};p,q;B).$ We define evaluation maps
$ev_{L^{(i)}}:\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)\to L^{(i)}$ by
$ev_{L^{(i)}}(\varphi)=\varphi(0,i),$ for $i=0,1$.

######  Lemma 12.5.

${\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)$ has an oriented Kuranishi structure
with corners. Its boundary is isomorphic to the union of the following three
kinds of fiber products as spaces with Kuranishi structure.

  1. (1)

${\mathcal{M}}(L^{(1)},L^{(0)};p,r;B^{\prime})\times{\mathcal{M}}(L^{(1)},L^{(0)};r,q;B^{\prime\prime}).$

Here $B^{\prime}\in\pi_{2}(L^{(1)},L^{(0)};p,r)$,
$B^{\prime\prime}\in\pi_{2}(L^{(1)},L^{(0)};r,q)$.

  2. (2)

${\mathcal{M}}_{1}(L(u);\beta^{\prime})\,\,{}_{ev_{0}}\times_{ev_{L^{(1)}}}\,\,\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B^{\prime\prime}).$

Here $\beta^{\prime}\in\pi_{2}(X;L^{(1)})\cong\pi_{2}(X;L(u))$,
$\beta^{\prime}\\#B^{\prime\prime}=B$. The fiber product is taken over
$L^{(1)}\cong L(u)$ by using $ev_{0}:{\mathcal{M}}(L(u);\beta^{\prime})\to
L(u)$ and
$ev_{L^{(1)}}:\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B^{\prime\prime})\to
L^{(1)}$.

  3. (3)

$\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B^{\prime})\,\,{}_{ev_{L^{(0)}}}\times_{ev_{0}}\,\,{\mathcal{M}}(L(u);\beta^{\prime\prime}).$

Here $\beta^{\prime}\in\pi_{2}(X;L^{(1)})\cong\pi_{2}(X;L(u))$,
$B^{\prime}\\#\beta^{\prime\prime}=B$. The fiber product is taken over
$L^{(0)}\cong L(u)$ by using
$ev_{0}:{\mathcal{M}}(L(u);\beta^{\prime\prime})\to L(u)$ and
$ev_{L^{(0)}}:\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B^{\prime})\to
L^{(0)}$.

We have

$\mathop{\kern 0.0pt{\rm
dim}}\nolimits{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)=\mu(B)-1$

where

$\displaystyle\mu(B_{1}\\#B_{2})$ $\displaystyle=\mu(B_{1})+\mu(B_{2}),$
$\displaystyle\mu(\beta^{\prime}\\#B^{\prime\prime})$
$\displaystyle=\mu(\beta^{\prime})+\mu(B^{\prime\prime}),\quad\mu(B^{\prime}\\#\beta^{\prime\prime})=\mu(B^{\prime})+\mu(\beta^{\prime\prime}).$

Here the notations are as in $(1)$, $(2)$, $(3)$ above.

Lemma 12.5 is proved in subsection 7.1.4 [FOOO3] (= section 29.4 [FOOO2]).

######  Lemma 12.6.

There exists a system of multisections on
${\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)$ such that

  1. (1)

It is transversal to $0$.

  2. (2)

It is compatible at the boundaries described in Lemma 12.5. Here we pull back
multisection of ${\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)$ to one on
$\widehat{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)$ and use the multisection in
Lemma 11.2 on ${\mathcal{M}}(L(u);\beta)$.

###### Proof.

We can find such system of multisections inductively over $\int_{B}\omega$ by
using the fact that $ev_{0}:{\mathcal{M}}(L(u);\beta)^{\mathfrak{s}}\to L(u)$
is a submersion and Lemma 12.5 (1) is a direct product. ∎

In the case when $\mu(B)=1$, we define

$n(B)=\\#{\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)^{\mathfrak{s}}\in{\mathbb{Q}}$

that is the number of zeros of our multisection counted with sign and
multiplicity. We use it to define Floer cohomology.

Let $CF(L^{(1)},L^{(0)})$ be the free $\Lambda_{0,nov}$ module generated by
$L^{(0)}\cap L^{(1)}$. We will define boundary operator on it.

Let $\rho_{i}:\pi_{1}(L^{(i)})\to{\mathbb{C}}^{*}$ be the representation. We
take harmonic $1$ form $h_{i}\in H^{1}(L^{(i)};{\mathbb{C}})$ such that

$\rho_{i}(\gamma)=\exp\left(\int_{\gamma}h_{i}\right).$

Let $b_{i,+}\in H^{1}(L^{(0)};\Lambda_{+})\subset\mathcal{M}(L^{(0)})$. An
element $B\in\pi_{2}(L^{(1)},L^{(0)};p,q)$ induces a path
$\partial_{i}:[0,1]\to L^{(i)}$ joining $p$ to $q$ in $L^{(i)}$ for $i=0,1$.
We define

$C(B;(h_{0},h_{1}),(b_{0,+},b_{1,+}))=\exp\left(\int_{\partial_{0}B}(h_{0}+b_{0,+})\right)\exp\left(-\int_{\partial_{1}B}(h_{1}+b_{1,+})\right).$

This is an element of $\Lambda_{0}\setminus\Lambda_{+}$. It is easy to see
that

$\displaystyle C(B_{1}\\#B_{2};(h_{0},h_{1}),(b_{0,+},b_{1,+}))$ (12.5)
$\displaystyle=C(B_{1};(h_{0},h_{1}),(b_{0,+},b_{1,+}))\,C(B_{2};(h_{0},h_{1}),(b_{0,+},b_{1,+})),$

and

$\displaystyle
C(\beta^{\prime}\\#B^{\prime\prime};(h_{0},h_{1}),(b_{0,+},b_{1,+}))$ (12.6)
$\displaystyle=C(B^{\prime\prime};(h_{0},h_{1}),(b_{0,+},b_{1,+}))\exp\left(\int_{\partial\beta^{\prime}}h_{1}+b_{0,+}\right).$

(Here $B^{\prime}$ and $\beta^{\prime\prime}$ are as in Lemma 12.5 (2).) Now
we define

$\displaystyle\langle\delta_{(h_{0},h_{1}),(b_{0,+},b_{1,+})}(p),q\rangle$
(12.7)
$\displaystyle=\sum_{B\in\pi_{2}(L^{(1)},L^{(0)};p,q);\mu(B)=1}n(B)C(B;(h_{0},h_{1}),(b_{0,+},b_{1,+})).$

For the case $h_{1}=\psi_{*}(h_{0})$, $b_{1,+}=\psi_{*}(b_{+})$,
$b_{0,+}=b_{+}$, we write $C(B;h_{0},b_{+})$ and $\delta^{h_{0},b_{+}}$, in
place of $C(B;(h_{0},h_{1}),(b_{0,+},b_{1,+}))$ and
$\delta_{(h_{0},h_{1}),(b_{0,+},b_{1,+})}$, respectively.

######  Lemma 12.7.

$\delta_{(h_{0},h_{1}),(b_{0,+},b_{1,+})}\circ\delta_{(h_{0},h_{1}),(b_{0,+},b_{1,+})}=(\mathfrak{PO}(b(1))-\mathfrak{PO}(b(0)))\cdot
id$

where $b(i)=h_{i}+b_{i,+}\in H^{1}(L(u);\Lambda_{0})$.

###### Proof.

Let $p,q\in L^{(1)}\cap L^{(0)}$. We consider
$B\in\pi_{2}(L^{(1)},L^{(0)};p,q)$ with $\mu(B)=2$. We consider the boundary
of ${\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)$. We put

$\delta_{B}=\begin{cases}1&\text{$B\in\pi_{2}(L^{(1)},L^{(0)};p,q)$, $p=q$,
and $B$ is the class of constant map,}\\\ 0&\text{otherwise.}\end{cases}$

Then using the classification of the boundary of
${\mathcal{M}}(L^{(1)},L^{(0)};p,q;B)$ in Lemma 12.5 we have the following
equality :

$0=\sum_{r,B,B^{\prime\prime}}n(B^{\prime})n(B^{\prime\prime})+\sum_{\beta^{\prime},B^{\prime\prime}}c_{\beta^{\prime}}\delta_{B^{\prime\prime}}-\sum_{\beta^{\prime\prime},B^{\prime}}c_{\beta^{\prime\prime}}\delta_{B^{\prime}},$
(12.8)

where the summention in the first, second, third terms of right hand side is
taken over the set described in Lemma 12.5 (1), (2), (3), respectively, and
$c_{\beta}$ is defined in Definition 11.6.

We multiply (12.8) by $C(B;(h_{0},h_{1}),(b_{0,+},b_{1,+}))$ and calculate the
right hand side by using Formulae (12.5), (12.6) and Lemma 11.8. It is easy to
see that the first term gives

$\langle\delta_{(h_{0},h_{1}),(h_{0,+},b_{1,+})}\circ\delta_{(h_{0},h_{1}),(b_{0,+},b_{1,+})}(p),q\rangle.$

The second term is $0$ if $p\neq q$, and is

$\sum_{\beta}c_{\beta}\exp\left(\int_{\partial\beta}h_{0}+b_{0,+}\right)=\mathfrak{PO}(b(0))$

if $p=q$. The third term gives $\mathfrak{PO}(b(1))\cdot id$ in a similar way.
∎

######  Definition 12.8.

Let $b(0)=h_{0}+b_{+}$ and $b(1)=\psi_{*}(b(0))$. We define

$HF((L^{(0)},b(0)),(L^{(1)},b(1));\Lambda_{0})\cong\frac{\text{Ker}\,\,\delta_{h_{0},h_{+}}}{\text{Im}\,\,\delta_{h_{0},h_{+}}}.$

This is well defined by Lemma 12.7 and by the identity
$\mathfrak{PO}(\psi_{*}(b(0)))=\mathfrak{PO}((b(0)))$.

Now we have

######  Lemma 12.9.

$HF((L^{(0)},b(0)),(L^{(1)},b(1));\Lambda_{0})\otimes_{\Lambda_{0}}\Lambda\cong\frac{\text{\rm
Ker}\,\,\mathfrak{m}_{1}^{\rho,can,b_{+}}}{\text{\rm
Im}\,\,\mathfrak{m}_{1}^{\rho,can,b_{+}}}\otimes_{\Lambda_{0}}\Lambda.$

Here $\rho(\gamma)=\exp\left(\int_{\gamma}h_{0}\right)$ and

$\mathfrak{m}_{1}^{\rho,can,b_{+}}(x)=\sum_{k_{1},k_{2}=0}^{\infty}\mathfrak{m}^{\rho,can}_{k_{1}+k_{2}+1}(b_{+}^{\otimes
k_{1}},x,b_{+}^{\otimes k_{2}}).$

Lemma 12.9 implies (the $\rho$ twisted version of) Theorem 3.11 in our case.
We omit the proof of Lemma 12.9 and refer section 5.3 [FOOO3] (= section 22
[FOOO2]) or section 8 [FOOO4].

##  13\. Floer cohomology at a critical point of potential function

In this section we prove Theorem 4.10 etc. and complete the proof of Theorem
1.5.

