A Theory of Algebraic Cocycles

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a “cohomology theory” for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a natural transformation to (singular) integral cohomology theory that preserves cup products. Computations in special cases are carried out. On a smooth variety it is proved that there are algebraic cocycles in each algebraic rational (p, p)-cohomology class. In this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle. Details will appear in [FL]. Our cohomology is a companion to the L-homology theory recently studied in [L, F, L-F1, L-F2, FM]. This homology is a bigraded theory based directly on the structure of the space of algebraic cycles. It admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class. Our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles. It carries a ring structure coming from the complex join (an elementary construction of projective geometry), and it admits a natural transformation Φ to integral cohomology. Chern classes are defined in the theory and transform under Φ to the usual ones. Our definition of cohomology is very far from a duality construction on L-homology. Nonetheless, there is a natural and geometrically defined Kronecker pairing between our “morphic cohomology” and L-homology. The foundation stone of our theory is the notion of an effective algebraic cocycle, which is of some independent interest. Roughly speaking, such a cocycle on a variety X , with values in a projective variety Y , is a morphism from X to the space of cycles on Y . When X is normal, this is equivalent (by “graphing”) to a cycle on X × Y with equidimensional fibres over X . Such cocycles abound in algebraic geometry and arise naturally in many circumstances. The simplest perhaps is that of a flat morphism f : X → Y whose corresponding cocycle associates to x ∈ X , the pullback cycle f({x}). Many more arise naturally from synthetic constructions in projective geometry. We show that every variety is rich in cocycles. Indeed if 1991 Mathematics Subject Classification. Primary 14F99, 14C05.