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

Rodolphe Richard, Emmanuel Ullmo with an appendix with Jiaming Chen

Publication date: 9 April 2021

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.

Mathematics Subject Classification ID

Structure of modular groups and generalizations; arithmetic groups (11F06) Variation of Hodge structures (algebro-geometric aspects) (14D07) Arithmetic problems in algebraic geometry; Diophantine geometry (14Gxx) Model theory of ordered structures; o-minimality (03C64)

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