Tame topology of arithmetic quotients and algebraicity of Hodge loci

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).

23 pages

References in corpus (1)

• Height bounds and the Siegel property

Cited by in corpus (2)

• Model Theory of Adeles and Number Theory
• On the geometric André-Oort conjecture for variations of Hodge structures