## Height pairing between algebraic cycles.(English)Zbl 0651.14002

$$K$$-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86, Lect. Notes Math. 1289, 1-26 (1987).
[For the entire collection see Zbl 0621.00010.]
Let K be a number field and X a smooth N-dimensional projective variety over K. The paper defines a height pairing between the subgroups CH i(X) 0 and $$CH^{N+1-i}(X)$$ 0 of the Chow groups of cycles homologous to zero in $$\ell$$-adic cohomology. The definition assumes two conjectures which have been verified in the good reduction case and for cycles algebraically equivalent to zero. The pairing is defined via local pairings.
The final section lists important conjectures and problems concerned with arithmetic matters, generalization of Tate-Néron height, K-theory and algebraic cycles, and Hodge cohomology.
Reviewer: G.Horrocks

### MSC:

 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14C05 Parametrization (Chow and Hilbert schemes)

Zbl 0621.00010