Algebraic, Homological, and Numerical Equivalence

Each k-dimensional complex variety V has a cycle class cl(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 $$ {H_{2k}}\left( V \right) \cong {H^{2n - 2k}}\left( {X,X - V} \right) $$ 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. If V and W are subvarieties of dimensions k and l of a non-singular n-dimensional variety X, a refined topological intersection product cl(V)·cl(W) is constructed in $$ {H_{2m}}\left( {V \cap W} \right) $$ , $$ m = k + l - n $$ If cl x V is the class in $$ {H^{2n - 2k}}\left( {X,X - V} \right) $$ dual to cl(V),and similarly for cl x (W), then cl(V)·cl(W) is defined to be the class dual to $$ c{l^x}\left( V \right) \cup c{l^x}\left( W \right) \in {H^{2n - 2k}}\left( {X,X - V \cap W} \right) $$ We show that the cycle map takes the refined intersection $$ V \cdot W \in {A_m}\left( {V \cap W} \right) $$ of Chap. 8 to the class cl(V)·cl(W). In particular, cl is a ring homomorphism from A * X toH * X More generally, if i: X→Y is a regular imbedding of codimension d, the cycle classes of the refined products i’ (α) of Chap. 6 are given by cap product with an orientation class uxy in H 2d (y,y-x) In the final section we discuss what is known about algebraic, homological, and numerical equivalence on non-singular projective varieties. Only a few salient facts are mentioned which relate most directly to other chapters, and few proofs are included. Together with the examples, this may serve as an introduction to the literature on the transcendental theory of algebraic cycles. Notation. Unless otherwise stated, all schemes in this chapter are assumed to be complex algebraic schemes which admit a closed imbedding into some non-singular complex variety. All topological spaces will be locally compact Hausdorff spaces which admit a closed imbedding into some Euclidean space. As in preceding chapters, a k-cycle on X is a formal sum of algebraic subvarieties of X.