Stringy Hodge numbers via crepant resolutions by Artin stacks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912100466556928 |
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| author | Satriano, Matthew Usatine, Jeremy |
| author_facet | Satriano, Matthew Usatine, Jeremy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23951 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stringy Hodge numbers via crepant resolutions by Artin stacks Satriano, Matthew Usatine, Jeremy Algebraic Geometry 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. |
| title | Stringy Hodge numbers via crepant resolutions by Artin stacks |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.23951 |