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Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture

  • Shuwen Chen
  • Fangyang Zheng
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Abstract

We prove that the Fino--Vezzoni conjecture holds on every compact complex manifold carrying a Hermitian connection with parallel torsion. More precisely, if a compact connected complex manifold carries a Hermitian metric g and a Hermitian connection D with DTD=0, then the coexistence of a balanced metric and a pluriclosed metric forces the existence of a D-parallel Kähler metric. The proof combines a holonomy symmetrization onto D-parallel forms with a finite-dimensional volume-maximization argument. In particular, the conjecture holds on every compact Bismut torsion-parallel manifold. In the non-balanced case, the obstruction of Zhao--Zheng yields the stronger conclusion that balanced and pluriclosed metrics cannot coexist.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026hermitian,
  title = {{Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture}},
  author = {Shuwen Chen and Fangyang Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27809},
  eprint = {2609.27809},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27809},
}
RIS
TY  - UNPB
TI  - Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture
AU  - Shuwen Chen
AU  - Fangyang Zheng
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.27809
UR  - https://arxiv.org/abs/2609.27809
AB  - We prove that the Fino--Vezzoni conjecture holds on every compact complex manifold carrying a Hermitian connection with parallel torsion. More precisely, if a compact connected complex manifold carries a Hermitian metric g and a Hermitian connection D with DT^D=0, then the coexistence of a balanced metric and a pluriclosed metric forces the existence of a D-parallel Kähler metric. The proof combines a holonomy symmetrization onto D-parallel forms with a finite-dimensional volume-maximization argument. In particular, the conjecture holds on every compact Bismut torsion-parallel manifold. In the non-balanced case, the obstruction of Zhao--Zheng yields the stronger conclusion that balanced and pluriclosed metrics cannot coexist.
ER  -