# Mirror symmetry and rigid structures of generalized K3 surfaces

Atsushi Kanazawa

###### Abstract.

The present article is concerned with mirror symmetry for generalized K3
surfaces, with particular emphasis on complex and Kähler rigid structures.
Inspired by the works of Dolgachev, Aspinwall–Morrison and Huybrechts, we
introduce a formulation of mirror symmetry for generalized K3 surfaces. As an
application, we settle the long-standing problem of mirror symmetry for the
Shioda–Inose K3 surfaces. Along the way, we investigate complex and Kähler
rigid structures of generalized K3 surfaces. The Kähler rigid structures
naturally appear when we consider generalized Calabi–Yau structures.

##  1\. Introduction

The present article is concerned with mirror symmetry for generalized K3
surfaces, with particular emphasis on complex and Kähler rigid structures. A
few important aspects of mirror symmetry for K3 surfaces are unified in the
framework of generalized K3 surfaces.

Mirror symmetry for a K3 surface $S$ is a very subtle problem as the complex
and Kähler structures are somewhat mixed as they both live in
$H^{2}(S,\mathbb{C})$. In the foundational article [2], Dolgachev formulated
mirror symmetry for lattice polarized K3 surfaces as a certain duality between
the algebraic and transcendental cycles. Although his formulation works
beautifully in many case, it cannot be a definitive one as there is an
artificial assumption which does not hold in general. Geometrically an obvious
exception corresponds to a Shioda–Inose K3 surface, which is a K3 surface with
the maximal Picard number $20$. It is also known as a rigid K3 surface in the
sense that it admits no deformation of complex structure keeping the maximal
Picard number. A satisfactory formulation of mirror symmetry for K3 surfaces,
which in particular should solve the puzzle of that for Shioda–Inose K3
surfaces, has been long-anticipated.

The celebrated work [1] of Aspinwall–Morrison is the key articles explaining
mirror symmetry for K3 surfaces from a physics point of view. They discussed
SCFTs on a K3 surface $S$ and showed that the moduli space
$\mathfrak{M}_{(2,2)}$ of $N=(2,2)$ SCFTs fibers over the moduli space
$\mathfrak{M}_{(4,4)}$ of $N=(2,2)$ SCFTs. Mathematically such moduli spaces
are related with space-like 4-spaces in
$H^{*}(S,\mathbb{R})\cong\mathbb{R}^{4,20}$ equipped with the Mukai pairing,
and $\mathfrak{M}_{(2,2)}$ and $\mathfrak{M}_{(4,4)}$ are identified with
$\mathrm{Gr}^{po}_{2,2}(H^{*}(M,\mathbb{R}))\cong\mathrm{O}(4,20)/(\mathrm{SO}(2)\times\mathrm{SO}(2)\times\mathrm{O}(20))$
and
$\mathrm{Gr}^{po}_{4}(H^{*}(M,\mathbb{R}))\cong\mathrm{O}(4,20)/(\mathrm{SO}(4)\times\mathrm{O}(20))$
respectively. Then mirror symmetry is realized as a certain discrete group
action on these moduli spaces. On the other hand, the moduli space
$\mathfrak{M}_{\mathrm{HK}}$ of $B$-field shifts of hyperKähler metrics can be
identified with an open dense subset of $\mathfrak{M}_{(4,4)}$ by the period
map $\mathfrak{per}_{\mathrm{HK}}$. Inspired by this fact, in the beautiful
article [5], Huybrechts showed that the counterpart of $\mathfrak{M}_{(2,2)}$
is given by the moduli space $\mathfrak{M}_{\mathrm{gK3}}$ of generalized K3
surfaces, which are the K3 version of the generalized Calabi–Yau (CY)
structures introduced by Hitchin [4]. The moduli space
$\mathfrak{M}_{\mathrm{K3}}\times H^{2}(M,\mathbb{R})$ of $B$-field shifts of
K3 surfaces endowed with a Ricci-flat metric are naturally contained in
$\mathfrak{M}_{\mathrm{gK3}}$ of real codimension $2$.

$\textstyle{\mathfrak{M}_{\mathrm{K3}}\times
H^{2}(M,\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\mathfrak{M}_{\mathrm{gK3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{per}_{\mathrm{gK3}}}$$\textstyle{\mathrm{Gr}^{po}_{2,2}(H^{*}(M,\mathbb{R}))=\mathfrak{M}_{(2,2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}_{\mathrm{HK}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{per}_{\mathrm{HK}}}$$\textstyle{\mathrm{Gr}^{po}_{4}(H^{*}(M,\mathbb{R}))=\mathfrak{M}_{(4,4)}}$

Then, as Huybrechts pointed out, it is natural to expect that points in
$\mathfrak{M}_{\mathrm{K3}}$ might be mirror symmetric to points that are no
longer in $\mathfrak{M}_{\mathrm{K3}}$. We will show that this is precisely
the source of the problem of mirror symmetry for the Shioda–Inose K3 surfaces.

There is a good chemistry between Hitchin’s generalized CY structures and
mirror symmetry as they both embrace A-structures (symplectic) and
B-structures (complex) on an equal footing. Inspired by the works of
Dolgachev, Aspinwall–Morrison and Huybrechts, we introduce a formulation of
mirror symmetry for generalized K3 surfaces (Section 4). The key features of
our formulation is twofold, and both are guided by Hitchin’s theory. The first
is to extend the scope of lattice polarizations to the Mukai lattice. Mixture
of degrees of cycles is indispensable for generalized K3 surfaces. The second
is to treat the A- and B-structures on a completely equal footing. The
Néron–Severi lattice and transcendental lattice are defined in the same
fashion and moreover the lattice polarizations are imposed on both.

Along the way, we investigate complex and Kähler rigid structures of
generalized K3 surfaces (Section 3). The notion of a complex rigid structure
is enhanced to the level of generalized K3 surfaces, and we give a
classification theorem which generalizes the famous Shioda–Inose theorem. A
rigid Kähler structure is defined by using a generalized CY structure, which
captures a hidden integral structure of the Kähler moduli space. Such a
structure has been anticipated for a long time also from the viewpoint of
mirror symmetry. We give a conjectural explicit mirror correspondence between
the complex rigid K3 surfaces and Kähler rigid K3 surfaces.

### Structure of article

Section 2 provides a brief summary of generalized K3 structures based on
Huybrechts’ work [5]. Section 3 investigates the complex and Kähler rigid
structures on generalized K3 surfaces. Section 4 is devoted to mirror symmetry
for generalized K3 surfaces in comparison with the classical formulation for
K3 surfaces. Section 5 settles the long-standing problem of mirror symmetry
for Shioda–Inose K3 surfaces.

### Acknowledgement

The author thanks Yu-Wei Fan and Shinobu Hosono for many valuable discussions
and helpful comments. Some parts of this work was done while the author was
affiliated by Kyoto University with Leading Initiative for Excellent Young
Researchers Grant. This work is partially supported by the JSPS Grant-in-Aid
Wakate(B)17K17817.

##  2\. Generalized K3 surfaces

Hitchin’s invention of generalized CY structures is the key to unify the
symplectic and complex structure [4]. Such structures have been extensively
studied in 2-dimensions by Huybrechts [5]. In this section, for the sake of
completeness, we provide a brief review of Huybrechts’ work [5].

