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Height pairings for algebraic cycles. (English) Zbl 0577.14004
Let X be a smooth projective variety of dimension d defined over a number field k. For $$p\leq d$$, let $$A^ p(X)$$ (resp. $$A_ p(X))$$ denote the group of algebraic cycles of codimension p (resp. dimension p) on X defined over k, cohomologous to 0 in $$H^{2p}(X_{\bar k},{\mathbb{Q}}_{\ell})$$ (resp. $$H^{2d-2p})$$, taken modulo rational equivalence. Let q be a geometric generic point. Let $$a\in K_ 0(X)_{{\mathbb{Q}}}$$ such that the Chern character $$ch_ q(a)=0$$. Assume that there exists $$a_ 0\in K_ 0(X)_{{\mathbb{Q}}}$$ lifting a such that $$ch_ p(a_ 0)=0$$ for all geometric points p of Sp $${\mathcal O}_ k$$. Let $$A^{'p}$$ be the subgroup of $$A^ p$$ generated by classes which can be lifted in such a way. The global height pairing $$<\cdot,\cdot >$$ is defined as a sum of local pairings $$<\cdot,\cdot >_ v$$, namely, $$[k:{\mathbb{Q}}]<a,b>=\sum_{v}<a,b>_ v.$$ Theorem: This pairing $$<\cdot,\cdot >: A^{'p}(X)\times A_{p-1}(X)\to {\mathbb{R}}$$ is a well- defined satisfying $$(i) <\cdot,\cdot >$$ coincides with the classical height pairing when $$p=d$$; (ii) given a correspondence C on $$X\times Y$$, we have $$<C^*a,b>=<a,C_*b>$$; $$(iii) <\cdot,\cdot >$$ is symmetric when defined.
Reviewer: K.Lai

##### MSC:
 14C15 (Equivariant) Chow groups and rings; motives 14C99 Cycles and subschemes 14G25 Global ground fields in algebraic geometry 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
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