Twisted Deformation Quantization of Algebraic Varieties (Survey)

TL;DR

This paper surveys the concept of twisted deformation quantization for algebraic varieties, establishing a canonical bijection between twisted Poisson and associative deformations, extending classical deformation theory.

Contribution

It introduces twisted deformation notions for algebraic varieties and proves a canonical bijection between twisted Poisson and associative deformations.

Findings

Established a twisted quantization map from Poisson to associative deformations.

Proved the bijectivity of the quantization map on equivalence classes.

Extended classical deformation theory to stack-like twisted deformations.

Abstract

Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on equivalence classes.