Twisted Deformation Quantization of Algebraic Varieties

Amnon Yekutieli, Ben Gurion University

Abstract. Let X be a smooth algebraic variety over a field of characteristic 0, with structure sheaf O_X. I will begin by explaining what are associative (resp. Poisson) deformations of O_X. These deformations form a stack of crossed groupoids on X.

Next I will introduce the concept of twisted object of a stack of crossed groupoids. This includes gerbes and stacks of algebroids as particular examples. A twisted deformation (associative or Poisson) is a twisted object of the stack of (associative or Poisson) deformations.

Our main result is the Twisted Quantization Theorem states that there is a canonical bijection between gauge equivalence classes of twisted Poisson deformations and twisted Poisson deformations. If time permits I will state several intermediate results, that lead to the proof of the main result.

There are lecture notes (pdf), a paper on this material arXiv:0905.0488, and also a survey article arxiv:0801.3233.

(updated 16 Sep 2013)