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Good reduction of K3 surfaces. (English) Zbl 1386.14138
Let \(K\) be the field of fractions of a local Henselian discrete valuation ring \(\mathcal{O}_K\) of characteristic zero with perfect residue field \(k\).
The authors show that an unramified Galois action on the second \(\ell\)-adic cohomology group of a \(K3\) surface \(X\) over \(K\) implies that the surface \(X\) has good reduction after a finite and unramified extension, under the assumption that over a finite extension of \(L/K\) the surface \(X_L\) admits a model that is a regular algebraic space with trivial canonical sheaf, and whose geometric special fiber is a normal crossing divisor.
The authors give examples where this unramified extension is really needed. Moreover, they give applications to good reduction after tame extensions and Kuga-Satake abelian varieties.
In the course of the proof the author settle the existence and termination of certain flops in mixed characteristic.

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
11G25 Varieties over finite and local fields
11F80 Galois representations
14E30 Minimal model program (Mori theory, extremal rays)
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