Quotient singularities in the Grothendieck ring of varieties

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 $Z$.

Theorems & Definitions (39)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Claim 2.3
• Claim 2.4
• Corollary 2.5