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I have a question similar to one given here.

What is the cohomology of the Lie algebra of polynomial vector fields on an affine space $\mathbb{A}^2$ over a field of characteristic $0?$ (I'm interested in the complex case, but I think there is no difference).

If I'm correct the answer to this question for $\mathbb{A}^1$ is Goncarova's theorem? However, I don't know if there is an answer for $\mathbb{A}^2.$ I found a remark in Fuchs's book on cohomology of infinite-dimensional Lie algebras that one should consider homology of polynomial vector fields instead of cohomology. I presume this is because the algebra of polynomial vector fields is dual to that of formal vector fields.

Is there an answer for cohomology as well or is it completely hopeless?

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