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In Kontsevich's Deformation quantization of Poisson manifolds, he gives an explicit formula for the star product: $$ f \star g = fg + \sum_{n=1}^\infty \hbar^n \sum_{\Gamma \in G_n} w_\Gamma B_{\Gamma} (f, g) \tag{$\ast$} $$ where $B_\Gamma (f, g)$ is the bilinear operator associated to a graph $\Gamma$ and $w_\Gamma$ is a "weight", constructed as the integral of some configuration space.

I understand that the generalization of Kontsevich formality to the algebraic setting (smooth algebraic variety, possibly non-affine) is more involved and I have the following questions:

  1. Is the formula for the weights in the (complex) algebraic context exactly the same as in the real and smooth case?

  2. Are the weights computed "globally" or chart-by-chart, so that one has to check compatibility/gluing in an additional step?

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  1. yes, the formula for the weights is the same in whatever setting (differentiable, holomorphic and algebro-geometric).

  2. weights are involved in a local formula for a start-product. Indeed, $B_\Gamma$ doesn't even make sense globally on a manifold (being differentiable, holomorphic, etc...). Hence, even in the differentiable setting, Kontsevich's local formula must be globalized. There are standard techniques to do this, but the resulting star-product is not quite explicit.

Let me add that in the holomorphic (or algebro-geometric) setting there is an additional issue: a global star-product may not exist. Indeed, it is not always possible to glue the local formula in order to get a sheaf of algebras deforming the structure sheaf of the complex manifold/algebraic variety. As it is explained in this other paper of Kontsevich, one can only hope to get an algebroid stack (this is a linearized analog of a gerbe). There has been a lot of work on these gadgets (by Yekutieli, Kashiwara-Schapira, and many others).

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    $\begingroup$ Thank you very much for your answer. Do you know of any papers that give explicit star products on non-local varieties? (I guess in the symplectic case such a quantization could be obtained for $T^* X$ as some form of $\mathscr D_X$, but I also haven't seen the corresponding star product being given explicitly.) $\endgroup$ – Earthliŋ Jan 22 '18 at 8:24
  • $\begingroup$ Consider a fibre bundle $E\to X$ together with a Poisson structure on its total space that is linear in the fibers. Then $E$ is dual to a Lie algebroid $L$. And the universal enveloping algebra of $L$ gives a deformation quantization of the fiberwise linear Poisson structure on $E$. $\endgroup$ – DamienC Feb 1 '18 at 11:59
  • $\begingroup$ Would you have a reference for this? My understanding is that as a deformation quantization with parameter $\hbar$, say, one wants to consider a "homogenized" universal enveloping algebra (homogenizing the relations using the parameter $\hbar$), giving the usual enveloping algebra for $\hbar = 1$. But I haven't seen an explicit description of the bidifferential operators appearing in the star product that describe this quantization. $\endgroup$ – Earthliŋ Feb 1 '18 at 12:48
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    $\begingroup$ I don't have a reference in mind. Of course one has to introduce $\hbar$ in the defining relations for the universal envelopping algebra: $e_1e_2-e_2e_1=\hbar[e_1,e_2]$, $ef-fe=\hbar\rho(e)(f)$. The fact that the coefficients of the deformation are bidifferential operators is a property that can be checked locally, and that is a consequence of the PBW theorem for Lie algebroids (see e.g. the paper of Nistor-Weinstein-Xu: arxiv.org/pdf/funct-an/9702004.pdf). $\endgroup$ – DamienC Feb 14 '18 at 18:31

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