Higher Picard varieties and the height pairing

Let X be a smooth projective variety which is defined over a number field. Beilinson and Bloch have defined under suitable asssumptions height pairings between Chow groups of homologically trivial cycles on X. Beilinson has also formulated a hard Lefschetz and a Hodge index conjecture for these Chow groups. We show that the restriction of the height pairing to cycles algebraically equivalent to zero can be computed via Abel-Jacobi maps in terms of the Néron-Tate height pairing on the higher Picard varieties of X. This description is used in the case where X is an abelian variety to prove a consequence of Beilinson conjectures. Namely, we prove a hard Lefschetz and a Hodge index theorem for the groups of cycles algebraically equivalent to zero modulo incidence equivalence.