Hölder estimates for degenerate complex Monge-Ampère equations

Abstract

Uniform $L^{\infty }$ and Hölder estimates were proved by the Kolodziej for complex Monge-Ampère equations on compact Kähler manifolds with $L^p$ volume measure with $p>1$. On the other hand, establishing Hölder estimates on singular Kähler varieties has remained open. In this paper, we establish uniform Hölder continuity for a family of complex Monge-Ampère equations on Kähler varieties, by developing a geometric regularization based on the partial $C^0$ estimate, i.e., quantitive Kodaira embeddings. As an application, we prove that local potentials of smoothable Kähler-Einstein varieties are Hölder continuous.