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Good reduction of K3 surfaces

  • Christian Liedtke (a1) and Yuya Matsumoto (a2)
Abstract

Let $K$ be the field of fractions of a local Henselian discrete valuation ring ${\mathcal{O}}_{K}$ of characteristic zero with perfect residue field $k$ . Assuming potential semi-stable reduction, we show that an unramified Galois action on the second $\ell$ -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.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
References
Hide All
[And96] André, Y., On the Shafarevich and Tate conjectures for hyperkähler varieties , Math. Ann. 305 (1996), 205248.
[Art70] Artin, M., Algebraization of formal moduli: II. Existence of modifications , Ann. of Math. (2) 91 (1970), 88135.
[Art73] Artin, M., Théorèmes de représentabilité pour les espaces algébriques, Séminaire de Mathématiques Supérieures, vol. 44 (Les Presses de l’Université de Montréal, Montréal, QC, 1973).
[Art74] Artin, M., Algebraic construction of Brieskorn’s resolutions , J. Algebra 29 (1974), 330348.
[Art77] Artin, M., Coverings of the rational double points in characteristic p , in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 1122.
[Băd01] Bădescu, L., Algebraic surfaces, Universitext (Springer, 2001).
[Bea85] Beauville, A., La théorème de torelliour les surfaces K3: fin de la démonstration , in Géométrie des surfaces K3: modules et périodes, Séminaire Palaiseau, October 1981–January 1982, Astérisque, vol. 126 (Société Mathématique de France, Paris, 1985), 111121.
[BM77] Bombieri, E. and Mumford, D., Enriques’ classification of surfaces in char. p. II , in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 2342.
[BLR90] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21 (Springer, Berlin, 1990).
[CJS13] Cossart, V., Jannsen, U. and Saito, S., Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, Preprint (2013), arXiv:0905.2191v2.
[CP14] Cossart, V. and Piltant, O., Resolution of singularities of arithmetical threefolds II, Preprint (2014), arXiv:1412.0868.
[CvS09] Cynk, S. and van Straten, D., Small resolutions and non-liftable Calabi–Yau threefolds , Manuscripta Math. 130 (2009), 233249.
[Del72] Deligne, P., La conjecture de Weil pour les surfaces K3 , Invent. Math. 15 (1972), 206226.
[HT17] Hassett, B. and Tschinkel, Y., Rational points on K3 surfaces and derived equivalence , in Brauer groups and obstruction problems, Progress in Mathematics, vol. 320 (Birkhäuser, Basel, 2017), 87113.
[Huy16] Huybrechts, D., Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158 (Cambridge University Press, Cambridge, 2016).
[Kaw94] Kawamata, Y., Semistable minimal models of threefolds in positive or mixed characteristic , J. Algebraic Geom. 3 (1994), 463491.
[KKMS73] Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973).
[Keu16] Keum, J., Orders of automorphisms of K3 surfaces , Adv. Math. 303 (2016), 3987.
[KM16] Kim, W. and Madapusi Pera, K., 2-adic integral canonical models and the Tate conjecture in characteristic 2 , Forum Math. Sigma 4 (2016), e28, 34 pp.
[KL13] Király, F. and Lütkebohmert, W., Group actions of prime order on local normal rings , Algebra Number Theory 7 (2013), 6374.
[Knu71] Knutson, D., Algebraic spaces, Lecture Notes in Mathematics, vol. 203 (Springer, Berlin, 1971).
[Kol89] Kollár, J., Flops , Nagoya Math. J. 113 (1989), 1536.
[KM98] Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).
[Kov09] Kovacs, S., Young person’s guide to moduli of higher dimensional varieties , in Algebraic geometry, Part 2, Seattle, 2005, Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, RI, 2009), 711743.
[KS67] Kuga, M. and Satake, I., Abelian varieties attached to polarized K3-surfaces , Math. Ann. 169 (1967), 239242.
