Subsection 2.5 (008F) · ScalingStacks

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2.5 Local Skoda estimate

We recall a basic version of the Skoda inequality:

Proposition 2.7. (cf. [22, Thm 3.1]) If $\phi$ is psh on $B_{2}\subset\mathbb{C}^{n}$ , with $\int_{B_{2}}|\phi|\omega_{E}^{n}\leq 1$ with respect to the standard Euclidean metric $\omega_{E}$ , then there are dimensional constants $\alpha$ , $C$ , such that

\int_{B_{1}}e^{-\alpha\phi}\omega_{E}^{n}\leq C.

Applying this with Lemma 2.6,

0084

Corollary 2.8. (Local Skoda estimate) Within every log scale, there are uniform positive constants $\alpha$ and $C$ , such that

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}e^{-\alpha u}\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.