###### Proof of Lemma 4.9.

Let $\beta\in H_{2}(X,L(u_{0}))$ with $\mu(\beta)=2$ and
$\mathcal{M}_{1}^{\text{\rm main}}(L(u_{0}),\beta)$ be nonempty. We have
$\beta=\sum_{i=1}^{m}c_{i}\beta_{i}+\sum_{j}\alpha_{j}$ by Theorem 11.1 (5).
Let $\rho$ be as in (4.15). We have
$\rho(\partial\beta)=\prod\rho(\partial\beta_{i})^{c_{i}}$. Note
$\partial\beta_{i}=\sum_{j}v_{i,j}\text{\bf e}_{j}^{*}$. Thus we have

$\left\\{\begin{aligned}
\rho(\partial\beta_{i})&=\mathfrak{y}_{1,0}^{v_{i,1}}\cdots\mathfrak{y}_{n,0}^{v_{i,n}},\\\
\rho(\partial\beta)&=\prod_{i}\prod_{j}\mathfrak{y}_{j,0}^{c_{i}v_{i,j}}.\end{aligned}\right.$
(13.1)

Therefore for $b\in H^{1}(L(u_{0});\Lambda_{+}^{{\mathbb{C}}})$ we have

$\displaystyle\sum_{k=0}^{\infty}\mathfrak{m}_{k,\beta}^{\rho,can}(b,\cdots,b)$
$\displaystyle=\sum_{k=0}^{\infty}\mathfrak{y}_{1,0}^{v_{i,1}}\cdots\mathfrak{y}_{n,0}^{v_{i,n}}\mathfrak{m}_{k,\beta}^{can}(b,\cdots,b)$
$\displaystyle=\sum_{k=0}^{\infty}e^{\mathfrak{x}_{1,0}v_{i,1}}\cdots
e^{\mathfrak{x}_{n,0}v_{i,n}}\frac{c_{\beta}}{k!}(b\cap\partial\beta)^{k}\cdot[PD(L)]$
$\displaystyle=\sum_{k=0}^{\infty}\mathfrak{m}^{can}_{k,\beta}\left(b+\sum_{j=1}^{n}\mathfrak{x}_{j,0}\text{\bf
e}_{j},\cdots,b+\sum_{j=1}^{n}\mathfrak{x}_{j,0}\text{\bf e}_{j}\right).$

On the other hand, it follows from Theorems 4.5 and 4.6 that the left and the
right sides of this identity correspond to those in Lemma 4.9 respectively.
This finishes the proof of Lemma 4.9. ∎

###### Proof of Theorem 4.10.

Let $x^{+}=(x^{+}_{1},\cdots,x^{+}_{n})$,
$x^{+}_{1},\cdots,x^{+}_{n}\in\Lambda_{+}$. We put

$x(x^{+})=\sum_{i}(\mathfrak{x}_{i,0}+x^{+}_{i})\text{\bf e}_{i},\quad
b(x^{+})=\sum_{i}x^{+}_{i}\text{\bf e}_{i}.$

From Lemma 4.9 we derive

$\displaystyle\mathfrak{PO}^{u_{0}}_{\rho}(b(x^{+}))$
$\displaystyle=\sum\mathfrak{m}_{k}^{\rho,can}(b(x^{+}),\cdots,b(x^{+}))\cap[L(u_{0})]$
$\displaystyle=\sum\mathfrak{m}_{k}^{can}(x(x^{+}),\cdots,x(x^{+}))\cap[L(u_{0})]=\mathfrak{PO}^{u_{0}}(x(x^{+})).$

Let $\mathfrak{x}$ be as in (4.13). Then we have

$\frac{\partial}{\partial
x^{+}_{i}}\mathfrak{PO}^{u_{0}}_{\rho}(b(x^{+}))\Big{|}_{x(x^{+})=\mathfrak{x}}=\frac{\partial}{\partial
x^{+}_{i}}\mathfrak{PO}^{u_{0}}(x(x^{+}))\Big{|}_{x(x^{+})=\mathfrak{x}}=\frac{\partial\mathfrak{PO}^{u_{0}}}{\partial
x_{i}}(\mathfrak{x})=0,$

where the last equality follows from the assumption (4.12). (In case when
(4.11) is assumed we have ‘$\equiv 0\mod T^{\mathcal{N}}$’ in place of
‘$=0$’.) On the other hand, we have

$\displaystyle\frac{\partial}{\partial x^{+}_{i}}$
$\displaystyle\mathfrak{PO}^{u_{0}}_{\rho}(b(x^{+}))\Big{|}_{x(x^{+})=\mathfrak{x}}$
(13.2)
$\displaystyle=\sum_{k}\sum_{\ell}\mathfrak{m}^{\rho,can}_{k}(b^{\otimes\ell},\text{\bf
e}_{i},b^{\otimes(k-\ell-1)})\cap[L(u_{0})]$
$\displaystyle=\mathfrak{m}_{1}^{\rho,can,b}(\text{\bf e}_{i})\cap[L(u_{0})].$

Note here and hereafter we write $b$ in place of $b(x^{+})$ with
$x(x^{+})=\mathfrak{x}$. Namely
$b=\mathfrak{x}-\sum\mathfrak{x}_{i,0}\text{\bf e}_{i}$.

Hence we obtain

$\mathfrak{m}_{1}^{\rho,can,b}(\text{\bf e}_{i})\begin{cases}=0&\text{if
(\ref{formula:critical0}) is satisfied}\\\ \equiv 0\mod
T^{{\mathcal{N}}}&\text{if (\ref{formula:criticalweak}) is
satisfied}.\end{cases}$ (13.3)

We remark that by the degree reason $\mathfrak{m}^{\rho,can,b}_{1}(\text{\bf
e}_{i})$ is proportional to $PD[L(u_{0})]$.

We next prove the vanishing of $\mathfrak{m}_{1}^{\rho,can,b}(\text{\bf f})$
for the classes f of higher degree. Namely we prove

######  Lemma 13.1.

For $\text{\bf f}\in H^{*}(L(u_{0});\Lambda_{0}^{{\mathbb{C}}})$ we have :

$\mathfrak{m}_{1,\beta}^{\rho,can,b}(\text{\bf f})\begin{cases}=0&\text{if
$(\ref{formula:critical0})$ is satisfied}\\\ \equiv 0\mod
T^{{\mathcal{N}}}&\text{if $(\ref{formula:criticalweak})$ is
satisfied}.\end{cases}$

###### Proof.

Let $d=\deg\text{\bf f}$ and $2\ell=\mu(\beta)$. We say
$(d,\ell)<(d^{\prime},\ell^{\prime})$ if $\ell<\ell^{\prime}$ or
$\ell=\ell^{\prime}$, $d<d^{\prime}$. We prove the lemma by upward induction
on $(d,\ell)$. The case $d=1$ is (13.3). We remark that
$\mathfrak{m}_{k,\beta}=0$ if $\mu(\beta)\leq 0$.

We assume that the lemma is proved for $(d^{\prime},\ell^{\prime})$ smaller
than $(d,\ell)$ and will prove the case of $(d,\ell)$. Since the case $d=1$ is
already proved, we may assume that $d\geq 2$. Let $\text{\bf f}=\text{\bf
f}_{1}\cup\text{\bf f}_{2}$ where $\deg\text{\bf f}_{i}\geq 1$. By the
$A_{\infty}$-relation we have

$\displaystyle\mathfrak{m}_{1,\beta}^{\rho,can,b}(\text{\bf
f}_{1}\cup\text{\bf f}_{2})=$
$\displaystyle\sum_{\beta_{1}+\beta_{2}=\beta}\pm\mathfrak{m}^{\rho,can,b}_{2,\beta_{1}}(\mathfrak{m}^{\rho,can,b}_{1,\beta_{2}}(\text{\bf
f}_{1}),\text{\bf f}_{2})$
$\displaystyle+\sum_{\beta_{1}+\beta_{2}=\beta}\pm\mathfrak{m}^{\rho,can,b}_{2,\beta_{1}}(\text{\bf
f}_{1},\mathfrak{m}_{1,\beta_{2}}(\text{\bf f}_{2}))$
$\displaystyle+\sum_{\beta_{1}+\beta_{2}=\beta,\beta_{2}\neq
0}\pm\mathfrak{m}^{\rho,can,b}_{1,\beta_{1}}(\mathfrak{m}_{2,\beta_{2}}(\text{\bf
f}_{1},\text{\bf f}_{2})).$

We remark that $\mathfrak{m}^{\rho,can,b}_{1,\beta_{0}}=0$ since we are
working on a canonical model.

The first two terms of the right hand side vanishes by the induction
hypothesis since $\deg\text{\bf f}_{i}<\deg\text{\bf f}$ and
$\mu(\beta_{i})\leq\mu(\beta)$. The third term also vanishes since
$\mu(\beta_{1})<\mu(\beta)$. The proof of Lemma 13.1 is complete. ∎

Lemma 13.1 immediately implies Theorem 4.10. ∎

###### Proof of Proposition 5.4.

Let us specialize to the case of 2 dimension. In case $\mathop{\kern 0.0pt{\rm
dim}}\nolimits L(u_{0})=2$, we can prove
$\mathfrak{m}_{1,\beta}^{\rho,can,b}=0$ for $\mu(\beta)\geq 4$ also by
dimension counting. We can use that to prove Proposition 5.4 in the same way
as above. ∎

###### Proof of Proposition 4.12.

Let $\omega_{i}$, $u_{i}$, $\mathfrak{x}_{i,\mathcal{N}}$ be as in Definition
4.11. We assume $\psi:X\to X$ does not satisfy (4.19) or (4.20) and will
deduce a contradiction. We use the same (time dependent) Hamiltonian as $\psi$
to obtain $\psi_{i}:(X,\omega_{i})\to(X,\omega_{i})$. Take an integer
${\mathcal{N}}$ such that $\|\psi_{i}\|<2\pi{\mathcal{N}}$ for large $i$. Then
for sufficiently large $i$, $L(u_{0}^{i})$ and $\psi_{i}$ does not satisfy
(4.19) or (4.20). In fact if $\psi(L(u_{0}))\cap L(u_{0})=\emptyset$ then for
sufficiently large $i$, we have $\psi_{i}(L(u^{i}_{0}))\cap
L(u^{i}_{0})=\emptyset$. If $\psi(L(u_{0}))$ is transversal to $L(u_{0})$ and
if (4.20) is not satisfied, then

$\\#(\psi(L(u_{0}))\cap
L(u_{0}))\geq\\#(\psi_{\varepsilon_{i}}(L(u^{i}_{0}))\cap L(u^{i}_{0})).$

On the other hand, by Theorem 4.10 we have

$HF((L(u_{i}),\mathfrak{x}_{i,k}),(L(u_{i}),\mathfrak{x}_{i,k});\Lambda_{0}^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))\cong
H(T^{n};\Lambda_{0}^{{\mathbb{C}}}/(T^{{\mathcal{N}}})).$

It follows from Universal Coefficient Theorem that

$HF((L(u_{i}),\mathfrak{x}_{i,k}),(L(u_{i}),\mathfrak{x}_{i,k});\Lambda_{0}^{{\mathbb{C}}})\cong\Lambda_{0}^{\oplus
a}\oplus\bigoplus_{i=1}^{b}\Lambda_{0}/(T^{c(i)})$ (13.4)

such that $c(i)\geq{\mathcal{N}}$ and $a+2b\geq 2^{n}$. This contradicts to
Theorem J [FOOO3]. (In fact Theorem J [FOOO3] (= Theorem 3.12) and (13.4)
imply that (4.19) and (4.20) hold for $L(u_{i})$ and $\psi_{i}$ with
$\|\psi_{i}\|<2\pi{\mathcal{N}}$.) Proposition 4.12 is proved. ∎

###### Proof of Theorem 5.11.

The proof is the same as the proof of Proposition 4.12 above. ∎

Now we are ready to complete the proof of Theorem 1.5.

###### Proof of Theorem 1.5.

In the case where the vertices of $P$ are contained in ${\mathbb{Q}}^{n}$,
Proposition 4.7 and Theorem 4.10 imply that $L(u_{0})$ is balanced in the
sense of Definition 4.11. Therefore Proposition 4.12 implies Theorem 1.5 in
this case. If the leading term equation is strongly nondegenerate, Theorem 1.5
also follows from Theorem 4.10, Theorem 10.4 and Proposition 4.12.

We finally present an argument to remove the rationality assumption. In view
of Lemma 4.12, it suffices to prove the following Proposition 13.2.

######  Proposition 13.2.

In the situation of Theorem 1.5, there exists $u_{0}$ such that $L(u_{0})$ is
a balanced Lagrangian fiber.

###### Proof.

Let $\pi:X\to P$ be as in Theorem 1.5. Let us consider $s_{k}$, $S_{k}$,
$P_{k}$ as in section 9. We obtain $u_{0}\in P$ such that $\\{u_{0}\\}=P_{K}$.
We will prove that $L(u_{0})$ is balanced.

We perturb the Kähler form $\omega$ of $X$ a bit so that we obtain
$\omega^{\prime}$. Let $P^{\prime}$ be the corresponding moment polytope and
$s_{k}^{\omega^{\prime}}$, $S_{k}^{\omega^{\prime}}$,
$P_{k}^{\omega^{\prime}}$, be the corresponding piecewise affine function,
number, subset of $P_{\omega^{\prime}}$ obtained for $\omega^{\prime}$,
$P_{\omega^{\prime}}$ as in section 9.

######  Proposition 13.3.

We can choose $\omega_{h}$ so that $\omega_{h}$ is rational and
$\lim_{h\to\infty}s_{k}^{\omega_{h}}=s_{k}$,
$\lim_{h\to\infty}S_{k}^{\omega_{h}}=S_{k}$,
$\lim_{h\to\infty}P_{k}^{\omega_{h}}=P_{k}$, $\mathop{\kern 0.0pt{\rm
dim}}\nolimits P_{k}^{\omega_{h}}=\mathop{\kern 0.0pt{\rm dim}}\nolimits
P_{k}$ for all $k$.

