A cohomological interpretation for stringy Hodge numbers
Jiahui Huang, Matthew Satriano, Jeremy Usatine
TL;DR
The paper provides a complete cohomological framework for stringy Hodge numbers by constructing a cohomology theory $H_{ ext{str}}^*(\mathcal{X})$ for smooth finite-type Artin stacks with affine diagonal and properly stable moduli space, and showing that its mixed Hodge structure recovers Batyrev's stringy Hodge–Deligne invariant $ ext{HD}_{\mathrm{str}}(Y)$ for log-terminal varieties $Y$ via crepant Artin-stack resolutions. The construction leverages motivic integration over twisted arc spaces, a careful decomposition of constructible sets, and the compactly supported cohomology of Artin stacks, with a key compatibility to orbifold cohomology in the Deligne–Mumford case. A weight-integrality criterion links the integrality of the stringy invariant to the weight function, while smooth truncation maps and co-unit fibrations enable precise cohomological manipulations. The results unify and extend known cases: when a crepant resolution by a DM stack exists, the stringy invariant equals orbifold cohomology, and in general the cohomology computes the full stringy Hodge numbers for all log-terminal varieties, establishing a broad cohomological interpretation and practical computational pathway.
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 $H_{\mathrm{str}}^*(\mathcal{X})$ 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, $H_{\mathrm{str}}^*(\mathcal{X})$ coincides with the orbifold cohomology of $\mathcal{X}$.