Une caractérisation des sous-variétés des variétés de Shimura qui contiennent un sous ensemble Zariski-dense de sous-variétés faiblement spéciales a été obtenue par le second auteur, il y a quelques années, en combinant des résultats d’-minimalité et des résultats de transcendance fonctionnelle. Nous obtenons dans ce texte une nouvelle preuve de cet énoncé par des techniques de dynamique sur les espaces homogènes dans l’esprit de travaux antérieurs de Clozel et du second auteur. La preuve utilise de la théorie ergodique à la Ratner, à la Mozes-Shah complétée par un résultat récent de Daw-Gorodnik et du second auteur. On obtient au passage des énoncés de dynamique homogène généraux valables sur des quotients arithmétiques arbitraires qui sont d’un intérêt indépendant, s’appliquant par exemple à l’étude des variations de structures de Hodge.
A characterization of subvarieties of Shimura varieties which contain a Zariski-dense subset of weakly special subvarieties was obtained by the second author a few years ago by combining -minimality and functional transcendence results. In this paper we obtain a new proof of this statement using dynamics techniques on homogeneous spaces in the spirit of earlier work by Clozel and the second author. The proof uses ergodic theory à la Ratner and a recent result by Daw-Gorodnik and the second author. In the process, general homogeneous dynamics statements valid on arbitrary arithmetic quotients are obtained which are of independent interest, applicable for example to the study of variations of Hodge structures.
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DOI : 10.5802/aif.3644
Mots-clés : Variétés de Shimura, Variations de structures de Hodge, André-Oort, $o$-minimalité, Théorèmes de Ratner
Keywords: Shimura varieties, variations of Hodge structures, André-Oort, $o$-minimality, Ratner’s theorems
Richard, Rodolphe 1 ; Ullmo, Emmanuel 2
CC-BY-ND 4.0
@article{AIF_2024__74_6_2667_0,
author = {Richard, Rodolphe and Ullmo, Emmanuel},
title = {\'Equidistribution de sous-vari\'et\'es faiblement sp\'eciales et $o$-minimalit\'e~: {Andr\'e-Oort} g\'eom\'etrique},
journal = {Annales de l'Institut Fourier},
pages = {2667--2721},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {74},
number = {6},
year = {2024},
doi = {10.5802/aif.3644},
mrnumber = {4810247},
zbl = {07929016},
language = {fr},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3644/}
}
TY - JOUR AU - Richard, Rodolphe AU - Ullmo, Emmanuel TI - Équidistribution de sous-variétés faiblement spéciales et $o$-minimalité : André-Oort géométrique JO - Annales de l'Institut Fourier PY - 2024 SP - 2667 EP - 2721 VL - 74 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3644/ DO - 10.5802/aif.3644 LA - fr ID - AIF_2024__74_6_2667_0 ER -
%0 Journal Article %A Richard, Rodolphe %A Ullmo, Emmanuel %T Équidistribution de sous-variétés faiblement spéciales et $o$-minimalité : André-Oort géométrique %J Annales de l'Institut Fourier %D 2024 %P 2667-2721 %V 74 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3644/ %R 10.5802/aif.3644 %G fr %F AIF_2024__74_6_2667_0
Richard, Rodolphe; Ullmo, Emmanuel. Équidistribution de sous-variétés faiblement spéciales et $o$-minimalité : André-Oort géométrique. Annales de l'Institut Fourier, Tome 74 (2024) no. 6, pp. 2667-2721. doi : 10.5802/aif.3644. https://aif.centre-mersenne.org/articles/10.5802/aif.3644/
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