On Hermitian manifolds with vanishing curvature

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.