###### Proof.

We write $I_{k}^{\omega^{\prime}}$ the set $I_{k}$ defined in (9.7) section 9
for $\omega^{\prime}$, $P^{\prime}$. We will prove the following lemma. We
remark that the set $\mathfrak{K}$ of $T^{n}$ invariant Kähler structure
$\omega^{\prime}$ is regarded as an open set of an affine space defined on
${\mathbb{Q}}$ (that is the Kähler cone). We may regard $\mathfrak{K}$ as a
moduli space of moment polytope as follows : We consider a polyhedron
$P^{\prime}$ each of whose edges is parallel to a corresponding edge of $P$.
Translation defines an ${\mathbb{R}}^{n}$ action on the set of such
$P^{\prime}$. The quotient space can be identified with $\mathfrak{K}$.

######  Lemma 13.4.

There exists a subset $\mathfrak{P}_{k}$ of $\mathfrak{K}$ which is a nonempty
open subset of an affine subspace defined over ${\mathbb{Q}}$ such that any
element $\omega^{\prime}\in\mathfrak{P}_{k}$ has the following properties :

  1. (1)

$\mathop{\kern 0.0pt{\rm dim}}\nolimits P_{l}^{\omega^{\prime}}=\mathop{\kern
0.0pt{\rm dim}}\nolimits P_{l}^{\omega}$ for $l\leq k$.

  2. (2)

$I_{l}^{\omega^{\prime}}=I_{l}^{\omega}$ for $l\leq k$.

######  Remark 13.5.

In the case of Example 8.1, the set $P^{\omega^{\prime}}_{k}$ etc. jumps at
the point $\alpha=1/3$ in the Kähler cone. Hence the set $\mathfrak{P}_{k}$
may have strictly smaller dimension than $\mathfrak{K}$.

###### Proof.

Let $A_{l}^{\omega^{\prime}}$ be the affine space defined in section 9. (We
put ${\omega^{\prime}}$ to specify the symplectic form.) We write
$\ell_{i}^{\omega}$, $\ell_{i}^{\omega^{\prime}}$ in place of $\ell_{i}$ to
specify symplectic form and moment polytope. We remark that the linear part of
$\ell_{i}^{\omega}$ is equal to the linear part of
$\ell_{i}^{\omega^{\prime}}$.

The proof of Lemma 13.4 is by induction on $k$. Let us first consider the case
$k=1$. We put

$\widehat{A}_{1}^{\omega^{\prime}}=\\{u\in
M_{\mathbb{R}}\mid\ell_{1,1}^{\omega^{\prime}}(u)=\cdots=\ell_{1,a_{1}}^{\omega^{\prime}}(u)\\}.$

We remark that
$\\{\ell^{\omega}_{1,1},\cdots,\ell^{\omega}_{1,a_{1}}\\}=I_{1}^{\omega}$ and
so $\widehat{A}_{1}^{\omega}=A_{1}^{\omega}$.

We put

$\mathfrak{P}^{\prime}_{1}=\\{\omega^{\prime}\mid\mathop{\kern 0.0pt{\rm
dim}}\nolimits\widehat{A}_{1}^{\omega^{\prime}}=\mathop{\kern 0.0pt{\rm
dim}}\nolimits A_{1}^{\omega}\\}.$

It is easy to see that $\mathfrak{P}^{\prime}_{1}$ is a nonempty affine subset
of $\mathfrak{K}$ and is defined over ${\mathbb{Q}}$.

######  Sublemma 13.6.

If $\omega^{\prime}\in\mathfrak{P}^{\prime}_{1}$ and is sufficiently close to
$\omega$, then $P_{1}^{\omega^{\prime}}$ is an equi-dimensional polyhedron in
$\widehat{A}_{1}^{\omega^{\prime}}$. In particular
$\widehat{A}_{1}^{\omega^{\prime}}=A_{1}^{\omega^{\prime}}$.

###### Proof.

The tangent space of $\widehat{A}_{1}^{\omega^{\prime}}$ is parallel to the
tangent space of $A_{1}^{\omega}$. Therefore $\ell^{\omega^{\prime}}_{1,j}$ is
constant on $\widehat{A}_{1}^{\omega^{\prime}}$. We put

$\widehat{S}_{1}^{\omega^{\prime}}=\ell^{\omega^{\prime}}_{1,1}(u)$

for some $u\in\widehat{A}_{1}^{\omega^{\prime}}$.

On the other hand, if $\ell_{i}^{\omega}\notin I_{1}^{\omega}$ then
$\ell_{i}^{\omega}(u)>S_{1}^{\omega}$ on $\text{Int}\,P_{1}^{\omega}$.
Therefore if $\omega^{\prime}$ is sufficiently close to $\omega$ we have
$\ell_{i}^{\omega^{\prime}}(u)>\widehat{S}_{1}^{\omega^{\prime}}$ on a
neighborhood of a compact subset of $\text{Int}\,\,P_{1}^{\omega}$, which we
identify with a subset of $P^{\prime}$. This implies the sublemma. ∎

The Condition (1), (2) of Lemma 13.3 in the case $k=1$ follows from Sublemma
13.6 easily.

Let us assume that Lemma 13.4 is proved up to $k-1$. We remark
$\\{\ell^{\omega}_{k,1},\cdots,\ell^{\omega}_{k,a_{k}}\\}=I_{k}^{\omega}$. We
put

$\widehat{A}_{k}^{\omega^{\prime}}=\\{u\in
A_{k-1}^{\omega^{\prime}}\mid\ell_{k,1}^{\omega^{\prime}}(u)=\cdots=\ell_{k,a_{k}}^{\omega^{\prime}}(u)\\}$

and

$\mathfrak{P}^{\prime}_{k}=\\{\omega^{\prime}\in\mathfrak{P}^{\prime}_{k-1}\mid\mathop{\kern
0.0pt{\rm dim}}\nolimits\widehat{A}_{k}^{\omega^{\prime}}=\mathop{\kern
0.0pt{\rm dim}}\nolimits A_{k}^{\omega}\\}.$

$\mathfrak{P}^{\prime}_{k}$ is a nonempty affine subset of $\mathfrak{K}$ and
is defined over ${\mathbb{Q}}$. We can show that a sufficiently small open
neighborhood $\mathfrak{P}_{k}$ of $\omega$ in $\mathfrak{P}^{\prime}_{k}$ has
the required properties in the same way as the first step of the induction.
The proof of Lemma 13.4 is complete. ∎

Proposition 13.3 follows immediately from Lemma 13.4. In fact the set of
rational points are dense in $\mathfrak{P}_{K}$. ∎

Proposition 13.2 follows from Proposition 13.3, Proposition 4.7 and Theorem
4.10. ∎

The proof of Theorem 1.5 is now complete. ∎

###### Proof of Proposition 10.8.

The proof is similar to the proof of Proposition 13.3. Let $I_{k}$ be as in
(10.2). We write it as $I_{k}(P,u_{0})$, where $P$ is the moment polytope of
$(X,\omega)$. We define $I_{k}(P^{\prime},u^{\prime}_{0})$ as follows.

Let $P^{\prime}$ be a polytope which is a small perturbation of $P$ and such
that each of the faces are parallel to the corresponding face of $P$. Let
$u^{\prime}_{0}\in\text{\rm Int}\,\,P^{\prime}$. Let us consider the set
$\mathfrak{K}^{+}$ of all such pairs $(P^{\prime},u^{\prime}_{0})$. It is an
open set of an affine space defined over ${\mathbb{Q}}$. Each of such
$P^{\prime}$ is a moment polytope of certain Kähler form on $X$. (We remark
that Kähler form on $X$ determine $P^{\prime}$ only up to translation.)

For each $P^{\prime}$, we take corresponding Kähler form on $X$ and it
determines a potential function. Therefore $I_{k}(P^{\prime},u^{\prime}_{0})$
is determined by (10.2). We define $A^{\perp}_{l}(P^{\prime},u^{\prime}_{0})$
in the same way as Definition 10.1.

######  Lemma 13.7.

There exists a subset $\mathfrak{Q}_{k}$ of $\mathfrak{K}^{+}$ which is a
nonempty open set of an affine subspace defined over ${\mathbb{Q}}$. All the
elements $(P^{\prime},u^{\prime}_{0})$ of $\mathfrak{Q}_{k}$ have the
following properties.

  1. (1)

$\mathop{\kern 0.0pt{\rm dim}}\nolimits
A^{\perp}_{l}(P^{\prime},u^{\prime}_{0})=\mathop{\kern 0.0pt{\rm
dim}}\nolimits A^{\perp}_{l}(P,u_{0})$ for $l\leq k$.

  2. (2)

$I_{l}(P^{\prime},u^{\prime}_{0})=I_{l}(P,u_{0})$ for $l\leq k$.

The proof is the same as the proof of Lemma 13.4 and is omitted.

Now we take a sequence of rational points
$(P_{h},u^{h}_{0})\in\mathfrak{Q}_{k}$ converging to $(P,u_{0})$. Lemma 13.7
(2) implies that the leading term equation at $u^{h}_{0}$ is the same as the
leading term equation at $u_{0}$. The proof of Proposition 10.8 is complete. ∎

###### Proof of Lemma 10.5.

Let $[\omega]\in H^{2}(X;{\mathbb{Q}})$. We may take the moment polytope $P$
such that its vertices are rational. (This time we do not change $P$
throughout the proof.) Let $u_{0}\in\text{\rm Int}\,P$ and we assume that
$\mathfrak{PO}_{0}^{u_{0}}$ has a nondegenerate critical point in
$(\Lambda_{0}\setminus\Lambda_{+})^{n}$.

We define $I_{k}(P,u)$ as above. In the same way as the proof of Lemma 13.7 we
can prove the following.

######  Sublemma 13.8.

The set $\mathcal{P}_{k}$ of all $u\in\text{\rm Int}\,P$ satisfying the
following conditions $(1)$,$(2)$ contains an open neighborhoods $u_{0}$ in
certain affine subspace $A$ of ${\mathbb{R}}^{n}$. $A$ is defined on
${\mathbb{Q}}$.

  1. (1)

$\mathop{\kern 0.0pt{\rm dim}}\nolimits A^{\perp}_{l}(P,u)=\mathop{\kern
0.0pt{\rm dim}}\nolimits A^{\perp}_{l}(P,u_{0})$ for all $l<k$.

  2. (2)

$I_{l}(P,u)=I_{l}(P,u_{0})$ for all $l<k$.

We omit the proof. We take $K$ such that $\\{d\ell_{i}\mid\ell_{i}\in
I_{1}(P,u_{0})\cup\cdots\cup I_{K}(P,u_{0})\\}$ generates $N_{{\mathbb{R}}}$.
(Note $P\subset N_{{\mathbb{R}}}=M_{{\mathbb{R}}}^{*}$.) By Sublemma 13.8
there exists a sequence $\\{u_{i}\\}_{i=1,2,\cdots}$ of rational points
$u_{i}$ in $\mathcal{P}_{K}$ which converges to $u_{0}$.

(1), (2) above implies $\mathfrak{PO}_{0}^{u_{0}}$ and
$\mathfrak{PO}_{0}^{u_{i}}$ has the same leading term equation. Therefore by
assumption $\mathfrak{PO}_{0}^{u_{i}}$ has a critical point on
$(\Lambda_{0}\setminus\Lambda_{+})^{n}$. Since
$Jac(\mathfrak{PO}_{0};\Lambda)$ is finite dimensional, it follows that we may
take a subsequence $u_{k_{i}}$ such that $u_{k_{1}}=u_{k_{2}}=\cdots$. Hence
$u_{0}=u_{k_{i}}$, is rational as required. ∎

######  Remark 13.9.

We can replace Definition 4.11 (3) by

$HF((L(u_{i}),\mathfrak{x}_{i,{\mathcal{N}}}),(L(u_{i}),\mathfrak{x}_{i,{\mathcal{N}}});\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))\supseteq\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}).$

In fact the following three conditions are equivalent to one another :

  1. (1)

$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))\cong
H(T^{n};{\mathbb{C}})\otimes\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}).$

  2. (2)

$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))\supseteq\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}).$

  3. (3)

$\frac{\partial\mathfrak{PO}^{u}}{\partial y_{k}}\equiv 0,\mod
T^{{\mathcal{N}}}\qquad k=1,\cdots,n,$ at $\mathfrak{x}$.

(1) $\Rightarrow$ (2) is obvious. (3) $\Rightarrow$ (1) is Theorem 4.10. Let
us prove (2) $\Rightarrow$ (3). Suppose (3) does not hold. We put
$\frac{\partial\mathfrak{PO}^{u}}{\partial y_{k}}\equiv cT^{\lambda}\mod
T^{\lambda}\Lambda_{+}$, where $c\in{\mathbb{C}}\setminus\\{0\\}$ and
$0\leq\lambda<N$. Then (13.2) implies $T^{{\mathcal{N}}-\lambda}PD[L(u)]=0$ in
$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda^{{\mathbb{C}}}/(T^{{\mathcal{N}}}))$.
Since $PD[L(u)]$ is a unit, (2) does not hold.

