### Higher Picard varieties and the height pairing, by Klaus Kuennemann

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
Neron-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.

This paper has appeared in American Journal of Mathematics 118 (1996) 781-797.

Klaus Kuennemann <kuenne@math.uni-muenster.de>