ScalingStacks

Proof of the diameter upper bound in Theorem 1.1. Let $\omega^{\prime}_{t}$ be the Kähler metric defined in Proposition 3.1, and let $\omega_{{\rm FS},t}=\frac{1}{|\log|t||}\omega_{\rm FS}|_{X_{t}},$ where $\omega_{\rm FS}$ is a suitable Fubini-Study metric on $\mathbb{P}^{N}$ scaled so that $\omega_{\rm FS}|_{X_{t}}\in c_{1}(\mathfrak{L})|_{X_{t}}$. Then the metrics $\omega^{\prime}_{t}$ and $\omega_{{\rm FS},t}$ are cohomologous, and on $B_{t}\subset X_{t}$ in any adapted coordinate chart we have

(4.1)

$$\omega^{\prime}_{t}+\omega_{{\rm FS},t}\geqslant C^{-1}\frac{i}{|\log|t||^{2}}\sum_{j=1}^{m}\frac{dz_{j}}{z_{j}}\wedge\frac{d\overline{z_{j}}}{\overline{z_{j}}}+C^{-1}\frac{i}{|\log|t||}\sum_{j=m+1}^{n}dz_{j}\wedge d\overline{z_{j}}.$$

Let us then fix a point $x\in E^{0}_{J}$, an adapted coordinate chart near $x$, and in these coordinates define a local Kähler metric $\tilde{\omega}_{t}$ on $X_{t}$ by the RHS of (4.1). Inside this coordinate chart intersected $X_{t}$, define also $\tilde{B}_{t}$ to be a Euclidean rectangle which is contained inside $B_{t}$ so that in the metric $\tilde{\omega}_{t}$, in the first $m$ complex directions $\tilde{B}_{t}$ has length $\sim 1$ in the radial directions and length $\sim|\log|t||^{-1}$ in the logarithmic angular directions, and in the other $n-m$ complex directions $\tilde{B}_{t}$ has length $\sim|\log|t||^{-\frac{1}{2}}$. Therefore, given any $x,y\in\tilde{B}_{t}$, if we denote by $\gamma_{x,y}$ the Euclidean straight line in $\tilde{B}_{t}$ joining them, parametrized linearly by $0\leqslant s\leqslant 1$, then $|\dot{\gamma}_{x,y}(s)|_{\tilde{\omega}_{t}}\leqslant C$ for a uniform constant $C$ independent of $t$ and $s$.