A cohomological interpretation for stringy Hodge numbers

Abstract

We obtain a cohomological interpretation for Batyrev's stringy Hodge numbers in the full generality in which they are defined. In a previous paper, the second and third authors used motivic integration to define the stringy Hodge--Deligne invariant of a smooth Artin stack $\mathcal{X}$ and proved that when $\mathcal{X}$ is a crepant resolution of a variety $Y$ with log-terminal singularities, the generating function for the stringy Hodge numbers of $Y$ is equal to the stringy Hodge--Deligne invariant of $\mathcal{X}$. In this paper, we introduce a cohomology theory $H_{\mathrm{str}}^{\ast }(\mathcal{X})$ that computes the stringy Hodge--Deligne invariant of $\mathcal{X}$. Since, by previous work of the second and third authors, all varieties with log-terminal singularities admit a crepant resolution by an Artin stack, this gives a cohomological interpretation for stringy Hodge numbers of any variety with log-terminal singularities. We also show that in the special case where $\mathcal{X}$ is Deligne--Mumford, $H_{\mathrm{str}}^{\ast }(\mathcal{X})$ coincides with the orbifold cohomology of $\mathcal{X}$.