On quantization of complex symplectic manifolds

Abstract

Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X+1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.