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 triviaĺ etale fundamental group, any isocrystal on X is constant. We prove this conjecture under certain additional assumptions.

Intermediate extensions and crystalline distribution algebras

“…It is conjectured by de Jong that, if X is a connected smooth projective variety over an algebraically closed field of characteristic p ą 0 with trivial étale fundamental group, then any isocrystal on X is constant. This conjecture is proved under certain additional assumptions by Esnault-Shiho in [17]. In our case, the fibration G Ñ G{B " P is a separable proper morphism with geometrically connected fibre between locally noetherian connected schemes.…”

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

“…In particular, it is also proven that rank 1 isocrystals (not necessarily convergent) are also trivial in this case [8,Theorem 1.2]. In the present work we want to extend the aforementioned results of [7] to the case of a rank one log extendable isocrystal on a non-proper variety with trivial étale fundamental group. Actually we just need to assume that its abelianized tame fundamental group is trivial.…”

“…The argument is the same as in [7, Proposition 2.10] in the local case. We can glue the local lattices as in [7,Lemma 2.11], because H 0 (X, O X ) log crys = H 0 (X, O X ) crys by [1, Section 6].…”

“…This question was addressed in great detail in [7] and in [8], where various subcases were considered. There the authors prove, among other results, that for a smooth connected projective variety over an algebraically closed field of positive characteristic, the triviality of the étale fundamental group implies that any convergent isocrystal, which is filtered so that the associated graded is a sum of rank 1 convergent isocrystals, is constant [7,Theorem 0.1]. In particular, it is also proven that rank 1 isocrystals (not necessarily convergent) are also trivial in this case [8,Theorem 1.2].…”

“…It then follows that we can assume (E log X , ∇ log X ) to be isomorphic to (O log X , d log ). In Theorem 5.1 we then use a deformation argument (adapted from [7]) to prove that L ⊗n is the trivial crystal and conclude that L and therefore also L are trivial, using Lemmas 5.2 and 5.3.…”