In this section we discuss a version of mirror symmetry for hyperkähler manifolds analogous to the one used for K3 surfaces in [18].

The situation for general hyperkähler manifolds is considerably less developed, however, and we shall have to make many assumptions in this discussion. The goal is to show, modulo these assumptions, that one obtains the expected Gromov-Hausdorff collapse at a large complex structure limit of hyperkähler manifolds, and that the limit can be identified using the main results of this paper. This is completely analogous to [18], and this discussion represents a summary of known results.

First we review known facts about periods of hyperkähler manifolds. Fix $M$ a manifold of real dimension $4n$ which supports a hyperkähler manifold structure with holonomy being the full group $Sp(n)$. (When a hyperkähler manifold has this full group as holonomy, it is said to be irreducible.) Set $L=H^{2}(M,\mathbb{Z})$, $L_{\mathbb{R}}:=L\otimes_{\mathbb{Z}}\mathbb{R}$, $L_{\mathbb{C}}:=L\otimes_{\mathbb{Z}}\mathbb{C}$. Then there is a real-valued non-degenerate quadratic form $q_{M}:L\rightarrow\mathbb{R}$, called the Beauville-Bogomolov form, with the property that there is a constant $c$ such that

$$q_{M}(\alpha)^{n}=c\int_{M}\alpha^{2n}$$

for $\alpha\in L$, of signature $(+,+,+,-,\cdots,-)$. We write $q_{M}(\cdot,\cdot)$ for the induced pairing, with $q_{M}(\alpha,\alpha)=q_{M}(\alpha)$.

We can define the period domain of $M$ to be

$$\mathcal{P}_{M}:=\{[\Omega]\in\mathbb{P}(L_{\mathbb{C}})\,|\,q_{M}(\Omega)=0,\quad q_{M}(\Omega,\bar{\Omega})>0\}.$$

The Teichmüller space of $M$, ${\bf Teich}_{M}$, is the set of hyperkähler complex structures on $M$ modulo elements of $\operatorname{Diff}_{0}(M)$, the diffeomorphisms of $M$ isotopic to the identity. By the Bogomolov-Tian-Todorov theorem, this is a (non-Hausdorff) manifold. There is a period map

$$\operatorname{Per}:{\bf Teich}_{M}\rightarrow\mathcal{P}_{M}$$

taking a complex structure on $M$ to the class of the line $H^{2,0}(M)$. Then $\operatorname{Per}$ is étale, and was proved to be surjective by Huybrechts in [21]. Although we shall not make use of this here, we note that recently Verbitsky [41] proved a suitably formulated global Torelli theorem. However, one must keep in mind that $\operatorname{Per}$ is not, in general, a diffeomorphism.

Next consider a complex structure on $M$ and Ricci–flat Kähler metric $\omega_{I}$ making $M$ hyperkähler. Then a choice of a holomorphic symplectic two-form $\Omega_{I}$, along with $\omega_{I}$, completely determines this structure. In particular, if we write $\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$, we can normalize $\Omega_{I}$ so that $q_{M}(\omega_{I})=q_{M}(\omega_{J})=q_{M}(\omega_{K})$. Furthermore, necessarily $q_{M}(\omega_{I},\omega_{J})=q_{M}(\omega_{I},\omega_{K})=q_{M}(\omega_{J},\omega_{K})=0$. The triple $\omega_{I},\omega_{J},\omega_{K}$ is called a hyperkähler triple. It gives rise to an $S^{2}$ worth of complex structures compatible with the same hyperkähler metric: in particular, one has the $J$ complex structure with holomorphic symplectic form $\Omega_{J}:=\omega_{K}+\sqrt{-1}\omega_{I}$ and Kähler form $\omega_{J}$, and the $K$ complex structure with holomorphic symplectic form $\Omega_{K}:=\omega_{I}+\sqrt{-1}\omega_{J}$ and Kähler form $\omega_{K}$.

