Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples

Abstract

Let $X$ be a compact Brody-hyperbolic complex manifold admitting a strictly positive real $(1,1)$-current $T$ with smooth $dT$ and $d{d}^{c}T\le 0$. We prove that the canonical bundle $K_X$ of $X$ is nef. If every positive Lelong level set of $T$ has complex dimension at most two, then $K_X$ is ample and $X$ admits a smooth plurinegative Hermitian metric. Strictly negative Chern holomorphic sectional curvature implies Brody hyperbolicity and, when constant, forces the given metric to be Kähler under the latter current hypotheses. We also establish Kählerness for nonpositive constant Chern holomorphic sectional curvature in Fujiki's class, and for nonzero constant curvature on compact complex threefolds and compact balanced fourfolds. In every complex dimension at least seven, we construct compact balanced metrics with zero Chern holomorphic sectional curvature, nonzero Chern curvature, vanishing of all four Chern--Ricci contractions, and a nowhere-vanishing Chern-parallel holomorphic volume form. Finally, for compact Hermitian manifolds with zero Chern holomorphic sectional curvature and vanishing first Chern--Ricci tensor, a smooth semipositive plurinegative $(1,1)$-form, positive definite at one point, forces the metric to be Kähler flat.