We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$ (a result originally due to Cattani-Deligne-Kaplan).

中文翻译：

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Hodge 轨迹的算术商和代数性的驯服拓扑

我们证明了任何算术商的统一映射，以及与任何纯极化 $\mathbb{Z}$-Hodge 结构 $\mathbb{V}$ 在平滑复拟射影变体 $ 上的变体相关的周期映射S$，在拓扑上是温和的。作为这些结果和 Peterzil-Starchenko 的 o-minimal GAGA 定理的一个简单推论，我们得到 $(S, \mathbb{V})$ 的 Hodge 轨迹是 $S$ 的代数子变体的可数并集（结果最初由于卡塔尼-德利涅-卡普兰）。