# Attractor mechanisms of moduli spaces of Calabi–Yau 3-folds

Yu-Wei Fan Atsushi Kanazawa

###### Abstract.

We investigate the complex and Kähler attractor mechanisms of moduli spaces of
Calabi–Yau 3-folds. The complex attractor mechanism was previously studied by
Ferrara–Kallosh–Strominger, Moore and others in string theory. It is concerned
with the minimizing problems of the normalized central charges of $3$-cycles
and defines a new interesting class of Calabi–Yau 3-folds called, the complex
attractor varieties. In light of mirror symmetry, we introduce the Kähler
attractor mechanism and define the Kähler attractor varieties. The complex and
Kähler attractor varieties are expected to possess very rich structures, in
particular certain complex and Kähler rigidities.

##  1\. Introduction

Let $X$ be a projective Calabi–Yau 3-fold. Let
$\pi:\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}\rightarrow\mathfrak{M}_{\mathrm{Cpx}}$
be the universal covering of the complex moduli space
$\mathfrak{M}_{\mathrm{Cpx}}$ of $X$. The normalized central charge of a
$3$-cycle $\gamma\in H_{3}(X,\mathbb{Z})$ is defined by

$Z(\Omega_{X_{z}},\gamma)=e^{\frac{K^{B}(z)}{2}}\int_{\gamma}\Omega_{X_{z}}$

where $\Omega_{X_{z}}$ is a holomorphic volume form of $X_{z}$ and $K^{B}(z)$
is the Weil–Petersson potential on $\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$.
It induces a function
$|Z(-,\gamma)|:\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}\rightarrow\mathbb{R}_{\geq
0}$, called the mass function of $\gamma$, and we are interested in its
stationary points, called the attractors. Our investigation is motivated by
the study of black holes in string theory (Ferrara–Kallosh–Strominger [9]),
where it is of great interest to find a $3$-cycle $\gamma\in
H_{3}(X,\mathbb{Z})$ which supports a BPS state. Finding stationary points of
the mass function $|Z(-,\gamma)|$ is a purely mathematical problem and can be
answered in parts by using the attractor mechanism investigated by Moore in
his unpublished article [23]. The Calabi–Yau 3-folds corresponding to the
attractors are called the attractor varieties. This new class of Calabi–Yau
3-folds are conjectured to posses very rich structures.

Moore’s article [23] is full of beautiful insights and he posed many
interesting mathematical questions (the attractor conjectures) pertaining to
the arithmetic nature of the attractor varieties. In fact, the attractor
varieties can be considered as a vast generalization of the rigid Calabi–Yau
3-folds.

In light of of mirror symmetry, which is a duality between complex and Kähler
(symplectic) geometries of distinct Calabi–Yau manifolds, a natural question
is, what is the mirror of the attractor mechanism? In this article, we
introduce the Kähler attractor mechanism of the Kähler moduli space and
develop parallel theories to the complex side. Moreover, the Kähler attractor
mechanism leads us to the idea of rigid Kähler structures, which should be
mirror to the rigid complex structures. This direction of research is further
carried out from the viewpoint of generalized Calabi–Yau geometry in a
separate article [15].

The present work is based on our previous work [8]. It investigates the
A-model Weil–Petersson geometry on the Kähler moduli space (or more precisely
the space of Bridgeland stability conditions), which is supposed to be mirror
to the classical Weil–Petersson geometry on the complex moduli spaces [26,
27].

The objective of this article is twofold. The first is to provide mathematical
foundations of the complex attractor mechanism (Section 2). The second is to
introduce the Kähler attractor mechanism inspired by mirror symmetry (Section
4).

(A-side) | (B-side)
---|---
Kähler moduli $\mathfrak{M}_{\mathrm{Kah}}$ | complex moduli $\mathfrak{M}_{\mathrm{Cpx}}$
Weil–Petersson metric $g^{A}$ | Weil–Petersson metric $g^{B}$
(Fan–Kanazawa–Yau [8]) | (Tian [26], Todorov [27])
Kähler attractor mechanism | complex attractor mechanism
(Fan–Kanazawa [present work]) | (Moore [23])

### Structure of article

Section 2 provides mathematical foundations of the complex attractor mechanism
based on [23]. Section 3 is a brief review of our previous work [8] on the
Weil–Petersson geometry by means of the Bridgeland stability conditions.
Section 4 introduces the Kähler attractor mechanism and develops parallel
theories to the complex side. We finally compare the complex and Kähler
attractor mechanisms from the view point of mirror symmetry.

### Notation and conventions

Throughout the article, we work over complex numbers $\mathbb{C}$. A
Calabi–Yau $n$-fold is an $n$-dimensional Kähler manifold whose canonical
bundle is trivial. $\mathrm{ch}(-)$ denotes the Chern character and
$\mathrm{Td}_{X}$ denotes the Todd class of $X$. For
$R=\mathbb{Z},\mathbb{Q},\mathbb{R}$, $H^{i,i}(X,R)$ denotes the intersection
$H^{i,i}(X)\cap H^{2i}(X,R)$. $\mathfrak{H}_{g}$ denotes the Siegel upper
half-space of degree $g$.

### Acknowledgement

First and foremost, we would like to express our gratitude to Shing-Tung Yau,
who first drew our attention to the attractor mechanism. Our thanks also go to
Shinobu Hosono and Hiroshi Iritani for very useful discussions. A.K. was
supported in part by the JSPS Grant-in-Aid Wakate(B)17K17817 and Leading
Initiative for Excellent Young Researchers Grant (Kyoto University).

##  2\. Complex attractor mechanism

###  2.1. Foundation of complex attractor mechanism

Let $X$ be a projective Calabi–Yau 3-fold and $\mathfrak{M}_{\mathrm{Cpx}}$
the complex moduli space of $X$. We consider the vector bundle
$\mathcal{H}=R^{3}\pi_{*}\underline{\mathbb{C}}\rightarrow\mathfrak{M}_{\mathrm{Cpx}}$
equipped with a natural Hodge filtration $F^{3}\mathcal{H}\subset\dots\subset
F^{0}\mathcal{H}$ of weight $3$. By the Calabi–Yau condition, the first piece
of the filtration defines a holomorphic line bundle
$\mathcal{L}=F^{3}\mathcal{H}\rightarrow\mathfrak{M}_{\mathrm{Cpx}}$, which we
call the Hodge bundle. It is classically known that
$\mathfrak{M}_{\mathrm{Cpx}}$ carries a natural Kähler metric, called the
Weil–Petersson metric $g^{B}$, whose Kähler potential is given by

(2.1)
$K^{B}(z)=-\log(\sqrt{-1}\int_{X_{z}}\Omega_{X_{z}}\wedge\overline{\Omega_{X_{z}}}),$

where $\\{\Omega_{X_{z}}\\}_{z}$ is a nowhere-zero (holomorphic) local section
of the Hodge bundle [26]. We call $K^{B}$ the Weil–Petersson potential. For
later use, we introduce the following useful formula for computing $K^{B}$.

######  Proposition 2.1 ([8]).

Assume that there exist formal sums of Lagrangian submanifolds $\\{L_{i}\\}$
representing a basis of
$H_{3}(X,\mathbb{Z})/\mathrm{tor}(H_{3}(X,\mathbb{Z}))$. Then

(2.2)
$K^{B}(z)=-\log(\sqrt{-1}\sum_{i,j}\chi_{\mathrm{Fuk}}^{i,j}\int_{L_{i}}\Omega_{X_{z}}\int_{L_{j}}\overline{\Omega_{X_{z}}}),$

where $(\chi_{\mathrm{Fuk}}^{i,j})=(\chi_{\mathrm{Fuk}}(L_{i},L_{j}))^{-1}$ is
the inverse matrix for the Euler paring $\chi_{\mathrm{Fuk}}$ of the Fukaya
category $\mathrm{D^{b}Fuk}(X)$.

We will develop the A-model Weil–Petersson geometry based upon this new
expression (Equation (2.2)).

######  Definition 2.2.

Let $\gamma\in H_{3}(X,\mathbb{Z})$ be a non-trivial $3$-cycle. Given an
identification $H_{3}(X,\mathbb{Z})\cong H_{3}(X_{z},\mathbb{Z})$, we define
the normalized central charge of $\gamma$ by

$Z(\Omega_{X_{z}},\gamma)=e^{\frac{K^{B}(z)}{2}}\int_{\gamma}\Omega_{X_{z}},$

where by abuse of notation $K^{B}(z)$ is given by the Equation (2.1) (it
depends not only on $z$ but also on $\Omega_{X_{z}}$).

Hereafter we always assume $\gamma\neq 0$. Let
$\pi:\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}\rightarrow\mathfrak{M}_{\mathrm{Cpx}}$
be the universal covering. Then $Z(-,\gamma)$ is a smooth function on the
total space of the pullback $\pi^{*}\mathcal{L}$ of the Hodge bundle
$\mathcal{L}$

$Z(-,\gamma):\pi^{*}\mathcal{L}\longrightarrow\mathbb{C}.$

We observe that the absolute value $|Z(\Omega_{X_{z}},\gamma)|$ is independent
of a choice of $\Omega_{X_{z}}\neq 0$. Therefore we obtain a function

$|Z(-,\gamma)|:\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}\longrightarrow\mathbb{R}_{\geq
0}.$

We call it the mass function of $\gamma\in H_{3}(X,\mathbb{Z})$.

######  Theorem 2.3.

A stationary point $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ of the mass
function
$|Z(-,\gamma)|:\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}\rightarrow\mathbb{R}_{\geq
0}$ with $Z(\Omega_{X_{z}},\gamma)\neq 0$ is characterized by the equation

(2.3) $\gamma^{PD}=\mathrm{Re}(C\Omega_{X_{z}}),\ \ \ (\exists
C\in\mathbb{C})$

in $H^{3}(X,\mathbb{Z})$, where $\gamma^{PD}$ denotes the Poincaré dual of
$\gamma$. We call the Equation (2.3) the complex attractor equation. It is
equivalent to the condition

$\gamma^{PD}\in H^{3,0}(X)\oplus H^{0,3}(X).$

###### Proof.

By the Bogomolov–Tian–Todorov Theorem, the Kodaira–Spencer map provides an
identification between an open neighborhood $U$ of
$z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ and an open neighborhood
$U^{\prime}$ of $0\in H^{1}(X,TX)\cong H^{2,1}(X)$. Therefore, for a basis
$\Omega_{1},\dots,\Omega_{k}$ of $H^{2,1}(X)$, the variation
$\Omega^{\prime}_{X_{z}}=\Omega_{X_{z}}+\sum_{i}\epsilon_{i}\Omega_{i}$ gives
a local coordinate $\epsilon=(\epsilon_{1},\dots,\epsilon_{n})$ of $U$. Then a
straightforward calculation shows

$\displaystyle\left.\frac{\partial}{\partial\epsilon_{i}}\right|_{\epsilon=0}|Z(\Omega^{\prime}_{X_{z}},\gamma)|^{2}=$
$\displaystyle\left.\frac{\partial}{\partial\epsilon_{i}}\right|_{\epsilon=0}\frac{|\int_{\gamma}\Omega^{\prime}_{X_{z}}|^{2}}{\sqrt{-1}\int_{X_{z}}\Omega^{\prime}_{X_{z}}\wedge\overline{\Omega^{\prime}_{X_{z}}}}$
$\displaystyle=$ $\displaystyle
e^{K^{B}(z)}\int_{\gamma}\overline{\Omega_{X_{z}}}\int_{\gamma}\Omega_{i}$

By the assumption that
$Z(\Omega_{X_{z}},\gamma)=e^{\frac{K^{B}(z)}{2}}\int_{\gamma}\Omega_{X_{z}}\neq
0$, $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ is a stationary point if and
only if $\int_{\gamma}\Omega_{i}=0\ (1\leq i\leq k)$. Since
$\Omega_{1},\dots,\Omega_{k}$ form a basis of $H^{2,1}(X)$, this condition is
equivalent to

$\gamma^{PD}\in H^{3,0}(X)\oplus H^{0,3}(X).$

Therefore, since $\gamma^{PD}\in H^{3}(X,\mathbb{Z})$,

$\gamma^{PD}=(C\Omega_{X_{z}}+\overline{C\Omega_{X_{z}}})=\mathrm{Re}(C\Omega_{X_{z}})$

for some $C\in\mathbb{C}$. ∎

######  Theorem 2.4.

A stationary point $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ of the mass
function $|Z(-,\gamma)|$ with $Z(\Omega_{X_{z}},\gamma)\neq 0$ is a local
minimizer. Moreover, such points are discrete.

###### Proof.

For a stationary point $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$, a
straightforward but tedious calculation shows

$\displaystyle\left.\frac{\partial^{2}}{\partial\epsilon_{i}\partial\overline{\epsilon_{j}}}\right|_{\epsilon=0}|Z(\Omega^{\prime}_{X_{z}},\gamma)|^{2}=$
$\displaystyle\left.\frac{\partial^{2}}{\partial\epsilon_{i}\partial\overline{\epsilon_{j}}}\right|_{\epsilon=0}\frac{|\int_{\gamma}\Omega^{\prime}_{X_{z}}|^{2}}{\sqrt{-1}\int_{X_{z}}\Omega^{\prime}_{X_{z}}\wedge\overline{\Omega^{\prime}_{X_{z}}}}$
$\displaystyle=$
$\displaystyle\frac{-\sqrt{-1}|\int_{\gamma}\Omega_{X_{z}}|^{2}\int_{X_{z}}\Omega_{i}\wedge\overline{\Omega_{j}}}{(\sqrt{-1}\int_{X_{z}}\Omega_{X_{z}}\wedge\overline{\Omega_{X_{z}}})^{2}}$
$\displaystyle=$ $\displaystyle
2e^{K^{B}(z)}|\int_{\gamma}\Omega_{X_{z}}|^{2}g^{B}_{i\bar{j}}(z).$

Therefore the complex Hessian of the function $|Z(-,\gamma)|^{2}$ at $z$ is
identified with the Weil–Petersson metric $g^{B}_{i\bar{j}}(z)$, rescaled by a
positive constant, and hence is positive definite. ∎

Theorem 2.4 asserts that there are 3 different types of the behavior of the
mass function $|Z(-,\gamma)|$ depending on the nature of $\gamma\in
H_{3}(X,\mathbb{Z})$.

