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
