Symplectic and odd orthogonal Pfaffian formulas for algebraic cobordism

Abstract

In this paper, we prove generalizations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of the symplectic/odd orthogonal Grassmann bundles. IntroductionThe r-th degeneracy locus of a morphism of vector bundles ϕ : E → F is the subvariety D r (ϕ) of the base scheme X formed by all the points x over which the image of the restriction ϕ(x) has dimension at most r. If, for simplicity, one assumes X to be a smooth quasi-projective scheme and ϕ to be sufficiently general, the classical Giambelli-Thom-Porteous formula describes the Chow ring fundamental class [D r (ϕ)] CH as a determinant in Chern classes. Similarly, it is possible to consider more restrictive settings in which ϕ is either skewsymmetric or symmetric, which means that one has F = E ∨ and ϕ ∨ = −ϕ (respectively ϕ = ϕ ∨ ). In both cases [D r (ϕ)] CH is given by a Pfaffian formula in the Chern classes of E.