Twisted deformation quantization of algebraic varieties

Abstract

Let X be a smooth algebraic variety over a field K containing the real numbers. 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 operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.

Non-commutative deformations and quasi-coherent modules

“…For instance, in [54,55], Yekutieli uses crossed groupoids as the technical tool to capture such twists on the level of deformation groupoids. Note that our approach is philosophically quite different, as it deals with twisted presheaves of algebras as algebraic objects in their own right, living a priori on an arbitrary base category U.…”

“…For instance, in smooth geometric setups-often considered in the context of deformation quantization-it is natural to replace Hochschild complexes by subcomplexes of polydifferential operators, in order to arrive at a sheaf of structured complexes which can be globalized [7,27,28,48,[53][54][55]. This method is detailed for instance in [48,Appendix 4], which also treats the relation with a construction by Hinich [22]-the most refined formality result in terms of higher structure being obtained by Calaque and Van den Bergh in [7].…”

“…Repeating the process by which a quasi-coherent sheaf is obtained by glueing modules on affine opens, we define the category of quasi-coherent modules over A to be . They play an important role in the context of deformation quantization, see [7,24,27,48,54,55]. In Sect.…”

Hochschild cohomology and deformation quantization of affine toric varieties

“…Kontsevich also extended the notion of deformation quantization into the algebro-geometric setting [19]. From Yekutieli's results [32], [33] it follows that on a smooth algebraic variety X (under certain cohomological restrictions) every Poisson structure admits a star product. As in Kontsevich's case, the construction is canonical and induces a bijection between the set of formal Poisson structures modulo gauge equivalence and the set of star products modulo gauge equivalence (see also Van den Bergh's paper [31]).…”

Flatness and Completion Revisited

“…Indeed, our paper [Ye1] was written as part of our project on deformation quantization, for which we needed sharper results on a-adically complete flat noncommutative central A-rings (and sheaves of this type). See the paper [Ye2] and its references.…”