Pink’s conjecture on unlikely intersections and families of semi-abelian varieties

Bertrand, Daniel; Edixhoven, Bas

doi:10.5802/jep.126

Pink’s conjecture on unlikely intersections and families of semi-abelian varieties
[Sur la conjecture de Pink sur les intersections exceptionnelles et les familles de variétés semi-abéliennes]

Bertrand, Daniel ¹ ; Edixhoven, Bas ²

¹ Institut de Mathématiques, Sorbonne Université Case 247, 4, Place Jussieu F-75252 Paris Cedex 05, France
² Mathematical Institute, Universiteit Leiden P.O. Box 9512, 2300 RA Leiden, The Netherlands

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 711-742.

Résumé
Abstract

Le torseur de Poincaré d’une famille de Shimura de variétés abéliennes s’interprète à la fois comme une famille de variétés semi-abéliennes et comme une variété de Shimura mixte. Nous montrons que ses sous-variétés spéciales en ce deuxième sens ne peuvent pas toutes se décrire en termes de sous-schémas en groupes. Cela donne un contre-exemple à la conjecture de Manin-Mumford relative, mais témoigne aussi de la pertinence de la conjecture de Pink sur les intersections exceptionnelles dans les variétés de Shimura mixtes. L’essentiel de l’article porte sur les structures de Hodge mixtes, mais d’autres approches, de nature plus géométrique, sont aussi abordées.

The Poincaré torsor of a Shimura family of abelian varieties can be viewed both as a family of semi-abelian varieties and as a mixed Shimura variety. We show that the special subvarieties of the latter cannot all be described in terms of the subgroup schemes of the former. This provides a counter-example to the relative Manin-Mumford conjecture, but also some evidence in favour of Pink’s conjecture on unlikely intersections in mixed Shimura varieties. The main part of the article concerns mixed Hodge structures and the uniformisation of the Poincaré torsor, but other, more geometric, approaches are also discussed.

Reçu le : 2019-11-23
Accepté le : 2020-03-29
Publié le : 2020-04-09

DOI : 10.5802/jep.126

Classification : 14K05, 14G35, 11G15, 14K30
Keywords: Semi-abelian varieties, Poincaré biextensions, mixed Shimura varieties, Manin-Mumford conjecture, André-Oort conjecture, Zilber-Pink conjecture
Mot clés : Variétés semi-abéliennes, bi-extensions de Poincaré, variétés de Shimura mixtes, conjecture de Manin-Mumford, conjecture d’André-Oort, conjecture de Zilber-Pink

Affiliations des auteurs :

Bertrand, Daniel ¹ ; Edixhoven, Bas ²

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Bertrand, Daniel; Edixhoven, Bas. Pink’s conjecture on unlikely intersections and families of semi-abelian varieties. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 711-742. doi : 10.5802/jep.126. http://www.numdam.org/articles/10.5802/jep.126/

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