which was introduced by the authors in section 3.8 [FOOO3] (= section 13
[FOOO2]). Using the class $z_{i}\in H^{2}(X;{\mathbb{Z}})$ the Poincaré dual
to $\pi^{-1}(\partial_{i}P)$ we put

$\psi_{u}(z_{i})=\sum_{k}\sum_{i=1}^{m}T^{\beta_{i}\cap\omega/2\pi}\int_{L(u)}\mathfrak{q}_{1,k;\beta_{i}}(z_{i}\otimes
x^{\otimes k}).$ (6.27)

Here we put $x=\sum x_{i}\text{\bf e}_{i}$ and the right hand side is a formal
power series of $x_{i}$ with coefficients in $\Lambda$.

Using the description of the moduli space defining the operators
$\mathfrak{q}_{1,k;\beta}$ (See section 11.) it is easy to see that the right
hand side of (6.27) coincides with the definition of $\overline{z}_{i}$ in the
current case, when $X$ is Fano toric. Extending the expression (6.27) to an
arbitrary homology class $z$ of arbitrary degree we obtain

$\psi_{u}(z)=\sum_{k}\sum_{\beta;\mu(\beta)=\deg
z}T^{\beta\cap\omega/2\pi}\int_{L(u)}\mathfrak{q}_{1,k;\beta}(z\otimes
x^{\otimes k}).$ (6.28)

Since $\mu(\beta)=\deg z$, $\mathfrak{q}_{1,k;\beta}(z\otimes x^{\otimes
k})\in H^{n}(L(u);{\mathbb{Q}})$. One can prove that (6.28) defines a ring
homomorphism from quantum cohomology to the Jacobian ring
$Jac(\mathfrak{PO}^{u})$. We may regard $Jac(\mathfrak{PO}^{u})$ as the moduli
space of deformations of Floer theories of Lagrangian fibers of $X$. (Note the
Jacobian ring parameterizes deformations of a holomorphic function up to an
appropriate equivalence. In our case the equivalence is the right equivalence,
that is, the coordinate change of the domain.)

Thus (6.28) is a particular case of the ring homomorphism

$QH(X)\to HH(\mathfrak{Lag}(X))$

where $HH(\mathfrak{Lag}(X))$ is the Hochschild cohomology of Fukaya category
of $X$. (We remark that Hochschild cohomology parameterizes deformations of
$A_{\infty}$ category.) Existence of such a homomorphism is a folk theorem
(See [Ko]) which is verified by various people in various favorable
situations. (See for example [Aur].) It is conjectured to be an isomorphism
under certain conditions by various people including P. Seidel and M.
Kontsevich.

This point of view is suitable for generalizing our story to more general $X$
(to non-Fano toric manifolds, for example) and also for including big quantum
cohomology group into our story. (We will then also need to use the operators
$\mathfrak{q}_{\ell,k}$ mentioned above for $\ell\geq 2$.)

These points will be discussed in subsequent papers in this series of papers.
In this paper we follow more elementary approach exploiting the known
calculation of quantum cohomology of toric manifolds, although it is less
conceptual.

######  Remark 6.16.

There are two other approaches towards a proof of Proposition 6.7 besides the
fixed point localization. One is written by Cieliebak and Salamon [CS] which
uses vortex equations (gauged sigma model) and the other is written by McDuff
and Tolman [MT] which uses Seidel’s result [Se1].

##  7\. Localization of quantum cohomology ring at moment polytope

In this section, we discuss applications of Theorem 1.9. In particular, we
prove Theorem 1.12. (Note Theorem 1.3 is a consequence of Theorem 1.12.) The
next theorem and Theorem 1.9 immediately imply (1) of Theorem 1.12.

######  Theorem 7.1.

There exists a bijection

$\text{\rm Crit}(\mathfrak{PO}_{0})\cong
Hom(Jac(\mathfrak{PO}_{0});\Lambda^{{\mathbb{C}}}).$

Here the right hand side is the set of unital $\Lambda^{{\mathbb{C}}}$-algebra
homomorphisms.

We start with the following definition

######  Definition 7.2.

For an element $x\in\Lambda\setminus\\{0\\}$, we define its valuation
$\mathfrak{v}_{T}(x)$ as the unique number $\lambda\in{\mathbb{R}}$ such that
$T^{-\lambda}x\in\Lambda_{0}\setminus\Lambda_{+}.$

We note that $\mathfrak{v}_{T}$ is multiplicative non-Archimedean valuation,
i.e., satisfies

$\displaystyle\mathfrak{v}_{T}(x+y)$ $\displaystyle\geq$
$\displaystyle\min(\mathfrak{v}_{T}(x),\mathfrak{v}_{T}(y)),$
$\displaystyle\mathfrak{v}_{T}(xy)$ $\displaystyle=$
$\displaystyle\mathfrak{v}_{T}(x)+\mathfrak{v}_{T}(y).$

######  Lemma 7.3.

For any $\varphi\in Hom(Jac(\mathfrak{PO}_{0});\Lambda^{{\mathbb{C}}})$ there
exists a unique $u\in M_{\mathbb{R}}$ such that

$\mathfrak{v}_{T}(\varphi(y_{j}(u)))=0$ (7.1)

for all $j=1,\cdots,n$.

###### Proof.

We still assume Condition 6.12. By definition (6.1) of $\overline{z}_{i}$,
homomorphism property of $\varphi$ and multiplicative property of valuation,
we obtain

$\mathfrak{v}_{T}(\varphi(\overline{z}_{i}))=\ell_{i}(u)+\sum_{j=1}^{n}v_{i,j}\mathfrak{v}_{T}(\varphi(y_{j}(u))),$
(7.2)

for $i=1,\cdots,m$. On the other hand, since $\ell_{i}(u)=\langle
u,v_{i}\rangle-\lambda_{i}$ and $(v_{i,j})_{i,j=1,\cdots,n}$ is invertible,
there is a unique $u$ that satisfies

$\mathfrak{v}_{T}(\varphi(\overline{z}_{i}))=\ell_{i}(u)$ (7.3)

for $i=1,\cdots,n$. But by the invertibility of $(v_{i,j})_{i,j=1,\cdots,n}$
and (7.2), this is equivalent to (7.1) and hence the proof. ∎

We remark that obviously by the above proof the formula (7.3) automatically
holds for $i=n+1,\cdots,m$ and $u$ in Lemma 7.3 as well.

###### Proof of Theorem 7.1.

Consider the maps

$\Psi_{1}(\varphi)=\sum_{i=1}^{n}(\log\varphi(y_{i}(u)))\text{\bf e}_{i}\in
H^{1}(L(u);\Lambda_{0}),\qquad\Psi_{2}(\varphi)=u\in M_{\mathbb{R}}$

where $u$ is obtained as in Lemma 7.3. Since
$y_{i}(u)\in\Lambda_{0}\setminus\Lambda_{+}$ it follows that we can define its
logarithm on $\Lambda_{0}$ as a convergent power series with respect to the
non-Archimedean norm.

Set $(\mathfrak{x},u)=(\Psi_{1}(\varphi),\Psi_{2}(\varphi))$. Since $\varphi$
is a ring homomorphism from $Jac(\mathfrak{PO}_{0})\cong
Jac(\mathfrak{PO}_{0}^{u})$ it follows from the definition of the Jacobian
ring that

$\frac{\partial\mathfrak{PO}_{0}^{u}}{\partial y_{i}}(\mathfrak{x})=0.$

Therefore by Theorem 4.10,
$HF((L(u),\mathfrak{x}),(L(u),\mathfrak{x});\Lambda)\neq 0$. We have thus
defined

$\Psi:Hom(Jac(\mathfrak{PO}_{0});\Lambda^{{\mathbb{C}}})\to\text{\rm
Crit}(\mathfrak{PO}_{0}).$

Let $(\mathfrak{x},u)\in\text{\rm Crit}(\mathfrak{PO}_{0})$. We put
$\mathfrak{x}=\sum\mathfrak{x}_{i}\text{\bf e}_{i}$. We define a homomorphism
$\varphi:Jac(\mathfrak{PO}_{0})\to\Lambda$ by assigning

$\varphi(y_{i}(u))=e^{\mathfrak{x}_{i}}.$

It is straightforward to check that $\varphi$ is well-defined. Then we define
$\Phi(\mathfrak{x},u):=\varphi$. It easily follows from definition that $\Phi$
is an inverse to $\Psi$. The proof of Theorem 7.1 is complete. ∎

We next work with the (Batyrev) quantum cohomology side.

######  Definition 7.4.

For each $z_{i}$, we define a $\Lambda$-linear map
$\widehat{z}_{i}:QH^{\omega}(X;\Lambda^{{\mathbb{C}}})\to
QH^{\omega}(X;\Lambda^{{\mathbb{C}}})$ by $\widehat{z}_{i}(z)=z_{i}\cup_{Q}z,$
where $\cup_{Q}$ is the product in $QH^{\omega}(X;\Lambda^{{\mathbb{C}}})$.

Since $QH^{\omega}(X;\Lambda)$ is generated by even degree elements it follows
that it is commutative. Therefore we have

$\widehat{z}_{i}\circ\widehat{z}_{j}=\widehat{z}_{j}\circ\widehat{z}_{i}.$
(7.4)

######  Definition 7.5.

For
$\mathfrak{w}=(\mathfrak{w}_{1},\cdots,\mathfrak{w}_{n})\in(\Lambda^{{\mathbb{C}}})^{n}$
we put

$QH^{\omega}(X;\mathfrak{w})=\\{x\in
QH^{\omega}(X;\Lambda^{{\mathbb{C}}})\mid(\widehat{z}_{i}-\mathfrak{w}_{i})^{N}x=0\quad\text{for
$i=1,\cdots,n$ and large $N$.}\\}$

We say that $\mathfrak{w}$ is a weight of $QH^{\omega}(X)$ if
$QH^{\omega}(X;\mathfrak{w})$ is nonzero. We denote by $W(X;\omega)$ the set
of weights of $QH^{\omega}(X)$ .

We remark that $\mathfrak{w}_{i}\neq 0$ since $z_{i}$ is invertible.
(Corollary 6.11.)

######  Remark 7.6.

Since $z_{i}$, $i=1,\cdots,n$ generates $QH^{\omega}(X;\Lambda)$ by Condition
6.12, we have the following. For each
$\mathfrak{w}=(\mathfrak{w}_{1},\cdots,\mathfrak{w}_{n})$, there exists
$\mathfrak{w}_{n+1},\cdots,\mathfrak{w}_{m}$ depending only on $\mathfrak{w}$
such that $(\widehat{z}_{i}-\mathfrak{w}_{i})^{N}x=0$ also holds for
$i=n+1,\cdots,m$, if $N$ is sufficiently large and $x\in
QH^{\omega}(X;\mathfrak{w})$.

######  Proposition 7.7.

  1. (1)

There exists a factorization of the ring

$QH^{\omega}(X;\Lambda^{{\mathbb{C}}})\cong\prod_{\mathfrak{w}\in
W(X;\omega)}QH^{\omega}(X;\mathfrak{w}).$

  2. (2)

There exists a bijection

$W(X;\omega)\cong Hom(QH^{\omega}(X;\Lambda);\Lambda^{{\mathbb{C}}}).$

  3. (3)

$QH^{\omega}(X;\mathfrak{w})$ is a local ring and $(1)$ is the factorization
to indecomposables.

###### Proof.

Existence of decomposition (1) as a $\Lambda^{{\mathbb{C}}}$-vector space is a
standard linear algebra, using the fact that $\Lambda^{{\mathbb{C}}}$ is an
algebraically closed field. (We will prove this fact in appendix as Lemma
14.1.) If $z\in QH^{\omega}(X;\mathfrak{w})$ and $z^{\prime}\in
QH^{\omega}(X;\mathfrak{w}^{\prime})$ then

$\displaystyle(z_{i}-\mathfrak{w}_{i})^{N}\cup_{Q}(z\cup_{Q}z^{\prime})$
$\displaystyle=((z_{i}-\mathfrak{w}_{i})^{N}\cup_{Q}z)\cup_{Q}z^{\prime}=0,$
$\displaystyle(z_{i}-\mathfrak{w}^{\prime}_{i})^{N}\cup_{Q}(z\cup_{Q}z^{\prime})$
$\displaystyle=((z_{i}-\mathfrak{w}^{\prime}_{i})^{N}\cup_{Q}z^{\prime})\cup_{Q}z=0.$

Therefore $z\cup_{Q}z^{\prime}\in QH^{\omega}(X;\mathfrak{w})\cap
QH^{\omega}(X;\mathfrak{w}^{\prime})$. This implies that the decomposition (1)
is a ring factorization.

Let $\varphi:QH^{\omega}(X;\mathfrak{w})\to\Lambda^{{\mathbb{C}}}$ be a unital
$\Lambda^{{\mathbb{C}}}$ algebra homomorphism. It induces a homomorphism
$QH^{\omega}(X;\Lambda)\to\Lambda^{{\mathbb{C}}}$ by (1). We denote this ring
homomorphism by the same letter $\varphi$. Let $z\in
QH^{\omega}(X;\mathfrak{w})$ be an element such that $\varphi(z)\neq 0$. Then
we have

$(\varphi(z_{i})-\mathfrak{w}_{i})^{N}\varphi(z)=\varphi((z_{i}-\mathfrak{w}_{i})^{N}\cup_{Q}z)=0.$

Therefore

$\mathfrak{w}_{i}=\varphi(z_{i}).$ (7.5)

Since $z_{i}$ generates $QH^{\omega}(X;\Lambda)$, it follows from (7.5) that
there is a unique $\Lambda^{{\mathbb{C}}}$ algebra homomorphism
$:QH^{\omega}(X;\mathfrak{w})\to\Lambda^{{\mathbb{C}}}$. (2) follows.

Since $QH^{\omega}(X;\mathfrak{w})$ is a finite dimensional
$\Lambda^{{\mathbb{C}}}$ algebra and $\Lambda^{{\mathbb{C}}}$ is algebraically
closed, we have an isomorphism

$\frac{QH^{\omega}(X;\mathfrak{w})}{\text{rad}}\cong(\Lambda^{{\mathbb{C}}})^{k}$
(7.6)

for some $k$. (Here $\text{rad}=\\{z\in QH^{\omega}(X;\mathfrak{w})\mid
z^{N}=0\quad\text{for some $N$.}\\}$) Since there is a unique unital
$\Lambda^{{\mathbb{C}}}$-algebra homomorphism
$:QH^{\omega}(X;\mathfrak{w})\to\Lambda^{{\mathbb{C}}}$, it follows that
$k=1$. Namely ${QH^{\omega}(X;\mathfrak{w})}$ is a local ring.

It also implies that ${QH^{\omega}(X;\mathfrak{w})}$ is indecomposable. ∎

The result up to here also works for the non-Fano case. But the next theorem
will require the fact that $X$ is Fano since we use the equality
$QH^{\omega}(X;\Lambda)\cong QH(X;\Lambda)$.

######  Theorem 7.8.

If $X$ is Fano then $\mbox{\rm
Crit}(\mathfrak{PO}_{0})=\mathfrak{M}(\mathfrak{Lag}(X))$.

###### Proof.

Let $\mathfrak{w}$ be a weight. We take $z\in
QH^{\omega}(X;\mathfrak{w})\subset H(X;\Lambda^{{\mathbb{C}}})\cong
H(X;{\mathbb{C}})\otimes\Lambda^{{\mathbb{C}}}$. We may take $z$ so that

$z\in(H(X;{\mathbb{C}})\otimes\Lambda_{0}^{{\mathbb{C}}})\,\,\setminus(H(X;{\mathbb{C}})\otimes\Lambda_{+}^{{\mathbb{C}}}).$

Since

$z_{i}\cup_{Q}z\equiv z_{i}\cup z\mod\Lambda_{+}^{{\mathbb{C}}},$

where $\cup$ is the classical cup product. (We use
$QH^{\omega}(X;\Lambda)=QH(X;\Lambda)$ here.) It follows that

$\mathfrak{w}_{i}^{n}z=(\widehat{z}_{i})^{n}(z)=(z_{i})^{n}\cup_{Q}z\equiv(z_{i})^{n}\cup
z\mod\Lambda_{+}^{{\mathbb{C}}}.$

Therefore $\mathfrak{w}_{i}\in\Lambda_{+}^{{\mathbb{C}}}$ as $(z_{i})^{n}\cup
z=0$. (7.3) and (7.5) then imply

$\ell_{i}(u)=\mathfrak{v}_{T}(\mathfrak{w}_{i})>0.$

Namely $u\in\text{Int}P$. ∎

We are now ready to complete the proof of Theorem 1.12. (1) is Theorem 7.1.
(2) is a consequence of Theorem 4.10. (3) is Theorem 7.8. If
$QH^{\omega}(X;\Lambda^{{\mathbb{C}}})$ is semi-simple, then (7.6) and $k=1$
there implies

$QH^{\omega}(X;\Lambda^{{\mathbb{C}}})\cong(\Lambda^{{\mathbb{C}}})^{\\#W(X;\omega)}$
(7.7)

as a $\Lambda^{{\mathbb{C}}}$ algebra. (4) follows from (7.7), Proposition 7.7
(2), and Theorem 7.1. The proof of Theorem 1.12 is complete. ∎

We next explain the factorization in Proposition 7.7 (1) from the point of
view of Jacobian ring. Let $(\mathfrak{x},u)\in\text{\rm
Crit}(\mathfrak{PO}_{0})$.

######  Definition 7.9.

We consider the ideal generated by

$\frac{\partial}{\partial
w_{i}}\mathfrak{PO}_{0}^{u}(\mathfrak{y}_{1}+w_{1},\cdots,\mathfrak{y}_{n}+w_{n})$
(7.8)

$i=1,\cdots,n$, in the ring $\Lambda[[w_{1},\cdots,w_{n}]]$ of formal power
series where $\mathfrak{x}=\sum\mathfrak{x}_{i}\text{\bf e}_{i}$ and
$\mathfrak{y}_{i}=e^{\mathfrak{x}_{i}}$. We denote its quotient ring by
$Jac(\mathfrak{PO}_{0};\mathfrak{x},u)$.

######  Proposition 7.10.

  1. (1)

There is a direct product decomposition :

$Jac(\mathfrak{PO}_{0})\cong\prod_{(\mathfrak{x},u)\in\text{\rm
Crit}(\mathfrak{PO}_{0})}Jac(\mathfrak{PO}_{0};\mathfrak{x},u),$

as a ring.

  2. (2)

If $(\mathfrak{x},u)\in\text{\rm Crit}(\mathfrak{PO}_{0})$ corresponds to
$\mathfrak{w}\in W(X;\omega)$ via the isomorphism given in Proposition 7.7
$(2)$ and Theorem 7.1, then $\psi_{u}$ induces an isomorphism

$\psi_{u}:QH^{\omega}(X;\mathfrak{w})\cong
Jac(\mathfrak{PO}_{0};\mathfrak{x},u).$

  3. (3)

$Jac(\mathfrak{PO}_{0};\mathfrak{x},u)$ is one dimensional (over $\Lambda$) if
and only if the Hessian

$\left(\frac{\partial^{2}\mathfrak{PO}_{0}^{u}}{\partial y_{i}\partial
y_{j}}\right)_{i,j=1,\cdots,n}$

is invertible over $\Lambda$ at $\mathfrak{x}$.

###### Proof.

We put $\mathfrak{x}=\sum\mathfrak{x}_{i}\text{\bf e}_{i}$ and
$\mathfrak{y}_{i}=e^{\mathfrak{x}_{i}}$. Let $\mathfrak{m}(\mathfrak{x},u)$ be
the ideal generated by $y_{i}-\mathfrak{y}_{i}$, in the ring :

$Jac(\mathfrak{PO}_{0})=\frac{\Lambda[y_{1}^{\pm},\cdots,y_{n}^{\pm}]}{(y_{i}\frac{\partial\mathfrak{PO}_{0}^{u}}{\partial
y_{i}})}.$

Since $Jac(\mathfrak{PO}_{0})$ is finite dimensional over $\Lambda$ it follows
that

$Jac(\mathfrak{PO}_{0})\cong\prod_{(\mathfrak{x},u)\in\text{\rm
Crit}(\mathfrak{PO}_{0})}Jac(\mathfrak{PO}_{0})_{\mathfrak{m}(\mathfrak{x},u)},$

where $Jac(\mathfrak{PO}_{0})_{\mathfrak{m}(\mathfrak{x},u)}$ is the
localization of the ring $Jac(\mathfrak{PO}_{0})$ at
$\mathfrak{m}(\mathfrak{x},u)$. Using finite dimensionality of
$Jac(\mathfrak{PO}_{0})$ again we have an isomorphism
$Jac(\mathfrak{PO}_{0})_{\mathfrak{m}(\mathfrak{x},u)}\cong
Jac(\mathfrak{PO}_{0};\mathfrak{x},u)$ which sends $y_{i}-\mathfrak{y}_{i}$ to
$w_{i}$. Here $\mathfrak{x}=\sum_{i}\mathfrak{x}_{i}\text{\bf e}_{i}$ and
$\mathfrak{y}_{i}=e^{\mathfrak{x}_{i}}$ (1) follows.

Now we prove (2). If $z\in QH^{\omega}(X;\mathfrak{w})$ then
$(\widehat{z}_{i}-\mathfrak{w}_{i})^{N}z=0$. Let
$\pi_{\mathfrak{x},u}:Jac(\mathfrak{PO}_{0})\to
Jac(\mathfrak{PO}_{0};\mathfrak{x},u)$ be the projection. We then have

$(T^{\ell_{i}(u)}\mathfrak{y}_{1}^{v_{i,1}}\cdots\mathfrak{y}_{n}^{v_{i,n}}-\mathfrak{w}_{i})^{N}\pi_{\mathfrak{x},u}(\psi_{u}(z))=0.$
(7.9)

We remark that

$\mathfrak{w}_{i}=T^{\ell_{i}(u^{\prime})}\mathfrak{y}_{1}^{\prime
v_{i,1}}\cdots\mathfrak{y}_{n}^{\prime v_{i,n}}$ (7.10)

if $\mathfrak{w}_{i}$ corresponds $(\mathfrak{x}^{\prime},u^{\prime})$ and
$\mathfrak{y}^{\prime}_{i}$ are exponential of the coordinates of
$\mathfrak{x}^{\prime}$. We define the operator
$\widehat{\mathfrak{y}}_{i}:Jac(\mathfrak{PO}_{0};\mathfrak{x},u)\to
Jac(\mathfrak{PO}_{0};\mathfrak{x},u)$ by

$\widehat{\mathfrak{y}}_{i}(c)={\mathfrak{y}}_{i}c.$

By definition of $Jac(\mathfrak{PO}_{0};\mathfrak{x},u)$ the eigenvalue of
$\widehat{\mathfrak{y}}_{i}$ is $\mathfrak{y}_{i}$. Therefore (7.9) and (7.10)
imply that $\pi_{\mathfrak{x},u}(\psi_{u}(z))=0$ unless
$(\mathfrak{x},u)=(\mathfrak{x}^{\prime},u^{\prime})$. (2) follows.

