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. We also prove that vanishing holomorphic sectional curvature forces the first Chern class to vanish. As a partial converse, we show that on a compact complex manifold with vanishing first Bott-Chern class, every Hermitian metric with nonnegative holomorphic sectional curvature must have vanishing holomorphic sectional curvature.