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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.

14J28 \(K3\) surfaces and Enriques surfaces
11G25 Varieties over finite and local fields
11F80 Galois representations
14E30 Minimal model program (Mori theory, extremal rays)
Full Text: DOI arXiv
[1] André, Y., On the Shafarevich and Tate conjectures for hyperkähler varieties, Math. Ann., 305, 205-248, (1996) · Zbl 0942.14018
[2] Artin, M., Algebraization of formal moduli: II. Existence of modifications, Ann. of Math. (2), 91, 88-135, (1970) · Zbl 0177.49003
[3] Artin, M., Théorèmes de représentabilité pour les espaces algébriques, (1973), Les Presses de l’Université de Montréal: Les Presses de l’Université de Montréal, Montréal, QC · Zbl 0323.14001
[4] Artin, M., Algebraic construction of Brieskorn’s resolutions, J. Algebra, 29, 330-348, (1974) · Zbl 0292.14013
[5] Artin, M., Coverings of the rational double points in characteristic p, in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 11-22. doi:10.1017/CBO9780511569197.003 · Zbl 0358.14008
[6] Bădescu, L., Algebraic surfaces, (2001), Springer
[7] 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, (Société Mathématique de France, Paris, 1985), 111-121.
[8] Bombieri, E. and Mumford, D., Enriques’ classification of surfaces in char. p. II, in Complex analysis and algebraic geometry (Iwanami Shoten, Tokyo, 1977), 23-42. doi:10.1017/CBO9780511569197.004 · Zbl 0348.14021
[9] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, (Springer, Berlin, 1990). doi:10.1007/978-3-642-51438-8
[10] 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.
[11] Cossart, V. and Piltant, O., Resolution of singularities of arithmetical threefolds II, Preprint (2014), arXiv:1412.0868. · Zbl 1308.13004
[12] Cynk, S. and Van Straten, D., Small resolutions and non-liftable Calabi-Yau threefolds, Manuscripta Math.130 (2009), 233-249. doi:10.1007/s00229-009-0293-0 · Zbl 1177.14081
[13] Deligne, P., La conjecture de Weil pour les surfaces K3, Invent. Math., 15, 206-226, (1972) · Zbl 0219.14022
[14] Hassett, B. and Tschinkel, Y., Rational points on K3 surfaces and derived equivalence, in Brauer groups and obstruction problems, (Birkhäuser, Basel, 2017), 87-113. doi:10.1007/978-3-319-46852-5_6 · Zbl 1386.14136
[15] Huybrechts, D., Lectures on K3 surfaces, (2016), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1360.14099
[16] Kawamata, Y., Semistable minimal models of threefolds in positive or mixed characteristic, J. Algebraic Geom., 3, 463-491, (1994) · Zbl 0823.14026
[17] Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, (Springer, Berlin, 1973). doi:10.1007/BFb0070318 · Zbl 0271.14017
[18] Keum, J., Orders of automorphisms of K3 surfaces, Adv. Math., 303, 39-87, (2016) · Zbl 1375.14125
[19] Kim, W. and Madapusi Pera, K., 2-adic integral canonical models and the Tate conjecture in characteristic 2, Forum Math. Sigma4 (2016), e28, 34 pp. doi:10.1017/fms.2016.23 · Zbl 1362.11059
[20] Király, F. and Lütkebohmert, W., Group actions of prime order on local normal rings, Algebra Number Theory7 (2013), 63-74. doi:10.2140/ant.2013.7.63 · Zbl 1273.14094
[21] Knutson, D., Algebraic spaces, (1971), Springer: Springer, Berlin · Zbl 0221.14001
[22] Kollár, J., Flops, Nagoya Math. J., 113, 15-36, (1989) · Zbl 0645.14004
[23] Kollár, J. and Mori, S., Birational geometry of algebraic varieties, (Cambridge University Press, Cambridge, 1998). doi:10.1017/CBO9780511662560
[24] Kovacs, S., Young person’s guide to moduli of higher dimensional varieties, in Algebraic geometry, Part 2, Seattle, 2005, (American Mathematical Society, Providence, RI, 2009), 711-743. · Zbl 1182.14034
[25] Kuga, M. and Satake, I., Abelian varieties attached to polarized K3-surfaces, Math. Ann.169 (1967), 239-242. doi:10.1007/BF01399540 · Zbl 0221.14019
[26] Kulikov, V., Degenerations of K3 surfaces and Enriques surfaces, Izv. Akad. Nauk. SSSR Ser. Mat., 41, 1008-1042, (1977) · Zbl 0367.14014
[27] Kulikov, V. and Kurchanov, P., Complex algebraic varieties: periods of integrals and Hodge structures, in Algebraic geometry III, (Springer, Berlin, 1998), 1-217. doi:10.1007/978-3-662-03662-4 · Zbl 0881.14003
[28] Lipman, J., Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math., 36, 195-279, (1969) · Zbl 0181.48903
[29] Liu, Q., Algebraic geometry and arithmetic curves, (2002), Oxford University Press · Zbl 0996.14005
[30] Liu, Y. and Zheng, W., Enhanced six operations and base change theorem for Artin stacks, Preprint (2014), arXiv:1211.5948v2.
