Hermitian threefolds with vanishing real bisectional curvature

TL;DR

This paper proves that compact Hermitian threefolds with zero real bisectional curvature are necessarily Chern flat, advancing understanding of curvature conditions in complex geometry beyond the K"ahler case.

Contribution

It establishes that in complex dimension three, zero real bisectional curvature implies Chern flatness for Hermitian manifolds, partially solving a higher-dimensional conjecture.

Findings

Hermitian threefolds with zero real bisectional curvature are Chern flat

The result extends known two-dimensional cases to three dimensions

Provides insights into curvature conditions in non-K"ahler Hermitian geometry

Abstract

We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such manifolds are conjectured to be K\"ahler (hence a complex space form) when the constant is non-zero and Chern flat (hence a quotient of a complex Lie group) when the constant is zero. The conjecture is known in complex dimension two but open in higher dimensions. In this paper, we establish a partial solution in complex dimension three by proving that any compact Hermitian threefold with zero real bisectional curvature must be Chern flat. Real bisectional curvature is a curvature notion introduced by Xiaokui Yang and the second named author in 2019, generalizing holomorphic sectional curvature. It is equivalent to the latter in the K\"ahler case and is slightly stronger in general.