In
mathematics, the
Porteous formula, or
Thom–Porteous formula, or
Giambelli–Thom–Porteous formula, is the expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of
. Giambelli's formula is roughly the special case when the vector bundles are sums of line bundles over
projective space. pointed out that the fundamental class must be a polynomial in the Chern classes and found this polynomial in a few special cases, and found the polynomial in general. proved a more general version, and generalized it further.
Statement
Given a morphism of vector bundles
E,
F of ranks
m and
n over a smooth variety, its
k-th degeneracy locus (
k ≤ min(
m,
n)) is the variety of points where it has rank at most
k. If all components of the degeneracy locus have the expected
codimension (
m –
k)(
n –
k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size
m –
k whose (
i,
j) entry is the Chern class
c n– k+ j– i(
F –
E).
