ScalingStacks

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.