Equidistribution of Hodge loci II

TL;DR

This paper characterizes the distribution of Hodge loci in variations of Hodge structure, showing they are either empty or equidistributed with respect to a natural form, with applications to Shimura varieties and homogeneous dynamics.

Contribution

It provides a complete description of the typical Hodge locus, establishing its equidistribution properties and connecting Hodge theory with homogeneous dynamics and Lie group actions.

Findings

Hodge loci are either empty or equidistributed with respect to the pull-push form.

In weight 2, the locus with Picard rank at least r is equidistributed with volume form c_q^r.

Results extend to Shimura varieties, including density criteria and distribution of CM points and Hecke translates.

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.