Hermitian connections with parallel torsion and constant holomorphic sectional curvature

@misc{chen2026hermitian,
  title = {{Hermitian connections with parallel torsion and constant holomorphic sectional curvature}},
  author = {Shuwen Chen and Fangyang Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33374},
  eprint = {2609.33374},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33374},
}

TY  - UNPB
TI  - Hermitian connections with parallel torsion and constant holomorphic sectional curvature
AU  - Shuwen Chen
AU  - Fangyang Zheng
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33374
UR  - https://arxiv.org/abs/2609.33374
AB  - A long-standing conjecture in non-Kähler geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature should be Kähler when the constant is nonzero and Chern flat when the constant is zero. In this article, we study the corresponding problem for a Hermitian connection ∇ with ∇ T=0. We first show that if the ∇-holomorphic sectional curvature is constant, then the (1,1)-part of the curvature is ∇-parallel. Our main result states that a nonzero constant forces T=0; consequently, the metric is Kähler and locally a complex space form. The proof is pointwise and requires neither compactness nor completeness. We also obtain applications to the Chern and Bismut connections.
ER  -