On Hermitian manifolds with vanishing curvature

TL;DR

This paper classifies Hermitian metrics with zero holomorphic or real bisectional curvature on various complex manifolds, showing they are often K"ahler and conformally balanced, and introduces the concept of 'altered' curvatures.

Contribution

It provides new classification results for Hermitian metrics with vanishing curvature on compact and Fujiki class C manifolds, and introduces the notion of 'altered' curvatures.

Findings

Hermitian metrics with vanishing holomorphic curvature are conformally balanced on certain manifolds.

Pluriclosed metrics with zero holomorphic curvature on compact K"ahler manifolds are K"ahler.

Metrics with zero real bisectional curvature on Fujiki class C manifolds are K"ahler.

Abstract

We show that Hermitian metrics with vanishing holomorphic curvature on compact complex manifolds with pseudoeffective canonical bundle are conformally balanced. Pluriclosed metrics with vanishing holomorphic curvature on compact K\"ahler manifolds are shown to be K\"ahler and hence, are completely classified. We prove that Hermitian metrics with vanishing real bisectional curvature on complex manifolds in the Fujiki class C are K\"ahler and thus fall under the same classification. Finally, we formalize the notion of `altered' curvatures, which force distinguished metric structures when mandated to coincide with their `standard' counterparts.