Streets-Tian Conjecture on several special types of Hermitian manifolds

Abstract

A Hermitian-symplectic metric is a Hermitian metric whose Kähler 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 Kählerian (i.e., admitting a Kähler metric). The conjecture is known to be true in complex dimension $2$ but is still open in complex dimensions $3$ or higher. In this article, we confirm the conjecture for some special types of compact Hermitian manifolds, including Chern Kähler-like manifolds, non-balanced Bismut torsion parallel (BTP) manifolds, and compact quotients of Lie groups whose Lie algebra contains a $J$-invariant abelian ideal of codimension $2$. The last type is a natural generalization to (compact quotients of) almost abelian Lie groups. The non-balanced BTP case contains all Vaisman manifolds and all Bismut Kähler-like manifolds as subsets. These results extend some of the earlier works on the topic by Fino, Kasuya, Vezzoni, Angella, Otiman, Paradiso, and others. Our approach is elementary in nature, by giving explicit descriptions of Hermitian-symplectic metrics on such spaces as well as the pathways of deforming them into Kähler ones, aimed at illustrating the algebraic complexity and subtlety of Streets-Tian Conjecture.