Motivic Chern classes and K-theoretic stable envelopes

TL;DR

This paper introduces a new K-theoretic characteristic class called the motivic Chern class, characterizes it axiomatically, and provides explicit formulas for various singular varieties, linking it to stable envelopes and quantum groups.

Contribution

It defines the equivariant motivic Chern class, characterizes it via axioms inspired by stable envelopes, and computes explicit formulas for several classes of singular varieties.

Findings

Explicit formulas for Schubert and matrix Schubert cells

Calculation of motivic Chern classes for A2 quiver orbits

Formulas for determinantal varieties and degeneracy loci

Abstract

We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formulas for the motivic Chern classes of determinantal varieties and more general degeneracy loci.