We use these facts to speculate on mirror symmetry for hyperkähler manifolds, starting with the Strominger-Yau-Zaslow point of view. Suppose that we are given a complex structure on $M$ such that there is a fibration $f:M\rightarrow N$, with fibers being holomorphic Lagrangian subvarieties of $M$. Suppose furthermore that $N$ is a Kähler manifold. Then by results of Matsushita (see [26] and [16], Proposition 24.8 for these results) the smooth fibers of $f$ are complex tori and $N$ is a Fano manifold with $b_{2}(N)=1$. Furthermore, if $M$ is projective of complex dimension $2n$ then $N=\mathbb{CP}^{n}$ by a result of Hwang [22]. Let $\omega_{I}$ be a Ricci–flat Kähler form on $M$. Write the holomorphic symplectic form $\Omega_{I}$ on $M$ as $\omega_{J}+\sqrt{-1}\omega_{K}$. Then after hyperkähler rotation, there is a a complex structure with holomorphic symplectic form $\Omega_{K}=\omega_{I}+\sqrt{-1}\omega_{J}$ and Kähler form $\omega_{K}$. If $M_{y}$ is a fiber of $f$, then $\omega_{J}|_{M_{y}}=\omega_{K}|_{M_{y}}=0$, from which it follows that $\operatorname{Im}(\Omega_{K}^{n})|_{M_{y}}=0$, so the fibers of $f$ are special Lagrangian.

The Strominger-Yau-Zaslow conjecture [36] predicts that mirror symmetry can be explained via dualizing such a special Lagrangian torus fibration. In a general situation, it can be hard to dualize torus fibrations, because of singular fibres. The case that $M$ is a K3 surface, treated in detail in [18], is rather special because Poincaré duality gives a canonical isomorphism between a two-torus and its dual.

With some additional assumptions, a similar situation holds in the hyperkähler case. Suppose that the Kähler form $\omega_{I}$ is integral, so that there is an ample line bundle $\mathcal{L}$ on $X$ whose first Chern class is represented by $\omega_{I}$. The restriction of this line bundle to a non-singular fiber $M_{y}$ then induces a polarization of some type $(d_{1},\ldots,d_{n})$. In particular there is a canonical map $M_{y}\rightarrow M_{y}^{\vee}$ given by

$$M_{y}\ni x\mapsto\mathcal{L}|_{M_{y}}\otimes t_{x}^{*}\mathcal{L}^{-1}|_{M_{y}}\in M_{y}^{\vee}.$$

Here $M_{y}^{\vee}$ is the dual abelian variety to $M_{y}$, classifying degree zero line bundles on $M_{y}$, and $t_{x}:M_{y}\rightarrow M_{y}$ is given by translation by $x$, which makes sense once one chooses an origin in $M_{y}$. The kernel of this map is $(\mathbb{Z}/d_{1}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/d_{n}\mathbb{Z})^{\oplus 2}$. In particular, if $f$ possesses a section $s:N\rightarrow M$, and $N_{0}:=N\setminus f(S)$ where $S$ is the critical locus of $f$, then the dual of $f^{-1}(N_{0})\rightarrow N_{0}$ can be described as a quotient map, given by dividing out by the kernel of the polarization on each fiber. One can then hope that this dual fibration can be compactified to a hyperkähler manifold.

In general, if $M_{y}$ carries a polarization of type $(d_{1},\ldots,d_{n})$, it is not difficult to check that the dual abelian variety $M_{y}^{\vee}$ carries a polarization of type $(d_{n}/d_{n},d_{n}/d_{n-1},\ldots,d_{n}/d_{1})$. Thus it is possible that the SYZ dual hyperkähler manifold need not be the same as $M$. There do indeed exist examples of abelian variety fibrations on hyperkähler manifolds which are not principally polarized; these were discovered by Justin Sawon, see Example 3.8 and Remark 3.9 of [31]. It is quite possible these fibrations do not have duals which are hyperkähler manifolds, as a natural compactification might be a holomorphic symplectic variety without a holomorphic symplectic resolution of singularities.

