Degeneration of Kummer surfaces

TL;DR

This paper proves that Kummer surfaces over certain valued fields admit strict Kulikov models after finite base change, confirming the semistable reduction conjecture for these surfaces and exploring their geometric and Galois properties.

Contribution

It constructs strict Kulikov models for Kummer surfaces over complete discretely valued fields, establishing the semistable reduction conjecture in this context and relating special fibre types to Abelian surface invariants.

Findings

Existence of strict Kulikov models after finite base change.

Relationship between special fibre type and Abelian surface toric rank.

Proof of the monodromy conjecture for Kummer surfaces in characteristic zero.

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 K\"unnemann, 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 $\ell$-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.