Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

TL;DR

This work extends the Riemann-Hilbert/Koszul duality paradigm to bionic symplectic varieties by constructing a deformation-quantization framework that localizes modules to the Lagrangian skeleton via a geometric category ${\mathcal O}$. It introduces a dg-algebra Omega from a local generator, establishing a derived equivalence between ${\mathcal W}$-modules and Omega-modules, and formulates an exotic/coderived perspective to realize Koszul duality in this noncommutative setting. The paper develops holonomic stability and recollment theory under open/closed inclusions and Hamiltonian reduction, and proves the existence of a holonomic local generator supported on $\Lambda^c$, with a precise reduction behavior on symplectic resolutions. Globally, it provides a concrete path from DQ-modules to Omega-modules, via a sheafified, skeleton-centered duality, enabling a Morse-theoretic, local-to-global description of module categories in geometric representation theory.

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.

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Lemma 2.4