Let us prove (3). We first remark $\Lambda=\Lambda^{{\mathbb{C}}}$ is an
algebraically closed field (Lemma 14.1). Therefore $\mathop{\kern 0.0pt{\rm
dim}}\nolimits_{\Lambda}Jac(\mathfrak{PO}_{0};\mathfrak{x},u)=1$ if and only
if the ideal generated by (7.8) (for $i=1,\cdots,n$) is the maximal ideal
$\mathfrak{m}=(w_{1},\cdots,w_{n})$. We remark that
$\mathfrak{m}/\mathfrak{m}^{2}=\Lambda^{n}$ and elements (7.8) reduces to

$\left(\frac{\partial^{2}\mathfrak{PO}_{0}^{u}}{\partial y_{i}\partial
y_{j}}\right)_{j=1,\cdots,n}\in\Lambda^{n},$

modulo $\mathfrak{m}^{2}$. (3) follows easily. ∎

We recall that a symplectic manifold $(X,\omega)$ is said to be (spherically)
monotone if there exists $\lambda>0$ such that
$c_{1}(X)\cap\alpha=\lambda\,[\omega]\cap\alpha$ for all
$\alpha\in\pi_{2}(X)$. Lagrangian submanifold $L$ of $(X,\omega)$ is said to
be monotone if there exists $\lambda>0$ such that
$\mu(\beta)=\lambda\omega(\beta)$ for any $\beta\in\pi_{2}(X,L)$. (Here $\mu$
is the Maslov index.) In the monotone case we have the following :

######  Theorem 7.11.

If $X$ is a monotone compact toric manifold then there exists a unique $u_{0}$
such that

$\mathfrak{M}(\mathfrak{Lag}(X))\subset\Lambda\times\\{u_{0}\\}$

i.e., whenever $(\mathfrak{x},u)\in\mathfrak{M}(\mathfrak{Lag}(X))$,
$u=u_{0}$. Moreover $L(u_{0})$ is monotone.

######  Remark 7.12.

Related results are discussed in [EP1].

###### Proof.

Since $X$ is Fano, we have $QH^{\omega}(X;\Lambda)=QH(X;\Lambda)$. We assume
$c_{1}(X)\cap\alpha=\lambda\,[\omega]\cap\alpha$ with $\lambda>0$. Let
$\cup_{\alpha}$ be the contribution to the moduli space of pseudo-holomorphic
curve of homology class $\alpha\in H_{2}(X;{\mathbb{Z}})$ in the quantum cup
product. (See (6.5).) We have a decomposition :

$x\cup_{Q}y=x\cup
y+\sum_{\alpha\in\pi_{2}(X)\setminus\\{0\\}}T^{\alpha\cap[\omega]/2\pi}x\cup_{\alpha}y.$

Then

$\deg(x\cup_{\alpha}y)=\deg x+\deg y-2c_{1}(X)\cap\alpha=\deg x+\deg
y-2\lambda\alpha\cap[\omega].$ (7.11)

We define

$\mathfrak{v}_{\deg}(T^{1/2\pi})=2\lambda,\quad\mathfrak{v}_{\deg}(x)=\deg
x\quad(\text{for $x\in H(X;{\mathbb{Q}})$}).$

$\mathfrak{v}_{\deg}$ is a multiplicative non-Archimedean valuation on
$QH(X;\Lambda)$ such that
$\mathfrak{v}_{\deg}(a\cup_{Q}b)=\mathfrak{v}_{\deg}(a)+\mathfrak{v}_{\deg}(b),$
by virtue of (7.11). Moreover for $c\in\Lambda$ and $a\in QH(X;\Lambda)$ we
have
$\mathfrak{v}_{\deg}(ca)=2\lambda\mathfrak{v}_{T}(c)+\mathfrak{v}_{\deg}(a).$
Now let $\mathfrak{w}$ be a weight and $x\in QH^{\omega}(X;\mathfrak{w})$.
Since $\mathfrak{v}_{\deg}(z_{i})=2$ it follows that

$2\lambda\mathfrak{v}_{T}(\mathfrak{w}_{i})+\mathfrak{v}_{\deg}(x)=\mathfrak{v}_{\deg}(z_{i}x)=2+\mathfrak{v}_{\deg}(x).$

Therefore if $(\mathfrak{x},u)$ corresponds to $\mathfrak{w}$ then
$\ell_{i}(u)=\mathfrak{v}_{T}(\mathfrak{w}_{i})=1/\lambda$. Namely $u$ is
independent of $\mathfrak{w}$. We denote it by $u_{0}$.

For $\beta_{i}\in H_{2}(X,L(u_{0}))$ ($i=1,\cdots,m$) given by (6.7), we have
$\omega(\beta_{i})=2\pi\ell_{i}(u_{0})=2\pi/\lambda$. Hence
$\mu(\beta_{i})=\lambda\omega(\beta_{i})/\pi$. Since $\beta_{i}$ generates
$H_{2}(X,L(u_{0}))$, it follows that $L(u_{0})$ is monotone, as required. ∎

So far we have studied Floer cohomology with
$\Lambda^{{\mathbb{C}}}$-coefficients. We next consider the case of
$\Lambda^{F}$ coefficient where $F$ is a finite Galois extension of
${\mathbb{Q}}$. We choose $F$ so that each of the weight $\mathfrak{w}$ lies
in $(\Lambda^{F}_{0})^{n}$. (Since every finite extension of
$\Lambda^{{\mathbb{Q}}}$ is contained in such $\Lambda^{F}$ we can always find
such an $F$. See appendix.) Then we have a decomposition

$QH^{\omega}(X;\Lambda^{F})\cong\prod_{\mathfrak{w}\in
W(X;\omega)}QH^{\omega}(X;\mathfrak{w};F).$ (7.12)

It follows that the Galois group $Gal(F/{\mathbb{Q}})$ acts on $W(X;\omega)$.
It induces a $Gal(F/{\mathbb{Q}})$ action on $\text{\rm
Crit}(\mathfrak{PO}_{0})$. (We use Remark 6.14 here.) We write it as
$(\mathfrak{x},u)\mapsto(\sigma(\mathfrak{x}),\sigma(u))$. We remark the
following:

######  Proposition 7.13.

  1. (1)

$\sigma(u)=u$.

  2. (2)

We write by $y_{i}(\mathfrak{x})$ the exponential of the coordinates of
$\mathfrak{x}$. Then $y_{i}(\mathfrak{x})\in\Lambda^{F}$ and
$y_{i}(\sigma(\mathfrak{x}))=\sigma(y_{i}(\mathfrak{x}))$.

  3. (3)

If $QH^{\omega}(X;\Lambda^{{\mathbb{Q}}})$ is indecomposable, there exists
$u_{0}$ such that whenever $(\mathfrak{x},u)\in\text{\rm
Crit}(\mathfrak{PO}_{0}))$, $u=u_{0}$.

###### Proof.

Let $\mathfrak{w}_{i}(\mathfrak{x})$ corresponds to $(\mathfrak{x},u)$. Then

$\ell_{i}(\sigma(u))=\mathfrak{v}_{T}(\mathfrak{w}_{i}(\sigma(\mathfrak{x},u)))=\mathfrak{v}_{T}(\sigma\mathfrak{w}_{i}(\mathfrak{x},u))=\mathfrak{v}_{T}(\mathfrak{w}_{i}(\mathfrak{x},u))=\ell_{i}(u).$

(1) follows. (2) follows from the definition and (1). (3) is a consequence of
(1). ∎

A monotone blow up of ${\mathbb{C}}P^{2}$ (at one point) gives an example
where the assumption of Proposition 7.13 (3) is satisfied.

It seems interesting to observe that the ring $QH(X;\Lambda^{{\mathbb{Q}}})$
jumps sometimes when we deform symplectic structure of $X$. The point where
this jump occurs is closely related to the point where the number of balanced
Lagrangian fibers jumps. In the case of Example 5.7 we have

$QH(X;\Lambda^{{\mathbb{Q}}})\cong\begin{cases}(\Lambda^{{\mathbb{Q}}})^{5}&\alpha>0,\\\
(\Lambda^{{\mathbb{Q}}})^{3}\times\Lambda^{{\mathbb{Q}}(\sqrt{-3})}&\alpha<0,\\\
\Lambda^{{\mathbb{Q}}(\sqrt{5})}\times\Lambda^{F}&\alpha=0,\end{cases}$

where $F={\mathbb{Q}}[x]/(x^{3}-x-1).$ We remark that
$x^{5}+x^{4}-2x^{3}-2x^{2}+1=(x^{2}+x-1)(x^{3}-x-1)$.

We also refer readers to Example 10.10 for further example.

######  Remark 7.14.

In sections 11 \- 13, we will use de Rham cohomology of Lagrangian submanifold
to define and study Floer cohomology. As a consequence, our results on Floer
cohomology is proved over $\Lambda^{{\mathbb{R}}}_{0}$ or
$\Lambda^{\mathbb{C}}_{0}$ but not over $\Lambda^{{\mathbb{Q}}}_{0}$ or
$\Lambda^{F}_{0}$. (The authors believe that those results can be also proved
over $\Lambda^{{\mathbb{Q}}}_{0}$ by using the singular cohomology version
developed in [FOOO3], although the detail of their proofs could be more
complicated.)

On the other hand, Proposition 6.8 and Theorem 6.1 are proved over
$\Lambda^{{\mathbb{Q}}}_{0}$. Therefore the discussion on quantum cohomology
here works over $\Lambda^{F}_{0}$.

We also remark that, though Proposition 7.13 (3) is related to Floer
cohomology, its proof given above does not use Floer cohomology over
$\Lambda^{{\mathbb{Q}}}_{0}$ but uses only Floer cohomology over
$\Lambda^{{\mathbb{C}}}_{0}$ and quantum cohomology over
$\Lambda^{{\mathbb{Q}}}_{0}$. In fact, the proof above implies the following :
If $u\in\text{\rm Int}P$ and $\mathfrak{x}\in
H^{1}(L(u);\Lambda^{{\mathbb{C}}}_{0})$ satisfy

$\frac{\partial\mathfrak{PO}^{u}_{0}}{\partial x_{i}}(\mathfrak{x})=0$

then $u=u_{0}$. This is because

$Jac(\mathfrak{PO}^{u}_{0};\Lambda^{{\mathbb{C}}}_{0})=Jac(\mathfrak{PO}^{u}_{0};\Lambda^{{\mathbb{Q}}}_{0})\otimes_{\Lambda^{{\mathbb{Q}}}_{0}}\Lambda^{{\mathbb{C}}}_{0}$

and

$Jac(\mathfrak{PO}^{u}_{0};\Lambda^{{\mathbb{Q}}}_{0})\cong
QH(X;\Lambda^{{\mathbb{Q}}}).$

##  8\. Further examples and remarks

In this section we show how we can use the argument of the last 2 sections to
illustrate calculations of $\mathfrak{M}(\mathfrak{Lag}(X))$ in examples.

######  Example 8.1.

We consider one point blow up $X$ of ${\mathbb{C}}P^{2}$. We choose its Kähler
form so that the moment polytope is

$P=\left\\{(u_{1},u_{2})\mid 0\leq u_{1},u_{2},\,\,u_{1}+u_{2}\leq
1,\,\,u_{2}\leq 1-\alpha\right\\},$

$0<\alpha<1$. The potential function is

$\mathfrak{PO}=y_{1}T^{u_{1}}+y_{2}T^{u_{2}}+(y_{1}y_{2})^{-1}T^{1-u_{1}-u_{2}}+y_{2}^{-1}T^{1-\alpha-
u_{2}}.$

We put $\overline{z}_{1}=y_{1}T^{u_{1}}$, $\overline{z}_{2}=y_{2}T^{u_{2}}$,
$\overline{z}_{3}=(y_{1}y_{2})^{-1}T^{1-u_{1}-u_{2}}$,
$\overline{z}_{4}=y_{2}^{-1}T^{1-\alpha-u_{2}}$.

The quantum Stanley-Reisner relation is

$\overline{z}_{1}\overline{z}_{3}=\overline{z}_{4}T^{\alpha},\quad\overline{z}_{2}\overline{z}_{4}=T^{1-\alpha},$
(8.1)

and linear relation is

$\overline{z}_{1}-\overline{z}_{3}=0,\quad\overline{z}_{2}-\overline{z}_{3}-\overline{z}_{4}=0.$
(8.2)

We put $X=\overline{z}_{1}$ and $Y=\overline{z}_{2}$ and solve (8.1), (8.2).
We obtain

$X^{3}(T^{\alpha}+X)=T^{1+\alpha},$ (8.3)

with $Y=X+T^{-\alpha}X^{2}$. We consider valuations of both sides of (8.3).
There are three different cases to consider.

Case 1; $\mathfrak{v}_{T}(X)>\alpha$ : (8.3) implies
$3\mathfrak{v}_{T}(X)+\alpha=1+\alpha$. Namely $\mathfrak{v}_{T}(X)=1/3$. So
$\alpha<1/3$. Moreover $\mathfrak{v}_{T}(Y)=1/3$. We have
$u_{1}=\mathfrak{v}_{T}(X)=1/3$, $u_{2}=\mathfrak{v}_{T}(Y)=1/3$. (See Lemma
7.3.) Writing $X=a_{1}T^{1/3}+a_{2}T^{\lambda}+$ higher order terms with
$\lambda>\frac{1}{3}$ and substituting this into (8.3), we obtain
$a_{1}^{3}=1$ which has 3 simple roots. Each of them corresponds to the
solution for $\mathfrak{x}$ by Hensel’s lemma (see Proposition 3 in p 144
[BGR], for example). (It also follows from Theorem 10.4 in section 10.)

Case 2; $\mathfrak{v}_{T}(X)<\alpha$ : By taking the valuation of (8.3) we
obtain $u_{1}=\mathfrak{v}_{T}(X)=(1+\alpha)/4$. Hence $\alpha>1/3$. Moreover
$u_{2}=\mathfrak{v}_{T}(Y)=(1-\alpha)/2$. In the same way as Case 1, we can
check that there are four solutions.

Case 3; $\mathfrak{v}_{T}(X)=\alpha$ : We put
$X=a_{1}T^{\alpha}+a_{2}T^{\lambda}+$ higher order terms where
$\lambda>\alpha$. (Case 3-1: $a_{1}\neq-1$) : By taking valuation of (8.3), we
obtain $u_{1}=\mathfrak{v}_{T}(X)=1/3$. Then $\alpha=1/3$ and
$u_{2}=\mathfrak{v}_{T}(Y)=1/3$. (8.3) becomes

$a_{1}^{4}+a_{1}^{3}-1=0.$ (8.4)

(In this case $X=a_{1}T^{\alpha}$ has no higher term.) There are four
solutions. We remark that (8.4) is irreducible over ${\mathbb{Q}}$, since it
is so over ${\mathbb{Z}}_{2}$. Namely the assumption of Proposition 7.13 (3)
is satisfied. Actually $X$ is monotone in the case $\alpha=1/3$. Hence the
same conclusion (uniqueness of $u$) follows from Theorem 7.11 also.

(Case 3-2: $a_{1}=-1$) : By taking valuation of (8.3), we obtain
$\lambda=1-2\alpha$. $\lambda>\alpha$ implies $\alpha<1/3$.
$u_{2}=\mathfrak{v}_{T}(Y)=1-2\alpha$. ($u_{1}=\mathfrak{v}_{T}(X)=\alpha$.)
There is one solution.

In summary, if $\alpha<1/3$ there are two choices of
$u=(\alpha,1-2\alpha),(1/3,1/3)$. On the other hand the numbers of choices of
$\mathfrak{x}$ are $1$ and $3$ respectively.

If $\alpha\geq 1/3$ there is the unique choice
$u=((1+\alpha)/4,(1-\alpha)/2)$. The number of choices of $\mathfrak{x}$ is
$4$.

We next study a non-Fano case. We will study Hirzebruch surface $F_{n}$. Note
$F_{1}$ is one point blow up of ${\mathbb{C}}P^{2}$ which we have already
studied. We leave the case $F_{2}$ to the reader.

######  Example 8.2.

We consider Hirzebruch surface $F_{n}$, $n\geq 3$. We take its Kähler form so
that the moment polytope is

$P=\left\\{(u_{1},u_{2})\mid 0\leq u_{1},u_{2},\,\,u_{1}+nu_{2}\leq
n,\,\,u_{2}\leq 1-\alpha\right\\},$

$0<\alpha<1$. The leading order potential function is

$\mathfrak{PO}_{0}=y_{1}T^{u_{1}}+y_{2}T^{u_{2}}+y_{1}^{-1}y_{2}^{-n}T^{n-u_{1}-nu_{2}}+y_{2}^{-1}T^{1-\alpha-
u_{2}}.$

We put $\overline{z}_{1}=y_{1}T^{u_{1}}$, $\overline{z}_{2}=y_{2}T^{u_{2}}$,
$\overline{z}_{3}=y_{1}^{-1}y_{2}^{-n}T^{n-u_{1}-nu_{2}}$,
$\overline{z}_{4}=y_{2}^{-1}T^{1-\alpha-u_{2}}$.

The quantum Stanley-Reisner relation and linear relation gives

$\displaystyle\overline{z}_{1}\overline{z}_{3}=\overline{z}_{4}^{n}T^{n\alpha},$
$\displaystyle\quad\overline{z}_{2}\overline{z}_{4}=T^{1-\alpha},$ (8.5)
$\displaystyle\overline{z}_{1}-\overline{z}_{3}=0,$
$\displaystyle\quad\overline{z}_{2}-n\overline{z}_{3}-\overline{z}_{4}=0.$
(8.6)

Let us assume $n$ is odd. We put

$\overline{z}_{1}=Z^{n},\qquad\overline{z}_{4}=Z^{2}T^{-\alpha}.$

(In case $n=2n^{\prime}$ is even we put $\overline{z}_{1}=Z^{n^{\prime}}$,
$\overline{z}_{4}=\pm ZT^{-\alpha}.$ The rest of the argument are similar and
is omitted.) Then $\overline{z}_{2}=T^{-\alpha}Z^{2}+nZ^{n}$ and

$Z^{4}(nZ^{n-2}+T^{-\alpha})=T.$ (8.7)

Case 1 ; $(n-2)\mathfrak{v}_{T}(Z)>-\alpha$ : In the first case, we have
$\mathfrak{v}_{T}(Z)=(\alpha+1)/4$. (Then $(n-2)\mathfrak{v}_{T}(Z)>-\alpha$
is automatically satisfied.) Therefore
$u_{1}=\mathfrak{v}_{T}(z_{1})=n(\alpha+1)/4$,
$u_{2}=\mathfrak{v}_{T}(z_{2})=(1-\alpha)/2$. We also can check that there are
$4$ solutions. We remark that we are using $\mathfrak{PO}_{0}$ in place of
$\mathfrak{PO}$. However we can use Corollary 10.6 to prove the following
lemma. This lemma in particular implies that $L(n(\alpha+1)/4,(1-\alpha)/2)$
is balanced which was already shown above in Example 8.1 for the case $n=1$.

######  Lemma 8.3.

Let $y^{(i)}\in\Lambda_{0}\times\Lambda_{0}$ ($i=1,\cdots,4$) be critical
points $\mathfrak{PO}_{0}^{u}$ for $u=(n(\alpha+1)/4,(1-\alpha)/2)$. Then
there exists $y^{(i)\prime}\in\Lambda_{0}\times\Lambda_{0}$ which is a
critical point of $\mathfrak{PO}^{u}$ and $y^{(i)}\equiv
y^{(i)\prime}\mod\Lambda_{+}$.

We will prove Lemma 8.3 in section 10.

Case 2 ; $(n-2)\mathfrak{v}_{T}(Z)<-\alpha$ : We have $v_{T}(Z)=1/(n+2)$. This
can never occur since $1/(n+2)>0>-\alpha/(n-2)$.

Case 3 ; $(n-2)\mathfrak{v}_{T}(Z)=-\alpha$ : We put
$Z=a_{1}T^{-\alpha/(n-2)}+a_{2}T^{\lambda}+$ higher order term.

(Case 3-1 : $na_{1}^{n-2}\neq-1$) : Then $\mathfrak{v}_{T}(Z)=(\alpha+1)/4$.
Since $(\alpha+1)/4\neq-\alpha/(n-2)$, this case never occur.

(Case 3-2 : $na_{1}^{n-2}=-1$) : We have $4v_{T}(Z)+(n-3)v_{T}(Z)+\lambda=1.$
Therefore

$\lambda=\frac{n-2+(n+1)\alpha}{n-2}.$

We have

$u_{1}=v_{T}(z_{1})=-\frac{n\alpha}{n-2},\qquad u_{2}=v_{T}(z_{2})=1-\alpha-
v_{T}(z_{4})=\frac{n-2+2\alpha}{n-2}.$

Thus $(u_{1},u_{2})$ is not in the moment polytope.

In Example 8.2, we have

$\mathfrak{M}(\mathfrak{Lag}(X))=\mathfrak{M}_{0}(\mathfrak{Lag}(X))\neq\text{\rm
Crit}(\mathfrak{PO}_{0}).$

On the other hand, the order of $\mathfrak{M}(\mathfrak{Lag}(X))$ is $4$ and
is equal to the sum of Betti numbers.

######  Remark 8.4.

In a sequel of this series of papers, we will prove the equality

$\sum_{d}\text{\rm
rank}\,H_{d}(X;{\mathbb{Q}})=\\#(\mathfrak{M}(\mathfrak{Lag}(X)))$

for any compact toric manifold $X$ (which is not necessarily Fano) such that
$QH(X;\Lambda)$ is semi-simple. If we count the right hand side with
multiplicity, the same equality holds without assuming semi-simplicity.

We next discuss the version of the above story where we substitute some
explicit number into the formal variable $T$. Let $u\in\mbox{Int}P$. We define
a Laurent polynomial

$\mathfrak{PO}^{u}_{0,T=t}\in{\mathbb{C}}[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y_{n}^{-1}]$

by substituting a complex number $t\in{\mathbb{C}}\setminus\\{0\\}$. In the
same way we define the algebra $QH^{\omega}(X;T=t;{\mathbb{C}})$ over
${\mathbb{C}}$ by substituting $T=t$ in the quantum Stanley-Reisner relation.
The argument of section 6 goes through to show

$QH^{\omega}(X;T=t;{\mathbb{C}})\cong Jac(\mathfrak{PO}^{u}_{0,T=t}).$ (8.8)

In particular the right hand side is independent of $u$ up to an isomorphism.
Here the ${\mathbb{C}}$-algebra in the right hand side of (8.8) is the
quotient of the polynomial ring
${\mathbb{C}}[y_{1},\cdots,y_{n},y_{1}^{-1},\cdots,y_{n}^{-1}]$ by the ideal
generated by $\partial\mathfrak{PO}_{0,T=t}/\partial y_{i}$. ($i=1,\cdots,n$.)

