Abelianness of Mumford-Tate groups associated to some unitary groups

Abstract

In this paper, we investigate the action of the Q-cohomology of the compact dual X^ of a compact Shimura Variety S(Γ) on the Q-cohomology of S(Γ) under a cup product. We use this to split the cohomology of S(Γ) into a direct sum of (not necessarily irreducible) Q-Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups U(p,q) arising from Hermitian forms over CM fields, the Mumford-Tate groups associated to certain holomorphic cohomology classes on S(Γ) are Abelian. As another application, we show that all classes of Hodge type (1,1) in H² of unitary four-folds associated to the group U(2,2) are algebraic.