  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.
