ScalingStacks

Theorem 4.5. (cf. [52, section 2.2]) Let $(Y,\omega)$ be a compact Kähler manifold, and $\phi\in PSH(Y,\omega)\cap C^{0}$, such that $\omega_{\phi}^{n}$ is an absolutely continuous measure. Assume there are positive constants $\alpha,A$, such that the Skoda type estimate holds with respect to $\omega_{\phi}^{n}$:

$$\int_{Y}e^{-\alpha u}\frac{\omega_{\phi}^{n}}{\text{Vol}(Y)}\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.$$

(11)

Then we have

• •

  If $\sup_{Y}\phi=0$, then $\left\lVert\phi\right\rVert_{C^{0}}\leq C(n,\alpha,A)$.

• •

  For fixed $n,\alpha,A$, there is number $B(n,\alpha,A)$, such that if $\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)}<(2B)^{-2n}$ for some $t_{0}$, then $\min\phi\geq-t_{0}-4B(\frac{\int_{\phi\leq-t_{0}}\omega_{\phi}^{n}}{\text{Vol}(Y)})^{1/2n}$.