Hermitian threefolds with constant holomorphic sectional curvature
Abstract
An old conjecture in non-Kähler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant c, then the metric must be Kähler when c≠ 0 and be Chern flat when c=0. The conjecture is known to be true in dimension two by the work of Balas--Gauduchon and Apostolov--Davidov--Muškarov in the 1980s and 1990s. Recently, Qin and Tian proved the conjecture in complex dimension three when c≠ 0, and the c=0 case was proved by Chen--Li under the additional assumption that the metric is balanced. In this article we remove this additional hypothesis and complete the confirmation of the conjecture in complex dimension three. The proof relies heavily on the fact that in dimension three the torsion $3$-tensor can be equivalently expressed as a $2$-tensor twisted by the canonical line bundle, plus Bochner type integration formulas. In particular, the method cannot be directly generalized to higher dimensions.
Related
- Hermitian connections with parallel torsion and constant holomorphic sectional curvature
- Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature
- Positivity of the canonical bundle and Hermitian metrics with quasi-negative holomorphic sectional curvature
- Constant Chern holomorphic sectional curvature on two-step solvmanifolds
- Biholomorphic rigidity and symmetry breaking for domains with Bergman metric of constant positive holomorphic sectional curvature
BibTeX
@misc{chen2026hermitian,
title = {{Hermitian threefolds with constant holomorphic sectional curvature}},
author = {Shuwen Chen and Fangyang Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39282},
eprint = {2609.39282},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39282},
}
RIS
TY - UNPB
TI - Hermitian threefolds with constant holomorphic sectional curvature
AU - Shuwen Chen
AU - Fangyang Zheng
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39282
UR - https://arxiv.org/abs/2609.39282
AB - An old conjecture in non-Kähler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant c, then the metric must be Kähler when c≠ 0 and be Chern flat when c=0. The conjecture is known to be true in dimension two by the work of Balas--Gauduchon and Apostolov--Davidov--Muškarov in the 1980s and 1990s. Recently, Qin and Tian proved the conjecture in complex dimension three when c≠ 0, and the c=0 case was proved by Chen--Li under the additional assumption that the metric is balanced. In this article we remove this additional hypothesis and complete the confirmation of the conjecture in complex dimension three. The proof relies heavily on the fact that in dimension three the torsion $3$-tensor can be equivalently expressed as a $2$-tensor twisted by the canonical line bundle, plus Bochner type integration formulas. In particular, the method cannot be directly generalized to higher dimensions.
ER -