Twisted motivic Chern class and stable envelopes

TL;DR

This paper introduces twisted motivic Chern classes for singular pairs, combining motivic and multiplier ideal constructions, and shows their relation to stable envelopes in K-theory.

Contribution

It defines twisted motivic Chern classes for singular pairs and connects them to stable envelopes, extending previous work on fundamental slopes.

Findings

Twisted motivic Chern classes are limits of elliptic classes.

They satisfy stable envelope axioms with suitable divisors.

The construction generalizes previous fundamental slope results.

Abstract

We present a definition of {\em twisted motivic Chern classes} for singular pairs $(X,\Delta)$ consisting of a singular space $X$ and a $\mathbb Q$-Cartier divisor containing the singularities of $X$. The definition is a mixture of the construction of motivic Chern classes previously defined by Brasselet-Sch{\"u}rmann-Yokura with the construction of multiplier ideals. The twisted motivic Chern classes are the limits of the elliptic classes defined by Borisov-Libgober. We show that with a suitable choice of the divisor $\Delta$ the twisted motivic Chern classes satisfy the axioms of the stable envelopes in the K-theory. Our construction is an extension of the results proven by the first author for the fundamental slope.