A priori estimates for Donaldson's equation over compact Hermitian manifolds

TL;DR

This paper establishes a priori estimates for Donaldson's equation on compact Hermitian manifolds, addressing a question posed by Tosatti and Weinkove, and advancing the understanding of complex geometric PDEs.

Contribution

It provides the first a priori estimates for Donaldson's equation on general Hermitian manifolds, extending previous results from Kähler to Hermitian settings.

Findings

Established a priori estimates for Donaldson's equation.

Answered a question posed by Tosatti and Weinkove.

Extended techniques to non-Kähler Hermitian manifolds.

Abstract

In this paper we prove a priori estimates for Donaldson's equation $\omega\wedge(\chi+\sqrt{-1}\partial\bar{\partial}\varphi)^{n-1}=e^{F}(\chi+\sqrt{-1}\partial\bar{\partial}\varphi)^{n}$ over a compact Hermitian manifold X of complex dimension n, where $\omega$ and $\chi$ are arbitrary Hermitian metrics. Our estimates answer a question of Tosatti-Weinkove.