## Height pairing between algebraic cycles.(English)Zbl 0624.14005

Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata/Calif. 1985, Contemp. Math. 67, 1-24 (1987).
[For the entire collection see Zbl 0615.00004.]
In the geometric case the author introduces both an intersection product on suitable Chow groups but also a height pairing on corresponding étale cohomology groups. The intersection product and height pairing are related. In the arithmetic case, over a number field, the ideas considered above are introduced by considering local indices over non- archimedean places and over $${\mathbb{R}}$$ or $${\mathbb{C}}$$. The intersection pairing for arithmetic varieties found by H. Gillet and Ch. Soulé does not fit in the given constructions. Numerous conjectures are given among which the following two:
(1) (Swinnerton-Dyer:) The groups $$CH^ 0(X)^ 0$$ are finitely generated and rk CH$${}^ 1(X)^ 0$$ is equal to the order of the zero of $$L(H^{2i-1}(X),s)$$ at $$s=1.$$
(2) (hard Lefschetz:) Let $$\ell \in CH^ i(X)$$ be the class of a hyperplane section. Then for $$i\leq (N+1)/2$$, the arrow $$\ell^{N+1-2i}: CH^ i(X)^ 0\otimes {\mathbb{Q}}\to CH^{N+1-i}(X)^ 0\otimes {\mathbb{Q}}$$ is an isomorphism. Here dim X$$=N+1$$.
Reviewer: P.Cherenack

### MSC:

 14C05 Parametrization (Chow and Hilbert schemes) 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry

Zbl 0615.00004