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 $E$ over a complex manifold $M$ a total Segre class $s(E)$ is the inverse to the total Chern class $c(E)$, see e.g.

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 $x_1, \dots x_k$