Motivic Classes of 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.

Arithmetic and metric aspects of open de Rham spaces

“…Following [14] and [12], in this section, we introduce Grothendieck rings with exponentials, a naive notion of motivic Fourier transform and convolution. Similar techniques were used in [52] to compute the motivic classes of Nakajima quiver varieties. Throughout this section, $$\mathbb {K}$$ will denote an arbitrary field; by a variety, we mean a separated scheme of finite type over $$\mathbb {K}$$; and by a morphism of varieties, we will mean a $$\mathbb {K}$$‐morphism.…”

“…The proof of Theorem 1.3.1 is first performed, as Theorem 4.3.1, in the case of $$k=0$$, that is, only irregular punctures. In this case, we can proceed by motivic Fourier transform, as in [52], and the result will be a motivic extension of Theorem 1.3.1. The general case — Theorems 5.1.6 and 5.2.3 — is then proved via the arithmetic harmonic analysis technique of [24].…”

Homotopy Spectra and Diophantine Equations

“…Proof. The argument is similar to Section 2 of [Wy17]. For (X, f ) V an object in C KExp V , the second iterate of the Fourier transform is given by…”

On the number of points of nilpotent quiver varieties over finite fields

“…(d) The formulas for the point count of Λ ♭ (v) in Theorem 1.4 also hold in the Grothendieck group of varieties (or more precisely stacks, see [3]) over an algebraically closed field. One simply replaces every occurence of q by the Lefschetz motive L. This is a consequence of the fact that the point count of the Nakajima quiver varieties performed in [11] admit a similar motivic lift, see [37].…”