######  Remark 13.10.

The proof of Theorem 4.10 (or of Lemma 13.1) does not imply

$\mathfrak{m}_{k,\beta}(\rho_{1},\cdots,\rho_{k})=0$ (13.5)

for $\mu(\beta)\geq 4$. So it does not imply that the numbers $c_{\beta}$
(Definition 11.6) determine the filtered $A_{\infty}$ algebra
$(H(L(u);\Lambda_{0}),\mathfrak{m})$ up to homotopy equivalence. We however
believe that this is indeed the case. In fact the homology group
$H(\mathcal{L}(T^{n});{\mathbb{Q}})$ of the free loop space
$\mathcal{L}(T^{n})$ is trivial of degree $>n$. On the other hand,
$\mathop{\kern 0.0pt{\rm dim}}\nolimits\mathcal{M}_{1}^{\text{\rm
main}}(L(u_{0});\beta)=n+\mu(\beta)-2$. Hence if $\mu(\beta)\geq 4$ there is
no nonzero homology class on the corresponding degree in the free loop space.
Using the argument of [Fu4] it may imply that the contribution of those
classes to the homotopy type of filtered $A_{\infty}$ structure is
automatically determined from the contribution of the classes of Maslov index
$2$.

On the other hand, pseudo-holomorphic disc with Maslov index $\geq 4$
certainly contributes to the operator $\mathfrak{q}_{\ell,k,\beta}$ introduced
in section 3.8 [FOOO3] (= section 13 [FOOO2]) : Namely
$\mathfrak{q}_{\ell,k,\beta}$ is the operator that involves a cohomology class
of the ambient symplectic manifold $X$. (See Remark 6.15) It seems that
tropical geometry will play a role in this calculation.

##  14\. Appendix 1 : Algebraically closedness of Novikov fields

######  Lemma 14.1.

If $F$ is an algebraically closed field of characteristic $0$, then
$\Lambda^{F}$ is algebraically closed.

###### Proof.

Let $\sum_{k=0}^{n}a_{k}x^{k}=0$ be a polynomial equation with $\Lambda^{F}$
coefficients. We will prove that it has a solution in $\Lambda^{F}$ by
induction on $n$. We may assume that $a_{n}=1$. Replacing $x$ by
$x-\frac{a_{n-1}}{n}$ we may assume $a_{n-1}=0$. (Here we use the fact that
the characteristic of $F$ is $0$.) We may assume furthermore $a_{0}\neq 0$,
since otherwise $0$ is a solution. We put

$c=\inf_{k=0,\cdots,n-2}\frac{\mathfrak{v}_{T}(a_{k})}{n-k}.$

We put $x=T^{c}y$, $b_{k}=T^{c(k-n)}a_{k}$. Then our equation is equivalent to
$P(y)=\sum_{k=0}^{n}b_{k}y^{k}=0$. We remark that $b_{n}=1$, $b_{n-1}=0$,
$b_{0}\neq 0$. Moreover

$\mathfrak{v}_{T}(b_{k})=c(k-n)+\mathfrak{v}_{T}(a_{k})\geq 0.$

Namely $b_{k}\in\Lambda_{0}$. We define $\overline{b}_{k}\in F$ to be the zero
order term of $b_{k}$ i.e., to satisfy
$b_{k}\equiv\overline{b}_{k}\mod\Lambda_{+}$. We consider the equation
$\overline{P}(\overline{y})=\sum_{k=0}^{n}\overline{b}_{k}\overline{y}^{k}=0.$
By our choice of $c$ there exists $k<n-1$ such that $\overline{b}_{k}\neq 0$
and $\overline{b}_{n}=1$, $\overline{b}_{n-1}=0$. Therefore $\overline{P}$ has
at least two roots. (We use the fact that the characteristic of $F$ is 0
here.) Since $F$ is algebraically closed we can decompose
$\overline{P}=\overline{Q}\overline{R}$ where $\overline{Q}$ and
$\overline{R}$ are monic, of nonzero degree and coprime. Therefore by Hensel’s
lemma, there exists $Q,R\in\Lambda_{0}[y]$ such that $P=QR$ and $\deg
Q=\deg\overline{Q}$, $Q\equiv\overline{Q}\mod\Lambda_{+}$,
$R\equiv\overline{R}\mod\Lambda_{+}$333A proof of Hensel’s lemma, in the case
when valuation is not necessarily discrete, is given, for example, in p 144
[BGR]. See also the proof of Lemma 8.5 given below..

Since the degree of $Q$ is smaller than the degree of $P$, it follows from
induction hypothesis that $Q$ has a root in $\Lambda^{F}$. The proof of Lemma
14.1 is now complete. ∎

By a similar argument we can show that if $F$ is characteristic $0$ then a
finite algebraic extension of $\Lambda^{F}$ is contained in $\Lambda^{K}$ for
some finite extension $K$ of $F$. (We used this fact in section 7.)

###### Proof of Lemma 8.5.

In view of the proof of Lemma 14.1, it suffices to show that
$\Lambda_{0}^{conv}$ is Henselian. (Namely Hensel’s lemma holds for it.) Let

$P(X)=\sum_{i=0}^{n-1}a_{i}X^{i}+X^{n}\in\Lambda_{0}^{conv}[X].$

We assume that its ${\mathbb{C}}$-reduction $\overline{P}\in{\mathbb{C}}[X]$
is decomposed as $\overline{P}=\overline{Q}\overline{R}$, where $\overline{Q}$
and $\overline{R}$ are monic and coprime. We put

$a_{i}=a_{i,0}+a_{i,+},\qquad a_{i,0}\in{\mathbb{C}},\quad
a_{i,+}\in\Lambda_{+}^{conv},$

and

$\widetilde{P}(X)=\sum_{i=0}^{n-1}(a_{i,0}+Z_{i})X^{i}+X^{n}\in{\mathbb{C}}[Z_{0},\cdots,Z_{n-1}][X],$

where $Z_{i}$ are formal variables.

The convergent power series ring ${\mathbb{C}}\\{Z_{0},\cdots,Z_{n-1}\\}$ is
Henselian (see section 45 [Na]). Therefore there exist monic polynomials
$\widetilde{Q},\widetilde{R}\in{\mathbb{C}}\\{Z_{0},\cdots,Z_{n-1}\\}[X]$ such
that

$\widetilde{Q}\widetilde{R}=\widetilde{P}$

and the ${\mathbb{C}}$-reduction of $\widetilde{Q}$, $\widetilde{R}$ are
$\overline{Q}$, $\overline{R}$ respectively. On the other hand, it is easy to
see that $Z_{i}\mapsto a_{i,+}$ induces a continuous ring homomorphism
${\mathbb{C}}\\{Z_{0},\cdots,Z_{n-1}\\}\to\Lambda_{0}^{conv}$. Thus
$\widetilde{Q}$, $\widetilde{R}$ induce $Q,R\in\Lambda_{0}^{conv}[X]$ such
that $QR=P$. Hence $\Lambda_{0}^{conv}$ is Henselian, as required. ∎

##  15\. Appendix 2 : $T^{n}$-equivariant Kuranishi structure

In this section we define the notion of $T^{n}$ equivariant Kuranishi
structure and prove that our moduli space $\mathcal{M}_{k+1}^{\text{\rm
main}}(\beta)$ has one. We also show the existence of $T^{n}$ equivariant
perturbation of the Kuranishi map.

Let $\mathcal{M}$ be a compact space with Kuranishi structure. The space
${\mathcal{M}}$ is covered by a finite number of Kuranishi charts
$(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$,
$\alpha\in\mathfrak{A}$ which satisfy the following :

######  Condition 15.1.

  1. (1)

$V_{\alpha}$ is a smooth manifold (with boundaries or corners) and
$\Gamma_{\alpha}$ is a finite group acting effectively on $V_{\alpha}$.

  2. (2)

$\text{pr}_{\alpha}:E_{\alpha}\to V_{\alpha}$ is a finite dimensional vector
bundle on which $\Gamma_{\alpha}$ acts so that $\text{pr}_{\alpha}$ is
$\Gamma_{\alpha}$\- equivariant.

  3. (3)

$s_{\alpha}$ is a $\Gamma_{\alpha}$ equivariant section of $E_{\alpha}$.

  4. (4)

$\psi_{\alpha}:s_{\alpha}^{-1}(0)/\Gamma_{\alpha}\to\mathcal{M}$ is a
homeomorphism to its image.

  5. (5)

The union of $\psi_{\alpha}(s_{\alpha}^{-1}(0)/\Gamma_{\alpha})$ for various
$\alpha$ is $\mathcal{M}$.

We assume that
$\\{(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})\mid\alpha\in\mathfrak{A}\\}$
is a good coordinate system, in the sense of Definition 6.1 [FO] or Lemma
A1.11 [FOOO3] (= Lemma A1.11 [FOOO2]). This means the following : The set
$\mathfrak{A}$ has a partial order $<$, where either
$\alpha_{1}\leq\alpha_{2}$ or $\alpha_{2}\leq\alpha_{1}$ holds for
$\alpha_{1},\alpha_{2}\in\mathfrak{A}$ if

$\psi_{\alpha_{1}}(s_{\alpha_{1}}^{-1}(0)/\Gamma_{\alpha_{1}})\cap\psi_{\alpha_{2}}(s_{\alpha_{2}}^{-1}(0)/\Gamma_{\alpha_{2}})\neq\emptyset.$

Let $\alpha_{1},\alpha_{2}\in\mathfrak{A}$ and $\alpha_{1}\leq\alpha_{2}$.
Then, there exists a $\Gamma_{\alpha_{1}}$-invariant open subset
$V_{\alpha_{2},\alpha_{1}}\subset V_{\alpha_{1}}$, a smooth embedding
$\varphi_{\alpha_{2},\alpha_{1}}:V_{\alpha_{2},\alpha_{1}}\to V_{\alpha_{2}}$
and a bundle map
$\widehat{\varphi}_{\alpha_{2},\alpha_{1}}:E_{\alpha_{1}}|_{V_{\alpha_{2},\alpha_{1}}}\to
E_{\alpha_{2}}.$ which covers $\varphi_{\alpha_{2},\alpha_{1}}$. Moreover
there exists an injective homomorphism
$\widehat{\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}:\Gamma_{\alpha_{1}}\to\Gamma_{\alpha_{2}}.$
We require that they satisfy the following

######  Condition 15.2.

  1. (1)

The maps $\varphi_{\alpha_{2},\alpha_{1}}$,
${\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}$ are
$\widehat{\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}$-equivariant.

  2. (2)

$\varphi_{\alpha_{2},\alpha_{1}}$ and
$\widehat{\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}$ induce an embedding of
orbifold

$\overline{\varphi}_{\alpha_{2},\alpha_{1}}:\frac{V_{\alpha_{2},\alpha_{1}}}{\Gamma_{\alpha_{1}}}\to\frac{V_{\alpha_{2}}}{\Gamma_{\alpha_{2}}}.$
(15.1)

  3. (3)

We have
$s_{\alpha_{2}}\circ\varphi_{\alpha_{2},\alpha_{1}}={\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}\circ
s_{\alpha_{1}}.$

  4. (4)

We have
$\psi_{\alpha_{2}}\circ\overline{\varphi}_{\alpha_{2},\alpha_{1}}=\psi_{\alpha_{1}}$
on $\frac{V_{\alpha_{2},\alpha_{1}}\cap
s_{\alpha_{1}}^{-1}(0)}{\Gamma_{\alpha_{1}}}.$

  5. (5)

If $\alpha_{1}<\alpha_{2}<\alpha_{3}$ then
$\varphi_{\alpha_{3},\alpha_{2}}\circ\varphi_{\alpha_{2},\alpha_{1}}=\varphi_{\alpha_{3},\alpha_{1}},$
on $\varphi_{\alpha_{2},\alpha_{1}}^{-1}(V_{\alpha_{3},\alpha_{2}})$.
${\widehat{\varphi}}_{\alpha_{3},\alpha_{2}}\circ{\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}={\widehat{\varphi}}_{\alpha_{3},\alpha_{1}},$
and

$\widehat{\widehat{\varphi}}_{\alpha_{3},\alpha_{2}}\circ\widehat{\widehat{\varphi}}_{\alpha_{2},\alpha_{1}}=\widehat{\widehat{\varphi}}_{\alpha_{3},\alpha_{1}},$

hold in the similar sense.

  6. (6)

$V_{\alpha_{2},\alpha_{1}}/\Gamma_{\alpha_{1}}$ contains
$\psi_{\alpha_{1}}^{-1}(\psi_{\alpha_{1}}(s_{\alpha_{1}}^{-1}(0)/\Gamma_{\alpha_{1}})\cap\psi_{\alpha_{2}}(s_{\alpha_{2}}^{-1}(0)/\Gamma_{\alpha_{2}}))$.

