Hermitian threefolds with constant holomorphic sectional curvature

Abstract

An old conjecture in non-K\"ahler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant $c$, then the metric must be K\"ahler when $c\neq 0$ and be Chern flat when $c=0$. The conjecture is known to be true in dimension two by the work of Balas--Gauduchon and Apostolov--Davidov--Mu\v{s}karov in the 1980s and 1990s. Recently, Qin and Tian proved the conjecture in complex dimension three when $c\neq 0$, and the $c=0$ case was proved by Chen--Li under the additional assumption that the metric is balanced. In this article we remove this additional hypothesis and complete the confirmation of the conjecture in complex dimension three. The proof relies heavily on the fact that in dimension three the torsion $3$-tensor can be equivalently expressed as a $2$-tensor twisted by the canonical line bundle, plus Bochner type integration formulas. In particular, the method cannot be directly generalized to higher dimensions.