Motivic classes and the integral Hodge Question

Abstract

We prove that the obstruction to the integral Hodge Question factors through the completion of the Grothendieck ring of varieties for the dimension filtration. As an application, combining work of Peyre, Colliot-Thélène and Voisin, we give the first example of a finite group $G$ such that the motivic class of its classifying stack $BG$ in Ekedahl’s Grothendieck ring of stacks over $\mathbb{C}$ is non-trivial and $BG$ has trivial unramified Brauer group.