Convergent isocrystals on simply connected varieties

Abstract

It is conjectured by de Jong that, if $X$ is a connected smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial \'etale fundamental group, any isocrystal on on $X/W$ is trivial. We prove this conjecture under two additional assumptions. Version 2: the main change is an addendum. We prove that if $X$ is a connected smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$ with trivial \'etale fundamental group, any infinitesimal isocrystal on $X/W$ is trivial. To this aim we wrote some general facts on such infinitesimal isocrystals over W which are missing in the literature.