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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: [email protected]
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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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