On balanced Hermitian threefolds with parallel Bismut torsion

TL;DR

This work classifies compact balanced BTP (Bismut-torsion-parallel) threefolds, separating possibilities by the rank of the B tensor into three families: Chern-flat quotients of ${SL}(2,\mathbb C)$ (r=3), the Wallach threefold (r=1), and middle-type manifolds (r=2) with a holomorphic line kernel $L$ satisfying $L^{\otimes 2}\cong \mathcal O_M$. The middle-type case is further refined into primary (abelian Bismut holonomy) and secondary (double cover to a primary) scenarios, with Vaisman companions arising as non-Kähler partners sharing the same Bismut connection. The Wallach threefold is shown to be balanced and BTP with positive Chern holomorphic sectional curvature and Einstein Riemannian metric; the r=3 case yields BAS (Bismut Ambrose Singer) locally homogeneous quotients of ${SL}(2,\mathbb C)$. The paper also develops a Lie-algebraic framework for middle-type BTP, identifies two families of unimodular examples ${A_{s,t}}$ and ${B_{z,t}}$, and demonstrates that balanced BTP threefolds do not admit pluriclosed metrics in the middle-type, thus confirming Fino–Vezzoni and Streets–Tian conjectures in this setting. Finally, it discusses prospects for higher-dimensional generalizations via splitting-type Vaisman manifolds and provides explicit Vaisman-companion constructions with abelian holonomy as illustrations.

Abstract

We continue our study on Hermitian manifolds that are {\em Bismut torsion parallel,} or {\em BTP} for brevity, which means that the Bismut connection has parallel torsion tensor. For $n\geq 3$, BTP metrics can be balanced (and non-Kähler). In this paper, we give a detailed description to characterize all compact, balanced BTP threefolds.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Lemma 2.1: ZhouZ
• proof
• Definition 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof