A Calabi-Yau manifold $M$ is a compact Kähler manifold with vanishing first Chern class $c_{1}(M)=0$ in $H^{2}(M,\mathbb{R})$. A fundamental theorem of Yau [45] says that on $M$ there exists a unique Ricci–flat Kähler metric in each Kähler class. If we move the Kähler class towards a limit class on the boundary of the Kähler cone, we get a family of Ricci–flat Kähler metrics which degenerates in the limit. The general question of understanding the geometric behaviour of these metrics was raised by Yau [46, 47], Wilson [44] and others, and much work has been devoted to it, see for example [18, 29, 30, 34, 37, 38] and references therein. In this paper, we study metric degenerations of Ricci–flat Kähler metrics whose Kähler classes approach semi-ample non-big classes.

The first useful observation is that the diameters of a family of Ricci–flat Kähler metrics $\tilde{\omega}_{t}$, $t\in(0,1]$, on a Calabi-Yau manifold $M$ are uniformly bounded if their Kähler classes $[\tilde{\omega}_{t}]$ tend to a limit class $\alpha$ on the boundary of the Kähler cone when $t\rightarrow 0$ [37, 48]. Another special feature of the Kähler case is that the volume of the Ricci–flat metrics can be computed cohomologically, and to determine whether it will approach zero or stay bounded away from it, it is enough to calculate the self-intersection $\alpha^{n}$ where $n=\dim_{\mathbb{C}}M$. If $\alpha^{n}$ is strictly positive, then it was proved by the second-named author [37] that the Ricci–flat metrics do not collapse, (i.e., there is a constant $\upsilon>0$ independent of $t$ such that each $\tilde{\omega}_{t}$ has a unit radius metric ball with volume bigger than $\upsilon$), and in fact converge smoothly away from a subvariety. If $\alpha^{n}$ is zero, then the total volume of the Ricci–flat metrics approaches zero, so one expects to have collapsing to a lower-dimensional space. This was shown to be the case for elliptically fibered $K3$ surfaces by Gross-Wilson [18], and later the second-named author considered the higher dimensional case when the Calabi-Yau manifold $M$ admits a holomorphic fibration to a lower-dimensional Kähler space, and the limit class is the pullback of a Kähler class [38]. The first goal of the present paper is to improve the convergence result in [38].

Let us now describe our first result in detail. Let $(M,\omega_{M})$ be a compact Calabi-Yau $n$-manifold which admits a holomorphic map $f:M\to Z$ where $(Z,\omega_{Z})$ is a compact Kähler manifold. Thanks to Yau’s theorem, we can assume that $\omega_{M}$ is Ricci–flat. Denote by $N=f(M)$ the image of $f$, and assume that $N$ is an irreducible normal subvariety of $Z$ with dimension $m$, $0<m<n$, and that the map $f:M\to N$ has connected fibers. Denote by $\omega_{0}=f^{*}\omega_{Z}$, which is a smooth nonnegative real $(1,1)$-form on $M$ whose cohomology class lies on the boundary of the Kähler cone of $M$, and denote also by $\omega_{N}$ the restriction of $\omega_{Z}$ to the regular part of $N$. For example, one can take either $Z=N$ (if $N$ is smooth), or $Z=\mathbb{CP}^{N}$ (if $N$ is an algebraic variety). This second case arises whenever we have a line bundle $L\to M$ which is semiample (some power is globally generated) and of Iitaka dimension $m<n$, so $L$ is not big.

In general, given a map $f:M\to N$ as above, there is a proper analytic subvariety $S\subset M$ such that $N\backslash f(S)$ is smooth and $f:M\backslash S\to N\backslash f(S)$ is a smooth submersion (the set $S$ is exactly where the differential $df$ does not have full rank $m$). For any $y\in N\backslash f(S)$ the fiber $M_{y}=f^{-1}(y)$ is a smooth Calabi-Yau manifold of dimension $n-m$, and it is equipped with the Kähler metric $\omega_{M}|_{M_{y}}$. The volume of the fibers $\int_{M_{y}}(\omega_{M}|_{M_{y}})^{n-m}$ is a homological constant that does not depend on $y$ in $N\backslash f(S)$, and we can assume that it equals $1$. Consider the Kähler metrics on $M$ given by $\omega_{t}=\omega_{0}+t\omega_{M}$, with $0<t\leqslant 1$, and call $\tilde{\omega}_{t}=\omega_{t}+\sqrt{-1}\partial\overline{\partial}\varphi_{t}$ the unique Ricci–flat Kähler metric on $M$ cohomologous to $\omega_{t}$, with potentials normalized by $\sup\limits_{M}\varphi_{t}=0$. They satisfy a family of complex Monge-Ampère equations

