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

Part of: (Co)homology theory

Published online by Cambridge University Press:  21 February 2011

Kiran S. Kedlaya
Kiran S. Kedlaya*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA (email: [email protected])
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Abstract

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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 zero. 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.

Keywords

rigid cohomologyoverconvergent isocrystalssemistable reductionnonmonomial valuations

MSC classification

Secondary: 14F30: $p$-adic cohomology, crystalline cohomology 14F40: de Rham cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 467 - 523
Copyright
Copyright © Foundation Compositio Mathematica 2011

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