ScalingStacks

Theorem 3.1. Let $(Y,\omega)$ be a compact Kähler manifold, and the Kähler potential $\phi$ solves the complex Monge-Ampère equation

$$\frac{(\omega+\sqrt{-1}\partial\bar{\partial}\phi)^{n}}{\int_{Y}\omega^{n}}=d\mu,\quad\sup\phi=0.$$

Assume there are positive constants $\alpha,A$, such that the Skoda type estimate (1) holds for $(Y,\omega,d\mu)$:

$$\int_{Y}e^{-\alpha u}d\mu\leq A,\quad\forall u\in PSH(Y,\omega)\text{ with }\sup_{Y}u=0.$$

Then $\left\lVert\phi\right\rVert_{C^{0}}\leq C(n,\alpha,A)$.