The Semistable Reduction Project

This page describes the components of my "semistable reduction project" for overconvergent F-isocrystals in p-adic cohomology. The four papers in the series have now appeared in Compositio Mathematica; however, they depend on other papers (and one book) of mine which appear elsewhere, and some of these are also described below. Go back to my preprints page for actual papers; see also my Strasbourg slides for a coherent summary. (This description last modified 15 Mar 11.)

"Semistable reduction I: Unipotence and logarithmic extensions" concerns constant and unipotent local monodromy of overconvergent F-isocrystals, and demonstrates an equivalence between the (logarithmic) extendability of an isocrystal to its having constant/unipotent local monodromy. Some of the results have been improved (and proofs simplified) by Atsushi Shiho; see his arXiv preprint. (There is an erratum to this paper appended to part IV.)
"Semistable reduction II: A valuation-theoretic approach" outlines an approach to the problem of semistable reduction for overconvergent F-isocrystals based on ideas from valuation theory. The idea is to use the quasicompactness of the Riemann-Zariski space associated to a finitely generated extension of fields to reduce the problem of semistable reduction to a question that is "local" around a particular valuation. (There is an erratum to this paper appended to part IV.)
"Semistable reduction III: Local semistable reduction at monomial valuations" treats the local semistable reduction problem at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree 0). The basis for this treatment is my version of the p-adic local monodromy theorem for fake annuli (see below), plus a notion of generic radius of convergence on a polyannulus (generalizing the usual notion on an annulus studied by Robba, Christol-Dwork, et al).
"Semistable reduction IV: Local semistable reduction at nonmonomial valuations" treats local semistable reduction at minimal nonmonomial valuations; by earlier results, this completes the proof of the semistable reduction theorem. (It also includes the case of a partially overconvergent F-isocrystal, in which one only desired logarithmic extension after altering the partial compactification.) This is accomplished by induction on transcendence defect (starting from the monomial case from the previous paper), using close analysis of some numerical invariants attached to (ordinary) p-adic differential equations. Some of the results used are taken from my course "p-adic differential equations" (see below). The appendix includes some errata for papers I and II.

I wrote several other papers (and one book) that bear some relation to this series.

The paper "Local monodromy of p-adic differential equations: an overview" is a relatively self-contained overview of what one knows about the monodromy of a differential equation on a rigid analytic annulus in one dimension, particularly in the presence of a Frobenius structure. Although this paper has been removed from the series for its independent interest, it is cited as a reference at several points in the series (although the results can all be found elsewhere too).
The paper "Slope filtrations revisited" gives a second-generation exposition of the slope filtration theorem, with a number of technical and expository simplifications and improvements. Although this paper has been removed from the series for its independent interest, the series depends logically on it (via the "fake annuli" paper below). I also have a third-generation exposition called "Slope filtrations for relative Frobenius", with an even more streamlined exposition, but that paper is not written in such a way as to cover the fake annuli situation (it could have been, but at the expense of some clarity).
The paper "The p-adic local monodromy theorem for fake annuli" uses the "Slope filtrations revisited" framework plus a careful version of Tsuzuki's unit-root argument to generalize the monodromy theorem to so-called "fake annuli". Although this paper has been removed from the series, the series does depend logically on it (via part III); there is a chance one can remove this dependence using "Swan conductors" (see below), but I have not looked into this in depth.
The papers "Swan conductors for p-adic differential modules I, II" define a Swan conductor for overconvergent F-isocrystals along boundary divisors, and collect some key properties. These include some continuity and convexity properties when varying the choice of the divisor on a surface, as well as a monotonicity result in the manner of Deligne-Laumon semicontinuity. Although the series depends only minimally on these papers, a similar point of view is adopted in part IV.
The book "p-adic differential equations" (Cambridge University Press, 2010) is a textbook/treatise developed from lecture notes from a graduate course I gave at MIT in fall 2007. It develops the foundations of the theory of ordinary p-adic differential equations more thoroughly than in most prior references. The series depends heavily on this book via part IV.

I also wrote some papers which were originally intended to pertain to this series, but no longer do.

The paper "The geometry of p-typical covers" describes some finite étale covers of varieties in characteristic p whose monodromy groups are p-groups. There was originally supposed to be a logical dependence on this paper, which has been removed.
There used to be a paper "Semistable reduction IV: Refining the local approach", which introduced the local problem of "local lifting of horizontal sections". If solved, this could be used to deduce local semistable reduction using an analogue of Mebkhout's proof of Crew's conjecture. However, I ended up not using this technique in the course of the proof of local semistable reduction; also, this paper included a faulty proof that the general local semistable reduction problem can be reduced to the case of transcendence defect 1. Consequently, this paper has been withdrawn. (Some technical results were absorbed into part IV.)