Deformation quantization modules

TL;DR

This paper explores modules over deformation quantization algebroids on complex Poisson manifolds, establishing key theorems on finiteness, duality, Hochschild classes, and holonomic modules, extending classical results to a non-commutative geometric framework.

Contribution

It introduces new finiteness, duality, and Hochschild class theorems for modules over deformation quantization algebroids, including a non-commutative Riemann-Roch type result.

Findings

Proved finiteness and duality theorems for modules

Constructed the Hochschild class of coherent modules

Established a non-commutative Riemann-Roch theorem

Abstract

We study modules over stacks of deformation quantization algebroids on complex Poisson manifolds. We prove finiteness and duality theorems in the relative case and construct the Hochschild class of coherent modules. We prove that this class commutes with composition of kernels, a kind of Riemann-Roch theorem in the non-commutative setting. Finally we study holonomic modules on complex symplectic manifolds and we prove in particular a constructibility theorem.