Schur's lemma (from Riemannian geometry)

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.

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

$c_1(E)=-s_1(E),c_2(E)=s_1(E{)}^2-s_2(E),\dots ,c_n(E)=-s_1(E)c_{n-1}(E)-s_2(E)c_{n-2}(E)-\cdots -s_n(E)$

Let $x_1,\dots ,x_k$ be Chern roots, i.e. formal eigenvalues of $\frac{i\mathrm{\Omega }}{2\pi }$ where $\mathrm{\Omega }$ 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

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