Good reduction of K3 surfaces

Abstract

Let$K$be the field of fractions of a local Henselian discrete valuation ring${\mathcal{O}}_{K}$of characteristic zero with perfect residue field$k$. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second$\ell$-adic cohomology group of a K3 surface over$K$implies that the surface has good reduction after a finite andunramifiedextension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.