Monge-Amp\`ere equations with prescribed singularities on compact Hermitian manifolds

TL;DR

This paper investigates the existence and uniqueness of weak solutions to degenerate Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds, extending previous results to more general settings.

Contribution

It generalizes prior work by establishing solutions with prescribed singularities on a broader class of Hermitian manifolds with semipositive, big reference forms.

Findings

Proved existence of weak solutions under new conditions.

Established uniqueness of solutions with prescribed singularities.

Extended the theory to more general Hermitian manifolds.

Abstract

Given a compact complex manifold $X$, we study the existence and the uniqueness of weak solutions to degenerate Monge-Amp\`ere equations on $X$ with prescribed singularities when the reference form is semipositive and big, while the right hand side is a non-pluripolar positive Radon measure. This generalizes our previous work to more general hermitian manifolds and also to the case of solutions with prescribed singularities.