Monodromy Criterion for the Good Reduction of K3 Surfaces

TL;DR

This paper presents a purely p-adic method to determine when a semistable K3 surface over a p-adic field has good reduction, avoiding complex Hodge theory or transcendental techniques.

Contribution

It introduces a new p-adic criterion for good reduction of K3 surfaces, utilizing log-family realizations and an arithmetic Clemens-Schmid sequence.

Findings

Established a criterion for good reduction of K3 surfaces in p-adic settings.

Developed a classification theorem in characteristic p for semistable K3 surfaces.

Provided a purely p-adic approach avoiding Hodge theory.

Abstract

We give a criterion for the good reduction of semistable $K3$ surfaces over $p$-adic fields using purely $p$-adic methods. We use neither $p$-adic Hodge theory nor transcendental methods as in the analogous proofs of criteria for good reduction of curves or $K3$ surfaces. We achieve our goal by realizing the special fiber $X_s$ of a semistable model $X$ of a $K3$ surface over the $p$-adic field $K$, $X_K$ as a special fiber of a log-family in characteristic $p$ and use an arithmetic version of the Clemens-Schimd exact sequence in order to obtain a Kulikov-Persson-Pinkham classification theorem in characteristic $p$.