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Good reduction criterion for \(K3\) surfaces. (English) Zbl 1317.14089
Let \(K\) be a complete discrete valuation field with perfect residue field of characteristic \(p\). The paper under review proves a Néron–Ogg–Shafarevich type criterion for good reduction of \(K3\) surfaces. More precisely, the main result is formulated in the following:
{ Theorem}. Let \(X\) be a \(K3\) surface over \(K\) which admits an ample line bundle \(L\) satisfying \(p>L^2+4\). Assume that one of the following conditions holds:
(a) For some prime \(\ell\neq p\), \(H^2_{\text{ét}}(X_{\bar K},{\mathbb{Q}}_{\ell})\) is unramified.
(b) \((K\) is of characteristic \(0\) and) \(H^2_{\text{ét}}(X_{\bar K},{\mathbb{Q}}_p)\) is crystalline.
Then \(X\) has potential good reduction with an algebraic space model, that is, for some finite extension \(K~{'}/K\), there exists an algebraic space smooth proper over \({\mathcal{O}}_K\) with generic fiber isomorphic to \(X_K\).
The model of the surface is not in general a scheme but an algebraic space.
There are two applications of the main result. The first is the surjectivity of the period map of \(K3\) surfaces in positive characteristic. The second application is that \(K3\) surfaces having semistable multiplication have potential good reduction.
The main theorem is proved using a method of D. Maulik [“Supersingular \(K3\) surfaces for large primes”, preprint, arXiv:1203.2889v2] studying reduction of \(K3\) surfaces, and comparison theorems for semistable algebraic spaces.

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
11G25 Varieties over finite and local fields
14G20 Local ground fields in algebraic geometry
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