where $c_{t}$ is a constant that has a positive limit as $t\to 0$ (see (4.28)). A general $C^{0}$ estimate $\|\varphi_{t}\|_{C^{0}}\leqslant C$ (independent of $t>0$) for such equations was proved by Demailly-Pali [9] and Eyssidieux-Guedj-Zeriahi [10], generalizing work of Kołodziej [24]. In the case under consideration, much more is true: the second-named author’s work [38] shows that there exists a smooth function $\varphi$ on $N\backslash f(S)$ so that as $t$ goes to zero we have $\varphi_{t}\to\varphi\circ f$ in $C^{1,\alpha}_{loc}(M\backslash S,\omega_{M})$ for any $0<\alpha<1$. Moreover $\omega=\omega_{N}+\sqrt{-1}\partial\overline{\partial}\varphi$ is a Kähler metric on $N\backslash f(S)$ with $\Ric(\omega)=\omega_{\rm WP}$. Here $\omega_{\rm WP}$ is the pullback of the Weil-Petersson metric from the moduli space of polarized Calabi-Yau fibers, which has appeared several times before in the literature [11, 18, 34, 38].

We now assume that the every fiber $M_{y}$ with $y\in N\backslash f(S)$ is biholomorphic to a complex torus (of course, it is enough to assume that just one smooth fiber is a complex torus). This is the case for example whenever $M$ is hyperkähler. We also assume that $M$ is projective, so we can take $[\omega_{M}]$ to be the first Chern class of an ample line bundle. In this case we can improve the above result, thus answering Questions 4.1 and 4.2 of [39] in our setting:

As remarked earlier, in the case of elliptically fibered $K3$ surfaces ($n=2,m=1$) this theorem follows from the work of Gross-Wilson [18]. In higher dimensions, in the very special case when $S$ is empty, the theorem (except (1.2)) also follows from the work of Fine [11]. Both these works take a different approach from us, by constructing the Ricci–flat metrics $\tilde{\omega}_{t}$ as small perturbations of semi-flat metrics (see section 3), which in [18] are glued to Ooguri-Vafa metrics near the singular fibers. By contrast, we work directly with the Ricci–flat metrics $\tilde{\omega}_{t}$ and prove that they satisfy a priori estimates away from the singular fibers, which then implies the convergence results. This was also the approach taken by the second-named author in [38], where the convergence $\tilde{\omega}_{t}\to f^{*}\omega$ was proved in a weaker topology (see also the work of Song-Tian [34] for the case of $K3$ surfaces).

The curvature bound (1.2) in Theorem 1.1 does not hold if the generic fibers are not tori, as one can see for example by taking the product of two non-flat Calabi-Yau manifolds with the product Ricci-flat Kähler metric and then scaling one factor to zero. On the other hand, we believe that the assumption in Theorem 1.1 that $M$ is projective is just technical and it should be possible to remove it.

We now describe the second main result of the paper, which concerns the Gromov-Hausdorff limit of our manifolds. The Gromov-Hausdorff distance $d_{GH}$ was introducted by Gromov in the 1980’s [15], and it defines a topology on the space of isometry classes of all compact metric spaces. For two compact metric spaces $X$ and $Y$, the Gromov-Hausdorff distance of $X$ and $Y$ is

where $Z$ is a metric space and $d_{H}^{Z}(X,Y)$ denotes the standard Hausdorff distance between $X$ and $Y$ regarded as subsets in $Z$ by the isometric embeddings (see for example [15, 28] for more background). The Gromov-Hausdorff topology provides a framework to study families of compact metric spaces or Riemannian manifolds. We would like to understand the Gromov-Hausdorff convergence of $(M,\tilde{\omega}_{t})$ in Theorem 1.1. Since the volume of the whole manifold goes to zero, the manifolds $(M,\tilde{\omega}_{t})$ are collapsing. Furthermore, from Theorem 1.1 we know that on a Zariski open set of $M$ the Ricci-flat metrics collapse with locally bounded curvature.

