Quotient singularities in the Grothendieck ring of varieties

TL;DR

This paper investigates the divisibility properties of quotient varieties in the Grothendieck ring, revealing conditions under which the difference between a quotient and its resolution is divisible by the affine line class, especially for abelian groups and symmetric group actions.

Contribution

It establishes new criteria for divisibility in the Grothendieck ring related to quotient singularities, including cases involving symmetric groups and configuration space compactifications.

Findings

Negative divisibility when BG is not stably rational

Affirmative divisibility for abelian groups

Progress on symmetric group actions and configuration spaces

Abstract

Let $G$ be a finite group, $X$ be a smooth complex projective variety with a faithful $G$-action, and $Y$ be a resolution of singularities of $X/G$. Larsen and Lunts asked whether $[X/G]-[Y]$ is divisible by $[\mathbb{A}^1]$ in the Grothendieck ring of varieties. We show that the answer is negative if $BG$ is not stably rational and affirmative if $G$ is abelian. The case when $X=Z^n$ for some smooth projective variety $Z$ and $G=S_n$ acts by permutation of the factors is of particular interest. We make progress on it by showing that $[Z^n/S_n]-[Z\langle n\rangle / S_n]$ is divisible by $[\mathbb{A}^1]$, where $Z\langle n\rangle$ is Ulyanov's polydiagonal compactification of the $n$-th configuration space of $Z$.