Height pairings for algebraic cycles
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.
group of algebraic cycles of codimension p
scientific article; zbMATH DE number 3922827
scientific article