No abstract

In this paper, we establish a criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic p > 0 to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor. This is a generalization of a result of Kedlaya, who treated the case of unipotent monodromy. Our result is regarded as a p-adic analogue of the theory of canonical extension of regular singular integrable connections on smooth varieties of characteristic 0.

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.

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.

Given a smooth scheme over Z/p n Z with a lift of relative Frobenius to Z/p n+1 Z, we construct a functor from the category of Higgs modules to that of modules with integrable connections as the composite of the level raising inverse image functors from the category of modules with integrable p mconnections to that of modules with integrable p m−1 -connections for 1 ≤ m ≤ n. In the case m = 1, we prove that the level raising inverse image functor is an equivalence when restricted to quasi-nilpotent objects, which generalizes a local result of Ogus-Vologodsky. We also prove that the above level raising inverse image functor for a smooth p-adic formal scheme induces an equivalence of Q-linearized categories for general m when restricted to nilpotent objects (in strong sense), under a strong condition on Frobenius lift. We also prove a similar result for the category of modules with integrable p m -Wittconnections.