On the distribution of the Hodge locus

Abstract: Given a polarizable $\mathbb{Z}$-variation of Hodge structures $\mathbb{V}$ over a complex smooth quasi-projective base $S$, a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$, called the special subvarieties for $\mathbb{V}$. Our main result in this paper is that, if the level of $\mathbb{V}$ is at least $3$, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension. For instance the Hodge locus of positive period dimension of the universal family of degree $d$ smooth hypersurfaces in $\mathbf{P}^{n+1}_\mathbb{C}$, $n\geq 3, d\geq 5$ and $(n,d)\neq (4,5)$, is algebraic. On the other hand we prove that in level $1$ or $2$, the Hodge locus is analytically dense in $S^{an}$ as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.