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Higher Picard varieties and the height pairing
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 118, Number 4, August 1996
- pp. 781-797
- 10.1353/ajm.1996.0033
- Article
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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.