A Kulikov-Type Classification Theorem for a One Parameter Family of K3-Surfaces Over a p-ADIC Field and a Good Reduction Criterion

Abstract

In this paper, we prove a $p$-adic analogous of the Kulikov-Persson-Pinkham classification theorem [Persson:1981wp] for the central fiber of a degeneration of $K3$-surfaces in terms of the nilpotency degree of the monodromy of the family. Namely, let $X_K$ be a be a smooth, projective $K3$-surface which has a minimal semi-stable model $X$ over $\mathcal O_K$. If we let $N_{st}$ be the monodromy operator on $D_{st}(H^2_{et}}(X_{\overline K},\mathbb Q_p))$, then we prove that the degree of nilpotency of $N_{st}$ determines the type of the special fiber of $X$. As a consequence we give a criterion for the good reduction of the semi-stable $K3$-surface $X_K$ over the $p$-adic field $K$ in terms of its $p$-adic representation $H^2_{\text{et}}(X_{\overline K},\mathbb Q_p)$, which is similar to the criterion of good reduction for $p$-adic abelian varieties and curves given by Coleman-Iovita and Andreatta-Iovita-Kim.