Quotient singularities in the Grothendieck ring of varieties

Abstract

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