Duality Relating Spaces of Algebraic Cocycles and Cycles, by Eric M. Friedlander and H. Blaine Lawson

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 $L^sH^k(X) \to L_{n-s}H_{2n-k}(X)$ for $2s\leq k$ which carry over via natural transformations to the Poincar\'e duality isomorphism $H^k(X;{\bf Z}) \to H_{2n-k}(X;{\bf Z})$. 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 $X\times Y$ 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.

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