  1. (1)

There exists no stationary point.

  2. (2)

There exists a stationary point $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$
with $Z(\Omega_{X_{z}},\gamma)=0$. In this case the equation
$\int_{\gamma}\Omega_{X_{z}}=0$ defines a divisor on
$\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$.

  3. (3)

There exists a stationary point $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$
with $Z(\Omega_{X_{z}},\gamma)\neq 0$.

######  Definition 2.5.

A stationary point $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ with
$Z(\Omega_{X_{z}},\gamma)\neq 0$ is called a complex attractor for $\gamma$.
The corresponding Calabi–Yau 3-fold $X_{z}$ is called a complex attractor
variety for $\gamma$. We denote by
$\widetilde{\mathrm{Attr}}_{\mathrm{Cpx}}(\gamma)\subset\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$
the set of complex attractors for $\gamma$. Then we define

$\mathrm{Attr}_{\mathrm{Cpx}}=\pi(\cup_{\gamma}\widetilde{\mathrm{Attr}}_{\mathrm{Cpx}}(\gamma))\subset\mathfrak{M}_{\mathrm{Cpx}},$

where $\gamma$ runs over $H^{3}(X,\mathbb{Z})$, and call it the complex
attractor constellation of $X$.

######  Remark 2.6.

The complex attractor equation is concerned with $\Omega_{X_{z}}$. It in
general does not determine a complex attractor
$z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ because a Torelli type theorem
fails in 3-dimensions.

It is natural to ask whether or not a complex attractor gives the global
minimum, but the situation is rather complicated partly due to the non-
compactness of the complex moduli space $\mathfrak{M}_{\mathrm{Cpx}}$. In
fact, it is claimed in [23, Section 9.2] that there exists a Calabi–Yau 3-fold
for which a single $\gamma$ leads to several distinct complex attractors with
different values of local minima.

Another important problem is to investigate the distribution of the complex
attractor constellation
$\mathrm{Attr}_{\mathrm{Cpx}}\subset\mathfrak{M}_{\mathrm{Cpx}}$. Note that it
is not an intrinsic property of the complex manifold
$\mathfrak{M}_{\mathrm{Cpx}}$. For example, many 1-parameter families of
Calabi–Yau 3-folds share their complex moduli
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$, but the complex constellations
should be different because they depend on the Calabi–Yau 3-folds they
parametrize (to be precise, the variation of Hodge structures).

######  Conjecture 2.7.

Let $\mathfrak{M}_{\mathrm{Cpx}}\subset\overline{\mathfrak{M}}_{\mathrm{Cpx}}$
be a (partial) compactification. The complex attractor constellation
$\mathrm{Attr}_{\mathrm{Cpx}}$ is dense near a large complex structure limit
$z\in\overline{\mathfrak{M}}_{\mathrm{Cpx}}$.

This conjecture is inspired by the observation that there seem infinitely many
Kähler attractor points (to be introduced in Section 4) near the large volume
limit of a Calabi–Yau 3-folds.

######  Remark 2.8.

A Calabi–Yau 3-fold $X$ is called rigid if $H^{2,1}(X)=0$. A rigid Calabi–Yau
3-fold $X$ is by definition a complex attractor variety for any $\gamma\in
H_{3}(X,\mathbb{Z})$. From this perspective, the complex attractor varieties
are a vast generalization of the rigid Calabi–Yau 3-folds, whose arithmetic
properties are of considerable interest. Indeed, in [23, Section 8.2], Moore
posed several interesting questions (the attractor conjectures) pertaining to
the arithmetic nature of the complex attractor varieties. This direction of
research has recently been carried out by Lam and Tripathy [19, 20].

We now take a closer look at the complex attractors. The plane

$V(z)=H^{3,0}(X_{z})\oplus H^{0,3}(X_{z})\subset H^{3}(X,\mathbb{C})$

varies as $z$ moves in $\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$, where we
have a natural identification $H^{3}(X_{z},\mathbb{C})\cong
H^{3}(X,\mathbb{C})$ for a reference $X$. The intersection with the real
$(2h^{2,1}+2)$-dimensional space $H^{3}(X,\mathbb{R})$ is the 2-plane
$V_{\mathbb{R}}(z)$ spanned over $\mathbb{R}$ by $\mathrm{Re}(\Omega_{X_{z}})$
and $\mathrm{Im}(\Omega_{X_{z}})$. For a generic
$z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$, the plane $V_{\mathbb{R}}(z)$
intersects $H^{3}(X,\mathbb{Z})\subset H^{3}(X,\mathbb{R})$ only in $0$.

######  Definition 2.9.

Let $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ be a complex attractor for
some $\gamma^{\prime}\in H_{3}(X,\mathbb{Z})$. There are two cases:

  1. (1)

The intersection $V_{\mathbb{R}}(z)\cap H^{3}(X,\mathbb{Z})$ is a lattice
line. The point $z$ is a complex attractor for any non-zero $\gamma^{PD}\in
V_{\mathbb{R}}(z)\cap H^{3}(X,\mathbb{Z})$. In this case $z$ is called a
complex attractor of rank 1.

  2. (2)

The intersection $V_{\mathbb{R}}(z)\cap H^{3}(X,\mathbb{Z})$ is a lattice
plane. Then there exist $\gamma_{1},\gamma_{2}\in H_{3}(X,\mathbb{Z})$ such
that the intersection $\gamma_{1}\cap\gamma_{2}\neq 0$ and $V(z)$ is the
complexification of the lattice
$\mathbb{Z}\gamma_{1}^{PD}+\mathbb{Z}\gamma_{2}^{PD}$. Therefore
$\gamma_{1},\gamma_{2}$ simultaneously satisfy the complex attractor
equations. In this case, $z$ is called a complex attractor of rank 2.

######  Proposition 2.10.

Let $z\in\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$ be a complex attractor of
rank 2 such that $\gamma_{1},\gamma_{2}\in H_{3}(X,\mathbb{Z})$ with
$\gamma_{1}\cap\gamma_{2}\neq 0$ simultaneously satisfy the complex attractor
equations

(2.4) $\gamma_{1}^{PD}=\mathrm{Re}(C_{1}\Omega_{X_{z}}),\ \ \
\gamma_{2}^{PD}=\mathrm{Re}(C_{2}\Omega_{X_{z}}).$

for some $C_{1},C_{2}\in\mathbb{C}$. Then

$\Omega_{X_{z}}=\frac{\sqrt{-1}}{\mathrm{Im}(C_{1}\overline{C_{2}})}(\overline{C_{1}}\gamma_{2}^{PD}-\overline{C_{2}}\gamma_{1}^{PD}).$

###### Proof.

Since $\Omega_{X_{z}}$ lies in $V(z)$, we can write
$\Omega_{X_{z}}=a_{1}\gamma_{1}^{PD}+a_{2}\gamma_{2}^{PD}$ for some
$a_{1},a_{2}\in\mathbb{C}$. By plugging this in the complex attractor
equations (2.4), we determine the coefficients $a_{1},a_{2}$. ∎

While the complex attractors of rank 1 are expected to be dense in the moduli
space, those of rank 2 are expected to be rare, as the underlying Calabi–Yau
3-fold in general need to satisfy very stringent conditions. In fact, Moore
showed by using mirror symmetry the complex attractors of rank 1 are dense
near a large complex structure limit (Conjecture 2.7) [23].

###  2.2. Complex attractor mechanism for torus

Let us consider a real $6$-dimensional torus
$X=\mathbb{C}^{3}/(\mathbb{Z}^{3}+\sqrt{-1}\mathbb{Z}^{3})$. We introduce a
complex structure on $X$ in such a way that

$dz_{i}=dx_{i}+\sum_{j=1}^{3}T^{ij}dy_{j}\ \ \ (1\leq i\leq 3)$

are holomorphic $1$-forms for a period matrix $T=(T^{ij})\in\mathfrak{H}_{3}$.
Such a complex torus is denoted by $X_{T}$. Then $X_{T}$ is biholomorphic to
$\mathbb{C}^{3}/(\mathbb{Z}^{3}+T\mathbb{Z}^{3})$ equipped with the natural
complex structure by the map

$\phi:\mathbb{C}^{3}/(\mathbb{Z}^{3}\oplus\sqrt{-1}\mathbb{Z}^{3})\longrightarrow\mathbb{C}^{3}/(\mathbb{Z}^{3}+T\mathbb{Z}^{3}),\
\ z=x+\sqrt{-1}y\mapsto z^{\prime}=x+Ty.$

We see $\phi$ is holomorphic because
$\phi^{*}(dz^{\prime})=d\phi^{*}(z^{\prime})=dx+Tdy$. We vary the complex
structure of $X_{T}$ by, not varying the lattice as usual, but by varying the
holomorphic volume form

$\Omega_{X_{T}}=dz_{1}\wedge dz_{2}\wedge dz_{3}.$

We fix a symplectic basis of $H^{3}(X_{T},\mathbb{Z})$ as follows

$\displaystyle\alpha_{0}$ $\displaystyle=dx_{1}\wedge dx_{2}\wedge dx_{3},$
$\displaystyle\alpha_{ij}$
$\displaystyle=\frac{1}{2}\sum_{l,m=1}^{3}\epsilon_{ilm}dx_{l}\wedge
dx_{m}\wedge dy_{j}\ \ (1\leq i,j\leq 3)$ $\displaystyle\beta^{0}$
$\displaystyle=-dy_{1}\wedge dy_{2}\wedge dy_{3},$ $\displaystyle\beta^{ij}$
$\displaystyle=\frac{1}{2}\sum_{l,m=1}^{3}\epsilon_{jlm}dx_{i}\wedge
dy_{l}\wedge dy_{m}\ \ (1\leq i,j\leq 3).$

where $\epsilon_{ilm}$ denotes the Levi–Civita symbol. With respect to this
basis, $\Omega_{X_{T}}$ has an expansion

$\Omega_{X_{T}}=\alpha_{0}+\sum_{i,j=1}^{3}T^{ij}\alpha_{ij}+\sum_{i,j=1}^{3}(\mathrm{Cof}(T)_{ij})\beta^{ij}-(\det(T))\beta^{0}$

where $\mathrm{Cof}(T)=(\mathrm{Cof}(T)_{ij})$ denotes the cofactor matrix of
$T$. We fix a $3$-cycle $\gamma\in H_{3}(X_{T},\mathbb{Z})$ and write it as

$\gamma\
=q_{0}A_{0}+\sum_{i,j=1}^{3}Q_{ij}A_{ij}+\sum_{i,j=1}^{3}P^{ij}B^{ij}+p^{0}B^{0}$

where $A_{0},A_{ij},B^{ij},B^{0}$ form a basis of $H_{3}(X_{T},\mathbb{Z})$
dual to the symplectic basis $\alpha_{0},\alpha_{ij},\beta^{ij},\beta^{0}$.
Then the normalized central charge of $\gamma$ reads

$\displaystyle Z(\Omega_{X_{T}},\gamma)$
$\displaystyle=e^{\frac{K^{B}(T)}{2}}\int_{\gamma}\Omega_{X_{T}}$
$\displaystyle=e^{\frac{K^{B}(T)}{2}}(q_{0}+\sum_{i,j=1}^{3}Q_{ij}T^{ij}+\sum_{i,j=1}^{3}P^{ij}(\mathrm{Cof}(T)_{ij})-p^{0}\det(T)).$

Therefore the complex attractor equation
$\mathrm{Re}(C\Omega_{X_{T}})=\gamma^{PD}$ is equivalent to the following
system of equations

$\displaystyle\mathrm{Re}(C)$ $\displaystyle=p^{0}$
$\displaystyle\mathrm{Re}(CT^{ij})$ $\displaystyle=P^{ij}$
$\displaystyle\mathrm{Re}(C\mathrm{Cof}(T)_{ij})$ $\displaystyle=-Q_{ij}$
$\displaystyle\mathrm{Re}(C\det(T))$ $\displaystyle=q_{0}.$

######  Theorem 2.11 (Moore [23]).

A complex attractor for $\gamma\in H_{3}(X,\mathbb{Z})$ exists if and only if
the coefficient matrices $P=(P^{ij}),Q=(Q_{ij})\in M_{3}(\mathbb{Z})$ are
symmetric. Then a complex attractor is unique and given by

$T=((2PQ-(p^{0}q_{0}+\mathrm{tr}(PQ)E_{3}))+\sqrt{-D}E_{3})(2R)^{-1}\in\mathfrak{H}_{3}$

where

$\displaystyle R$ $\displaystyle=\mathrm{Cof}(P)+p^{0}Q,$ $\displaystyle D$
$\displaystyle=((\mathrm{tr}(PQ))^{2}-\mathrm{tr}((PQ)^{2}))-(p^{0}q_{0}+\mathrm{tr}(PQ))^{2}+4(p^{0}\det(Q)-q_{0}\det(P)).$

###### Proof.

We provide in Appendix A a rigorous and accessible proof based on Moore’s
original argument. One of our contributions is to show that there is no
complex attractor if $P,Q$ are not symmetric. ∎

The complex attractor variety
$X_{T}\cong\mathbb{C}^{3}/(\mathbb{Z}^{3}+T\mathbb{Z}^{3})$ has the following
interesting property. The lattice embedding

$\mathbb{Z}^{3}+T(2R)\mathbb{Z}^{3}\hookrightarrow\mathbb{Z}^{3}+T\mathbb{Z}^{3}$

induces an isogeny

$\phi:(E_{\sqrt{-D}})^{3}\cong\mathbb{C}^{3}/(\mathbb{Z}^{3}+T(2R)\mathbb{Z}^{3})\longrightarrow
X_{T}.$

In other words, the complex attractor variety $X_{T}$ is isogenous to the
self-product $(E_{\sqrt{-D}})^{3}$ of the elliptic curves $E_{\sqrt{-D}}$ with
complex multiplication by the covering map $\phi$ of degree $8\det(R)$. In
particular, $X_{T}$ is defined over a finite extension of the field
$\mathbb{Q}(\sqrt{-D})$.

