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Torsion algebras of Hermitian manifolds and rigidity

  • Wangyang Lin
  • Yibo Ren
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Abstract

We regard the Chern torsion of a Hermitian manifold as a skew-symmetric complex-bilinear product on its holomorphic tangent bundle. For a compact connected Chern--Kähler-like Hermitian manifold, we prove that this product satisfies the Jacobi identity pointwise. If, in addition, the Chern holomorphic sectional curvature is strongly quasi-positive, we show that the resulting torsion Lie algebra is nilpotent and must be abelian. Consequently, the metric is Kähler. The underlying complex manifold is therefore projective and rationally connected. For general Hermitian manifolds, we construct a metric with positive real bisectional curvature on a Hopf surface which is neither simply connected nor rationally connected.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026torsion,
  title = {{Torsion algebras of Hermitian manifolds and rigidity}},
  author = {Wangyang Lin and Yibo Ren},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.29916},
  eprint = {2608.29916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29916},
}
RIS
TY  - UNPB
TI  - Torsion algebras of Hermitian manifolds and rigidity
AU  - Wangyang Lin
AU  - Yibo Ren
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.29916
UR  - https://arxiv.org/abs/2608.29916
AB  - We regard the Chern torsion of a Hermitian manifold as a skew-symmetric complex-bilinear product on its holomorphic tangent bundle. For a compact connected Chern--Kähler-like Hermitian manifold, we prove that this product satisfies the Jacobi identity pointwise. If, in addition, the Chern holomorphic sectional curvature is strongly quasi-positive, we show that the resulting torsion Lie algebra is nilpotent and must be abelian. Consequently, the metric is Kähler. The underlying complex manifold is therefore projective and rationally connected. For general Hermitian manifolds, we construct a metric with positive real bisectional curvature on a Hopf surface which is neither simply connected nor rationally connected.
ER  -