A note on convergent isocrystals on simply connected varieties

TL;DR

This paper investigates de Jong's conjecture that convergent isocrystals on simply connected, liftable varieties over fields of characteristic p are trivial, proving the conjecture under specific stability conditions.

Contribution

It proves the triviality of convergent isocrystals on liftable, simply connected varieties in characteristic p under certain (semi)stability assumptions, advancing understanding of their structure.

Findings

Convergent isocrystals are trivial on liftable, simply connected varieties under stability conditions.

Supports de Jong's conjecture in specific cases.

Provides conditions under which isocrystals are trivial.

Abstract

It is conjectured by de Jong that, if $X$ is a connected projective smooth variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial etale fundamental group, any convergent isocrystal $\mathcal{E}$ on $X$ is trivial. We discuss this conjecture when $X$ is liftable to characteristic zero, and prove the triviality of $\mathcal{E}$ in this case under certain conditions on (semi)stability.