Kontsevich weights in the complex algebraic setting 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:


*

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

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

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

*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). 
