Singular Gauduchon Conjecture

TL;DR

This paper investigates a singular version of Gauduchon's conjecture, establishing $C^{0}$-estimates and smoothness results for solutions on certain complex manifolds, extending previous work on Ricci curvature and Monge-Ampère equations.

Contribution

It introduces a singular framework for Gauduchon metrics, providing new estimates and smoothness results that generalize prior solutions to the conjecture.

Findings

Established $C^{0}$-estimates without gradient terms

Proved smoothness of solutions on holomorphic Kähler families

Extended previous results to singular settings

Abstract

In 1984 Gauduchon conjectured that one can find Gauduchon metrics with prescribed Ricci curvature on all compact complex manifolds. This conjecture was settled by Sz\'ekelyhidi-Tosatti-Weinkove (TW17, TW19, STW17) by the study of the Monge-Amp\`ere equation for $(n-1)$-plurisubharmonic functions with a gradient term. In this paper we study a singular version of this conjecture. We obtain a $C^{0}$-estimate for this problem, without gradient terms, in smoothable hermitian variaties by adapting a recent technique of Guedj-Lu. We also prove the smoothness of solutions on holomorphic K\"ahler families, generalizing TW17.