A monodromy criterion for the good reduction of $K3$ surfaces

Abstract

We give a criterion for the good reduction of semistable $K3$ surfaces over $p$-adic fields. 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 $K$, as a special fiber of a log-family in characteristic $p$ and use an arithmetic version of the Clemens–Schmid exact sequence in order to obtain a Kulikov–Persson–Pinkham classification theorem in characteristic $p$.