Moving algebraic cycles of bounded degree (1998) [12 citations — 9 self]
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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 eective s-cycles of degree e on X so that each such cycle meets every eective 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 dened 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
