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.

中文翻译：

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Kummer 表面退化

我们证明了在完全严格 Henselian 离散值域上定义的 Kummer 曲面ķ与 2 不同的残差特征在有限基数变化后承认严格的 Kulikov 模型。我们构建的 Kulikov 模型将是方案，因此我们的结果表明，即使在方案类别中，半稳态归约猜想对于该设置中的 Kummer 曲面也是正确的。我们构建的 Kulikov 模型与早期构建的 Künnemann 密切相关，后者产生了阿贝尔变体的半稳定模型。众所周知，严格库利科夫模型的特殊纤维属于三种类型之一，我们将证明库默曲面的严格库利科夫模型的特殊纤维的类型和对应的阿贝尔曲面的复曲面秩为由彼此决定。我们还研究了这个不变量与第二个上的伽罗瓦表示之间的关系ℓ- Kummer 曲面的进数上同调。最后，我们应用我们的结果，连同 Halle-Nicaise 的早期工作，来证明 Kummer 曲面在等特征零上的单调猜想。