Duality relating spaces of algebraic cocycles and cycles

Abstract

In this paper a fundamental duality is established between algebraic cycles and algebraic cocycles on a smooth projective variety. The proof makes use of a new Chow moving lemma for families. If X is a smooth projective variety of dimension n, our duality map induces isomorphisms L s H k ( X) → L n − s H 2 n − k ( X) for 2 s ⩽ k which carry over via natural transformations to the Poincaré duality isomorphism H k ( X; Z) → H 2 n − k ( X; Z). More generally, for smooth projective varieties X and Y the natural graphing homomorphism sending algebraic cocycles on X with values in Y to algebraic cycles on the product X × Y is a weak homotopy equivalence. The main results have a wide variety of applications. Among these are the determination of the homotopy type of certain algebraic mapping complexes and a computation of the group of algebraic s-cocycles modulo algebraic equivalence on a smooth projective variety.