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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2408.15786 |
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| _version_ | 1866912254177312768 |
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| author | Hennecart, Lucien |
| author_facet | Hennecart, Lucien |
| contents | In this paper, we establish the sheafified version of the cohomological integrality conjecture for stacks obtained as a quotient of a smooth affine symmetric algebraic variety by a reductive algebraic group equipped with an invariant function. A crucial step is the definition of the BPS sheaf as a complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf when the situation arises from a smooth affine weakly symplectic algebraic variety with a weak moment map. This situation gives local models for 1-Artin derived stacks with self-dual cotangent complex. We then apply these results to prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore homology of $0$-shifted symplectic stacks (or more generally, derived stacks with self-dual cotangent complex) having a proper good moduli space. One striking application is the purity of the Borel--Moore homology of the moduli stack of principal Higgs bundles over a smooth projective curve for a reductive group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomological integrality for symmetric quotient stacks Hennecart, Lucien Algebraic Geometry In this paper, we establish the sheafified version of the cohomological integrality conjecture for stacks obtained as a quotient of a smooth affine symmetric algebraic variety by a reductive algebraic group equipped with an invariant function. A crucial step is the definition of the BPS sheaf as a complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf when the situation arises from a smooth affine weakly symplectic algebraic variety with a weak moment map. This situation gives local models for 1-Artin derived stacks with self-dual cotangent complex. We then apply these results to prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore homology of $0$-shifted symplectic stacks (or more generally, derived stacks with self-dual cotangent complex) having a proper good moduli space. One striking application is the purity of the Borel--Moore homology of the moduli stack of principal Higgs bundles over a smooth projective curve for a reductive group. |
| title | Cohomological integrality for symmetric quotient stacks |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2408.15786 |