Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples
We study the relation between canonical positivity, rigidity of a prescribed Hermitian metric, and the curvature not detected by Chern holomorphic sectional curvature. A compact Brody-hyperbolic complex manifold admitting an independent smooth plurinegative Hermitian metric is projective with ample canonical bundle. In Fujiki's class, nonpositive constant Chern holomorphic sectional curvature forces the original metric to be Kähler. On compact complex threefolds, without an auxiliary metric or a Fujiki-class assumption, a nonzero constant forces Kählerness. We also give an alternative proof of Chen--Zheng's Chern-flatness theorem for zero-HSC threefolds, by proving local rank-two differential compatibility and localizing their curvature bound under $\partialη=0$. In contrast, for every complex dimension at least seven we give compact balanced zero-HSC metrics with nonzero Chern curvature. Their octonionic curvature formula also implies the vanishing of all four Chern--Ricci contractions, and an explicit invariant volume form is Chern parallel. Finally, an auxiliary semipositive plurinegative $(1,1)$-form, positive definite at some point, restores flat Kählerness under zero HSC and vanishing first Chern--Ricci tensor. A mixed Ricci integral criterion explains the role of this last curvature hypothesis.