A theory of algebraic cocycles

Abstract

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.

5 pages

Cited by in corpus (18)

• Chow quotients and projective bundle formulas for Euler-Chow series
• The homology of moduli stacks of complexes
• An approach to intersection theory on singular varieties using motivic complexes
• Algebraic cycles and the classical groups II: Quaternionic cycles
• Lawson homology, morphic cohomology and Chow motives
• An algebraic geometric realization of the Chern character
• Real rectifiable currents, holomorphic chains and algebraic cycles
• The Generalized Hodge conjecture for 1-cycles and codimension two algebraic cycles
• Grothendieck standard conjectures, morphic cohomology and Hodge index theorem
• Birational invariants defined by Lawson homology
• A homology and cohomology theory for real projective varieties
• Algebraic Cycles and the Classical Groups - Part I, Real Cycles
• Holomorphic K-theory, algebraic co-cycles, and loop groups
• Families of algebraic varieties parametrized by topological spaces
• Euler-Chow series for scrolls and ruled surfaces
• Morphic cohomology of toric varieties
• Semi-topological K-theory for certain projective varieties
• Semi-topologization in motivic homotopy theory and applications