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On Hermitian manifolds with constant mixed curvature

Shuwen Chen, Fangyang Zheng

TL;DR

The paper advances Tang's conjecture on constant mixed curvature by establishing it for four Lie-Hermitian classes: complex nilmanifolds, solvable algebras with complex commutators, almost abelian Lie algebras, and algebras with a $J$-invariant abelian ideal of codimension $2$, showing that any nonzero constant mixed curvature $\mathcal{C}_{\alpha,\beta}=c$ forces the metric to be Kähler (in practice, $c=0$ under these hypotheses implies Chern flatness). It also analyzes Bismut torsion-parallel manifolds, proving that balanced BTP threefolds cannot have nonzero constant mixed curvature and detailing the sharp outcomes (Chern flat or middle-type under specific conditions) in dimension three, with a separate result that non-balanced BTP manifolds cannot admit nonzero constant mixed curvature. The work thus provides substantial partial evidence toward Tang's conjecture in structured homogeneous settings and expands the landscape of non-Kähler Hermitian geometries where constant mixed curvature enforces strong rigidity.

Abstract

In a recent work, Kai Tang conjectured that any compact Hermitian manifold with non-zero constant mixed curvature must be Kähler. He confirmed the conjecture in complex dimension $2$ and for Chern Kähler-like manifolds in general dimensions. In this paper, we verify his conjecture for several special types of Hermitian manifolds, including complex nilmanifolds, solvmanifolds with complex commutators, almost abelian Lie groups, and Lie algebras containing a $J$-invariant abelian ideal of codimension $2$. We also verify the conjecture for all compact balanced threefolds when the Bismut connection has parallel torsion. These results provide partial evidence towards the validity of Tang's conjecture.

On Hermitian manifolds with constant mixed curvature

TL;DR

The paper advances Tang's conjecture on constant mixed curvature by establishing it for four Lie-Hermitian classes: complex nilmanifolds, solvable algebras with complex commutators, almost abelian Lie algebras, and algebras with a -invariant abelian ideal of codimension , showing that any nonzero constant mixed curvature forces the metric to be Kähler (in practice, under these hypotheses implies Chern flatness). It also analyzes Bismut torsion-parallel manifolds, proving that balanced BTP threefolds cannot have nonzero constant mixed curvature and detailing the sharp outcomes (Chern flat or middle-type under specific conditions) in dimension three, with a separate result that non-balanced BTP manifolds cannot admit nonzero constant mixed curvature. The work thus provides substantial partial evidence toward Tang's conjecture in structured homogeneous settings and expands the landscape of non-Kähler Hermitian geometries where constant mixed curvature enforces strong rigidity.

Abstract

In a recent work, Kai Tang conjectured that any compact Hermitian manifold with non-zero constant mixed curvature must be Kähler. He confirmed the conjecture in complex dimension and for Chern Kähler-like manifolds in general dimensions. In this paper, we verify his conjecture for several special types of Hermitian manifolds, including complex nilmanifolds, solvmanifolds with complex commutators, almost abelian Lie groups, and Lie algebras containing a -invariant abelian ideal of codimension . We also verify the conjecture for all compact balanced threefolds when the Bismut connection has parallel torsion. These results provide partial evidence towards the validity of Tang's conjecture.

Paper Structure

This paper contains 6 sections, 11 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solvable Lie algebras with complex commutators
  4. Almost abelian Lie algebras
  5. Lie algebras with $J$-invariant abelian ideal of codimension $2$
  6. Bismut torsion-parallel manifolds

Key Result

Theorem 1

Let $(M^n, g)$ be a Lie-Hermitian manifold with universal cover $(G, J, g)$. Assume that the Lie algebra ${\mathfrak g}$ of $G$ satisfies one of the following: If $\mathcal{C}_{\alpha,\beta} = c$, then $c=0$. If $\mathcal{C}_{\alpha,\beta} =0$ and $\beta \neq 0$, then $g$ must be Chern flat.

Theorems & Definitions (21)

  • Conjecture 1: Constant Holomorphic Sectional Curvature Conjecture
  • Conjecture 2: Tang
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 3
  • Lemma 1
  • Lemma 2
  • ...and 11 more