ScalingStacks

Lemma 4.18. (Local Skoda estimate) Consider any $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ normalised to $\sup_{X_{s}}\varphi=0$. There are uniform positive constants $\alpha$, $C$, such that the local potentials $\phi$ satisfy

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  In a log scale in the toric region,

  $$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}e^{-\alpha\sqrt{s}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-6.57559pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-4.84631pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.71837pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.36836pt}}\!\int\phi)}d\mu_{s}\leq C.$$

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  In a boundary type chart,

  $$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{U_{P}}e^{-\alpha\sqrt{s}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-6.57559pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-4.84631pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.71837pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.36836pt}}\!\int\phi)}d\mu_{s}\leq C.$$