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PreprintCC BY 4.0arXiv

RC-positivity of complex surfaces

  • You-Cheng Chou
  • Kuang-Ru Wu
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Abstract

Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.

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Cite

BibTeXRIS
BibTeX
@misc{chou2026rcpositivity,
  title = {{RC-positivity of complex surfaces}},
  author = {You-Cheng Chou and Kuang-Ru Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33656},
  eprint = {2609.33656},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33656},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - RC-positivity of complex surfaces
AU  - You-Cheng Chou
AU  - Kuang-Ru Wu
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33656
UR  - https://arxiv.org/abs/2609.33656
AB  - Given a compact complex surface with a nonflat Ricci-flat Kähler metric, we show that its holomorphic tangent bundle admits an RC-positive Hermitian metric. The proof relies on a characterization of RC-positivity through the anti-self-dual part of the Weyl operator, the Weitzenböck formula on the four dimensional Einstein manifold, and a conformal perturbation on the Ricci-flat metric. As a consequence, we prove that RC-positivity is not preserved after taking tensor, exterior, or symmetric power. Moreover, we show that RC-positivity is a strictly weaker notion than uniform RC-positivity, and that RC-positivity of holomorphic tangent bundle does not necessarily imply rational connectedness of the base manifold.
ER  -