A note on rank 1 log extendable isocrystals on simply connected open varieties

TL;DR

This paper proves de Jong's $p$-adic Gieseker conjecture for rank 1 log extendable isocrystals on simply connected, non-proper varieties with trivial tame fundamental group, extending previous results to a broader class.

Contribution

It establishes the conjecture for non-proper varieties with trivial tame fundamental group, specifically for rank 1 log extendable isocrystals, broadening the scope of prior proofs.

Findings

The conjecture holds for non-proper varieties with trivial tame fundamental group.

Rank 1 log extendable isocrystals are shown to be constant in this setting.

Extends previous results from proper to certain non-proper varieties.

Abstract

In 2010 de Jong proposed a $p$-adic version of Gieseker's conjecture: if $X$ is a smooth, simply connected projective variety, then any isocrystal on $X$ is constant. This was proven by Esnault and Shiho under some additional assumptions. We show that the conjecture holds in the case of a non-proper variety with trivial tame fundamental group and for rank 1 log extendable isocrystals.