Bismut torsion parallel metrics with constant holomorphic sectional curvature

TL;DR

This paper proves that compact Hermitian manifolds with constant holomorphic sectional curvature and Bismut torsion parallel condition are K"ahler or Chern flat, confirming a long-standing conjecture in this specific class.

Contribution

It confirms the conjecture for all non-balanced Bismut torsion parallel manifolds, including Vaisman manifolds, extending known results to higher dimensions.

Findings

Confirms the conjecture for non-balanced BTP manifolds

Includes all Vaisman manifolds as special cases

Extends validity to higher complex dimensions

Abstract

An old conjecture in non-K\"ahler geometry states that, if a compact Hermitian manifold has constant holomorphic sectional curvature, then the metric must be K\"ahler (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 K\"ahler 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.