Abstract
In this paper, we investigate the action of the ℚ-cohomology of the compact dual \(\widehat X\) 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.
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Venkataramana, T.N. Abelianness of Mumford–Tate Groups Associated to Some Unitary Groups. Compositio Mathematica 122, 223–242 (2000). https://doi.org/10.1023/A:1002011002167
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DOI: https://doi.org/10.1023/A:1002011002167