Motivic classes of generalized Kummer schemes via relative power structures

Abstract

We develop a power structure over the Grothendieck ring of varieties relative to an abelian monoid, which allows us to compute the motivic class of the generalized Kummer scheme. We obtain a generalized version of Cheah's formula for the Hilbert scheme of points, which specializes to Gulbrandsen's conjecture for Euler characteristics. Moreover, in the surface case we prove a conjecture of G\"{o}ttsche for geometrically ruled surfaces, and we obtain an explicit formula for the virtual motive of the generalized Kummer scheme in dimension three.