For a projective complex manifold $X$ the Lefschetz $(1,1)$-theorem asserts
that the Néron–Severi group $NS(X)=H^{2}(X,\mathbb{Z})\cap H^{1,1}(X)$. The
rank $\rho(X)$ of the Néron–Severi group, the so-called Picard number,
satisfies the inequality $1\leq\rho(X)\leq h^{1,1}(X)$.

######  Theorem 2.12 ([13, Theorem 2.1]).

Let $A$ be an abelian variety of dimension $g$. The following are equivalent.

  1. (1)

The Picard number is maximal, i.e. $\rho(A)=g^{2}$;

  2. (2)

$A$ is isogenous to the self-product of an elliptic curve $E$ with complex
multiplication, i.e. $A\sim E^{g}$;

  3. (3)

$A$ is isomorphic to the product of some pairwise isogenous elliptic curves
$E_{1},\dots,E_{g}$ with complex multiplication, i.e. $A\cong
E_{1}\times\cdots\times E_{g}$.

Theorem 2.12 points out how the Picard number forces the structure of an
abelian variety to be rigid. It is classically known that the algebraic
varieties with the maximum Picard number possible often possess interesting
arithmetic and geometric properties.

######  Corollary 2.13.

The complex attractor variety $X_{T}$ has the maximal Picard number
$\rho(X_{T})=9$, and hence is of rank 2. Moreover, $X_{T}\cong E_{1}\times
E_{2}\times E_{3}$ for some pairwise isogenous elliptic curves
$E_{1},E_{2},E_{3}$ with complex multiplication.

In fact, the converse is also true and we have the following.

######  Theorem 2.14.

The complex constellation $\mathrm{Attr}_{\mathrm{Cpx}}$ bijectively
corresponds to the abelian 3-folds with Picard number $9$.

###### Proof.

It suffices to show that an abelian 3-fold
$X_{T}=\mathbb{C}^{3}/(\mathbb{Z}^{3}+T\mathbb{Z}^{3})$ with $\rho(X_{T})=9$
is a complex attractor variety for some $\gamma\in H_{3}(X_{T},\mathbb{Z})$. A
proof is based on a straightforward but tedious computation, and we leave it
in Appendix B. ∎

###  2.3. Complex attractor mechanism for $E\times S$

Let $E=\mathbb{C}/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})$ be a real 2-dimensional
torus. We put a complex structure on $E$ in such a way that $dz=dx+\tau dy$ is
holomorphic for $\tau\in\mathbb{H}$ so that
$E\cong\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ as a complex manifold. Let $S$
be a K3 surface equipped with a holomorphic volume form $\Omega_{S}\in
H^{2,0}(S)$. We consider the product Calabi–Yau 3-fold $X=E\times S$, which
carries a natural holomorphic volume form

$\Omega_{X}=dz\wedge\Omega_{S}.$

Note that $dx,dy$ form a symplectic basis of $H^{1}(E,\mathbb{Z})$. By the
Künneth theorem, we have the identification $H^{3}(X,\mathbb{Z})\cong
H^{1}(E,\mathbb{Z})\otimes_{\mathbb{Z}}H^{2}(S,\mathbb{Z})$. Therefore the
Poincaré dual of a 3-cycle $\gamma\in H_{3}(X,\mathbb{Z})$ can be expressed as

$\gamma^{PD}=dx\otimes u_{1}+dy\otimes u_{2},$

for some $u_{1},u_{2}\in H^{2}(S,\mathbb{Z})$. We define
$D_{u_{1},u_{2}}=u_{1}^{2}u_{2}^{2}-(u_{1},u_{2})^{2}\in\mathbb{Z}$.

Then the complex attractor equation $\mathrm{Re}(C\Omega_{X})=\gamma^{PD}$ is
equivalent to the following system of equations

$\displaystyle\mathrm{Re}(C\Omega_{S})=u_{1}$
$\displaystyle\mathrm{Re}(C\tau\Omega_{S})=u_{2}$

Before solving the complex attractor equation, we introduce some notations.
The Néron–Severi lattice of $S$ is $NS(S)=H^{2}(X,\mathbb{Z})\cap H^{1,1}(S)$
equipped with the cup product. It is of signature $(1,\rho(S)-1)$. The
transcendental lattice is its complement $T(S)=NS(S)^{\perp}\subset
H^{2}(S,\mathbb{Z})$. It is of signature $(2,20-\rho(S))$ and characterized as
the minimal sublattice of $H^{2}(S,\mathbb{Z})$ whose complexification
contains $\Omega_{S}$.

######  Theorem 2.15 (Moore [23]).

A complex attractor for $\gamma\in H_{3}(X,\mathbb{Z})$ exists if and only if
the lattice $\mathbb{Z}u_{1}+\mathbb{Z}u_{2}$ is positive definite. Moreover,
if it exits, it is uniquely determined by the following periods

$\tau=\frac{(u_{1},u_{2})+\sqrt{-D_{u_{1},u_{2}}}}{u_{1}^{2}},\ \ \
\Omega_{S}=-\sqrt{-1}(\overline{\tau}u_{1}-u_{2}).$

In particular, it is of rank 2.

###### Proof.

By Proposition 2.10, we obtain

$\Omega_{S}=-\sqrt{-1}(\overline{\tau}u_{1}-u_{2}).$

The global Torelli theorem asserts that the K3 surface $S$ is uniquely
determined by $\Omega_{S}\in H^{2,0}(S)$, up to isomorphism. On the other
hand, the Hodge–Riemann bilinear relation $\Omega_{S}\wedge\Omega_{S}=0$
implies

$u_{1}^{2}\tau^{2}-2(u_{1},u_{2})\tau+u_{2}^{2}=0.$

If $u_{1}^{2}=(u_{1},u_{2})=0$, then the transcendental lattice
$T(S)\subset\mathbb{Z}u_{1}+\mathbb{Z}u_{2}$ is degenerate and this is a
contradiction. If $u_{1}^{2}=0$ and $(u_{1},u_{2})\neq 0$, then
$\tau=\frac{u_{2}^{2}}{2(u_{1},u_{2})}\notin\mathbb{H}$ and this is a
contradiction. Hence $u_{1}^{2}\neq 0$ and we get

$\tau=\frac{(u_{1},u_{2})\pm\sqrt{-D_{u_{1},u_{2}}}}{u_{1}^{2}}.$

We need $D_{u_{1},u_{2}}>0$ in order for $\tau$ to lie in $\mathbb{H}$. If
$u_{1}^{2}<0$, $T(S)\subset\mathbb{Z}u_{1}+\mathbb{Z}u_{2}$ is negative
definite and this is a contradiction. Therefore we conclude that
$\mathbb{Z}u_{1}+\mathbb{Z}u_{2}$ is positive definite. In this case, we have

$\tau=\frac{(u_{1},u_{2})+\sqrt{-D_{u_{1},u_{2}}}}{u_{1}^{2}}$

Note that a change of the symplectic basis $dx,dy$ by an element of
$\mathrm{SL}(2,\mathbb{Z})$ does not change $\Omega_{S}$ and $\tau$. ∎

At the complex attractor point, the Néron–Severi lattice is
$NS(S)=(\mathbb{Z}u_{1}+\mathbb{Z}u_{2})^{\perp}$ and the Picard number
$\rho(S)$ obtains its maximal possible value 20. A K3 surface with
$\rho(S)=20$ is known as a singular K3 surface. It admits a rational map of
degree 2 to a Kummer surface constructed from the product of two isogenous
elliptic curves which have complex multiplications (a Shioda–Inose structure)
[24]. Theorem 2.15 shows that any singular K3 surface appears as the second
factor of some complex attractor variety $X=E\times S$.

######  Remark 2.16.

A singular K3 surface is known as a rigid K3 surface as it does not admit any
complex deformation keeping the property $\rho(S)=20$.

######  Remark 2.17.

In the foundational article [7], Dolgachev formulated mirror symmetry for
lattice polarized K3 surfaces. Although his formulation works beautifully in
many case, a singular K3 surface has been an exception. The long-standing
problem of mirror symmetry for singular K3 surfaces is recently settled in
[15] inspired by the results in this article (cf. Section 4.5)

######  Theorem 2.18 ([24]).

There is a bijective correspondence between the isomorphism classes of
singular K3 surfaces and the isomorphism classes of positive definite even
lattices of rank 2.

The bijective correspondence is given by associating a singular K3 surface $S$
with its transcendental lattice $T(S)$.

######  Lemma 2.19.

Let $L$ be a positive definite even lattices of rank 2. Then

$Q_{L}=\Big{\\{}\frac{(u_{1},u_{2})+\sqrt{-D_{u_{1},u_{2}}}}{u_{1}^{2}}\Big{\\}}_{u_{1},u_{2}}\subset\mathbb{H}$

is dense, where $u_{1},u_{2}$ run over $\mathbb{Z}$-linearly independent
vectors in $L$.

###### Proof.

Let us write
$\tau_{u_{1},u_{2}}=\frac{(u_{1},u_{2})+\sqrt{-D_{u_{1},u_{2}}}}{u_{1}^{2}}$.
Then for $k,l\in\mathbb{Z}$ we have the elementary identities:

$\tau_{ku_{1},lu_{2}}=\frac{l}{k}\tau_{u_{1},u_{2}},\ \ \
\tau_{u_{1},ku_{1}+u_{2}}=k+\tau_{u_{1},u_{2}}.$

They show that $Q_{L}\subset\mathbb{H}$ is dense. ∎

######  Theorem 2.20.

For $X=E\times S$, the complex constellation $\mathrm{Attr}_{\mathrm{Cpx}}$ is
dense in the complex moduli space $\mathfrak{M}_{\mathrm{Cpx}}$.

###### Proof.

By a result of Beauville [2], we have a canonical isomorphism
$\mathrm{Aut}(E\times S)=\mathrm{Aut}(E)\times\mathrm{Aut}(S)$. Then there is
a natural fibration
$p:\mathfrak{M}_{\mathrm{Cpx}}\rightarrow\mathfrak{M}_{\mathrm{Cpx}}^{K3}$,
where $\mathfrak{M}_{\mathrm{Cpx}}^{K3}$ denotes the complex moduli space of
K3 surfaces. The image
$p(\mathrm{Attr}_{\mathrm{Cpx}})\subset\mathfrak{M}_{\mathrm{Cpx}}^{S}$
corresponds to the isomorphism classes of singular K3 surfaces and is hence
dense. Therefore it suffices to show that
$p^{-1}(s)\cap\mathrm{Attr}_{\mathrm{Cpx}}\subset
p^{-1}(s)\cong\mathrm{SL}(2,\mathbb{Z})\backslash\mathbb{H}$ is dense where
$s$ corresponds to a singular K3 surface $S$. This assertion follows from the
fact that $Q_{T(S)}\subset\mathbb{H}$ is dense (Lemma 2.19). ∎

The results presented in Sections 2.2 and 2.3 imply that the complex
attractors have maximal Picard numbers possible. This kind of results do not
hold for the Calabi–Yau 3-folds with $h^{2,1}(X)=0$ as their Picard numbers
are topological. Nevertheless, the complex attractors are discrete and the
complex attractor varieties possess complex rigidity.

##  3\. Weil–Petersson geometry on Bridgeland stability space

We provide a brief review of our previous work [8]. It introduced a
provisional mirror Weil–Petersson geometry on the space of Bridgeland
stability conditions $\mathrm{Stab}(\mathcal{D}_{X})$, which can be thought of
as an approximation of the Kähler moduli space $\mathfrak{M}_{\mathrm{Kah}}$.

###  3.1. Bridgeland stability conditions

Let $X$ be a smooth projective variety of dimension $n$. We define
$\mathcal{D}_{X}=\mathrm{D^{b}Coh}(X)$ to be the bounded derived category of
coherent sheaves on $X$.

The numerical Grothendieck group
$\mathcal{N}(\mathcal{D}_{X})=K(\mathcal{D}_{X})/K(\mathcal{D}_{X})^{\perp_{\chi}}$
is the quotient group of the Grothendieck group $K(\mathcal{D}_{X})$ by the
null group $K(\mathcal{D}_{X})^{\perp_{\chi}}$ of the Euler form $\chi$. It is
a free abelian group is of rank $\sum_{i=0}^{n}h^{i,i}(X)$.

######  Definition 3.1 ([4]).

A (Bridgeland) stability condition $\sigma=(\mathcal{Z},\mathcal{P})$ on
$\mathcal{D}_{X}$ consists of a group homomorphism
$\mathcal{Z}:\mathcal{N}(\mathcal{D}_{X})\rightarrow\mathbb{C}$ and a
collection $\mathcal{P}=\\{\mathcal{P}(\phi)\\}_{\phi\in\mathbb{R}}$ of full
additive subcategories of $\mathcal{D}_{X}$ parametrized by
$\phi\in\mathbb{R}$ such that:

  1. (1)

If $0\neq F\in\mathcal{P}(\phi)$, then $\mathcal{Z}(F)\in\mathbb{R}_{>0}\
e^{\sqrt{-1}\pi\phi}$.

  2. (2)

$\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]$.

  3. (3)

If $\phi_{1}>\phi_{2}$ and $A_{i}\in\mathcal{P}(\phi_{i})$, then
$\mathrm{Hom}_{\mathcal{D}_{X}}(A_{1},A_{2})=0$.

  4. (4)

For every $0\neq F\in\mathcal{D}_{X}$, there exists a sequence of exact
triangles

$\textstyle{0=F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

such that $A_{i}\in\mathcal{P}(\phi_{i})$ and
$\phi_{1}>\phi_{2}>\cdots>\phi_{k}$.

  5. (5)

(Support property [18]) There exist a constant $C>0$ and a norm
$|\mkern-2.0mu|-|\mkern-2.0mu|$ on
$\mathcal{N}(\mathcal{D}_{X})\otimes_{\mathbb{Z}}\mathbb{R}$ such that
$|\mkern-2.0mu|F|\mkern-2.0mu|\leq C|\mathcal{Z}(F)|$ for any semistable
object $E$.

$\mathcal{Z}$ is called a central charge and an element
$A\in\mathcal{P}(\phi)$ is called a semistable object of phase $\phi$.