###  2.1. Generalized CY structures

Let $M$ be a differentiable manifold underlying a K3 surface and
$A^{2*}_{\mathbb{C}}(M)=\oplus_{i=0}^{2}A_{\mathbb{C}}^{2i}(M)$ the space of
even differential forms with $\mathbb{C}$-coefficients. Let $\varphi_{i}$
denote the degree $i$ part of $\varphi\in A^{2*}_{\mathbb{C}}(M)$. The Mukai
pairing for $\varphi,\varphi^{\prime}\in A_{\mathbb{C}}^{2*}(M)$ is defined as

$\langle\varphi,\varphi^{\prime}\rangle=\varphi_{2}\wedge\varphi^{\prime}_{2}-\varphi_{0}\wedge\varphi^{\prime}_{4}-\varphi_{4}\wedge\varphi^{\prime}_{0}\in
A_{\mathbb{C}}^{4}(M)$

######  Definition 2.1 (generalized CY structure).

A generalized CY structure on $M$ is a closed form $\varphi\in
A^{2*}_{\mathbb{C}}(M)$ such that

$\langle\varphi,\varphi\rangle=0,\ \ \
\langle\varphi,\overline{\varphi}\rangle>0.$

The special appeal of generalized CY manifolds resides in the fact that they
embrace complex and symplectic structures on an equal footing. There are two
fundamental CY structures:

  1. (1)

A symplectic structure $\omega$ on $M$ induces a generalized CY structure

$\varphi=e^{\sqrt{-1}\omega}=1+\sqrt{-1}\omega-\frac{1}{2}\omega^{2}.$

  2. (2)

A complex structure $J$ of $M$ makes $M$ a K3 surface $M_{J}$. A holomorphic
$2$-form $\sigma$ of $M_{J}$, which is unique up to scaling, defines a
generalized CY structure $\varphi=\sigma$. We also write $M_{\sigma}=M_{J}$.

For $B\in A^{2}_{\mathbb{C}}(M)$, $e^{B}$ acts on $A^{2*}_{\mathbb{C}}(M)$ by
exterior product

$e^{B}\varphi=(1+B+\frac{1}{2}B\wedge B)\wedge\varphi.$

This action is orthogonal with respect to the Mukai pairing

$\langle
e^{B}\varphi,e^{B}\varphi^{\prime}\rangle=\langle\varphi,\varphi^{\prime}\rangle.$

A real closed 2-form is called a $B$-field. For a $B$-field $B$ and a
generalized CY structure $\varphi$, the $B$-field transform $e^{B}\varphi$ is
a generalized CY structure.

We will later show that a $B$-field is indispensable when we view complex and
symplectic structures as special instance of a more general notion. The
following shows that a generalized CY structure is a $B$-field transform of
either one of the fundamental CY structures.

######  Proposition 2.2 ([4]).

Let $\varphi$ be a generalized CY structure.

  1. (1)

If $\varphi_{0}\neq 0$, then

$\varphi=\varphi_{0}e^{B+\sqrt{-1}\omega}$

with a symplectic $\omega$ and a $B$-field $B$.

  2. (2)

If $\varphi_{0}=0$, then

$\varphi=e^{B}\sigma=\sigma+B^{0,2}\wedge\sigma$

with is a holomorphic 2-form $\sigma$ with respect to a complex structure $J$
and a $B$-field $B$.

######  Definition 2.3.

Generalized CY structure of (1) and (2) in Proposition 2.2 are called type $A$
and type $B$ respectively.

We consider the group $\mathrm{Diff}_{*}(M)$ of all diffeomorphisms
$f:M\rightarrow M$ such that the induced action
$f^{*}:H^{2}(M,\mathbb{Z})\rightarrow H^{2}(M,\mathbb{Z})$ is trivial.
Generalized CY structures $\varphi$ and $\varphi^{\prime}$ are called
isomorphic if there exists an exact $B$-field $B$ and a diffeomoprhism
$f\in\mathrm{Diff}_{*}(M)$ such that $\varphi=e^{B}f^{*}\varphi^{\prime}$.

The most fascinating aspects of generalized CY structures is the occurrence of
the classical CY structure $\sigma$ (type $B$) as well as of symplectic
generalized CY structure $e^{\sqrt{-1}\omega}$ (type $A$) in the same moduli
space. This allows us to pass from the symplectic to the complex world in a
continuous fashion.

######  Example 2.4 ([4]).

For a complex structure $\sigma$, the real and imaginary parts
$\mathrm{Re}(\sigma),\mathrm{Im}(\sigma)$ are themselves real symplectic
forms. A family of generalized CY structures of type $A$

$\varphi_{t}=te^{\frac{1}{t}(\mathrm{Re}(\sigma)+\sqrt{-1}\mathrm{Im}(\sigma))}$

converges, as $t$ goes to $0$, to the generalized CY structure $\sigma$ of
type $B$. In this way, the $B$-fields interpolate between generalized
Calabi–Yau structures of type $A$ and $B$.

###  2.2. (hyper)Kähler structures

######  Definition 2.5.

Let $\varphi$ be a generalized CY structure. Then $P_{\varphi}\subset
A^{*}(M)$ and $P_{[\varphi]}\subset H^{*}(M,\mathbb{R})$ denote the real
$2$-planes

$\displaystyle P_{\varphi}$
$\displaystyle=\mathbb{R}\mathrm{Re}\varphi\oplus\mathbb{R}\mathrm{Im}\varphi\subset
A^{*}(M),$ $\displaystyle P_{[\varphi]}$
$\displaystyle=\mathbb{R}[\mathrm{Re}\varphi]\oplus\mathbb{R}[\mathrm{Im}\varphi]\subset
H^{*}(M,\mathbb{R})$

spanned by the real and imaginary parts of $\varphi$ and $[\varphi]$
respectively. They are positive plane with respect to the Makai pairings.

We say that two generalized CY structures $\varphi$ and $\varphi^{\prime}$ are
orthogonal if $P_{\varphi}$ and $P_{\varphi^{\prime}}$ are pointwise
orthogonal. The orthogonality of $2$-planes $P_{\varphi}$ and
$P_{\varphi^{\prime}}$ is in general a stronger condition than just
$\langle\varphi,\varphi^{\prime}\rangle=0$.

######  Definition 2.6 (Kähler).

A CY structure $\varphi$ is called Kähler if there exists another generalized
CY structure $\varphi^{\prime}$ orthogonal to $\varphi$. In this case,
$\varphi^{\prime}$ is called a Kähler structure for $\phi$.

Let $\varphi=\sigma$ a holomorphic $2$-form for a complex structure $J$ on
$M$. If $\varphi^{\prime}$ is a Kähler structure for $\varphi$, then it is of
the form $\varphi^{\prime}=\varphi^{\prime}_{0}e^{B+\sqrt{-1}\omega}$ (type
$A$) for some $B$-field $B$ and symplectic form $\omega$. The orthogonality is
equivalent to

$\sigma\wedge B=\sigma\wedge\omega=0.$

Therefore $B$ is a closed real $(1,1)$-form and $\pm\omega$ is a Kähler form
with respect to $J$.

