Tame topology of arithmetic quotients and algebraicity of Hodge loci

Bruno Klingler

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arXiv · 1803.09384

Tame topology of arithmetic quotients and algebraicity of Hodge loci

Bruno Klingler

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Abstract

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).

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Bruno Klingler. 2018-03-26. Tame topology of arithmetic quotients and algebraicity of Hodge loci. https://arxiv.org/abs/1803.09384

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