刚性空间的驯服基本群
Tame fundamental groups of rigid spaces
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中文总结 AI 辅助
该研究定义了刚性空间的驯服平展基本群，证明其在特定条件下拓扑有限生成或有限表现，为相关几何研究提供了关键有限性结论。
中文摘要 AI 辅助
我们引入了非阿基米德域K上刚性空间X的驯服平展基本群π₁ᵗ(X/K)。证明了若X为拟紧拟分离（qcqs）且K具有拓扑有限生成的驯服伽罗瓦群（例如代数闭域或局部域），则π₁ᵗ(X/K)拓扑有限生成；若X还是严格半稳定形式概形的刚性一般纤维，且其特殊纤维的光滑轨迹存在射影正常交叉（snc）紧化，则π₁ᵗ(X/K)拓扑有限表现。证明依赖于对数几何技术（扩展至有限生成幺半群的常规范围之外），尤其涉及驯服对数平展基本群的类似有限性结论，以及仿射空间映射的“垂直紧化”方法。
英文摘要
We introduce the tame étale fundamental group $π_1^t(X/K)$ of a rigid space X over a non-archimedean field K. We show that if X is qcqs and K has topologically finitely generated tame Galois group (e.g. algebraically closed or a local field), then $π_1^t(X/K)$ is topologically finitely generated. If X is moreover the rigid generic fibre of a strictly semistable formal scheme such that the smooth locus of its special fibre admits a projective snc compactification, then $π_1^t(X/K)$ is topologically finitely presented. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids), in particular on an analogous finiteness statement for the tame log étale fundamental group, and on the 'vertical compactification' of a map of adic spaces.
