Algebraic, Homological, and Numerical Equivalence

Summary

Each k-dimensional complex variety V has a cycle class c l(V) in H _2k V, where H _* denotes homology with locally finite supports (Borel-Moore homology). If V is a subvariety of an n-dimensional complex manifold X, then H _2k (V) ≅ H ^2n−2k(X, X − V). The resulting homomorphism from cycles to homology passes to algebraic equivalence. There results in particular a cycle map

$$cl:A_* X \to H_* X$$

for complex schemes X, which is covariant for proper morphisms, and compatible with Chern classes of vector bundles.