Arithmetic monodromy of hyper-K\"ahler varieties over $p$-adic fields

Kazuhiro Ito; Tetsushi Ito; Teruhisa Koshikawa; Teppei Takamatsu; Haitao Zou

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arXiv · 2507.13713

Arithmetic monodromy of hyper-K\"ahler varieties over $p$-adic fields

Kazuhiro Ito·Tetsushi Ito·Teruhisa Koshikawa·Teppei Takamatsu·Haitao Zou

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Abstract

In this paper, we study the $p$-adic and $\ell$-adic monodromy operators associated with hyper-K\"ahler varieties over $p$-adic fields, in connection with Looijenga-Lunts-Verbitsky Lie algebras. We investigate a conjectural relation between the nilpotency indices of these monodromy operators on higher-degree cohomology groups and on the second cohomology, which may be viewed as an arithmetic analogue of Nagai's conjecture for degenerations of hyper-K\"ahler manifolds over a disk. We verify this arithmetic version of Nagai's conjecture for hyper-K\"ahler varieties over $p$-adic fields, assuming they belong to one of the four known deformation types. As part of our approach, we introduce a new method to analyze the $p$-adic cohomology of hyper-K\"ahler varieties via Sen's theory.

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Kazuhiro Ito, Tetsushi Ito, Teruhisa Koshikawa, Teppei Takamatsu, Haitao Zou. 2025-07-18. Arithmetic monodromy of hyper-K\"ahler varieties over $p$-adic fields. https://arxiv.org/abs/2507.13713

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