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Moscow Mathematical Journal

Volume 21, Issue 4, October–December 2021  pp. 807–830.

Hodge Numbers of Generalized Kummer Schemes via Relative Power Structures

Authors:  Andrew Morrison (1) and Junliang Shen (2)
Author institution:(1) Departement Mathematik, ETH Zürich
(2) Department of Mathematics, Yale University


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.

2020 Math. Subj. Class. Primary: 14C05; Secondary: 14K05.

Keywords: Power structure, Hodge polynomial, Donaldson–Thomas invariant, generalized Kummer scheme.

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