AbstractGiven $${{\mathbb {V}}}$$
V
a polarizable variation of $${{\mathbb {Z}}}$$
Z
-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$
V
⊗
is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$
V
s
has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$
V
⊗
is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$
V
. Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$
V
is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$
V
whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$
A
g
of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$
A
g
is either a closed algebraic subvariety of S or is Zariski-dense in S.
Hodge Structures on Abelian Varieties of Type III