A hyperKähler structure is then defined as a special instance of a Kähler
structure. Recall first that a Kähler form $\omega$ on a K3 surface is a
hyperKähler form if

$2\omega^{2}=C\sigma\wedge\overline{\sigma}$

for some $C\in\mathbb{R}$. We may assume $C=1$ by rescaling $\sigma$.

######  Definition 2.7 (hyperKähler).

A generalized CY structure $\varphi$ is hyperKähler if there exists another
generalized CY structure $\varphi^{\prime}$ such that

  1. (1)

$\varphi$ and $\varphi^{\prime}$ are orthogonal

  2. (2)

$\langle\varphi,\overline{\varphi}\rangle=\langle\varphi^{\prime},\overline{\varphi^{\prime}}\rangle$.

Such $\varphi^{\prime}$ is called a hyperKähler structure for $\varphi$.

######  Remark 2.8.

  1. (1)

The definition of a (hyper)Kähler structure is symmetric for $\varphi$ and
$\varphi^{\prime}$.

  2. (2)

Let $\varphi$ be a generalized CY structure and $\varphi^{\prime}$ a
(hyper)Kähler for $\varphi$. Then $e^{B}\varphi^{\prime}$ is a (hyper)Kähler
structure for $e^{B}\varphi$.

We give a classification of the hyperKähler structures for a generalized CY
structure $\varphi$. By remark 2.8, we may assume that $\varphi$ is either of
the form $\varphi=\lambda e^{\sqrt{-1}\omega}$ (type $A$) or $\varphi=\sigma$
(type $B$).

  1. (1)

If $\varphi=\sigma$, then a hyperKähler structure is a generalized CY
structure $\varphi^{\prime}=\lambda e^{B+\sqrt{-1}\omega}$ of type $A$, where
$B$ is a closed $(1,1)$-form and $\pm\omega$ is a hyperKähler form such that
$2|\lambda|^{2}\omega^{2}=\sigma\wedge\overline{\sigma}$.

  2. (2)

If $\varphi=\lambda e^{\sqrt{-1}\omega}$, then a hyperKähler structure is
either

    1. (a)

a generalized CY structure $\varphi^{\prime}=\sigma$ of type $B$, where
$\pm\omega$ is a hyperKähler form,

    2. (b)

a generalized CY structure
$\varphi^{\prime}=\lambda^{\prime}e^{B^{\prime}+\sqrt{-1}\omega^{\prime}}$ of
type $A$ such that

      1. (i)

$\omega\wedge\omega^{\prime}=\omega\wedge B^{\prime}=\omega^{\prime}\wedge
B=0$, $B^{\prime 2}=\omega^{2}+\omega^{\prime 2}$,

      2. (ii)

$|\lambda|^{2}\omega^{2}=|\lambda^{\prime}|^{2}\omega^{\prime 2}$.

###  2.3. Generalized K3 surfaces

######  Definition 2.9 (K3 surface).

A K3 surface is a pair $(\sigma,\omega)$ of a closed 2-form $\sigma\in
A_{\mathbb{C}}^{2}(M)$ with $\sigma\wedge\sigma=0$ and a symplectic form
$\omega\in A^{2}(M)$ such that

  1. (1)

$\omega\wedge\sigma=0$

  2. (2)

$2\omega^{2}=\sigma\wedge\overline{\sigma}>0$.

A K3 surface $(\sigma,\omega)$ is simply a classical K3 surface $M_{\sigma}$
with a chosen hyperKähler structure $\omega$. To avoid confusion, we usually
denote by $M_{\sigma}$ or $S$ a classical K3 surface.

######  Definition 2.10 (generalized K3 surface).

A generalized K3 surface is a pair $(\varphi_{A},\varphi_{B})$ of generalized
CY structures such that $\varphi_{A}$ is a hyperKähler structure for
$\varphi_{B}$.

A K3 surface $(\sigma,\omega)$ is often identified with a generalized K3
surface $(e^{\sqrt{-1}\omega},\sigma)$. A $B$-field transform
$e^{B}(\varphi_{A},\varphi_{B})=(e^{B}\varphi_{A},e^{B}\varphi_{B})$ of a
generalized K3 surface $(\varphi_{A},\varphi_{B})$ is a generalized K3
surface. Generalized K3 surfaces $(\varphi_{A},\varphi_{B})$ and
$(\psi_{A},\psi_{B})$ are called isomorphic if there exists a diffeomorphism
$f\in\mathrm{Diff}_{*}(M)$ and an exact real 2-form $B\in A^{2}(M)$ such that
$(\varphi_{A},\varphi_{B})=e^{B}f^{*}(\psi_{A},\psi_{B})=(e^{B}f^{*}\psi_{A},e^{B}f^{*}\psi_{B})$.

######  Definition 2.11.

We associate to a generalized K3 surface $(\varphi_{A},\varphi_{B})$ the
oriented positive 4-spaces

$\displaystyle\Pi_{(\varphi_{A},\varphi_{B})}$
$\displaystyle=P_{\varphi_{A}}\oplus P_{\varphi_{B}}\subset A^{2*}(M),$
$\displaystyle\Pi_{([\varphi_{A}],[\varphi_{B}])}$
$\displaystyle=P_{[\varphi_{A}]}\oplus P_{[\varphi_{B}]}\subset
H^{*}(M,\mathbb{R}).$

The set $T_{\Pi}$ of the generalized K3 surface $(\varphi_{A},\varphi_{B})$
with fixed positive 4-space $\Pi$ is naturally isomorphic to the Grassmannian
of oriented planes

$\mathrm{Gr}^{o}_{2}(\Pi)=S^{2}\times
S^{2}=\mathbb{P}^{1}\times\mathbb{P}^{1},$

which is identified with a quadric
$Q_{\Pi}\subset\mathbb{P}(\Pi_{\mathbb{C}})$.

For a K3 surface $(e^{\sqrt{-1}\omega},\sigma)$, the 4-space
$\Pi_{(e^{\sqrt{-1}\omega},\sigma)}$ is spanned by the oriented basis

$1-\frac{1}{2}\omega^{2},\ \omega,\ \mathrm{Re}(\sigma),\
\mathrm{Im}(\sigma).$

We may write $\omega_{I}=\omega$, $\omega_{J}=\mathrm{Re}(\sigma)$,
$\omega_{K}=\mathrm{Im}(\sigma)$, where $I,J,K$ are the complex structures
induced by the hyperKähler form. Then there is a 2-sphere

$S^{2}=\\{aI+bJ+cK\ |\ a^{2}+b^{2}+c^{2}=1\\}$

of complex structures with respect to which the metric is Kähler. This
$S^{2}\cong\mathbb{P}^{1}$ is identified with the hyperplane section
$Q_{\Pi}\cap\mathbb{P}(\mathbb{C}\langle\omega_{I},\omega_{J},\omega_{K}\rangle)$.
The other generalized K3 surfaces parametrized by
$\mathbb{P}^{1}\times\mathbb{P}^{1}\setminus S^{2}$ are not realized as the
$B$-field transforms of points in $S^{2}$.

