Motivic classes of Nakajima quiver varieties

TL;DR

This paper extends Hausel's formula for counting rational points of Nakajima quiver varieties over finite fields to a localized Grothendieck ring setting, using advanced motivic and harmonic analysis techniques.

Contribution

It generalizes the point-counting formula to a motivic context by employing Grothendieck rings with exponentials, advancing the understanding of quiver varieties in algebraic geometry.

Findings

Hausel's formula holds in a localized Grothendieck ring setting.

Uses Grothendieck rings with exponentials for motivic harmonic analysis.

Provides a new framework for studying Nakajima quiver varieties.

Abstract

We prove, that Hausel's formula for the number of rational points of a Nakajima quiver variety over a finite field also holds in a suitable localization of the Grothendieck ring of varieties. In order to generalize the arithmetic harmonic analysis in his proof we use Grothendieck rings with exponentials as introduced by Cluckers-Loeser and Hrushovski-Kazhdan.