Rob De Jeu, James D. Lewis, Masanori Asakura
Journal of K-Theory 11 (2) 243 - 282 1865-2433 2013/04
[Refereed][Not invited] Let U/C be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, and cl r,m: CHr (U,m) âŠ-â"š → homMHS (â"š(0), H 2r-m (U, â"š(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. © 2013 ISOPP.