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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

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