Geodesic
Special Subvarieties in Mumford-Tate Varieties
Documenta mathematica, Tome 24 (2019), pp. 523-544.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $X=\Gamma \backslash D$ be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain $D=G(\mathcal{R})/V$ by a discrete subgroup $\Gamma$. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety $Y \subset X$ (of Shimura type), and formulate necessary criteria for $Y$ to be special. Our method consists in looking at finitely many compactified special curves $C_i$ in $Y$, and testing whether the inclusion $\bigcup_i C_i \subset Y$ 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 $X=A_g$, the moduli space of principally polarized abelian varieties.
Classification : 14G35
Keywords: André-Oort conjecture, period domain, Shimura variety, Higgs bundle 
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     author = {Mohajer, Abolfazl and M\"uller-Stach, Stefan and Zuo, Kang},
     title = {Special {Subvarieties} in {Mumford-Tate} {Varieties}},
     journal = {Documenta mathematica},
     pages = {523--544},
     publisher = {mathdoc},
     volume = {24},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a48/}
}
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Mohajer, Abolfazl; Müller-Stach, Stefan; Zuo, Kang. Special Subvarieties in Mumford-Tate Varieties. Documenta mathematica, Tome 24 (2019), pp. 523-544. http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a48/