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Degeneration of Kummer surfaces

Part of: Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  16 April 2020

OTTO OVERKAMP
OTTO OVERKAMP*
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, 30167Hannover e-mail: otto.overkamp@googlemail.com
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Abstract

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle–Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

MSC classification

Primary: 14G20: Local ground fields
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 65 - 97
Copyright
© Cambridge Philosophical Society 2020

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References

Bădescu, L. S. Algebraic Surfaces (Universitext. Springer-Verlag, New York, 2001).CrossRefGoogle Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models. Ergeb. Math. Grenzgeb. (3), (Springer-Verlag, 1990).Google Scholar
Breuil, C., Groupes p-divisibles, groupes finis et modules filtrés. Ann. of Math. 152 (2000), pp. 489–549.CrossRefGoogle Scholar
Chiarellotto, B. and Lazda, C., Combinatorial degenerations of surfaces and Calabi–Yau threefolds. Algebra and Number Theory 10(10), (2016).CrossRefGoogle Scholar
Coleman, R. and Iovita, A., The Frobenius and monodromy operator for curves and Abelian varieties. Duke Math. J., 97, No. 1, (1999).10.1215/S0012-7094-99-09708-9CrossRefGoogle Scholar
Faltings, G. and Chai, C.-L., Degeneration of Abelian varieties. Ergeb. Math. Grenzgeb. (3), (Springer-Verlag, 1990).CrossRefGoogle Scholar
Fontaine, J.-M., Le corps des périodes p-adiques. Périodes p-adiques (Séminaire de Bures, 1988), Asterisque, 223 (1994), pp. 59–101.Google Scholar
Grothendieck, A., Le groupe de Brauer III. Dix exposes sur la cohomologie des schémas. Grothendieck, A., Kuiper, N. H. eds. (North-Holland Publishing Company, 1968).Google Scholar
Grothendieck, A., Elements du géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie. Publ. Math. IHES, 11 (1961), pp. 5–167.Google Scholar
Grothendieck, A., Deligne, P., Katz, N., with Raynaud, M., and Rim, D. S., Groupes de monodromie en géométrie algébrique. Lecture Notes in Math. 269, 270, 305 (1972–73).Google Scholar
Hartshorne, R., Stable reflexive sheaves. Math. Ann. 254, (1980), pp. 121–176.CrossRefGoogle Scholar
Halle, L. H. and Nicaise, J., Motivic zeta functions of degenerating Calabi–Yau varieties. Math. Ann. 370, (2018), pp. 1277–1320.10.1007/s00208-017-1578-3CrossRefGoogle Scholar
Huybrechts, D., Lectures on K3 surfaces. Camb. Stud. Adv. Math (158), (Cambridge University Press, 2016).CrossRefGoogle Scholar
Ito, K., Unconditional construction of K3 surfaces over finite fields with given L-function in large characteristic. Preprint; available at https://arxiv.org/pdf/1612.05382.pdf.Google Scholar
Kato, K., Semistable reduction and p-adic étale cohomology. Périodes p-adiques (Séminaire de Bures, 1988), Asterisque, 223, (1994), pp. 269–293.Google Scholar
Kleiman, S., Algebraic Cycles and the Weil conjectures. Dix exposes sur la cohomologie des schémas. Grothendieck, A., Kuiper, N. H. eds. (North-Holland Publishing Company, 1968).Google Scholar
Kulikov, V. S., Degenerations of K3 surfaces and Enriques surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 5, pp.1008–1042 Google Scholar
Künnemann, K., Projective regular models for Abelian varieties, semistable reduction, and the height pairing. Duke Math. J. 95, No. 1, (1998).10.1215/S0012-7094-98-09505-9CrossRefGoogle Scholar
Liedtke, C., Matsumoto, Y., Good reduction of K3 surfaces. Compositio Math. 154 (2018), pp. 1–35.10.1112/S0010437X17007400CrossRefGoogle Scholar
Loerke, K., Reduction of Abelian varieties and Grothendieck’s pairing. Preprint, (2009).Google Scholar
Matsumoto, Y., On good reduction of some K3 surfaces related to Abelian surfaces. Tohoku Math. J. 67 (2015), pp. 83–104 10.2748/tmj/1429549580CrossRefGoogle Scholar
Matsumoto, Y., A good reduction criterion for K3 surfaces. Math. Z. 279 (2015), pp. 241–266.10.1007/s00209-014-1365-8CrossRefGoogle Scholar
Maulik, D., Supersingular K3 surfaces for large primes. Duke Math. J. 163, No. 13, (2014).10.1215/00127094-2804783CrossRefGoogle Scholar
Mumford, D., An analytic construction of degenerating Abelian varieties over complete rings. Cs. Math. 24, Fasc. 3 (1972), pp. 239–272.Google Scholar
Nakayama, C., Nearby cycles for log smooth families. Cs. Math. 112 (1998), pp. 45–75.Google Scholar
Persson, U., On degenerations of algebraic surfaces. Mem. Amer. Math. Soc. 11 (1977), no. 189.Google Scholar
Persson, U. and Pinkham, H., Degenerations of surfaces with trivial canonical bundle. Ann. of Math. 113 (1981), pp. 45–66.CrossRefGoogle Scholar
Rapoport, M. and Zink, T., Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math. 68, pp. 21–101, (1982).CrossRefGoogle Scholar
Raynaud, M., Faisceaux amples sur les schémas en groupes at les espaces homogènes. Lecture Notes in Math. 119, (Springer-Verlag, 1970).Google Scholar
Saito, T., Log smooth extension of a family of curves and semi-stable reduction. J. Alg. Geom. 13 (2004), pp. 287–321.CrossRefGoogle Scholar
Skorobogatov, A. and Zarhin, Y., Kummer varieties and their Brauer groups. To appear in Pure Appl. Math. Quart.Google Scholar
Tsuji, T., p-adic étale and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137, (1999), pp. 233–411.CrossRefGoogle Scholar