Cut-by-curves criterion for the log extendability of overconvergent isocrystals

TL;DR

This paper establishes a criterion for extending overconvergent isocrystals logarithmically to compactifications, paralleling complex-analytic criteria, and explores related solvability and ramification properties in the p-adic setting.

Contribution

It introduces a cut-by-curves criterion for the log extendability of overconvergent isocrystals, advancing p-adic geometric analysis.

Findings

Proved a cut-by-curves criterion for log extendability of overconvergent isocrystals.

Established criteria for solvability, ramification break, and exponent of $ abla$-modules.

Extended complex-analytic concepts to the p-adic context.

Abstract

In this paper, we prove a `cut-by-curves 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, under certain assumption. This is a $p$-adic analogue of a version of cut-by-curves criterion for regular singuarity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of $\nabla$-modules.