We remark that right hand side of (8.8) is always nonzero, for small $t$, by
Proposition 4.7. It follows that the equation

$\frac{\partial\mathfrak{PO}^{u}_{0,T=t}}{\partial y_{i}}=0$ (8.9)

has a solution $y_{i}\neq 0$ for any $u$. Namely, as far as the Floer
cohomology after $T=t$ substituted, there always exists $b\in
H^{1}(X;{\mathbb{C}})$ with nonvanishing Floer cohomology
$HF((L(u),b),(L(u),b);{\mathbb{C}})$ for any $u\in\mbox{Int}P$. Since the
version of Floer cohomology after substituting $T=t$ is not invariant under
the Hamiltonian isotopy, this is not useful for the application to symplectic
topology. (Compare this with section 14.2 [CO].)

The relation between the set of solutions of (8.9) and that of (4.10) is
stated as follows : Let $(y^{(c)}_{1}(t;u),\cdots,y^{(c)}_{n}(t;u))$ be a
branch of the solutions of (8.9) for $t\neq 0$ where $c$ is an integer with
$1\leq c\leq l$ for some $l\in{\mathbb{N}}$. We can easily show that it is a
holomorphic function of $t$ on ${\mathbb{C}}\setminus{\mathbb{R}}_{-}$. We
consider its behavior as $t\to 0$. For generic $u$ the limit either diverges
or converges to $0$. However if
$(\mathfrak{x},u)\in\mathfrak{M}_{0}(\mathfrak{Lag}(X))$ and
$\mathfrak{x}=\sum x_{i}\text{\bf e}_{i}$ then there is some $c$ such that

$\lim_{t\to 0}y^{(c)}_{i}(t;u)\in{\mathbb{C}}\setminus\\{0\\}\quad\text{and
that}\quad y^{(c)}_{i}(t;u)=e^{x_{i}(t)}.$

To prove this claim it suffices to show that if $(\mathfrak{x},u)\in\text{\rm
Crit}(\mathfrak{PO}_{0})$ and $\mathfrak{x}=\sum x_{i}\text{\bf e}_{i}$,
$y_{i}=e^{x_{i}}$, $y_{i}=\sum_{j}y_{ij}T^{\lambda_{ij}}$ then
$\sum_{j}y_{ij}t^{\lambda_{ij}}$ converges for $0<|t|<\epsilon$. (Here
$\epsilon$ is sufficiently small positive number.) This follows from the
following Lemma 8.5. Let $\Lambda_{0}^{conv}$ be the ring

$\left\\{\sum_{i}a_{i}T^{\lambda_{i}}\in\Lambda_{0}^{{\mathbb{C}}}\,\left|\,\exists\epsilon>0\,\,\text{such
that}\,\,\sum_{i}|a_{i}||t|^{\lambda_{i}}\,\,\text{\rm converges for
$|t|<\epsilon$.}\right\\}\right.$

and $\Lambda^{conv}$ be its field of fractions. We put
$\Lambda_{+}^{conv}=\Lambda_{0}^{conv}\cap\Lambda_{+}$.

######  Lemma 8.5.

The field $\Lambda^{conv}$ is algebraically closed.

We will prove Lemma 8.5 in section 14.

We go back to the discussion on the difference between two sets
$\mathfrak{M}_{0}(\mathfrak{Lag}(X))$ and $\text{\rm
Crit}(\mathfrak{PO}_{0})$. (See Definition 1.11 for its definition.) The rest
of this section owes much to the discussion with H. Iritani and also to his
papers [Iri1], [Iri2]. The results we describe below will not be used in the
other part of this paper.

We recall that we did not take closure of the ideal $(P(X)+SR_{\omega}(X))$ in
section 6. This is actually the reason why we have
$\mathfrak{M}_{0}(\mathfrak{Lag}(X))\neq\text{\rm Crit}(\mathfrak{PO}_{0})$.
More precisely we have the following Proposition 8.6.

We consider the polynomial ring $\Lambda[z_{1},\cdots,z_{m}]$. We define its
norm $\|\cdot\|$ so that

$\left\|\sum_{\vec{i}}a_{\vec{i}}z_{1}^{i_{1}}\cdots
z_{m}^{i_{m}}\right\|=\exp\left(-\inf_{\vec{i}}\mathfrak{v}_{T}(a_{\vec{i}})\right).$

We take the closure of the ideal $(P(X)+SR_{\omega}(X))$ with respect to this
norm and denote it by $\text{\rm Clos}(P(X)+SR_{\omega}(X))$. We put

$\overline{QH}^{\omega}(X;\Lambda)=\frac{\Lambda[z_{1},\cdots,z_{m}]}{\text{\rm
Clos}(P(X)+SR_{\omega}(X))}.$ (8.10)

Let $W^{\text{\rm geo}}(X;\omega)$ be the set of all weight such that the
corresponding $(\mathfrak{x},u)$ satisfies $u\in\text{\rm Int}\,P$. We remark
that $\mathfrak{w}\in W^{\text{\rm geo}}(X;\omega)$ if and only if
$\mathfrak{v}_{T}(\mathfrak{w}_{i})>0$ for all $i$.

######  Proposition 8.6 (Iritani).

There exists an isomorphism

$\overline{QH}^{\omega}(X;\Lambda^{{\mathbb{C}}})\cong\prod_{\mathfrak{w}\in
W^{\text{\rm geo}}(X;\omega)}QH^{\omega}(X;\mathfrak{w}).$

###### Proof.

Let $\mathfrak{w}\in W(X;\omega)\setminus W^{\text{\rm geo}}(X;\omega)$. We
first assume $\mathfrak{v}_{T}(\mathfrak{w}_{i})=-\lambda<0$. (The case
$\mathfrak{v}_{T}(\mathfrak{w}_{i})=0$ will be discussed at the end of the
proof.)

Then, there exists $f\in\Lambda_{0}\setminus\Lambda_{+}$ such that
$T^{\lambda}f\mathfrak{w}_{i}=1$. Let $x\in QH^{\omega}(X;\mathfrak{w})$. We
assume $x\neq 0$. We take $k$ such that
$(\widehat{z}_{i}-\mathfrak{w}_{i})^{k}x\neq 0$,
$(\widehat{z}_{i}-\mathfrak{w}_{i})^{k+1}x=0$ and replace $x$ by
$(\widehat{z}_{i}-\mathfrak{w}_{i})^{k}x$. We then have
$T^{\lambda}f\widehat{z}_{i}x=x$. Since
$\lim_{N\to\infty}\|(fz_{i}T^{\lambda})^{N}\|=0$, it follows that $x=0$ in
$\overline{QH}^{\omega}(X;\Lambda^{{\mathbb{C}}})$. This is a contradiction.

We next assume $\mathfrak{v}_{T}(\mathfrak{w}_{i})>0$ for all $i$. We consider
the homomorphism

$\varphi:\Lambda[z_{1},\cdots,z_{m}]\to
Hom_{\Lambda}(QH^{\omega}(X;\mathfrak{w}),QH^{\omega}(X;\mathfrak{w})),$

defined by

$\varphi(z_{i})(x)=z_{i}\cup_{Q}x.$

We have $\varphi(P(X)+SR_{\omega}(X))=0$. We may choose the basis of
$QH^{\omega}(X;\mathfrak{w})$ so that $\varphi(z_{i})$ is upper triangular
matrix whose diagonal entries are all $\mathfrak{w}_{i}$ and whose off
diagonal entries are all $0$ or $1$. We use it and
$\mathfrak{v}_{T}(\mathfrak{w}_{i})>0$ to show that $\varphi(\text{\rm
Clos}(P(X)+SR_{\omega}(X)))=0$. Namely $\varphi$ induces a homomorphism from
$\overline{QH}^{\omega}(X;\Lambda)$. It follows easily that the restriction of
the projection
${QH}^{\omega}(X;\Lambda^{{\mathbb{C}}})\to\overline{QH}^{\omega}(X;\Lambda^{{\mathbb{C}}})$
to $QH^{\omega}(X;\mathfrak{w})$ is an isomorphism to its image.

We finally show that for $u\in\partial P$, there is no critical point of
$\mathfrak{PO}_{0}$ on $(\Lambda_{0}\setminus\Lambda_{+})^{n}$. Let

$u\in\bigcup_{i\in I}\partial_{i}P\setminus\bigcup_{i\notin
I}\partial_{i}P_{i}.$

Then

$\mathfrak{PO}_{0}^{u}\equiv\sum_{i\in I}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}\mod\Lambda_{+}.$

We remark that $v_{i}$ ($i\in I$) is a part of the ${\mathbb{Z}}$ basis of
${\mathbb{Z}}^{n}$, since $X$ is nonsingular toric. Hence by changing the
variables to appropriate $y^{\prime}_{i}$ it is easy to see that there is no
nonzero critical point of $\sum_{i\in I}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}=\sum_{i\in I^{\prime}}y^{\prime}_{i}$. The proof of
Proposition 8.6 is now complete. ∎

To further discuss the relationship between the contents of sections 6 and 7
and those in [Iri2], we compare the coefficient rings used here and in [Iri2].
In [Iri2] (like many of the literatures on quantum cohomology such as [Gi1])
the formal power series ring ${\mathbb{Q}}[[q_{1},\cdots,q_{m-n}]]$ is taken
as the coefficient ring. ($m-n$ is the rank of $H^{2}(X;{\mathbb{Q}})$ and we
choose a basis of it.) The superpotential in [Iri2] (which is the same as the
one used in [Gi1]) is given as 222We change the notation so that it is
consistent to ours. $m,n,v_{i,j}$ here corresponds to $r+N$, $r$, $x_{i,b}$ in
[Iri2], respectively.

$F_{q}=\sum_{i=1}^{m}\left(\prod_{a=1}^{m-n}q_{a}^{l_{a,i}}\prod_{j=1}^{n}s_{j}^{v_{i,j}}\right).$
(8.11)

Here $l_{a,i}$ is a matrix element of a splitting of $H_{2}(X;{\mathbb{Z}})\to
H_{2}(X,T^{n};{\mathbb{Z}})$. We will show that (8.11) pulls back to our
potential function $\mathfrak{PO}^{u}_{0}$ after a simple change of variables.
Let $\alpha_{a}\in H_{2}(X;{\mathbb{Z}})$ be the basis we have chosen. (We
choose it so that $[\omega]\cap\alpha_{a}$ is positive.)

######  Lemma 8.7.

There exists $f_{j}(u)\in{\mathbb{R}}$ ($j=1,\cdots,n$) such that

$\frac{1}{2\pi}\sum_{a}l_{a,i}[\omega]\cap\alpha_{a}=\ell_{i}(u)-\sum_{j=1}^{n}v_{i,j}f_{j}(u).$

###### Proof.

We consider the exact sequence

$0\longrightarrow
H_{2}(X;{\mathbb{Z}})\overset{i_{*}}{\longrightarrow}H_{2}(X,L(u);{\mathbb{Z}})\longrightarrow
H_{1}(L(u);{\mathbb{Z}})\longrightarrow 0.$

$(c_{1},\cdots,c_{m})\in H_{2}(X,L(u);{\mathbb{Z}})$ is in the image of
$H_{2}(X;{\mathbb{Z}})$ if and only if $\sum_{i}c_{i}v_{i}=0$. (Here
$v_{i}=(v_{i,1},\cdots,v_{i,n})\in{\mathbb{Z}}^{n}$.) For given $\alpha\in
H_{2}(X,{\mathbb{Z}})$ denote $i_{*}(\alpha)=(c_{1},\cdots,c_{m})$. Then we
have

$\sum_{a}[\omega]\cap c_{i}l_{a,i}\alpha_{a}=[\omega]\cap\alpha=2\pi\sum
c_{i}\ell_{i}(u).$

This implies the lemma. ∎

We now put

$q_{a}=T^{[\omega]\cap\alpha_{a}/2\pi},\qquad s_{j}(u)=T^{f_{j}(u)}y_{j}.$
(8.12)

We obtain the identity

$F_{q}(s_{1}(u),\cdots,s_{n}(u))=\mathfrak{PO}_{0}^{u}(y_{1},\cdots,y_{n}).$
(8.13)

We remark that if we change the choice of Kähler form then the identification
(8.12) changes. In other words, the story over
${\mathbb{Q}}[[q_{1},\cdots,q_{m-n}]]$ corresponds to studying all the
symplectic structures simultaneously, while the story over $\Lambda$ focuses
on one particular symplectic structure.

In [Iri2] Corollary 5.12, Iritani proved semi-simplicity of quantum cohomology
ring of toric manifold with coefficient ring
${\mathbb{Q}}[[q_{1},\cdots,q_{m-n}]]$. It does not imply the semi-simplicity
of our $QH^{\omega}(X;\Lambda)$ since the semi-simplicity in general is not
preserved by the pull-back. (On the other way round, semi-simplicity follows
from semi-simplicity of the pull-back.) However it is preserved by the pull-
back at a generic point. Namely we have:

######  Proposition 8.8.

The set of $T^{n}$-invariant symplectic structures on $X$ for which
$Jac(\mathfrak{PO}_{0}^{u})$ is semi-simple is open and dense.

###### Proof.

We give a proof for completeness, following the argument in the proof of
Proposition 5.11 [Iri2]. Consider the polynomial

$F_{w_{1},\cdots,w_{m}}=\sum_{i=1}^{m}w_{i}y_{1}^{v_{i,1}}\cdots
y_{n}^{v_{i,n}}$

where $w_{i}\in{\mathbb{C}}\setminus\\{0\\}$. By Kushnirenko’s theorem [Ku]
the Jacobian ring of $F_{w_{1},\cdots,w_{m}}$ is semi-simple for a generic
$w_{1},\cdots,w_{m}$. We put

$w_{i}=\exp\left(\frac{1}{2\pi}\sum_{a}l_{a,i}[\omega]\cap\alpha+\sum_{j}v_{i,j}f_{j}(u)\right).$

It is easy to see that when we move $[\omega]\cap\alpha_{a}$ and $u$ (there
are $m-n$, $n$ parameters respectively) then $w_{i}$ moves in an arbitrary
way. Therefore for generic choice of $\omega$ and $u$, the Jacobian ring
$Jac(\mathfrak{PO}_{0}^{u})$ is semi-simple. Since
$Jac(\mathfrak{PO}_{0}^{u})$ is independent of $u$ up to isomorphism, the
proposition follows. ∎

######  Remark 8.9.

Combined with Theorem 1.9, this proposition gives a partial answer to Question
in section 3 [EP2].

##  9\. Variational analysis of potential function

In this section, we prove Proposition 4.7. Let $\mathfrak{PO}$ be defined as
in (4.7).

We define

$s_{1}(u)=\inf\\{\ell_{i}(u)\mid i=1,\cdots,m\\}.$

$s_{1}$ is a continuous, piecewise affine and convex function and $s_{1}\equiv
0$ on $\partial P$. Recall if $u\in\partial_{i}P$ then $\ell_{i}(u)=0$ by
definition.

We put

$\displaystyle S_{1}$ $\displaystyle=\sup\\{s_{1}(u)\mid u\in P\\},$
$\displaystyle P_{1}$ $\displaystyle=\\{u\in P\mid s_{1}(u)=S_{1}\\}.$

######  Proposition 9.1.

There exist $s_{k}$, $S_{k}$, and $P_{k}$ with the following properties.

  1. (1)

$P_{k+1}$ is a convex polyhedron in $M_{\mathbb{R}}$. $\mathop{\kern 0.0pt{\rm
dim}}\nolimits P_{k+1}\leq\mathop{\kern 0.0pt{\rm dim}}\nolimits P_{k}$.

  2. (2)

$s_{k+1}:P_{k}\to{\mathbb{R}}$ is a continuous, convex piecewise affine
function.

  3. (3)

$s_{k+1}(u)=\inf\\{\ell_{i}(u)\mid\ell_{i}(u)>S_{k}\\}$ for $u\in\text{\rm
Int}P_{k}$.

  4. (4)

$s_{k+1}(u)=S_{k}$ for $u\in\partial P_{k}$.

  5. (5)

$S_{k+1}=\sup\\{s_{k+1}(u)\mid u\in P_{k}\\}.$

  6. (6)

$P_{k+1}=\\{u\in P_{k}\mid s_{k+1}(u)=S_{k+1}\\}.$

  7. (7)

$P_{k+1}\subset\text{\rm Int}P_{k}$.

  8. (8)

$s_{k}$, $S_{k}$, $P_{k}$ are defined for $k=1,2,\cdots,K$ for some
$K\in{\mathbb{Z}}_{+}$ and $P_{K}$ consists of a single point.

######  Example 9.2.

Let $P=[0,a]\times[0,b]$ ($a<b$.) Then
$s_{1}(u_{1},u_{2})=\inf\\{u_{1},u_{2},a-u_{1},b-u_{2}\\}.$ $S_{1}=a/2$,
$P_{1}=\\{(a/2,u_{2})\mid a/2\leq u_{2}\leq b-a/2\\}$,
$s_{2}(1/2,u_{2})=\inf\\{u_{2},b-u_{2}\\}$, $S_{2}=b/2$,
$P_{2}=\\{(a/2,b/2)\\}$.

###### Proof.

We define $s_{k}$, $S_{k}$, $P_{k}$ inductively over $k$. We assume that
$s_{k}$, $S_{k}$, $P_{k}$ are defined for $k=1,\cdots,k_{0}$ so that (1) - (7)
of Proposition 9.1 are satisfied for $k=1,\cdots,k_{0}-1$.

We define $s_{k_{0}+1}$ by (3) and (4). We will prove that it satisfies (2).
We use the following lemma for this purpose.

######  Lemma 9.3.

Let $u_{j}\in\text{\rm Int}P_{k_{0}}$ and
$\lim_{j\to\infty}u_{j}=u_{\infty}\in\partial P_{k_{0}}$. Then

$\lim_{j\to\infty}s_{k_{0}+1}(u_{j})=S_{k_{0}}.$

###### Proof.

We put

$I_{k_{0}}^{\prime}=\\{\ell_{i}|\ell_{i}(u_{\infty})=S_{k_{0}}\\}.$ (9.1)

By (6) for $k=k_{0}-1$, we find that $s_{k_{0}}(u_{\infty})=S_{k_{0}}$. Then
(3) for $k=k_{0}-1$ implies that there is $\ell_{i}$ such that
$\ell_{i}(u_{\infty})=S_{k_{0}}$. Thus $I_{k_{0}}^{\prime}$ is non-empty. We
take the affine space $A_{k_{0}}\subset M_{\mathbb{R}}$ such that $\text{\rm
Int}P_{k_{0}}$ is relatively open in $A_{k_{0}}$.

Now since $u_{\infty}\in\partial P_{k_{0}}$, we can take a $\vec{u}\in
T_{u_{\infty}}A_{k_{0}}$ such that $u_{\infty}+\epsilon\vec{u}\notin
P_{k_{0}}$ for any sufficiently small $\epsilon>0$. It follows from (7) for
$k=k_{0}-1$ that $u_{\infty}+\epsilon\vec{u}\in\operatorname{Int}P_{k_{0}-1}$,
and hence $u+\epsilon\vec{u}\in\operatorname{Int}P_{k_{0}-1}\setminus
P_{k_{0}}$.

By definition, we also have $s_{k_{0}}(u)\leq S_{k_{0}}$ for all $u\in
P_{k_{0}-1}$. Therefore we have

$s_{k_{0}}(u_{\infty}+\epsilon\vec{u})<S_{k_{0}}.$

It follows that there exists $\ell_{i}\in I^{\prime}_{k_{0}}$ such that

$\ell_{i}(u_{\infty}+\epsilon\vec{u})<\ell_{i}(u_{\infty})=S_{k_{0}}<\ell_{i}(u_{\infty}-\epsilon\vec{u}).$
(9.2)

Since (9.2) holds for any $\vec{u}\in T_{u_{\infty}}A_{k_{0}}$ with
$u_{\infty}+\epsilon\vec{u}\notin P_{k_{0}}$, it holds for
$\epsilon\vec{u}:=u_{j}-u_{\infty}$ for any sufficiently large $j$. We note
that since $u_{j}\in\operatorname{Int}P_{k_{0}}\subset A_{k_{0}}$,
$\vec{u}_{j}=u_{\infty}-u_{j}$ is an ‘outward’ vector as a tangent vector in
$T_{u_{\infty}}A_{k_{0}}$ at $u_{\infty}\in\partial P_{k_{0}}$. Therefore we
have $u_{\infty}+\vec{u}_{j}\notin P_{k_{0}}$. Because
$u_{j}=u_{\infty}-\vec{u}_{j}$, it follows from (9.2) that

$\ell_{i}(u_{j})>S_{k_{0}}$ (9.3)

for any sufficiently large $j$. Therefore we have

$s_{k_{0}+1}(u_{j})=\inf\\{\ell_{i}(u_{j})\mid\ell_{i}\in
I^{\prime}_{k_{0}},\,\,\ell_{i}(u_{j})>S_{k_{0}}\\}$ (9.4)

and
$\lim_{j\to\infty}s_{k_{0}+1}(u_{j})=\lim_{j\to\infty}\ell_{i}(u_{j})=\ell_{i}(u_{\infty})=S_{k_{0}}$.
This finishes the proof of the lemma. ∎

Lemma 9.3 implies that $s_{k_{0}+1}$ is continuous and piecewise linear in a
neighborhood of $\partial P_{k_{0}}$. We can then check (2) easily.

We define $S_{k_{0}+1}$ by (5). Then we can define $P_{k_{0}+1}$ by (6). (In
other words the right hand side of (6) is nonempty.) (7) is a consequence of
Lemma 9.3. We can easily check that $P_{k_{0}+1}$ satisfies (1).

We finally prove that $P_{K}$ becomes a point for some $K$. Let $\text{\bf
u}_{k}\in\text{\rm Int}\,P_{k}$ and put

$I_{k}=\\{\ell_{i}\mid\ell_{i}(\text{\bf u}_{k})=S_{k}\\}.$ (9.5)

Here $k=1,\cdots,K$. We remark that $I_{k}$ is independent of the choice of
$\text{\bf u}_{k}\in\text{\rm Int}\,P_{k}$.

Note we defined $I^{\prime}_{k_{0}}$ by the formula (9.1). We have
$I_{k_{0}}\subseteq I^{\prime}_{k_{0}}$. But the equality may not hold in
general. In fact $u_{\infty}$ in the boundary of $P_{k_{0}}$ but $\text{\bf
u}_{k_{0}}$ is an interior point of $P_{k_{0}}$. Therefore if $\ell_{i}\in
I_{k_{0}}$ then $\ell_{i}$ is constant on $P_{k_{0}}$. But element of
$I^{\prime}_{k_{0}}$ may not have this property.

