Motivic Classes of Commuting Varieties via Power Structures

TL;DR

This paper refines and generalizes a classical formula for the class of commuting varieties in the Grothendieck group, using power structures on stacks, with applications to motivic Donaldson-Thomas theory.

Contribution

It introduces a new method using power structures on the Grothendieck group of stacks to extend the Feit-Fine formula for commuting varieties.

Findings

Proved a generalized formula for commuting varieties in the Grothendieck group.

Developed a new approach using power structures on stacks.

Applied results to motivic Donaldson-Thomas theory.

Abstract

We prove a formula, originally due to Feit and Fine, for the class of the commuting variety in the Grothendieck group of varieties. Our method, which uses a power structure on the Grothendieck group of stacks, allows us to prove several refinements and generalizations of the Feit-Fine formula. Our main application is to motivic Donaldson-Thomas theory.