Theorem 2.9 (0085) · ScalingStacks

Verified tagged author-source HTML · 2006.16961v1 · cited publication edition alignment unverified.

Theorem 2.9. (Uniform Skoda estimate) Suppose the measures $d\mu_{t}$ on $X_{t}$ satisfy a uniform upper bound of class $(a_{i})$ . Then there are uniform positive constants $\alpha$ and $A$ , such that

\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in PSH(X_{t},\omega_{t})\text{ with }\sup_{X_{t}}u=0.

0086

Proof. We choose the charts so that each point on $X_{t}$ is covered by $\leq C$ log scales. Summing over the local Skoda estimates from all log scales, $\int_{X_{t}}e^{-\alpha u}d\mu_{t}$ is bounded by

\begin{split}&\frac{C}{|\log|t||^{m}}\sum_{\text{log scales}}\int_{loc}|z_{0}|^{2a_{0}}\ldots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{k}\wedge d\bar{z}_{k}\\ &\leq C.\end{split}

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