Algebraic cocycles on normal, quasi-projective varieties

Abstract

This paper extends to quasi-projective varieties earlier work by the author and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid $C_r(Y)(U)$ of effective cocycles on a normal, quasi-projective variety $U$ with values in a projective variety $Y$ consists of algebraic cycles on $U×Y$ equi-dimensional of relative dimension r over $U$. A careful choice of topology enables the establishment of various good properties: the definition is essentially algebraic, the group completion $Z_r(Y)(U)$ has “sensible” homotopy groups, the construction is contravariant with respect to $U$, convariant with respect to $Y$, and there is a natural “quality map” to the topological group of cycles on $U×Y$. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map $Z_r(Y)(U)$ to $Z_r+m(U × Y)$ is a homotopy equivalence provided that both $U$ and $Y$ are smooth (where $m=dim U$). Various application are given, especially the determination of the homotopy types of certain topological groups of algeb raic morphisms.