Bismut torsion parallel metrics with constant holomorphic sectional curvature
Shuwen Chen, Fangyang Zheng
TL;DR
The paper investigates whether compact Hermitian manifolds with constant holomorphic sectional curvature must be Kähler or Chern flat, focusing on the Bismut torsion-parallel (BTP) class. By deriving curvature-torsion relations for the Chern, Bismut, and Gauduchon connections, and analyzing both non-balanced and balanced cases, the authors show that no nontrivial constant holomorphic sectional curvature can occur for non-balanced BTP manifolds, and, in balanced BTP threefolds, the same rigidity holds except in the trivial Chern-flat scenario. This establishes the Constant Holomorphic Sectional Conjecture within these broad classes, including all Vaisman manifolds as a corollary. The results connect BTP geometry with complex space form rigidity in higher dimensions and clarify the role of Gauduchon connections in this rigidity. Overall, the work extends known low-dimensional results to a substantial non-Kähler setting and provides a clear obstruction framework for constant holomorphic sectional curvature in BTP geometry.
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 (when the constant is positive). In dimension $3$ or higher, the conjecture is only known in some special cases, such as the locally conformally Kähler case (when the constant is negative or zero) by the work of Chen, Chen and Nie, or for complex nilmanifolds with nilpotent $J$ by the work of Li and the second named author. In this note, we confirm the above conjecture for all non-balanced Bismut torsion parallel (BTP) manifolds. Here the BTP condition means that the Bismut connection has parallel torsion. In particular, the conjecture is valid for all Vaisman manifolds.