Cycles over Fields of Transcendence Degree 1

where (a) the group of cycles Z(V ) is the free abelian group on scheme-theoretic points of V of codimension p and (b) rational equivalence R(V ) is the subgroup generated by cycles of the form divW(f ), where W is a subvariety of V of codimension p − 1 and f is a nonzero rational function on it. There is a natural cycle class map clp : CH (V ) → H(V ), where the latter denotes the singular cohomology group H2p(V(C),Z) with the (mixed) Hodge structure given by Deligne (see [D]). The kernel of clp is denoted by F 1 CH(V ). There is an Abel–Jacobi map (see [G]) p : F 1 CH(V ) → IJp(H2p−1(V )),