Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Ito, Kazuhiro, Ito, Tetsushi, Koshikawa, Teruhisa, Takamatsu, Teppei, Zou, Haitao
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416280644
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