In case some $\ell_{i}\in I_{k_{0}+1}$ is not constant on $P_{k_{0}}$ it is
easy to see that $\mathop{\kern 0.0pt{\rm dim}}\nolimits
P_{k_{0}+1}<\mathop{\kern 0.0pt{\rm dim}}\nolimits P_{k_{0}}$. We remark that
there exists some $\ell_{j}\notin\bigcup_{k\leq k_{0}}I_{k_{0}}$ which is not
constant on $S_{k_{0}}$ unless $S_{k_{0}}$ is a point. Therefore if
$\mathop{\kern 0.0pt{\rm dim}}\nolimits S_{k_{0}}\neq 0$, there exists
$k^{\prime}>k_{0}$ such that $\mathop{\kern 0.0pt{\rm dim}}\nolimits
P_{k^{\prime}}<\mathop{\kern 0.0pt{\rm dim}}\nolimits P_{k_{0}}$. Therefore
there exists $K$ such that $P_{K}$ becomes $0$-dimensional (namely a point).
Hence we have achieved (8). The proof of Proposition 9.1 is now complete. ∎

######  Remark 9.4.

In a recent preprint [Mc], McDuff pointed out an error in the statement (1) of
Proposition 9.1 in the previous version of this paper. We have corrected the
statement and modified the last paragraph of its proof, following the
corresponding argument in section 2.2 of [Mc]. We thank her for pointing out
this error.

The next lemma easily follows from construction.

######  Lemma 9.5.

If all the vertices of $P$ lie in ${\mathbb{Q}}^{n}$ then
$u_{0}\in{\mathbb{Q}}^{n}$. Here $\\{u_{0}\\}=P_{K}$.

By parallelly translating the polytope, we may assume, without loss of
generality, that $u_{0}=\text{\bf 0}$, the origin. In the rest of this
subsection, we will prove that $\mathfrak{PO}^{\text{\bf 0}}$ has a critical
point on $(\Lambda_{0}\setminus\Lambda_{+})^{n}$. More precisely we prove
Proposition 4.7 for $u_{0}=\text{\bf 0}$. (We remark that if $P$ and
$\ell_{i}$ are given we can easily locate $u_{0}$.)

######  Example 9.6.

Let us consider Example 8.1 in the case $\alpha>1/3$. At
$u_{0}=((1+\alpha)/4,(1-\alpha)/2)$ we have

$\mathfrak{PO}^{u_{0}}=(y_{2}+y_{2}^{-1})T^{(1-\alpha)/2}+(y_{1}+(y_{1}y_{2})^{-1})T^{(1+\alpha)/4}.$

Therefore the constant term $\mathfrak{y}_{i;0}$ of the coordinate $y_{i}$ of
the critical point is given by

$1-\mathfrak{y}_{2;0}^{-2}=0,\quad
1-\mathfrak{y}_{1;0}^{-2}\mathfrak{y}_{2;0}^{-1}=0.$ (9.6)

Note the first equation comes from the term of the smallest exponent and
contains only $\mathfrak{y}_{2;0}$. The second equation comes from the term
which has second smallest exponent and contains both $\mathfrak{y}_{1;0}$ and
$\mathfrak{y}_{2;0}$. So we need to solve the equation inductively according
to the order of the exponent. This is the situation we want to work out in
general.

We remark that the affine space $A_{i}$ defined above in the proof of Lemma
9.3

$M_{\mathbb{R}}=A_{0}\supseteq A_{1}\supseteq\cdots\supseteq A_{K-1}\supseteq
A_{K}=\\{\text{\bf 0}\\}$

is a nonincreasing sequence of linear subspaces such that $\text{\rm Int
}P_{k}$ is an open subset of $A_{k}$. Let

$A_{l}^{\perp}\subset(M_{\mathbb{R}})^{*}\cong N_{\mathbb{R}}$

be the annihilator of $A_{l}\subset M_{\mathbb{R}}$. Then we have

$\\{\text{\bf 0}\\}=A_{0}^{\perp}\subseteq
A_{1}^{\perp}\subseteq\cdots\subseteq A_{K-1}^{\perp}\subseteq
A_{K}^{\perp}=N_{\mathbb{R}}.$

We recall:

$I_{k}=\\{\ell_{i}\mid\ell_{i}(\text{\bf 0})=S_{k}\\},$ (9.7)

for $k=1,\cdots,K$. In fact $\text{\bf 0}\in P_{k+1}\subseteq\text{\rm
Int}\,P_{k}$ for $k<K$.

We renumber each of $I_{k}$ in (9.7) so that

$\\{\ell_{k,j}\mid j=1,\cdots,a(k)\\}=I_{k}.$ (9.8)

By construction

$s_{k}(u)=\inf_{j}\ell_{k,j}(u)$ (9.9)

in a neighborhood of 0 in $P_{k-1}$. In fact $s_{k-1}(\text{\bf
0})=S_{k-1}<S_{k}=s_{k}(\text{\bf 0})$ and

$\\{\ell_{i}(\text{\bf 0})\mid i=1,\cdots,m\\}\cap(S_{k-1},S_{k})=\emptyset.$

######  Lemma 9.7.

If $u\in A_{k}$ then $\ell_{k,j}(u)=S_{k}$.

###### Proof.

We may assume $k<K$. Hence $\text{\bf 0}\in\text{\rm Int}P_{k}$. We regard
$u\in A_{k}=T_{\text{\bf 0}}A_{k}$. By (9.9), we have

$s_{k}(\varepsilon u)=\inf\\{\ell_{k,j}(\epsilon u)\mid j=1,\cdots,a(k)\\}.$

Since $s_{k}(\varepsilon u)=S_{k}$ for $\varepsilon u\in P_{k}$ it follows
that $\ell_{k,j}(u)=S_{k}$. ∎

Lemma 9.7 implies that the linear part $d\ell_{k,j}$ of $\ell_{k,j}$ is an
element of $A_{k}^{\perp}\subset\mathfrak{t}=N_{\mathbb{R}}$. In fact if
$\ell_{k,j}=\ell_{i}$, we have $d\ell_{k,j}=v_{i}$ from the definition of
$\ell_{i}$, $\ell_{i}(u)=\langle u,v_{i}\rangle-\lambda_{i}$ given in Theorem
2.13.

######  Lemma 9.8.

For any $v\in A_{k}^{\perp}$, there exist nonnegative real numbers $c_{j}\geq
0$, $j=1,\cdots,a(k)$ such that

$v-\sum_{j=1}^{a(k)}c_{j}d\ell_{k,j}\in A_{k-1}^{\perp}.$

###### Proof.

Suppose to the contrary that

$\left\\{v-\sum_{j=1}^{a(k)}c_{j}d\ell_{k,j}\,\Big{|}\,c_{j}\geq
0,\,\,j=1,\cdots,a(k)\right\\}\cap A_{k-1}^{\perp}=\emptyset.$

Then we can find $u\in A_{k-1}\setminus A_{k}$ such that

$d\ell_{k,j}(u)\geq 0$ (9.10)

for all $j=1,\cdots,a(k)$.

Since $\varepsilon u\in A_{k-1}\setminus A_{k}$ it follows that

$s_{k}(\varepsilon u)<S_{k}$

for a sufficiently small $\varepsilon$. On the other hand, (9.10) implies
$d\ell_{k,j}(\varepsilon u)\geq 0$ for all $\varepsilon>0$ and so
$\ell_{k,j}(\varepsilon u)\geq\ell_{k,j}(\text{\bf 0})=S_{k}$. Therefore by
definition of $s_{k}$ in Proposition 9.1 we have

$\displaystyle s_{k}(\varepsilon u)$ $\displaystyle\geq$
$\displaystyle\inf\\{\ell_{k,j}(\varepsilon u)\mid j=1,\cdots,a(k)\\}$
$\displaystyle\geq$ $\displaystyle\inf\\{\ell_{k,j}(\text{\bf 0})\mid
j=1,\cdots,a(k)\\}=S_{k}.$

This is a contradiction. ∎

Applying Lemma 9.8 inductively downwards starting from $\ell=k$ ending at
$\ell=1$, we immediately obtain the following

######  Corollary 9.9.

For any $v\in A_{k}^{\perp}$, there exist $c_{l,j}\geq 0$ for $l=1,\cdots,k$,
$j=1,\cdots,a(l)$ such that

$v=\sum_{l=1}^{k}\sum_{j=1}^{a(l)}c_{l,j}d\ell_{l,j}.$

We denote

$\mathfrak{I}=\\{\ell_{i}\mid i=1,\cdots,m\\}\setminus\bigcup_{k=1}^{K}I_{k}.$
(9.11)

It is easy to see that

$\ell\in\mathfrak{I}\Rightarrow\ell(\text{\bf 0})>S_{K}.$ (9.12)

Now we go back to the situation of (4.7). We use the notation of (4.7). In
this case, for each $k=1,\cdots,K$, we also associate, in Definition 9.10, a
set $\mathfrak{I}_{k}$ consisting of pairs $(\ell,\rho)$ with an affine map
$\ell:M_{\mathbb{R}}\to{\mathbb{R}}$ and $\rho\in{\mathbb{R}}_{+}$.

######  Definition 9.10.

We say that a pair $(\ell,\rho)=(\ell^{\prime}_{j},\rho_{j})$ is an element of
$\mathfrak{I}_{k}$ if the following holds :

  1. (1)

If $e_{j}^{i}\neq 0$ then $\ell_{i}\in\bigcup_{l=1}^{k}I_{l}$. (Note
$\ell^{\prime}_{j}=\sum_{i}e_{j}^{i}\ell_{i}$.)

  2. (2)

(1) does not hold for some $i,j$ if we replace $k$ by $k-1$.

A pair $(\ell,\rho)=(\ell^{\prime}_{j},\rho_{j})$ as in (4.7) is, by
definition, an element of $\mathfrak{I}_{K+1}$ if it is not contained in any
of $\mathfrak{I}_{k}$, $k=1,\cdots,K$.

######  Lemma 9.11.

  1. (1)

If $(\ell,\rho)\in\mathfrak{I}_{k}$ then $d\ell\in A_{k}^{\perp}$.

  2. (2)

If $(\ell,\rho)\in\mathfrak{I}_{k}$ then $\ell(\text{\bf 0})+\rho>S_{k}$.

  3. (3)

If $(\ell,\rho)\in\mathfrak{I}_{K+1}$ then $\ell(\text{\bf 0})+\rho>S_{K}$.

###### Proof.

(1) follows from Definition 9.10 (1) and Lemma 9.7.

If $(\ell,\rho)=(\ell_{j}^{\prime},\rho_{j})\in\mathfrak{I}_{k}$ then there
exists $e_{j}^{i}\neq 0$, $\ell_{i}=\ell_{k,j^{\prime}}$. Then

$\ell(\text{\bf 0})+\rho\geq e^{i}_{j}\ell_{i}(\text{\bf
0})+\rho_{j}>\ell_{i}(\text{\bf 0})=S_{k}.$

(2) follows. The proof of (3) is the same. ∎

######  Lemma 9.12.

The vector space $A_{k}$ is defined over ${\mathbb{Q}}$.

###### Proof.

$A_{k}$ is defined by equalities of the type $\ell_{i}=S_{k}$ on $A_{k-1}$.
Since the linear part of $\ell_{i}$ has integer coefficients, the lemma
follows by induction on $k$. ∎

We put $d(k)=\mathop{\kern 0.0pt{\rm dim}}\nolimits A_{k-1}-\mathop{\kern
0.0pt{\rm dim}}\nolimits A_{k}=\mathop{\kern 0.0pt{\rm dim}}\nolimits
A^{\perp}_{k}-\mathop{\kern 0.0pt{\rm dim}}\nolimits A^{\perp}_{k-1}$. We
choose $\text{\bf e}_{i,j}^{*}\in Hom(M_{\mathbb{Q}},{\mathbb{Q}})\cong
N_{\mathbb{Q}}$ ($i=1,\cdots,K$, $j=1,\cdots,d(k)$) such that the following
condition holds. Here $M_{\mathbb{Q}}=M\otimes{\mathbb{Q}}$ and
$N_{\mathbb{Q}}=N\otimes{\mathbb{Q}}$

######  Condition 9.13.

  1. (1)

$\text{\bf e}_{1,1}^{*},\cdots,\text{\bf e}_{k,d(k)}^{*}$ is a ${\mathbb{Q}}$
basis of $A^{\perp}_{k}\cap N_{{\mathbb{Q}}}$.

  2. (2)

$d\ell_{k,j}=\sum_{k^{\prime},j^{\prime}}v_{(k,j),(k^{\prime},j^{\prime})}\text{\bf
e}_{k^{\prime},j^{\prime}}^{*}$ with
$v_{(k,j),(k^{\prime},j^{\prime})}\in{\mathbb{Z}}$.

  3. (3)

If $(\ell,\rho)\in\mathfrak{I}_{k}$ or $\ell\in\mathfrak{I}$, then
$d\ell=\sum_{k^{\prime},j^{\prime}}v_{\ell,(k^{\prime},j^{\prime})}\text{\bf
e}_{k^{\prime},j^{\prime}}^{*}$ with
$v_{\ell,(k^{\prime},j^{\prime})}\in{\mathbb{Z}}$.

Note $d(k)=0$ if $A_{k}=A_{k-1}$.

We identify ${\mathbb{R}}^{n}$ with $H^{1}(L(u);{\mathbb{R}}^{n})$ in the same
way as Lemma 4.4 and let $x_{k,j}\in
Hom(H^{1}(L(u);{\mathbb{R}}),{\mathbb{R}})$ be the element corresponding to
$\text{\bf e}_{k,j}^{*}$ by this identification. In other words, if

$\text{\bf e}_{k,j}^{*}=\sum_{i}a_{(k,j);i}\text{\bf e}_{i}^{*},$

where $\text{\bf e}_{i}^{*}$ is as in Lemma 4.4, then we have

$x_{k,j}=\sum_{i}a_{(k,j);i}x_{i}.$

We put $y_{k,j}=e^{x_{k,j}}.$ We define

$Y(k,j)=\prod_{k^{\prime}=1}^{K}\prod_{j^{\prime}=1}^{d(k^{\prime})}y_{k^{\prime},j^{\prime}}^{v_{(k,j),(k^{\prime},j^{\prime})}}.$
(9.13)

And for $(\ell,\rho)\in\mathfrak{I}_{k}$ or $\ell\in\mathfrak{I}$, we define

$Y(\ell)=\prod_{k=1}^{K}\prod_{j=1}^{d(k)}y_{k,j}^{v_{\ell,(k,j)}}.$ (9.14)

By Theorem 4.6 there exists $c_{(\ell,\rho)}\in{\mathbb{Q}}$ such that :

$\displaystyle\mathfrak{PO}^{\text{\bf 0}}=$
$\displaystyle\sum_{k=1}^{K}\left(\sum_{j=1}^{a(k)}Y(k,j)\right)T^{S_{k}}+\sum_{\ell\in\mathfrak{I}}Y(\ell)T^{\ell(\text{\bf
0})}$ (9.15)
$\displaystyle+\sum_{k=1}^{K+1}\sum_{(\ell,\rho)\in\mathfrak{I}_{k}}c_{(\ell,\rho)}Y(\ell)T^{\ell(\text{\bf
0})+\rho}$

where $\mathfrak{PO}^{\text{\bf 0}}$ is $\mathfrak{PO}^{u}$ with $u=\text{\bf
0}$.

######  Lemma 9.14.

  1. (1)

If $k^{\prime}<k$ then

$\frac{\partial Y(k^{\prime},j^{\prime})}{\partial y_{k,j}}=0.$ (9.16)

  2. (2)

If $(\ell,\rho)\in\mathfrak{I}_{k^{\prime}}$, $k^{\prime}<k$ then

$\frac{\partial Y(\ell)}{\partial y_{k,j}}=0.$ (9.17)

  3. (3)

If $(\ell,\rho)\in\mathfrak{I}_{k}$ then $\ell(\text{\bf 0})+\rho>S_{k}$.

  4. (4)

If $(\ell,\rho)\in\mathfrak{I}_{K+1}$ then $\ell(\text{\bf 0})+\rho>S_{K}$.

  5. (5)

If $\ell\in\mathfrak{I}$ then $\ell(\text{\bf 0})>S_{K}$.

###### Proof.

Since $d\ell_{k^{\prime},j^{\prime}}\in A^{\perp}_{k^{\prime}}$ by Lemma 9.7
it follows that $v_{(k^{\prime},j^{\prime}),(k,j)}=0$ for $k>k^{\prime}$. (1)
follows. (2) follows from Lemma 9.11 (1) in the same way. (3) follows from
Lemma 9.11 (2). (4) follows from Lemma 9.11 (3). (5) follows from (9.12). ∎

Now equation (4.10) becomes

$0=\frac{\partial\mathfrak{PO}^{\text{\bf 0}}}{\partial y_{k,j}}.$

We calculate this equation using Lemma 9.14 to find that it is equivalent to :

$\displaystyle 0=$ $\displaystyle\sum_{j^{\prime}=1}^{a(k)}\frac{\partial
Y(k,j^{\prime})}{\partial
y_{k,j}}+\sum_{k^{\prime}>k}\sum_{j^{\prime}=1}^{a(k^{\prime})}\frac{\partial
Y(k^{\prime},j^{\prime})}{\partial y_{k,j}}T^{S_{k^{\prime}}-S_{k}}$ (9.18)
$\displaystyle+\sum_{k^{\prime}=k}^{K+1}\sum_{(\ell,\rho)\in\mathfrak{I}_{k^{\prime}}}c_{(\ell,\rho)}\frac{\partial
Y(\ell)}{\partial y_{k,j}}T^{\ell(\text{\bf 0})+\rho-
S_{k}}+\sum_{\ell\in\mathfrak{I}}\frac{\partial Y(\ell)}{\partial
y_{k,j}}T^{\ell(\text{\bf 0})-S_{k}}.$

Note the exponents of $T$ in the second, third, and fourth terms of (9.18) are
all strictly positive. So after putting $T=0$ we have

$0=\sum_{j^{\prime}=1}^{a(k)}\frac{\partial Y(k,j^{\prime})}{\partial
y_{k,j}}.$ (9.19)

Note that the equation (9.19) does not involve $T$ but becomes a numerical
equation. We call (9.19) the leading term equation.

######  Lemma 9.15.

There exist positive real numbers $\mathfrak{y}_{k,j;0}$, $k=1,\cdots,K$,
$j=1,\cdots,d(k)$, solving the leading term equations for $k=1,\cdots,K$.

###### Proof.

We remark the leading term equation for $k,j$ contain the monomials involving
only $y_{k^{\prime},j}$ for $k^{\prime}\leq k$. We first solve the leading
term equation for $k=1$. Denote

$f_{1}(x_{1,1},\cdots,x_{1,d(1)})=\sum_{j=1}^{a(1)}Y(1,j).$

It follows from Corollary 9.9 that for any $(x_{1,1},\cdots,x_{1,d(1)})\neq
0$, there exists $j$ such that

$d\ell_{1,j}(x_{1,1},\cdots,x_{1,d(1)})>0.$

Therefore, we have

$\lim_{t\to\infty}f_{1}(tx_{1,1},\cdots,tx_{1,d(1)})\geq\lim_{t\to\infty}C\exp(td\ell_{1,j}(x_{1,1},\cdots,x_{1,d(1)}))=+\infty.$

Hence $f_{1}(x_{1,1},\cdots,x_{1,d(1)})$ attains its minimum at some point of
${\mathbb{R}}^{d(1)}$. Taking its exponential, We obtain
$\mathfrak{y}_{1,j;0}\in{\mathbb{R}}_{+}$.

Suppose we have already found $\mathfrak{y}_{k^{\prime},j;0}$ for
$k^{\prime}<k$. Then we put

$F_{k}(x_{1,1},\cdots,x_{k,1},\cdots,x_{k,d(k)})=\sum_{j=1}^{a(k)}Y(k,j)$

and

$f_{k}(x_{k,1},\cdots,x_{k,d(k)})=F_{k}(\mathfrak{x}_{1,1;0},\cdots,\mathfrak{x}_{k-1,d(k-1);0},x_{k,1},\cdots,x_{k,d(k)})$

where $\mathfrak{x}_{k^{\prime},j;0}=\log\mathfrak{y}_{k^{\prime},j;0}$. Again
using Corollary 9.9, we find

$\lim_{t\to\infty}f_{k}(tx_{k,1},\cdots,tx_{k,d(k)})=+\infty.$

for any $(x_{k,1},\cdots,x_{k,d(k)})\neq 0$. Hence
$f_{k}(x_{k,1},\cdots,x_{k,d(k)})$ attains a minimum and we obtain
$\mathfrak{y}_{k,j;0}$. Lemma 9.15 now follows by induction. ∎

We next find the solution of our equation (4.11) or (4.12). We take a
sufficiently large ${\mathcal{N}}$ and put

$\displaystyle\mathfrak{PO}^{\text{\bf 0}}_{k,{\mathcal{N}}}=$
$\displaystyle\sum_{j=1}^{a(k)}Y(k,j)+\sum_{k^{\prime}>k}\sum_{j^{\prime}=1}^{a(k^{\prime})}\frac{\partial
Y(k^{\prime},j^{\prime})}{\partial y_{k,j}}T^{S_{k^{\prime}}-S_{k}}$ (9.20)
$\displaystyle+\sum_{\ell\in\mathfrak{I},\,\,\ell(\text{\bf
0})\leq{\mathcal{N}}}Y(\ell)T^{\ell(\text{\bf 0})-S_{k}}$
$\displaystyle+\sum_{k^{\prime}=k+1}^{K+1}\sum_{(\ell,\rho)\in\mathfrak{I}_{k^{\prime}},\ell(\text{\bf
0})+\rho\leq{\mathcal{N}}}c_{(\ell,\rho)}Y(\ell)T^{\ell(\text{\bf 0})+\rho-
S_{k}}.$

We remark that (4.11) is equivalent to

$\frac{\partial\mathfrak{PO}_{k,{\mathcal{N}}}^{\text{\bf 0}}}{\partial
y_{k,j}}(\mathfrak{y}_{1},\cdots,\mathfrak{y}_{n})\equiv 0\mod
T^{{\mathcal{N}}-S_{k}}\qquad k=1,\cdots,K,\,j=1,\cdots,a(k).$ (9.21)

We also put

$\overline{\mathfrak{PO}}^{\text{\bf 0}}_{k}=\sum_{j=1}^{a(k)}Y(k,j).$

It satisfies

$\overline{\mathfrak{PO}}^{\text{\bf 0}}_{k}\equiv\mathfrak{PO}^{\text{\bf
0}}_{k,{\mathcal{N}}}\mod\Lambda_{+}.$ (9.22)

For given positive numbers $R(1),\cdots,R(K)$ we define the discs

$D(R(k))=\\{(x_{k,1}\cdots,x_{k,d(k)})\mid
x_{k,1}^{2}+\cdots+x_{k,d(k)}^{2}\leq R(k)\\}\subset{\mathbb{R}}^{d(k)}$

and the poly-discs

$\displaystyle D(R(\cdot))$ $\displaystyle=$
$\displaystyle\prod_{k=1}^{K}D(R(k))$ $\displaystyle=$
$\displaystyle\\{(x_{1,1}\cdots,x_{K,d(K)})\mid
x_{k,1}^{2}+\cdots+x_{k,d(k)}^{2}\leq R(k),\,k=1,\cdots,K\\}.$

We factorize

${\mathbb{R}}^{n}=\prod_{k=1}^{K}{\mathbb{R}}^{d(k)}.$

Then we consider the Jacobian of $\overline{\mathfrak{PO}}^{\text{\bf 0}}_{k}$

$\nabla\overline{\mathfrak{PO}}^{\text{\bf
0}}_{k}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{d(k)}$

i.e., the map

$(\mathfrak{x}_{1,1},\cdots,\mathfrak{x}_{K,d(K)})\mapsto\left(\frac{\partial\overline{\mathfrak{PO}}^{\text{\bf
0}}_{k}}{\partial
x_{k,j}}(\mathfrak{x}_{1,1},\cdots,\mathfrak{x}_{K,d(K)})\right)_{j=1,\cdots,d(k)}.$
(9.23)

We remark that $\nabla\overline{\mathfrak{PO}}^{\text{\bf 0}}_{k}$ depends
only on ${\mathbb{R}}^{d(1)}\times\cdots\times{\mathbb{R}}^{d(k)}$ components.

