Algebraic Cocycles on Normal, Quasi-Projective Varieties

Blaine Lawson and the author introduced algebraic cocycles on complex algebraic varieties in [FL-1] and established a duality theorem relating spaces of algebraic cocycles and spaces of algebraic cycles in [FL-2]. This theorem has non-trivial (and perhaps surprising) applications in several contexts. In particular, duality enables computations of “algebraic mapping spaces” consisting of algebraic morphisms. Moreover, duality appears to be an important property in motivic cohomology/homology (cf. [F-V]). In this paper, we extend the theory of [FL-1], [FL-2] to quasi-projective varieties. (Indeed, our duality theorem is an assertion of a natural homotopy equivalence from cocycle spaces to cycle spaces and thus is a refinement of the duality theorem of [FL-2] when specialized to projective varieties.) One can view this work as developing an algebraic bivariant theory for complex quasi-projective varieties which is closely based on algebraic cycles. On the other hand, one can also view the resulting spaces of algebraic cocycles as function complexes equipped with a natural topology. Thus, the theory of cycle spaces, cocycle spaces, and duality has both a formal role in providing invariants for algebraic varieties (closely related to classical invariants and problems as seen in [F-2]) and a more explicit role in the analysis of heretofore inaccessible function complexes. Our consideration of quasi-projective varieties enables computations as exemplified in §7. Many local calculations, useful even for projective varieties, should now be accessible. Other applications of this theory in the quasi-projective context can be found in §6. Duality for cocycle and cycle spaces should be viewed as a somewhat sophisticated generalization of the comparison of Cartier and Weil divisors on a (smooth) variety. From this point of view, one does indeed expect that the theory developed for projective varieties to extend to quasi-projective varieties. The essential difficulty in providing such an