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Streets-Tian Conjecture on several special types of Hermitian manifolds

Yuqin Guo, Fangyang Zheng

TL;DR

The paper addresses the Streets-Tian conjecture, which asserts that a compact complex manifold admitting a Hermitian-symplectic metric must be Kähler. It develops explicit, constructive descriptions of Hermitian-symplectic metrics and deformation pathways to Kähler metrics for several special types of manifolds: (i) compact Chern Kähler-like manifolds, (ii) non-balanced BTP manifolds, and (iii) compact quotients of Lie groups whose Lie algebra contains a $J$-invariant abelian ideal of codimension $2$. A key methodological theme is reducing to invariant metrics on Lie groups via averaging, and employing algebraic criteria (on structure constants and torsion) to preclude HS-structures unless a Kähler metric exists. The results extend prior work by showing ST holds in broader solvable and Lie-algebraic settings, and provide an elementary framework to study the conjecture through explicit descriptions and deformations. Overall, the paper clarifies how algebraic restrictions in Lie-algebras and curvature-like conditions interact with the HS condition to force Kähler geometry in these special cases.

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.

Streets-Tian Conjecture on several special types of Hermitian manifolds

TL;DR

The paper addresses the Streets-Tian conjecture, which asserts that a compact complex manifold admitting a Hermitian-symplectic metric must be Kähler. It develops explicit, constructive descriptions of Hermitian-symplectic metrics and deformation pathways to Kähler metrics for several special types of manifolds: (i) compact Chern Kähler-like manifolds, (ii) non-balanced BTP manifolds, and (iii) compact quotients of Lie groups whose Lie algebra contains a -invariant abelian ideal of codimension . A key methodological theme is reducing to invariant metrics on Lie groups via averaging, and employing algebraic criteria (on structure constants and torsion) to preclude HS-structures unless a Kähler metric exists. The results extend prior work by showing ST holds in broader solvable and Lie-algebraic settings, and provide an elementary framework to study the conjecture through explicit descriptions and deformations. Overall, the paper clarifies how algebraic restrictions in Lie-algebras and curvature-like conditions interact with the HS condition to force Kähler geometry in these special cases.

Abstract

A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is given by the -part of a closed -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 but is still open in complex dimensions 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 -invariant abelian ideal of codimension . 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.
Paper Structure (5 sections, 15 theorems, 63 equations)

This paper contains 5 sections, 15 theorems, 63 equations.

Table of Contents

  1. Introduction and statement of results
  2. Non-balanced Bismut torsion parallel manifolds
  3. Hermitian-symplectic metrics on Lie algebras
  4. The almost abelian case revisted
  5. Lie algebras with $J$-invariant abelian ideal of codimension $2$

Key Result

Proposition 1

Let $(M^n,g)$ be a compact Hermitian manifold. If $g$ is Chern Kähler-like but not Kähler, then $M^n$ cannot admit any Hermitian-symplectic metric or any astheno-Kähler metric.

Theorems & Definitions (22)

  • Conjecture 1: Streets-Tian ST
  • Proposition 1
  • Proposition 2
  • Theorem 1: Enrietti-Fino-Vezzoni EFV
  • Theorem 2: Fino-Kasuya-Vazzoni FKV
  • Theorem 3: Fino-Kasuya FK
  • Proposition 3
  • Conjecture 2: Streets-Tian $\cap$ Fino-Vezzoni
  • proof : Proof of Proposition \ref{['prop0']}.
  • Lemma 1: ZhaoZ24
  • ...and 12 more