On quantizations of complex contact manifolds

TL;DR

This paper studies the classification and existence of quantizations of complex contact manifolds, showing they are classified by a specific cohomology group and exploring algebraic quantizations.

Contribution

It provides a classification of holomorphic quantizations of complex contact manifolds using cohomology and investigates algebraic quantization existence.

Findings

Quantizations are classified by the first cohomology group with values in a sheaf of forms.

Existence and classification of algebraic quantizations are addressed.

The work extends Kashiwara's results on canonical quantizations.

Abstract

A (holomorphic) quantization of a complex contact manifold is a filtered algebroid stack which is locally equivalent to the ring E of microdifferential operators and which has trivial graded. The existence of a canonical quantization has been proved by Kashiwara. In this paper we consider the classification problem, showing that the above quantizations are classified by the first cohomology group with values in a certain sheaf of homogeneous forms. Secondly, we consider the problem of existence and classification for quantizations given by algebras.