Porteous formula
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 Chern classes. Giambelli's formula is roughly the special case when the vector bundles are sums of line bundles over projective space. {{harvs|txt|authorlink=René Thom|last=Thom|year=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 {{harvs|txt|last=Porteous|year=1971|authorlink=Ian R. Porteous}} found the polynomial in general. {{harvtxt|Kempf|Laksov|1974}} proved a more general version, and {{harvtxt|Fulton|1992}} 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 cn–k+j–i(F – E).
References
- {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician)| title=Flags, Schubert polynomials, degeneracy loci, and determinantal formulas | doi=10.1215/S0012-7094-92-06516-1 |mr=1154177 | year=1992 | journal=Duke Mathematical Journal | issn=0012-7094 | volume=65 | issue=3 | pages=381–420}}
- {{Citation | last1=Kempf | first1=G. | last2=Laksov | first2=D. | title=The determinantal formula of Schubert calculus | doi=10.1007/BF02392111 |mr=0338006 | year=1974 | journal=Acta Mathematica | issn=0001-5962 | volume=132 | pages=153–162| doi-access=free }}
- {{Citation | last1=Porteous | first1=Ian R. | title=Proceedings of Liverpool Singularities Symposium, I (1969/70) | publisher=Springer-Verlag | location=Berlin, New York | series= Lecture Notes in Mathematics | doi=10.1007/BFb0066829 |mr=0293646 | year=1971 | volume=192 | chapter=Simple singularities of maps | pages=286–307|origyear=1962| isbn=978-3-540-05402-3 }}
- {{Citation | last1=Thom | first1=René | title=Les ensembles singuliers d'une application différentiable et leurs propriétés homologiques | series=Séminaire de Topologie de Strasbourg | year=1957}}