Porteous formula


In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus of a morphism of vector bundles in terms of Chern classes. 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 is the variety of points where it has rank at most k. If all components of the degeneracy locus have the expected codimension then Porteous's formula states that its fundamental class is the determinant of the matrix of size m – k whose entry is the Chern class cn–k+j–i.