We denote by $\mathrm{Stab}(\mathcal{D}_{X})$ the set of stability conditions
on $\mathcal{D}_{X}$. There is a nice topology on it such that the forgetful
map

$\mathrm{Stab}(\mathcal{D}_{X})\longrightarrow\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C}),\
\ \ \sigma=(\mathcal{Z},\mathcal{P})\mapsto\mathcal{Z}$

is a local homeomorphism [4, 18]. In other words, the deformations of the
central charge lift uniquely to deformations of the stability condition.
Therefore $\mathrm{Stab}(\mathcal{D}_{X})$ naturally becomes a complex
manifold, locally modelled on the $\mathbb{C}$-vector space
$\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})\cong\mathbb{C}^{\sum_{i=0}^{n}h^{i,i}(X)}$.

Moreover, $\mathrm{Stab}(\mathcal{D}_{X})$ naturally carries a right action of
the group $\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$, the universal covering
of the group of orientation-preserving linear transformations
$\mathrm{GL}^{+}(2,\mathbb{R})$, as well as a left action of the group
$\mathrm{Aut}(\mathcal{D}_{X})$ of autoequivalences of $\mathcal{D}_{X}$. The
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-action is given by post-
composition on the central charge
$\mathcal{Z}:\mathcal{N}(\mathcal{D}_{X})\rightarrow\mathbb{C}\cong\mathbb{R}^{2}$
with a suitable relabelling of the phases $\phi$. We often restrict this
action to the subgroup
$\mathbb{C}\subset\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$ which acts
freely.

###  3.2. Central charge via twisted Mukai pairing

The Mukai pairing on $H^{*}(X,\mathbb{C})$ is defined, for $v,w\in
H^{*}(X,\mathbb{C})$,

$\langle v,w\rangle=\int_{X}e^{\frac{c_{1}(X)}{2}}v^{\vee}w,$

where $v=\sum_{j}v_{j}\in\oplus_{j}H^{j}(X,\mathbb{C})$ and its Mukai dual
$v^{\vee}=\sum_{j}\sqrt{-1}^{j}v_{j}$. Note it differs from the usual Mukai
pairing for K3 surfaces by a sign. We define a twisted Mukai vector of
$F\in\mathcal{D}_{X}$ by

$v_{\Lambda}(F)=\mathrm{ch}(F)\sqrt{\mathrm{Td}_{X}}e^{\sqrt{-1}\Lambda}$

for any $\Lambda\in H^{*}(X,\mathbb{C})$ such that $\Lambda^{\vee}=-\Lambda$.
A twisted Mukai pairing is compatible with the Euler pairing; by the
Hirzebruch–Riemann–Roch theorem,

(3.1)
$\chi(E,F)=\int_{X}\mathrm{ch}(E^{\vee})\mathrm{ch}(F)\mathrm{Td}_{X}=\langle
v_{\Lambda}(E),v_{\Lambda}(F)\rangle.$

A geometric twisting $\Lambda_{X}$ compatible with the integral structure on
the quantum cohomology was introduced by Iritani [14] and
Katzarkov–Kontsevich–Pantev [17].

It is called the the log Gamma class and, in the Calabi–Yau case, we can
explicitly write it as

$\Lambda_{X}=-\frac{\zeta(3)}{(2\pi)^{3}}c_{3}(X)+\frac{\zeta(5)}{(2\pi)^{5}}(c_{5}(X)-c_{2}(X)c_{3}(X))+\dots$

For K3 and abelian surfaces, there is no effect of twisting as
$\Lambda_{X}=0$. For Calabi–Yau 3-folds, the modification is given by the
first term, which is familiar in the B-model period computations.

######  Definition 3.2.

We define $v_{X}(F)$ to be the twisted Mukai vector of
$F\in\mathcal{N}(\mathcal{D}_{X})$ associated to the log Gamma class
$\Lambda_{X}$, namely

$v_{X}(F)=\mathrm{ch}(F)\sqrt{\mathrm{Td}_{X}}e^{\sqrt{-1}\Lambda_{X}}$

Let $X$ be a projective Calabi–Yau manifold equipped with a complexified
Kähler parameter

$\omega=B+\sqrt{-1}\kappa\in H^{1,1}(X,\mathbb{C}),$

where $\kappa$ is a Kähler class. The set of such classes is the complexified
Kähler cone $\mathcal{K}^{\mathbb{C}}_{X}$. We set
$q=e^{2\pi\sqrt{-1}\omega}$.

######  Conjecture 3.3 (Bridgeland [4]).

Let $\mathfrak{M}_{\mathrm{Cpx}}$ be the Kähler moduli space of a projective
Calabi–Yau manifold $X$. Then there exists an embedding

$\mathfrak{per}^{\vee}:\mathfrak{M}_{\mathrm{Kah}}\hookrightarrow\mathrm{Aut}(\mathcal{D}_{X})\backslash\mathrm{Stab}(\mathcal{D}_{X})/\mathbb{C}.$

The complexified Kähler cone $\mathcal{K}_{X}^{\mathbb{C}}$ gives a local
chart of $\mathfrak{M}_{\mathrm{Kah}}$ and, near the large volume limit, there
exists a stability condition $\sigma_{\omega}$ with central charge of the form

(3.2)
$\mathcal{Z}_{\sigma_{\omega}}(F)=-\left\langle(2\pi\sqrt{-1})^{-\frac{\deg}{2}}J(-2\pi\sqrt{-1}\omega),v_{X}(F)\right\rangle.$

Here $J(\tau)=J(\tau,1)$ denotes the $J$-function of $X$ evaluated at the
spectral parameter $z=1$ and $\deg$ is the degree operator defined by
$\deg(\alpha)=2p\alpha$ for $\alpha\in H^{p,p}(X)$. This expression was
introduced by Iritani in his study of integral structures of quantum
cohomology [14]. $\mathcal{Z}_{\sigma_{\omega}}$ is called the quantum
cohomology central charge (cf. Hosono [11]). The embedding
$\mathfrak{per}^{\vee}$ is comparable with a period map in mirror symmetry.

The asymptotic behavior of the quantum cohomology central charge
$\mathcal{Z}_{\sigma_{\omega}}(F)$ is given by

$\mathcal{Z}_{\sigma_{\omega}}(F)\sim-\int_{X}e^{-\omega}v_{X}(F)+O(q).$

The existence of a stability condition with the asymptotic central charge
given by the leading term has been proven for various examples including K3
surfaces, abelian surfaces [5], and abelian 3-folds [1, 22].

###  3.3. Weil–Petersson geometry

In this subsection we assume $X$ is a projective Calabi–Yau $n$-fold. Then the
Serre duality implies that for $E,F\in\mathcal{D}_{X}$, there is a natural
functorial isomorphism

$\mathrm{Hom}^{*}_{\mathcal{D}_{X}}(E,F)\cong\mathrm{Hom}^{*}_{\mathcal{D}_{X}}(F,E[n])^{\vee}.$

An important consequence of the Calabi–Yau condition is that the Euler form on
$\mathcal{N}(\mathcal{D}_{X})$ is (skew-) symmetric if $n$ is even (odd). The
following bilinear form is inspired by Proposition 2.1.

######  Definition 3.4.

Let $\\{F_{i}\\}$ be a basis of $\mathcal{N}(\mathcal{D}_{X})$. We define a
bilinear form
$\mathfrak{b}:\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})^{2}\rightarrow\mathbb{C}$
by

$(\mathcal{Z}_{1},\mathcal{Z}_{2})\mapsto\sum_{i,j}\chi^{i,j}\mathcal{Z}_{1}(F_{i})\mathcal{Z}_{2}(F_{j}),$

where $(\chi^{i,j})=(\chi(F_{i},F_{j}))^{-1}$. Then $\mathfrak{b}$ is
independent of the choice of a basis.

We think of $\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})$ as the
tangent space of $\mathrm{Stab}(\mathcal{D}_{X})$ at a point. Therefore
$\mathfrak{b}$ defines a holomorphic symplectic structure on
$\mathrm{Stab}(\mathcal{D}_{X})$ for odd $n$.

######  Definition 3.5.

We define
$\mathrm{Stab}^{+}(\mathcal{D}_{X})\subset\mathrm{Stab}(\mathcal{D}_{X})$ by

$\mathrm{Stab}^{+}(\mathcal{D}_{X})=\\{\sigma=(\mathcal{Z},\mathcal{P})\ |\
\mathfrak{b}(\mathcal{Z},\mathcal{Z})=0,\
(\sqrt{-1})^{-n}\mathfrak{b}(\mathcal{Z},\overline{\mathcal{Z}})>0\\}.$

If $n$ is odd, the first condition is vacuous as $\mathfrak{b}$ is skew-
symmetric.

######  Remark 3.6.

It is worth mentioning that $\mathrm{Stab}^{+}(\mathcal{D}_{X})$ is an
analogue of a period domain in the Hodge theory and the defining equations are
an analogue of the Hodge–Riemann bilinear relations. The natural free
$\mathbb{C}$-action on $\mathrm{Stab}(\mathcal{D}_{X})$ preserves
$\mathrm{Stab}^{+}(\mathcal{D}_{X})$.

######  Definition 3.7.

Let $s=(\mathcal{Z}_{\bar{\sigma}},\mathcal{P}_{\bar{\sigma}})$ be a
holomorphic local section of the $\mathbb{C}$-torsor
$\mathrm{Stab}^{+}(\mathcal{D}_{X})\rightarrow\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$.
Then

(3.3)
$K^{A}(\bar{\sigma})=-\log\left((\sqrt{-1})^{-n}\mathfrak{b}(\mathcal{Z}_{\bar{\sigma}},\overline{\mathcal{Z}_{\bar{\sigma}}})\right)$

defines a local smooth function on
$\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$. We call $K^{A}$ the A-model
Weil–Petersson potential.

######  Proposition 3.8 ([8, Proposition 3.5]).

The complex Hessian
$g^{A}=\frac{\sqrt{-1}}{2}\partial\overline{\partial}K^{A}$ of the A-model
Weil–Petersson potential $K^{A}$ is independent of the choice of a local
section $s$. Moreover, it descends to the quotient

$\mathrm{Aut}(\mathcal{D})\backslash\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$

away from the singular loci.

We call $g^{A}$ the A-model Weil–Petersson metric on
$\mathrm{Aut}(\mathcal{D}_{X})\backslash\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$.
Note that $g^{A}$ is in general a degenerate metric. The following examples
are discussed in [8].

######  Example 3.9.

Let $X$ an elliptic curve. Since
$\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})$-action on
$\mathrm{Stab}(\mathcal{D}_{X})$ is free and transitive, we observe

$\mathrm{Stab}^{+}(\mathcal{D}_{X})=\mathrm{Stab}(\mathcal{D}_{X})\cong\widetilde{\mathrm{GL}}^{+}(2,\mathbb{R})\cong\mathbb{C}\times\mathbb{H}.$

Therefore we conclude

$\mathrm{Aut}(\mathcal{D}_{X})\backslash\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}\cong\mathrm{SL}(2,\mathbb{Z})\backslash\mathbb{H}.$

This is indeed the expected Kähler moduli space of $X$. Up to the
$\mathbb{C}$-action, the central charge at $\tau\in\mathbb{H}$ is given by

$\mathcal{Z}(F)=-\deg(F)+\tau\ \mathrm{rank}(F).$

Since
$K(\mathcal{D}_{X})=\mathbb{Z}\mathcal{O}_{X}\oplus\mathbb{Z}\mathcal{O}_{p}$,
the A-model Weil–Petersson potential is

$\displaystyle K^{A}(\tau)$
$\displaystyle=-\log\left((\sqrt{-1})^{-1}(\mathcal{Z}(\mathcal{O}_{p})\overline{\mathcal{Z}}(\mathcal{O}_{X})-\mathcal{Z}(\mathcal{O}_{X})\overline{\mathcal{Z}}(\mathcal{O}_{p}))\right)$
$\displaystyle=-\log(\mathrm{Im}(\tau))-\log 2.$

This is the Poincaré potential on $\mathbb{H}$ and descends to
$\mathrm{SL}(2,\mathbb{Z})\backslash\mathbb{H}$. Therefore $g^{A}$ is the
Poincaré metric. This computation is compatible with the fact that a mirror of
an elliptic curve is an elliptic curve.

######  Example 3.10.

Let $X$ be the self-product $E_{\tau}\times E_{\tau}$ of an elliptic curve
$E_{\tau}=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$. Then there is an
identification

$\overline{\mathrm{Aut}}_{\mathrm{CY}}(\mathcal{D}_{X})\backslash\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}^{\times}\cong\mathrm{Sp}(4,\mathbb{Z})\backslash\mathfrak{H}_{2}$

where $\overline{\mathrm{Aut}}_{\mathrm{CY}}(\mathcal{D}_{X})$ is an
appropriate group induced by the group of autoequivalences. Moreover, the
A-model Weil–Petersson metric on the LHS is identified with the Bergman metric
on the RHS (a Siegel modular variety). This result is compatible with the
mirror symmetry between $X$ and a principally polarized abelian surface. The
complex moduli space of the latter is given by
$\mathrm{Sp}(4,\mathbb{Z})\backslash\mathfrak{H}_{2}$.