######  Proposition 2.12.

Let $(\varphi,\varphi^{\prime})$ be a generalized K3 surface. Then there
exists a K3 surface $(\sigma,\omega)$ and a closed $B$-field $B$ such that

$\Pi_{(\varphi,\varphi^{\prime})}=e^{B}\Pi_{(e^{\sqrt{-1}\omega},\sigma)}.$

###  2.4. Moduli spaces and Torelli theorems

Let $\mathfrak{M}_{\mathrm{K3}}$ and $\mathfrak{M}_{\mathrm{gK3}}$ be the
moduli spaces of K3 surfaces and generalized K3 structures respectively. We
also define the moduli space of the $B$-field shifts of the hyperKähler
metrics on $M$

$\mathfrak{M}_{\mathrm{HK}}=\left(\mathrm{Met}^{\mathrm{HK}}(M)/\mathrm{Diff}_{*}(M)\right)\times
H^{2}(M,\mathbb{R}).$

Proposition 2.12 implies that there exists a natural $S^{2}\times
S^{2}$-fibration

$\mathfrak{M}_{\mathrm{gK3}}\longrightarrow\mathfrak{M}_{\mathrm{HK}},\
(\varphi,\varphi^{\prime})\mapsto(g,B),$

where $B$ is given by
$\Pi_{(\varphi,\varphi^{\prime})}=e^{B}\Pi_{(e^{\sqrt{-1}\omega},\sigma)}$ and
$g$ is the hyperKähler metric associated with $\sigma$ and $\omega$.

We have the canonical inclusion

$\iota:\mathfrak{M}_{\mathrm{K3}}\times
H^{2}(M,\mathbb{R})\longrightarrow\mathfrak{M}_{\mathrm{gK3}},\
(\sigma,\omega,B)\mapsto e^{B}(\sigma,e^{\sqrt{-1}\omega}).$

######  Definition 2.13.

The period of a generalized K3 surface $(\varphi,\varphi^{\prime})$ is the
orthogonal pair of positive oriented planes

$(P_{[\varphi]},P_{[\varphi^{\prime}]})\in\mathrm{Gr}^{po}_{2,2}(H^{*}(M,\mathbb{R})).$

Then the period map is defined by

$\mathfrak{per}_{\mathrm{gK3}}:\mathfrak{M}_{\mathrm{gK3}}\longrightarrow\mathrm{Gr}^{po}_{2,2}(H^{*}(M,\mathbb{R})),\
(\varphi,\varphi^{\prime})\mapsto(P_{[\varphi]},P_{[\varphi^{\prime}]}).$

######  Definition 2.14.

The period of a $B$-field shift of hyperKähler metric $(g,B)$ is the positive
4-space

$\Pi_{([\varphi],[\varphi^{\prime}])}\in\mathrm{Gr}^{po}_{4}(H^{*}(M,\mathbb{R}))$

where $g$ is the hyperKähler metric associated with $\sigma$ and $\omega$ and
$\Pi_{(\varphi,\varphi^{\prime})}=e^{B}\Pi_{(e^{\sqrt{-1}\omega},\sigma)}$.
Then the period map is defined by

$\mathfrak{per}_{\mathrm{HK}}:\mathfrak{M}_{\mathrm{HK}}\longrightarrow\mathrm{Gr}^{po}_{4}(H^{*}(M,\mathbb{R})),\
(g,B)\mapsto\Pi_{([\varphi],[\varphi^{\prime}])}.$

To summarize, we obtain the following commutative diagram:

$\textstyle{\mathfrak{M}_{\mathrm{K3}}\times
H^{2}(M,\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{S^{2}}$$\textstyle{\mathfrak{M}_{\mathrm{gK3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{2}\times
S^{2}}$$\scriptstyle{\mathfrak{per}_{\mathrm{gK3}}}$$\textstyle{\mathrm{Gr}^{po}_{2,2}(H^{*}(M,\mathbb{R}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{S^{2}\times
S^{2}}$$\textstyle{\mathfrak{M}_{\mathrm{HK}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{per}_{\mathrm{HK}}}$$\textstyle{\mathrm{Gr}^{po}_{4}(H^{*}(M,\mathbb{R}))}$

where $S^{2}$ and $S^{2}\times S^{2}$ denote the fibers of corresponding the
vertical arrows. An important consequence of the diagram is that
$\mathfrak{per}_{\mathrm{gK3}}$ is an immersion with dense image.

Finally we compare the moduli space
$\mathfrak{N}_{\mathrm{gCY}}=\\{\mathbb{C}\varphi\\}/\cong$ of the generalized
CY structures of hyperKähler type, and the moduli space
$\mathfrak{N}_{\mathrm{K3}}=\\{\mathbb{C}\sigma\\}/\mathrm{Diff}_{*}(M)$ of
the classical marked K3 surfaces.

######  Theorem 2.15 ([5]).

The classical period map $\mathfrak{per}_{\mathrm{K3}}$ naturally extends to
the period map $\mathfrak{per}_{\mathrm{gCY}}$ for the generalized CY
structures of hyperKähler type.

$\textstyle{\mathfrak{N}_{\mathrm{gCY}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\
}$$\scriptstyle{\bigcup}$$\scriptstyle{\mathfrak{per}_{\mathrm{gCY}}}$$\scriptstyle{\mathbb{C}\varphi\to[\phi]}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{\mathfrak{D}}=\\{[\varphi]\in\mathbb{P}(H^{*}(M,\mathbb{C}))\
|\
\langle\varphi,\varphi\rangle=0,\langle\varphi,\overline{\varphi}\rangle>0\\}}$$\scriptstyle{\bigcup}$$\textstyle{\mathfrak{N}_{\mathrm{K3}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
\
}$$\scriptstyle{\mathfrak{per}_{\mathrm{K3}}}$$\scriptstyle{\mathbb{C}\sigma\to[\sigma]}$$\textstyle{\mathfrak{D}=\\{[\sigma]\in\mathbb{P}(H^{2}(M,\mathbb{C}))\
|\ \langle\sigma,\sigma\rangle=0,\langle\sigma,\overline{\sigma}\rangle>0\\}}$

$\mathfrak{per}_{\mathrm{gCY}}$ is étale surjective, and bijective over the
complement of the hyperplane section $\mathbb{P}(H^{2}(M,\mathbb{C})\oplus
H^{4}(M,\mathbb{C}))\cap\widetilde{\mathfrak{D}}$.

Therefore the generalized K3 surfaces could be considered as geometric
realizations of points in the extended period domain
$\widetilde{\mathfrak{D}}$. This Torelli theorem will be a foundation of our
formulation of mirror symmetry in Section 4.

Figure 1. Moduli space $\mathfrak{N}_{\mathrm{gCY}}$

##  3\. Rigid structures

###  3.1. Néron–Severi lattice and transcendental lattice

We begin by introducing the Néron–Severi lattice and the transcendental
lattice of a generalized K3 surface.

######  Definition 3.1.

The Néron–Severi lattice and transcendental lattices of
$X=(\varphi_{A},\varphi_{B})$ are defined respectively by

$\displaystyle\widetilde{NS}(X)=\\{\delta\in H^{*}(M,\mathbb{Z})\ |\
\langle\delta,\varphi_{B}\rangle=0\\},$
$\displaystyle\widetilde{T}(X)=\\{\delta\in H^{*}(M,\mathbb{Z})\ |\
\langle\delta,\varphi_{A}\rangle=0\\}.$

It is worth mentioning that $\widetilde{NS}(X)\cap\widetilde{T}(X)$ can be
non-trivial. Moreover, neither $\varphi_{A}\in\widetilde{NS}(X)_{\mathbb{C}}$
nor $\varphi_{B}\in\widetilde{T}(X)_{\mathbb{C}}$ holds in general. Our
definition of the transcendental lattice $\widetilde{T}(X)$ differs from
Huybrechts’ [5] (the orthogonal complement of $\widetilde{NS}(X)$). We define
$\widetilde{NS}(X)$ and $\widetilde{T}(X)$ on a completely equal footing.

