Segre class

In mathematics, the Segre class is a characteristic class used in the study of singular 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 singular vector bundles, while the Chern class does not. The Segre class is named after Beniamino Segre.

Definition

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.