The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917629985292288 |
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| author | Gómez, Tomás L. Herrero, Andres Fernandez Zamora, Alfonso |
| author_facet | Gómez, Tomás L. Herrero, Andres Fernandez Zamora, Alfonso |
| contents | 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 by Alper, Halpern-Leistner and Heinloth 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 as constructed by Schmitt and Gómez-Langer-Schmitt-Sols.
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 reductions previously considered at the level of points by Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_03918 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations Gómez, Tomás L. Herrero, Andres Fernandez Zamora, Alfonso Algebraic Geometry 14D23 (Primary) 14D20, 14J60, 14L24, 14F06 (Secondary) 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 by Alper, Halpern-Leistner and Heinloth 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 as constructed by Schmitt and Gómez-Langer-Schmitt-Sols. 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 reductions previously considered at the level of points by Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure. 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. |
| title | The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations |
| topic | Algebraic Geometry 14D23 (Primary) 14D20, 14J60, 14L24, 14F06 (Secondary) |
| url | https://arxiv.org/abs/2107.03918 |