For later use, we introduce one more notation. For $\psi\in
H^{*}(M,\mathbb{C})$, we denote by $L_{\psi}$ the smallest sublattice
$L\subset H^{*}(M,\mathbb{Z})$ such that $\psi\in L_{\mathbb{C}}$. Then we may
write

$\widetilde{NS}(X)=L_{\varphi_{B}}^{\perp},\ \ \
\widetilde{T}(X)=L_{\varphi_{A}}^{\perp}.$

This in particular shows that the signatures of $\widetilde{NS}(X)$ and
$\widetilde{T}(X)$ are of the form $(t,*)$ with $0\leq t\leq 2$.

It is instructive to compute basic examples. Let $\delta_{i}$ denote the
degree $i$ part of $\delta\in H^{*}(M,\mathbb{Z})$.

  1. (1)

If $\varphi_{B}=\sigma+B^{0,2}\wedge\sigma$, a twist of $\sigma$ by a
$B$-field $B\in H^{2}(M,\mathbb{R})$, then

$\widetilde{NS}(X)=\\{\delta_{0}+\delta_{2}\ |\
\int_{M}\delta_{2}\wedge\sigma=\delta_{0}\int_{M}B^{0,2}\wedge\sigma\\}\oplus
H^{4}(M,\mathbb{Z}).$

There is a natural inclusion $H^{4}(M,\mathbb{Z})\oplus
NS(M_{\sigma})\subset\widetilde{NS}(S)$ (the inclusion is strict if $B\in
H^{2}(M,\mathbb{Q})$). In particular, for the classical case
$\varphi_{B}=\sigma$, we recover

$\displaystyle\widetilde{NS}(X)$ $\displaystyle=H^{0}(M,\mathbb{Z})\oplus
NS(M_{\sigma})\oplus H^{4}(M,\mathbb{Z})$ $\displaystyle\cong
NS^{\prime}(M_{\sigma})$

  2. (2)

If $\varphi_{A}=e^{B+\sqrt{-1}\omega}$ for $B\in H^{2}(M,\mathbb{R})$ and a
symplectic form $\omega$, the orthogonality condition
$\langle\delta,\varphi_{A}\rangle=0$ is equivalent to

$\displaystyle\int_{M}\delta_{2}\wedge
B-\frac{\delta_{0}}{2}\int_{M}(B^{2}-\omega^{2})+\int_{M}\delta_{4}=0,$
$\displaystyle\int_{M}\delta_{2}\wedge\omega-\delta_{0}\int_{M}B\wedge\omega=0.$

In particular, if a $B$-field is absent, i.e.
$\varphi_{A}=e^{\sqrt{-1}\omega}$, then

$\widetilde{T}(X)=H^{2}(M,\mathbb{Z})_{\omega}\oplus\\{\delta_{0}+\delta_{4}\
|\ \delta_{0}\int_{M}\omega^{2}=2\int_{M}\delta_{4}\\},$

where $H^{2}(M,\mathbb{Z})_{\omega}$ is the space of $\omega$-primitive
classes.

###  3.2. Complex rigidity

The Shioda–Inose K3 surfaces are known as complex rigid K3 surfaces in the
sense that they do not admit any complex deformation keeping the maximal
Picard number $20$. We first give a generalization of complex rigid K3
surfaces.

######  Definition 3.2.

A generalized K3 surface $X=(\varphi_{A},\varphi_{B})$ is called complex rigid
if

  1. (1)

$\varphi_{B}$ is of type $B$,

  2. (2)

$\mathrm{rank}(\widetilde{NS}(X))=22$.

######  Theorem 3.3.

A complex rigid generalized K3 surface is of the form $X=(\lambda
e^{B+B^{\prime}+\sqrt{-1}\omega},\sigma+B^{\prime}\wedge\sigma)$, where

  1. (1)

$M_{\sigma}$ is a complex rigid K3 surface,

  2. (2)

$B\in H^{1,1}(M_{\sigma},\mathbb{R})$,

  3. (3)

$B^{\prime}\in H^{2}(M,\mathbb{Q})$,

  4. (4)

$\pm\omega$ is a hyperKähler form for $\sigma$.

In other words, it is a rational $B$-field $B^{\prime}$ shift of a rigid K3
surface equipped with a complexified Kähler structure $(\lambda
e^{B\pm\sqrt{-1}\omega},\sigma)$.

###### Proof.

Let $X=(\varphi_{A},\varphi_{B})$ be a complex rigid K3 surface. Since
$H^{2}(M,\mathbb{Z})$ is of signature $(3,19)$, $\varphi_{A}$ must be of type
$A$. Hence we can write $X=e^{B^{\prime}}(\lambda
e^{B+\sqrt{-1}\omega},\sigma)$. Then $Y=(\lambda
e^{B+\sqrt{-1}\omega},\sigma)$ is a generalized K3 surface, i.e. $B$ is a
closed real $(1,1)$-form and $\pm\omega$ is a hyperKähler form for $\sigma$.
On the other hand, for

$\varphi_{B}=\sigma+B^{\prime}\wedge\sigma,$

$\mathrm{rank}(L_{\varphi_{B}})=2$ if and only if
$\mathrm{rank}(T(M_{\sigma}))=2$ and $B^{\prime}$ is rational. ∎

The following is a generalization of the celebrated Shioda–Inose theorem
concerning the classification of rigid K3 surfaces [8].

######  Theorem 3.4.

Let $\mathfrak{L}^{ev}_{(2,0)}$ be the set of isomorphism classes of positive
definite even lattices of rank $2$. Let
$\mathfrak{M}_{\mathrm{Cpx}}^{\mathrm{rigid}}$ be the set of isomorphism
classes of complex rigid generalized K3 surfaces. Then

$\mathfrak{M}_{\mathrm{Cpx}}^{\mathrm{rigid}}\longrightarrow\mathfrak{L}^{ev}_{(2,0)}\times
H^{2}(M,\mathbb{Q}),\ \ \ X\mapsto(T(M_{\sigma}),B^{\prime})$

where $X=e^{B^{\prime}}(\lambda e^{B+\sqrt{-1}\omega},\sigma)$ as in Theorem
3.3, is surjective. The fiber over $(T(M_{\sigma}),B^{\prime})$ is given by
the 2 copies of complexified Kähler cone
$H^{1,1}(M_{\sigma})+\sqrt{-1}\mathcal{K}({M_{\sigma}})$

###### Proof.

The Shioda–Inose theorem [8] asserts that there is a bijective correspondence
between the isomorphism classes of rigid K3 surfaces and the set
$\mathfrak{L}^{ev}_{(2,0)}$ of isomorphism classes of positive definite even
lattices of rank $2$. The correspondence is given by $S\to T(S)$. Then the
assertion follows from Theorem 3.3. ∎

###  3.3. Kähler rigidity

In light of mirror symmetry, a rigid Kähler structure has been anticipated for
a long time. A Kähler rigidity comparable with a complex rigidity naturally
appears in the framework of generalized CY structures.

