In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Segre class is a characteristic class

Characteristic class

In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...

 used in the study of singular vector bundle

Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...

s. The total Segre class is inverse to the total Chern class

Chern class

In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to singular vector bundles, while the Chern class does not.

Definition

For a holomorphic vector bundle over a complex manifold a total Segre class is the inverse to the total Chern class

Chern class

In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

 , see e.g. .

Explicitly, for a total Chern class

Chern class

In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

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

Elementary symmetric polynomial

In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial P can be expressed as a polynomial in elementary symmetric polynomials: P can be given by an...

 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

Complete homogeneous symmetric polynomial

In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials...

of

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