Combining all $\nabla\overline{\mathfrak{PO}}^{\text{\bf 0}}_{k}$,
$k=1,\cdots,K$ (9.23) induces a map

$\nabla\overline{\mathfrak{PO}}^{\text{\bf
0}}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$

defined by

$\nabla\overline{\mathfrak{PO}}^{\text{\bf
0}}=(\nabla\overline{\mathfrak{PO}}^{\text{\bf
0}}_{1},\cdots,\nabla\overline{\mathfrak{PO}}^{\text{\bf 0}}_{K}).$

The next lemma is closely related to Lemma 9.15.

######  Lemma 9.16.

We may choose the positive numbers $R(k)$ for $k=1,\cdots,K$ such that the
following holds :

  1. (1)

$\nabla\overline{\mathfrak{PO}}^{\text{\bf 0}}$ is nonzero on
$\partial(D(R(\cdot)))$.

  2. (2)

The map $:\partial(D(R(\cdot)))\to S^{n-1}$

$\mathfrak{x}\mapsto\frac{\nabla\overline{\mathfrak{PO}}^{\text{\bf
0}}}{\|\nabla\overline{\mathfrak{PO}}^{\text{\bf 0}}\|}$

has degree $1$.

###### Proof.

We first prove the following sublemma by an upward induction on $k_{0}$.

######  Sublemma 9.17.

There exist $R(k)$’s for $1\leq k\leq K$ such that for any given $1\leq
k_{0}\leq K$ we have

$\sum_{j=1}^{d(k_{0})}x_{k_{0},j}\frac{\partial\overline{\mathfrak{PO}}^{\text{\bf
0}}_{k_{0}}}{\partial x_{k_{0},j}}(x_{1,1},\cdots,x_{k_{0},d(k_{0})})>0$
(9.24)

if $(x_{k,1},\cdots,x_{k,d(k)})\in D(R(k))$ for all $1\leq k\leq k_{0}-1$ and
$(x_{k_{0},1},\cdots,x_{k_{0},d(k_{0})})\in\partial D(R(k_{0}))$.

###### Proof.

In case $k_{0}=1$ the existence of $R(1)$ satisfying (9.24) is a consequence
of Corollary 9.9. We assume that the sublemma is proved for
$1,\cdots,k_{0}-1$.

For each fixed $\text{\bf x}=(x_{1,1},\cdots,x_{k_{0}-1,d(k_{0}-1)})$ we can
find $R(k_{0})_{\text{\bf x}}$ such that (9.24) holds for
$(x_{k_{0},1},\cdots,x_{k_{0},d(k_{0})})\in{\mathbb{R}}^{d(k_{0})}\setminus
D(R(k_{0})_{\text{\bf x}}/2)$. This is also a consequence of Corollary 9.9.

We take supremum of $R(k_{0})_{\text{\bf x}}$ over the compact set $\text{\bf
x}\in\prod_{k=1}^{k_{0}-1}D(R(k))$ and obtain $R(k_{0})$. The proof of
Sublemma 9.17 is complete. ∎

It is easy to see that Lemma 9.16 follows from Sublemma 9.17. ∎

We now use our assumption that the vertices of $P$ lies in
$M_{\mathbb{Q}}={\mathbb{Q}}^{n}$ and that $\rho_{j}\in{\mathbb{Q}}$.
Replacing $T$ by $T^{1/\mathcal{C}}$ if necessary, we may assume that all the
exponents of $y_{k,j}$ and $T$ appearing in (9.20) are integers. Then

$\mathfrak{PO}^{\text{\bf 0}}_{k,{\mathcal{N}}}=\mathfrak{PO}^{\text{\bf
0}}_{k,{\mathcal{N}}}(y_{1,1},\cdots,y_{K,d(K)};T)$

are polynomials of $y_{k,j}$, $y_{k,j}^{-1}$ and $T$. Define the set
$\mathfrak{X}$ by the set consisting of

$(\mathfrak{y}_{1,1},\cdots,\mathfrak{y}_{K,d(K)};q)\in({\mathbb{R}}_{+})^{n}\times{\mathbb{R}}$

that satisfy

$\frac{\partial\mathfrak{PO}^{\text{\bf 0}}_{k,{\mathcal{N}}}}{\partial
y_{k,j}}(\mathfrak{y}_{1,1},\cdots,\mathfrak{y}_{K,d(K)};q)=0,$ (9.25)

for $k=1,\cdots,K$, $j=1,\cdots,d(k)$. Clearly $\mathfrak{X}$ is a real affine
algebraic variety. (Note the equation for $y_{i}$ are polynomials. So we need
to regard $y_{i}$ (not $x_{i}$) as variables to regard $\mathfrak{X}$ as a
real affine algebraic variety. )

Consider the projection

$\pi:\mathfrak{X}\to{\mathbb{R}},\quad\pi(\mathfrak{y}_{1,1},\cdots,\mathfrak{y}_{K,d(K)};q)=q$

which is a morphism of algebraic varieties.

######  Lemma 9.18.

There exists a sufficiently small $\epsilon>0$ such that if $|q|<\epsilon$
then

$\pi^{-1}(q)\cap\\{(e^{x_{1}},\cdots,e^{x_{n}})\mid(x_{1},\cdots,x_{n})\in
D(R(\cdot))\\}\neq\emptyset.$

###### Proof.

We consider the real analytic $q$-family of polynomials

$\mathfrak{PO}^{\text{\bf
0}}_{k,{\mathcal{N}},q}(y_{1,1},\cdots,y_{K,d(K)})=\mathfrak{PO}^{\text{\bf
0}}_{k,{\mathcal{N}}}(y_{1,1},\cdots,y_{K,d(K)};q).$

Then

$\mathfrak{PO}^{\text{\bf
0}}_{k,{\mathcal{N}},0}=\overline{\mathfrak{PO}}^{\text{\bf
0}}_{k,{\mathcal{N}}}.$ (9.26)

Replacing $\overline{\mathfrak{PO}}^{\text{\bf 0}}_{k,{\mathcal{N}}}$ by
$\mathfrak{PO}^{\text{\bf 0}}_{k,{\mathcal{N}},q}$, we can repeat construction
of the map

$\nabla\mathfrak{PO}^{\text{\bf
0}}_{{\mathcal{N}},q}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$

for each fixed $q\in{\mathbb{R}}$ in the same way as we defined
$\nabla\overline{\mathfrak{PO}}^{\text{\bf 0}}$. Then the conclusion of Lemma
9.16 holds for $\nabla\mathfrak{PO}^{\text{\bf 0}}_{{\mathcal{N}},q}$ if $|q|$
is sufficiently small. (This is a consequence of Lemma 9.16 and (9.26).) Lemma
9.18 follows from elementary algebraic topology. ∎

Lemma 9.18 implies that we can find

$\mathfrak{y}_{0}=(\mathfrak{y}_{1,1;0},\cdots,\mathfrak{y}_{K,d(K);0})\in{\mathbb{R}}_{+}^{n}$

and a sequence

$(\mathfrak{y}_{h},q_{h})=(\mathfrak{y}^{h}_{1,1;0},\cdots,\mathfrak{y}^{h}_{K,d(K);0};q_{h})\in\mathfrak{X}\subset{\mathbb{R}}^{n+1}$

$h=1,2,\cdots$ such that $q_{h}>0$ and
$\lim_{h\to\infty}(\mathfrak{y}_{h},q_{h})=(\mathfrak{y}_{0},0).$ Therefore by
the curve selection lemma (Lemma 3.1 [Mi]) there exists a real analytic map

$\gamma:[0,\epsilon)\to\mathfrak{X}$

such that $\gamma(0)=(\mathfrak{y}_{0},0)$ and $\pi(\gamma(t))>0$ for $t>0$.
We reparameterize $\gamma(t)$, so that its $q$-component is $t^{a/b}$, where
$a$ and $b$ are relatively prime integers. We put $T=t^{a/b}$ i.e.,
$t=T^{b/a}$ and denote the $y_{k,j}$-components of $\gamma(t)$ by

$\mathfrak{y}_{k,j}=\mathfrak{y}_{k,j;0}+\sum_{\ell=1}^{\infty}\mathfrak{y}_{k,j;\ell}T^{b\ell/a}.$

Since $\gamma(t)\in\mathfrak{X}$, the element
$(\mathfrak{y}_{k,j})_{k,j}\in(\Lambda^{{\mathbb{R}}}_{0}\setminus\Lambda^{{\mathbb{R}}}_{+})^{n}$
is the required solution of (4.11).

Since $\mathfrak{PO}_{0}$ contains only a finite number of summands, we can
take $\mathfrak{PO}_{0,{\mathcal{N}}}=\mathfrak{PO}_{0}$. Therefore we can
find a solution of (4.12) for $\mathfrak{PO}_{0}$.

The proof of Proposition 4.7 is now complete. ∎

##  10\. Elimination of higher order term in nondegenerate cases

In this section, we prove a rather technical (but useful) result, which shows
that solutions of the leading term equation (9.19) correspond to actual
critical points under certain non-degeneracy condition. For this purpose, we
slightly modify the argument of the last part of section 9. This result will
be useful to determine $u\in\text{\rm Int}\,P$ such that
$HF((L(u);\mathfrak{x}),(L(u);\mathfrak{x});\Lambda_{0})\neq 0$ for some
$\mathfrak{x}$ in non-Fano cases. (In other words we study the image of
$\mathfrak{M}(\mathfrak{Lag}(X))\to\text{\rm Int}\,P$ by the map
$(\mathfrak{x},u)\mapsto u$.) In fact it shows that we can use
$\mathfrak{PO}_{0}$ in place of $\mathfrak{PO}$ in most practical cases. We
remark that we explicitly calculate $\mathfrak{PO}_{0}$ but do not know the
precise form of $\mathfrak{PO}$ in non-Fano cases.

In order to state the result in a general form, we prepare some notations. Let
$u_{0}\in\text{\rm Int}\,P$. (In section 9, $u_{0}$ is determined as the
unique element of $P_{K}$ defined in Proposition 9.1. The present situation is
more general.)

We define positive real numbers $S_{1}<S_{2}<\cdots$ by

$\\{\ell_{i}(u_{0})\mid
i=1,\cdots,m\\}=\\{S_{1},S_{2},\cdots,S_{m^{\prime}}\\}$ (10.1)

and the sets

$I_{k}=\\{\ell_{i}\mid\ell_{i}(u_{0})=S_{k}\\},$ (10.2)

for $k=1,\cdots$. We renumber each of $I_{k}$ so that

$\\{\ell_{k,j}\mid j=1,\cdots,a(k)\\}=I_{k}.$ (10.3)

######  Definition 10.1.

Let $A^{\perp}_{l}$ be the linear subspace of $N_{\mathbb{R}}$ spanned by
$d\ell_{k,j}$ $k\leq l$, $j\leq a(k)$. We define $K$ to be the smallest number
such that $A^{\perp}_{K}=N_{\mathbb{R}}$.

Note our notations here are consistent with one in section 9 in case
$\\{u_{0}\\}=P_{K}$. We define $\mathfrak{I}$ and $\mathfrak{I}_{k}$ by (9.11)
and Definition 9.10. Then Lemma 9.11 and (9.12) hold. We choose $\text{\bf
e}_{i,j}^{*}\in Hom(M_{\mathbb{Q}},{\mathbb{Q}})$ such that Condition 9.13 is
satisfied. (Note $A^{\perp}_{l}$ is defined over ${\mathbb{Q}}$.) $x_{i,j}$
and $y_{i,j}$ then are defined in the same way as section 9. We define
$Y(k,j)$ by (9.13) and $Y(\ell)$ by (9.14). Then (9.15) and Lemma 9.14 hold.

We remark that Corollary 9.9 does not hold in general in the current
situation. In fact we can write

$v=\sum_{l=1}^{k}\sum_{j=1}^{a(l)}c_{l,j}d\ell_{l,j}$

under the assumption of Corollary 9.9 but we may not be able to ensure
$c_{l,j}\geq 0$.

######  Definition 10.2.

  1. (1)

We call

$0=\sum_{j^{\prime}=1}^{a(k)}\frac{\partial Y(k,j^{\prime})}{\partial
y_{k,j}},\quad k=1,\cdots,K,\,\,j=1,\cdots,d(k)$

the leading term equation at $u_{0}$. We regard it as a polynomial equation
for $\mathfrak{y}_{k,j}\in{\mathbb{C}}\setminus\\{0\\}$, $k=1,\cdots,K$,
$j=1,\cdots,d(k)$.

  2. (2)

A solution
$\mathfrak{y}^{0}=(\mathfrak{y}_{k,j;0})_{k=1,\cdots,K,\,j=1,\cdots,d(k)}$ of
leading term equation is said to be weakly nondegenerate if it is isolated in
the set of solutions.

  3. (3)

A solution
$\mathfrak{y}^{0}=(\mathfrak{y}_{k,j;0})_{k=1,\cdots,K,\,j=1,\cdots,d(k)}$ of
leading term equation is said to be strongly nondegenerate if the matrices

$\left(\sum_{j^{\prime}=1}^{a(k)}\frac{\partial^{2}Y(k,j^{\prime})}{\partial
y_{k,j_{1}}\partial y_{k,j_{2}}}\right)_{j_{1},j_{2}=1,\cdots,a(k)}$

are invertible for $k=1,\cdots,K$, at $\mathfrak{y}^{0}$.

  4. (4)

We define the multiplicity of leading term equation in the standard way of
algebraic geometry, in the weakly nondegenerate case.

######  Example 10.3.

In Example 9.6, the equation (9.6) is the leading term equation.

Let $\mathfrak{PO}^{u_{0}}_{*}$ be either $\mathfrak{PO}^{u_{0}}_{0}$ or
$\mathfrak{PO}^{u_{0}}$.

######  Theorem 10.4.

For any strongly nondegenerate solution
$\mathfrak{y}^{0}=(\mathfrak{y}_{k,j;0})$ of leading term equation, there
exists a solution
$\mathfrak{y}=(\mathfrak{y}_{k,j})\in(\Lambda^{{\mathbb{C}}}_{0}\setminus\Lambda^{{\mathbb{C}}}_{+})^{n}$
of

$\frac{\partial\mathfrak{PO}^{u_{0}}_{*}}{\partial y_{k,j}}(\mathfrak{y})=0$
(10.4)

such that
$\mathfrak{y}_{k,j}\equiv\mathfrak{y}_{k,j;0}\mod\Lambda^{{\mathbb{C}}}_{+}.$

If all the vertices of $P$ and $u_{0}$ are rational, the same conclusion holds
for weakly nondegenerate $\mathfrak{y}^{0}$.

We prove the following at the end of section 13.

######  Lemma 10.5.

We assume $[\omega]\in H^{2}(X;{\mathbb{Q}})$ and choose the moment polytope
$P$ such that its vertices are all rational. Let $u_{0}\in\text{\rm Int}P$
such that $\mathfrak{PO}_{0}^{u_{0}}$ has weakly nondegenerate critical point
in $(\Lambda_{0}\setminus\Lambda_{+})^{n}$. Then $u_{0}$ is rational.

The following corollary is an immediate consequence of Theorem 10.4 and Lemma
10.5.

######  Corollary 10.6.

Let $(\mathfrak{x},u)\in\mathfrak{M}_{0}(\mathfrak{Lag}(X))$ and
$u\in\text{\rm Int}\,P$. Assume one of the following conditions :

  1. (1)

The corresponding solution of the leading term equation is strongly
nondegenerate.

  2. (2)

$[\omega]\in H^{2}(X;{\mathbb{R}})$ is rational and the corresponding solution
of leading term equation is weakly nondegenerate.

Then there exists $\mathfrak{x}^{\prime}$ such that
$(\mathfrak{x}^{\prime},u)\in\mathfrak{M}(\mathfrak{Lag}(X))$ and
$\mathfrak{x}^{\prime}\equiv\mathfrak{x}\mod\Lambda^{{\mathbb{C}}}_{+}$.

######  Remark 10.7.

  1. (1)

Using Proposition 10.8 below, we can also apply Theorem 10.4 and Corollary
10.6 to study non-displacement of Lagrangian fibers for the weakly
nondegenerate case, without assuming rationality. See the last step of the
proof of Theorem 1.5 given at the end of section 13.

  2. (2)

The conclusion of Theorem 10.4 does not hold in general without weakly
nondegenerate assumption. We give an example (Example 10.17) where both the
assumption of weak nondegeneracy and the conclusion of Theorem 10.4 fail to
hold.

  3. (3)

In this section we work with $\Lambda^{{\mathbb{C}}}$ coefficients, while in
the last section we work with $\Lambda^{{\mathbb{R}}}$ coefficients. We also
remark that in the last section, we did not assume the weak nondegeneracy
condition.

  4. (4)

If we define the multiplicity of the element of
$\mathfrak{M}_{0}(\mathfrak{Lag}(X))$ as the dimension of the Jacobian ring
$Jac(\mathfrak{PO}_{0};\mathfrak{x},u_{0})$ in Definition 7.9 (namely as the
Milnor number) then the sum of the multiplicities of the solutions of (10.4)
converging to $\mathfrak{y}^{0}$ as $T\to 0$, is equal to the multiplicity of
$\mathfrak{y}^{0}$.

  5. (5)

In the strongly nondegenerate case, the solution of (10.4) with the given
leading term is unique.

######  Proposition 10.8.

Let $(X,\omega)$ be a compact toric manifold with moment polytope $P$ and
$u_{0}\in\text{\rm Int}\,P$. Then there exist $(X,\omega^{h})$ with moment
polytope $P^{h}$ and $u_{0}^{h}\in\text{\rm Int}\,P^{h}$ such that the
following holds:

  1. (1)

$\lim_{h\to\infty}\omega^{h}=\omega$. $\lim_{h\to\infty}u^{h}_{0}=u_{0}$.

  2. (2)

The vertices of $P^{h}$ and $u_{0}^{h}$ are rational.

  3. (3)

The leading term equation at $u_{0}^{h}$ is the same as the leading term
equation at $u_{0}$.

We prove Proposition 10.8 at the end of section 13.

We first derive Theorem 1.14 and Lemma 8.3 from Theorem 10.4 before proving
Theorem 10.4.

###### Proof of Theorem 1.14.

We start with ${\mathbb{C}}P^{2}$ and blow up a $T^{2}$ fixed point to obtain
${\mathbb{C}}P^{2}\\#(-{\mathbb{C}}P^{2})$. We take a Kähler form so that the
volume of the exceptional ${\mathbb{C}}P^{1}$ is $2\pi\epsilon_{1}$ which is
small. We next blow up again at one of the fixed points so that the volume of
the exceptional ${\mathbb{C}}P^{1}$ is $\epsilon_{2}$ and is small compared
with $\epsilon_{1}$. We continue $k$ times to obtain $X(k)$, whose Kähler
structure depends on $\epsilon_{1},\cdots,\epsilon_{k}$. Note $X(k)$ is non-
Fano for $k>3$.

Let $P(k)$ be the moment polytope of $X(k)$ and $\mathfrak{PO}_{0,k}$ be the
leading order potential function of $X(k)$. We remark that $P(k)$ is obtained
by cutting out a vertex of $P(k-1)$. (See [Ful].)

######  Lemma 10.9.

We may choose $\epsilon_{i}$ ($i=1,\cdots,k$) so that the following holds for
$l\leq k$.

  1. (1)

The number of balanced fibers of $P(l)$ is $l+1$. We write them as
$L(u^{(l,i)})$ $i=0,\cdots,l$.

  2. (2)

$u^{(l-1,i)}=u^{(l,i)}$ for $i\leq l-1$. $u^{(l,0)}=(1/3,1/3)$.

  3. (3)

$u^{(l,l)}$ is in an $o(\epsilon_{l})$ neighborhood of the vertices
corresponding to the point we blow up.

  4. (4)

The leading term equation of $\mathfrak{PO}_{0,l-1}$ at $u^{(l-1,i)}$ is the
same as the leading term equation of $\mathfrak{PO}_{0,l}$ at $u^{(l,i)}$ for
$i\leq l-1$.

  5. (5)

The leading term equations are all strongly nondegenerate.

###### Proof.

The proof is by induction on $k$. There is nothing to show for $k=0$. Suppose
that we have proved Lemma 10.9 up to $k-1$. Let $w$ be the vertex of the
polytope we cut out which corresponds to the blow up of $X(k-1)$. Let
$\ell_{i}$, $\ell_{i^{\prime}}$ be the affine functions associated to the two
edges containing $w$. It is easy to see that

$P(k)=\\{u\in P(k-1)\mid\ell_{i}(u)+\ell_{i^{\prime}}(u)\geq\epsilon_{k}\\}.$

We also have :

$\mathfrak{PO}_{0,k}=\mathfrak{PO}_{0,k-1}+T^{\ell_{i}(u)+\ell_{i^{\prime}}(u)-\epsilon_{k}}y_{1}^{v_{i,1}+v_{i^{\prime},1}}y_{2}^{v_{i,2}+v_{i^{\prime},2}}.$

Therefore if we choose $\epsilon_{k}$ sufficiently small, the leading term
equation at $u^{(k-1,i)}$ does not change.

