Beilinson's Hodge conjecture for smooth varieties

Let U /ℂ be a smooth quasi-projective variety of dimension d , CH r ( U,m ) Bloch's higher Chow group, and cl r,m : CH r ( U,m ) ⊗ ℚ → hom MHS (ℚ(0), H 2 r−m ( U , ℚ( r ))) the cycle class map. Beilinson once conjectured cl r,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of cl r,m in more detail (as well as at the “generic point” of U ) in terms of kernels of Abel-Jacobi mappings. When r = m , we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of cl m,m at the generic point is the same for integral or rational coefficients.