Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields
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arXiv
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| Format: | Preprint |
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2025
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| author | Ito, Kazuhiro Ito, Tetsushi Koshikawa, Teruhisa Takamatsu, Teppei Zou, Haitao |
| author_facet | Ito, Kazuhiro Ito, Tetsushi Koshikawa, Teruhisa Takamatsu, Teppei Zou, Haitao |
| contents | In this paper, we study the $p$-adic and $\ell$-adic monodromy operators associated with hyper-Kähler 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-Kähler manifolds over a disk. We verify this arithmetic version of Nagai's conjecture for hyper-Kähler 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-Kähler varieties via Sen's theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13713 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields Ito, Kazuhiro Ito, Tetsushi Koshikawa, Teruhisa Takamatsu, Teppei Zou, Haitao Algebraic Geometry Number Theory 14J42, 14G20, 14F30 In this paper, we study the $p$-adic and $\ell$-adic monodromy operators associated with hyper-Kähler 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-Kähler manifolds over a disk. We verify this arithmetic version of Nagai's conjecture for hyper-Kähler 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-Kähler varieties via Sen's theory. |
| title | Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields |
| topic | Algebraic Geometry Number Theory 14J42, 14G20, 14F30 |
| url | https://arxiv.org/abs/2507.13713 |