######  Condition 15.3.

$\mathcal{M}$ has a tangent bundle : i.e., the differential of
$s_{\alpha_{2}}$ in the direction of the normal bundle induces a bundle
isomorphism

$ds_{\alpha_{2}}:\frac{{\varphi}_{\alpha_{2},\alpha_{1}}^{*}TV_{\alpha_{2}}}{TV_{\alpha_{2},\alpha_{1}}}\to\frac{\widehat{\varphi}_{\alpha_{2},\alpha_{1}}^{*}E_{\alpha_{2}}}{E_{\alpha_{1}}}.$

We say $\mathcal{M}$ is oriented if $V_{\alpha}$, $E_{\alpha}$ is oriented,
the $\Gamma_{\alpha}$ action is orientation preserving, and $ds_{\alpha}$ is
orientation preserving.

######  Definition 15.4.

Suppose that $\mathcal{M}$ has a $T^{n}$ action. We say our Kuranishi
structure on $\mathcal{M}$ is $T^{n}$ equivariant in the strong sense if the
following holds :

  1. (1)

$V_{\alpha}$ has a $T^{n}$ action which commutes with the given
$\Gamma_{\alpha}$ action.

  2. (2)

$E_{\alpha}$ is a $T^{n}$ equivariant bundle.

  3. (3)

The Kuranishi map $s_{\alpha}$ is $T^{n}$ equivariant and $\psi_{\alpha}$ is a
$T^{n}$ equivariant map.

  4. (4)

The coordinate changes $\varphi_{\alpha_{2},\alpha_{1}}$ and
$\varphi_{\alpha_{2},\alpha_{1}}$ are $T^{n}$ equivariant.

######  Remark 15.5.

We remark that Condition (1) above is more restrictive than the condition that
the orbifold $V_{\alpha}/\Gamma_{\alpha}$ has a $T^{n}$ action. This is the
reason why we use the phrase in the strong sense in the above definition.
Hereafter we will say $T^{n}$ equivariant instead for simplicity.

Let $L$ be a smooth manifold. A strongly continuous smooth map
$ev:\mathcal{M}\to L$ is a family of $\Gamma_{\alpha}$ invariant smooth maps

$ev_{\alpha}:V_{\alpha}\to L$ (15.2)

which induce $\overline{ev}_{\alpha}:V_{\alpha}/\Gamma_{\alpha}\to L$ such
that
$\overline{ev}_{\alpha_{2}}\circ\overline{\varphi}_{\alpha_{2},\alpha_{1}}=\overline{ev}_{\alpha_{1}}$
on $V_{\alpha_{2},\alpha_{1}}/\Gamma_{\alpha}$. (Note $\Gamma_{\alpha}$ action
on $L$ is trivial.)

We say that $ev$ is weakly submersive if each of ${ev}_{\alpha}$ in (15.2) is
a submersion.

######  Definition 15.6.

Assume that there exists $T^{n}$ actions on $L$ and $\mathcal{M}$. We say that
$ev:\mathcal{M}\to L$ is $T^{n}$ equivariant if ${ev}_{\alpha}$ in (15.2) is
$T^{n}$ equivariant.

Now we show

######  Proposition 15.7.

The moduli space $\mathcal{M}_{k+1}(\beta)$ has a $T^{n}$ equivariant
Kuranishi structure such that $ev_{0}:\mathcal{M}_{k+1}(\beta)\to L$ is
$T^{n}$ equivariant strongly continuous weakly submersive map.

###### Proof.

Except the $T^{n}$ equivariance this is proved in section 7.1 [FOOO3] (=
section 29 [FOOO2]). Below we explain how we choose our Kuranishi structure so
that it is $T^{n}$ equivariant. We also include the case of the moduli space
$\mathcal{M}_{k+1,\ell}(\beta)$ with interior marked points.

We first review the construction of the Kuranishi neighborhood from section
7.1 [FOOO3] (= section 29 [FOOO2]).

Let $\text{\bf x}=((\Sigma,\vec{z}),w)\in\mathcal{M}_{k+1,\ell}(\beta)$. Let
$\Sigma_{a}$ be an irreducible component of $\Sigma$. We consider the operator

$D_{w,a}\overline{\partial}:W^{1,p}(\Sigma_{a};w^{*}(TX);L,\vec{z}_{a})\to
W^{0,p}(\Sigma_{a};w^{*}(TX)\otimes\Lambda^{0,1}).$ (15.3)

Here $W^{1,p}(\Sigma_{a};w^{*}(TX);L,\vec{z}_{a})$ is the space of section $v$
of $w^{*}(TX)$ of $W^{1,p}$ class with the following properties.

  1. (1)

The restriction of $v$ to $\partial\Sigma_{a}$ is in $w^{*}(TL)$.

  2. (2)

$\vec{z}_{a}$ is the set of $\Sigma$ which is either singular or marked. We
assume that $v$ is $0$ at those points.

$\Lambda^{0,1}$ is the bundle of $(0,1)$ forms on $\Sigma_{a}$ and
$W^{0,p}(\Sigma_{a};w^{*}(TX)\otimes\Lambda^{0,1})$ is the set of sections of
$W^{0,p}$ class of $w^{*}(TX)\otimes\Lambda^{0,1}$.
$D_{w,a}\overline{\partial}$ is the linearization of (nonlinear) Cauchy-
Riemann equation. (See [Fl].) The operator (15.3) is Fredholm by ellipticity
thereof.

We choose open subsets $W_{a}$ of $\Sigma_{a}$ whose closure is disjoint from
the boundary of each of the irreducible component $\Sigma_{a}$ of $\Sigma$ and
from the singular points and marked points. By the unique continuation
theorem, we can choose a finite dimensional subset $E_{0,a}$ of
$C_{0}^{\infty}(W_{a};w^{*}TX)$ (the set of smooth sections of compact support
in $W_{a}$) such that

$\text{\rm
Im}D_{w,a}\overline{\partial}+E_{0,a}=W^{0,p}(\Sigma_{a};w^{*}(TX)\otimes\Lambda^{0,1}).$

When x has nontrivial automorphisms, we choose $\bigoplus_{a}E_{0,a}$ to be
invariant under the automorphisms.

We next associate a finite dimensional subspace
$E_{0,a}((\Sigma,\vec{z}),w^{\prime})$ for $w^{\prime}$ which is ‘$C^{0}$
close to $w$.’ We need some care to handle the case where some component
$(\Sigma_{a},\vec{z}_{a})$ is not stable, that is the case for which the
automorphism group $G_{a}$ of $(\Sigma_{a},\vec{z}_{a})$ is of positive
dimension. (Note $G_{a}$ is different from the automorphism group of
$((\Sigma_{a},\vec{z}_{a}),w|_{\Sigma_{a}})$. The latter group is finite.) We
explain this choice of $E_{0,1}$ below following Appendix [FO].

For each unstable components $\Sigma_{a}$, pick points $z_{a,i}\in\Sigma$,
$(i=1,\cdots,d_{a})$ with the following properties.

######  Condition 15.8.

  1. (1)

$w$ is immersed at $z_{a,i}$.

  2. (2)

$z_{a,i}$ in the interior of $\Sigma_{a}$, $z_{a,i}\neq z_{a,j}$ for $i\neq j$
and $z_{a,i}\notin\vec{z}$.

  3. (3)

$(\Sigma_{a};(\vec{z}\cap\Sigma_{a})\cup(z_{a,1},\cdots,z_{a,d_{a}}))$ is
stable.

  4. (4)

Let $\Gamma=\Gamma(\text{\bf x})$ be the group of automorphisms of $\text{\bf
x}=((\Sigma,\vec{z}),w)$. If $\gamma\in\Gamma$,
$\gamma(\Sigma_{a})=\Sigma_{a^{\prime}}$ then $d_{a}=d_{a^{\prime}}$ and
$\\{\gamma(z_{a,i})\mid i=1,\cdots,d_{a}\\}=\\{z_{a^{\prime},i^{\prime}}\mid
i^{\prime}=1,\cdots,d_{a}\\}$.

For each $a,i$ we choose a submanifold $N_{a,i}$ of codimension $2$ in $X$
that transversely intersects with $(\Sigma_{a},w)$ at $w_{a}(z_{a,i})$. In
relation to Condition 15.8 (4) above we choose
$N_{a,i}=N_{a^{\prime},i^{\prime}}$ if
$\gamma(z_{a,i})=z_{a^{\prime},i^{\prime}}$.

We add extra interior marked points $\\{z_{a,i}\mid a,i\\}$ in addition to
$\vec{z}$ on $(\Sigma,\vec{z})$, and obtain a stable curve
$(\Sigma,\vec{z}^{+})$. (Namely $\vec{z}^{+}=\vec{z}\sqcup\\{z_{a,i}\mid
a,i\\}$.) (We choose an order of the added marked points and fix it.)

We consider a neighborhood $\mathfrak{U}_{0}$ of $[\Sigma,\vec{z}^{+}]$ in
$\mathcal{M}_{k+1,\ell^{\prime}}^{\text{\rm main}}$, that is the moduli space
of bordered stable curve of genus $0$ with $k+1$ boundary and $\ell^{\prime}$
interior marked points. Let $\Gamma_{0}$ be the group of automorphisms of
$((\Sigma,\vec{z}^{+}),w)$. Now both $\Gamma$ and $\Gamma_{0}$ are finite
groups and $\Gamma\supseteq\Gamma_{0}$.

An element $\gamma\in\Gamma$ induces an automorphism $\gamma:\Sigma\to\Sigma$
which fixes marked points in $\vec{z}$ and permutes $\ell-\ell^{\prime}$
marked points $\\{z_{a,i}\mid a,i\\}$, by Condition 15.8 (4). Therefore we
obtain an element of $[\gamma_{*}(\Sigma,\vec{z}^{+})]$ that is different from
$[\Sigma,\vec{z}^{+}]$ only by reordering of marked points. We take the union
of neighborhoods of $[\gamma_{*}(\Sigma,\vec{z}^{+})]$ for various
$\gamma\in\Gamma$ in $\mathcal{M}_{k+1,\ell^{\prime}}^{\text{\rm main}}$ and
denote it by $\mathfrak{U}$.

$\mathfrak{U}_{0}$ is written as $\mathfrak{V}_{0}/\Gamma_{0}$ where
$\mathfrak{V}_{0}$ is a manifold. Moreover there exists a manifold
$\mathfrak{V}$ on which $\Gamma$ acts such that
$\mathfrak{V}/\Gamma_{0}=\mathfrak{U}$,
$\mathfrak{V}/\Gamma\cong\mathfrak{U}_{0}$.

Furthermore there is a ‘universal family’ $\mathfrak{M}\to\mathfrak{V}$ where
the fiber $\Sigma(\text{\bf v})$ of $\text{\bf v}\in\mathfrak{V}$ is
identified with the bordered marked stable curve that represents the element
$[\text{\bf
v}]\in\mathfrak{V}/\Gamma\subset\mathcal{M}_{k+1,\ell^{\prime}}^{\text{\rm
main}}$. There is a $\Gamma$ action on $\mathfrak{M}$ such that
$\mathfrak{M}\to\mathfrak{V}$ is $\Gamma$ equivariant.

By the construction, each member $\Sigma(\text{\bf v})$ of our universal
family is diffeomorphic to $\Sigma$ away from singularity. More precisely we
have the following :

Let $\Sigma_{0}=\Sigma\setminus S$ where $S$ is a small neighborhood of the
union of the singular point sets and the marked point sets. Then for each v
there exists a smooth embedding $i_{\text{\bf
v}}:\Sigma_{0}\to\Sigma(\text{\bf v})$. The error of this embedding for
becoming a biholomorphic map goes to $0$ as v goes to $0$. We may assume
$\text{\bf v}\mapsto i_{\text{\bf v}}$ is $\Gamma$ invariant in an obvious
sense and $i_{\text{\bf v}}$ depends smoothly on v.

We may choose $W_{a}$ so that $W_{a}\subset\Sigma_{0}$ for each $a$. We
moreover assume that $\bigoplus_{a}E_{0,a}$ is invariant under the $\Gamma$
action in the following sense : if $\gamma\in\Gamma$ and
$\Sigma_{a^{\prime}}=\gamma(\Sigma_{a})$ then the isomorphism induced by
$\gamma$ sends $E_{0,a}$ to $E_{0,a^{\prime}}$.

Now we consider a pair $(w^{\prime},\text{\bf v})$ where

$w^{\prime}:(\Sigma(\text{\bf v}),\partial\Sigma(\text{\bf v}))\to(X,L).$

We assume :

######  Condition 15.9.

There exists $\epsilon>0$ depending only on x such that the following holds.

