Semistable reduction for overconvergent F-isocrystals, IV: Local semistable reduction at nonmonomial valuations

TL;DR

This paper proves that overconvergent F-isocrystals can be locally extended with controlled singularities after suitable modifications, using valuation theory and invariants of p-adic differential modules.

Contribution

It completes the proof of local semistable reduction for overconvergent F-isocrystals at nonmonomial valuations, introducing new invariants and inductive techniques.

Findings

Established local semistable reduction at nonmonomial valuations.

Developed invariants analogous to irregularity for p-adic differential modules.

Reduced the problem to valuations of height 1 and residual transcendence degree 0.

Abstract

We complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree 0. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.