Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

TL;DR

This paper establishes a Koszul duality equivalence between categories of DQ-modules and dg-modules supported on Lagrangian sets in bionic symplectic varieties with specific group actions.

Contribution

It introduces a new Koszul duality framework for DQ-modules on bionic symplectic varieties with elliptic and Hamiltonian actions, generalizing classical D-module dualities.

Findings

Existence of a local generator in geometric category O

Derived equivalence between dg endomorphism rings and DQ-modules

Generalization of D-module and de Rham complex duality

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.