Moving algebraic cycles of bounded degree

Abstract

The Chow Moving Lemma is a theorem which asserts that a given algebraic s-cycle on a smooth algebraic variety X can be moved within its rational equivalence class to intersect properly a given r-cycle on X provided that r + s ≥ dim(X) (cf. [Chow], [S2]). In the past few years, there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation. With this in mind, it is natural to ask whether one can move a given “bounded family” of s-cycles on the smooth variety X to intersect properly a given “bounded family” of r-cycles. The main point of this paper is to formulate and prove just such a result. In Theorem 3.1, we demonstrate that for any integer e and any smooth projective variety X, one can simultaniously and algebraically “move” all effective s-cycles of degree ≤ e on X so that each such cycle meets every effective r-cycle of degree ≤ e on X in proper dimension. The primary motivation for this Moving Lemma for Cycles of Bounded Degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces. Using Theorem 3.1, we have proved such a duality theorem for complex quasi-projective varieties in [F-L2]. We prove our Moving Lemma for varieties over an arbitary infinite field, permitting a proof in [F-V] of a duality theorem for “motivic cohomology and homology”. The reader will find that our Moving Lemma has numerous good properties. First of all, the move is given as an algebraic move (parametrized by a punctured projective line) on Chow varietes. Although this move is “good” only for s-cycles of bounded degree, it is defined on all effective s-cycles. Moreover, the move starts at “time 0” by expressing an effective s-cycle Z as a difference of effective s-cycles both of which have intersection properties no worse than Z. Finally, our Moving Lemma is applicable to smooth quasiprojective varieties, for it is stated for a possibly singular projective variety X resulting in a conclusion of proper intersection off the singular locus of X. The classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence, thereby establishing the Chow ring A∗(X). Some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move (e.g., [Chow], [R], [S2]; on the other hand, cf. [Chev], [S3])). One direct consequence of our Moving Lemma is a proof for smooth quasi-projective varieties that the intersection product is indeed well defined independent of the choice of move (Theorem 3.4). Of course, the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to Fulton