######  Example 3.11.

Let $X\subset\mathbb{P}^{4}$ be a quintic Calabi–Yau 3-fold, for which the
existence of a stability condition is recently proven by Li [21]. Let $\tau
H\in H^{2}(X,\mathbb{C})$ be the complexified Kähler class, where $H$ is the
hyperplane class and $\tau\in\mathbb{H}$. Then we have

$J(2\pi\sqrt{-1}\tau H,z)=e^{\frac{2\pi\sqrt{-1}\tau
H}{z}}(1+\frac{1}{z^{2}}\sum_{d\geq
0}N_{d}^{X}q^{d}d\frac{H^{2}}{5}-\frac{2}{z^{3}}\sum_{d\geq
0}N_{d}^{X}q^{d}\frac{H^{3}}{5})$

where $q=e^{2\pi\sqrt{-1}\tau}$ and $N_{d}^{X}$ denotes the genus 0
Gromov–Witten invariant of $X$ of degree $d$ (cf. [6]). Then we have

$\displaystyle J(2\pi\sqrt{-1}\tau H)=$ $\displaystyle e^{2\pi\sqrt{-1}\tau
H}(1+\frac{1}{5}\sum_{d\geq 0}N_{d}^{X}q^{d}dH^{2}-\frac{2}{5}\sum_{d\geq
0}N_{d}^{X}q^{d}H^{3})$ $\displaystyle=$ $\displaystyle 1+2\pi\sqrt{-1}\tau
H+(\frac{1}{2}(2\pi\sqrt{-1}\tau)^{2}+\frac{1}{5}\sum_{d\geq
0}N_{d}^{X}q^{d}d)H^{2}$
$\displaystyle+(\frac{1}{6}(2\pi\sqrt{-1}\tau)^{3}+\frac{1}{5}2\pi\sqrt{-1}\tau\sum_{d\geq
0}N_{d}^{X}q^{d}d-\frac{2}{5}\sum_{d\geq 0}N_{d}^{X}q^{d})H^{3}.$

Therefore the quantum central charge reads

$\displaystyle\mathcal{Z}_{\sigma_{\tau H}}(F)=$
$\displaystyle-\left\langle(2\pi\sqrt{-1})^{-\frac{\deg}{2}}J(-2\pi\sqrt{-1}\tau),v_{X}(F)\right\rangle$
$\displaystyle=$ $\displaystyle-\int_{X}e^{-\tau
H}v_{X}(F)+2\mathrm{ch}_{0}(F)(\pi\sqrt{-1}\tau\sum_{d\geq
0}N_{d}^{X}q^{d}d+\sum_{d\geq 0}N_{d}^{X}q^{d})$
$\displaystyle+\frac{1}{5}\sum_{d\geq
0}N_{d}^{X}q^{d}d\int_{X}\mathrm{ch}_{1}(F)H^{2}$

Hence, near the large volume limit, the A-model Weil–Petersson potential is
given by

$\displaystyle K^{A}(\tau)=-\log(\frac{2^{3}\cdot
5}{3!}\mathrm{Im}(\tau)^{3})+O(q).$

and hence $g^{A}$ is a quantum deformation of the Poincaré metric. In
particular, for sufficiently small $q$, it is non-degenerate and the
Weil–Petersson distance to the large volume limit is infinite.

##  4\. Kähler attractor mechanism

We will introduce the provisional definition of the Kähler attractor mechanism
mirror to the complex attractor mechanism. Throughout this section $X$ is a
projective Calabi–Yau 3-fold. Let $k=\dim H^{1,1}(X)$ be the expected
dimension of the complexified Kähler moduli space. We define
$H^{ev}(X,\mathbb{C})=\oplus_{i=0}^{3}H^{i,i}(X,\mathbb{C})$.

###  4.1. Kähler attractor mechanism of stability space

We first investigate some fundamental structures on
$\mathrm{Stab}^{+}(\mathcal{D}_{X})$.

######  Definition 4.1.

For $\sigma=(\mathcal{Z},\mathcal{P})\in\mathrm{Stab}^{+}(\mathcal{D}_{X})$,
we define the normalized Kähler central charge of
$F\in\mathcal{N}(\mathcal{D}_{X})$ by

$V(\sigma,F)=e^{\frac{K^{A}(\sigma)}{2}}\mathcal{Z}(F),$

where by abuse of notation $K^{A}(\sigma)$ is given by the Equation (3.3)
($K^{A}(\sigma)$ really depends on $\sigma$, not on the class $\bar{\sigma}$).

Then $V(-,F)$ defines a smooth function

$V(-,F):\mathrm{Stab}^{+}(\mathcal{D}_{X})\longrightarrow\mathbb{C}.$

Since the absolute value $W(\sigma,F)$ is invariant under the
$\mathbb{C}$-action, we obtain a function

$|V(-,F)|:\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}\longrightarrow\mathbb{R}_{\geq
0}.$

We call it the Kähler mass function of $F\in\mathcal{N}(\mathcal{D}_{X})$.

Let us recall some basic facts about
$\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$, a projectivization of a
holomorphic symplectic manifold $\mathrm{Stab}^{+}(\mathcal{D}_{X})$. The
tangent space of $\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$ at
$\sigma=(\mathcal{Z},\mathcal{P})$ is naturally identified with
$\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})/\mathbb{C}\mathcal{Z}$,
and hence $\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$ is a holomorphic
contact manifold with the canonical contact form

$\theta=\mathfrak{b}(d\mathcal{Z},\mathcal{Z}).$

There is a precise relation between Legendrian and Lagrangian submanifolds:
the lift of a Legendrian submanifold in
$\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$ is a Lagrangian submanifold in
$\mathrm{Stab}^{+}(\mathcal{D}_{X})$. Moreover, a Legendrian subspace $L$
defines a polarized Hodge structure of weight 3 on
$\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{Z})$ equipped with
$\mathfrak{b}$, namely

$H^{3,0}=\mathbb{C}\mathcal{Z},\ \ \ H^{3,0}\oplus H^{2,1}=\pi^{-1}(L),$

where
$\pi:\mathrm{Stab}^{+}(\mathcal{D}_{X})\rightarrow\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$
is the quotient map.

######  Theorem 4.2.

Let $F\in\mathcal{N}(\mathcal{D}_{X})$ and
$\sigma=(\mathcal{Z},\mathcal{P})\in\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$
such that $\mathcal{Z}(F)\neq 0$. Given a Legendrian subspace
$L\subset\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})/\mathbb{C}\mathcal{Z}$,
then $\sigma$ is a stationary point of the Kähler mass function $|V(-,F)|$
along the $L$-direction if and only if

$\chi(F,-)=\mathrm{Re}(C\mathcal{Z}(-))\ \ \ (\exists C\in\mathbb{C})$

holds in $\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})$.

###### Proof.

The proof is parallel to that of Theorem 2.3. Let
$L_{1},\ldots,L_{k}\in\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})$
be lifts of a basis of $L$. By Bridgeland’s result [4], the deformation of the
central charge

$\mathcal{Z}_{\epsilon}=\mathcal{Z}+\sum_{i=1}^{k}\epsilon_{i}L_{i}\in\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})$

for small $\epsilon=(\epsilon_{i})_{i=1}^{k}\in\mathbb{C}^{k}$ induces a
unique deformation $\sigma_{\epsilon}$ of the stability condition $\sigma$.
Then a straightforward calculation shows

$\left.\frac{\partial}{\partial\epsilon_{i}}\right|_{\epsilon=0}|V(\sigma_{\epsilon},F)|^{2}=e^{K^{A}(\sigma)}\overline{\mathcal{Z}(F)}L_{i}(F).$

By the assumption that $\mathcal{Z}(F)\neq 0$, $\sigma$ is a stationary point
of $|V(-,E)|^{2}$ along the $L$-direction if and only if $L_{i}(F)=0$ for
$1\leq i\leq k$. Then the Legendrian property of $L$ implies that
$\chi(F,-)\in\mathbb{C}\mathcal{Z}\oplus\mathbb{C}\overline{\mathcal{Z}}$.
Since $\chi(E,-)\in\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{Z})$, we
have

$\chi(F,-)=(C\mathcal{Z}+\overline{C\mathcal{Z}})=\mathrm{Re}(C\mathcal{Z}(-))$

for some $C\in\mathbb{C}$. ∎

###  4.2. Kähler attractor mechanism of complexified Kähler cone

We defined the normalized Kähler central charges on
$\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$. However,
$\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}$ is in general conjectured to
be much larger than (the universal covering of) the Kähler moduli space
$\mathfrak{M}_{\mathrm{Kah}}$ (Conjecture 3.3). Therefore it is reasonable to
restrict ourselves to a submanifold of the expected dimension $k$. In this
section, we will consider the complexified Kähler cone
$\mathcal{K}^{\mathbb{C}}_{X}$, a natural candidate of such a submanifold.

Let $\omega=B+\sqrt{-1}\kappa\in\mathcal{K}^{\mathbb{C}}_{X}$ be a
complexified Kähler class of $X$. Henceforth we consider the quantum
cohomology central charge (Equation (3.2))

$\mathcal{Z}_{\sigma_{\omega}}(F)=-\left\langle\widetilde{J}(\omega),v_{X}(F)\right\rangle.$

where we write
$\widetilde{J}(\omega)=(2\pi\sqrt{-1})^{-\frac{\deg}{2}}J(-2\pi\sqrt{-1}\omega)$
for the sake of shorthand.

Let $\phi_{1},\dots,\phi_{k}\in H^{1,1}(X)$ be a basis and $t_{1},\dots,t_{k}$
the linear coordinate system of $H^{1,1}(X)$ dual to the basis, i.e. we may
write $\omega=\sum_{i=1}^{k}t_{i}\phi_{i}$. Let $L(\omega)$ be the fundamental
solution of the quantum differential equation, that is the
$\mathrm{End}(H^{ev}(X,\mathbb{C}))$-valued function satisfying

$\nabla^{A}L(\omega)=0,\ \ \ L(\omega)=\mathrm{id}+O(\omega),$

where $\nabla^{A}=d+\sum_{i=1}^{k}(\phi_{i}*)dt_{i}$ denotes the Dubrovin
connection on $H^{ev}(X,\mathbb{C})$. Then the $J$-function is obtained by
applying the fundamental solution $L(\omega)$ to $1\in H^{0}(X,\mathbb{Z})$,
i.e. $J(\omega)=L(\omega)1$.

######  Proposition 4.3.

Near the large volume limit (i.e. for sufficiently small $q$),
$\sqrt{-1}\mathfrak{b}(\mathcal{Z}_{\sigma_{\omega}},\overline{\mathcal{Z}_{\sigma_{\omega}}})>0$
holds.

###### Proof.

This is a quantum corrected version of [8, Proposition 4.6], where
$\widetilde{J}(\omega)$ is replaced by $e^{\omega}$. Since
$\widetilde{J}(\omega)=e^{\omega}+O(q)$, the assertion follows. ∎

######  Theorem 4.4.

The conjectural embedding (Conjecture 3.3)

$\iota:\mathcal{K}^{\mathbb{C}}_{X}\longrightarrow\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C},\
\ \
\omega\mapsto\sigma_{\omega}=(\mathcal{Z}_{\sigma_{\omega}},\mathcal{P}_{\sigma_{\omega}})$

is Legendrian (if it exists).

###### Proof.

It suffices to check that

$\mathbb{C}\langle\frac{\partial}{\partial
t_{1}}\mathcal{Z}_{\sigma_{\omega}},\dots,\frac{\partial}{\partial
t_{k}}\mathcal{Z}_{\sigma_{\omega}}\rangle\subset\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})/\mathbb{C}\mathcal{Z}_{\sigma_{\omega}}$

is a Legendrian subspace. This follows from the following calculation.

$\displaystyle\mathfrak{b}(\frac{\partial}{\partial
t_{i}}\mathcal{Z}_{\sigma_{\omega}},\frac{\partial}{\partial
t_{j}}\mathcal{Z}_{\sigma_{\omega}})$
$\displaystyle=-\langle\frac{\partial}{\partial
t_{i}}\widetilde{J}(\omega),\frac{\partial}{\partial
t_{j}}\widetilde{J}(\omega)\rangle$
$\displaystyle=\frac{-1}{(-2\pi\sqrt{-1})^{3}}\langle\frac{\partial}{\partial
t_{i}}J(-2\pi\sqrt{-1}\omega),\frac{\partial}{\partial
t_{j}}J(-2\pi\sqrt{-1}\omega)\rangle$
$\displaystyle=\frac{1}{2\pi\sqrt{-1}}\langle
L(-2\pi\sqrt{-1}\omega)\phi_{i},L(-2\pi\sqrt{-1}\omega)\phi_{j}\rangle$
$\displaystyle=\frac{1}{2\pi\sqrt{-1}}\langle\phi_{i},\phi_{j}\rangle$
$\displaystyle=0$

The first equality is due to [8, Lemma 4.4]. Although [8, Lemma 4.4] is
classical (no quantum correction), an identical proof works. The third
equality follows from the definition of the $J$-function. The fourth equality
follows from [14, Proposition 4.2] (essentially the Frobenius property of the
quantum product). ∎

Motivated by the Theorem 4.4, we define the normalized Kähler central charge
of $F\in\mathcal{N}(\mathcal{D}_{X})$ on $\mathcal{K}^{\mathbb{C}}_{X}$ by

$W(\omega,F)=e^{\frac{K^{A}(\omega)}{2}}\mathcal{Z}_{\sigma_{\omega}}(F),$

where $K^{A}$ denotes the A-model Weil–Petersson potential. Then $W(-,F)$ is a
smooth function

$W(-,F):\mathcal{K}^{\mathbb{C}}_{X}\longrightarrow\mathbb{C}$

and the Kähler mass function of $F\in\mathcal{N}(\mathcal{D}_{X})$ is defined
by

$|W(-,F)|:\mathcal{K}^{\mathbb{C}}_{X}\longrightarrow\mathbb{R}_{\geq 0}.$

To summarize, we have the following commutative diagram

$\textstyle{\mathcal{K}^{\mathbb{C}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{|W(-,F)|}$$\scriptstyle{\iota}$$\textstyle{\mathrm{Stab}^{+}(\mathcal{D}_{X})/\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{|V(-,F)|}$$\textstyle{\mathbb{R}_{\geq
0}}$

where $\iota$ is in general hypothetical, but the Kähler mass functions are
well-defined.

######  Theorem 4.5.

For $F\in\mathcal{N}(\mathcal{D}_{X})$, a stationary point
$\omega\in\mathcal{K}^{\mathbb{C}}_{X}$ of the Kähler mass function such that
$W(\omega,F)\neq 0$ is characterized by the equation

(4.1) $\chi(F,-)=\mathrm{Re}(C\mathcal{Z}_{\sigma_{\omega}}(-)),\ \ \ (\exists
C\in\mathbb{C})$

in $\mathrm{Hom}(\mathcal{N}(\mathcal{D}_{X}),\mathbb{C})$. We call Equation
(4.1) the Kähler attractor equation.

###### Proof.

The assertion follows from Theorem 4.2 and Theorem 4.4. Note that we do not
used $\mathcal{P}_{\sigma_{\omega}}$ but only $\mathcal{Z}_{\sigma_{\omega}}$
in Theorem 4.4. ∎

######  Definition 4.6.

A stationary point $\omega\in\mathcal{K}^{\mathbb{C}}_{X}$ with
$W(\omega,F)\neq 0$ is called a Kähler attractor for $F$. The corresponding
Calabi–Yau 3-fold $(X,\omega)$ is called a Kähler attractor variety for $F$.

