The complex Monge-Amp\`ere type equation on compact Hermitian manifolds and Applications

TL;DR

This paper establishes existence and uniqueness of solutions to a complex Monge-Ampère type equation on compact Hermitian manifolds, extending previous results and applying them to important conjectures in complex geometry.

Contribution

It generalizes key results of Monge-Ampère equations to Hermitian manifolds outside the Fujiki class, with significant applications to longstanding conjectures.

Findings

Proved existence and uniqueness of solutions for $L^p$ right-hand sides.

Extended results of Eyssidieux, Guedj, and Zeriahi to broader Hermitian settings.

Provided partial progress on conjectures by Tosatti-Weinkove and Demailly-Paun.

Abstract

We prove the existence and uniqueness of continuous solutions to the complex Monge-Amp\`ere type equation with the right hand side in $L^p$, $p>1$, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi \cite{EGZ09, EGZ11} to compact Hermitian manifolds which {\em a priori} are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove \cite{TW12a} and on a weak form of a conjecture of Demailly and Paun \cite{DP04}.