Chern classes of crystals

TL;DR

This paper proves that crystalline Chern classes vanish for locally free crystals on smooth varieties over perfect fields, leading to new insights into de Jong's conjecture and the triviality of isocrystal categories.

Contribution

It establishes the vanishing of crystalline Chern classes for locally free crystals and derives new cases of de Jong's conjecture relating fundamental groups and isocrystals.

Findings

Crystalline Chern classes vanish for locally free crystals.

New cases of de Jong's conjecture are established.

Discussion of the Gau{df}-Manin convergent isocrystal.

Abstract

The crystalline Chern classes of the value of a locally free crystal vanish on a smooth variety defined over a perfect field. Out of this we conclude new cases of de Jong's conjecture relating the geometric \'etale fundamental group of a smooth projective variety defined over a perfect field and the triviality of its category of isocrystals. We also discuss the case of the Gau{\ss}-Manin convergent isocrystal.