Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909459648872448 |
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| author | Bellamy, Gwyn Dodd, Christopher McGerty, Kevin Nevins, Thomas |
| author_facet | Bellamy, Gwyn Dodd, Christopher McGerty, Kevin Nevins, Thomas |
| contents | We consider the category of modules over sheaves of Deformation-Quantization (DQ) algebras on bionic symplectic varieties. These spaces are equipped with both an elliptic $\mathbb{G}_m$-action and a Hamiltonian $\mathbb{G}_m$-action, with finitely many fixed points. On these spaces one can consider geometric category $\mathcal{O}$: the category of (holonomic) modules supported on the Lagrangian attracting set of the Hamiltonian action. We show that there exists a local generator in geometric category $\mathcal{O}$ whose dg endomorphism ring, cohomologically supported on the Lagrangian attracting set, is derived equivalent to the category of all DQ-modules. This is a version of Koszul duality generalizing the equivalence between D-modules on a smooth variety and dg-modules over the de Rham complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13286 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties Bellamy, Gwyn Dodd, Christopher McGerty, Kevin Nevins, Thomas Algebraic Geometry Representation Theory Symplectic Geometry We consider the category of modules over sheaves of Deformation-Quantization (DQ) algebras on bionic symplectic varieties. These spaces are equipped with both an elliptic $\mathbb{G}_m$-action and a Hamiltonian $\mathbb{G}_m$-action, with finitely many fixed points. On these spaces one can consider geometric category $\mathcal{O}$: the category of (holonomic) modules supported on the Lagrangian attracting set of the Hamiltonian action. We show that there exists a local generator in geometric category $\mathcal{O}$ whose dg endomorphism ring, cohomologically supported on the Lagrangian attracting set, is derived equivalent to the category of all DQ-modules. This is a version of Koszul duality generalizing the equivalence between D-modules on a smooth variety and dg-modules over the de Rham complex. |
| title | Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties |
| topic | Algebraic Geometry Representation Theory Symplectic Geometry |
| url | https://arxiv.org/abs/2407.13286 |