Chern classes of crystals

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 étale fundamental group of a smooth projective variety defined over an algebraically closed field and the constancy of its category of isocrystals. We also discuss the case of the Gauß–Manin convergent isocrystal.

Constancy of Newton Polygons of -Isocrystals on Abelian Varieties and Isotriviality of Families of Curves

“…We will use the following proposition to reduce the problem in the case where the field k of definition is a finite field. This similar argument is discussed in [19,Section 5.4].…”

“…In order to prove (1) it is sufficient to prove the assertion when S is simply connected, i.e., the geometrically etale fundamental group π et 1 (S k ) is trivial. Then any geometric convergent isocrystal on S/K is constant by [19,Theorem 1.3], so that it is also constant as a convergent F -isocrystal. The isotriviality follows from Theorem 4.5.…”

“…Concerning to Problem 1.2 (1), H.Esnault and A.Shiho studied the constant problem, called de Jong conjecture [18, Conjecture 2.1], in the case of convergent isocrystals (without Frobenius structures). They proved the de Jong conjecture under certain conditions [40] [18] [19]. Moreover the constancy of geometric convergent isocrystals (see Definition 2.10) are proved in [19,Theorem 1.3].…”

“…They proved the de Jong conjecture under certain conditions [40] [18] [19]. Moreover the constancy of geometric convergent isocrystals (see Definition 2.10) are proved in [19,Theorem 1.3]. As mathematical statements, the case of convergent isocrystals is much stronger than that of convergent F -isocrystals.…”

Etale and crystalline companions, II

“…We now specialize to curves and combine with the theory of vector bundles to obtain the crucial uniformity. In passing, we mention that some related results have been obtained by Esnault-Shiho [24] and Bhatt-Lurie [8] from a somewhat different point of view. Hypothesis 5.4.1.…”