Vector Bundles and Chern Classes

William Fulton ()
Additional contact information
William Fulton: University of Michigan, Department of Mathematics

Chapter Chapter 3 in Intersection Theory, 1998, pp 47-69 from Springer

Abstract: Abstract 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).

Keywords: Exact Sequence; Vector Bundle; Line Bundle; Normal Bundle; Chern Class (search for similar items in EconPapers)
Date: 1998
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-1700-8_4

Ordering information: This item can be ordered from
http://www.springer.com/9781461217008

DOI: 10.1007/978-1-4612-1700-8_4

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().