[31] Madapusi Pera, K., The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math., 201, 625-668, (2015) · Zbl 1329.14079
[32] Matsumoto, Y., Good reduction criterion for K3 surfaces, Math. Z., 279, 241-266, (2015) · Zbl 1317.14089
[33] Matsusaka, T. and Mumford, D., Two fundamental theorems on deformations of polarized varieties, Amer. J. Math.86 (1964), 668-684. doi:10.2307/2373030 · Zbl 0128.15505
[34] Maulik, D., Supersingular K3 surfaces for large primes, Duke Math. J., 163, 2357-2425, (2014) · Zbl 1308.14043
[35] Morrison, D. R., Semistable degenerations of Enriques’ and hyperelliptic surfaces, Duke Math. J., 48, 197-249, (1981) · Zbl 0476.14015
[36] Mumford, D., Abelian varieties, (1970), Tata Institute of Fundamental Research Studies in Mathematics, Oxford University Press · Zbl 0198.25801
[37] Nakayama, C., Degeneration of -adic weight spectral sequences, Amer. J. Math., 122, 721-733, (2000) · Zbl 1033.14012
[38] Nakkajima, Y., Liftings of simple normal crossing log K3 and log Enriques surfaces in mixed characteristics, J. Algebraic Geom., 9, 355-393, (2000) · Zbl 0972.14029
[39] Ochiai, T., l-independence of the trace of monodromy, Math. Ann., 315, 321-340, (1999) · Zbl 0980.14014
[40] Oda, T., A note on ramification of the Galois representation on the fundamental group of an algebraic curve. II, J. Number Theory, 53, 342-355, (1995) · Zbl 0844.14013
[41] Ogus, A., Supersingular K3 crystals, Astérisque, 64, 3-86, (1979) · Zbl 0435.14003
[42] Persson, U., On degenerations of algebraic surfaces, Mem. Amer. Math. Soc., 11, (1977) · Zbl 0368.14008
[43] Persson, U. and Pinkham, H., Degeneration of surfaces with trivial canonical bundle, Ann. of Math. (2)113 (1981), 45-66. doi:10.2307/1971133 · Zbl 0426.14015
[44] Rapoport, M. and Zink, T., Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math.68 (1982), 21-101. doi:10.1007/BF01394268 · Zbl 0498.14010
[45] Rizov, J., Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic, J. Reine Angew. Math., 648, 13-67, (2010) · Zbl 1208.14031
[46] Rudakov, A. N. and Shafarevich, I. R., Inseparable morphisms of algebraic surfaces, Izv. Akad. Nauk SSSR40 (1976), 1269-1307.
[47] Grothendieck, A., Théorie des topos et cohomologie étale des schémas, SGA 4, (1973), Springer: Springer, Berlin
[48] Deligne, P., Cohomologie étale, SGA 4½, (1977), Springer: Springer, Berlin · Zbl 0345.00010
[49] Serre, J.-P. and Tate, J., Good reduction of abelian varieties, Ann. of Math. (2)88 (1968), 492-517. doi:10.2307/1970722 · Zbl 0172.46101
[50] Steenbrink, J. H. M., Limits of Hodge structures, Invent. Math., 31, 229-257, (1976) · Zbl 0303.14002
[51] Van Luijk, R., K3 surfaces with Picard number one and infinitely many rational points, Algebra Number Theory, 1, 1-15, (2007) · Zbl 1123.14022
[52] Washington, L. C., Introduction to cyclotomic fields, , second edition (Springer, New York, 1982). doi:10.1007/978-1-4684-0133-2 · Zbl 0484.12001
[53] Wewers, S., Regularity of quotients by an automorphism of order\(p\), Preprint (2010), arXiv:1001.0607.
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