Good reduction of K3 surfaces

Abstract

Let K be the field of fractions of a local Henselian discrete valuation ring 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 -adic cohomology group 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 flops in mixed characteristic, and study group actions and their quotients on models of varieties.

A Néron–Ogg–Shafarevich criterion for K3 surfaces

“…The equivalences were already established in and Matsumoto showed that when is complete, (4) implies good reduction after some finite, but possibly ramified, extension of . Remark Using the examples constructed in , we see that the implication requires in general a non‐trivial unramified extension: more precisely, for every prime , there exists a K3 surface over that satisfies conditions (2)–(4) of Theorem but that does not have good reduction over , see Theorem .…”

“…We now come to our first improvement of Theorem , which yields a necessary and sufficient criterion for good reduction of a K3 surface over . This result also ‘explains’ the counterexamples from . Theorem Let be a K3 surface over that satisfies and the equivalent conditions of Theorem .…”

“…More precisely, if is a K3 surface over , then a Kulikov model is a flat and proper algebraic space with generic fibre , whose special fibre is a strict normal crossing divisor, and whose relative canonical divisor is trivial. Following the discussion in [, Section 3.1], we make the following assumption. Assumption A K3 surface over satisfies () if there exists a finite field extension such that admits a Kulikov model.…”

“…In this article, we study criteria for good reduction of a K3 surface over in terms of the ‐representation on . If , then such a criterion was already established in . Our first result is the following extension to the ‐adic case.…”

“…Thus, although we usually do not have good reduction over in the situation of Theorem , we may appeal to to construct certain ‘RDP models’ of K3 surfaces over , which are models that have reasonably mild singularities.…”