We take $u^{(k,k)}$ such that

$\ell_{i}(u^{(k,k)})=\ell_{i^{\prime}}(u^{(k,k)})=\epsilon_{k}.$

It is easy to see that there exists such $u^{(k,k)}$ uniquely if
$\epsilon_{k}$ is sufficiently small. We put

$y^{\prime}_{1}=y_{1}^{v_{i,1}}y_{2}^{v_{i,2}},\quad
y^{\prime}_{2}=y_{1}^{v_{i^{\prime},1}}y_{2}^{v_{i^{\prime},2}}.$

(We remark that $v_{i}$ and $v_{i^{\prime}}$ are ${\mathbb{Z}}$ basis of
${\mathbb{Z}}^{2}$, since $X(k-1)$ is smooth toric.) Then we have

$\mathfrak{PO}^{u^{(k,k)}}_{0,k}\equiv(y_{1}^{\prime}+y^{\prime}_{2}+y^{\prime}_{1}y^{\prime}_{2})T^{\epsilon_{k}}\mod
T^{\epsilon_{k}}\Lambda_{+}.$

Therefore the leading term equation is

$1+y^{\prime}_{1}=1+y^{\prime}_{2}=0$

and hence has a unique solution $(-1,-1)$. In particular it is strongly
nondegenerate. We can also easily check that there is no other solution of
leading term equation. The proof of Lemma 10.9 now follows by Theorem 10.4. ∎

Theorem 1.14 immediately follows from Lemma 10.9. ∎

Note that Theorem 1.14 can be generalized to ${\mathbb{C}}P^{n}$ by the same
proof.

###### Proof of Lemma 8.3.

Let $u_{0}=(n(\alpha+1)/4,(1-\alpha)/2)$. We calculate

$\mathfrak{PO}_{0}^{u_{0}}=(y_{2}+y_{2}^{-1})T^{(1-\alpha)/2}+(y_{1}+y_{1}^{-1}y_{2}^{-1})T^{n(\alpha+1)/4}.$

The leading equation is

$1-y^{-2}_{2}=0,\quad 1-y_{1}^{-2}y_{2}^{-1}=0.$

Its solution are $(1,1),(-1,1),(\sqrt{-1},-1),(-\sqrt{-1},-1)$, all of which
are strongly nondegenerate. The lemma them follows from Corollary 10.6. ∎

We give another example which demonstrates the way how one can use the leading
term equation and Theorem 10.4 to locate balanced fibers.

######  Example 10.10.

Let us consider ${\mathbb{C}}P^{n}$ with moment polytope
$P=\\{(u_{1},\cdots,u_{n})\mid u_{i}\geq 0,\sum u_{i}\leq 1\\}$. We take
${\mathbb{C}}P^{n-\ell}\subset{\mathbb{C}}P^{n}$ corresponding to
$u_{1}=\cdots=u_{\ell}=0$. ($\ell\geq 2$.) We blow up ${\mathbb{C}}P^{n}$
along the center ${\mathbb{C}}P^{n-\ell}$ and denote the blow-up by $X$. (The
case $\ell=n=2$ is Example 8.1.) We take $\alpha\in(0,1)$ so that the moment
polytope of $X$ is

$P_{\alpha}=\left\\{(u_{1},\cdots,u_{n})\in
P~{}\Big{|}~{}\sum_{i=1}^{\ell}u_{i}\geq\alpha\right\\}.$

Below we use $\mathfrak{PO}_{0}$ in place of the potential function
$\mathfrak{PO}$. Since the all the critical points of $\mathfrak{PO}_{0}$ are
weakly nondegenerate, they correspond to the critical points of
$\mathfrak{PO}$. (We thank D. McDuff for pointing out that this example is not
Fano.) The function $\mathfrak{PO}_{0}$ is given by

$\mathfrak{PO}_{0}=\sum_{i=1}^{n}T^{u_{i}}y_{i}+T^{1-\sum_{i=1}^{n}u_{i}}(y_{1}\cdots
y_{n})^{-1}+T^{\sum_{i=1}^{\ell}u_{i}-\alpha}y_{1}\cdots y_{\ell}.$

We denote

$z_{i}=T^{u_{i}}y_{i},\quad z_{0}=T^{1-\sum_{i=1}^{n}u_{i}}(y_{1}\cdots
y_{n})^{-1},\quad z=T^{\sum_{i=1}^{\ell}u_{i}-\alpha}y_{1}\cdots y_{\ell}.$

Then the quantum Stanley-Reisner relations are :

$z_{1}\cdots z_{n}z_{0}=T,\,\,\,z_{1}\cdots z_{\ell}=zT^{\alpha}.$

By computing the derivatives $y_{i}\frac{\partial\mathfrak{PO}_{0}}{\partial
y_{i}}$, we obtain the linear relations which can be written as

$\displaystyle z_{i}-z_{0}+z$ $\displaystyle=$ $\displaystyle
0\,\quad\mbox{for }\,i\leq\ell$ $\displaystyle z_{i}-z_{0}$ $\displaystyle=$
$\displaystyle 0\,\quad\mbox{for }\,i>\ell.$

Putting $X=z_{0}-z$, $Y=z$, we obtain

$z_{i}=\begin{cases}X&\quad\mbox{for $1\leq i\leq\ell$}\\\ X+Y&\quad\mbox{for
$\,i>\ell$ or $i=0$}.\end{cases}$

We also have $X^{\ell}=YT^{\alpha}$ and

$X^{n+1}(X^{\ell-1}T^{-\alpha}+1)^{n-\ell+1}=T.$ (10.5)

Case 1; $(\ell-1)\mathfrak{v}_{T}(X)<\alpha$ : In this case, we obtain

$\mathfrak{v}_{T}(X)=\frac{1+\alpha(n-\ell+1)}{n+1+(n-\ell+1)(\ell-1)}.$

The condition $(\ell-1)\mathfrak{v}_{T}(X)<\alpha$ then is equivalent to

$\alpha>\frac{\ell-1}{n+1}.$

And we have

$\mathfrak{v}_{T}(X+Y)=\mathfrak{v}_{T}(Y)=\ell\mathfrak{v}_{T}(X)-\alpha<\mathfrak{v}_{T}(X)$

for $i>\ell$. At the point $\text{\bf u}=(u_{1},\cdots,u_{n})$ with
$u_{i}=\mathfrak{v}_{T}(X)$ for $1\leq i\leq\ell$ and
$u_{i}=\mathfrak{v}_{T}(X+Y)$ for $i\geq\ell$, we have

$\mathfrak{PO}_{0}^{\text{\bf
u}}=T^{\mathfrak{v}_{T}(Y)}(y_{\ell+1}+\cdots+y_{n}+(y_{1}\cdots
y_{n})^{-1}+y_{1}\cdots
y_{\ell})+T^{\mathfrak{v}_{T}(X)}(y_{1}+\cdots+y_{\ell}).$

To obtain the leading term equation, it is useful to make a change of
variables from $y_{1},\cdots,y_{n}$ to $y_{2},\cdots,y_{n},y$ with
$y=y_{1}\cdots y_{\ell}$.

In fact, using the notation which we introduced at the beginning of this
section, we have

$S_{1}=\mathfrak{v}_{T}(Y),\quad S_{2}=\mathfrak{v}_{T}(X),$

and $A^{\perp}_{1}$ is the vector space generated by $\partial/\partial
u_{i}$, $(i=\ell+1,\cdots,n)$ and $\partial/\partial
u_{1}+\cdots+\partial/\partial u_{\ell}$. We also have
$A^{\perp}_{2}={\mathbb{R}}^{n}$. Therefore a basis satisfying Condition 9.13
is

$\frac{\partial}{\partial u_{2}},\quad\cdots,\quad\frac{\partial}{\partial
u_{n}},\quad\frac{\partial}{\partial u_{1}}+\cdots+\frac{\partial}{\partial
u_{\ell}}.$

The variables corresponding to this basis is $y_{2},\cdots,y_{n},y$. In these
variables, $\mathfrak{PO}_{0}^{\text{\bf u}}$ has the form

$\displaystyle\mathfrak{PO}_{0}^{\text{\bf u}}$ $\displaystyle=$
$\displaystyle
T^{\mathfrak{v}_{T}(Y)}(y_{\ell+1}+\cdots+y_{n}+(yy_{\ell+1}\cdots
y_{n})^{-1}+y)$ $\displaystyle\quad+T^{\mathfrak{v}_{T}(X)}(y(y_{2}\cdots
y_{\ell})^{-1}+y_{2}+\cdots+y_{\ell}).$

Then the leading term equation becomes :

$\begin{cases}0=1-y^{-1}(yy_{\ell+1}\cdots y_{n})^{-1}\\\
0=1-y_{i}^{-1}(yy_{\ell+1}\cdots y_{n})^{-1}&\mbox{for }i>\ell\\\
0=1-y_{i}^{-1}y(y_{2}\cdots y_{\ell})^{-1}\,&\mbox{for }\,2\leq
i\leq\ell.\end{cases}$

Its solutions are

$\displaystyle y_{\ell+1}=\cdots=y_{n}=y=\theta,$
$\displaystyle\qquad\theta^{n-\ell+2}=1$ $\displaystyle
y_{2}=\cdots=y_{\ell}=\rho,$ $\displaystyle\qquad\rho^{\ell}=\theta.$

It follows that this leading term equation has $\ell(n-\ell+2)$ solutions, all
of which are strongly nondegenerate.

Case 2; $(\ell-1)\mathfrak{v}_{T}(X)>\alpha$ : We have

$\mathfrak{v}_{T}(X)=\frac{1}{n+1}.$

We also have
$\mathfrak{v}_{T}(Y)=\ell\mathfrak{v}_{T}(X)-\alpha>\mathfrak{v}_{T}(X)$ and
hence

$\mathfrak{v}_{T}(X+Y)=\mathfrak{v}_{T}(X)=\frac{1}{n+1}.$

The condition $(\ell-1)\mathfrak{v}_{T}(X)>\alpha$ becomes
$\alpha<\frac{\ell-1}{n+1}.$ If we consider the point $\text{\bf
u}=(u_{1},\cdots,u_{n})$ with $u_{i}=\mathfrak{v}_{T}(X)=\frac{1}{n+1}$ for
$i=1,\cdots n$, $\mathfrak{PO}_{0}^{\text{\bf u}}$ has the form

$\mathfrak{PO}_{0}^{\text{\bf u}}=T^{1/n+1}(y_{1}+\cdots+y_{n}+(y_{1}\cdots
y_{n})^{-1})+T^{\frac{\ell}{n+1}-\alpha}y_{1}\cdots y_{\ell}$

and so the leading term equation is obtained by differentiating

$y_{1}+\cdots+y_{n}+(y_{1}\cdots y_{n})^{-1}.$

Its solutions are

$y_{1}=\cdots=y_{n}=\theta,\qquad\theta^{n+1}=1.$

There are $n+1$ solutions all of which are strongly nondegenerate.

Case 3; $(\ell-1)\mathfrak{v}_{T}(X)=\alpha$ :

(Case 3-1; $-X^{\ell-1}\not\equiv T^{\alpha}\mod T^{\alpha}\Lambda_{+}$.) We
have $u_{i}=\mathfrak{v}_{T}(X)=\mathfrak{v}_{T}(Y)=1/(n+1)$ ($i=1,\cdots,n$),
$\alpha=(\ell-1)/(n+1)$. In this case

$\mathfrak{PO}_{0}^{\text{\bf u}}=T^{1/(n+1)}(y_{1}+\cdots+y_{n}+(y_{1}\cdots
y_{n})^{-1}+y_{1}\cdots y_{\ell}).$

This formula implies that the symplectic area of all discs with Maslov index
$2$ are $2\pi/(n+1)$.

The leading term equation is

$\begin{cases}1-\frac{1}{y_{i}}((y_{1}\cdots y_{n})^{-1}-y_{1}\cdots
y_{\ell})=0,&i=1,\cdots,\ell\\\ 1-\frac{1}{y_{i}}((y_{1}\cdots
y_{n})^{-1})=0,&i=\ell+1,\cdots,n.\end{cases}$

Its solutions are $y_{1}=\cdots=y_{\ell}=\rho$,
$y_{\ell+1}=\cdots=y_{n}=\theta$ with

$\rho^{\ell}(\rho+\rho^{\ell})^{n-\ell+1}=1,$ (10.6)

and $\theta=\rho+\rho^{\ell}$. It looks rather cumbersome to check whether
(10.6) has multiple root. Certainly all the solutions are weakly
nondegenerate. The number of solutions counted with multiplicity is
$\ell(n-\ell+2)$.

(Case 3-2; $-X^{\ell-1}\equiv T^{\alpha}\mod T^{\alpha}\Lambda_{+}$.) We have

$z_{0}=X+Y=T^{-\alpha}X(T^{\alpha}+X^{\ell-1}).$

We put

$\mathfrak{v}_{T}(z_{0})=\lambda>\mathfrak{v}_{T}(X)=\frac{\alpha}{\ell-1}.$

Using (10.5) we obtain

$\lambda=\frac{\ell-1-\ell\alpha}{(\ell-1)(n-\ell+1)}.$

The condition $\lambda>\frac{\alpha}{\ell-1}$ becomes
$\alpha<\frac{\ell-1}{n+1}$. We have
$u_{i}=\mathfrak{v}_{T}(X)=\alpha/(\ell-1)$, $i\leq\ell$ and
$u_{i}=\mathfrak{v}_{T}(Y)=\lambda$, $i>\ell$. We have

$\mathfrak{PO}_{0}^{\text{\bf
u}}=T^{\mathfrak{v}_{T}(X)}(y_{1}+\cdots+y_{\ell}+y_{1}\cdots
y_{\ell})+T^{\mathfrak{v}_{T}(Y)}(y_{\ell+1}+\cdots+y_{n}+(y_{1}\cdots
y_{n})^{-1}).$

The first term gives the leading term equation :

$1+y_{1}\cdots\widehat{y}_{i}\cdots y_{\ell}=0,\qquad i=1,\cdots,\ell.$

It solution is $y_{1}=\cdots=y_{\ell}=\rho$ and $\rho^{\ell-1}=-1$. The second
term of $\mathfrak{PO}_{0}^{\text{\bf u}}$ gives the leading term equation

$1-y_{i}^{-1}\rho^{-\ell}(y_{\ell+1}\cdots y_{n})^{-1}=0,\qquad
i=\ell+1,\cdots,n.$

Its solutions are $y_{\ell+1}=\cdots y_{n}=\theta,$ with

$\rho^{\ell}\theta^{n-\ell+1}=1.$

Thus the leading term equation has $(\ell-1)(n+1-\ell)$ solutions all of which
are strongly nondegenerate.

We remark that $(\ell-1)(n+1-\ell)+(n+1)=\ell(n-\ell+2)$. Hence the number of
solutions are always $\ell(n-\ell+2)$ which coincides with the Betti number of
$X$. There are $2$ balanced fibers for $\alpha<(\ell-1)/(n+1)$ and $1$
balanced fiber for $\alpha\geq(\ell-1)/(n+1)$.

By the above discussion and Theorem 6.1 (see also Remark 6.14), we can
calculate $QH(X;\Lambda^{{\mathbb{Q}}})$ as follows :

$QH(X;\Lambda^{{\mathbb{Q}}})=\begin{cases}\Lambda^{R_{1}}&\alpha>(\ell-1)/(n+1),\\\
\Lambda^{R_{2}}&\alpha=(\ell-1)/(n+1),\\\
\Lambda^{R_{3}}\times\Lambda^{R_{4}}&\alpha<(\ell-1)/(n+1),\end{cases}$ (10.7)

where

$\displaystyle R_{1}={\mathbb{Q}}[Z]/(Z^{\ell(n-\ell+2)}-1),$ $\displaystyle
R_{2}={\mathbb{Q}}[Z]/(Z^{\ell}(Z+Z^{\ell})^{n-\ell+1}-1),$ $\displaystyle
R_{3}={\mathbb{Q}}[Z]/(Z^{n+1}-1),$ $\displaystyle
R_{4}={\mathbb{Q}}[Z_{1},Z_{2}]/(Z_{1}^{\ell-1}+1,Z_{1}^{\ell}Z_{2}^{n-\ell+1}-1).$

Here we assume (10.6) has only a simple root in case $\alpha=(\ell-1)/(n+1)$.
We use the next lemma to show (10.7).

######  Lemma 10.11.

Let $\mathfrak{x}=\sum\mathfrak{x}_{i}\text{\bf e}_{i}$ be a critical point of
$\mathfrak{PO}^{\text{\bf u}}_{*}$. We assume
$\mathfrak{y}_{i;0}=e^{\mathfrak{x}_{i;0}}$ (where
$\mathfrak{x}_{i}=\mathfrak{x}_{i;0}\mod\Lambda_{+}$) is a strongly
nondegenerate solution of leading term equation. We also assume that
$\mathfrak{y}_{i;0}\in F$ where $F\subset{\mathbb{C}}$ is a field.

Then $\mathfrak{y}_{i}=e^{\mathfrak{x}_{i}}$ is an element of
$\Lambda_{0}^{F}$.

We prove Lemma 10.11 later this section.

We are now ready to give the proof of Theorem 10.4.

###### Proof of Theorem 10.4.

We first consider the weakly nondegenerate case. Let $\mathfrak{m}$ be the
multiplicity of $\mathfrak{y}^{0}$. We choose $\delta$ such that the ball
$B_{\delta}(\mathfrak{y}^{0})$ centered at $\mathfrak{y}^{0}$ and of radius
$\delta$ does not contain a solution of the leading term equation other than
$\mathfrak{y}^{0}$. For $y\in\partial B_{\delta}(\mathfrak{y}^{0})$ we define

$\nabla\overline{\mathfrak{PO}}(y)=\left(\sum_{j^{\prime}=1}^{a(k)}\frac{\partial
Y(k,j^{\prime})}{\partial
y_{k,j}}(y)\right)_{k=1,\cdots,K,\,j=1,\cdots,d(k)}\in{\mathbb{C}}^{n}.$

The map

$y\mapsto\frac{\nabla\overline{\mathfrak{PO}}(y)}{\|\nabla\overline{\mathfrak{PO}}(y)\|}\in
S^{2n-1}$

is well defined and of degree $\mathfrak{m}\neq 0$ by the definition of
multiplicity.

We define $\mathfrak{PO}^{u_{0}}_{*,k,{\mathcal{N}}}$ in the same way as
(9.20). For $q\in{\mathbb{C}}$, we define
$\mathfrak{PO}^{u_{0}}_{*,k,{\mathcal{N}}}(\cdots;q)$ by substituting $q$ to
$T$. Then in the same way as the proof of Lemma 9.18 we can prove the
following.

######  Lemma 10.12.

There exists $\epsilon>0$ such that if $|q|<\epsilon$, the equation

$0=\frac{\partial}{\partial
y_{k,j}}\mathfrak{PO}^{u_{0}}_{*,k,{\mathcal{N}}}(\cdots;q)$ (10.8)

has a solution in $B_{\delta}(\mathfrak{y}^{0})$. The sum of multiplicities of
the solutions of $(\ref{eq:orderN})$ converging to $\mathfrak{y}_{k,j;0}$ is
$\mathfrak{m}$.

(10.8) is a polynomial equation. Hence multiplicity of its solution is well-
defined in the standard sense of algebraic geometry.

Now we assume that all the vertices of $P$ and $u_{0}$ are contained in
${\mathbb{Q}}^{n}$. (10.8) also depends polynomially on
$q^{\prime}=T^{\prime}$, where $T^{\prime}=T^{1/\mathcal{C}!}$ for a
sufficiently large integer $\mathcal{C}$. (We remark that $\mathcal{C}$ is
determined by the denominators of the coordinates of the vertices of $P$ and
of $u_{0}$. In particular it can be taken to be independent of
${\mathcal{N}}$.)

We denote $y=(y_{1},\cdots,y_{n})$ and put

$\mathfrak{X}=\\{(y,q^{\prime})\mid y\in
B_{\delta}(\mathfrak{y}^{0}),\,q^{\prime}\,\text{with $|q^{\prime}|<\epsilon$
and $q=(q^{\prime})^{\mathcal{C}!}$ satisfying (\ref{eq:orderN})}\\}.$

We consider the projection

$\pi_{q^{\prime}}:\mathfrak{X}\to\\{q^{\prime}\in{\mathbb{C}}\mid|q^{\prime}|<\epsilon\\}.$
(10.9)

By choosing a sufficiently small $\epsilon>0$, we may assume that (10.9) is a
local isomorphism on the punctured disc $\\{q^{\prime}\mid
0<|q^{\prime}|<\epsilon\\}$. Namely, $\pi_{q^{\prime}}$ is an étale covering
over the punctured disc.

We remark that for each $q^{\prime}$ the fiber consists of at most
$\mathfrak{m}$ points, since the multiplicity of the leading term equation is
$\mathfrak{m}$. We put $q^{\prime\prime}=(q^{\prime})^{1/\mathfrak{m}!}$. Then
the pull-back

$\pi_{q^{\prime\prime}}:\mathfrak{X}^{\prime}\to\\{q^{\prime\prime}\in{\mathbb{C}}\mid
0<|q^{\prime\prime}|<\epsilon\\}$ (10.10)

of (10.9) becomes a trivial covering space. Namely there exists a single
valued section of $\pi_{q^{\prime\prime}}$ on $\\{q^{\prime\prime}\mid
0<|q^{\prime\prime}|<\epsilon\\}$. It extends to a holomorphic section of
$\\{q^{\prime\prime}\mid|q^{\prime\prime}|<\epsilon\\}$.

In other words there exists a holomorphic family of solutions of (10.8) which
is parameterized by
$q^{\prime\prime}\in\\{q^{\prime\prime}\mid|q^{\prime\prime}|<\epsilon\\}$. We
put $T^{\prime\prime}=(T^{\prime})^{1/\mathfrak{m}!}$. Then by taking the
Taylor series of the $q^{\prime\prime}$-parameterized family of solutions at
$0$, we obtain the following :

######  Lemma 10.13.

If all the vertices of $P$ and $u_{0}$ are rational, then for each
${\mathcal{N}}$ there exists
$\mathfrak{y}^{({\mathcal{N}})}=(\mathfrak{y}^{({\mathcal{N}})}_{k,j})$

$\mathfrak{y}^{({\mathcal{N}})}_{k,j}=\sum_{l=0}^{{\mathcal{N}}}\mathfrak{y}^{({\mathcal{N}})}_{k,j;l}(T^{\prime\prime})^{l}$

$(\mathfrak{y}^{({\mathcal{N}})}_{k,j;l}\in{\mathbb{C}})$ such that

$\frac{\partial\mathfrak{PO}^{u_{0}}_{*}}{\partial
y_{k,j}}(\mathfrak{y}^{({\mathcal{N}})}_{k,j})\equiv
0\mod(T^{\prime\prime})^{{\mathcal{N}}+1}$ (10.11)

and that $\mathfrak{y}^{({\mathcal{N}})}_{k,j;0}\equiv\mathfrak{y}_{k,j;0}.$

We remark that Lemma 10.13 is sufficient for most of the applications. In fact
it implies that $L(u_{0})$ is balanced if there exists a weakly nondegenerate
solution of leading term equation at $u_{0}$. Hence we can apply Lemma 4.12.