  1. (1)

$\sup_{x\in\Sigma_{0}}\text{\rm dist}(w^{\prime}(i_{\text{\bf
v}}(x)),w(x))\leq\epsilon$.

  2. (2)

For any connected component $D_{c}$ of $\Sigma(\text{\bf v})\setminus\text{\rm
Im}(i_{\text{\bf v}})$, the diameter of $w^{\prime}(D_{c})$ in $X$ (with a
fixed Riemannian metric on it) is smaller than $\epsilon$.

For each point $x\in W_{a}$ we use the parallel transport to make the
identification

$T_{w(x)}X\otimes\Lambda^{0,1}_{x}(\Sigma)\equiv T_{w^{\prime}(i_{\text{\bf
v}}(x))}X\otimes\Lambda^{0,1}_{i_{\text{\bf v}}(x)}(\Sigma(\text{\bf v})).$

Using this identification we obtain an embedding

$I_{(\text{\bf v},w^{\prime})}:\bigoplus_{a}E_{0,a}\longrightarrow
w^{\prime*}(TX)\otimes\Lambda^{0,1}(\Sigma(\text{\bf v})).$

Via this embedding $I_{(\text{\bf v},w^{\prime})}$, we consider the equation

$\overline{\partial}w^{\prime}\equiv 0\mod\bigoplus_{a}I_{(\text{\bf
v},w^{\prime})}(E_{0,a}),$ (15.4)

together with the additional conditions

$w^{\prime}(z_{a,i})\in N_{a,i}.$ (15.5)

Let $V(\text{\bf x})$ be the set of solution of (15.4), (15.5). This is a
smooth manifold (with boundary and corners) by the implicit function theorem
and a gluing argument. (See [FOOO3] section A1.4 = [FOOO2] section A1.4 for
the smoothness at boundary and corner.) Since we can make all the construction
above invariant under the $\Gamma(\text{\bf x})$ action, the space
$V(\text{\bf x})$ has a $\Gamma(\text{\bf x})$ action. (Note that we may
choose $N_{a,i}$ so that $\\{N_{a,i}\mid a,i\\}$ is invariant under the action
of $\Gamma(\text{\bf x})$.)

The obstruction bundle $E$ is the space $\bigoplus_{a}E_{0,a}$ at x and
$\bigoplus_{a}I_{(\text{\bf v},w^{\prime})}(E_{0,a})$ at in $(\text{\bf
v},w^{\prime})$.

We omit the construction of coordinate changes. See [FOOO3] section 7.1 (=
[FOOO2] section 29) which in turn is similar to section 15 [FO].

The Kuranishi map is given by

$((\Sigma^{\prime},\vec{z}^{\prime}),w^{\prime})\mapsto\overline{\partial}w^{\prime}\in\bigoplus_{a}E_{0,a}.$
(15.6)

We have thus finished our review of the construction of Kuranishi charts.

Now we assume that $X$ has a $T^{n}$ action, which preserves both the complex
and the symplectic structures on $X$. We also assume that $L$ is a $T^{n}$
orbit.

We want to construct a family of vector spaces $(\text{\bf
v},w^{\prime})\mapsto\bigoplus_{a}I_{(\text{\bf v},w^{\prime})}(E_{0,a})$ so
that it is invariant under the $T^{n}$ action. We need to slightly modify the
above construction for this purpose. In fact it is not totally obvious to make
the condition (15.5) $T^{n}$-invariant.

For this purpose we proceed in the following way. We fix a point $p_{0}\in L$
and consider an element

$\text{\bf x}\in ev_{0}^{-1}(p_{0})\cap\mathcal{M}_{k+1,\ell}(\beta).$

We are going to construct a $T^{n}$ equivariant Kuranishi neighborhood of the
$T^{n}$ orbit of x. Let $\text{\bf v}\in\mathfrak{V}$ and
$w^{\prime}:(\Sigma(\text{\bf v}),\partial\Sigma(\text{\bf v}))\to(X,L)$ be as
before. We replace Condition 15.9 by the following.

######  Condition 15.10.

Let $z_{0}$ be the 0-th boundary marked point and let $g\in T^{n}$ be the
unique element satisfying $w^{\prime}(z_{0})=g(p_{0})$. There exists
$\epsilon>0$ depending only on x such that the following holds.

  1. (1)

$\sup_{x\in\Sigma_{0}}\operatorname{dist}(w^{\prime}(i_{\text{\bf
v}}(x)),g(w(x)))\leq\epsilon$.

  2. (2)

For any connected component $D_{c}$ of $\Sigma(\text{\bf v})\setminus\text{\rm
Im}(i_{\text{\bf v}})$, the diameter of $w^{\prime}(D_{c})$ in $X$ (with a
fixed Riemannian metric on it) is smaller than $\epsilon$.

Now we define an embedding

$I_{(\text{\bf v},w^{\prime})}:\bigoplus_{a}E_{0,a}\longrightarrow
w^{\prime*}(TX)\otimes\Lambda^{0,1}(\Sigma(\text{\bf v}))$

as follows : We first use the $g$ action to define an isomorphism

$g_{*}:T_{w(x)}X\otimes\Lambda^{0,1}_{x}(\Sigma)\cong
T_{g(w(x))}X\otimes\Lambda^{0,1}_{x}(\Sigma).$

Then we use the parallel transport in the same way as before to define

$\bigoplus_{a}g_{*}(E_{0,a})\longrightarrow
w^{\prime*}(TX)\otimes\Lambda^{0,1}(\Sigma(\text{\bf v})).$

By composing the two we obtain the embedding $I_{(\text{\bf v},w^{\prime})}$.
Now we consider the equation

$\overline{\partial}w^{\prime}\equiv 0\mod\bigoplus_{a}I_{(\text{\bf
v},w^{\prime})}(E_{0,a}),$ (15.7)

together with the additional conditions

$w^{\prime}(z_{a,i})\in g(N_{a,i})$ (15.8)

as before. Clearly these equations are $T^{n}$ invariant. It is also
$\Gamma(\text{\bf x})$ invariant.

Note the action of the automorphism group $\Gamma(\text{\bf x})$ of x of our
Kuranishi structure, which is a finite group, acts on the source while $T^{n}$
acts on the target. Therefore it is obvious that two actions commute.

By definition the obstruction bundle has a $T^{n}$ action. Moreover (15.6) is
$T^{n}$ equivariant. It is fairly obvious from the construction that
coordinate changes of the constructed Kuranishi structure is also $T^{n}$
equivariant.

The proof of Proposition 15.7 is now complete. ∎

######  Remark 15.11.

  1. (1)

From the above construction, it is easy to see that our $T^{n}$ equivariant
Kuranishi structure is compatible with the gluing at the boundary marked
points. Namely under the embedding

$\mathcal{M}_{k_{1}+1}(\beta_{1}){}_{ev_{0}}\times_{ev_{i}}\mathcal{M}_{k_{2}+1}(\beta_{2})\subset\mathcal{M}_{k_{1}+k_{2}}(\beta_{1}+\beta_{2})$

the restriction of the Kuranishi structure of the right hand side coincides
with the fiber product of the Kuranishi structure of the left hand side. More
precisely speaking, we can construct the system of our Kuranishi structures so
that this statement holds : this Kuranishi structure is constructed
inductively over the number of disc components and the energy of $\beta$.

  2. (2)

On the other hand, when we construct the $T^{n}$ invariant Kuranishi structure
in the way we described above it may not be compatible with the gluing at the
interior marked point. Namely by the embedding

$\mathcal{M}_{1}(\alpha)\times_{X}\mathcal{M}_{k+1,1}(\beta)\subset\mathcal{M}_{k+1}(\beta+\alpha)$
(15.9)

the restriction of the Kuranishi structure of the right hand sides may not
coincide with the fiber product of the Kuranishi structure of the left hand
side. This compatibility is not used in this paper and hence not required. See
Remark 11.4 where similar point is discussed for the choice of multisections.

However contrary to the choice of multisections mentioned in Remark 11.4, we
remark that it is possible to construct $T^{n}$ equivariant Kuranishi
structure compatible with (15.9). Since we do not use this point in the paper,
we will not elaborate it here.

We next review on the multisections. (See section 3 [FO].) Let
$(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$ be a
Kuranishi chart of $\mathcal{M}$. For $x\in V_{\alpha}$ we consider the fiber
$E_{\alpha,x}$ of the bundle $E_{\alpha}$ at $x$. We take its $l$ copies and
consider the direct product $E_{\alpha,x}^{l}$. We take the quotient thereof
by the action of symmetric group of order $l!$ and let
$\mathcal{S}^{l}(E_{\alpha,x})$ be the quotient space. There exists a map
$tm_{m}:\mathcal{S}^{l}(E_{\alpha,x})\to\mathcal{S}^{lm}(E_{\alpha,x}),$ which
sends $[a_{1},\cdots,a_{l}]$ to $[\,\underbrace{a_{1},\cdots,a_{1}}_{\text{$m$
copies}},\cdots,\underbrace{a_{l},\cdots,a_{l}}_{\text{$m$ copies}}].$ A
smooth multisection $s$ of the bundle $E_{\alpha}\to V_{\alpha}$ consists of
an open covering $\bigcup_{i}U_{i}=V_{\alpha}$ and $s_{i}$ which maps $x\in
U_{i}$ to $s_{i}(x)\in\mathcal{S}^{l_{i}}(E_{\alpha,x})$. They are required to
have the following properties :

######  Condition 15.12.

  1. (1)

$U_{i}$ is $\Gamma_{\alpha}$-invariant. $s_{i}$ is
$\Gamma_{\alpha}$-equivariant. (We remark that there exists an obvious map
$\gamma:\mathcal{S}^{l_{i}}(E_{\alpha,x})\to\mathcal{S}^{l_{i}}(E_{\alpha,\gamma
x})$ for each $\gamma\in\Gamma_{\alpha}$.)

  2. (2)

If $x\in U_{i}\cap U_{j}$ then we have

$tm_{l_{j}}(s_{i}(x))=tm_{l_{i}}(s_{j}(x))\in\mathcal{S}^{l_{i}l_{j}}(E_{\alpha,\gamma
x}).$

  3. (3)

$s_{i}$ is liftable and smooth in the following sense. For each $x$ there
exists a smooth section $\tilde{s}_{i}$ of
$\underbrace{E_{\alpha}\oplus\cdots\oplus E_{\alpha}}_{\text{$l_{i}$ times}}$
in a neighborhood of $x$ such that

$\tilde{s}_{i}(y)=(s_{i,1}(y),\cdots,s_{i,l_{i}}(y)),\quad
s_{i}(y)=[s_{i,1}(y),\cdots,s_{i,l_{i}}(y)].$ (15.10)

We identify two multisections $(\\{U_{i}\\},\\{s_{i}\\},\\{l_{i}\\})$,
$(\\{U^{\prime}_{i}\\},\\{s^{\prime}_{i}\\},\\{l^{\prime}_{i}\\})$ if

$tm_{l_{i}}(s_{i}(x))=tm_{l^{\prime}_{j}}(s^{\prime}_{j}(x))\in\mathcal{S}^{l_{i}l^{\prime}_{j}}(E_{\alpha,\gamma
x})$

on $U_{i}\cap U^{\prime}_{j}$. We say $s_{i,j}$ to be a branch of $s_{i}$ in
the situation of (15.10).

We next prove the following lemma which we used in section 11.

######  Lemma 15.13.

Let $\mathcal{M}$ have a $T^{n}$ action and a $T^{n}$ equivariant Kuranishi
structure. Suppose that the $T^{n}$ action on each of the Kuranishi
neighborhood is free. Then we can descend the Kuranishi structure to
$\mathcal{M}/T^{n}$ in a canonical way.

###### Proof.

Let $(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$ be a
Kuranishi chart. Since $T^{n}$ action on $V_{\alpha}$ is free
$V_{\alpha}/T^{n}$ is a smooth manifold and $E_{\alpha}/T^{n}\to
V_{\alpha}/T^{n}$ is a vector bundle. Since $\Gamma_{\alpha}$ action commutes
with $T^{n}$ action, it follows that it acts on this vector bundle. The
$T^{n}$ equivariance of $s_{\alpha}$ implies that we have a section
$\overline{s}_{\alpha}$ of $E_{\alpha}/T^{n}\to V_{\alpha}/T^{n}$. Moreover
since $\psi_{\alpha}:s_{\alpha}^{-1}(0)\to\mathcal{M}$ is $T^{n}$ equivariant
it induces a map $\overline{s}_{\alpha}^{-1}(0)\to\mathcal{M}/T^{n}$. Thus we
obtain a Kuranishi chart. It is easy to define the coordinate changes. ∎

######  Definition 15.14.

In the situation of Proposition 15.13, we say that a system of multisections
of the Kuranishi structure of $\mathcal{M}$ is $T^{n}$ equivariant, if it is
induced from the multisection of the Kuranishi structure on
$\mathcal{M}/T^{n}$ in an obvious way.

