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].