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, endowed with a Poisson bracket. A quantization 
of this Poisson bracket is a formal associative deformation of 
the structure sheaf O_X, which realizes the Poisson bracket as 
its first order commutator. More generally one can consider Poisson 
deformations of O_X and their quantizations. 

I will explain what these deformations are. Then I'll state a 
theorem which says that when X is affine, there is a canonical 
quantization map (up to gauge equivalence). This is an algebro-
geometric analogue of the celebrated result of Kontsevich (which 
talks about differentiable manifolds). 

In the second half of the lecture I'll talk about twisted Poisson
deformations, and twisted associative deformations (aka stacks of
algebroids). There is a canonical twisted quantization map, and I 
will describe this result. I will end with a question regarding 
twisted quantization of symplectic Poisson brackets. 

Some of this work is joint with F. Leitner. 

There is a paper on this material: arXiv:0905.0488, and also a 
survey article: arxiv:0801.3233.
  
(updated 10 Dec 2010)