Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples
Year: 2026
Type: preprint
Abstract: We prove that a compact Brody-hyperbolic manifold carrying a strictly positive quasi-closed plurinegative $(1,1)$-current has nef canonical bundle, which is ample if every positive Lelong level set has complex dimension at most two. We also establish Kähler rigidity of the prescribed Hermitian metric for nonpositive constant Chern holomorphic sectional curvature on compact manifolds in Fujiki's class and for nonzero constant Chern holomorphic sectional curvature on compact complex threefolds. In complex dimension four, we prove that every balanced Hermitian metric with nonzero constant Chern holomorphic sectional curvature is Kähler, without assuming compactness or completeness. Under the same curvature assumption, dimension inequalities for the subspaces spanned by the image of the $(1,0)$ covariant derivative of the Chern torsion and its iterated images under Chern curvature endomorphisms imply Kählerness in dimensions at least five for balanced metrics and at least four without balancedness. At zero Chern holomorphic sectional curvature, we construct compact balanced examples in every complex dimension at least seven that are not Chern flat. However, on a compact connected Hermitian manifold with zero Chern holomorphic sectional curvature and vanishing first Chern--Ricci tensor, the prescribed metric is Kähler flat whenever the manifold admits a smooth semipositive plurinegative $(1,1)$-form positive definite at some point.
Source: arXiv (Cornell University)
Authors: Qin, Xiangsen, Tian, Yuanhong
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214081508