Work

Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

Year: 2025

Type: article

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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf104

Authors: Gwyn Bellamy, Christopher Dodd, Kevin McGerty, Thomas Nevins

Institutions: University of Glasgow, University of Illinois Urbana-Champaign, University of Oxford

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,534

Funders: National Science Foundation, Leverhulme Trust, Engineering and Physical Sciences Research Council