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

TL;DR

This paper explores the $p$-adic monodromy operators of hyper-K"ahler varieties over $p$-adic fields, proposing and verifying an arithmetic analogue of Nagai's conjecture using Sen's theory.

Contribution

It introduces a new method to analyze $p$-adic cohomology of hyper-K"ahler varieties and verifies a conjecture relating monodromy nilpotency indices across cohomology degrees.

Findings

Confirmed the arithmetic Nagai's conjecture for known deformation types.

Developed a novel approach using Sen's theory for $p$-adic cohomology analysis.

Established a conjectural relation between monodromy operators on different cohomology groups.

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.