On the other hand, if $\omega_{I}$ induces a principal polarization on each fiber $M_{y}$, i.e., the map $M_{y}\rightarrow M_{y}^{\vee}$ is an isomorphism, then the SYZ dual of the fibration $f^{-1}(N_{0})\rightarrow N_{0}$, assuming again the existence of a section, can be canonically identified with $f^{-1}(N_{0})\rightarrow N_{0}$, and thus it is natural to consider $f:M\rightarrow N$ to be a self-dual fibration, at least at the purely topological level. In this case, and only in this case, SYZ mirror symmetry predicts that hyperkähler manifolds are self-mirror. The idea that hyperkähler manifolds should be self-mirror was first suggested and explored by Verbitsky in [40].

In this case only, we can be more explicit about mirror symmetry. We summarize our assumptions so far:

Thus, with these assumptions, it is natural to assume that mirror symmetry exchanges complex and Kähler moduli for the fixed underlying space $M$. This can be described at the level of period domains as follows.

Let $\sigma\in L_{\mathbb{R}}$ be the class represented by $\omega_{I}$. Fix an integral Kähler class $\omega_{N}$ on $N$, and let $E\in L$ be represented by $f^{*}\omega_{N}$, so that $q_{M}(E)=0$.

Denote by $E^{\perp}\subseteq L_{\mathbb{R}}$ the orthogonal complement of $E$ under $q_{M}$, and denote by $E^{\perp}/E$ the quotient space $E^{\perp}/\mathbb{R}E$. Then $q_{M}$ induces a quadratic form on $E^{\perp}/E$. Let

$$\mathcal{C}(M):=\{x\in E^{\perp}/E\,|\,q_{M}(x)>0\},$$

and define the complexified Kähler moduli space of $M$ to be

$$\mathcal{K}(M):=E^{\perp}/E\oplus i\mathcal{C}(M)\subseteq(E^{\perp}/E)\otimes\mathbb{C}.$$

We then have an isomorphism

$$m_{E,\sigma}:\mathcal{K}(M)\rightarrow\mathcal{P}_{M}\setminus E^{\perp}$$

via, representing an element of $(E^{\perp}/E)\otimes\mathbb{C}$ by $\alpha\in E^{\perp}\otimes\mathbb{C}$,

$$\alpha\mapsto\left[\frac{1}{q_{M}(E,\sigma)}\sigma+\alpha-\frac{1}{2}\left(\frac{q_{M}(\sigma)}{q_{M}(E,\sigma)^{2}}+q_{M}(\alpha)+2\frac{q_{M}(\alpha,\sigma)}{q_{M}(E,\sigma)}\right)E\right].$$

Indeed, one first checks that this is independent of which representative $\alpha$ is chosen. Then one notes that the coefficient of $E$ is chosen so that $q_{M}(m_{E,\sigma}(\alpha))=0$, and $q_{M}(m_{E,\sigma}(\alpha),m_{E,\sigma}(\bar{\alpha}))=2q_{M}(\operatorname{Im}\alpha)>0$ by assumption that $\alpha\in\mathcal{K}(M)$. Further, $m_{E,\sigma}$ is clearly injective, since $\alpha=m_{E,\sigma}(\alpha)-\sigma/q_{M}(E,\sigma)\mod E$. It is surjective, since given $[\Omega]\in\mathcal{P}_{M}\setminus E^{\perp}$, we can rescale $\Omega$ so that $q_{M}(\Omega,E)=1$, and then $[\Omega]=m_{E,\sigma}(\Omega-\sigma/q_{M}(E,\sigma)\mod E)$.

We can then view the mirror map $m_{E,\sigma}$ described above as realising mirror symmetry on the level of period domains as follows, defining an exchange of data

$$(M,\Omega,{\bf B}+\sqrt{-1}\omega)\leftrightarrow(M,\check{\Omega},\check{\bf B}+\sqrt{-1}\check{\omega}).$$

