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]:

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

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].

In this section we recall some background on the asymptotic behavior of the relative Calabi-Yau volume forms on a polarized Calabi-Yau degeneration family, and set up some notation for later use.

As in the introduction, we assume we have a polarized Calabi-Yau family $\pi:X\to\Delta^{*}$ with relative polarization $L$, and we fix a trivializing section $\Omega$ of $K_{X}$ and define trivializations $\Omega_{t}$ of $K_{X_{t}}$ by $\Omega=dt\wedge\Omega_{t}$ along $X_{t}$. Up to passing to a finite base change, we may assume 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. In this case we have

and letting $\kappa=\min_{i\in I}a_{i}$, up to replacing $\Omega_{t}$ by $t^{-\kappa}\Omega_{t}$, we may assume without loss that $\kappa=0$.

Recall that $\omega_{t}$ denotes the Calabi-Yau metric on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)|_{X_{t}}$, and that the dimension $m$ of the essential skeleton of $X$ is assumed to be stricty positive (and necessarily $m\leqslant n$). Denote by $\mu_{t}$ the Calabi-Yau volume form on $X_{t}$ normalized to be a probability measure, i.e.

Let us now recall the asymptotic behavior of the integrals $\int_{X_{t}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}$, largely following [3]. For any $J\subset I$ we denote by $E_{J}=\bigcap_{j\in J}E_{j}$. As in [13, 14], we fix a Kähler metric on $\mathfrak{X}$ and for $\varepsilon>0$ small and $J\subset I$ with $E_{J}\neq\emptyset$ we define

For any given $x\in E_{J}\subset\mathfrak{X}$ let $p=|J|-1$ and pick local coordinates $z_{0},\dots,z_{n}$ on $x\in V\subset\mathfrak{X}$, defined in the unit polydisc, such that $z_{0},\dots,z_{p}$ are defining equations for $E_{j},j\in J$, so that in these coordinates we have $t=z_{0}\cdots z_{p}$. We shall call these adapted coordinates. We can then write

where $f_{J}$ is a local non-vanishing holomorphic function. Since $dt\wedge\Omega_{t}=\Omega$ along $X_{t}$, on $E^{0}_{J}$ we get

from which using polar coordinates $z_{j}=\exp(x_{j}\log|t|+i\theta_{j}),j\in J,$ one can easily see as in [3] that

where

while

where

The local logarithmic variables $x_{j}=\frac{\log|z_{j}|}{\log|t|}$ vary in the standard simplex

and in this way one obtains a map $\mathrm{Log}_{t}:V\backslash\bigcup_{j}E_{j}\to\Delta_{J}$, see [3].

For each $i\in I$ we fix now a defining section $\sigma_{i}\in H^{0}(\mathfrak{X},\mathcal{O}(E_{i}))$ and a Hermitian metric $h_{i}$ on $\mathcal{O}(E_{i})$, so that $r_{i}:=|\sigma_{i}|^{2}_{h_{i}}$ is a smooth nonnegative function of $\mathfrak{X}$ which vanishes precisely along $E_{i}$ and is uniformly comparable to $|z_{i}|^{2}$ in any adapted coordinate chart as above where $z_{i}$ is the local defining equation of $E_{i}$. In particular,

is now defined on the whole $\mathfrak{X}\backslash\bigcup_{j}E_{j}$, and in the adapted coordinates as above it is equal to $2x_{i}$ up to very small errors (as $t$ approaches $0$). It follows that on $X_{t}$ (for $|t|$ sufficiently small) in an adapted coordinate chart near a point of $E^{0}_{J}$ as above, the point $\left(\frac{1}{2}\tilde{x}_{0},\dots,\frac{1}{2}\tilde{x}_{p}\right)$ will lie very close to $\Delta_{J}$.

In this section we prove the diameter lower bound in Theorem 1.1.

The setting is the same as in the previous section, so $X\to\Delta^{*}$ is a polarized Calabi-Yau degeneration family with $m=\dim\mathrm{Sk}(X)>0$, with a semistable model $\mathfrak{X}\to\Delta$ with $X_{0}=\sum_{i\in I}E_{i}$ and $K_{\mathfrak{X}/\Delta}=\sum_{i\in I}a_{i}E_{i}$. We fix also an embedding of the family $\mathfrak{X}\hookrightarrow\mathbb{P}^{N}\times\Delta$ and denote by $\mathfrak{L}\to\mathfrak{X}$ the restriction of the hyperplane bundle.

We choose a nonempty $E_{J}$ which realizes the maximum in (2.2), with $m=|J|-1$, and relabel so that $J=\{0,\dots,m\}$. We also denote by $U$ an open neighborhood of $E_{J}$ in $\mathfrak{X}$ which can be covered by finitely many adapted coordinate charts as above. In particular, in these charts we have

We need the following construction:

For ease of notation, in the rest of the paper we will denote by $C>0$ a generic uniform constant, independent of $t$, which may vary from line to line.

We can now give the proof of the diameter lower bound in Theorem 1.1:

Here we prove the diameter upper bound in Theorem 1.1.