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Peer-reviewedCC BY 4.0Algebraic Geometry

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

  • Zhiyuan Li
  • Lie Fu
  • Teppei Takamatsu
  • Haitao Zou
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Abstract

We prove the unpolarized Shafarevich conjecture for hyper-Kähler varieties of a fixed deformation type, unifying earlier results for K3 surfaces and polarized hyper-Kähler varieties. We also study a cohomological variant in which good reduction is replaced by unramifiedness of cohomology; the resulting finiteness statements depend on the faithfulness of the automorphism action on cohomology. For hyper-Kähler varieties of CM type, we prove finiteness of geometric isomorphism classes in a fixed deformation type over number fields of bounded degree, extending a theorem of Orr and Skorobogatov for K3 surfaces. Our main tool is a uniform Kuga-Satake map, whose arithmetic properties are developed along the way.

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Cite

BibTeXRIS
BibTeX
@article{li2026unpolarized,
  title = {{Unpolarized Shafarevich conjectures for hyper-Kähler varieties}},
  author = {Zhiyuan Li and Lie Fu and Teppei Takamatsu and Haitao Zou},
  journal = {Algebraic Geometry},
  year = {2026},
  month = sep,
  doi = {10.14231/ag-2026-018},
  url = {https://hal.science/hal-05750255v1},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Unpolarized Shafarevich conjectures for hyper-Kähler varieties
AU  - Zhiyuan Li
AU  - Lie Fu
AU  - Teppei Takamatsu
AU  - Haitao Zou
PY  - 2026
DA  - 2026/09/01
JO  - Algebraic Geometry
DO  - 10.14231/ag-2026-018
UR  - https://hal.science/hal-05750255v1
AB  - We prove the unpolarized Shafarevich conjecture for hyper-Kähler varieties of a fixed deformation type, unifying earlier results for K3 surfaces and polarized hyper-Kähler varieties. We also study a cohomological variant in which good reduction is replaced by unramifiedness of cohomology; the resulting finiteness statements depend on the faithfulness of the automorphism action on cohomology. For hyper-Kähler varieties of CM type, we prove finiteness of geometric isomorphism classes in a fixed deformation type over number fields of bounded degree, extending a theorem of Orr and Skorobogatov for K3 surfaces. Our main tool is a uniform Kuga-Satake map, whose arithmetic properties are developed along the way.
SN  - 2214-2584
SN  - 2313-1691
ER  -