Quasi-plurisubharmonic envelopes 3: Solving Monge-Amp\`ere equations on hermitian manifolds

TL;DR

This paper introduces a novel approach to solving degenerate complex Monge-Ampère equations on hermitian manifolds, providing new a priori estimates and existence results by leveraging compactness and envelope properties of quasi-plurisubharmonic functions.

Contribution

It extends previous methods to the hermitian setting, offering new estimates and solutions for Monge-Ampère equations on hermitian manifolds, which were previously studied mainly in Kähler geometry.

Findings

Developed new $L^{ abla}$$^{ ext{infty}}$-a priori estimates for Monge-Ampère equations.

Extended techniques to hermitian manifolds, previously limited to Kähler cases.

Produced several existence results for degenerate Monge-Ampère equations on hermitian manifolds.

Abstract

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel \cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in K\"ahler geometry. In \cite{GL21b} we have studied the behavior of Monge-Amp\`ere volumes on hermitian manifolds. We extend here the techniques of \cite{GL21a} to the hermitian setting and use the bounds established in \cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Amp\`ere equations on compact hermitian manifolds.