Streets-Tian Conjecture on several special types of Hermitian manifolds

Abstract

A Hermitian-symplectic metric is a Hermitian metric whose K\"ahler form is given by the $(1,1)$-part of a closed $2$-form. Streets-Tian Conjecture states that a compact complex manifold admitting a Hermitian-symplectic metric must be K\"ahlerian (i.e., admitting a K\"ahler metric). The conjecture is known to be true in dimension $2$ but is still open in dimensions $3$ or higher. In this article, we confirm the conjecture for some special types of compact Hermitian manifolds, including the Chern flat manifolds, non-balanced Bismut torsion parallel manifolds (which contains Vaisman manifolds as a subset), and quotients of Lie groups which are either almost ableian or whose Lie algebra contains a codimension $2$ abelian ideal that is $J$-invariant. The last case presents adequate algebraic complexity which illustrates the subtlety and intricacy of Streets-Tian Conjecture.