The collapsing of Einstein manifolds and Riemannian manifolds with definite curvature bounds in the Gromov-Hausdorff sense has been extensively studied from different viewpoints, see for example [2, 5, 6, 7, 8, 12, 18, 27, 28, 33] and the reference therein. These general theories provide us with results which are particularly strong in the case of Riemannian manifolds with bounded sectional curvature and Einstein manifolds of dimension $4$. The first detailed analysis of the collapsing of geometrically interesting families of Einstein $4$-manifolds was done by Anderson in [2]. More recently, a result of Cheeger-Tian [7] shows that on any sufficiently collapsed Ricci–flat Einstein $4$-manifold with volume $1$ there is a large open set $U$ where the sectional curvature is bounded by a universal constant, and $U$ admits an $\mathcal{F}$-structure, which is a generalization of torus fibration. Furthermore, by [27], the collapsed limits of Ricci–flat Einstein $4$-manifolds with bounded Euler numbers are smooth Riemannian orbifolds away from a finite number of points. The metric structure of collapsed limits of higher-dimensional Einstein $n$-manifolds (and more generally manifolds with a uniform lower bound on the Ricci curvature) was extensively studied by Cheeger-Colding [5] and collaborators. Regarding the collapsed Gromov-Hausdorff limit of the Ricci–flat metrics in Theorem 1.1, we have the following result.

First of all, thanks to [37, 48] we know that the diameter of $(M,\tilde{\omega}_{t})$ satisfies

for some constant $D$ and for all $t>0$. Furthermore, since $\tilde{\omega}_{t}\to f^{*}\omega$ and the base $N$ is not a point, we also have that $\mathrm{diam}_{\tilde{\omega}_{t}}(M)\geqslant D^{-1}.$ Given any sequence $t_{k}\rightarrow 0$, Gromov’s precompactness theorem shows that a subsequence of $(M,\tilde{\omega}_{t_{k}})$ converges to some compact path metric space $(X,d_{X})$ in the Gromov-Hausdorff topology. Note that because of the upper and lower bounds for the diameter, if we rescaled the metrics $\tilde{\omega}_{t_{k}}$ to have diameter equal to one, the Gromov-Hausdorff limit (modulo subsequences) would be isometric to $(X,d_{X})$ after a rescaling.

In fact we prove that $X\backslash X_{0}$ has measure zero with respect to the renormalized limit measure of [5], which implies that $X_{0}$ is dense in $X$. It would be interesting to prove that the metric completion of $(N\backslash f(S),\omega)$ is isometric to $(X,d_{X})$. In the case of $K3$ surfaces this was proved by Gross-Wilson [18].

As an application of Theorem 1.1 and Theorem 1.2 we study the metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In [36], Stominger, Yau and Zaslow proposed a conjecture about constructing the mirror manifold of a given Calabi-Yau manifold via special Lagrangian fibrations. This became known as the SYZ conjecture, and has generated an immense amount of work, see for example [16, 17, 18, 25] and references therein. Later another version of the SYZ conjecture was proposed by Gross-Wilson [18], Kontsevich-Soibelman [25] and Todorov via degenerations of Ricci–flat Kähler-Einstein metrics. The conjecture says that if $\{M_{t}\}$, $t\in\Delta\backslash\{0\}\subset\mathbb{C}$, is a family of polarized Calabi-Yau $n$-manifolds, $\omega_{t}$ is the Ricci–flat Kähler-Einstein metric representing the polarization on $M_{t}$, and the complex structure of $M_{t}$ tends to a large complex structure limit point in the deformation moduli space of $M_{t}$ when $t\rightarrow 0$, then after rescaling $(M_{t},\omega_{t})$ to have diameter $1$, they collapse to a compact metric space $(X,d_{X})$ in the Gromov-Hausdorff sense. Furthermore, a dense open subset $X_{0}\subset X$ is a smooth manifold of real dimension $n$, and the codimension of $X\backslash X_{0}$ is bigger or equal to $2$. This conjecture holds trivially for tori, and was verified for $K3$ surfaces by Gross-Wilson in [18].

