In this paper, which is a continuation of [To2], we study the behaviour of Ricci-flat Kähler metrics on a compact Calabi-Yau manifold when the Kähler class degenerates to the boundary of the Kähler cone. Given a compact Calabi-Yau manifold, a fundamental theorem of Yau [Y1] says that there exists a unique Ricci-flat Kähler metric in each Kähler class. If we now move the Kähler class inside the Kähler cone, the corresponding Ricci-flat metrics vary smoothly, as long as the class does not approach the boundary of the Kähler cone. The question that we want to address is to understand what happens to the Ricci-flat metrics if the class goes to the boundary of the Kähler cone. This question has been raised by Yau and reiterated by others, see for example [Y3, W, McM]. In our previous work [To2] we have studied the case when the limit class has positive volume. In this paper we consider the case when the limit volume is zero, and we focus on the situation when the Calabi-Yau manifold admits a holomorphic fibration to a lower dimensional space such that the limit class is the pullback of a Kähler class from the base.

To state our result, let us introduce some notation. Let $(X,\omega_{X})$ be a compact Kähler $n$-manifold with $c_{1}(X)=0$ in $H^{2}(X,\mathbb{R})$. The condition that $c_{1}(X)=0$ is equivalent to the requirement that the canonical bundle of $X$ be torsion (see [To2]), and we call $X$ a Calabi-Yau manifold. We assume that there is a holomorphic map $f:X\to Z$ where $(Z,\omega_{Z})$ is another compact Kähler manifold. We denote by $Y$ the image of $X$ under $f$, and we assume that $Y$ is an irreducible normal subvariety of $Z$ of dimension $m$ with $0<m<n$, and that the map $f:X\to Y$ has connected fibers. Then $\omega_{0}=f^{*}\omega_{Z}$ is a smooth nonnegative $(1,1)$ form on $X$, whose cohomology class lies on the boundary of the Kähler cone. We will also denote by $\omega_{Y}$ the restriction of $\omega_{Z}$ to the regular part of $Y$. The map $f:X\to Y$ is an “algebraic fiber space” in the sense of [La] and we can find a subvariety $S\subset X$ such that $Y\backslash f(S)$ is smooth and $f:X\backslash S\to Y\backslash f(S)$ is a smooth submersion ($S$ consists of singular fibers, as well as fibers of dimension strictly larger than $n-m$). Then for any $y\in Y\backslash f(S)$ the fiber $X_{y}=f^{-1}(y)$ is a smooth $(n-m)$-manifold, equipped with the Kähler form $\omega_{X}|_{X_{y}}$. Notice that since $f$ is a submersion near $X_{y}$, we have that $c_{1}(X_{y})=c_{1}(X)|_{X_{y}},$ and so the fibers $X_{y}$ with $y\in Y\backslash f(S)$ are themselves Calabi-Yau. Yau’s theorem [Y1] says that in each Kähler class of $X$ there is a unique Kähler metric with Ricci curvature identically zero. For each $0<t\leq 1$ we call $\tilde{\omega}_{t}$ the Ricci-flat Kähler metric cohomologous to $[\omega_{0}]+t[\omega_{X}]$, and we wish to study the behaviour of these metrics when $t$ goes to zero. First of all in [To2] we proved the following

The volume of any fiber $X_{y}$ with respect to $\tilde{\omega}_{t}$ is comparable to $t^{n-m}$, and the Ricci-flat metrics $\tilde{\omega}_{t}$ approach an “adiabatic limit”. We will show that the metrics $\tilde{\omega}_{t}$ collapse to a Kähler metric on $Y\backslash f(S)$.

Our main theorem is the following:

This result generalizes work of Gross and Wilson [GW], who considered the case when $f:X\to Y=\mathbb{P}^{1}$ is an elliptically fibered $K3$ surface with $24$ singular fibers of type $I_{1}$ (see section 5). They achieved their result by writing down explicit approximations of the Ricci-flat metrics near the adiabatic limit. A similar approach in higher dimensions seems out of reach at present. Instead, our main technical tool are some new general a priori estimates for complex Monge-Ampère equations on the total space of a holomorphic fibration.

Let us briefly explain the meaning of $\omega_{WP}$, referring the reader to section 4 for more details. We have already remarked that the smooth fibers $X_{y}$ of $f$ are themselves Calabi-Yau $(n-m)$-manifolds, polarized by $\omega_{X}|_{X_{y}}$. If the canonical bundles of the fibers are actually trivial we get a map from $Y\backslash f(S)$ to the moduli space of polarized Calabi-Yau manifolds and by pulling back the Weil-Petersson metric we get a smooth nonnegative form $\omega_{WP}$ on $Y\backslash f(S)$ (a similar construction goes through in the case when the fibers have torsion canonical bundle). Notice that $\omega_{WP}$ is identically zero precisely when the complex structure of the fibers doesn’t change. The appearance of the Weil-Petersson metric in the more general setting of adiabatic limits of constant scalar curvature Kähler metrics was observed by Fine [Fi] and further studied by Stoppa [St].

There are two possible situations that we have in mind for our setup: in one case $Y$ is smooth, and then we can just take $Z=Y$. In the second case we take $Z=\mathbb{P}^{N}$ with $\omega_{Z}$ the Fubini-Study metric, and then $Y$ is an algebraic variety. A natural class of examples where this situation arises is the following: $X$ is a a projective Calabi-Yau manifold and $L$ is a semiample line bundle over $X$ with Iitaka dimension $\kappa(X,L)=m<n$. Then a classical construction of Iitaka (see 2.1.27 in [La]) gives a holomorphic map $f:X\to\mathbb{P}^{N}$ exactly as in the setup. Note that if the log Abundance Conjecture holds then every line bundle $L$ with cohomology class on the boundary of the Kähler cone and with $\kappa(X,L)=m<n$ is automatically semiample (see [To2]). This is known to hold if $n=2$.

The organization of the paper is the following. In section 2 we set up the problem as a family of degenerating complex Monge-Ampère equations and state our estimates that imply the main result. In section 3 we prove a priori $C^{2}$ estimates for these equations as well as $C^{3}$ estimates along the fibers. In section 4 we use the estimates to prove our main theorem, and in section 5 we provide a few examples.

Acknowledgments. I would like to thank my advisor Shing-Tung Yau for suggesting this problem and for constant support. I also thank Chen-Yu Chi, Jian Song, Gábor Székelyhidi and Ben Weinkove for very useful discussions. I was partially supported by a Harvard Merit Fellowship. These results are part of my PhD thesis at Harvard University [To3].