Stringy Hodge numbers via crepant resolutions by Artin stacks

TL;DR

This paper establishes a precise correspondence between log-terminal singularities and crepant resolutions via smooth Artin stacks, and provides a new formula for stringy Hodge numbers using motivic integration, advancing understanding of their cohomological interpretation.

Contribution

It proves the equivalence between log-terminal singularities and crepant resolutions by smooth Artin stacks, and introduces a formula for stringy Hodge numbers via motivic integration.

Findings

Proves the if and only if condition for crepant resolutions by smooth Artin stacks.

Derives a formula for stringy Hodge numbers using motivic integration over twisted arcs.

Defines stringy Hodge numbers for smooth Artin stacks, generalizing orbifold Hodge numbers.

Abstract

In a previous paper we showed that any variety with log-terminal singularities admits a crepant resolution by a smooth Artin stack. In this paper we prove the converse, thereby proving that a variety admits a crepant resolution by a smooth Artin stack if and only if it has log-terminal singularities. Furthermore if $\mathcal{X} \to Y$ is such a resolution, we obtain a formula for the stringy Hodge numbers of $Y$ in terms of (motivically) integrating an explicit weight function over twisted arcs of $\mathcal{X}$. That weight function takes only finitely many values, so we believe this result provides a plausible avenue for finding a long-sought cohomological interpretation for stringy Hodge numbers. Using that the resulting integral is defined intrinsically in terms of $\mathcal{X}$, we also obtain a notion of stringy Hodge numbers for smooth Artin stacks, that in particular, recovers Chen and Ruan's notion of orbifold Hodge numbers.