Beilinson-Hodge cycles on semiabelian varieties

Beilinson conjectured that all rational cycles of type (q,q) on the qth cohomology of a smooth complex algebraic variety should come from motivic cohomology. The purpose of this note is to prove this when the variety is a semiabelian variety or a product of curves. The proof is based on the study of invariants under the Mumford-Tate group.