Here $[\Omega],[\check{\Omega}]\in\mathcal{P}_{M}$, with $q_{M}(E,\Omega),q_{M}(E,\check{\Omega})\not=0$, so that we can assume $\Omega$ and $\check{\Omega}$ are normalized with $q_{M}(E,\Omega)=q_{M}(E,\check{\Omega})=1$. Furthermore, ${\bf B},\check{\bf B}\in E^{\perp}/E$ and $\omega,\check{\omega}\in E^{\perp}$ satisfy $q_{M}(\omega,\Omega)=q_{M}(\check{\omega},\check{\Omega})=0$ and $q_{M}(\omega),q_{M}(\check{\omega})>0$. The relationship between the two triples is that $\check{\Omega}=m_{E,\sigma}({\bf B}+\sqrt{-1}\omega)$ and $\check{\bf B},\check{\omega}$ are the unique cohomology classes satisfying the above conditions and $\Omega=m_{E,\sigma}(\check{\bf B}+\sqrt{-1}\check{\omega})$. Indeed, $\check{\bf B}$ and $\check{\omega}$ exist, since as $q_{M}(E,\Omega)=1$, we can write $\Omega=\frac{1}{q_{M}(E,\sigma)}\sigma+\check{\bf B}+\sqrt{-1}\check{\omega}\mod E$, and replacing a chosen representative $\check{\omega}$ with $\check{\omega}-(q_{M}(\check{\omega},\sigma)/q_{M}(E,\sigma)-q_{M}(\check{\omega},{\bf B}))E$, one guarantees that $q_{M}(\check{\Omega},\check{\omega})=0$.

This mirror symmetry on the level of period domains doesn’t quite give an exact mirror symmetry on the level of moduli spaces, since global Torelli does not in general hold for hyperkähler manifolds, so there might be a number of choices of complex structure on $M$ with period $[\Omega]$. In addition, $\omega$ or $\check{\omega}$ need not represent a Kähler form except for very general choices of complex structure.

Nevertheless, this allows us to identify a large complex structure limit as being mirror to a large Kähler limit. The family

$$\left(M,\Omega=\frac{1}{q_{M}(E,\sigma)}\sigma+\check{\bf B}+\sqrt{-1}\check{\omega}\bmod E,s\omega\right),$$

represents a large Kähler limit, with the Kähler class moving off to infinity while the complex structure is fixed, and this is mirror to the triple

$$\left(M,\check{\Omega}_{s}=\frac{1}{q_{M}(E,\sigma)}\sigma+\sqrt{-1}s\omega\bmod E,\check{\bf B}+\sqrt{-1}\check{\omega}\right).$$

If for each $s$, we have an actual hyperkähler manifold with period $\check{\Omega}_{s}$ and Kähler form $\check{\omega}$, we would like to understand the limiting metric behaviour.

To do so, we use hyperkähler rotation, and to do this we need to normalize the holomorphic symplectic form, defining

$$\check{\Omega}_{s}^{{\operatorname{nor}}}=s^{-1}\sqrt{\frac{q_{M}(\check{\omega})}{q_{M}(\omega)}}\check{\Omega}_{s}.$$

Then we have $q_{M}(\operatorname{Re}\check{\Omega}^{{\operatorname{nor}}}_{s})=q_{M}(\operatorname{Im}\check{\Omega}^{{\operatorname{nor}}}_{s})=q_{M}(\check{\omega})$. So $\operatorname{Re}\check{\Omega}^{{\operatorname{nor}}}_{s}$, $\operatorname{Im}\check{\Omega}^{{\operatorname{nor}}}_{s}$ and $\check{\omega}$ form a hyperkähler triple, and hence we can hyperkähler rotate to obtain a hyperkähler manifold with holomorphic two-form

$$\check{\Omega}_{s,J}:=\operatorname{Im}\check{\Omega}^{{\operatorname{nor}}}_{s}+\sqrt{-1}\check{\omega}=\sqrt{\frac{q_{M}(\check{\omega})}{q_{M}(\omega)}}\left(\omega-\frac{q_{M}(\omega,\sigma)}{q_{M}(E,\sigma)}E\right)+\sqrt{-1}\check{\omega}$$

and Kähler form

$$\check{\omega}_{s,J}=\operatorname{Re}\check{\Omega}^{{\operatorname{nor}}}_{s}=\sqrt{\frac{q_{M}(\check{\omega})}{q_{M}(\omega)}}\left[\frac{1}{s}\left(\frac{1}{q_{M}(E,\sigma)}\sigma-\frac{1}{2}\frac{q_{M}(\sigma)}{q_{M}(E,\sigma)^{2}}E\right)+\frac{s}{2}q_{M}(\omega)E\right].$$

