Vector Bundles and Chern Classes

We will construct, for any vector bundle E on a scheme X, Chern class operations $$ {c_i}\left( E \right) \cap \_:{A_k}X \to {A_{k - 1}}X $$ satisfying properties expected from topology. From the special case of line bundles done in § 2.5, we first construct inverse Chern classes, or Segre classes, which are then inverted to produce Chern classes. The first Chern class operations are also used to describe A * E and A * P(E) in terms of A * X Chern classes will be used later for one of the constructions of general intersection products. Although Chern classes are not absolutely needed for intersection theory, they are used in most applications. For the quickest route to intersection theory proper, the reader will need only Proposition 3.1(a) and Theorem 3.3. For vector bundles, Chern classes and Segre classes determine each other; Chern classes are preferred since they vanish beyond the rank of the bundle. We will see in the next chapter that for cones — “singular vector bundles” — there is a natural analogue of Segre classes, but not Chern classes. Segre classes for normal cones have other remarkable properties not shared by Chern classes (cf. § 4.2).