On extension of overconvergent log isocrystals on log smooth varieties

TL;DR

This paper generalizes the extension theorem of overconvergent log isocrystals from smooth varieties with normal crossing divisors to more general log smooth varieties, broadening the scope of such extensions under certain conditions.

Contribution

It extends the known results on overconvergent log isocrystals to a wider class of log smooth varieties with specific conditions, beyond the classical normal crossing divisor case.

Findings

Extension of overconvergent log isocrystals to broader log smooth varieties.

Generalization of monodromy conditions for extension.

Broader applicability of extension theorems in p-adic cohomology.

Abstract

By works of Kedlaya and Shiho, it is known that, for a smooth variety $\overline{X}$ over a field of positive characteristic and its simple normal crossing divisor $Z$, an overconvergent isocrystal on the compliment of $Z$ satisfying a certain monodromy condition can be extended to a convergent log isocrystal on $\left(\overline{X}, \mathcal{M}_Z\right)$, where $\mathcal{M}_Z$ is the log structure associated to $Z$. We prove a generalization of this result: for a log smooth variety $\left(\overline{X},\mathcal{M}\right)$ satisfying some conditions, an overconvergent log isocrystal on the trivial locus of a direct summand of $\mathcal{M}$ satisfying a certain monodromy condition can be extended to a convergent log isocrystal on $\left(\overline{X}, \mathcal{M}\right)$.