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

## TL;DR

This paper extends H"older continuity results for complex Monge-Amp ext{e}re equations to singular K"ahler varieties, using geometric regularization and partial $C^0$ estimates, with applications to K"ahler-Einstein varieties.

## Contribution

It introduces a new approach to establish H"older estimates on singular K"ahler varieties via geometric regularization and partial $C^0$ estimates, advancing the understanding of complex Monge-Amp ext{e}re equations.

## Key findings

- Established uniform H"older continuity for Monge-Amp ext{e}re equations on K"ahler varieties.
- Proved local potentials of smoothable K"ahler-Einstein varieties are H"older continuous.
- Developed a geometric regularization method based on partial $C^0$ estimates.

## Abstract

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.