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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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