We note that the period $\check{\Omega}_{s,J}$ is in fact independent of $s$, so we can fix the complex structure on $M$ independent of $s$. Assume that $E$ is the first Chern class of a nef line bundle on $M$ with respect to a complex structure with period $\check{\Omega}_{s,J}$, and $\check{\omega}_{s,J}$ is a Kähler class with respect to this complex structure if $s\geqslant s_{0}$, for some $s_{0}\gg 0$. We now take $s=s_{0}\sqrt{\frac{t+1}{t}}$, so that as $t$ goes to zero, $s$ goes to infinity and we define the rescaled metrics

$$\check{\omega}_{t,J}^{\mathrm{nor}}=\sqrt{t(t+1)}\check{\omega}_{s(t),J}=t\check{\omega}_{s_{0},J}+\frac{s_{0}}{2}\sqrt{q_{M}(\check{\omega})q_{M}(\omega)}E.$$

So as $t\rightarrow 0$, $\check{\omega}_{t,J}^{\mathrm{nor}}$ moves on a straight line towards $\frac{s_{0}}{2}\sqrt{q_{M}(\check{\omega})q_{M}(\omega)}E$, and $\check{\omega}_{s_{0},J}$ is Kähler.

To relate this to the results of this paper, we have the following conjecture, stated in [19, 42]:

If such a map exists, it is necessarily a holomorphic Lagrangian fibration. If furthermore $M$ is projective then $N^{\prime}=\mathbb{CP}^{n}$ by [22]. This conjecture follows from the log abundance conjecture if some multiple of $\mathcal{L}$ is effective, and has been studied for example in [1, 4, 19, 42].

Let us suppose this conjecture holds. By choosing $s_{0}$ properly, we assume that $\frac{s_{0}}{2}\sqrt{q_{M}(\check{\omega})q_{M}(\omega)}$ is a integer, and thus $\frac{s_{0}}{2}\sqrt{q_{M}(\check{\omega})q_{M}(\omega)}E=f^{\prime*}\alpha$ for an ample class $\alpha$ on $N^{\prime}$, where $f^{\prime}$ and $N^{\prime}$ are obtained by Conjecture 2.3. Because of the hyperkähler rotation, the Riemannian metrics defined by $(\check{\Omega}^{{\operatorname{nor}}}_{s},\check{\omega})$ and by $(\check{\Omega}_{s,J},\check{\omega}_{s,J})$ are the same. Therefore, to understand the Gromov-Hausdorff limit of the large complex structure limit $(M,\check{\Omega}^{{\operatorname{nor}}}_{s},\check{\omega})$ (this is the same that appears in the statement of Theorem 1.3) we can instead consider $(M,\check{\Omega}_{s,J},\check{\omega}_{s,J})$. Now $\check{\Omega}_{s,J}$ is independent of $s$, so we are simply changing the Kähler class, and the rescaled metrics $\check{\omega}_{t,J}=\sqrt{t(t+1)}\check{\omega}_{s(t),J}=t\check{\omega}_{s_{0},J}+f^{\prime*}\alpha$ move towards $f^{\prime*}\alpha$ along a straight line. Therefore we are exactly in the setting of Theorem 1.1 and Theorem 1.2, which describe the Gromov-Hausdorff limit of $(M,\check{\omega}_{t,J})$ as $t$ goes to zero. But as remarked in the Introduction, we also have that the diameter of $\check{\omega}_{t,J}$ is bounded uniformly away from zero and infinity, so if we further rescale the metrics $\check{\omega}_{t,J}$ to have diameter $1$, then up to a subsequence the Gromov-Hausdorff limit only changes by a rescaling, and Theorem 1.3 follows.