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 (or determinantal variety) 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. Thom ( 1957) pointed out that the fundamental class must be a polynomial in the Chern classes and found this polynomial in a few special cases, and Porteous ( 1971) found the polynomial in general. Kempf & Laksov (1974) proved a more general version, and Fulton (1992) generalized it further.
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 cn–k+j–i(F – E).
- Fulton, William (1992), "Flags, Schubert polynomials, degeneracy loci, and determinantal formulas", Duke Mathematical Journal, 65 (3): 381–420, doi: 10.1215/S0012-7094-92-06516-1, ISSN 0012-7094, MR 1154177
- Kempf, G.; Laksov, D. (1974), "The determinantal formula of Schubert calculus", Acta Mathematica, 132: 153–162, doi: 10.1007/BF02392111, ISSN 0001-5962, MR 0338006
- Porteous, Ian R. (1971) [1962], "Simple singularities of maps", Proceedings of Liverpool Singularities Symposium, I (1969/70), Lecture Notes in Mathematics, vol. 192, Berlin, New York: Springer-Verlag, pp. 286–307, doi: 10.1007/BFb0066829, ISBN 978-3-540-05402-3, MR 0293646
- Thom, René (1957), Les ensembles singuliers d'une application différentiable et leurs propriétés homologiques, Séminaire de Topologie de Strasbourg