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arXiv · 2610.07890

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 $\mathbb{Q}$. The proof establishes semisimplicity of $\ell$-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 $b_2\geq 4$, 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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Kazuhiro Ito, Haitao Zou. 2026-10-06. Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties. https://arxiv.org/abs/2610.07890

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