Segre class

In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Beniamino Segre (1953).

Definition

Suppose C is a cone over X, q is the projection from the projective completion P(C+1) of C to X and O(1) is the canonical line bundle on P(C+1). The Chern class c1(O(1)) is a group endomorphism of the Chow ring of P(C+1). The Segre classes are given by q*( (c1(O(1)))i [P(C+1)]) for various integers i. The total Segre class is the sum of the Segre classes for each integer i.

The reason for using P(C+1) rather than P(C) is that this makes the total Segre class stable under addition of the trivial bundle 1.

Relation to Chern classes for vector bundles

For a holomorphic vector bundle over a complex manifold a total Segre class is the inverse to the total Chern class , see e.g.[1]

Explicitly, for a total Chern class

one gets the total Segre class

where

Let be Chern roots, i.e. formal eigenvalues of where is a curvature of a connection on .

While the Chern class s(E) is written as

where is an elementary symmetric polynomial of degree in variables

the Segre for the dual bundle which has Chern roots is written as

Expanding the above expression in powers of one can see that is represented by a complete homogeneous symmetric polynomial of

References

1. Fulton W. (1998). Intersection theory, p.50. Springer, 1998.
• Segre, Beniamino (1953), "Nuovi metodi e resultati nella geometria sulle varietà algebriche", Ann. Mat. Pura Appl. (in Italian), 35 (4): 1–127, MR 0061420