Weak quantization of Poisson structures

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.

“…Theorem 8.8 (for artinian R and S) is similar to [CH,Theorem 3.5]. But [CH, Theorem 3.5] is not actually proved in that paper (there appear to be holes in the arguments, cf.…”

Section: Lie Descent Datamentioning

confidence: 94%

Twisted deformation quantization of algebraic varieties

Yekutieli

2015

“…Global formality on the sheaf level is important for deformation theory. See for example [5,19,21,34,40].…”

Section: Introductionmentioning

confidence: 99%

Hochschild cohomology and Atiyah classes

Calaque

Bergh

2010

Self Cite

146

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.Comment: Reference to work of Cattaneo, Felder and Willwacher adde

“…(ii) Let f : P * M − → P * N be a contact transformation. By a result of [28] (see also [17,20]), there exists an invertible 7 where the right-hand side is defined byČech cohomology.…”

Section: Quantization Algebrasmentioning

confidence: 99%

On quantizations of complex contact manifolds

Polesello¹

2015

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.