ScalingStacks

Proof. By the local Skoda estimate and the Remark above, for both a log scale in the toric region, and a boundary type chart, the local average

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int e^{\alpha\sqrt{s}(-\varphi+\sup_{loc}\varphi)}d\mu_{s}\leq C,$$

(28)

so in particular $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int e^{\alpha(-\varphi+\sup_{loc}\varphi)}\leq C.$ But we have already achieved a $C^{0}$-bound on local average functions, and in particular a lower bound on local suprema. Thus

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int e^{-\alpha\varphi}d\mu_{s}\leq Ce^{-\alpha\sup_{loc}\varphi}\leq C,$$

or equivalently $\int_{loc}e^{-\alpha\varphi}d\mu_{s}\leq C\int_{loc}d\mu_{s}$ for local integrals. To pass from this to the global Skoda estimate, we need to take a large collection of log scales and boundary type charts and sum over the estimates:

$$\int_{X_{s}}e^{-\alpha\varphi}d\mu_{s}\leq C\sum\int_{loc}d\mu_{s}.$$

The only problem is to ensure that the local charts can be chosen without substantially overcounting the measure. For points on $X_{s}$ whose $\text{Log}_{s}$ image is at $O(s^{-1})$ Euclidean distance to $\partial\Delta_{\lambda}^{\vee}$, it is easy to choose the charts so that each point is contained in $O(1)$ number of charts. Away from $\partial\Delta_{\lambda}^{\vee}$, the points deep inside the boundary type charts in general do not have this local finiteness property, but this is compensated by the fact that the measure $d\mu_{s}$ decays exponentially away from $\partial\Delta_{\lambda}^{\vee}$ (cf. (16)). The conclusion is that

$$\sum\int_{loc}d\mu_{s}\leq C\int_{X_{s}}d\mu_{s},$$

whence the global Skoda estimate. ∎