Equidistribution of Hodge loci II

Abstract

Let $\mathbb V$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, \emph{the pull-push form}. In particular, it is always analytically dense when the pull-push form does not vanish. When the weight is $2$, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$\textsuperscript{th} Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal{A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal A_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull-push form appear in this greater generality and we provide several tools to determine it and we compute it in many examples.