Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y , one may endow the set of morphisms, Mor(X, Y ), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of “smooth curves”) for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including: (1) the existence of cap and join product pairings in topological cycle theory, (2) the agreement of cup product and intersection product for topological cycle theory, (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product, and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles. At first glance, imposing a topology on the set Mor(X,Y ) of morphisms between two complex algebraic varieties seems unnatural. Nevertheless, just such a construction applied to the set of morphisms from X to certain Chow varieties of cycles in projective space leads to the “morphic cohomology” of X as introduced in [FL-1]. In this paper, we show that, in general, the “topology of bounded convergence” (introduced in [FL-2]) on Mor(X,Y ) has a natural algebraic description arising from the enriched structure on Mor(X,Y ) as a contravariant functor on the category of smooth curves. This functorial interpretation leads to a convenient formulation of the technique of demonstrating “uniqueness of specialization” introduced in [F-1] for the construction of continuous algebraic maps. We use this new technique to establish the continuity of various constructions and pairings involving the “function spaces”Mor(X,Y ), where X and Y are complex (but not necessarily projective) varieties. More generally, we introduce the notion of a “proper, constructible presentation” of a functor (cf. Definition 2.1), a property which provides a natural topological realization of a contravariant functor on smooth curves. This point of view facilitates (cf. Theorem 2.6) a careful proof of the continuity of the slant product pairing of [FL-1] and the cap product pairing relating Lawson homology and morphic cohomology which plays a central role in [F-3]. Indeed, our techniques provide, not merely a pairing on the level of homology groups, but pairings (in the derived category) of the presheaves of chain complexes used to define Lawson homology 1991 Mathematics Subject Classification. Primary 14E99, 19E20, 14F99.