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PreprintCC BY 4.0arXiv

Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties

  • Kazuhiro Ito
  • Haitao Zou
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Abstract

We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over ℚ. The proof establishes semisimplicity of ℓ-adic cohomology by eliminating the unipotent radical of the algebraic monodromy group of total cohomology. We also prove the weight-monodromy conjecture for hyper-Kähler varieties over p-adic fields. For hyper-Kähler varieties over number fields with b2≥ 4, we establish, after suitable finite extensions, strong compatibility of the associated Weil-Deligne representations valued in Mumford-Tate groups and its integral refinement away from finitely many primes.

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Cite

BibTeXRIS
BibTeX
@misc{ito2026semisimplicity,
  title = {{Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties}},
  author = {Kazuhiro Ito and Haitao Zou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07890},
  eprint = {2610.07890},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07890},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties
AU  - Kazuhiro Ito
AU  - Haitao Zou
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07890
UR  - https://arxiv.org/abs/2610.07890
AB  - We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over ℚ. The proof establishes semisimplicity of ℓ-adic cohomology by eliminating the unipotent radical of the algebraic monodromy group of total cohomology. We also prove the weight-monodromy conjecture for hyper-Kähler varieties over p-adic fields. For hyper-Kähler varieties over number fields with b_2≥ 4, we establish, after suitable finite extensions, strong compatibility of the associated Weil-Deligne representations valued in Mumford-Tate groups and its integral refinement away from finitely many primes.
ER  -