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?



Kontsevich weights in the complex algebraic setting
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](https://arxiv.org/abs/math/0106006), 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). 


