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In this paper we prove, assuming the Generalized Riemann Hypothesis, the André-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH.

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.

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