Correspondences

A correspondence from X to Y, denoted $$ \alpha :X \vdash Y $$ is a subvariety, cycle, or equivalence class of cycles on X x Y. The graph of a morphism, or the closure of the graph of a rational map, are basic examples, but more general correspondences have played an important role in the development of algebraic geometry. On complete non-singular varieties correspondences have a product ß ° α, and a correspondence $$ \alpha :X \vdash Y $$ determines homomorphisms α*from A(X) to A(Y), and α* from A(Y) to A(X), these notions generalizing composition, push-forward, and pull-back for morphisms. The basic algebra of correspondences is deduced easily from the general theory of Chap. 8. If X = Y has dimension n, and T is an n-dimensional correspondence, then the degree of the intersection class T·Δ of T with the diagonal is the virtual number of fixed points of T. In case there are non-isolated fixed points, the excess intersection formulas can be applied. (If T = V x W, with V, W sub-varieties of X, T·Δ= V·W is the intersection class studied in Chap. 8.) When one has explicit formulas for the equivalence class of [T] or of [Δ] on X x X, fixed point formulas for T·Δcan be deduced. Notation. Unless otherwise stated, all ambient varieties X,Y,Z,… in this chapter are assumed to be complete and non-singular, i.e., proper and smooth over the given ground field.