TY - JOUR

T1 - Degeneration of Kummer surfaces

AU - Overkamp, Otto

N1 - Funding Information: Acknowledgements. The author is very grateful to Professor Alexei Skorobogatov for suggesting that he study degenerations of K3 surfaces, as well as for reading an earlier version of this paper and suggesting various changes which greatly improved the presentation of this paper. He would also like to express his gratitude to Professor Johannes Nicaise for several helpful conversations, and particularly for pointing out to him that something along the lines of Corollary 2·8 might be true. Finally, the author would like to thank the anonymous referee for carefully reading the manuscript and making numerous helpful suggestions. This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], and the EPSRC Centre for Doctoral Training in Geometry and Number Theory (London School of Geometry and Number Theory), University College London.

PY - 2021/7

Y1 - 2021/7

N2 - 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 .,"-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.

AB - 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 .,"-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.

KW - 14G20

KW - 2010 Mathematics Subject Classification:

KW - 2010 Mathematics Subject Classification: 14G20

UR - http://www.scopus.com/inward/record.url?scp=85083366426&partnerID=8YFLogxK

U2 - 10.1017/S0305004120000067

DO - 10.1017/S0305004120000067

M3 - Article

AN - SCOPUS:85083366426

VL - 171

SP - 65

EP - 97

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -