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".

I understand that the generalization of Kontsevich formality to the algebraic setting (smooth algebraic variety, possibly non-affine) is more involved.

1. Is the formula for the weights in the (complex) algebraic context exactly the same as $(\ast)$?

2. If the first term of a star product is the coefficient of a (holomorphic) Poisson bracket, then the first "weight" in the algebraic context is not a constant, but the coefficient of a (global) section of the second exterior power of the holomorphic tangent bundle, which even in local coordinates would not necessarily be constant. (If it is a constant everywhere, then we would be dealing with a holomorphic symplectic form, which is a special case.) Should I expect the weights to be holomorphic/algebraic functions?

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

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?

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".

I understand that the generalization of Kontsevich formality to the algebraic setting (smooth algebraic variety, possibly non-affine) is more involved.

Is the formula for the weights in the (complex) algebraic context exactly the same as $(\ast)$?

If the first term of a star product is the coefficient of a (holomorphic) Poisson bracket, then the first "weight" in the algebraic context is not a constant, but the coefficient of a (global) section of the second exterior power of the holomorphic tangent bundle, which even in local coordinates would not necessarily be constant. (If it is a constant everywhere, then we would be dealing with a holomorphic symplectic form, which is a special case.) Should I expect the weights to be holomorphic/algebraic functions?

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".

I understand that the generalization of Kontsevich formality to the algebraic setting (smooth algebraic variety, possibly non-affine) is more involved.

1. Is the formula for the weights in the (complex) algebraic context exactly the same as $(\ast)$?

2. If the first term of a star product is the coefficient of a (holomorphic) Poisson bracket, then the first "weight" in the algebraic context is not a constant, but the coefficient of a (global) section of the second exterior power of the holomorphic tangent bundle, which even in local coordinates would not necessarily be constant. (If it is a constant everywhere, then we would be dealing with a holomorphic symplectic form, which is a special case.) Should I expect the weights to be holomorphic/algebraic functions?

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