Hodge numbers of generalized Kummer schemes via relative power structures

We develop a power structure over the Grothendieck ring of varieties relative to an abelian monoid, which provides a systematic method to compute the class of the generalized Kummer scheme in the Grothendieck ring of Hodge structures. 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 Göttsche for geometrically ruled surfaces.