Good reduction criterion for K3 surfaces

Abstract

We prove a Neron--Ogg--Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its $l$-adic cohomology group is unramified. We also prove a $p$-adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic.

31 Pages, Accepted version (plus minor modifications on Remark 1.2(2), Proposition 2.2(4), Section 5.3, Remark 6.2)

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• A Néron-Ogg-Shafarevich criterion for K3 surfaces
• Degeneration of Kummer surfaces
• Twisted Derived Equivalences and Isogenies between K3 Surfaces in Positive Characteristic
• On a cohomological generalization of the Shafarevich conjecture for K3 surfaces
• A Kulikov-Type Classification Theorem for a One Parameter Family of K3-Surfaces Over a p-ADIC Field and a Good Reduction Criterion