  Let U be a smooth quasiprojective complex variety and CH^r(U,1) a special
instance of Bloch's higher Chow groups. Jannsen was the first to show that the
cycle class map cl_{r,1} from CH^r(U,1) (tensored with Q) to hom_{MHS}(Q(0),
H^{2r-1}(U,Q(r)) is not in general surjective, contradicting an earlier
conjecture of Beilinson. In this paper, we give a refinement of Jannsen's
counterexample, and further show that the aforementioned cycle class map
becomes surjective at the generic point.
