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.