Cycle Spaces and Intersection Theory

This paper constitutes a preliminary discussion of joint work in progress as presented by the first author at Stony Brook in June 1991 at the symposium in honor of John Milnor. Our results include a construction of an intersection pairing on spaces of algebraic cycles on a given smooth complex quasi-projective variety, thereby providing a ring structure in “Lawson homology.” We verify that the Lawson homology for quasi-projective varieties satisfies sufficiently many of the “standard properties” of a good homology theory that it admits a theory of Chern classes from algebraic K-theory. The reader familiar with higher Chow groups of S. Bloch might find it useful to view Lawson homology as a topological analogue of that theory. In fact, we exhibit a tantalizing map from Bloch’s higher Chow groups to Lawson homology. Our subject is algebraic geometry. Until we discuss “algebraic bivariant cycle theory” in section 4, our ground field will always be the complex numbers C. We shall consider projective varieties, reduced schemes over C which admit a closed embedding in some (complex) projective space P ; as such, a projective variety is the zero locus of a family of homogeneous polynomials {Fα(X0, ..., XN )}. The fact that we consider polynomial equations characterizes our study as algebraic geometry. More generally, we shall consider quasi-projective varieties which are complements of (algebraic) embeddings of one projective variety in another. Our program is to study invariants of a given quasi-projective variety X using the group of all algebraic cycles on X of some fixed dimension r. The Lawson homology groups, LrH∗(X), are defined to be the homotopy groups of the “space of algebraic r-cycles on X.” Recall that an algebraic r−cycle on X is a formal sum Z = ΣmiYi , mi ∈ Z