mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...

es.

Giambelli's formula In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes in terms of special Schubert classes, or Schur functions in terms of complete symmetric functions. It states :\displaystyle \sigma_\lambda= \det(\si ...

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'').

References

* * * *{{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 Theorems in algebraic geometry