For completeness we prove the slightly stronger statement made for the weakly
nondegenerate case in Theorem 10.4. The argument is similar to one in
subsection 7.2.11 [FOOO3] (= subsection 30.11 [FOOO2]).

For each ${\mathcal{N}}$, we denote by
$\widetilde{\mathfrak{M}}((\mathfrak{y}_{k,j;0});{\mathcal{N}})$ the set of
all
$(\mathfrak{y}_{k,j;l}^{({\mathcal{N}})})_{k,j;l}\in{\mathbb{C}}^{n{\mathcal{N}}}$,
where $k=1,\cdots,K$, $j=1,\cdots,a(k)$, $l=1,\cdots,{\mathcal{N}}$, such that

$\mathfrak{y}^{({\mathcal{N}})}_{k,j}=\mathfrak{y}_{k,j;0}+\sum_{l=1}^{{\mathcal{N}}}\mathfrak{y}^{({\mathcal{N}})}_{k,j;l}(T^{\prime\prime})^{l}$

satisfies (10.11).

By definition,
$\widetilde{\mathfrak{M}}((\mathfrak{y}_{k,j;0});{\mathcal{N}})$ is the set of
${\mathbb{C}}$-valued points of certain complex affine algebraic variety (of
finite dimension). Lemma 10.13 implies that
$\widetilde{\mathfrak{M}}((\mathfrak{y}_{k,j;0});{\mathcal{N}})$ is nonempty.
For ${\mathcal{N}}_{1}>{\mathcal{N}}_{2}$ there exists an obvious morphism

$I_{{\mathcal{N}}_{1},{\mathcal{N}}_{2}}:\widetilde{\mathfrak{M}}((\mathfrak{y}_{k,j;0});{\mathcal{N}}_{1})\to\widetilde{\mathfrak{M}}((\mathfrak{y}_{k,j;0});{\mathcal{N}}_{2})$

of complex algebraic variety.

To complete the proof of Theorem 10.4 in the weakly nondegenerate case, it
suffices to show that the projective limit

$\lim_{\longleftarrow}(\widetilde{\mathfrak{M}}(\mathfrak{y}_{k,j;0};{\mathcal{N}}))$
(10.12)

is nonempty.

######  Lemma 10.14.

$\bigcap_{{\mathcal{N}}>1}\text{\rm Im}I_{{\mathcal{N}},1}\neq\emptyset.$

###### Proof.

By a theorem of Chevalley (see Chapter 6 [Mat]), each $\text{\rm
Im}I_{{\mathcal{N}},1}$ is a constructible set. It is nonempty and its
dimension $\mathop{\kern 0.0pt{\rm dim}}\nolimits\text{\rm
Im}I_{{\mathcal{N}},1}$ is nonincreasing as ${\mathcal{N}}\to\infty$.
Therefore we may assume $\mathop{\kern 0.0pt{\rm dim}}\nolimits\text{\rm
Im}I_{{\mathcal{N}},1}=d$ for ${\mathcal{N}}\geq{\mathcal{N}}_{1}$.

We consider the number of $d$ dimensional irreducible components of $\text{\rm
Im}I_{{\mathcal{N}},1}$. This number is nonincreasing for
${\mathcal{N}}\geq{\mathcal{N}}_{1}$. Therefore, there exists
${\mathcal{N}}_{2}$ such that for ${\mathcal{N}}\geq{\mathcal{N}}_{2}$ the
number of $d$ dimensional irreducible components of $\text{\rm
Im}I_{{\mathcal{N}},1}$ is independent of ${\mathcal{N}}$. It follows that
there exists $X_{{\mathcal{N}}}$ a sequence of $d$ dimensional irreducible
components of $\text{\rm Im}I_{{\mathcal{N}},1}$ such that
$X_{{\mathcal{N}}+1}\subset X_{{\mathcal{N}}}$. Since $\mathop{\kern 0.0pt{\rm
dim}}\nolimits(X_{{\mathcal{N}}}\setminus X_{{\mathcal{N}}+1})<d$, it follows
from Baire’s category theorem that
$\cap_{{\mathcal{N}}}X_{{\mathcal{N}}}\neq\emptyset$. Hence the lemma. ∎

######  Lemma 10.15.

There exists a sequence $(\mathfrak{y}_{k,j;l}^{(n)})_{k,j;l}$
$n=1,2,3,\cdots\,m$ such that

$I_{n,n-1}((\mathfrak{y}_{k,j;l}^{(n)})_{k,j;l}=(\mathfrak{y}_{k,j;l}^{(n-1)})_{k,j;l})$

for $n=2,\cdots,m$ and that

$(\mathfrak{y}_{k,j;l}^{(m)})_{k,j;l}\in\bigcap_{{\mathcal{N}}>m}\text{\rm
Im}I_{{\mathcal{N}},m}.$

###### Proof.

The proof is by induction on $m$. The case $m=1$ is Lemma 10.14. Each of the
inductive step is similar to the proof of Lemma 10.14 and so it omitted. ∎

Lemma 10.15 implies that the projective limit (10.12) is nonempty. The proof
of weakly nondegenerate case of Theorem 10.4 is complete.

We next consider the strongly nondegenerate case. We prove the following lemma
by induction on ${\mathcal{N}}$. Let $G$ be a submonoid of
$({\mathbb{R}}_{\geq 0},+)$ generated by the numbers appearing in the exponent
of (9.18). Namely it is generated by

$\displaystyle S_{k^{\prime}}-S_{k}\,\,(k^{\prime}>k),$
$\displaystyle\ell(u_{0})+\rho-
S_{k}\,\,((\ell,\rho)\in\mathfrak{I}_{k^{\prime}},\,\,k^{\prime}\geq k),$
(10.13) $\displaystyle\ell(u_{0})-S_{k}\,\,(\ell\in\mathfrak{I}).$

We define $0<\lambda_{1}<\lambda_{2}<\cdots$ by

$\\{\lambda_{i}\mid i=1,2,\dots\\}=G.$

######  Lemma 10.16.

We assume that
$\mathfrak{y}^{0}=(\mathfrak{y}_{k,j;0})_{k=1,\cdots,K,\,j=1,\cdots,d(k)}$ is
a strongly nondegenerate solution of the leading term equation. Then, there
exists

$\mathfrak{y}^{({\mathcal{N}})}_{k,j}=\mathfrak{y}_{k,j;0}+\sum_{l=1}^{{\mathcal{N}}}\mathfrak{y}_{k,j;l}T^{\lambda_{l}}$

such that

$\sum_{j^{\prime}=1}^{a(k)}\frac{\partial Y(k,j^{\prime})}{\partial
y_{k,j}}(\mathfrak{y}^{({\mathcal{N}})}_{k,1},\cdots,\mathfrak{y}^{({\mathcal{N}})}_{K,d(K)})\equiv
0\mod T^{\lambda_{{\mathcal{N}}+1}}.$ (10.14)

Moreover we may choose $\mathfrak{y}^{({\mathcal{N}})}_{k,j}$ so that

$\mathfrak{y}^{({\mathcal{N}})}_{k,j}\equiv\mathfrak{y}^{({\mathcal{N}}+1)}_{k,j}\mod
T^{\lambda_{{\mathcal{N}}+1}}.$

###### Proof.

The proof is by induction on ${\mathcal{N}}$. There is nothing to show in the
case ${\mathcal{N}}=0$. Assume we have proved the lemma up to
${\mathcal{N}}-1$. Then we have

$\sum_{j^{\prime}=1}^{a(k)}\frac{\partial Y(k,j^{\prime})}{\partial
y_{k,j}}(\mathfrak{y}^{({\mathcal{N}}-1)}_{k,1},\cdots,\mathfrak{y}^{({\mathcal{N}}-1)}_{K,d(K)})\equiv
c_{k,j,M}T^{{\mathcal{N}}}\mod T^{\lambda_{{\mathcal{N}}+1}}.$

Consider $\mathfrak{y}^{({\mathcal{N}})}_{k,j}$ of the form

$\mathfrak{y}^{({\mathcal{N}})}_{k,j}=\mathfrak{y}^{({\mathcal{N}}-1)}_{k,j}+\Delta_{k,j,{\mathcal{N}}}T^{\lambda_{{\mathcal{N}}}}$

for some $\Delta_{k,j,{\mathcal{N}}}$. Then we can write

$\displaystyle\sum_{j^{\prime}=1}^{a(k)}$ $\displaystyle\frac{\partial
Y(k,j^{\prime})}{\partial
y_{k,j}}(\mathfrak{y}^{(M-1)}_{k,1}+\Delta_{k,1,{\mathcal{N}}}T^{\lambda_{{\mathcal{N}}}},\cdots,\mathfrak{y}^{({\mathcal{N}}-1)}_{K,d(K)}+\Delta_{K,d(K),{\mathcal{N}}}T^{\lambda_{{\mathcal{N}}}})$
(10.15)
$\displaystyle\equiv\left(c_{k,j,{\mathcal{N}}}+\sum_{j^{\prime},j^{\prime\prime}=1}^{a(k)}\frac{\partial^{2}Y(k,j^{\prime})}{\partial
y_{k,j}\partial
y_{k,j^{\prime\prime}}}\Delta_{k,j^{\prime\prime},{\mathcal{N}}}\right)T^{\lambda_{\mathcal{N}}}\mod
T^{\lambda_{{\mathcal{N}}+1}}.$

Since
$\mathfrak{y}^{0}=(\mathfrak{y}_{k,j;0})_{k=1,\cdots,K,\,j=1,\cdots,d(k)}$ is
strongly nondegenerate, we can find
$\Delta_{k,j^{\prime\prime},{\mathcal{N}}}\in{\mathbb{C}}$ so that the right
hand side become zero module $T^{\lambda_{{\mathcal{N}}+1}}$. The proof of
Lemma 10.16 is complete. ∎

By Lemma 10.16, the limit
$\lim_{{\mathcal{N}}\to\infty}\mathfrak{y}^{({\mathcal{N}})}_{k,j}$ exists. We
set

$\mathfrak{y}_{k,j}:=\lim_{{\mathcal{N}}\to\infty}\mathfrak{y}^{({\mathcal{N}})}_{k,j}.$

This is the required solution of (10.4). The proof of Theorem 10.4 is complete
∎

###### Proof of Lemma 10.11.

We put

$\mathfrak{y}_{k,j}=\mathfrak{y}_{k,j;0}+\sum_{l=1}^{\infty}\mathfrak{y}_{k,j;l}T^{\lambda_{l}}.$

By assumption $\mathfrak{y}_{k,j;0}\in F$. We remark that (10.15) gives a
linear equation which determines $\mathfrak{y}_{k,j;l}$ inductively on $l$. We
use it to show $\mathfrak{y}_{k,j;l}\in F$ inductively on $l$. ∎

We next give an example where weakly nondegeneracy condition is not satisfied.

######  Example 10.17.

Consider the 2-point blow up $X(\alpha,\beta)$ of ${\mathbb{C}}P^{2}$ with its
moment polytope given by

$P=\\{(u_{1},u_{2})\mid 0\leq u_{1}\leq 1,\,\,0\leq u_{2}\leq
1-\alpha,\,\,\beta\leq u_{1}+u_{2}\leq 1\\}.$

We consider the case when $1-\alpha$ is sufficiently small. The potential
function is

$\mathfrak{PO}=T^{u_{1}}y_{1}+T^{u_{2}}y_{2}+T^{1-\alpha-
u_{2}}y_{2}^{-1}+T^{1-u_{1}-u_{2}}y_{1}^{-1}y_{2}^{-1}+T^{u_{1}+u_{2}-\beta}y_{1}y_{2}.$

(We remark that $X$ is Fano.) We fix $\alpha$ and move $\beta$ starting from
zero. When $\beta$ is small compared to $1-\alpha$, there are two balanced
fibers.

One is located at $((1+\alpha)/4,(1-\alpha)/2)$. This corresponds to the
location of the balanced fiber of the one point blow up, which is nothing but
the case $\beta=0$. The other appears near the origin and is $(\beta,\beta)$.
The leading term equation at the first point is

$1-y_{2}^{-2}=0,\quad 1-y_{1}^{-2}y_{2}^{-1}=0.$

The solutions are $(y_{1},y_{2})=(\pm 1,1),(\pm\sqrt{-1},-1)$, all of which
are strongly nondegenerate. The leading term equation at the second point is

$1+y_{1}=1+y_{2}=0.$

$(-1,-1)$ is the nondegenerate solution. Thus we have $5$ solutions.

The situation jumps when $\beta=(1-\alpha)/2$. Denote $\beta_{0}=(1-\alpha)/2$
for the simplicity of notation. In that case the potential function at
$(\beta_{0},\beta_{0})$ becomes

$T^{\beta_{0}}(y_{1}+y_{2}+y_{1}y_{2}+y_{2}^{-1})+T^{1-2\beta_{0}}y_{1}^{-1}y_{2}^{-1}.$

The leading term equation is

$1+y_{2}=0,\quad 1+y_{1}-y_{2}^{-2}=0.$

Its solution is $(0,-1)$. Since $y_{1}=0$ it follows that there is no solution
in $(\Lambda_{0}\setminus\Lambda_{+})^{2}$. Hence there is no weak bounding
cochain $\mathfrak{x}$ for which the Floer cohomology
$HF((L(\beta_{0},\beta_{0}),\mathfrak{x}),(L(\beta_{0},\beta_{0}),\mathfrak{x});\Lambda)$
is nontrivial. In other words, the fiber $L(\beta_{0},\beta_{0})$ in
$X(\alpha,\beta_{0})$ is not strongly balanced.

On the other hand, this fiber $L(\beta_{0},\beta_{0})$ in
$X(\alpha,\beta_{0})$ is balanced because by choosing $\beta$ arbitrarily
close to $\beta_{0}$ and $\beta<\beta_{0}$, we can approximate it by the
fibers

$L(\beta,\beta)\subset X(\alpha,\beta)$

for which the Floer cohomology
$HF((L(\beta,\beta),\mathfrak{x}),(L(\beta,\beta),\mathfrak{x}))$ is non-
trivial. In particular $L(\beta_{0},\beta_{0})$ in $X(\alpha,\beta_{0})$ is
not displaceable.

We can also verify that

$\overline{\mathfrak{E}}(\beta_{0},\beta_{0})=\infty\quad\mbox{in
}\,X(\alpha,\beta_{0})$ (10.16)

while $\mathfrak{E}(\beta_{0},\beta_{0})=\beta_{0}$ in
$X(\alpha_{0},\beta_{0})$.

Now we examine where the missing solutions at $\beta=\beta_{0}$ have gone. We
consider $(u_{1},(1-\alpha)/2)$ where
$\beta_{0}=(1-\alpha)/2<u_{1}<(1+\alpha)/4$. The potential function is

$T^{\beta_{0}}(y_{2}+y_{2}^{-1})+T^{\beta_{0}+\lambda_{1}}(y_{1}+y_{1}y_{2})+T^{\beta_{0}+\lambda_{2}}y_{1}^{-1}y_{2}^{-1}.$
(10.17)

Here

$\lambda_{1}=u_{1}-\beta_{0}<\lambda_{2}=(1+\alpha)/2-u_{1}-\beta_{0}.$

The leading term equation is

$1-y_{2}^{-2}=0,\quad 1+y_{2}=0.$ (10.18)

The solution is $y_{2}=-1$ and $y_{1}$ is arbitrary. Thus there are infinitely
many solutions of the leading term equation. Therefore these solutions of
(10.18) are not weakly nondegenerate.

So we need to study the critical point of (10.17) more carefully. The
condition that $(y_{1},y_{2})$ is a critical point of (10.17) is written as

$\left\\{\begin{aligned}
&1-y_{2}^{-2}+T^{\lambda_{1}}y_{1}-T^{\lambda_{2}}y_{1}^{-1}y_{2}^{-2}=0\\\
&1+y_{2}-T^{\lambda_{2}-\lambda_{1}}y_{1}^{-2}y_{2}^{-1}=0.\end{aligned}\right.$
(10.19)

The leading order term of $y_{2}$ should be $-1$. We need to study also the
second order term. We can write

$y_{2}=-1+cT^{\mu},\quad y_{1}=d,$

where $c,d\in\Lambda_{0}\setminus\Lambda_{+}$. Then we have

$\displaystyle-2cT^{\mu}+dT^{\lambda_{1}}\equiv 0\mod
T^{\min\\{\mu,\lambda_{1}\\}}\Lambda_{+},$ (10.20) $\displaystyle
cT^{\mu}+d^{-2}T^{\lambda_{2}-\lambda_{1}}\equiv 0\mod
T^{\min\\{\mu,\lambda_{2}-\lambda_{1}\\}}\Lambda_{+}.$ (10.21)

(10.20) implies $\mu=\lambda_{1}$. (10.21) then implies
$\lambda_{2}-\lambda_{1}=\lambda_{1}$. It implies $u_{1}=1/3$. Furthermore

$c^{3}\equiv-1/4\mod\Lambda_{+},\quad d\equiv 2c\mod\Lambda_{+}.$ (10.22)

Since the three solutions of the ${\mathbb{C}}$-reduction of (10.22) are all
simple, we can show, by the same way as that of the proof of Theorem 10.4,
that all solutions correspond to solutions of the equation (10.19). Therefore,
$L(1/3,\beta_{0})$ is a strongly balanced fiber.

We remark that solutions of the leading term equation (10.18) do not lift to
solutions of (10.19) unless $u_{1}=1/3$ and $y_{1}=-1$. This shows that the
weakly nondegeneracy assumption in Theorem 10.4 is essential.

We remark that at $((1+\alpha)/4,(1-\alpha)/2)$ the leading term equation
becomes

$1-y_{2}^{-2}=0,\quad 1+y_{2}-y_{1}^{-2}y_{2}^{-1}=0.$

Its solutions in $({\mathbb{C}}\setminus\\{0\\})^{2}$ are $(\pm
1/\sqrt{2},1)$. The number of solutions jumps from 4 to 2 here. $2+3=5$. So
this is consistent with Theorem 1.3.

In summary for the case of $(\alpha,\beta_{0})$ with $\beta_{0}=(1-\alpha)/2$,
there are 3 balanced fibers $(1/3,\beta_{0})$, $((1+\alpha)/4,\beta_{0})$ and
$(\beta_{0},\beta_{0})$. The first two of them are strongly balanced and the
last is not strongly balanced.

The balanced fiber $L(1/3,\beta_{0})\subset X(\alpha,\beta_{0})$ disappears as
we deform $X(\alpha,\beta_{0})$ to $X(\alpha,\beta)$ as $\beta_{0}$ moves to
nearby $\beta$. To see this let us take $\beta$ which is slightly bigger than
$\beta_{0}=(1-\alpha)/2$. Then $((\alpha+\beta)/2,(1-\alpha)/2)$, and
$(1-\alpha-\beta,\beta)$ are the balanced fibers. The leading term equation at
the first point is

$1-y_{2}^{-2}=0,\quad y_{2}-y_{1}^{-2}y_{2}^{-1}=0.$

Hence there are 4 solutions $(\pm 1,\pm 1)$. The leading term equation at the
second point is

$1+y_{2}=0,\quad y_{1}-y_{2}^{-2}=0.$

Hence the solution is $(1,-1)$. Total number is again 5.

We remark that the ${\mathbb{Q}}$-structure of quantum cohomology also jumps
at $\beta=(1-\alpha)/2$. Namely

$QH(X(\alpha,\beta);\Lambda^{{\mathbb{Q}}})=\begin{cases}\Lambda^{{\mathbb{Q}}(\sqrt{-1})}\times(\Lambda^{{\mathbb{Q}}})^{3}&\text{$\beta$
is slightly smaller than $(1-\alpha)/2$},\\\
\Lambda^{{\mathbb{Q}}(\sqrt{2})}\times\Lambda^{{\mathbb{Q}}((-2)^{1/3})}&\text{$\beta=(1-\alpha)/2$},\\\
(\Lambda^{{\mathbb{Q}}})^{5}&\text{$\beta$ is slightly larger than
$(1-\alpha)/2$}.\end{cases}$

######  Remark 10.18.

In a sequel of this series of papers, we will prove that
$L(u_{1},(1-\alpha)/2))$ is not displacable for any
$u_{1}\in((1-\alpha)/2,1/3)\cup(1/3,(1+\alpha)/4)$, in the case
$\beta=\beta_{0}$. We will use the bulk deformation introduced by [FOOO3]
section 3.8 (= [FOOO2] section 13) to prove it.

The next example shows that Theorems 1.3, 1.4 and 10.4 can not be generalized
to the case of a positive characteristic.

######  Example 10.19.

Consider the $2$-point blow up $X$ of ${\mathbb{C}}P^{2}$ with moment polytope

$P=\\{(u_{1},u_{2})\mid 0\leq u_{i}\leq 1-\epsilon,\sum u_{i}\leq 1\\}.$

Since $X$ is monotone for $\epsilon=1/3$, it follows that $X$ is Fano. We will
assume $\epsilon>0$ is sufficiently small. Then the fiber at $u_{0}=(1/3,1/3)$
is balanced.

Now we consider the Novikov ring $\Lambda^{F}$ with $F=\mathbb{F}_{3}$ a field
of characteristic $3$. We will prove that there exists no element
$\mathfrak{x}\in H(L(u);\Lambda_{0}^{F})$ such that
$HF((L(u_{0}),\mathfrak{x}),(L(u_{0}),\mathfrak{x});\Lambda^{F})\neq 0$.

The potential function at $u_{0}$ is

$\mathfrak{PO}^{u_{0}}=T^{1/3}(y_{1}+y_{2}+1/(y_{1}y_{2}))+T^{2/3-\epsilon}(y_{1}^{-1}+y_{2}^{-1}).$

Therefore the critical point equation is given by

$1-1/(y_{i}y_{1}y_{2})-ty_{i}^{-2}=0\quad i=1,2$ (10.23)

where $t=T^{1/3-\epsilon}$. From this it follows that $y_{i}\equiv
1\mod\Lambda_{+}$. In fact the leading term equation is $a_{i}a_{1}a_{2}=1$
for $i=1,2$ which is reduced to

$a_{1}=a_{2}=a,\quad a^{3}=1.$

Obviously this equation has the unique solution $a_{1}=a_{2}=1$ in
$\mathbb{F}_{3}$.

Going back to the study of solutions of the critical point equation (10.23),
we first prove $y_{1}=y_{2}$. We put $z_{i}=y_{i}^{-1}$. We assume
$z_{i}-z_{j}\neq 0$ and put $z_{i}-z_{j}\equiv t^{\lambda}c\mod
t^{\lambda}\Lambda_{+}$ with $c\in\mathbb{F}_{3}\setminus\\{0\\}$. Then by
(10.23) we have

$(z_{i}-z_{j})z_{1}z_{2}+t(z_{i}-z_{j})(z_{i}+z_{j})=0.$

This is a contradiction since the left hand side is congruent to
$ct^{\lambda}$ modulo $t^{\lambda}\Lambda_{+}^{\mathbb{F}_{3}}$.

