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: | , , |
|---|---|
| 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 |
| Sumario: | 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. |
|---|