Bismut-Torsion-Parallel Hermitian Manifolds With Constant Chern Holomorphic Sectional Curvature

Abstract

A well-known conjecture in complex geometry states that a compact Hermitian manifold with constant Chern holomorphic sectional curvature must be Kähler when the constant is nonzero and Chern flat when the constant is zero. The conjecture is known in complex dimension two and in several special classes in higher dimensions. For Hermitian metrics with Bismut-parallel torsion, the non-balanced case and the balanced threefold case were established by Chen--Zheng, while the balanced fourfold case was settled recently by Wang--Zheng. In this article, we prove the nonzero case for balanced Bismut-torsion-parallel Hermitian manifolds in every complex dimension. As a corollary, we confirm that every BTP Hermitian manifold with Chern holomorphic sectional curvature is a nonzero constant, then $g$ is Kähler.