Abelianness of Mumford–Tate Groups Associated to Some Unitary Groups

Abstract

In this paper, we investigate the action of the ${\Bbb Q}$-cohomology of the compact dual $\widehat {X}$ of a compact Shimura Variety $S(Γ)$ on the ${\Bbb Q}$-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) ${{\Bbb Q}}$-Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups ${\rm U}(p,q)$ 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 ${\rm U}(2,2)$ are algebraic.