Beilinson's Hodge conjecture for smooth varieties

Consider the cycle class map cl_{r,m} : CH^r(U,m;\Q) \to ΓH^{2r-m}(U,\Q(r)), where CH^r(U,m;\Q) is Bloch's higher Chow group (tensored with \Q) of a smooth complex quasi-projective variety U, and H^{2r-m}(U,\Q(r)) is singular cohomology. We study the image of cl_{r,m} in terms of kernels of Abel-Jacobi maps. When r=m, we deduce from the Bloch-Kato theorem that the cokernel of cl_{r,m} at the generic point of U is the same for integral or rational coefficients.

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