We now put $x=y_{i}$ and obtain

$x^{3}-tx-1=0.$ (10.24)

We prove:

######  Lemma 10.20.

$(\ref{eq:x})$ has no solution in $\Lambda_{0}^{\mathbb{F}_{3}}$.

###### Proof.

We put $x=1+t^{1/3}x^{\prime}$ and obtain

$(x^{\prime})^{3}-t^{1/3}x^{\prime}-1=0.$

This equation resembles (10.24) except that $t$ is replaced by $t^{1/3}$. We
now put

$x_{N}\equiv 1+\sum_{k=1}^{N}t^{\sum_{i=1}^{k}3^{-i}}.$

Then $x_{N}^{3}=1+tx_{N-1}.$ Therefore

$(x_{N})^{3}-tx_{N}-1=-t^{1+\sum_{i=1}^{N}3^{-i}}.$

Namely $x_{N}$ is a solution of (10.24) modulo $t^{1+\sum_{i=1}^{N}3^{-i}}$.
It is easy to see that there are no other solution of (10.24) modulo
$t^{1+\sum_{i=1}^{N}3^{-i}}$.

However since

$1+\sum_{i=1}^{\infty}3^{-i}=\frac{3}{2}<\infty,$

it follows that

$\lim_{N\to\infty}x_{N}$

does not converge in $\Lambda_{0}^{\mathbb{F}_{3}}$. Thus there is no solution
of (10.24) over a field of characteristic $3$. ∎

Lemma 10.20 implies that the field of fraction of the Puiseux series ring with
coefficients in an algebraically closed field of _positive_ characteristic is
not algebraically closed. It is well known that this phenomenon does not occur
in the case of characteristic zero. See, for example, Corollary 13.15 [Ei].
Since we could not find a proof of a similar result for universal Novikov ring
in the literature, we will provide its proof in the appendix for completeness.
(We used it in the proof of Theorem 1.12 in section 7.)

##  11\. Calculation of potential function

In this section, we prove Theorems 4.5 and 4.6. We begin with a review of
[CO]. Let $\pi:X\to P$ be the moment map and $\partial
P=\bigcup_{i=1}^{m}\partial_{i}P$ be the decomposition of the boundary of $P$
into $n-1$ dimensional faces. Let $\beta_{i}\in H_{2}(X,L(u);{\mathbb{Z}})$ be
the elements such that

$\beta_{i}\cap[\pi^{-1}(\partial P_{j})]=\begin{cases}1&\text{if $i=j$,}\\\
0&\text{if $i\neq j$.}\end{cases}$

The Maslov index $\mu(\beta_{i})$ is 2. (Theorem 5.1 [CO].)

Let $\beta\in\pi_{2}(X,L(u))$ and $\mathcal{M}_{k+1}^{\text{\rm
main}}(L(u),\beta)$ be the moduli space of stable maps from bordered Riemann
surfaces of genus zero with $k+1$ boundary marked points in homology class
$\beta$. (See [FOOO1] section 3 = [FOOO3] subsection 2.1.2. We require the
boundary marked points to respect the cyclic order of $S^{1}=\partial D^{2}$.
(In other words we consider the main component in the sense of [FOOO1] section
3.)) Let $\mathcal{M}^{\text{\rm main},\text{\rm reg}}_{k+1}(L(u),\beta)$ be
its subset consisting of all maps from a disc. (Namely the stable map without
disc or sphere bubble.) The next theorem easily follows from the results of
[CO]. In (3) Theorem 11.1 we use the spin structure of $L(u)$ which is induced
by the diffeomorphism of $L(u)\cong T^{n}$ by the $T^{n}$ action and the
standard trivialization of the tangent bundle of $T^{n}$.

######  Theorem 11.1.

  1. (1)

If $\mu(\beta)<0$, or $\mu(\beta)=0$, $\beta\neq 0$, then
$\mathcal{M}^{\text{\rm main},\text{\rm reg}}_{k+1}(L(u),\beta)$ is empty.

  2. (2)

If $\mu(\beta)=2$, $\beta\neq\beta_{1},\cdots,\beta_{m}$, then
$\mathcal{M}^{{\text{\rm main}},\text{\rm reg}}_{k+1}(L(u),\beta)$ is empty.

  3. (3)

For $i=1,\cdots,m$, we have

$\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{1}(L(u),\beta_{i})=\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i}).$
(11.1)

Moreover $\mathcal{M}_{1}^{\text{\rm main}}(L(u),\beta_{i})$ is Fredholm
regular. Furthermore the evaluation map

$ev:\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i})\to L(u)$

is an orientation preserving diffeomorphism.

  4. (4)

For any $\beta$, the moduli space $\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{1}(L(u),\beta)$ is Fredholm regular. Moreover

$ev:\mathcal{M}^{\text{\rm main,reg}}_{1}(L(u),\beta)\to L(u)$

is a submersion.

  5. (5)

If $\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta)$ is not empty then there
exists $k_{i}\in{\mathbb{Z}}_{\geq 0}$ and $\alpha_{j}\in
H_{2}(X;{\mathbb{Z}})$ such that

$\beta=\sum_{i}k_{i}\beta_{i}+\sum_{j}\alpha_{j}$

and $\alpha_{j}$ is realized by holomorphic sphere. There is at least one
nonzero $k_{i}$.

###### Proof.

For reader’s convenience and completeness, we explain how to deduce Theorem
11.1 from the results in [CO].

By Theorems 5.5 and 6.1 [CO], $\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{k+1}(L(u),\beta)$ is Fredholm regular for any $\beta$. Since the
complex structure is invariant under the $T^{n}$ action and $L(u)$ is $T^{n}$
invariant, it follows that $T^{n}$ acts on $\mathcal{M}^{{\text{\rm
main}},\text{\rm reg}}_{k+1}(L(u),\beta)$ and

$ev:\mathcal{M}^{{\text{\rm main}},\text{\rm reg}}_{k+1}(L(u),\beta)\to L(u)$

is $T^{n}$ equivariant. Since the $T^{n}$ action on $L(u)$ is free and
transitive, it follows that $ev$ is a submersion if $\mathcal{M}^{{\text{\rm
main}},\text{\rm reg}}_{k+1}(L(u),\beta)$ is nonempty. (4) follows.

We assume $\mathcal{M}^{{\text{\rm main}},\text{\rm reg}}_{k+1}(L(u),\beta)$
is nonempty. Since $ev$ is a submersion it follows that

$n=\mathop{\kern 0.0pt{\rm dim}}\nolimits L(u)\leq\mathop{\kern 0.0pt{\rm
dim}}\nolimits\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{k+1}(L(u),\beta)=n+\mu(\beta)-2$

if $\beta\neq 0$. Therefore $\mu(\beta)\geq 2$. (1) follows.

We next assume $\mu(\beta)=2$, and $\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{k+1}(L(u),\beta)$ is nonempty. Then by Theorem 5.3 [CO], we find
$\beta=\beta_{i}$ for some $i$. (2) follows.

We next prove (5). It suffices to consider

$[f]\in\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)\setminus\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{1}(L(u),\beta).$

We decompose the domain of $u$ into irreducible components and restrict $f$
there. Let $f_{j}:D^{2}\to M$ and $g_{k}:S^{2}\to M$ be the restriction of $f$
to disc or sphere components respectively. We have

$\beta=\sum[f_{j}]+\sum[g_{k}].$

Theorem 5.3 [CO] implies that each of $f_{j}$ is homologous to the sum of the
element of $\beta_{i}$. It implies (5).

We finally prove (3). The fact that $ev$ is a diffeomorphism for
$\beta=\beta_{i}$, follows dirctly from Theorem 5.3 [CO]. We next prove that
$ev$ is orientation preserving. Since $L(u),u\in\text{\rm Int}\,P,$ is a
principal homogeneous space of $T^{n}$, the tangent bundle $TL(u)$ is
trivialized once we fix an isomorphism $T^{n}\equiv S^{1}\times\dots\times
S^{1}$. Using the orientation and the spin structure on $L(u)$ induced by such
a trivialization, we orient the moduli space $\mathcal{M}_{1}(\beta)$ of
holomorprhic discs. If we change the identification $T^{n}\equiv
S^{1}\times\dots\times S^{1}$ by an orientation preserving, (resp. reversing),
isomorphism, the corresponding orientations on $L(u)$ and
$\mathcal{M}_{1}(\beta)$ is preserved, (resp. reversed). Therefore whether
$ev:\mathcal{M}(\beta_{i})\to L(u)$ is orientation preserving or not does not
depend on the choice of the identification $T^{n}$ and $S^{1}\times\dots\times
S^{1}$.

For each $i=1,\dots,m$, we can find an automorphism $\phi$ of
$({\mathbb{C}}^{*})^{n}$ and a biholomorphic map $f:X\setminus\cup_{j\neq
i}\pi^{-1}(\partial_{j}P)\to{\mathbb{C}}\times({\mathbb{C}}^{*})^{n-1}$ such
that

  1. (1)

$f$ is $\phi$-equivariant,

  2. (2)

$f(L(u))=L_{\text{\rm std}}$, where $L_{\text{\rm
std}}=\\{(w_{1},\dots,w_{n})\in({\mathbb{C}})^{n}||w_{1}|=\dots=|w_{n}|=1\\}.$

Under this identification, $\mathcal{M}_{1}(\beta_{i})$ is identified with the
space of holomorphic discs

$z\in D^{2}\mapsto(\zeta\cdot
z,w_{2},\dots,w_{n})\in{\mathbb{C}}\times({\mathbb{C}}^{*})^{n-1},\quad\zeta\in
S^{1}\subset{\mathbb{C}}^{*},$

where $w_{k}\in{\mathbb{C}},k=2,\dots,n$ with $|w_{k}|=1$. Therefore it is
enough to check the statement that $ev$ is orientation preserving in a single
example. Cho [Cho] proved it in the case of the Clifford torus in
${\mathbb{C}}P^{n}$, hence the proof.

To prove (11.1) and complete the proof of Theorem 11.1, it remains to prove
$\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{1}(L(u),\beta_{i_{0}})=\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta_{i_{0}})$. (Here $i_{0}\in\\{1,\cdots,m\\}$.) Let
$[f]\in\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i_{0}})$. We take $k_{i}$
and $\alpha_{j}$ as in (5). (Here $\beta=\beta_{i_{0}}$). We have

$\partial\beta_{i_{0}}=\sum_{i}k_{i}\partial\beta_{i}.$

Using the convexity of $P$, (5) and $k_{i}\geq 0$, we show the inequality

$\beta_{i_{0}}\cap\omega\leq\sum_{i}k_{i}\beta_{i}\cap\omega$ (11.2)

holds and that the equality holds only if $k_{i}=0$ ($i\neq i_{0}$),
$k_{i_{0}}=1$, as follows : By (5) we have

$\ell_{i_{0}}=\sum_{i=1}^{m}k_{i}\ell_{i}+c$

where $c$ is a constant. Since $k_{i}\geq 0$ and $\ell_{i_{0}}(u^{\prime})=0$
for $u^{\prime}\in\partial_{i_{0}}P$, it follows that $c\leq 0$. (Note
$\ell_{i}\geq 0$ on $P$.) Since $\beta_{i}\cap\omega=\ell_{i}(u)$, we have the
inequality (11.2). Let us assume that the equality holds. If there exists
$i\neq j$ with $k_{i},k_{j}>0$ then

$\partial_{i_{0}}\,P=\\{u^{\prime}\in
P\mid\ell_{i_{0}}(u^{\prime})=0\\}\subseteq\\{u^{\prime}\in
P\mid\ell_{i}(u^{\prime})=\ell_{j}(u^{\prime})=0\\}\subseteq\partial_{i}P\cap\partial_{j}P.$

This is a contradiction since $\partial_{i_{0}}P$ is codimension 1. Therefore
there is only one nonzero $k_{i}$. It is easy to see that $i=i_{0}$ and
$k_{i_{0}}=1$.

On the other hand since $\alpha_{j}\cap\omega>0$ it follows that

$\beta_{i_{0}}\cap\omega\geq\sum_{i}k_{i}\beta_{i}\cap\omega.$

Therefore there is no sphere bubble (that is $\alpha_{j}$). Moreover the
equality holds in (11.2). Hence the domain of our stable map is irreducible.
Namely

$\mathcal{M}^{\text{\rm main},\text{\rm
reg}}_{1}(L(u),\beta_{i_{0}})=\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta_{i_{0}}).$

The proof of Theorem 11.1 is now complete. ∎

Next we discuss one delicate point to apply Theorem 11.1 to the proofs of
Theorems 4.5 and 4.6. (This point was already mentioned in section 16 [CO].)
Let us consider the case where there exists a holomorphic sphere $g:S^{2}\to
X$ with

$c_{1}(X)\cap g_{*}[S^{2}]<0.$

We assume moreover that there exists a holomorphic disc $f:(D^{2},\partial
D^{2})\to(X,L(u))$ such that

$f(0)=g(1).$

We glue $D^{2}$ and $S^{2}$ at $0\in D^{2}$ and $1\in S^{2}$ to obtain
$\Sigma$. $f$ and $g$ induce a stable map
$h:(\Sigma,\partial\Sigma)\to(X,L(u))$.

In general $h$ will not be Fredholm regular since $g$ may not be Fredholm
regular or the evaluation is not transversal at the interior nodes. In other
words, elements of $\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)\setminus\mathcal{M}^{{\text{\rm main}},\text{\rm
reg}}_{1}(L(u),\beta)$ may not be Fredholm regular in general. Moreover
replacing $g$ by its multiple cover, we obtain an element of
$\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta)\setminus\mathcal{M}^{{\text{\rm
main}},\text{\rm reg}}_{1}(L(u),\beta)$ such that $\mu(\beta)$ is negative.
Theorem 11.1 says that all the holomorphic disc without any bubble are
Fredholm regular. However we can not expect that all stable maps in
$\mathcal{M}^{{\text{\rm main}}}_{1}(L(u),\beta)$ are Fredholm regular.

In order to prove Theorem 4.6, we need to find appropriate perturbations of
those stable maps. For this purpose we use the $T^{n}$ action and proceed as
follows. (We remark that many of the arguments below are much simplified in
the Fano case, where there exists no holomorphic sphere $g$ with $c_{1}(M)\cap
g_{*}[S^{2}]\leq 0.$ )

We equip each of $\mathcal{M}_{1}(L(u),\beta)$ with Kuranishi structure. (See
[FO] for the general theory of Kuranishi structure and section 17-18 [FOOO1]
(or section 7.1 [FOOO3] = section 29 [FOOO2]) for its construction in the
context we currently deal with.) We may construct Kuranishi neighborhoods and
obstruction bundles that carry $T^{n}$ actions induced by the $T^{n}$ action
on $X$, and choose $T^{n}$-equivariant Kuranishi maps. (See Definition 15.4.)
We note that the evaluation map

$ev:\mathcal{M}_{1}(L(u),\beta)\to L(u)$

is $T^{n}$-equivariant. We use the fact that the complex structure of $X$ is
$T^{n}$-invariant and $L(u)$ is a free $T^{n}$-orbit to find such a Kuranishi
structure. (See Proposition 15.7 for the detail.)

We remark that the $T^{n}$ action on the Kuranishi neighborhood is free since
the $T^{n}$ action on $L(u)$ is free and $ev$ is $T^{n}$ equivariant. We take
a perturbation (that is, a multisection) of the Kuranishi map that is $T^{n}$
equivariant. We can find such a multisection which is also transversal to $0$
as follows : Since the $T^{n}$ action is free, we can take the quotient of
Kuranishi neighborhood, obstruction bundle etc. to obtain a space with
Kuranishi structure. Then we take a transversal multisection of the quotient
Kuranishi structure and lift it to a multisection of the Kuranishi
neighborhood of $\mathcal{M}_{1}(L(u),\beta)$. (See Corollary 15.15 for
detail.) Let $\mathfrak{s}_{\beta}$ be such a multisection and let
$\mathcal{M}_{1}(L(u),\beta)^{\mathfrak{s}_{\beta}}$ be its zero set. We
remark that the evaluation map

$ev:\mathcal{M}_{1}(L(u),\beta)^{\mathfrak{s}_{\beta}}\to L(u)$ (11.3)

is a submersion. This follows from the $T^{n}$ equivariance. This makes our
construction of system of multisections much simpler than the general one in
section 7.2 [FOOO3] (= section 30 [FOOO2]) since the fiber product appearing
in the inductive construction is automatically transversal. (See subsection
7.2.2 [FOOO3] = section 30.2 [FOOO2]) for the reason why this is crucial.)
More precisely we prove the following Lemma 11.2. Let

$\mathfrak{forget}_{0}:\mathcal{M}^{\text{\rm
main}}_{k+1}(L(u),\beta)\to\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta)$
(11.4)

be the forgetful map which forgets the first, …, $k$-th marked points. (In
other words, only the $0$-th marked point remains.) We can construct our
Kuranishi structure so that it is compatible with $\mathfrak{forget}_{0}$ in
the same sense as Lemma 7.3.8 [FOOO3] (= Lemma 31.8 [FOOO2]).

######  Lemma 11.2.

For each given $E>0$, there exists a system of multisections
$\mathfrak{s}_{\beta,k+1}$ on $\mathcal{M}^{\text{\rm
main}}_{k+1}(L(u),\beta)$ for $\beta\cap\omega<E$ with the following
properties :

  1. (1)

They are transversal to $0$.

  2. (2)

They are invariant under the $T^{n}$ action.

  3. (3)

The multisection $\mathfrak{s}_{\beta,k+1}$ is the pull-back of the
multisection $\mathfrak{s}_{\beta,1}$ by the forgetful map $(\ref{37.183})$.

  4. (4)

The restriction of $\mathfrak{s}_{\beta,1}$ to the boundary of
$\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta)$ is the fiber product of the
multisections $\mathfrak{s}_{\beta^{\prime},k^{\prime}}$ with respect to the
identification of the boundary

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

  5. (5)

We do not perturb $\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i})$ for
$i=1,\cdots,m$.

###### Proof.

We construct multisections inductively over $\omega\cap\beta$. Since (2)
implies that fiber products of the perturbed moduli spaces which we have
already constructed in the earlier stage of induction are automatically
transversal, we can extend them so that (1), (2), (3), (4) are satisfied by
the method we already explained above. We recall from Theorem 11.1 (3) that

$\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i})=\mathcal{M}_{1}^{{\text{\rm
main}},\text{\rm reg}}(L(u),\beta_{i})$

and it is Fredholm regular and its evaluation map is surjective to $L(u)$.
Therefore when we perturb the multisection we do not need to worry about
compatibility of it with other multisections we have already constructed in
the earlier stage of induction. This enable us to leave the moduli space
$\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta_{i})$ unperturbed for all
$\beta_{i}$. The proof of Lemma 11.2 is complete. ∎

######  Remark 11.3.

We need to fix $E$ and stop the inductive construction of multisections at
some finite stage. Namely we define $s_{\beta,k+1}$ only for $\beta$ with
$\beta\cap\omega<E$. The reason is explained in subsection 7.2.3 [FOOO3] (=
section 30.3 [FOOO2]). We can go around this trouble in the same way as
explained in section 7.2 [FOOO3] (= section 30 [FOOO2]). See Remark 11.11.

######  Remark 11.4.

We explain one delicate point of the proof of Lemma 11.2. Let
$\alpha\in\pi_{2}(X)$ be represented by a holomorphic sphere with
$c_{1}(X)\cap\alpha<0$. We consider the moduli space $\mathcal{M}_{1}(\alpha)$
of holomorphic sphere with one marked point and in homology class $\alpha$.
Let us consider $\beta\in\pi_{2}(X;L(u))$ and the moduli space
$\mathcal{M}_{k+1,1}^{\text{\rm main}}(\beta)$ of holomorphic discs with one
interior and $k+1$ boundary marked points and of homotopy class $\beta$. The
fiber product

$\mathcal{M}_{1}(\alpha)\times_{X}\mathcal{M}_{k+1,1}^{\text{\rm
main}}(\beta)$

taken by the evaluation maps at interior marked points are contained in
$\mathcal{M}_{k+1,1}^{\text{\rm main}}(\beta+\alpha)$. If we want to define a
multisection compatible with the embedding

$\mathcal{M}_{1}(\alpha)\times_{X}\mathcal{M}_{k+1,1}^{\text{\rm
main}}(\beta)\subset\mathcal{M}_{k+1}^{\text{\rm main}}(\beta+\alpha)$ (11.5)

then it is impossible to make it both transversal and $T^{n}$ equivariant in
general : This is because the nodal point of such a singular curve could be
contained in the part of $X$ with non-trivial isotropy group.

Our perturbation constructed above satisfies (1) and (2) of Lemma 11.2 and so
may not be compatible with the embedding (11.5). Our construction of the
perturbation given in Lemma 11.2 exploits the fact that the $T^{n}$ action
acts freely on the Lagrangian fiber $L(u)$ and carried out by induction on the
number of disc components (and of energy) only, regardless of the number of
sphere components.

The following corollary is an immediate consequence of Lemma 11.2.

######  Corollary 11.5.

If $\mu(\beta)<0$ or $\mu(\beta)=0$, $\beta\neq 0$, then
$\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta)^{s_{\beta}}$ is empty.

Now we consider $\beta\in\pi_{2}(X;L)$ with $\mu(\beta)=2$ and
$\beta\cap\omega<E$, where $E$ is as in Lemma 11.2. One immediate consequence
of Corollary 11.5 is that the virtual fundamental chain of
$\mathcal{M}^{\text{\rm main}}_{1}(L(u),\beta)$ becomes a _cycle_. More
precisely, we introduce

######  Definition 11.6.

Let $\beta\in\pi_{2}(X;L)$ with $\mu(\beta)=2$ and $\beta\cap\omega<E$, where
$E$ is as in Lemma 11.2. We define a homology class $c_{\beta}\in
H_{n}(L(u);{\mathbb{Q}})\cong{\mathbb{Q}}$ by the pushforward

$c_{\beta}=ev_{*}([\mathcal{M}^{\text{\rm
main}}_{1}(L(u),\beta)^{s_{\beta}}]).$

######  Lemma 11.7.

The number $c_{\beta}$ is independent of the choice of the system of
multisections $\mathfrak{s}_{\beta,k+1}$ satisfying $(1)$ \- $(5)$ of
Proposition 11.2.

###### Proof.

If there are two such systems, we can find a $T^{n}$ invariant homotopy
between them which is also transversal to $0$. By a dimension counting
argument applied to the parameterized version of $\mathcal{M}_{1}^{\text{\rm
main}}(L(u),\beta)$ and its perturbation, we will have the parameterized
version of Corollary 11.5. This in turn implies that the perturbed
(parameterized) moduli space defines a compact cobordism between the perturbed
moduli spaces of $\mathcal{M}_{1}^{\text{\rm main}}(\beta)$ associated to the