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.
Révisé le :
Accepté le :
Publié le :
Mot 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},
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/
[1] Mumford–Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math., Volume 82 (1992) no. 1, pp. 1-24 | Numdam | Zbl
[2] Tame topology of arithmetic quotients and algebraicity of the Hodge loci, J. Am. Math. Soc., Volume 33 (2020) no. 4, pp. 917-939 | DOI | Zbl
[3] Measure theory. Vol. I, II, Springer, 2007, xviii+500 et xiv+575 pages | DOI
[4] Weak convergence of measures, Mathematical Surveys and Monographs, 234, American Mathematical Society, 2018, xii+286 pages | DOI
[5] Introduction aux groupes arithmétiques, Actualités Scientifiques et Industrielles, 1341, Hermann, 1969, 125 pages
[6] Linear Algebraic Groups, Graduate Texts in Mathematics, 126, Springer, 1991
[7] Arithmetic subgroups of algebraic groups, Ann. Math., Volume 75 (1962), pp. 485-535 | DOI | Zbl
[8] Compactifications of symmetric and locally symmetric spaces, Mathematics : Theory & Applications, Birkhäuser, 2006, xvi+479 pages
[9] Elements of Mathematics. Integration. II. Chapters 7–9, Springer, 2004, viii+326 pages (translated from the 1963 and 1969 French originals by Sterling K. Berberian)
[10] On the locus of Hodge classes, J. Am. Math. Soc., Volume 8 (1995) no. 2, pp. 483-506 | DOI | Zbl
[11] On the geometric André–Oort conjecture for variations of Hodge structures, J. Reine Angew. Math., Volume 776 (2021), pp. 295-317 | DOI | Zbl
[12] Équidistribution de sous-variétés spéciales, Ann. Math., Volume 161 (2005) no. 3, pp. 1571-1588 | DOI | Zbl
[13] An introduction to o-minimal geometry (1999) (accessible en ligne à https://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf)
[14] Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci., Math. Sci., Volume 101 (1991) no. 1, pp. 1-17 | DOI | Zbl
[15] Convergence of measures on compactifications of locally symmetric spaces, Math Zeitchrift, Volume 297 (2021) no. 3-4, pp. 1293-1328 | Zbl
[16] The space of homogeneous probability measures on is compact, Math. Ann. (2022) | DOI | Zbl
[17] Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, Cambridge University Press, 1998, x+180 pages | DOI
[18] Limit sets in o-minimal structures, O-minimal Structures : Lisbon 2003 ; Proceedings of a Summer School by the European Research and Training Network RAAG (2005)
[19] Subvarieties of Shimura varieties, Ann. Math., Volume 157 (2003) no. 2, pp. 621-645 | DOI | Zbl
[20] Urysohn–Brouwer lemma (https://www.encyclopediaofmath.org/index.php title=Urysohn-Brouwer_lemma&oldid=23095)
[21] Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 48-80 | DOI | Zbl
[22] The shaping of the Riesz representation theorem : a chapter in the history of analysis, Arch. Hist. Exact Sci., Volume 31 (1984) no. 2, pp. 127-187 | DOI | Zbl
[23] Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math., Inst. Hautes Étud. Sci., Volume 38 (1970), pp. 125-180 | DOI | Numdam | Zbl
[24] Hodge locus and atypical intersections : conjectures (à paraitre dans Motives and complex multiplication)
[25] On the closure of the Hodge locus of positive period dimension, Invent. Math., Volume 225 (2021) no. 3, pp. 857-883 | DOI | Zbl
[26] The hyperbolic Ax–Lindemann–Weierstrass conjecture, Publ. Math., Inst. Hautes Étud. Sci., Volume 123 (2016), pp. 333-360 | DOI | Numdam | Zbl
[27] A geometric proof of the definability of Hausdorff limits, Sel. Math., New Ser., Volume 10 (2004) no. 3, pp. 377-390 | DOI
[28] Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 17, Springer, 1991, x+388 pages | DOI
[29] Linearity properties of Shimura varieties. I, J. Algebr. Geom., Volume 7 (1998) no. 3, pp. 539-567 | Zbl
[30] Ratner’s theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, 2005, xii+203 pages
[31] On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dyn. Syst., Volume 15 (1995) no. 1, pp. 149-159 | DOI | Zbl
[32] The Andre–Oort conjecture for the moduli space of Abelian surfaces, Compos. Math., Volume 149 (2013) no. 2, pp. 204-216 | DOI | Zbl
[33] Algebraic groups and number theory, Pure and Applied Mathematics, 139, Academic Press Inc., 1994, xii+614 pages (translated from the 1991 Russian original by Rachel Rowen)
[34] Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer, 1972, ix+227 pages | DOI
[35] On Raghunathan’s measure conjecture, Ann. Math., Volume 134 (1991) no. 3, pp. 545-607 | DOI | Zbl
[36] Real and complex analysis, McGraw-Hill, 1966, xi+412 pages
[37] Equidistribution de sous-variétés spéciales. II, J. Reine Angew. Math., Volume 606 (2007), pp. 193-216 | Zbl
[38] Applications du théorème d’Ax–Lindemann hyperbolique, Compos. Math., Volume 150 (2014) no. 2, pp. 175-190 | DOI | Zbl
[39] Hyperbolic Ax–Lindemann theorem in the cocompact case, Duke Math. J., Volume 163 (2014) no. 2, pp. 433-463 | Zbl
[40] Algebraic Flows on Shimura Varieties, Manuscr. Math., Volume 155 (2018) no. 3-4, pp. 355-367 | DOI | Zbl
[41] Topology of definable Hausdorff limits, Discrete Comput. Geom., Volume 33 (2005) no. 3, pp. 423-443 | DOI | Zbl
Cité par Sources :