We begin with a simple observation. Let $S$ be a K3 surface with Picard number
$1$. We write $NS(S)=\mathbb{Z}H$ with $H^{2}=2n>0$, and consider

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

Then

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

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
$\mathrm{rank}(L_{e^{\sqrt{-1}\epsilon H}})=2$. This calculation illustrates
that a generalized CY structure $e^{B+\sqrt{-1}\omega}$ is able to detect a
hidden integral structure of the Kähler moduli space.

######  Definition 3.5.

A symplectic manifold $(M,\omega)$ is called Kähler rigid if $\omega^{2}\in
H^{4}(M,\mathbb{Q})$.

As the previous calculation shows, a symplectic manifold $(M,\omega)$ is
Kähler rigid if and only if $\mathrm{rank}(L_{e^{\sqrt{-1}\omega}})=2$.

######  Definition 3.6.

A generalized K3 surface $X=(\varphi_{A},\varphi_{B})$ is called Kähler rigid
if

  1. (1)

$\varphi_{A}$ is of type $A$,

  2. (2)

$\mathrm{rank}(\widetilde{T}(X))=22$.

We provide a characterization of the Kähler rigid generalized K3 surfaces
$(\varphi_{A},\varphi_{B})$ as follows. In contrast to the complex rigid case,
the partner $\varphi_{B}$ can be either of type $A$ or $B$.

######  Theorem 3.7.

A Kähler rigid generalized K3 surface is of the form $X=(\lambda
e^{B+\sqrt{-1}\omega},\varphi_{B})$, where

  1. (1)

$B\in H^{2}(M,\mathbb{Q})$,

  2. (2)

$\omega^{2}\in H^{4}(M,\mathbb{Q})$.

In other words, it is a rational $B$-field $B$ shift of a Kähler rigid
symplectic manifold equipped with a hyperKähler structure $(\lambda
e^{\sqrt{-1}\omega},\varphi_{B})$.

###### Proof.

Let $X=(e^{B+\sqrt{-1}\omega},\varphi_{B})$ be a Kähler rigid K3 surface. We
consider an existence condition of a rank $2$ sublattice $L\subset
H^{*}(M,\mathbb{Z})$ such that

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

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

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

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

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

contains $e^{B+\sqrt{-1}\kappa H}$. ∎

By analogy with the complex rigid case, we conjecture the following mirror
assertion of the Shioda–Inose theorem.

######  Conjecture 3.8.

There is a bijective correspondence between the isomorphism classes of Kähler
rigid structure on $M$ and the set $\mathfrak{L}^{ev}_{(2,0)}$ of isomorphism
classes of positive definite even lattices of rank $2$. The correspondence is
given by $(M,\omega)\to L_{e^{\sqrt{-1}\omega}}$.

The conjecture in particular implies a natural bijective correspondence
between the complex rigid structures and the Kahler rigid symplectic
structures on $M$ (and between their generalized versions by the rational
$B$-field shifts). This mirror correspondence is essentially given by
comparison of the Néron–Severi lattices and transcendental lattices.

$\textstyle{\\{\text{Complex
rigid}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$mirror$\scriptstyle{\omega\mapsto
L_{e^{\sqrt{-1}\omega}}}$$\textstyle{\\{\text{K\"{a}hler
rigid}\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma\mapsto
L_{\sigma}}$$\textstyle{\mathfrak{L}^{ev}_{(2,0)}}$

######  Remark 3.9.

Clearly, there exist parallel theories for $T^{4}$, the $4$-manifold
underlying a $2$-dimensional complex torus.

##  4\. Mirror symmetry for generalized K3 surfaces

Our formulation of mirror symmetry for generalized K3 surfaces builds upon the
combination of two novel ideas: (1) lattice polarizations of cycles (Dolgachev
[2]) and (2) generalized K3 structures (Hitchin [4] and Huybrechts [5]). As is
discussed Section 3, lattices are sensitive to the integral structures
associated with generalized K3 structures.

###  4.1. Lattices

The hyperbolic lattice $U$ is the rank $2$ even unimodular lattice defined by
the Gram matrix $\begin{bmatrix}0&1\\\ 1&0\\\ \end{bmatrix}$. $E_{8}$ is the
rank $8$ even unimodular lattices defined by the corresponding Cartan matrix.
Let $\Lambda_{K3}=U^{\oplus 3}\oplus E_{8}^{\oplus 2}$ be the K3 lattice.

###  4.2. Mirror symmetry for K3 surfaces ala Dolgachev

Let $S$ be a K3 surface. The Néron-Severi lattice and transcendental lattice
are defined respectively by

$NS(S)=H^{1,1}(S,\mathbb{R})\cap H^{2}(S,\mathbb{Z}),\ \ \
T(S)=NS(S)^{\perp},$

where the orthogonal complement is taken in
$H^{2}(S,\mathbb{Z})\cong\Lambda_{K3}$. The Picard number is
$\rho(S)=\mathrm{rank}(NS(S))$. The extended Néron-Severi lattice
$NS^{\prime}(S)$ by the sublattice

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

of the Mukai lattice $H^{*}(S,\mathbb{Z})\cong U^{\oplus 4}\oplus
E_{8}^{\oplus 2}$.

######  Definition 4.1.

Let $L$ be an even non-degenerate lattice of signature $(1,*)$. An
$L$-polarized K3 surface is a pair $(S,i)$ of a K3 surface $S$ and a primitive
lattice embedding $i:L\hookrightarrow NS(S)$. It is called ample if $i(L)$
contains a Kähler class.

There may be several families of ample $L$-polarized K3 surfaces, and we
usually choose one for the purpose of mirror symmetry. In the foundational
article [2], Dolgachev formulated mirror symmetry for lattice polarized K3
surfaces as follows.

######  Definition 4.2 ([2]).

For a sublattice $L\subset\Lambda_{K3}$ of signature $(1,t)$, assume there
exists a lattice $N$ such that $L^{\perp}=N\oplus U$. Then

  1. (1)

a family of ample $L$-polarized K3 surfaces $\\{S\\}$

  2. (2)

a family of ample $N$-polarized K3 surfaces $\\{S^{\vee}\\}$

are mirror symmetric.

In fact, when the assumption holds, for generic $L$-polarized K3 surface $S$
and $N$-polarized K3 surface $S^{\vee}$, we have

$NS^{\prime}(S)\cong L\oplus U\cong T(S^{\vee}),\ \ \ T(S)\cong N\oplus U\cong
NS^{\prime}(S^{\vee}).$

These are considered as a duality between algebraic cycles of $S$ and
transcendental cycles of $S^{\vee}$, and vice versa. Mirror duality can also
be investigated at the level of the moduli spaces. We refer the reader to the
original article [2] for more details.