######  Corollary 15.15.

In the situation of Proposition 15.13, we assume that we have a $T^{n}$
equivariant multisection at the boundary of $\mathcal{M}$, which is
transversal to $0$. Then it extends to a $T^{n}$ equivariant multisection of
$\mathcal{M}$.

This is an immediate consequence of Lemma 3.14 [FO], that is the non-
equivariant version.

##  16\. Appendix 3 : Smooth correspondence via the zero set of multisection

In this section we explain the way how we use the zero set of multisection to
define smooth correspondence, when appropriate submersive properties are
satisfied.

Such a construction is a special case of the techniques of using a continuous
family of multisections and integration along the fiber on their zero sets so
that smooth correspondence by spaces with Kuranishi structure induces a map
between de Rham complex.

This (more general) technique is not new and is known to some experts. In fact
[Ru], section 16 [Fu2] use a similar technique and section 7.5 [FOOO3] (=
section 33 [FOOO2]), [Fu4], [Fu5] contain the details of this more general
technique.

We explain the special case (namely the case we use a single multisection) in
our situation of toric manifold, for the sake of completeness and of reader’s
convenience.

Let $\mathcal{M}$ be a space with Kuranishi structure and
$ev_{s}:\mathcal{M}\to L_{s}$, $ev_{t}:\mathcal{M}\to L_{t}$ be strongly
continuous smooth maps. (Here $s$ and $t$ stand for source and target,
respectively.) We assume our smooth manifolds $L_{s},L_{t}$ are compact and
oriented without boundary. We also assume $\mathcal{M}$ has a tangent bundle
and is oriented in the sense of Kuranishi structure.

Suppose that $L_{t}$ has a free and transitive $T^{n}$ action, and $L_{s}$ and
$\mathcal{M}$ have $T^{n}$ action. We assume that the Kuranishi structure on
$\mathcal{M}$ is $T^{n}$ equivariant and the maps $ev_{s}$, $ev_{t}$ are
$T^{n}$ equivariant.

Let $\\{(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})\\}$
be a $T^{n}$ equivariant Kuranishi coordinate system (good coordinate system)
of $\mathcal{M}$. We use Corollary 15.15 to find a $T^{n}$ equivariant
(system) of multisection $\mathfrak{s}_{\alpha}:V_{\alpha}\to E_{\alpha}$
which is transversal to $0$.

Let $\theta_{\alpha}$ be a smooth differential form of compact support on
$V_{\alpha}$. We assume that $\theta_{\alpha}$ is $\Gamma_{\alpha}$-invariant.
Let $f_{\alpha}:V_{\alpha}\to L_{s}$ be a $\Gamma_{\alpha}$ equivariant
submersion. (The $\Gamma_{\alpha}$ action on $L_{s}$ is trivial.) We next
define integration along the fiber

$((V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha}),\mathfrak{s}_{\alpha},f_{\alpha})_{*}(\theta_{\alpha})\in\Omega^{\deg\theta_{\alpha}+\mathop{\kern
0.0pt{\rm dim}}\nolimits L_{t}-\mathop{\kern 0.0pt{\rm
dim}}\nolimits\mathcal{M}}(L_{t}).$

We first fix $\alpha$. Let $(U_{i},\mathfrak{s}_{\alpha,i})$ be a
representative of $\mathfrak{s}_{\alpha}$. Namely $\\{U_{i}\mid i\in I\\}$ is
an open covering of $V_{\alpha}$ and $\mathfrak{s}_{\alpha}$ is represented by
$\mathfrak{s}_{\alpha,i}$ on $U_{i}$. By the definition of the multisection,
$U_{i}$ is $\Gamma_{\alpha}$-invariant. We may shrink $U_{i}$, if necessary,
so that there exists a lifting
$\tilde{\mathfrak{s}}_{\alpha,i}=(\tilde{\mathfrak{s}}_{\alpha,i,1},\cdots,\tilde{\mathfrak{s}}_{\alpha,i,l_{i}})$
as in (15.10).

Let $\\{\chi_{i}\mid i\in I\\}$ be a partition of unity subordinate to the
covering $\\{U_{i}\mid i\in I\\}$. By replacing $\chi_{i}$ with its average
over $\Gamma_{\alpha}$ we may assume $\chi_{i}$ is
$\Gamma_{\alpha}$-invariant.

We put

$\tilde{\mathfrak{s}}_{\alpha,i,j}^{-1}(0)=\\{x\in
U_{i}\mid\tilde{\mathfrak{s}}_{\alpha,i,j}(x)=0\\}.$ (16.1)

By assumption $\tilde{\mathfrak{s}}_{\alpha,i,j}^{-1}(0)$ is a smooth
manifold. Since the $T^{n}$ action on $L_{t}$ is free and transitive it
follows that

$ev_{t,\alpha}|_{\tilde{\mathfrak{s}}_{\alpha,i,j}^{-1}(0)}:\tilde{\mathfrak{s}}_{\alpha,i,j}^{-1}(0)\to
L_{t}$ (16.2)

is a submersion.

######  Definition 16.1.

We define a differential form on $L_{t}$ by

$\displaystyle((V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha}),\mathfrak{s}_{\alpha},ev_{t,\alpha})_{*}(\theta_{\alpha})$
(16.3)
$\displaystyle=\frac{1}{\\#\Gamma_{\alpha}}\sum_{i=1}^{I}\sum_{j=1}^{l_{i}}\frac{1}{l_{i}}(ev_{t,\alpha})_{!}\left(\chi_{i}\theta_{\alpha}|_{\tilde{\mathfrak{s}}_{\alpha,i,j}^{-1}(0)}\right).$

Here $(ev_{t,\alpha})_{!}$ is the integration along fiber of the smooth
submersion (16.2).

######  Lemma 16.2.

The right hand side of $(\ref{intfibermainformula})$ depends only on
$(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$,
$\mathfrak{s}_{\alpha}$, $ev_{t,\alpha}$, and $\theta_{\alpha}$ but
independent of the following choices :

  1. (1)

The choice of representatives $(\\{U_{i}\\},\mathfrak{s}_{\alpha,i})$ of
$\mathfrak{s}_{\alpha}$.

  2. (2)

The partition of unity $\chi_{i}$.

###### Proof.

The proof is straightforward generalization of the proof of well-definedness
of integration on manifold, which can be found in the text book of manifold
theory, and is left to the leader. ∎

So far we have been working on one Kuranishi chart
$(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$. We next
describe the compatibility conditions of multisections for various $\alpha$.
During the construction we need to shrink $V_{\alpha}$ a bit several times. We
will not explicitly mention this point henceforth.

Let $\alpha_{1}<\alpha_{2}$. For $\alpha_{1}<\alpha_{2}$, we take the normal
bundle $N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}$ of
$\varphi_{\alpha_{2},\alpha_{1}}(V_{\alpha_{2},\alpha_{1}})$ in
$V_{\alpha_{2}}$. We use an appropriate $\Gamma_{\alpha_{2}}$ invariant
Riemannian metric on $V_{\alpha_{2}}$ to define the exponential map

$\text{Exp}_{\alpha_{2},\alpha_{1}}:B_{\varepsilon}N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}\to
V_{2}.$ (16.4)

(Here $B_{\varepsilon}N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}$ is the
$\epsilon$ neighborhood of the zero section of
$N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}$.)

Using $\text{Exp}_{\alpha_{2},\alpha_{1}}$, we identify
$B_{\varepsilon}N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}/{\Gamma_{\alpha_{1}}}$
to an open subset of $V_{\alpha_{1}}/{\Gamma_{\alpha_{1}}}$ and denote it by
$U_{\epsilon}({V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}})$.

Using the projection

$\text{\rm
Pr}_{V_{\alpha_{2},\alpha_{1}}}:U_{\epsilon}({V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}})\to{V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}}$

we extend the orbibundle $E_{\alpha_{1}}$ to
$U_{\epsilon}({V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}})$. Also we
extend the embedding
$E_{\alpha_{1}}\to{\widehat{\varphi}_{\alpha_{2},\alpha_{1}}^{*}E_{\alpha_{2}}}$,
(which is induced by $\widehat{\varphi}_{\alpha_{2},\alpha_{1}}$) to
$U_{\epsilon}({V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}})$.

We fix a $\Gamma_{\alpha}$-invariant inner product of the bundles
$E_{\alpha}$. We then have a bundle isomorphism

$E_{\alpha_{2}}\cong
E_{\alpha_{1}}\oplus\frac{\widehat{\varphi}_{\alpha_{2},\alpha_{1}}^{*}E_{\alpha_{2}}}{E_{\alpha_{1}}}$
(16.5)

on $U_{\varepsilon}(V_{\alpha_{2},\alpha_{1}}/\Gamma_{\alpha_{1}})$. We can
use Condition 15.3 to modify $\text{Exp}_{\alpha_{2},\alpha_{1}}$ in (16.4) so
that the following is satisfied.

######  Condition 16.3.

If $y=\text{Exp}_{\alpha_{2},\alpha_{1}}(\tilde{y})\in
U_{\epsilon}({V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}})$ then

$ds_{\alpha_{2}}(\tilde{y}\mod TV_{\alpha_{1}})\equiv s_{\alpha_{2}}(y)\mod
E_{\alpha_{1}}.$ (16.6)

Let us explain the notation of (16.6). We remark that $\tilde{y}\in
T_{{\varphi}_{\alpha_{2},\alpha_{1}}(x)}V_{\alpha_{2}}$ for $x=\text{\rm
Pr}(\tilde{y})\in V_{\alpha_{2},\alpha_{1}}$. Hence

$\tilde{y}\mod
TV_{\alpha_{1}}\in\frac{T_{{\varphi}_{\alpha_{2},\alpha_{1}}(x)}V_{\alpha_{2}}}{T_{x}V_{\alpha_{1}}}.$

Therefore

$ds_{\alpha_{2}}(\tilde{y}\mod
TV_{\alpha_{1}})\in\frac{(E_{\alpha_{2}})_{{\varphi}_{\alpha_{2},\alpha_{1}}(x)}}{(E_{\alpha_{1}})_{x}}.$

(16.6) claims that it coincides with $s_{\alpha_{2}}$ modulo
$(E_{\alpha_{1}})_{x}$.

We remark that Condition 15.3 implies that

$[\tilde{y}]\mapsto ds_{\alpha_{2}}(\tilde{y}\mod
TV_{\alpha_{1}}):\frac{T_{{\varphi}_{\alpha_{2},\alpha_{1}}(x)}V_{\alpha_{2}}}{T_{x}V_{\alpha_{1}}}\longrightarrow\frac{(E_{\alpha_{2}})_{{\varphi}_{\alpha_{2},\alpha_{1}}(x)}}{(E_{\alpha_{1}})_{x}}$

is an isomorphism. Therefore we can use the implicit function theorem to
modify $\text{Exp}_{\alpha_{2},\alpha_{1}}$ so that Condition 16.3 holds.

######  Definition 16.4.

A multisection $\mathfrak{s}_{\alpha_{2}}$ of $V_{\alpha_{2}}$ is said to be
compatible with $\mathfrak{s}_{\alpha_{1}}$ if the following holds for each
$y=\text{Exp}_{\alpha_{2},\alpha_{1}}(\tilde{y})\in
U_{\epsilon}({V_{\alpha_{2},\alpha_{1}}}/{\Gamma_{\alpha_{1}}})$.

$\mathfrak{s}_{\alpha_{2}}(y)=\mathfrak{s}_{\alpha_{1}}(\text{\rm
Pr}(\tilde{y}))\oplus ds_{\alpha_{2}}(\tilde{y}\mod TV_{\alpha_{1}}).$ (16.7)

We remark that $\mathfrak{s}_{\alpha_{1}}(w,\text{\rm Pr}(\tilde{y}))$ is a
multisection of $\pi_{\alpha_{1}}^{*}E_{\alpha_{1}}$ and
$ds_{\alpha_{2}}(\tilde{y}\mod TV_{\alpha_{1}})$ is a (single valued) section.
Therefore via the isomorphism (16.5) the right hand side of (16.7) defines an
element of $\mathcal{S}^{l_{i}}(E_{\alpha_{2}})_{x}$ ($x=\text{\rm
Pr}(\tilde{y})$), and hence is regarded as a multisection of
$\pi_{\alpha_{2}}^{*}E_{\alpha_{2}}$. In other words, we omit
$\widehat{\varphi}_{\alpha_{2},\alpha_{1}}$ in (16.7).

