Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
Abstract
An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (in general). In dimension $3$ or higher, the conjecture is only known in some special cases, such as for all twistor spaces by the work of Davidov, Grantcharov, and Muskarov, or for the locally conformally Kähler case (when the constant is negative or zero) by the work of H. Chen, L. Chen and Nie, or for all non-balanced Bismut torsion parallel (BTP) manifolds by the work of S. Chen and Zheng, where they also showed that the conjecture holds for all balanced BTP threefolds, utilizing a classification result by Zhao and Zheng. In this article, we show that the conjecture is valid for all balanced BTP manifolds in complex dimension $4$. The interesting part is that while balanced BTP manifolds are highly restrictive, the classification is still lacking in dimensions $4$ or higher, and this study might shed some light on the structure of balanced BTP manifolds in dimension $4$.
Related
- Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature
- Hermitian connections with parallel torsion and constant holomorphic sectional curvature
- 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
BibTeX
@misc{wang2026balanced,
title = {{Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature}},
author = {Qingsong Wang and Fangyang Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.00457},
eprint = {2608.00457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00457},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Balanced Bismut torsion-parallel fourfolds with constant holomorphic sectional curvature
AU - Qingsong Wang
AU - Fangyang Zheng
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2608.00457
UR - https://arxiv.org/abs/2608.00457
AB - An old conjecture in non-Kähler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be Kähler (when the constant is non-zero) or Chern flat (when the constant is zero). It is known to be true in complex dimension $2$ by the work of Balas and Gauduchon in 1985 (when the constant is negative or zero) and Apostolov, Davidov and Muskarov in 1996 (in general). In dimension $3$ or higher, the conjecture is only known in some special cases, such as for all twistor spaces by the work of Davidov, Grantcharov, and Muskarov, or for the locally conformally Kähler case (when the constant is negative or zero) by the work of H. Chen, L. Chen and Nie, or for all non-balanced Bismut torsion parallel (BTP) manifolds by the work of S. Chen and Zheng, where they also showed that the conjecture holds for all balanced BTP threefolds, utilizing a classification result by Zhao and Zheng. In this article, we show that the conjecture is valid for all balanced BTP manifolds in complex dimension $4$. The interesting part is that while balanced BTP manifolds are highly restrictive, the classification is still lacking in dimensions $4$ or higher, and this study might shed some light on the structure of balanced BTP manifolds in dimension $4$.
ER -