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 ddcT≤0. We prove that the canonical bundle KX of X is nef. If every positive Lelong level set of T has complex dimension at most two, then KX 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.
Related
- Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature
- Constant Chern holomorphic sectional curvature on two-step solvmanifolds
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- Hermitian connections with parallel torsion and constant holomorphic sectional curvature
- Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature
BibTeX
@misc{qin2026chern,
title = {{Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples}},
author = {Xiangsen Qin and Yuanhong Tian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25762},
eprint = {2609.25762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25762},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Chern Holomorphic Sectional Curvature: Canonical Ampleness, Rigidity, and Flatness Counterexamples
AU - Xiangsen Qin
AU - Yuanhong Tian
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.25762
UR - https://arxiv.org/abs/2609.25762
AB - Let X be a compact Brody-hyperbolic complex manifold admitting a strictly positive real (1,1)-current T with smooth dT and dd^cT≤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.
ER -