Streets-Tian Conjecture on several special types of Hermitian manifolds

TL;DR

This paper proves the Streets-Tian Conjecture for certain special classes of compact Hermitian manifolds, showing that Hermitian-symplectic metrics imply K"ahler metrics in these cases, extending previous results.

Contribution

It confirms the conjecture for specific types of Hermitian manifolds, including Chern K"ahler-like, non-balanced BTP, and certain Lie group quotients, using explicit metric descriptions.

Findings

Confirmed the conjecture for Chern K"ahler-like manifolds.

Extended results to non-balanced BTP and Lie group quotients.

Provided explicit deformation pathways from Hermitian-symplectic to K"ahler metrics.

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 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 K\"ahler-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 K\"ahler-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 K\"ahler ones, aimed at illustrating the algebraic complexity and subtlety of Streets-Tian Conjecture.