Good Reduction of K3 Surfaces

Abstract

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. 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 semi-stable flops in mixed characteristic, and study group actions and their quotients on models of varieties.

40 pages, final version

References in corpus (4)

• Good reduction criterion for K3 surfaces
• Resolution of Singularities of Arithmetical Threefolds II
• 2-adic integral canonical models and the Tate conjecture in characteristic 2
• Rational points on K3 surfaces and derived equivalence

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• A census of zeta functions of quartic K3 surfaces over F_2
• On a cohomological generalization of the Shafarevich conjecture for K3 surfaces
• On the Shafarevich conjecture for Enriques surfaces
• Reduction of Kummer surfaces modulo 2 in the non-supersingular case
• Twisted Derived Equivalences and Isogenies between K3 Surfaces in Positive Characteristic
• Monodromy Criterion for the Good Reduction of K3 Surfaces
• On log minimality of weak K-moduli compactifications of Calabi-Yau varieties
• A Kulikov-Type Classification Theorem for a One Parameter Family of K3-Surfaces Over a p-ADIC Field and a Good Reduction Criterion
• Non-liftability of automorphism groups of a K3 surface in positive characteristic