K-theoretic Chern class formulas for vexillary degeneracy loci

Abstract

Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank. In an appendix, we construct a K-theoretic Euler class for even-rank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.

48 pages; v2 is a substantial revision, including formulas for all classical types, as well as an expanded appendix on K-theory and a new appendix introducing an Euler class for even orthogonal bundles; v3 corrects the entries of the Pfaffians in Theorems 2, 3, and 4, and includes other minor improvements