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).
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 c 1 ( 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 ( O ( 1 ) ) i ∩ [ 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 O .
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 ) + ⋯ one gets the total Segre class
s ( E ) = 1 + s 1 ( E ) + s 2 ( E ) + ⋯ where
c 1 ( E ) = − s 1 ( E ) , c 2 ( E ) = s 1 ( E ) 2 − s 2 ( E ) , … , c n ( E ) = − s 1 ( E ) c n − 1 ( E ) − s 2 ( E ) c n − 2 ( E ) − ⋯ − s n ( E ) Let x 1 , … , x k be Chern roots, i.e. formal eigenvalues of i Ω 2 π where Ω is a curvature of a connection on E .
While the Chern class s(E) is written as
c ( E ) = ∏ i = 1 k ( 1 + x i ) = c 0 + c 1 + ⋯ + c k where c i is an elementary symmetric polynomial of degree i in variables x 1 , … , x k
the Segre for the dual bundle E ∨ which has Chern roots − x 1 , … , − x k is written as
s ( E ) = ∏ i = 1 k 1 1 − x i = s 0 + s 1 + ⋯ Expanding the above expression in powers of x 1 , … x k one can see that s i ( E ∨ ) is represented by a complete homogeneous symmetric polynomial of x 1 , … x k
