In this paper we prove the following results:
1
)
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
)
2)
We prove that the period map associated to any pure polarized variation of integral Hodge structures
V
\mathbb {V}
on a smooth complex quasi-projective variety
S
S
is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure.
3
)
3)
As a corollary of
2
)
2)
and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of
(
S
,
V
)
(S, \mathbb {V})
is a countable union of algebraic subvarieties of
S
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
S
L
2
SL_2
-orbit theorem of Cattani-Kaplan-Schmid.