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Equidistribution of Hodge loci II

Part of: Arithmetic algebraic geometry Families, fibrations Locally compact groups and their algebras

Published online by Cambridge University Press:  12 January 2023

Salim Tayou  and
Nicolas Tholozan
Salim Tayou
Affiliation:
Department of Mathematics, Harvard University, 1 Oxford St, Cambridge, MA 02138, USA tayou@math.harvard.edu
Nicolas Tholozan
Affiliation:
DMA – UMR8553, École Normale Supérieure, CNRS – PSL Research University, 45 rue d'Ulm, 75230, Paris Cedex 5, France nicolas.tholozan@ens.fr
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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.

Keywords

Hodge locusequidistributionLie groupshomogeneous spaces

MSC classification

Primary: 14D07: Variation of Hodge structures
Secondary: 22D40: Ergodic theory on groups 11G15: Complex multiplication and moduli of abelian varieties
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 1 , January 2023 , pp. 1 - 52
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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