######  Remark 4.7.

Near the large volume limit, we have an asymptotic expansion
$K^{A}(\omega)=-\log\mathrm{Im}(\omega)^{3}+O(q)$ (up to constant term). Then
the A-model Weil–Petersson metric $g^{A}$ is positive definite and we can show
that the Kähler attractors are discrete by an almost identical argument to the
complex side (Theorem 2.4). Note that this sort of asymptotic metric has
previously been investigated by Trenner and Wilson [28]. Our work [8] can be
considered as a globalization of their pioneering work.

###  4.3. Kähler attractor mechanism for torus

Let us consider a complex 3-torus
$Y=\mathbb{C}^{3}/(\mathbb{Z}^{3}+\sqrt{-1}\mathbb{Z}^{3})$. We choose a
symplectic basis of
$H^{ev}(Y,\mathbb{Z})=\oplus_{i=0}^{3}H^{i,i}(Y,\mathbb{Z})$ as follows

$\displaystyle\delta_{0}$ $\displaystyle=1\ \in H^{0,0}(Y,\mathbb{Z}),$
$\displaystyle\delta_{ij}$ $\displaystyle=\frac{\sqrt{-1}}{2}dz_{i}\wedge
d\bar{z_{j}}\ \in H^{1,1}(Y,\mathbb{Z})\ \ \ (1\leq i,j\leq 3)$
$\displaystyle\epsilon^{0}$
$\displaystyle=(\frac{\sqrt{-1}}{2})^{3}dz_{1}\wedge d\bar{z_{1}}\wedge
dz_{2}\wedge d\bar{z_{2}}\wedge dz_{3}\wedge d\bar{z_{3}}\in
H^{3,3}(Y,\mathbb{Z})$ $\displaystyle\epsilon^{ij}$
$\displaystyle=(\frac{\sqrt{-1}}{2})^{-1}\frac{\partial}{\partial
z_{i}}\lrcorner(\frac{\partial}{\partial\bar{z_{j}}}\lrcorner\epsilon^{0})\in
H^{2,2}(Y,\mathbb{Z})\ \ \ (1\leq i,j\leq 3).$

We introduce a complexified Kähler structure on $Y$ by

$\omega=B+\sqrt{-1}\kappa=\sum_{i,j=1}^{3}\omega^{ij}\delta_{ij}\in
H^{1,1}(Y,\mathbb{C}).$

and identify $\omega$ with the matrix
$\Omega=(\omega^{ij})\in\mathfrak{H}_{3}$ for the sake of convenience.

The twisted Mukai vector of $F\in\mathcal{N}(\mathcal{D}_{Y})$ has an
expansion

$v_{Y}(F)=v^{0}\delta_{0}+\sum_{i,j}^{3}v^{ij}\delta_{ij}+\sum_{i,j}^{3}u_{ij}\epsilon^{ij}+u_{0}\epsilon^{0}.$

There is no quantum correction ($\mathcal{Z}_{\sigma_{\omega}}(F)=-\langle
e^{\omega},F\rangle$), and hence the Kähler attractor equation
$\chi(F,-)=\mathrm{Re}(C\mathcal{Z}_{\sigma_{\omega}}(-))$ is equivalent to
the following system of equations

$\displaystyle\mathrm{Re}(C)$ $\displaystyle=-v^{0}$
$\displaystyle\mathrm{Re}(C\omega^{ij})$ $\displaystyle=-v^{ij}$
$\displaystyle\mathrm{Re}(C\mathrm{Cof}(\Omega)_{ij})$ $\displaystyle=u_{ij}$
$\displaystyle\mathrm{Re}(C\det(\Omega))$ $\displaystyle=-u_{0}.$

They are parallel to the complex attractor equation for $T^{6}$. We are able
to solve the equations to obtain the solutions in an explicit form.

######  Theorem 4.8.

Assume that the coefficient matrices $V=(v^{ij}),U=(u_{ij})\in
M_{3}(\mathbb{Z})$ are symmetric. There exists a unique Kähler attractor

$\Omega=((2VU-(v^{0}u_{0}+\mathrm{tr}(VU)E_{3}))+\sqrt{-D}E_{3})(2R)^{-1}\in\mathfrak{H}_{3}$

where

$\displaystyle R$ $\displaystyle=\mathrm{Cof}(V)+v^{0}U,$ $\displaystyle D$
$\displaystyle=((\mathrm{tr}(VU))^{2}-\mathrm{tr}((VU)^{2}))-(v^{0}u_{0}+\mathrm{tr}(VU))^{2}+4(v^{0}\det(U)-u_{0}\det(V)).$

###### Proof.

The proof is based on a step-by-step explicit calculation given in Appendix A
and is parallel to the complex attractor case. ∎

In light of the B-model side, we introduce a covering of the Kähler attractor
variety $(Y,\omega)$. The lattice embedding

$\mathbb{Z}^{3}+\sqrt{-1}(2R)\mathbb{Z}^{3}\hookrightarrow\mathbb{Z}^{3}+\sqrt{-1}\mathbb{Z}^{3}$

induces a covering map

$\phi^{\vee}:Y^{\prime}=\mathbb{C}^{3}/(\mathbb{Z}^{3}+\sqrt{-1}(2R)\mathbb{Z}^{3})\longrightarrow
Y=\mathbb{C}^{3}(\mathbb{Z}^{3}+\sqrt{-1}\mathbb{Z}^{3}).$

Then $Y^{\prime}$ carries a natural complexified Kähler structure
$\omega^{\prime}$ given by the pullback

$\Omega^{\prime}=(\phi^{\vee})^{*}\Omega=(2VU-(v^{0}u_{0}+\mathrm{tr}(VU)E_{3}))+\frac{\sqrt{-D}}{2}E_{3}$

If we regard the complexified Kähler structures as elements of
$H^{2}(Y^{\prime},\mathbb{C})/H^{2}(Y^{\prime},\mathbb{Z})$, then the B-field
$\mathrm{Re}(\omega^{\prime})$ becomes trivial and the Kähler structure reads

$\mathrm{Im}(\omega^{\prime})=\frac{\sqrt{D}}{2}\sum_{1\leq i,j\leq
3}\delta_{ij}dz_{i}\wedge d\bar{z_{j}}=\sqrt{D}\sum_{1\leq i\leq
3}dx_{i}\wedge dy_{i}$

Hence $Y^{\prime}$ is a principally polarized abelian 3-fold with the Kähler
structure $\sqrt{D}$. This computation is compatible with the fact that the
product $X^{\prime}=(E_{\sqrt{-D}})^{3}$ of an elliptic curve $E_{\sqrt{-D}}$
is mirror symmetric to a principally polarized abelian 3-fold $Y^{\prime}$
(c.f. [16]).

We conclude that a complex attractor variety $X$ (resp. $X^{\prime}$) is
mirror symmetric to a Kähler attractor variety $Y$ (resp. $Y^{\prime}$)
provided that $\gamma\in H_{3}(X,\mathbb{Z})$ and
$F\in\mathcal{N}(\mathcal{D}_{Y})$ are mirror cycles.

$\textstyle{(E_{\sqrt{-D}})^{3}\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$mirror$\textstyle{\
(\mathbb{C}^{3}/(\mathbb{Z}^{3}+\sqrt{-1}(2R)\mathbb{Z}^{3}),\omega^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\vee}}$$\textstyle{\mathbb{C}/(\mathbb{Z}^{3}+T\mathbb{Z}^{3})}$$\textstyle{(\mathbb{C}^{3}/(\mathbb{Z}^{3}+\sqrt{-1}\mathbb{Z}^{3}),\omega)}$

######  Remark 4.9.

It is expected that if $\Omega$ is a complex attractor of a charge $\gamma$,
then $\gamma$ supports BPS states with respect to $\Omega$ [23, Section 2.6].
In terms of Kähler attractors and stability conditions on the mirror side, one
expects that if $\sigma$ is a Kähler attractor of a class $v$, then $v$ should
support a Bridgeland semistable object with repsect to $\sigma$. The space of
stability conditions on abelian threefolds has been studied in [1]. It would
be interesting to verify that the Kähler attractors found in Theorem 4.8 do
indeed support semistable objects.

###  4.4. Kähler attractor mechanism for $E\times S$

For an elliptic curve $E=\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$ and a K3
surface $S$, we consider the product Calabi–Yau 3-fold $Y=S\times E$. In the
following, we use $\left<-,-\right>$ for the Mukai pairing on
$H^{*}(Y,\mathbb{Z})$, $\left<-,-\right>_{S}$ for the Mukai pairing on
$H^{*}(S,\mathbb{Z})$, and $(-,-)$ for the cup pairing on
$H^{2}(S,\mathbb{Z})$. We also define the algebraic lattice by

$NS^{\prime}(S)=H^{0}(S,\mathbb{Z})\oplus NS(S)\oplus H^{4}(S,\mathbb{Z}).$

The twisted Mukai vector $v_{Y}(F)$ of $F\in\mathcal{N}(\mathcal{D}_{Y})$ can
be written as

$v_{Y}(F)=v_{1}+v_{2}\frac{\sqrt{-1}}{2}dz\wedge d\bar{z}\in
H^{ev}(Y,\mathbb{Z})$

where for $i=1,2$

$v_{i}=(r_{i},D_{i},s_{i})\in NS^{\prime}(S).$

We would like to find $\omega_{S}\in NS(S)_{\mathbb{C}}$ and
$\omega_{E}\in\mathbb{H}$ such that

$\omega=\omega_{S}+\omega_{E}\frac{\sqrt{-1}}{2}dz\wedge d\bar{z}\in
H^{1,1}(Y)$

satisfies the Kähler attractor equation

$\mathrm{Re}(CZ_{\omega}(-))=\left<v_{Y}(F),-\right>$

for some $C\in\mathbb{C}$, where we rewrite the equation by the
Hirzebruch–Riemann–Roch theorem (Equation (3.1)).

######  Lemma 4.10.

The Kähler attractor equation is equivalent to the following system of
equations

$\displaystyle\mathrm{Re}(C\left<\delta,-\right>_{S})$
$\displaystyle=\left<v_{1},-\right>_{S},$
$\displaystyle\mathrm{Re}(C\omega_{E}\left<\delta,-\right>_{S})$
$\displaystyle=\left<v_{2},-\right>_{S},$

where $\delta=e^{\omega_{S}}$.

###### Proof.

Recall first that there is no quantum correction for $Y$, and hence we have
$\widetilde{J}(\omega)=e^{\omega}$. We plug
$\alpha\in\oplus_{i=0}^{2}H^{i,i}(S)$ in the Kähler attractor equation to
obtain

$\left<v_{2},\alpha\right>_{S}=\left<v_{Y}(F),\alpha\right>=\mathrm{Re}(C\left<e^{\omega},\alpha\right>)=\mathrm{Re}(C\omega_{E}\left<\delta,\alpha\right>).$

Similarly for $\beta dz\wedge d\bar{z}$ where
$\beta\in\oplus_{i=0}^{2}H^{i,i}(S)$ we obtain obtain

$\left<v_{1},\beta\right>_{S}=\left<v_{Y}(F),\beta\right>=\mathrm{Re}(C\left<e^{\omega},\beta\right>)=\mathrm{Re}(C\left<\delta,\beta\right>).$

∎

Let us write $D_{v_{1},v_{2}}=v_{1}^{2}v_{2}^{2}-\langle
v_{1},v_{2}\rangle^{2}_{S}\in\mathbb{Z}$, where $v_{i}^{2}=\langle
v_{i},v_{i}\rangle_{S}$.

######  Proposition 4.11.

There exist $\omega_{S}\in H^{1,1}(S)$ and $\omega_{E}\in\mathbb{H}$
satisfying the Kähler attractor equation for $F$ if and only if the lattice
$\mathbb{Z}v_{1}+\mathbb{Z}v_{2}$ is positive definite. Moreover, they are
unique and given by

$\delta=\frac{-\sqrt{-1}}{C\
\mathrm{Im}(\omega_{E})}(v_{2}-\overline{\omega}_{E}v_{1}),\ \ \
\omega_{E}=\frac{\left<v_{1},v_{2}\right>_{S}+\sqrt{-D_{v_{1},v_{2}}}}{v_{1}^{2}}.$

where the constant $C$ is chosen so that the degree 0 part of $\delta$ is
$1\in H^{0}(S,\mathbb{C})$.

###### Proof.

First, by Proposition 2.10, we can solve the Kähler attractor equation to
obtain

$\delta=\frac{\sqrt{-1}}{\mathrm{Im}(|C|^{2}\overline{\omega}_{E})}(\overline{C}v_{2}-\overline{C\omega_{E}}v_{1})=\frac{-\sqrt{-1}}{C\
\mathrm{Im}(\omega_{E})}(v_{2}-\overline{\omega}_{E}v_{1}).$

Assume that there exist $\omega_{S}\in H^{1,1}(S)$ and
$\omega_{E}\in\mathbb{H}$ satisfying the Kähler attractor equation. The
condition that $\delta$ is of the form $\delta=e^{\omega_{S}}$ implies that
$\left<\delta,\delta\right>=0$ and hence

$\omega_{E}^{2}v_{1}^{2}-2\omega_{E}\left<v_{1},v_{2}\right>_{S}+v_{2}^{2}=0.$

By a parallel argument to the proof of Proposition 2.15, we conclude that the
lattice $\mathbb{Z}v_{1}+\mathbb{Z}v_{2}$ is positive definite and

$\omega_{E}=\frac{\left<v_{1},v_{2}\right>_{S}+\sqrt{-D_{v_{1},v_{2}}}}{v_{1}^{2}}\in\mathbb{H}.$

On the other hand, assume that $\mathbb{Z}v_{1}+\mathbb{Z}v_{2}$ is positive
definite, then $(r_{1},r_{2})\neq(0,0)$. Moreover, we can choose a constant
$C$ so that the degree 0 part of

$\delta=\frac{-\sqrt{-1}}{C\
\mathrm{Im}(\omega_{E})}(v_{2}-\overline{\omega}_{E}v_{1})$

is $1$, and then $\delta=e^{\omega_{S}}$ can be solved for
$\omega_{S}=\log\delta\in H^{1,1}(S)$. $\omega_{E}$ is uniquely determined in
a similar manner. ∎

Note that $\omega_{S}\in H^{1,1}(S)$ in Proposition 4.11 is not necessarily a
complexified Kähler class. The best we could prove is the following (cf.
Remark 4.16).

