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Good reduction criterion for K3 surfaces

Abstract

We prove a Néron–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.

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Notes

  1. 1.

    A representation of the absolute Galois group of a discrete valuation field is said to be unramified if the inertia subgroup acts trivially.

  2. 2.

    This paper of Ito is unpublished, but it is included in [23, Theorem 1.18] as an appendix.

  3. 3.

    This numbering is the same to that of Saito [43]. Some authors (e.g. [10, 26, 38]) write \(-^{(p+1)}\) for this space.

  4. 4.

    This liftability assumption is satisfied for any projective SNC log K3 surface [29, Corollary 6.9].

  5. 5.

    We fixed a small error in the corresponding formula in [43, Corollary 2.8]: his \(i-1 \ge \max \{0,-p\}\) should be \(i-1 \ge \max \{0,-p-2\}\).

  6. 6.

    In detail: Let \(C = (f=0) \subset {\mathbb {P}}^2\) be the ramification divisor of \(X \rightarrow {\mathbb {P}}^2\). Since \(p > B^2 + 4 = 6 = \deg f\) (since \(B\) is the pullback of \({\mathcal {O}}(1)\)) and \(C\) is smooth, \(C\) has only finitely many inflection points. Take a point on \({\mathbb {P}}^2\) which is not on the union of \(C\) and the tangent lines at the inflection points, and take the projection from that point. Then all the fibers of the resulting fibration \(X' \rightarrow {\mathbb {P}}^1\) are nodal and general fibers are smooth irreducible. Now use Saito’s result similarly.

  7. 7.

    One should be careful since the notation differs in these papers. We mainly follow that of [22], but in order to avoid collision of notation we use \({\mathbb {K}}\) instead of his \(K\) and \(\Lambda _{d}\) instead of his \(L_{d}\).

  8. 8.

    A groupoid is a category such that all morphisms are isomorphisms. A set can be naturally regarded as a groupoid, but the groupoids \(M_{2d}(S)\) and \(M^\circ _{2d}(S)\) are not of that kind.

  9. 9.

    To be precise, the period map is defined only on a suitable double covering \(\tilde{M}_{2d}\) of \(M_{2d}\). However, if \({\mathbb {K}}\) is neat, then \(\tilde{M}_{2d, {\mathbb {K}}} \rightarrow M_{2d, {\mathbb {K}}}\) admits a (non-canonical) section, and the level structured period map \(\iota _{\mathbb {K}}\) is indeed defined on \(M_{2d,{\mathbb {K}}}\) via that section. Since we actually use only \(\iota _{\mathbb {K}}\) for such \({\mathbb {K}}\)’s, we omit this tilde for simplicity.

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Acknowledgments

The author expresses his sincere gratitude to his advisor Atsushi Shiho for supporting him in many ways. The author also thanks Takuma Hayashi, Tetsushi Ito, Teruhisa Koshikawa, Keerthi Madapusi Pera, Yukiyoshi Nakkajima, Takeshi Saito, Naoya Umezaki, and Kohei Yahiro for giving him helpful comments. This work was supported by Grant-in-Aid for JSPS Fellows Grant Number 12J08397.

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Correspondence to Yuya Matsumoto.

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Matsumoto, Y. Good reduction criterion for K3 surfaces. Math. Z. 279, 241–266 (2015). https://doi.org/10.1007/s00209-014-1365-8

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Keywords

  • K3 surfaces
  • Good reduction
  • Galois representations
  • Period map
  • Complex multiplication

Mathematics Subject Classification

  • 14J28
  • 11G25
  • 14G20