Let be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain by a discrete subgroup . Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety (of Shimura type), and formulate necessary criteria for to be special. Our method consists in looking at finitely many compactified special curves in , and testing whether the inclusion 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 . In this way, we give necessary and sufficient criteria for a subvariety of to be a special subvariety of Shimura type in the sense of the André-Oort conjecture. We discuss in detail the important case where , the moduli space of principally polarized abelian varieties.
Cite this article
Abolfazl Mohajer, Stefan Müller-Stach, Kang Zuo, Special Subvarieties in Mumford-Tate Varieties. Doc. Math. 24 (2019), pp. 523–544DOI 10.4171/DM/687