Gauduchon metrics with prescribed volume form

TL;DR

This paper proves that on any compact complex manifold, it is possible to find Gauduchon metrics with a specified volume form, thereby solving a longstanding conjecture and enabling control over Chern-Ricci curvature.

Contribution

It establishes the existence of Gauduchon metrics with prescribed volume form on all compact complex manifolds, confirming a conjecture by Gauduchon from 1984.

Findings

Existence of Gauduchon metrics with any given volume form

Solution to Gauduchon's 1984 conjecture

Equivalence to prescribing Chern-Ricci curvature

Abstract

We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.