Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields

In this paper, we study the $p$-adic and $\mathrm{\ell }$-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.