Section 1. (00FC) · ScalingStacks

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1. Introduction

Einstein metrics, namely metrics with constant Ricci curvature, have been an important subject of study in the field of differential geometry since the early days. The solution of the Calabi Conjecture given by Yau [Y1] in 1976 provided a very powerful existence theorem for Kähler-Einstein metrics with negative or zero Ricci curvature (the negative case was also done independently by Aubin [Au]). This produced a number of nonhomogeneous examples of Ricci-flat manifolds. These spaces have been named Calabi-Yau manifolds by the physicists in the Eighties, and have been throughly studied in several different areas of mathematics and physics. Prompted by the physical intuition of mirror symmetry, mathematicians have studied the ways in which Calabi-Yau manifolds can degenerate when they are moving in families. In general both the complex and symplectic (Kähler) structure are changing, and the general behaviour is not well understood. In this paper we will consider the case when the complex structure is fixed, and so we will be looking at a single compact projective Calabi-Yau manifold. The Kähler class is then allowed to vary inside the ample cone. As long as the class stays inside the cone, the corresponding Ricci-flat metrics vary smoothly, but they will degenerate when the class approaches the boundary of the cone. We will try to understand this degeneration process and see what the limiting space looks like.

To introduce our results, let us fix some notation first. Let $X$ be a compact projective Calabi-Yau manifold, of complex dimension $n$ . The real Neron-Severi space is by definition

N^{1}(X)_{\mathbb{R}}=(H^{2}(X,\mathbb{Z})_{free}\cap H^{1,1}(X))\otimes\mathbb{R}=N^{1}(X)_{\mathbb{Z}}\otimes\mathbb{R},

and we assume that

\dim N^{1}(X)_{\mathbb{R}}=\rho(X)>1.

This cohomology space contains $\mathcal{K}_{NS}$ the ample cone, which is open. Its closure $\overline{\mathcal{K}}_{NS}$ is the nef cone. Fix a nonzero class $\alpha\in\overline{\mathcal{K}}_{NS}\backslash\mathcal{K}_{NS}$ , which exists precisely when $\rho(X)>1$ , and a smooth path $\alpha_{t}:[0,1]\to\overline{\mathcal{K}}_{NS}$ such that $\alpha_{t}\in\mathcal{K}_{NS}$ for $t<1$ and $\alpha_{1}=\alpha$ . For any $t<1$ Yau’s Theorem [Y2] gives us a unique Ricci-flat Kähler metric $\omega_{t}\in\alpha_{t}$ . Fixing a smooth path of reference metrics in $\alpha_{t}$ , it can be verified that the Ricci-flat metrics $\omega_{t}$ vary smoothly, as long as $t<1$ . We have the following very natural

Question 1: What is the behaviour of the metrics $\omega_{t}$ as $t\to 1$ ?

This question has a long history: it is a special case of a problem by Yau [Y3], [Y4], where the complex structure is also allowed to vary; it has been stated explicitly in this form by McMullen [McM] and Wilson [W2]. Physicists have also looked at this question, roughly predicting the behaviour that we will describe in Theorem 1.1 (see e.g. [HW]). One of the reasons that makes this question interesting is that the Ricci-flat metrics are not known explicitly, except in very few cases.

A nef class $\alpha\in N^{1}(X)_{\mathbb{R}}$ is called big if $\alpha^{n}>0.$ Classical results of Anderson [An], Bando-Kasue-Nakajima [BKN], Tian [Ti] and more recent results of Cheeger-Colding-Tian [CCT] give a partial answer to this question when $\alpha$ is big (we’ll explain this in section 3). Our main theorem, which does not rely on the previous results just quoted, gives a satisfactory answer to Question 1 in this case (see section 2 for definitions).

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Theorem 1.1. Let $X$ be a compact projective Calabi-Yau manifold, and let $\alpha\in N^{1}(X)_{\mathbb{R}}$ be a big and nef class that is not ample. Then there exist a proper analytic subvariety $E\subset X$ and a smooth incomplete Ricci-flat Kähler metric $\omega_{1}$ on $X\backslash E$ , that depend only on $\alpha$ , such that for any smooth path $\alpha_{t}\in\mathcal{K}_{NS}$ with $\alpha_{1}=\alpha$ , the Ricci-flat metrics $\omega_{t}\in\alpha_{t}$ converge to $\omega_{1}$ in the $C^{\infty}$ topology on compact sets of $X\backslash E$ . Moreover $\omega_{1}$ extends to a closed positive current with continuous potentials on the whole of $X$ , which lies in $\alpha$ . If $\alpha\in N^{1}(X)_{\mathbb{Z}}$ , that is if $\alpha=c_{1}(L)$ for some line bundle $L$ , then $E$ is the null locus of $L$ and $\omega_{1}$ is the pullback of a singular Ricci-flat Kähler metric on a Calabi-Yau model of $X$ obtained from the contraction map of $L$ .

There are many interesting concrete examples of our theorem, and we will examine a few of them in section 5. Roughly speaking, the case when $\alpha$ is nef and big corresponds to a “non-collapsing” sequence of metrics, meaning that the Gromov-Hausdorff limit has the same dimension. The “collapsing” case, when $\alpha$ is nef but not big, is much harder and we will briefly discuss it at the end of the paper. We state and prove our results for a path of classes $\alpha_{t}$ , but it’s immediate to see that the same results hold if instead we look at a sequence of classes $\alpha_{i}$ that converge to $\alpha$ . On the other hand, our result doesn’t say anything about the case when the classes $\alpha_{t}$ approach the boundary of the ample cone without converging to a limiting class, but moving out to infinity in $N^{1}(X)_{\mathbb{R}}$ . This case is relevant for mirror symmetry, as it should sometimes be the mirror of a large complex structure limit. Finally let us remark that the projectivity assumptions are only technical, and that we expect that a similar result holds when $X$ is just assumed to be Kähler, and the ample cone is replaced by the Kähler cone (see section 6).

The paper is organized as follows. In section 2 we recall some definition and results from algebraic geometry. In section 3 we prove a uniform diameter bound and we compare our results with previous literature. In section 4 we prove our main Theorem 1.1. The main analytic tool comes from pluripotential theory, and was developed in [Koł], [EGZ]. In section 5 we give some examples where our results apply, and recover in particular a result of Kobayashi and Todorov [KT]. Finally in section 6 we discuss some further directions for research.

Acknowledgements. I would like to thank my advisor Prof. Shing-Tung Yau for suggesting this problem and for constant support. I also thank Prof. Curt McMullen and Prof. M.S. Narasimhan for inspiring conversations, and Chen-Yu Chi, Jian Song, Gábor Székelyhidi and Ben Weinkove for useful comments.