Tame topology of arithmetic quotients and algebraicity of Hodge loci

TL;DR

This paper proves that the uniformizing and period maps of arithmetic quotients and Hodge structures are topologically tame, leading to the algebraicity of Hodge loci as countable unions of algebraic subvarieties.

Contribution

It establishes the topological tameness of uniformizing and period maps, and derives the algebraicity of Hodge loci using o-minimal GAGA techniques.

Findings

Uniformizing maps are topologically tame.

Period maps are topologically tame.

Hodge loci are countable unions of algebraic subvarieties.

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).