We next choose a partition of unity $\chi_{\alpha}$ subordinate to our
Kuranishi charts. To define the notion of partition of unity, we need some
notation. Let
$\text{Pr}_{\alpha_{2},\alpha_{1}}:N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}\to
V_{\alpha_{2},\alpha_{1}}$ be the projection. We fix a
$\Gamma_{\alpha_{1}}$-invariant positive definite metric of
$N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}$ and let
$r_{\alpha_{2},\alpha_{1}}:N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}\to[0,\infty)$
be the norm with respect to this metric. We fix a sufficiently small $\delta$
and let $\chi^{\delta}:{\mathbb{R}}\to[0,1]$ be a smooth function such that

$\chi^{\delta}(t)=\begin{cases}0&t\geq\delta\\\ 1&t\leq\delta/2.\end{cases}$

Let $U_{\delta}(V_{\alpha_{2},\alpha_{1}}/\Gamma_{\alpha_{1}})$ be the image
of the exponential map. Namely

$U_{\delta}(V_{\alpha_{2},\alpha_{1}}/\Gamma_{\alpha_{1}})=\\{\text{Exp}(v)\mid
v\in N_{V_{\alpha_{2},\alpha_{1}}}V_{\alpha_{2}}/\Gamma_{\alpha_{1}}\mid
r_{\alpha_{2},\alpha_{1}}(v)<\delta\\}.$

We push out our function $r_{\alpha_{2},\alpha_{1}}$ to
$U_{\delta}(V_{\alpha_{2},\alpha_{1}}/\Gamma_{\alpha_{1}})$ and denote it by
the same symbol. We call $r_{\alpha_{2},\alpha_{1}}$ a tubular distance
function. We require $r_{\alpha_{2},\alpha_{1}}$ to satisfy the compatibility
conditions for various tubular neighborhoods and tubular distance functions,
which is formulated in sections 5 and 6 in [Ma].

Let $x\in V_{\alpha}$. We put

$\displaystyle\mathfrak{A}_{x,+}$ $\displaystyle=\\{\alpha_{+}\mid x\in
V_{\alpha_{+},\alpha},\,\alpha_{+}>\alpha\\}$
$\displaystyle\mathfrak{A}_{x,-}$
$\displaystyle=\\{\alpha_{-}\mid[x\mod\Gamma_{\alpha}]\in
U_{\delta}(V_{\alpha,\alpha_{-}}/\Gamma_{\alpha_{-}}),\,\alpha_{-}<\alpha\\}.$

For $\alpha_{-}\in\mathfrak{A}_{x,-}$ we take $x_{\alpha_{-}}$ such that
$\text{Exp}(x_{\alpha_{-}})=x$.

######  Definition 16.5.

A system $\\{\chi_{\alpha}\mid\alpha\in\mathfrak{A}\\}$ of
$\Gamma_{\alpha}$-equivariant smooth functions
$\chi_{\alpha}:V_{\alpha}\to[0,1]$ of compact support is said to be a
partition of unity subordinate to our Kuranishi chart if :

$\chi_{\alpha}(x)+\sum_{\alpha_{-}\in\mathfrak{A}_{x,-}}\chi^{\delta}(r_{\alpha,\alpha_{-}}(x_{\alpha_{-}}))\chi_{\alpha_{-}}(\text{Pr}_{\alpha,\alpha_{-}}(x_{\alpha_{-}}))+\sum_{\alpha_{+}\in\mathfrak{A}_{x,+}}\chi_{\alpha_{+}}(\varphi_{\alpha_{+},\alpha}(x))=1.$

######  Lemma 16.6.

There exists a partition of unity subordinate to our Kuranishi chart. We may
choose them so that they are $T^{n}$ equivariant.

###### Proof.

We may assume that $\mathfrak{A}$ is a finite set since $\mathcal{M}$ is
compact. By shrinking $V_{\alpha}$ if necessary we may assume that there
exists $V^{-}_{\alpha}$ such that $V^{-}_{\alpha}$ is a relatively compact
subset of $V_{\alpha}$ and that $E_{\alpha}$,
$\varphi_{\alpha_{2},\alpha_{1}}$, $s_{\alpha}$, etc restricted to
$V^{-}_{\alpha}$ still defines a good coordinate system. We take a
$\Gamma_{\alpha}$ invariant smooth function $\chi^{\prime}_{\alpha}$ on
$V_{\alpha}$ which has compact support and satisfies
$\chi^{\prime}_{\alpha}=1$ on $V^{-}_{\alpha}$. We define

$h_{\alpha}(x)=\chi^{\prime}_{\alpha}(x)+\sum_{\alpha_{-}\in\mathfrak{A}_{x,-}}\chi^{\delta}(r_{\alpha,\alpha_{-}}(x_{\alpha_{-}}))\chi^{\prime}_{\alpha_{-}}(\text{Pr}_{\alpha,\alpha_{-}}(x_{\alpha_{-}}))+\sum_{\alpha_{+}\in\mathfrak{A}_{x,+}}\chi^{\prime}_{\alpha_{+}}(\varphi_{\alpha_{+},\alpha}(x)).$

Using compatibility of tubular neighborhoods and tubular distance functions,
we can show that $h_{\alpha}$ is $\Gamma_{\alpha}$ invariant and

$h_{\alpha_{2}}(\varphi_{\alpha_{2},\alpha_{1}}(x))=h_{\alpha_{1}}(x)$

if $x\in V_{\alpha_{2},\alpha_{1}}$. Therefore

$\chi_{\alpha}(x)=\chi^{\prime}_{\alpha}(x)/h_{\alpha}(x)$

has the required properties. ∎

Now we consider the situation we start with. Namely we have two strongly
continuous $T^{n}$ equivariant smooth maps

$ev_{s}:\mathcal{M}\to L_{s},\qquad ev_{t}:\mathcal{M}\to L_{t}$

and $ev_{t}$ is weakly submersive. (In fact $T^{n}$ action on $L_{t}$ is
transitive and free.)

Let a differential form $h$ on $L_{s}$ be given. We choose a $T^{n}$
equivariant good coordinate system
$\\{(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})\\}$ of
$\mathcal{M}$ and a $T^{n}$ equivariant multisection represented by
$\\{\mathfrak{s}_{\alpha}\\}$ in this Kuranishi chart. Assume that the
multisection is transversal to zero.

We also choose a partition of unity $\\{\chi_{\alpha}\\}$ subordinate to our
Kuranishi chart. Then we put

$\theta_{\alpha}=\chi_{\alpha}(ev_{s,\alpha})^{*}h$ (16.8)

which is a differential form on $V_{\alpha}$.

######  Definition 16.7.

Define

$(\mathcal{M};ev_{s},ev_{t})_{*}(h)=\sum_{\alpha}((V_{\alpha},\Gamma_{\alpha},E_{\alpha},\psi_{\alpha},s_{\alpha}),\mathfrak{s}_{\alpha},ev_{t,\alpha})_{*}(\theta_{\alpha}).$
(16.9)

This is a smooth differential form on $L_{t}$. It is $T^{n}$ equivariant if
$h$ is $T^{n}$ equivariant.

######  Remark 16.8.

  1. (1)

Actually the right hand side of (16.9) depends on the choice of
$(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$,
$\mathfrak{s}_{\alpha}$. We write $\mathfrak{s}$ to demonstrate this choice
and write $(\mathcal{M};ev_{s},ev_{t},\mathfrak{s})_{*}(h)$.

  2. (2)

The right hand side of (16.9) is independent of the choice of partition of
unity. The proof is similar to the well-definiedness of integration on
manifolds.

In case $\mathcal{M}$ has a boundary $\partial\mathcal{M}$, the choices
$(V_{\alpha},E_{\alpha},\Gamma_{\alpha},\psi_{\alpha},s_{\alpha})$,
$\mathfrak{s}_{\alpha}$ on $\mathcal{M}$ induces one for
$\partial\mathcal{M}$. We then have the following :

######  Lemma 16.9 (Stokes’ theorem).

We have

$d((\mathcal{M};ev_{s},ev_{t},\mathfrak{s})_{*}(h))=(\mathcal{M};ev_{s},ev_{t},\mathfrak{s})_{*}(dh)+(\partial\mathcal{M};ev_{s},ev_{t},\mathfrak{s})_{*}(h).$
(16.10)

We will discuss the sign at the end of this section.

###### Proof.

Using the partition of unity $\chi_{\alpha}$ it suffices to consider the case
when $\mathcal{M}$ has only one Kuranishi chart $V_{\alpha}$. We use the open
covering $U_{i}$ of $V_{\alpha}$ and the partition of unity again to see that
we need only to study on one $U_{i}$. In that case (16.10) is immediate from
the usual Stokes’ formula. ∎

We next discuss composition of smooth correspondences. We consider the
following situation. Let

$ev_{s;st}:\mathcal{M}_{st}\to L_{s},\qquad ev_{t;st}:\mathcal{M}_{st}\to
L_{t}$

be as before such that $T^{n}$ action on $L_{t}$ is free and transitive. Let

$ev_{r;rs}:\mathcal{M}_{rs}\to L_{r},\qquad ev_{s;rs}:\mathcal{M}_{rs}\to
L_{s}$

be a similar diagram such that $T^{n}$ on $L_{s}$ is free and transitive.
Using the fact that $ev_{s;rs}$ is weakly submersive, we define the fiber
product

$\mathcal{M}_{rs}\,{}_{ev_{s;rs}}\times_{ev_{s;st}}\mathcal{M}_{st}$

as a space with Kuranishi structure. We write it as $\mathcal{M}_{rt}$. We
have a diagram of strongly continuous smooth maps

$ev_{r;rt}:\mathcal{M}_{rt}\to L_{r},\qquad ev_{t;rt}:\mathcal{M}_{rt}\to
L_{t}.$

We next make choices $\mathfrak{s}^{st}$, $\mathfrak{s}^{rs}$ for
$\mathcal{M}_{st}$ and $\mathcal{M}_{rs}$. It is easy to see that it
determines a choice $\mathfrak{s}^{rt}$ for $\mathcal{M}_{rt}$.

Now we have :

######  Lemma 16.10 (Composition formula).

We have the following formula for each differential form $h$ on $L_{r}$.

$\displaystyle(\mathcal{M}_{rt};ev_{r;rt},ev_{t;,rt},\mathfrak{s}^{rt})_{*}(h)$
(16.11)
$\displaystyle=((\mathcal{M}_{st};ev_{s;st},ev_{t;,st},\mathfrak{s}^{st})_{*}\circ(\mathcal{M}_{rs};ev_{r;rs},ev_{s;,rs},\mathfrak{s}^{rs})_{*})(h).$

###### Proof.

Using a partition of unity it suffices to study locally on $\mathcal{M}_{rs}$,
$\mathcal{M}_{st}$. In that case it suffices to consider the case of usual
manifold, which is well-known. ∎

We finally discuss the signs in Lemmas 16.9 and 16.10. It is rather cumbersome
to fix appropriate sign convention and show those lemmata with sign. So,
instead, we use the trick of subsection 8.10.3 [FOOO3] (= section 53.3
[FOOO2]) (see also section 13 [Fu5]) to reduce the orientation problem to the
case which is already discussed in Chapter 8 [FOOO3] (= Chapter 9 [FOOO2]), as
follows.

The correspondence $h\mapsto(\mathcal{M};ev_{s},ev_{t},\mathfrak{s})_{*}(h)$
extends to the currents $h$ that satisfy appropriate transversality properties
about its wave front set. (See [Hö].) We can also represent the smooth form
$h$ by an appropriate average (with respect to certain smooth measure) of a
family of currents realized by smooth singular chains. So, as far as sign
concerns, it suffices to consider a current realized by a smooth singular
chain. Then the right hand side of (16.3) turn out to be a current realized by
a smooth singular chain which is obtained from a smooth singular chain on
$L_{s}$ by a transversal smooth correspondence. In fact, we may assume that
all the fiber products appearing here are transversal, since it suffices to
discuss the sign in the generic case. Thus the problem reduces to find a sign
convention (and orientation) for the correspondence of the singular chains by
a smooth manifold. In the situation of our application, such sign convention
(singular homology version) was determined and analyzed in detail in Chapter 8
[FOOO3] (= Chapter 9 [FOOO2]). Especially, existence of an appropriate
orientation that is consistent with the sign appearing in $A_{\infty}$
formulae etc. was proved there. Therefore we can prove that there is a sign
(orientation) convention which induces all the formulae we need with sign, in
our de Rham version, as well. See subsection 8.10.3 [FOOO3] (= section 53.3
[FOOO2]) or section 13 [Fu5] for the detail.

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