On Hermitian manifolds with constant mixed curvature

Abstract

In a recent work, Kai Tang conjectured that any compact Hermitian manifold with non-zero constant mixed curvature must be K\"ahler. He confirmed the conjecture in complex dimension $2$ and for Chern K\"ahler-like manifolds in general dimensions. In this paper, we verify his conjecture for several special types of Hermitian manifolds, including complex nilmanifolds, solvmanifolds with complex commutators, almost abelian Lie groups, and Lie algebras containing a $J$-invariant abelian ideal of codimension $2$. We also verify the conjecture for all compact balanced threefolds when the Bismut connection has parallel torsion. These results provide partial evidence towards the validity of Tang's conjecture.