Relative Chow Correspondences and the Griffiths Group

In the monograph [FM-1], the author and Barry Mazur introduce a filtration on algebraic cycles on a (complex) projective variety which we called the topological filtration. This filtration, defined using a fundamental operation on the homotopy groups of cycle spaces, has an interpretation in terms of “Chow correspondences.” The purpose of this paper is to give examples in which specific levels of this filtration are non-trivial. Thus, we obtain examples of cycles which lie in different levels of a naturally defined filtration of the Griffiths group (of cycles homologically equivalent to 0 modulo cycles algebraically equal to 0). Our examples are cycles on general complete intersections analyzed by Madhav Nori by means of his (rational) Lefschetz hyperplane theorem [N]. The relevance of Nori’s examples is suggested by a description given in [F3] of the topological filtration closely resembling the filtration on cycles that Nori considers. Nori’s theorem is a result about cohomology and Nori’s application to his filtration on cycles involves working with cycle classes in cohomology; our topological filtration lends itself less easily to a cohomological analysis. One difficulty we face is that the topological filtration of a smooth variety involves cycles on singular varieties. This provides considerable awkwardness for cycles on singular varieties need not have cycle classes in cohomology. Another difficulty is that Nori’s application of his Lefschetz theorem to cycles involves the consideration of families of varieties over a quasi-projective base variety, whereas the machinery for studying the topological filtration has been formulated in the context of projective varieties. Thus, we are led to consider “relative Chow correspondences.” We briefly sketch the organization of the paper. Section 1 summarizes the context and results of Nori’s paper which we shall use. In section 2, we extend to quasiprojective varieties the construction of Chow correspondences and graph mappings. More importantly, we interpret the Chow correspondence homomorphism of [FM1] in terms of slant product, a familiar operation in homology theory. Section 3 presents our results, most notably Theorem 3.4 which is our strengthened version of one aspect of Nori’s theorems about the Griffiths group. A few corollaries are given enabling us to obtain examples of varieties with non-trivial layers in the topological filtration on algebraic cycles. In section 4, we develop relative Chow correspondences in order to work with families of varieties as we encounter in Nori’s context. Finally, section 5 completes the proof of Theorem 3.4. As we have observed previously (cf. [FM-1], [F2]), one of the intriguing aspects of geometric techniques involving cycle spaces is that these techniques rarely require