Hermitian manifolds with constant holomorphic sectional curvature or real bisectional curvature

Abstract

An old conjecture in Hermitian geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature is K\"ahler when the constant is nonzero and Chern flat when the constant is zero. Yang--Zheng introduced the real bisectional curvature as a generalization of the holomorphic sectional curvature and conjectured that, a compact Hermitian manifold with constant real bisectional curvature has zero constant and is Chern flat. They proved that the constant cannot be positive. In this paper, we first prove their conjecture. For a negative constant, we show that the metric is pluriclosed and use comparison with a negative K\"ahler--Einstein metric to obtain a contradiction. For the zero constant, we combine a torsion identity of Lin--Ren with the Bochner formula of Zhou--Zheng to prove Chern flatness. We also prove that a pluriclosed metric with constant negative holomorphic sectional curvature on a compact K\"ahler manifold is K\"ahler.