Chern class formulas for classical-type degeneracy loci

TL;DR

This paper extends geometric methods to derive Pfaffian formulas for degeneracy loci in all classical types, including isotropic Grassmannians, connecting to theta- and eta-polynomials with simplified proofs.

Contribution

It refines Kazarian's geometric approach to produce unified Pfaffian formulas for degeneracy loci across all classical types, including new cases involving isotropic Grassmannians.

Findings

Derived Pfaffian formulas for degeneracy loci in types B, C, D

Connected formulas to theta- and eta-polynomials of Buch et al.

Provided simplified, parallel proofs for all four classical types.

Abstract

In previous work, we employed a geometric method of Kazarian to prove Pfaffian formulas for a certain class of degeneracy loci in types B, C, and D. Here we refine that approach to obtain formulas for more general loci, including those coming from all isotropic Grassmannians. In these cases, the formulas recover the remarkable theta- and eta-polynomials of Buch, Kresch, Tamvakis, and Wilson. The streamlined geometric approch yields simple and direct proofs, which proceed in parallel for all four classical types. In an appendix, we develop some foundational algebra and prove several Pfaffian identities. Another appendix establishes a basic formula for classes in quadric bundles.