Tame topology of arithmetic quotients and algebraicity of Hodge loci

TL;DR

This paper establishes a semi-algebraic structure on arithmetic quotients, proves the definability of period maps in o-minimal structures, and simplifies the proof that Hodge loci are algebraic, advancing understanding of Hodge theory and algebraic geometry.

Contribution

It introduces a natural semi-algebraic structure on arithmetic quotients and simplifies the proof of algebraicity of Hodge loci without relying on complex orbit theorems.

Findings

Arithmetic quotients admit semi-algebraic structures

Period maps are definable in o-minimal structures

Hodge loci are countable unions of algebraic subvarieties

Abstract

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. $3)$ As a corollary of $2)$ and of Peterzil-Starchenko's o-minimal Chow theorem we recover 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. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable $SL_2$-orbit theorem of Cattani-Kaplan-Schmid.