The crucial assumption in Dolgachev’s formulation is the existence of a
decomposition of the form $L^{\perp}=N\oplus U$ for $L\subset\Lambda_{K3}$ (or
more generally $L^{\perp}=N\oplus U(k)$ for some $k\in\mathbb{N}$). There are
geometric explanations for such a decomposition; the hyperbolic lattice $U$
corresponds to

  1. (1)

the standard cusps of the Baily–Borel compactification of the period domains
[7],

  2. (2)

the lattice spanned by the fiber and section classes of the SYZ fibrations on
a mirror K3 surface.

Although this formulation works beautifully in many case, it cannot be a
definitive one as this artificial assumption does not hold in general. A
prototypical example of such K3 surfaces is a Shioda–Inose K3 surface, which
is a K3 surface $S$ with the maximal Picard number $20$. Hence, by the Hodge
index theorem, $T(S)$ is of signature $(2,0)$ and the assumption is never
satisfied. A natural question ”How can we understand mirror symmetry for
Shioda–Inose K3 surfaces?” is a long-standing puzzle in the mirror symmetry
community. Indeed, its Kähler moduli space is of dimension 20 and complex
moduli space is of dimension 0, and there seems no mirror partner for a
Shioda–Inose K3 surface.

| Shioda–Inose K3 surface |  mirror ??
---|---|---
Kähler | 20-dim | 0-dim
complex | 0-dim | 20-dim

We will provide an answer to this puzzling problem in the framework of mirror
symmetry for generalized K3 surfaces.

###  4.3. Lattice polarization via Mukai lattice

For integers $\kappa,\lambda\geq 2$ such that $\kappa+\lambda=24$, let $K$ and
$L$ be even lattice of signature $(2,\kappa-2)$ and $(2,\lambda-2)$
respectively.

######  Definition 4.3.

A $(K,L)$-polarized generalized K3 surface is a pair $(X,i)$ of a generalized
K3 surface $X=(\varphi_{A},\varphi_{B})$ and a primitive embedding $i:K\oplus
L\hookrightarrow H^{*}(M,\mathbb{Z})$ such that

  1. (1)

$K\subset\widetilde{NS}(X)$ and $K_{\mathbb{C}}$ contains a generalized CY
structure of type $A$,

  2. (2)

$L\subset\widetilde{T}(X)$ and $L_{\mathbb{C}}$ contains a generalized CY
structure of type $B$.

The containment of a generalized CY structure of type $A$ or type $B$ is
comparable with the ampleness condition for the conventional lattice
polarizations. Without these, swapping $\varphi_{A}$ and $\varphi_{B}$
produces a trivial mirror families.

######  Remark 4.4.

In the conventional $L$-polarization, there is asymmetry between algebraic and
transcendental cycles; the inclusions $L\subset NS(S)$ and $L^{\perp}\supset
T(S)$ are not symmetric. On the other hand, a $(K,L)$-polarization imposes the
inclusions $K\subset\widetilde{NS}(X)$ and $L\subset\widetilde{T}(X)$ evenly.

###  4.4. Mirror symmetry for generalized K3 surfaces

As in the case of K3 surfaces, there may be several families of
$(K,L)$-polarized generalized K3 surfaces, and we choose one for the purpose
of mirror symmetry.

######  Definition 4.5.

For integers $\kappa,\lambda\geq 2$ such that $\kappa+\lambda=24$, let $K,L$
be even lattice of signature $(2,\kappa-2)$ and $(2,\lambda-2)$ respectively.
Then

  1. (1)

a family of $(K,L)$-polarized generalized K3 surfaces $\\{X\\}$

  2. (2)

a family of $(L,K)$-polarized generalized K3 surfaces $\\{X^{\vee}\\}$

are mirror symmetric.

The fundamental idea is comparable with Dolgachev’s formulation. A major
difference is the mixture of the degrees of cycles. For example,
$\widetilde{NS}(X)$ may not contain a pure zero cycle (a point is no longer
algebraic in this sense). Another notable difference is that there is no
artificial assumption on the lattices.

Let $X=(\varphi_{A},\varphi_{B})$ be a $(K,L)$-polarized generalized K3
surface. Note that the inclusions $K\subset\widetilde{NS}(X)$ and
$L\subset\widetilde{T}(X)$ are equivalent to $\varphi_{B}\in L_{\mathbb{C}}$
and $\varphi_{A}\in K_{\mathbb{C}}$ respectively.

######  Definition 4.6.

A deformation of $\mathbb{C}\varphi_{A}$ keeping $X$ as a $(K,L)$-polarized
generalized K3 surface is called an $A$-deformation. The $A$-moduli space
$\mathfrak{M}_{A}^{X}$ is defined as the space of $A$-deformations. A
$B$-deformation and the $B$-moduli space $\mathfrak{M}_{B}^{X}$ are defined
similarly.

By the Torelli theorem (Theorem 2.15), $\dim\mathfrak{M}_{A}^{X}=\kappa-2$ and
$\dim\mathfrak{M}_{B}^{X}=\lambda-2$. The construction of moduli spaces works
in largely the same way as that for lattice polarized K3 surfaces (see
Dolgachev’s work [2] for details). We first define

$\mathfrak{D}_{K}=\\{[\varphi]\in\mathbb{P}(K_{\mathbb{C}})\ |\
\langle\sigma,\sigma\rangle=0,\langle\varphi,\overline{\varphi}\rangle>0\\},$

which can be identified with
$\mathrm{Gr}^{po}_{2}(K_{\mathbb{R}})\cong\mathrm{O}(2,\kappa-2)/(\mathrm{SO}(2)\times\mathrm{O}(\kappa-2))$.
It is a complex manifold of dimension $\kappa-2$ with two connected components
$\mathfrak{D}_{L}^{+}$ and $\mathfrak{D}_{L}^{-}$, each of which is a
symmetric domain of type IV. Independent of the choice of lattices, the
$(K,L)$-polarized generalized K3 surface $X$ form a $20$-dimensional family.
By the Torelli theorem, the moduli space
$\mathfrak{M}_{A}^{X}\times\mathfrak{M}_{B}^{X}$ is identified with an open
subset of
$\mathfrak{D}_{(K,L)}=\mathfrak{D}_{K}^{+}\times\mathfrak{D}_{L}^{+}$.

###  4.5. Comparison: classical and new

We will next confirm that our formulation of mirror symmetry naturally
includes Dolgachev’s.

Let $K^{\prime}\subset\Lambda_{K3}$ be a sublattice of signature $(1,\rho-1)$.
Assume that there exists a decomposition
$(K^{\prime})^{\perp}=L^{\prime}\oplus U$ for a lattice $L^{\prime}$ of
signature $(1,19-\rho)$. Mirror symmetry for lattice polarized K3 surfaces
asserts that the $K^{\prime}$-polarized K3 surfaces and $L^{\prime}$-polarized
K3 surfaces are mirror symmetric to each other.

