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, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, 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 $q$ 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 appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

On the distribution of the Hodge locus

“…16 We do not address here the question of whether the typical Hodge locus is equidistributed in level 1 or 2. This interesting question has been investigated in more details for the typical Hodge locus of zero period dimension by Tayou and Tholozan in [55,56]. In particular Theorem 3.14 is also a corollary of their work.…”

Existence and density of typical Hodge loci

“…Combining recent work of Tayou-Tholozan and the density of HLpS, V b q typ in S that we prove here, one should be able to obtain that this locus is even equidistributed in S for a natural measure. We refer the interested reader to their paper [21].…”