The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations
Let X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation ρ of G into a product of general linear groups, we define a moduli stack of principal ρ-sheaves that compactifies the stack of G-bundles on X....
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/381580 |
| Acceso en línea: | http://hdl.handle.net/10261/381580 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85195669684&doi=10.1007%2fs00209-024-03497-6&partnerID=40&md5=7748fde71d79edd69f4b2acdcb60d5a1 |
| Access Level: | acceso abierto |
| Palabra clave: | 14D20 14D23 14F06 14J60 14L24 Gieseker stability Harder–Narasimhan filtration Moduli spaces Principal bundles Principal ρ-sheaves Stacks Θ-stratifications |
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The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrationsGómez, T.L.Herrero, A.F.Zamora, A.14D2014D2314F0614J6014L24Gieseker stabilityHarder–Narasimhan filtrationModuli spacesPrincipal bundlesPrincipal ρ-sheavesStacksΘ-stratificationsLet X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation ρ of G into a product of general linear groups, we define a moduli stack of principal ρ-sheaves that compactifies the stack of G-bundles on X. We apply the theory developed in Alper et al. (Invent Math 234(3):949–1038, 2023), Halpern-Leistner D (On the structure of instability in moduli theory, http://arxiv.org/abs/1411.0627, 2014) and Halpern-Leistner D et al. (Moduli spaces of sheaves via affine Grassmannians, http://arxiv.org/abs/2107.02172v1, 2021) to construct a moduli space of Gieseker semistable principal ρ-sheaves. This provides an intrinsic stack-theoretic construction of the moduli space of semistable singular principal bundles (Gómez et al. in Adv Math 219(4):1177–1245, 2008; Schmitt Int Math Res Not 23:1183–2002). Our second main result is the definition of a schematic Gieseker–Harder–Narasimhan filtration for ρ-sheaves, which induces a stratification of the stack by locally closed substacks. This filtration for a general reductive group G is a refinement of the canonical slope parabolic reduction previously considered at the level of points in Anchouche et al. (Math Ann 323(4):693–712, 2002) and as a stratification of the stack in Gurjar and Nitsure (Harder–Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics, http://arxiv.org/abs/1605.08997, 2016) and Gurjar and Nitsure (Math Z 289(3-4):1121-1142, 2018). In an appendix, we apply the same techniques to define Gieseker–Harder–Narasimhan filtrations in arbitrary characteristic and show that they induce a stratification of the stack by radicial morphisms. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.: This project was started during the workshop “Moduli problems beyond geometric invariant theory” at the American Institute of Mathematics. We would like to thank the American Institute of Mathematics and the organizers that made the workshop possible. We would like to thank V. Balaji, Federico Fuentes, Daniel Halpern-Leistner 5 and Jochen Heinloth for helpful remarks. We would also like to thank an anonymous referee for thoughtful comments on the manuscript. This work is supported by grants CEX2019-000904-S and PID2019-108936GB-C21 (funded by MCIN/AEI/ 10.13039/501100011033), and NSF grants DMS-1454893 and DMS-2001071.Peer reviewedSpringer NatureMinisterio de Ciencia e Innovación (España)Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]202520252024info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Preprintinfo:eu-repo/semantics/submittedVersionhttp://hdl.handle.net/10261/381580https://www.scopus.com/inward/record.uri?eid=2-s2.0-85195669684&doi=10.1007%2fs00209-024-03497-6&partnerID=40&md5=7748fde71d79edd69f4b2acdcb60d5a1reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)InglésSíinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/3815802026-05-22T06:33:51Z |
| dc.title.none.fl_str_mv |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| title |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| spellingShingle |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations Gómez, T.L. 14D20 14D23 14F06 14J60 14L24 Gieseker stability Harder–Narasimhan filtration Moduli spaces Principal bundles Principal ρ-sheaves Stacks Θ-stratifications |
| title_short |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| title_full |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| title_fullStr |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| title_full_unstemmed |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| title_sort |
The moduli stack of principal ρ-sheaves and Gieseker–Harder–Narasimhan filtrations |
| dc.creator.none.fl_str_mv |
Gómez, T.L. Herrero, A.F. Zamora, A. |
| author |
Gómez, T.L. |
| author_facet |
Gómez, T.L. Herrero, A.F. Zamora, A. |
| author_role |
author |
| author2 |
Herrero, A.F. Zamora, A. |
| author2_role |
author author |
| dc.contributor.none.fl_str_mv |
Ministerio de Ciencia e Innovación (España) Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72] |
| dc.subject.none.fl_str_mv |
14D20 14D23 14F06 14J60 14L24 Gieseker stability Harder–Narasimhan filtration Moduli spaces Principal bundles Principal ρ-sheaves Stacks Θ-stratifications |
| topic |
14D20 14D23 14F06 14J60 14L24 Gieseker stability Harder–Narasimhan filtration Moduli spaces Principal bundles Principal ρ-sheaves Stacks Θ-stratifications |
| description |
Let X be a smooth projective variety and let G be a connected reductive group, both defined over a field of characteristic 0. Given a faithful representation ρ of G into a product of general linear groups, we define a moduli stack of principal ρ-sheaves that compactifies the stack of G-bundles on X. We apply the theory developed in Alper et al. (Invent Math 234(3):949–1038, 2023), Halpern-Leistner D (On the structure of instability in moduli theory, http://arxiv.org/abs/1411.0627, 2014) and Halpern-Leistner D et al. (Moduli spaces of sheaves via affine Grassmannians, http://arxiv.org/abs/2107.02172v1, 2021) to construct a moduli space of Gieseker semistable principal ρ-sheaves. This provides an intrinsic stack-theoretic construction of the moduli space of semistable singular principal bundles (Gómez et al. in Adv Math 219(4):1177–1245, 2008; Schmitt Int Math Res Not 23:1183–2002). Our second main result is the definition of a schematic Gieseker–Harder–Narasimhan filtration for ρ-sheaves, which induces a stratification of the stack by locally closed substacks. This filtration for a general reductive group G is a refinement of the canonical slope parabolic reduction previously considered at the level of points in Anchouche et al. (Math Ann 323(4):693–712, 2002) and as a stratification of the stack in Gurjar and Nitsure (Harder–Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics, http://arxiv.org/abs/1605.08997, 2016) and Gurjar and Nitsure (Math Z 289(3-4):1121-1142, 2018). In an appendix, we apply the same techniques to define Gieseker–Harder–Narasimhan filtrations in arbitrary characteristic and show that they induce a stratification of the stack by radicial morphisms. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024. |
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2024 |
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2024 2025 2025 |
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