QUOTIENT SINGULARITIES IN THE GROTHENDIECK RING OF VARIETIES

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 [A¹] 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 = Zⁿ for some smooth projective variety Z and G = Sn acts by permutation of the factors is of particular interest. We make progress on it by showing that [Zⁿ/Sn] − [Z〈n〉/Sn] is divisible by [A¹], where Z〈n〉 is Ulyanov’s polydiagonal compactification of the nth configuration space of Z.