A Lefschetz theorem for crystalline representations

TL;DR

This paper establishes an arithmetic analog of Simpson's Lefschetz theorem for complex Hodge structures using $p$-adic nonabelian Hodge theory, with extensions to a logarithmic setting under certain conditions.

Contribution

It introduces an arithmetic version of the Lefschetz theorem for crystalline representations, advancing the application of $p$-adic nonabelian Hodge theory in arithmetic geometry.

Findings

Proves an arithmetic Lefschetz theorem for crystalline representations

Extends results to a logarithmic setting under certain assumptions

Connects nonabelian Hodge theory with arithmetic and $p$-adic contexts

Abstract

As a corollary of nonabelian Hodge theory, Simpson proved a strong Lefschetz theorem for complex polarized variations of Hodge structure. We show an arithmetic analog. Our primary technique is $p$-adic nonabelian Hodge theory. Conditional on certain foundational results in \emph{logarithmic} $p$-adic Hodge theory, we also show a logarithmic analog.