Let $M_{\sigma}$ be a generic $K^{\prime}$-polarized K3 surface equipped with
a $B$-field $B$ shift of a hyperKähler form $\omega$ ($B,\omega\in
K^{\prime}_{\mathbb{R}}$ are also taken generically). It defines a generalized
K3 surface

$X=(\varphi_{A},\varphi_{B})=(e^{B+\sqrt{-1}\omega},\sigma).$

and we have

$\displaystyle\widetilde{NS}(X)$ $\displaystyle=NS^{\prime}(M_{\sigma})\cong
K^{\prime}\oplus U,$ $\displaystyle\widetilde{T}(X)$
$\displaystyle=T(M_{\sigma})\cong L^{\prime}\oplus U.$

Similarly, let $M_{\sigma^{\vee}}$ be a generic $L^{\prime}$-polarized K3
surface equipped with a $B$-field $B^{\vee}$ shift of a hyperKähler form
$\omega^{\vee}$. It defines a generalized K3 surface

$X^{\vee}=(\varphi_{A},\varphi_{B})=(e^{B^{\vee}+\sqrt{-1}\omega^{\vee}},\sigma^{\vee}).$

and we have

$\displaystyle\widetilde{NS}(X^{\vee})$
$\displaystyle=NS^{\prime}(M_{\sigma^{\vee}})\cong L^{\prime}\oplus U,$
$\displaystyle\widetilde{T}(X^{\vee})$
$\displaystyle=T(M_{\sigma^{\vee}})\cong K\oplus U.$

By setting $K=K^{\prime}\oplus U$ and $L=L^{\prime}\oplus U$, $X$ is a
$(K,L)$-polarized generalized K3 surface and $X^{\vee}$ is a $(L,K)$-polarized
generalized K3 surface. Therefore the classical formulation of mirror symmetry
is naturally considered as a special case of ours.

##  5\. Mirror symmetry for Shioda–Inose K3 surfaces

###  5.1. Mirror symmetry for Shioda–Inose K3 surfaces

Our formulation of mirror symmetry will really comes into its own when
classical formulation fails. We will finally discuss mirror symmetry for
Shioda–Inose (complex rigid) K3 surfaces in the framework of mirror symmetry
for generalized K3 surfaces.

For an integer $n>0$, we define

$K=\langle-2n\rangle^{\oplus 2}\oplus U\oplus E_{8}^{\oplus 2},\ \ \ L=\langle
2n\rangle^{\oplus 2},$

where $\langle k\rangle$ denotes the lattice of rank $1$ generated by $v$ with
$v^{2}=k$.

  1. (1)

A family of the $(K,L)$-polarized generalized K3 surfaces is given by the
family of the Shioda–Inose K3 surfaces
$\\{X=(e^{B+\sqrt{-1}\omega},\sigma)\\}$, where $\sigma$ is the complex
structure of $M$ such that $T(M_{\sigma})=L$ (such a complex structure is
unique up to isomorphism) and $B,\omega\in NS(M_{\sigma})_{\mathbb{R}}$. For a
generic $X$, we have

$\widetilde{NS}(X)=NS^{\prime}(M_{\sigma}),\ \ \
\widetilde{T}(X)=T(M_{\sigma}),$

which are isomorphic to $K$ and $L$ respectively.

  2. (2)

A family of the $(L,K)$-polarized generalized K3 surfaces contains the family
of the classical K3 surfaces of the form
$\\{X^{\vee}=(e^{\sqrt{-1}H},\sigma^{\vee})\\}$ as a subfamily. Here
$\sigma^{\vee}$ is a complex structure of $M$ such that
$NS(M_{\sigma^{\vee}})=\mathbb{Z}H,H^{2}=2n>0$. For a generic $X^{\vee}$, we
have

$\widetilde{NS}(X^{\vee})\subset NS^{\prime}(M_{\sigma^{\vee}}),\ \ \
\widetilde{T}(X^{\vee})\supset T(M_{\sigma^{\vee}})$

and $\widetilde{NS}(X^{\vee})=L$, $\widetilde{T}(X^{\vee})=K$.

We observe that the dimensions of the A- and B-moduli spaces are interchanged
between the mirror families. (cf. table in Section 4.2)

|  $(K,L)$-polarization |  $(L,K)$-polarization
---|---|---
$\mathfrak{M}_{A}$ | 20-dim | 0-dim
$\mathfrak{M}_{B}$ | 0-dim | 20-dim

To summarize, the mirror partner of a Shioda–Inose K3 surfaces is in general
given by a generalized K3 surface. The family of the K3 surfaces of the form
$\\{X^{\vee}=(e^{\sqrt{-1}H},\sigma^{\vee})\\}$ is a 19-dimensional family
contained in the genuine 20-dimensional mirror family. In order to understand
mirror symmetry for K3 surfaces, we need to incorporate the deformations of
generalized K3 surfaces in general.

######  Remark 5.1.

The combination of generalized CY geometry and lattice porarlization via the
Muaki lattice is also the correct framework for mirror symmetry for $4$-tori
$T^{4}$.

###  5.2. Attractor mechanisms

The initial motivation of the present article comes from the attractor
mechanisms on moduli spaces of CY 3-folds [6, 3]. Let $X$ be a projective CY
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 normal 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

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

on $\widetilde{\mathfrak{M}}_{\mathrm{Cpx}}$. It induces a function

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

called the mass function of $\gamma$. The stationary points of the mass
functions are called the complex attractors and corresponding CY 3-folds are
called the complex attractor varieties. This new class of CY 3-folds are a
vast generalization of rigid CY 3-folds and expected to posses rich
structures.

In light of mirror symmetry, the Kähler attractor mechanisms are developed in
our recent article [3]. The Kähler attractor varieties are defined in the same
fashion (they correspond to the stationary points of the Kähler mass
functions). If $X$ and $Y$ are mirror CY 3-folds, then the mirror map should
induce a bijective correspondence between the set
$\mathrm{Attr}_{\mathrm{Cpx}}^{X}$ of complex attractors of $X$ and the set
$\mathrm{Attr}_{\mathrm{Kah}}^{Y}$ of Kähler attractors 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}}}$

An interesting observation is obtained when $X$ and $Y$ are the product of a
K3 surface and an elliptic curve. On the complex side, a complex attractor
variety is the product of a Shioda–Inose K3 surface $M_{\sigma}$ and an
elliptic curve. On the Kähler side, a Kähler attractor variety is the product
of a Kähler rigid symplectic manifold $(M,\omega)$ and an elliptic curve. This
observation led us to the hidden integral structure on the Kähler moduli space
and the idea of mirror symmetry in the framework of generalized K3 surfaces.

## References

  * [1] P. Aspinwal and D. Morriron, String theory on K3 surfaces, Mirror symmetry, II, AMS/IP Stud. Adv. Math., vol. 1, Amer. Math. Soc., Providence, RI, 1997, 703-716.
  * [2] I. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci. 81 (1996), no. 3, 2599-2630, Algebraic geometry, 4.
  * [3] Y.-W. Fan and A. Kanazawa, Attractor mechanisms on moduli spaces of Calabi–Yau 3-folds, preprint.
  * [4] N. Hitchin, Generalized Calabi–Yau manifolds, Quart. J. Math. Oxford Ser. 54: 281-308, 2003.
  * [5] D. Huybrechts, Generalized Calabi–Yau structures, K3 surfaces, and $B$-fields, Int. J. Math. 16 (2005), 13-36.
  * [6] G. Moore, Arithmetic and Attractors, hep-th/9807087.
  * [7] F. Scattone, On the compactification of moduli spaces for algebraic K3 surfaces, Mem. AMS 70 (1987) No.374.
  * [8] T. Shioda and H. Inose, On singular K3 surfaces, Complex analysis and algebraic geometry, Cambridge University Press, Cambridge, 1977.

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

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