Motivic Classes of Isotropic Degeneracy Loci and Symmetric Orbit Closures

TL;DR

This work provides explicit formulas for motivic Chern and Hirzebruch classes of isotropic (type C) and odd orthogonal (type B) degeneracy loci, extending the ACT-type results beyond type A. It develops a resolution-based, stratification-driven method, leveraging raising operators and theta-polynomials to express pushforwards as Pfaffian/Schur-Pfaffian forms in terms of Chern data, with clear specializations to CSM and L-classes. The results yield concrete formulas for orthogonal and symplectic orbit closures in flag varieties, particularly for vexillary involutions, enabling computation of their motivic invariants and providing a unified framework across isotropic degeneracy loci and symmetric orbit closures. The methods offer computational tools for Schubert-type problems in isotropic geometries and contribute to the understanding of characteristic classes of singular symmetric varieties.

Abstract

We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern--Schwartz--MacPherson classes, K-theory classes, and Cappell--Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our result is the analogue of the result of Anderson--Chen--Tarasca for the degeneracy loci from the ordinary Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag varieties.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2: ACT
• Proposition 3.1
• proof
• Lemma 3.2
• proof