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 ${\mathcal {O}}(1)$ is the canonical line bundle on $P(C+1)$ . The Chern class $c_{1}({\mathcal {O}}(1))$ is a group endomorphism of the Chow ring of $P(C+1)$ . The $i$ th Segre class of $C$ is given by $s_{i}(C)=q_{*}(c_{1}({\mathcal {O}}(1))^{i}\cap [P(C+1)])$ for non-negative integers $i$ . The total Segre class is the sum of the Segre classes.

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 ${\mathcal {O}}$ .

Relation to Chern classes for vector bundles

For a holomorphic vector bundle $E$ over a complex manifold $M$ a total Segre class $s(E)$ is the inverse to the total Chern class $c(E)$ , see e.g.^[1]

Explicitly, for a total Chern class

c(E)=1+c_{1}(E)+c_{2}(E)+\cdots \,

one gets the total Segre class

s(E)=1+s_{1}(E)+s_{2}(E)+\cdots \,

where

c_{1}(E)=-s_{1}(E),\quad c_{2}(E)=s_{1}(E)^{2}-s_{2}(E),\quad \dots ,\quad c_{n}(E)=-s_{1}(E)c_{{n-1}}(E)-s_{2}(E)c_{{n-2}}(E)-\cdots -s_{n}(E)

Let $x_{1},\dots ,x_{k}$ be Chern roots, i.e. formal eigenvalues of ${\frac {i\Omega }{2\pi }}$ where $\Omega$ is a curvature of a connection on $E$ .

While the Chern class s(E) is written as

c(E)=\prod _{{i=1}}^{{k}}(1+x_{i})=c_{0}+c_{1}+\cdots +c_{k}\,

where $c_{i}$ is an elementary symmetric polynomial of degree $i$ in variables $x_{1},\dots ,x_{k}$

the Segre for the dual bundle $E^{\vee }$ which has Chern roots $-x_{1},\dots ,-x_{k}$ is written as

s(E)=\prod _{{i=1}}^{{k}}{\frac {1}{1-x_{i}}}=s_{0}+s_{1}+\cdots

Expanding the above expression in powers of $x_{1},\dots x_{k}$ one can see that $s_{i}(E^{\vee })$ is represented by a complete homogeneous symmetric polynomial of $x_{1},\dots x_{k}$

References

↑ 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

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