Duality Relating Spaces of Algebraic Cocycles and Cycles

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 LH(X) → Ln−sH2n−k(X) for 2s ≤ k which carry over via natural transformations to the Poincaré duality isomorphism H(X;Z)→ H2n−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. Among applications presented 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.