######  Proposition 4.12.

Assume the lattice $\mathbb{Z}v_{1}+\mathbb{Z}v_{2}$ is positive definite. For
$\omega_{S}=\log\delta\in H^{1,1}(S)$, we have
$\mathrm{Im}(\omega_{S})^{2}>0$.

###### Proof.

Recall that we write $v_{i}=(r_{i},D_{i},s_{i})\in NS^{\prime}(S)$. A
straightforward computation of shows that
$\mathrm{Im}(\omega_{S})^{2}=(r_{2}D_{1}-r_{1}D_{2})^{2}$. Since
$\mathbb{Z}v_{1}+\mathbb{Z}v_{2}$ is positive definite,
$(r_{1},r_{2})\neq(0,0)$. Assume first that $r_{1}r_{2}\neq 0$. Then, since
$\mathbb{Z}v_{1}+\mathbb{Z}v_{2}$ is positive definite

$0<(\frac{1}{r_{1}}v_{1}-\frac{1}{r_{2}}v_{2})^{2}=(0,\frac{1}{r_{1}}D_{1}-\frac{1}{r_{2}}D_{2},\frac{s_{1}}{r_{1}}-\frac{s_{2}}{r_{2}})^{2}=(\frac{1}{r_{1}}D_{1}-\frac{1}{r_{2}}D_{2})^{2}.$

Therefore we have

$\mathrm{Im}(\omega_{S})^{2}=(r_{1}r_{2})^{2}(\frac{1}{r_{1}}D_{1}-\frac{1}{r_{2}}D_{2})^{2}>0.$

Assume next that $r_{1}=0$ and $r_{2}\neq 0$. Then we have

$\mathrm{Im}(\omega_{S})^{2}=r_{2}^{2}D_{1}^{2}=r_{2}^{2}v_{1}^{2}>0.$

Similarly for $r_{1}\neq 0$ and $r_{2}=0$. ∎

######  Example 4.13.

Let $S$ be a K3 surface such that $NS(S)=\mathbb{Z}H$, where $H$ is ample with
$H^{2}=2n>0$. It is known that for $v\in NS^{\prime}(S)$ with $v^{2}>0$ there
is a sheaf $\mathcal{E}$ such that $v_{S}(\mathcal{E})=v$. In particular there
are sheaves $\mathcal{E}_{1},\mathcal{E}_{2}$ whose twisted Mukai vectors are

$v_{1}=(1,0,-n),\ v_{2}=(0,-H,0)\in NS^{\prime}(S).$

For $Y=S\times E$, let us consider

$F=\mathcal{E}_{1}\boxtimes\mathcal{O}_{E}\oplus\mathcal{E}_{2}\boxtimes\mathcal{O}_{p}\in\mathcal{N}(Y).$

Then

$e^{\sqrt{-1}H}\in(\mathbb{Z}v_{1}+\mathbb{Z}v_{2})_{\mathbb{C}},\ \ \
\omega_{E}=\sqrt{-1}\in\mathbb{H}$

satisfy the Kähler attractor equation for $F$.

Let us take a close look at Example 4.13.

$\displaystyle e^{\sqrt{-1}H}$ $\displaystyle=(1,\sqrt{-1}H,-n)$
$\displaystyle=v_{1}+\sqrt{-1}v_{2}\in(\mathbb{Z}v_{1}+\mathbb{Z}v_{2})_{\mathbb{C}}\subsetneq
NS^{\prime}(S)_{\mathbb{C}}$

On the other hand, for $\epsilon^{2}\notin\mathbb{Q}$,

$\displaystyle e^{\sqrt{-1}\epsilon H}$ $\displaystyle=(1,\sqrt{-1}\epsilon
H,-\epsilon^{2}n)$
$\displaystyle=(1,0,-\epsilon^{2}n)+\sqrt{-1}\epsilon(0,H,0)$
$\displaystyle=(1,0,0)-\epsilon^{2}(0,0,n)+\sqrt{-1}\epsilon(0,H,0)\in
NS^{\prime}(S)_{\mathbb{C}}.$

Hence there is no proper sublattice $L\subsetneq NS^{\prime}(S)$ such that
$e^{\sqrt{-1}\epsilon H}\in L_{\mathbb{C}}$. Therefore the Kähler structure
$H$ is not deformable in such a way that $e^{\omega_{S}}\in L_{\mathbb{C}}$
for some lattice $L\subset NS^{\prime}(S)$ of rank 2. This calculation
illustrates that $e^{B+\sqrt{-1}\omega}$ is able to detect a fine integral
structure of the Kähler moduli space.

######  Definition 4.14.

A complexified Kähler structure $\omega_{S}$ is called Kähler rigid if there
exists a rank $2$ lattice $L\subset NS^{\prime}(S)$ such that
$e^{\omega_{S}}\in L_{\mathbb{C}}$.

######  Theorem 4.15 (cf. [15]).

A complexified Kähler structure $B+\sqrt{-1}\kappa\in H^{1,1}(S)$ is Kähler
rigid if and only if $B\in H^{1,1}(S,\mathbb{Q})$ and $\kappa^{2}\in
H^{4}(S,\mathbb{Q})$

###### Proof.

We consider an existence condition of a rank $2$ sublattice $L\subset
H^{*}(M,\mathbb{Z})$ such that

$e^{B+\sqrt{-1}\kappa}=1+B+\frac{1}{2}(B^{2}-\kappa^{2})+\sqrt{-1}(\kappa+B\wedge\kappa)\in
L_{\mathbb{C}}.$

First, $B$ needs to be rational, and hence so is $\kappa^{2}$. Then we may
write $\kappa=kH$ for $k^{2}\in\mathbb{Q}$ and $H\in H^{2}(S,\mathbb{Z})$ with
$H^{2}>0$. Indeed, in this case, there exist $m,n\in\mathbb{N}$ such that

$m\mathrm{Re}(e^{B+\sqrt{-1}kH}),\ n\mathrm{Im}(e^{B+\sqrt{-1}kH})\in
H^{*}(S,\mathbb{Z}).$

and the complexification $L_{\mathbb{C}}$ of the lattice

$L=\mathbb{Z}m\mathrm{Re}(e^{B+\sqrt{-1}kH})+\mathbb{Z}n\mathrm{Im}(e^{B+\sqrt{-1}kH})\subset
H^{*}(S,\mathbb{Z})$

contains $e^{B+\sqrt{-1}kH}$. ∎

It is natural to expect that a Kähler rigid K3 surface is mirror to a singular
K3 surface. However, there is an obvious puzzle. The dimension of the Kähler
moduli space of a singular K3 surface is $20$, while the dimension of the
complex moduli space of a Kähler rigid K3 surface is at most $19$. It turns
out that the correct framework of mirror symmetry for K3 surfaces is the
generalized Calabi–Yau structures developed by Hitchin [10] and Huybrechts
[12]. To solve the above puzzle, we need to incorporate deformations as a
generalized K3 surface (namely a Kähler rigid K3 surface should be defined as
a generalized K3 surface). A recent article [15] investigates mirror symmetry
for generalized K3 surfaces with particular emphasis on complex and Kähler
rigid structures.

######  Remark 4.16.

As to Proposition 4.12, the reason why the imaginary part of
$\omega_{S}=\log\delta\in H^{1,1}(S)$ is not necessarily Kähler but merely
positive is also well-explained from the viewpoint of the generalized
Calabi–Yau structures.

###  4.5. Kähler constellation

From the perspective of homological mirror symmetry, the derived equivalent
Calabi–Yau 3-folds share the same mirror Calabi–Yau 3-fold (remember that
birationality implies derived equivalence for Calabi–Yau 3-folds [3]). It is a
folklore conjecture that the complexified Kähler cones of the derived
equivalent Calabi–Yau 3-folds give local charts of the Kähler moduli space
$\mathfrak{M}_{\mathrm{Kah}}$ (Conjecture 3.3). Indeed, the union of the
Kähler cones of birational Calabi–Yau 3-folds form a cone, known as the
movable cone, and has been extensively studied in birational geometry. From
this point of view, the Kähler constellation $\mathrm{Attr}_{\mathrm{Kah}}$ of
a Calabi–Yau 3-fold $Y$ is defined as the union of the Kähler attractors of
Calabi–Yau 3-folds derived equivalent to $Y$.

In light of mirror symmetry, if $X$ and $Y$ are mirror Calabi–Yau 3-folds,
then the mirror map should induce a bijective correspondence between the
complex constellation $\mathrm{Attr}_{\mathrm{Cpx}}^{X}$ of $X$ and the Kähler
constellation $\mathrm{Attr}_{\mathrm{Kah}}^{Y}$ of $Y$.

$\textstyle{\mathfrak{M}_{\mathrm{Cpx}}^{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\cong}$$\textstyle{\mathfrak{M}_{\mathrm{Kah}}^{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bigcup}$$\textstyle{\mathrm{Attr}^{X}_{\mathrm{Cpx}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cong}$$\textstyle{\mathrm{Attr}^{Y}_{\mathrm{Kah}}}$

The mirror correspondence of the complex rigid and Kähler rigid K3 surfaces is
one occurrence of such [15] (cf. Section 2.3 and Section 4.4).

## Appendix A

Let $p^{0},q_{0}\in\mathbb{R}$ and $P=(P^{ij}),Q=(Q_{ij})\in
M_{3}(\mathbb{R})$. We investigate conditions on $p^{0},q_{0},P,Q$ under which
there exist $C\in\mathbb{C}$ and $A=(A^{ij})\in M_{3}(\mathbb{C})$ such that
the following system of equations hold.

(A.1) $\displaystyle\mathrm{Re}(C)$ $\displaystyle=p^{0},$ (A.2)
$\displaystyle\mathrm{Re}(CA^{ij})$ $\displaystyle=P^{ij},$ (A.3)
$\displaystyle\mathrm{Re}(C\mathrm{Cof}(A)_{ij})$ $\displaystyle=-Q_{ij},$
(A.4) $\displaystyle\mathrm{Re}(C\det(A))$ $\displaystyle=q_{0}.$

Here $\mathrm{Cof}(A)_{ij}$ denotes the $(i,j)$-th entry of the cofactor
matrix of $A$.

We first define

$\displaystyle R\coloneqq$ $\displaystyle\mathrm{Cof}(P)+p^{0}Q,$
$\displaystyle M\coloneqq$ $\displaystyle
2\det(P)+(p^{0})^{2}q_{0}+p^{0}\mathrm{tr}(P^{T}Q),$ $\displaystyle
D\coloneqq$ $\displaystyle
2\left((\mathrm{tr}P^{T}Q)^{2}-\mathrm{tr}((P^{T}Q)^{2})\right)-(p^{0}q_{0}+\mathrm{tr}(P^{T}Q))^{2}$
$\displaystyle\ +4(p^{0}\det(Q)-q_{0}\det(P)).$

######  Lemma A.1.

We have the identity

(A.5) $4\det(R)-M^{2}=(p^{0})^{2}D.$

###### Proof.

We first assume that $P$ is invertible. Then $\det(R)$ can be expressed as

$\displaystyle\det\left(\mathrm{Cof}(P)+p^{0}Q\right)$
$\displaystyle=\det(P)^{-1}\det\left(P^{T}\mathrm{Cof}(P)+p^{0}P^{T}Q\right)$
$\displaystyle=\det(P)^{2}\det\left(E_{3}+\frac{p^{0}}{\det(P)}P^{T}Q\right)$

where $E_{3}\in M_{3}(\mathbb{C})$ denotes the identity matrix. Using the
standard formula

$\det(E_{3}+B)=1+\mathrm{tr}(B)+\frac{1}{2}\left((\mathrm{tr}B)^{2}-\mathrm{tr}(B^{2})\right)+\det(B)$

for $B\in M_{3}(\mathbb{C})$, we can check by direct computation that the
Equation (A.5) holds.

Since the Equation (A.5) is an algebraic identity for
$(p^{0},q_{0},P,Q)\in\mathbb{R}^{2}\times M_{3}(\mathbb{R})^{2}$ which holds
in the open dense subset $\\{\det(P)\neq 0\\}\subset\mathbb{R}^{2}\times
M_{3}(\mathbb{R})^{2}$, it remain valid in $\mathbb{R}^{2}\times
M_{3}(\mathbb{R})^{2}$. ∎

######  Theorem A.2.

Suppose that $p^{0},q_{0},P,Q$ are not all zero. The following two statements
are equivalent:

  1. (1)

There exists $(C,A)$ satisfying (A.1)-(A.4) in which $\mathrm{Im}(A)$ is
invertible;

  2. (2)

$\det(R)>0$ and $D>0$.

In this case, there are exactly two solutions $(C,A)$, given by

$\displaystyle C$ $\displaystyle=p^{0}\pm\sqrt{-1}\frac{M}{\sqrt{D}},$
$\displaystyle A$
$\displaystyle=\left(2PQ^{T}-(p^{0}q_{0}+\mathrm{tr}(P^{T}Q))E_{3}\mp\sqrt{-D}\right)(2R)^{-1,T}.$

Moreover, the following two statements are equivalent:

  1. (1)

There exists $(C,A)$ satisfying (A.1)-(A.4) in which $A$ is symmetric and
$\mathrm{Im}(A)$ is positive definite;

  2. (2)

$P$ and $Q$ are symmetric, $R$ is positive definite, and $D>0$.

