In mathematics, the Porteous formula, or ThomâPorteous formula, or GiambelliâThomâPorteous 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 â k)(n â k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size m â k whose (i, j) entry is the Chern class cnâk+jâi(F â E).
In mathematics, the Porteous formula, or ThomâPorteous formula, or GiambelliâThomâPorteous 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 â k)(n â k) then Porteous's formula states that its fundamental class is the determinant of the matrix of size m â k whose (i, j) entry is the Chern class cnâk+jâi(F â E).