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arXiv · 2407.13286

Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

Gwyn Bellamy·Christopher Dodd·Kevin McGerty·Thomas Nevins
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Abstract

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.

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Gwyn Bellamy, Christopher Dodd, Kevin McGerty, Thomas Nevins. 2024-07-18. Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties. https://arxiv.org/abs/2407.13286

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