Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

Abstract Weconsider 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 $\mathcal{D}$-modules on a smooth variety and dg-modules over the de Rham complex.