A cohomological interpretation for stringy Hodge numbers

TL;DR

This paper provides a cohomological framework for understanding stringy Hodge numbers, extending previous motivic integration results to a broader class of varieties with log-terminal singularities.

Contribution

It introduces a new cohomology theory that computes stringy Hodge--Deligne invariants for Artin stacks, generalizing the orbifold cohomology to all varieties with log-terminal singularities.

Findings

Cohomology theory $H_{str}^*( ext{X})$ computes stringy Hodge--Deligne invariants.

For Deligne--Mumford stacks, $H_{str}^*( ext{X})$ matches orbifold cohomology.

Provides a cohomological interpretation for stringy Hodge numbers in full generality.

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 $\mathcal{X}$.