Geodesic
Special Subvarieties in Mumford-Tate Varieties
Documenta mathematica, Tome 24 (2019), pp. 523-544
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Let X=Γ\D be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D=G(R)/V by a discrete subgroup Γ. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y⊂X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves Ci​ in Y, and testing whether the inclusion ⋃i​Ci​⊂Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties X. In this way, we give necessary and sufficient criteria for a subvariety Y of X to be a special subvariety of Shimura type in the sense of the André-Oort conjecture. We discuss in detail the important case where X=Ag​, the moduli space of principally polarized abelian varieties.
DOI : 10.4171/dm/687
Classification : 14G35
Mots-clés : Shimura variety, Higgs bundle, André-Oort conjecture, period domain 
@article{10_4171_dm_687,
     author = {Abolfazl Mohajer and Stefan M\"uller-Stach and Kang Zuo},
     title = {Special {Subvarieties} in {Mumford-Tate} {Varieties}},
     journal = {Documenta mathematica},
     pages = {523--544},
     year = {2019},
     volume = {24},
     doi = {10.4171/dm/687},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/687/}
}
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Abolfazl Mohajer; Stefan Müller-Stach; Kang Zuo. Special Subvarieties in Mumford-Tate Varieties. Documenta mathematica, Tome 24 (2019), pp. 523-544. doi: 10.4171/dm/687

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