Abstract

In this thesis we prove a $p$-adic analogous of the Kulikov-Persson-Pinkham classification theorem for the central fibre of a degeneration of $K3$-surfaces in terms of the nilpotency degree of the monodromy of the family \citep{Persson:1981wp}.

Namely, let $X_K$ be a be a smooth, projective $K3$-surface which has a minimal semistable model $X$ over $\mcal O_K$.
If we let $N_{st}$ be the monodromy operator on $D_{st}(H^2_{\et}(X_{\oln K},\Qp))$, then we prove that the degree of nilpotency of $N_{st}$ determines the type of the special fibre 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 $H^2_{\et}(X_{\oln K},\Qp)$, which is similar to the criterion of good reduction for $p$-adic abelian varieties and curves given by \citep{Coleman:1997vg} and \citep{Iovita:2013ts}.