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Deformations of Kähler and Balanced Hyperbolicity

  • Jixiang Fu
  • Jingcao Wu
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Abstract

We study the deformation stability of Kähler and balanced hyperbolicity. We prove that Kähler hyperbolicity is open under arbitrary small holomorphic deformations, but need not be closed, even in families of smooth projective surfaces. By contrast, balanced hyperbolicity is not open in general: in every complex dimension at least five, we construct a balanced hyperbolic central fibre with non-balanced nearby fibres. We establish criteria for the deformation stability of balanced hyperbolicity using Aeppli cohomology and Lefschetz theory. For principal holomorphic torus bundles over products of curves of genus at least two, we compute the relevant hyperbolic cohomology, obtain a numerical characterization of balanced hyperbolicity, and prove its openness under small deformations.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026deformations,
  title = {{Deformations of Kähler and Balanced Hyperbolicity}},
  author = {Jixiang Fu and Jingcao Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.04816},
  eprint = {2609.04816},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04816},
}
RIS
TY  - UNPB
TI  - Deformations of Kähler and Balanced Hyperbolicity
AU  - Jixiang Fu
AU  - Jingcao Wu
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.04816
UR  - https://arxiv.org/abs/2609.04816
AB  - We study the deformation stability of Kähler and balanced hyperbolicity. We prove that Kähler hyperbolicity is open under arbitrary small holomorphic deformations, but need not be closed, even in families of smooth projective surfaces. By contrast, balanced hyperbolicity is not open in general: in every complex dimension at least five, we construct a balanced hyperbolic central fibre with non-balanced nearby fibres. We establish criteria for the deformation stability of balanced hyperbolicity using Aeppli cohomology and Lefschetz theory. For principal holomorphic torus bundles over products of curves of genus at least two, we compute the relevant hyperbolic cohomology, obtain a numerical characterization of balanced hyperbolicity, and prove its openness under small deformations.
ER  -