On $$p$$ -adic differential equations on semistable varieties

Abstract

-In this paper we prove a comparison theorem between the category of certain modules with integrable connection on the complement of a normal crossing divisor of the generic fiber of a proper semistable variety over a DVR and the category of certain log overconvergent isocystrals on the special fiber of the same open.

On p-adic differential equations on semistable varieties II

“…Now we recall the theorem of logarithmic extension for log overconvergent isocrystals in semistable situation proven in theorem 5, section 12 of [DP12], which generalizes of the main theorem of [Shi10b]. We start recalling the notion of log overconvergent isocrystal with Σ-unipotent monodromy, and the notion of log convergent isocrystal with exponents in Σ.…”

“…However, this is not our concern here.) In [DP12] the author generalized this notion to the present situation and constructed a fully faithful algebraization functor from the category of log overconvergent isocrystals on U k with Σ-unipotent monodromy to the category of modules with integrable connection on U K , regular along the generic fiber D K of D.…”

“…To construct the algebraization functor in this setting, we choose a section τ : Z p /Z → Z p of the canonical projection and we proceed as in [DP12]: It is obtained as the composite of the following four functors.…”

“…In the second section, we give preliminaries and fix some notations on our geometric setting. In the third section, we recall the definitions and the results in [DP12] concerning log-∇-modules on rigid analytic spaces and isocrystals. In the fourth section, we recall the definitions and the results in [DP12], [AB01] concerning modules with integrable connection on (formal log) schemes.…”

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This paper is a complement to the paper [DP12]. Given an open variety over a DVR with semistable reduction, the author constructed in [DP12] a fully faithful algebraization functor from the category of certain log overconvergent isocrystals on the special fiber to the category of modules with regular integrable connection on the generic fiber.

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