On logarithmic extension of overconvergent isocrystals

TL;DR

This paper develops a criterion for extending overconvergent isocrystals logarithmically to smooth compactifications with normal crossing divisors, generalizing Kedlaya's unipotent monodromy result and paralleling complex regular singular connection theory.

Contribution

It introduces a new criterion for logarithmic extension of overconvergent isocrystals, broadening Kedlaya's unipotent monodromy case to more general scenarios.

Findings

Established a criterion for logarithmic extension of overconvergent isocrystals.

Generalized Kedlaya's unipotent monodromy result.

Analogous to the complex theory of regular singular connections.

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.