In the third main result of this paper we consider this conjecture for higher dimensional hyperkähler manifolds. We will describe it briefly here, and give a more complete description in Section 2. Let $(M,I)$ be a compact complex manifold of complex dimension $2n$ with a Ricci–flat Kähler metric $\omega_{I}$ with holonomy the full group $Sp(n)$. In particular $M$ is Calabi-Yau (in our definition), and furthermore it has a hyperkähler structure. We assume that there is an ample line bundle over $M$ with the first Chern class $[\omega_{I}]$, that we have a holomorphic fibration $f:M\to N$ as before with $N$ a projective variety, and that there is a holomorphic section $s:N\to M$. Under these assumptions, it is known that $N=\mathbb{CP}^{n}$ [22], and that the smooth fibers of $f$ are complex Lagrangian tori [26]. If we perform a hyperkähler rotation of the complex structure, the fibers become special Lagrangian, and we are exactly in the setup of Strominger, Yau and Zaslow [36]. We furthermore assume that the polarization induced on the torus fibers is principal. In this case, the SYZ mirror symmetry picture predicts that $M$ is mirror to itself, and that a large complex structure limit is mirror to a large Kähler structure limit. We use this as our definition of large complex structure limit, so we have a family of polarized hyperkähler structures $(M,\check{\Omega}_{s})$ with Ricci-flat Kähler metric $\check{\omega}$ which approach a large complex structure limit as $s\to\infty$. By assuming the validity of a standard conjecture on hyperkähler manifolds (Conjecture 2.3), we can performe a hyperkähler rotation and a normalization to reduce exactly to the setup covered by Theorems 1.1 and 1.2, and we can prove:

This proves the conjecture of Gross-Wilson [18], Kontsevich-Soibelman [25] and Todorov in our situation, modulo these assumptions, except for the statement that $\mathrm{codim}_{\mathbb{R}}(X\backslash X_{0})\geqslant 2$ where more arguments are needed. Again, this was proved by Gross-Wilson [18] in the case of $K3$ surfaces.

This paper is organized as follows. In Section 2 we study SYZ mirrors of some hyperkähler manifolds, and derive Theorem 1.3 as a consequence of Theorems 1.1 and 1.2. In Section 3 we construct semi-flat background metrics on the total space of a holomorphic torus fibration. Theorem 1.1 is proved in Section 4 while Theorem 1.2 is proved in Section 5.

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

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

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

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

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

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

We then have an isomorphism

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

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

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

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

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

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

and Kähler form

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

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.

In this section we discuss semi-flat forms and metrics, extending some results in [18, 20] to our setting.

In general a closed real $(1,1)$-form $\omega_{SF}$ on an open set $U\subset M\backslash S$ will be called semi-flat if its restriction to each torus fiber $M_{y}\cap U$ with $y\in f(U)$ is a flat metric, which we will always assume to be cohomologous to $\omega_{M}|_{M_{y}}$. If $\omega_{SF}$ is also Kähler then we will call it a semi-flat metric. Semi-flat forms can also be defined when the fibers $M_{y}$ are not tori but general Calabi-Yau manifolds, by requiring that the restriction to each fiber be Ricci–flat (see [34, 38]). They were first introduced by Greene-Shapere-Vafa-Yau in [14].

