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<p>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 <i>J</i>-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 toward the validity of Tang’s conjecture.</p>

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On Hermitian Manifolds with Constant Mixed Curvature

  • 期刊论文

  • 发表日期：

    2025年9月4日

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  • Shuwen Chen,
  • Fangyang Zheng

摘要

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 toward the validity of Tang’s conjecture.