In this case, the unique solution is given by

$\displaystyle C$ $\displaystyle=p^{0}-\sqrt{-1}\frac{M}{\sqrt{D}},$
$\displaystyle A$
$\displaystyle=\left(2PQ-(p^{0}q_{0}+\mathrm{tr}(PQ))E_{3}+\sqrt{-D}\right)(2R)^{-1}.$

###### Proof.

We first prove the theorem under the assumption that $p^{0}\neq 0$ and $M\neq
0$. Suppose that $(C,A)$ satisfies the Equation (A.1)-(A.4) and
$\mathrm{Im}(A)$ is invertible. By the Equation (A.1), $C\in\mathbb{C}$ can be
written as

$C=p^{0}+i\zeta^{0}$

for some $\zeta^{0}\in\mathbb{R}$. We denote

$X\coloneqq\mathrm{Im}(A)\in M_{3}(\mathbb{R}).$

The Equation (A.2) gives

$\displaystyle P^{ij}$
$\displaystyle=\mathrm{Re}\left((p^{0}+i\zeta^{0})(\mathrm{Re}(A^{ij})+\sqrt{-1}\mathrm{Im}(A^{ij}))\right)$
$\displaystyle=p^{0}\mathrm{Re}(A^{ij})-\zeta^{0}\mathrm{Im}(A^{ij}).$

Since we assume $p^{0}\neq 0$, therefore

$\mathrm{Re}(A^{ij})=\frac{1}{p^{0}}(P^{ij}+\zeta^{0}X^{ij}).$

Using these expressions of the real and imaginary parts of $A$, we have

$\displaystyle\mathrm{Re}(CA^{ij}A^{kl})$
$\displaystyle=\mathrm{Re}\left((p^{0}+i\zeta^{0})(\mathrm{Re}(A^{ij})+\sqrt{-1}\mathrm{Im}(A^{ij}))(\mathrm{Re}(A^{kl})+\sqrt{-1}\mathrm{Im}(A^{kl}))\right)$
$\displaystyle=\frac{1}{p^{0}}(P^{ij}P^{kl}-|C|^{2}X^{ij}X^{kl})$

for any $1\leq i,j,k,l\leq 3$. Together with the Equation (A.3), we have

$-Q=\mathrm{Re}(C\mathrm{Cof}(A))=\frac{1}{p^{0}}(\mathrm{Cof}(P)-|C|^{2}\mathrm{Cof}(X)).$

Hence

(A.6)
$\mathrm{Cof}(X)=\frac{1}{|C|^{2}}(\mathrm{Cof}(P)+p^{0}Q)=\frac{1}{|C|^{2}}R.$

Since $X=\mathrm{Im}(A)$ is invertible, we have

$\det(\mathrm{Cof}(X))=\det(X)^{2}>0$

therefore $\det(R)>0$.

Using again the expressions of $\mathrm{Re}(A)$ and $\mathrm{Im}(A)$, we have

$\mathrm{Re}(CA^{ij}A^{kl}A^{mn})=\frac{1}{(p^{0})^{2}}\Big{(}-2\zeta^{0}|C|^{2}X^{ij}X^{kl}X^{mn}+P^{ij}P^{kl}P^{mn}\\\
-|C|^{2}(P^{ij}X^{kl}X^{mn}+P^{kl}X^{ij}X^{mn}+P^{mn}X^{ij}X^{kl})\Big{)}$

Together with the Equation (A.4) and (A.6), we obtain

$\displaystyle q_{0}$ $\displaystyle=\mathrm{Re}(C\det(A)$
$\displaystyle=\frac{1}{(p^{0})^{2}}\left(-2\zeta^{0}|C|^{2}\det(X)+\det(P)-|C|^{2}\mathrm{tr}(P^{T}\mathrm{Cof}(X))\right)$
$\displaystyle=\frac{1}{(p^{0})^{2}}\left(-2\zeta^{0}|C|^{2}\det(X)+\det(P)-\mathrm{tr}\left(P^{T}(\mathrm{Cof}(P)+p^{0}Q)\right)\right)$
$\displaystyle=\frac{1}{(p^{0})^{2}}\left(-2\zeta^{0}|C|^{2}\det(X)-2\det(P)-p^{0}\mathrm{tr}(P^{T}Q)\right).$

Hence

(A.7) $2\zeta^{0}|C|^{2}\det(X)=-M$

by the definition of $M$. Since we assume that $M\neq 0$, therefore
$\zeta^{0}\neq 0$. Using again Equation (A.6), we have

$\frac{M^{2}}{4(\zeta^{0})^{2}|C|^{4}}=\det(X)^{2}=\det(\mathrm{Cof}(X))=\frac{1}{|C|^{6}}\det(R).$

By Lemma A.1,

$(p^{0})^{2}D=4\det(R)-M^{2}=M^{2}\left(\frac{|C|^{2}}{(\zeta^{0})^{2}}-1\right)=\frac{M^{2}(p^{0})^{2}}{(\zeta^{0})^{2}}.$

Hence $D>0$ and

$\zeta^{0}=\pm\frac{M}{\sqrt{D}}.$

By Equation (A.6) and (A.7), we have

$X=|C|^{2}\det(X)R^{-1,T}=\mp\sqrt{D}(2R)^{-1,T}.$

Hence

$\displaystyle\mathrm{Re}(A)$ $\displaystyle=\frac{1}{p^{0}}(P+\zeta^{0}X)$
$\displaystyle=\frac{1}{p^{0}}(P-\frac{M}{2}R^{-1,T})=\frac{1}{p^{0}}(PR^{T}-\frac{M}{2}E_{3})R^{-1,T}$
$\displaystyle=\left(PQ^{T}-\frac{1}{2}\left(p^{0}q_{0}+\mathrm{tr}\left(P^{T}Q\right)\right)E_{3}\right)R^{-1,T}.$

Therefore

$A=\left(2PQ^{T}-(p^{0}q_{0}+\mathrm{tr}(P^{T}Q))E_{3}\mp\sqrt{-D}\right)(2R)^{-1,T}.$

This proves the statement under the assumption that $p^{0}\neq 0$ and $M\neq
0$.

Next we assume that $p^{0}\neq 0$ and $M=0$. Then the above argument up to the
Equation A.7 is still valid. By Lemma A.1 and (A.6), we have

$D=\frac{4\det(R)}{(p^{0})^{2}}>0.$

Now assume that $p^{0}=0$. First we observe that the solutions are still valid
in this case. Hence it suffices to show that they are the only two solutions.
Since $p^{0},q_{0},P,Q$ are not all zero, we have $c=i\zeta^{0}\neq 0$. Then
Equation (A.2) gives

$X\coloneqq\mathrm{Im}(A)=\frac{-1}{\zeta^{0}}P.$

Note that $P$ is invertible since $X=\mathrm{Im}(A)$ is assumed to be
invertible. We denote

$Y\coloneqq\mathrm{Re}(A).$

Then

$\displaystyle\mathrm{Re}(CA^{ij}A^{kl})$
$\displaystyle=\mathrm{Re}\left((i\zeta^{0})(\mathrm{Re}(A^{ij})+\sqrt{-1}\mathrm{Im}(A^{ij}))(\mathrm{Re}(A^{kl})+\sqrt{-1}\mathrm{Im}(A^{kl}))\right)$
$\displaystyle=P^{ij}Y^{kl}+P^{kl}Y^{ij}.$

By Equation (A.3), we have

$\begin{pmatrix}0&0&0&0&P^{33}&-P^{32}&0&-P^{23}&P^{22}\\\
0&0&0&-P^{33}&0&P^{31}&P^{23}&0&-P^{21}\\\
0&0&0&P^{32}&-P^{31}&0&-P^{22}&P^{21}&0\\\
0&-P^{33}&P^{32}&0&0&0&0&P^{13}&-P^{12}\\\
P^{33}&0&-P^{31}&0&0&0&-P^{13}&0&P^{11}\\\
-P^{32}&P^{31}&0&0&0&0&P^{12}&-P^{11}&0\\\
0&P^{23}&-P^{22}&0&-P^{13}&P^{12}&0&0&0\\\
-P^{23}&0&P^{21}&P^{13}&0&-P^{11}&0&0&0\\\
P^{22}&-P^{21}&0&-P^{12}&P^{11}&0&0&0&0\\\
\end{pmatrix}\begin{pmatrix}Y^{11}\\\ Y^{12}\\\ Y^{13}\\\ Y^{21}\\\ Y^{22}\\\
Y^{23}\\\ Y^{31}\\\ Y^{32}\\\ Y^{33}\\\
\end{pmatrix}=-\begin{pmatrix}Q_{11}\\\ Q_{12}\\\ Q_{13}\\\ Q_{21}\\\
Q_{22}\\\ Q_{23}\\\ Q_{31}\\\ Q_{32}\\\ Q_{33}\\\ \end{pmatrix}$

The $9\times 9$ matrix on the left hand side has determinant
$-2(\det(P))^{3}\neq 0$, hence $Y$ can solved uniquely.

$\mathrm{Re}(CA^{ij}A^{kl}A^{mn})=(P^{ij}Y^{kl}Y^{mn}+P^{kl}Y^{ij}Y^{mn}+P^{mn}Y^{ij}Y^{kl})-\frac{1}{(\zeta^{0})^{2}}P^{ij}P^{kl}P^{mn}.$

By Equation (A.4),

$q_{0}=\mathrm{tr}(P^{T}\mathrm{Cof}(Y))-\frac{1}{(\zeta^{0})^{2}}\det(P).$

This solves $\zeta^{0}$ up to sign. This proves that there are no other
solutions. ∎

## Appendix B

######  Theorem B.1 (Thoerem 2.14).

The complex constellation $\mathrm{Attr}_{\mathrm{Cpx}}$ bijectively
corresponds to the abelian 3-folds with Picard number $9$.

###### Proof.

It suffices to show that an abelian 3-fold
$X_{T}=\mathbb{C}^{3}/(\mathbb{Z}^{3}+T\mathbb{Z}^{3})$ with $\rho(X_{T})=9$
is a complex attractor variety. By Theorem 2.12, $X_{T}$ is isogenous to
$(E_{\sqrt{-D}})^{3}$ for some $D\in\mathbb{N}$. Therefore there exists a
lattice embedding
$\mathbb{Z}^{3}+\sqrt{-D}\mathbb{Z}^{3}\hookrightarrow\mathbb{Z}^{3}+T\mathbb{Z}^{3}$
induced by $R\in M_{3}(\mathbb{Z})$ such that

$TR\in M_{3}(\mathbb{Z})+\sqrt{-D}E_{3}.$

Then $R=\sqrt{D}\mathrm{Im}(T)^{-1}$ is symmetric and positive definite. We
will find a $3$-cycle $\gamma\in H_{3}(X_{T},\mathbb{Z})$ for which $T$ is a
complex attractor point. We write $\gamma$ as

$\gamma\
=q_{0}A_{0}+\sum_{i,j=1}^{3}Q_{ij}A_{ij}+\sum_{i,j=1}^{3}P^{ij}B^{ij}+p^{0}B^{0}$

as in Section 2.2. We introduce

  * •

$n\coloneqq(D+1)\det(R)\in\mathbb{N}$.

  * •

$M\coloneqq 2n\det(R)\in\mathbb{N}$.

  * •

$S\coloneqq 2n(TR-\sqrt{-D}E_{3})=2n\mathrm{Re}(T)R\in M_{3}(\mathbb{Z})$.

  * •

$p^{0}\coloneqq\det(R)\in\mathbb{Z}$.

  * •

$P\coloneqq(p^{0}S+ME_{3})(2nR)^{-1}\in M_{3}(\mathbb{Q})$.

  * •

$Q\coloneqq\frac{1}{p^{0}}(nR-\mathrm{Cof}(P))\in M_{3}(Q)$.

  * •

$q_{0}\coloneqq\frac{1}{(p^{0})^{2}}(2n\det(R)-2\det(P)-p^{0}\mathrm{tr}(PQ))\in\mathbb{Q}$.

Now we show that $T$ is an attractor of $\gamma$ given by $(p^{0},P,Q,q_{0})$.
Define as in Appendix A:

$\displaystyle\widetilde{R}\coloneqq$ $\displaystyle\mathrm{Cof}(P)+p^{0}Q,$
$\displaystyle\widetilde{M}\coloneqq$ $\displaystyle
2\det(P)+(p^{0})^{2}q_{0}+p^{0}\mathrm{tr}(PQ),$
$\displaystyle\widetilde{D}\coloneqq$ $\displaystyle
2\left((\mathrm{tr}PQ)^{2}-\mathrm{tr}((PQ)^{2})\right)-(p^{0}q_{0}+\mathrm{tr}(PQ))^{2}$
$\displaystyle\ +4(p^{0}\det(Q)-q_{0}\det(P)).$

Then we have

$\widetilde{R}=nR,\ \ \ \widetilde{M}=2n\det(R)=M.$

Moreover, by Lemma A.1,

$\displaystyle\widetilde{D}$
$\displaystyle=\frac{1}{(p^{0})^{2}}(4\det(\widetilde{R})-\widetilde{M}^{2})$
$\displaystyle=\frac{1}{\det(R)^{2}}(4n^{3}\det(R)-4n^{2}\det(R)^{2})$
$\displaystyle=4(D+1)^{2}\det(R)(n-\det(R))$
$\displaystyle=4D(D+1)^{2}\det(R)^{2}$ $\displaystyle=4n^{2}D>0.$

By Theorem A.2, an attractor is given by

$\widetilde{T}=\left(2PQ-(p^{0}q_{0}+\mathrm{tr}(PQ))E_{3}+\sqrt{-\widetilde{D}}\right)(2\widetilde{R})^{-1}.$

Hence

$\mathrm{Im}(\widetilde{T})=\sqrt{\widetilde{D}}(2\widetilde{R})^{-1}=\sqrt{D}R^{-1}=\mathrm{Im}(T),$

and

$\displaystyle 2n\mathrm{Re}(\widetilde{T})R$
$\displaystyle=2PQ-(p^{0}q_{0}+\mathrm{tr}(PQ))E_{3}$
$\displaystyle=\frac{2}{p^{0}}(P\widetilde{R}-\det(P)E_{3})-\frac{1}{p^{0}}(\widetilde{M}-2\det(P))E_{3}$
$\displaystyle=\frac{1}{p^{0}}(2P\widetilde{R}-\widetilde{M}E_{3})$
$\displaystyle=\frac{1}{p^{0}}(2(p^{0}S+ME_{3})(2nR)^{-1}(nR)-ME_{3})$
$\displaystyle=S=2n\mathrm{Re}(T)R$

since $P\widetilde{R}=\det(P)E_{3}+p^{0}PQ$. Therefore we have
$\widetilde{T}=T$, so $T$ is an attractor. ∎

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Department of Mathematics, Evans Hall
University of California, Berkeley
Berkeley, CA 94720

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Faculty of Policy Management, Keio University
Endo 5322, Fujisawa, Kanagawa, 252-0882, Japan

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