Hermitian manifolds with nonpositive holomorphic sectional curvature

Abstract

We study compact Kähler manifolds admitting Hermitian metrics with nonpositive holomorphic sectional curvature. We prove that the canonical bundle of such a manifold is nef, removing the pluriclosed assumption from the corresponding nefness result of Broder-Stanfield \cite{BroderStanfield}. In complex dimension two, we further show that negative holomorphic sectional curvature implies the ampleness of the canonical bundle. In complex dimension three, the same conclusion holds if the Hermitian metric is additionally assumed to be balanced. We also prove that vanishing holomorphic sectional curvature forces the first Chern class to vanish.

This version includes a new result: an ampleness criterion for the canonical bundle of compact Kähler threefolds admitting balanced Hermitian metrics with negative holomorphic sectional curvature