Relative étale slices and cohomology of moduli spaces
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914545504616448 |
|---|---|
| author | de Cataldo, Mark Andrea Herrero, Andres Fernandez Núñez, Andrés Ibáñez |
| author_facet | de Cataldo, Mark Andrea Herrero, Andres Fernandez Núñez, Andrés Ibáñez |
| contents | We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has equisingular fibers. As an application, we show that any two fibers have isomorphic $\ell$-adic cohomology rings and intersection cohomology groups. If we work over the complex numbers, we show that the family is topologically locally trivial on the base, and that the intersection cohomology groups of the fibers fit into a polarizable variation of pure Hodge structures. We apply these results to derive some consequences for the moduli spaces of $G$-bundles on smooth projective curves, and for certain moduli spaces of sheaves on del Pezzo surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_00350 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative étale slices and cohomology of moduli spaces de Cataldo, Mark Andrea Herrero, Andres Fernandez Núñez, Andrés Ibáñez Algebraic Geometry 14D23 (Primary) 14B25, 14J60, 14D07, 14F45 (Secondary) We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has equisingular fibers. As an application, we show that any two fibers have isomorphic $\ell$-adic cohomology rings and intersection cohomology groups. If we work over the complex numbers, we show that the family is topologically locally trivial on the base, and that the intersection cohomology groups of the fibers fit into a polarizable variation of pure Hodge structures. We apply these results to derive some consequences for the moduli spaces of $G$-bundles on smooth projective curves, and for certain moduli spaces of sheaves on del Pezzo surfaces. |
| title | Relative étale slices and cohomology of moduli spaces |
| topic | Algebraic Geometry 14D23 (Primary) 14B25, 14J60, 14D07, 14F45 (Secondary) |
| url | https://arxiv.org/abs/2307.00350 |