H\"older estimates for degenerate complex Monge-Amp\`ere equations

Uniform $L^\infty$ and H\"older estimates were proved by the Kolodziej for complex Monge-Amp\`ere equations on compact K\"ahler manifolds with $L^p$ volume measure with $p>1$. On the other hand, establishing H\"older estimates on singular K\"ahler varieties has remained open. In this paper, we establish uniform H\"older continuity for a family of complex Monge-Amp\`ere equations on K\"ahler 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 K\"ahler-Einstein varieties are H\"older continuous.