Equidistribution de sous-vari\'et\'es sp\'eciales et o-minimalit\'e: Andr\'e-Oort g\'eom\'etrique

TL;DR

This paper provides a new proof of a characterization of subvarieties in Shimura varieties containing dense sets of weakly special subvarieties, using dynamics on homogeneous spaces and o-minimality.

Contribution

It introduces a novel proof combining ergodic theory and o-minimality, and establishes new homogeneous dynamics results applicable to arithmetic quotients.

Findings

New proof of André-Oort geometric statement

Homogeneous dynamics results on arithmetic quotients

Applications to variations of Hodge structures

Abstract

A characterization of subvarieties of Shimura varieties which contain a Zariski dense subset of weakly special subvarieties has been proved by the second author, by combining o-minimality results and functional transcendence results. In this paper, we obtain a new proof of this statement by dynamics techniques on homogeneous spaces in the spirit of the earlier work of Clozel and the second author. The proof combines ergodic theory \`a la Ratner, with a statement on the dimension of a Hausdorff limit of a sequence of definable subsets (in an o-minimal theory) extracted from a definable family. One obtains in passing general homogeneous dynamics statements valid on arbitrary arithmetic quotients which are of independent interest, that can be applied in the study of variations of Hodge structures and their associated period domains.