[Kul77] Kulikov, V., Degenerations of K3 surfaces and Enriques surfaces , Izv. Akad. Nauk. SSSR Ser. Mat. 41 (1977), 10081042.
[KK98] Kulikov, V. and Kurchanov, P., Complex algebraic varieties: periods of integrals and Hodge structures , in Algebraic geometry III, Encyclopaedia of Mathematical Sciences, vol. 36 (Springer, Berlin, 1998), 1217.
[Lip69] Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization , Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195279.
[Liu02] Liu, Q., Algebraic geometry and arithmetic curves (Oxford University Press, 2002).
[LZ14] Liu, Y. and Zheng, W., Enhanced six operations and base change theorem for Artin stacks, Preprint (2014), arXiv:1211.5948v2.
[Mad15] Madapusi Pera, K., The Tate conjecture for K3 surfaces in odd characteristic , Invent. Math. 201 (2015), 625668.
[Mat15] Matsumoto, Y., Good reduction criterion for K3 surfaces , Math. Z. 279 (2015), 241266.
[MM64] Matsusaka, T. and Mumford, D., Two fundamental theorems on deformations of polarized varieties , Amer. J. Math. 86 (1964), 668684.
[Mau14] Maulik, D., Supersingular K3 surfaces for large primes , Duke Math. J. 163 (2014), 23572425.
[Mor81] Morrison, D. R., Semistable degenerations of Enriques’ and hyperelliptic surfaces , Duke Math. J. 48 (1981), 197249.
[Mum70] Mumford, D., Abelian varieties (Tata Institute of Fundamental Research Studies in Mathematics, Oxford University Press, 1970).
[Naka00] Nakayama, C., Degeneration of -adic weight spectral sequences , Amer. J. Math. 122 (2000), 721733.
[Nakk00] Nakkajima, Y., Liftings of simple normal crossing log K3 and log Enriques surfaces in mixed characteristics , J. Algebraic Geom. 9 (2000), 355393.
[Och99] Ochiai, T., l-independence of the trace of monodromy , Math. Ann. 315 (1999), 321340.
[Oda95] Oda, T., A note on ramification of the Galois representation on the fundamental group of an algebraic curve. II , J. Number Theory 53 (1995), 342355.
[Ogu79] Ogus, A., Supersingular K3 crystals , Astérisque 64 (1979), 386.
[Per77] Persson, U., On degenerations of algebraic surfaces , Mem. Amer. Math. Soc. 11 (1977).
[PP81] Persson, U. and Pinkham, H., Degeneration of surfaces with trivial canonical bundle , Ann. of Math. (2) 113 (1981), 4566.
[RZ82] Rapoport, M. and Zink, T., Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik , Invent. Math. 68 (1982), 21101.
[Riz10] Rizov, J., Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic , J. Reine Angew. Math. 648 (2010), 1367.
[RS76] Rudakov, A. N. and Shafarevich, I. R., Inseparable morphisms of algebraic surfaces , Izv. Akad. Nauk SSSR 40 (1976), 12691307.
[SGA4] Grothendieck, A. et al. , Théorie des topos et cohomologie étale des schémas, SGA 4, Lecture Notes in Mathematics vols. 269, 270, 305 (Springer, Berlin, 1973).
[SGA4½] Deligne, P., Cohomologie étale, SGA 4½, Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977).
[ST68] Serre, J.-P. and Tate, J., Good reduction of abelian varieties , Ann. of Math. (2) 88 (1968), 492517.
[Ste76] Steenbrink, J. H. M., Limits of Hodge structures , Invent. Math. 31 (1976), 229257.
[vanL07] van Luijk, R., K3 surfaces with Picard number one and infinitely many rational points , Algebra Number Theory 1 (2007), 115.
[Was82] Washington, L. C., Introduction to cyclotomic fields, Graduate Text in Mathematics, vol. 83, second edition (Springer, New York, 1982).
[Wew10] Wewers, S., Regularity of quotients by an automorphism of order , Preprint (2010), arXiv:1001.0607.
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Compositio Mathematica
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