Fix now a small ball $B\subset N\backslash f(S)$ with coordinates $y=(y_{1},\dots y_{m})$, and consider the preimage $f:U=f^{-1}(B)\to B$. This is a holomorphic family of complex tori, and if $B$ is small enough it has a holomorphic section $\sigma_{0}$, which we also fix. We can then define a complex Lie group structure on each fiber $M_{y}=f^{-1}(y)$ with unit $\sigma_{0}(y)$. We claim that this family is locally isomorphic to a family of the form $f^{\prime}:(B\times\mathbb{C}^{n-m})/\Lambda\to B,$ where $h:\Lambda\to B$ is a lattice bundle with fiber $h^{-1}(y)=\Lambda_{y}\cong\mathbb{Z}^{2n-2m}$, so that $M_{y}\cong\mathbb{C}^{n-m}/\Lambda_{y}.$ To see this, note that each fiber $M_{y}=f^{-1}(y)$ is a torus biholomorphic to $\mathbb{C}^{n-m}/\Lambda_{y}$ for some lattice $\Lambda_{y}$ that varies holomorphically in $y$. We choose a basis $v_{1}(y),\dots,v_{2n-2m}(y)$ of this lattice, which varies holomorphically in $y$. Given these lattices we can construct the family $f^{\prime}$ by taking the quotient of $B\times\mathbb{C}^{n-m}$ by the $\mathbb{Z}^{2n-2m}$-action given by $(n_{1},\dots,n_{2n-2m})\cdot(y,z)=(y,z+\sum_{i}n_{i}v_{i}(y))$, where $z=(z_{1},\dots,z_{n-m})\in\mathbb{C}^{n-m}$. Note that different choices of generators give isomorphic quotients. By construction the fiber $f^{\prime-1}(y)$ is biholomorphic to $f^{-1}(y)$ for all $y\in B$. A theorem of Kodaira-Spencer [23] (see also [43, Satz 3.6]) then implies that the families $f$ and $f^{\prime}$ are locally isomorphic, so up to shrinking $B$ there exists a biholomorphism $(B\times\mathbb{C}^{n-m})/\Lambda\to U$ compatible with the projections to $B$, proving our claim. With this identification, the section $\sigma_{0}:B\to U$ is induced by the map $B\to B\times\mathbb{C}^{n-m}$ given by $y\mapsto(y,0)$.

Composing this biholomorphism with the quotient map $B\times\mathbb{C}^{n-m}\to(B\times\mathbb{C}^{n-m})/\Lambda$ by the $\mathbb{Z}^{2n-2m}$-action we get a holomorphic map $p:B\times\mathbb{C}^{n-m}\to U$ such that $f\circ p(y,z)=y$ for all $(y,z)$, and $p$ is a local isomorphism (the map $p$ is also the universal covering map of $U$).

We now assume that $M$ is projective and $[\omega_{M}]$ is an integral class, so each complex torus fiber $M_{y}$, $y\in B$, can be polarized by $[\omega_{M}]$, which gives an ample polarization of type $(d_{1},\ldots,d_{n-m})$ for some sequence of integers $d_{1}|d_{2}|\cdots|d_{n-m}$. By [3], Proposition 8.1.1, one can then assume that $\Lambda$ is generated by $d_{1}e_{1},\ldots,d_{n-m}e_{n-m},Z_{1},\ldots,Z_{n-m}\in\mathbb{C}^{n-m}$, where $e_{1},\ldots,e_{n-m}$ is the standard basis for $\mathbb{C}^{n-m}$. Furthermore, the matrix $Z$ with columns $Z_{1},\ldots,Z_{n-m}$ must satisfy $Z=Z^{t}$ and ${\rm Im}Z$ positive definite. Also, on the fibre $M_{y}$, the Kähler form $\sum_{i,j}\sqrt{-1}(\operatorname{Im}Z)_{ij}dz^{i}\wedge d\bar{z}^{j}$ is cohomologous to $\omega_{M}|_{M_{y}}$. Let

Note that $Z$ depends on $y\in B$, as does $g_{ij}$. Recall that we have the fiber coordinates $z_{1},\dots,z_{n-m}$. Consider the function

We would first like to show that $\sqrt{-1}\partial\bar{\partial}\eta$ is invariant under translation by flat sections of the Gauss-Manin connection on $B\times\mathbb{C}^{n-n}$ (this is the connection on this bundle such that sections of $\Lambda$ are flat sections of the bundle). It is enough to check invariance under translation by $\lambda s$ for $s$ one of the generators of $\Lambda$, $\lambda\in\mathbb{R}$. First, consider the composition of $\eta$ with a general translation $z_{i}\mapsto z_{i}+\tau_{i}(y)$:

the last equality by the symmetry $g_{ij}=g_{ji}$. We now consider two cases. If $\tau_{i}=\lambda\delta_{ik}$ for some $k$, so that $\tau_{i}$ is real, then in fact the above formula reduces to $\eta$, so $\eta$ is itself invariant under this translation. Secondly, if we take $\tau_{i}=\lambda Z_{ik}$ for some $k$, $\lambda\in\mathbb{R}$, we obtain

Applying $\partial\bar{\partial}$ kills the correction term, so $\sqrt{-1}\partial\bar{\partial}\eta$ is invariant under this action. This means that $\sqrt{-1}\partial\overline{\partial}\eta$ is the pullback under $p$ of a two-form $\omega_{SF}$ on $U$