Degeneration of Kummer surfaces

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.

On The Two Types Of Affine Structures For Degenerating Kummer Surfaces -Non-Archimedean VS Gromov-Hausdorff Limits-

“…However, we do not consider the quotient X under this setting (5.2) in this paper. It is because we are not sure that an analog of what Overkamp proved on Kummer surfaces in [Ove21] also works. Proposition 5.5.…”

“…To construct them, we shall use the category C introduced by [Kün98, §3] (cf. [Ove21]). Objects of the category C are tuples (M, L, φ, a, b), where M and L are free Abelian groups of the same finite rank, φ : L → M is an injective homomorphism, a : L → Z is a function with a(0) = 0, and b : L×M → Z is a bilinear pairing such that b(−, φ(−)) is symmetric, positive definite, and satisfies a(l…”

Motivic zeta function of the Hilbert schemes of points on a surface

“…• Assume that X is a Kummer surface constructed from an abelian surface A; then Overkamp proved in [18] that Z X,ω has a unique pole.…”

“…The monodromy conjecture has been proven in several classes of varieties: Halle and Nicaise proved it for Abelian varieties, [10], while Jaspers proved it in [12] when X is a K3 surfaces admitting a Crauder-Morrison model and Overkamp proved it for Kummer K3 surfaces in [18]. Yet we do not know whether all the K3 surfaces satisfy the Monodromy conjecture.…”

Reduction of Kummer surfaces modulo 2 in the non-supersingular case

“…We can then form the quotient A/ι, and blowup the singular subscheme to obtain a smooth model of Kum(A) over O K , cf. [Ove21,Proposition 3.11].…”