Section 1. (00DN) · ScalingStacks

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1. Introduction

The main objects of study in this note are Ricci-flat Kähler metrics on Calabi-Yau manifolds whose complex structure degenerates. More precisely, we assume that we have $\pi:X\to\Delta^{*}\subset\mathbb{C}$ a projective holomorphic submersion with connected fibers of relative dimension $n$ with $K_{X/\Delta^{*}}\cong\mathcal{O}_{X}$ and which is meromorphic at $0$ (in the sense of [3]), meaning that it extends to a proper flat map $\pi:\mathfrak{X}\to\Delta$ with $\mathfrak{X}$ normal. We also fix a relative polarization $L\to X$ , and we will refer to this data as a polarized Calabi-Yau degeneration family, often without mentioning $L$ explicitly. The fibers $X_{t}$ for $t\in\Delta^{*}$ are thus polarized Calabi-Yau $n$ -folds.

A choice of $\mathfrak{X}$ as above will be called a model of $X$ . Models are highly non-unique, and in particular up to passing to a finite base change, we may assume by [10] that $X$ admits a semistable model, where $\mathfrak{X}$ is smooth and $X_{0}=\sum_{i\in I}E_{i}$ is reduced and has simple normal crossings. One can then apply a relative MMP to a semistable model and obtain a relatively minimal dlt model [15, 17], which is unique up to applying sequences of flops on the central fiber [2, 9]. Taking the dual intersection complex of the central fiber of any such minimal dlt model, one obtains a simplicial complex $\mathrm{Sk}(X)$ , the essential skeleton of $X$ , which was introduced in this context by Kontsevich-Soibelman [11] with a different but equivalent definition (cf. [16]), whose homeomorphism type is well-defined independent of any choice of models [17].

In this note we will not make direct use of the skeleton $\mathrm{Sk}(X)$ itself, but only of its real dimension which will be denoted by $m$ , and which appears naturally [3, 11] as the power of logarithmic blowup of the fiberwise integrals of the Calabi-Yau volume forms, as we will recall in Section 2 below. As shown in [17] we always have $m\leqslant n$ , and the case when $m=0$ happens if and only if (after possibly a finite base change) $X$ admits a semistable model with central fiber $X_{0}$ which is a Calabi-Yau variety with klt singularities. Furthermore, the case $m=n$ is equivalent to the monodromy transformation around $0$ acting on $H^{n}(X_{t},\mathbb{C})$ having a Jordan block of size $n+1$ . This is the familiar notion of a “large complex structure limit” from mirror symmetry, see e.g. [7], where these polarized Calabi-Yau degeneration families play a crucial role.

Our main interest is in the behavior as $t\to 0$ of the Ricci-flat Kähler metrics $\omega_{t}$ on $X_{t}$ in the scaled class $\frac{1}{|\log|t||}c_{1}(L)|_{X_{t}}$ , whose existence is guaranteed by Yau’s Theorem [26]. In [11, Conjecture 1] Kontsevich-Soibelman conjectured that if $X\to\Delta^{*}$ is a large complex structure limit of Calabi-Yau manifolds, then the diameter of $(X_{t},\omega_{t})$ is bounded away from zero and infinity (note that there is a typo in the statement of their conjecture), and furthermore they, and independently also Gross-Wilson [8] and Todorov, conjectured that the collapsed Gromov-Hausdorff limit of $(X_{t},\omega_{t})$ is a half-dimensional affine manifold with singularities in codimension $2$ , which is homeomorphic to $\mathrm{Sk}(X)$ , and which is expected to be the base of the Strominger-Yau-Zaslow fibration of $X_{t}$ [20], see e.g. [1, §7] and [24] for surveys of these and related topics.

The main theorem of this note is to prove the conjectured sharp diameter bound in [11, Conjecture 1], for all polarized Calabi-Yau degeneration families with $m>0$ , thus also settling [24, Conjecture 4.7]:

00DI

Theorem 1.1. Let $\pi:X\to\Delta^{*}$ be a polarized Calabi-Yau degeneration family, suppose that the dimension $m$ of the essential skeleton $\mathrm{Sk}(X)$ is positive, and let $\omega_{t}$ be the Ricci-flat Kähler metric on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)|_{X_{t}}$ , for $t\in\Delta^{*}.$ Then there is $C>0$ such that

C^{-1}\leqslant\mathrm{diam}(X_{t},\omega_{t})\leqslant C,

for all $t\in\Delta^{*}$ with $|t|$ sufficiently small.

The assumption that $m>0$ is necessary, since when $m=0$ we can find a semistable model $\mathfrak{X}$ with central fiber $X_{0}$ a Calabi-Yau variety with klt singularities, as mentioned above, and then it is known by work of Rong-Zhang [18] that we have instead $\mathrm{diam}(X_{t},\omega_{t})\sim|\log|t||^{-\frac{1}{2}}$ .

Despite several recent works addressing diameter bounds for Kähler-Einstein metrics under various assumptions, see e.g. [6, 12, 19], the only previously known general results in the direction of our main theorem are the following. First, tracing through the arguments given in [18, Theorem 2.1] (see also [21, Proposition 4.2]) gives the upper bound

\mathrm{diam}(X_{t},|\log|t||\omega_{t})\leqslant C|\log|t||^{m},

which is worse than the one provided by Theorem 1.1. And secondly, it follows from the earlier works [25, 23, 21] that when $m>0$ we necessarily have $\mathrm{diam}(X_{t},|\log|t||\omega_{t})\to\infty$ (see also the exposition in [27]), but the arguments there do not provide any explicit lower bound.

The rough idea of our proof is the following: as we will recall in Section 2, a well-known computation in polar coordinates (cf. [3]) reveals that most of the mass of the Calabi-Yau volume forms $\omega_{t}^{n}$ on $X_{t}$ is carried by “very small” regions which are near certain intersections of $m+1$ irreducible components of the central fiber. In Section 3 we then construct a Kähler metric $\omega^{\prime}_{t}$ cohomologous to $\omega_{t}$ which behaves like a toric metric in this good region (in the directions $z_{1},\dots,z_{m}$ normal to these components). Using $\omega^{\prime}_{t}$ we then obtain a uniform $L^{1}$ bound for $|d\rho_{t}|_{\omega_{t}}$ where $\rho_{t}$ looks like a paraboloid in the logarithmic coordinates $x_{1},\dots,x_{m}$ in our good region, and using Cheeger-Colding’s segment inequality [4] we deduce the diameter lower bound. Lastly, in Section 4 we again use $\omega^{\prime}_{t}$ to produce a unit-size $\omega_{t}$ -geodesic ball in $X_{t}$ whose volume is a definite fraction of the total, from which the diameter upper bound follows from an argument of Yau, as in [22, 18].

Acknowledgments. The first-named author is a 2020 Clay Research Fellow, currently based at the Institute for Advanced Study, supported by the Zurich Insurance Company Membership. He thanks Song Sun for earlier discussions. The second-named author would like to thank S.Takayama and Y.Zhang for earlier discussions on these topics. He was partially supported by NSF grant DMS-1903147, and this article was written during his visit at the Department of Mathematics and at the Center for Mathematical Sciences and Applications at Harvard University, which he would like to thank for the hospitality.