[K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I".


Background
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Let $X$ be a smooth affine variety (over $\mathbb{C}$ or maybe a field of characteristic zero) or resp. a smooth (compact?) real manifold. Let $A = \Gamma(X; \mathcal{O}\_X)$ or resp. $C^\infty(X)$.


Denote the dg *Lie algebra* of polyvector fields on $X$ (with Schouten-Nijenhuis bracket and zero differential) by $T$. Denote the dg *Lie algebra* of the shifted Hochschild cochain complex of $A$ (with Gerstenhaber bracket and Hochschild differential) by $D$.


Then the Hochschild-Konstant-Rosenberg theorem states that there is a quasi-isomorphism of dg *vector spaces* from $T$ to $D$. However, the HKR map is *not* a map of dg *Lie algebras*. It is *not* a map of dg *algebras*, either (where the multiplication on $T$ is given by the wedge product and the multiplication on $D$ is given by the cup product of Hochschild cochains).


I believe "Kontsevich formality" refers to the statement that, while the HKR map is not a quasi-isomorphism --- or even a morphism --- of dg *Lie algebras*, there is an $L\_\infty$ quasi-isomorphism $U$ from $T$ to $D$, and therefore $D$ is in fact formal as a dg *Lie algebra*.


The first "Taylor coefficient" of the $L\_\infty$ morphism $U$ is precisely the HKR map (see section 4.6.2 of [K]).


Moreover, this quasi-isomorphism $U$ is compatible with the dg *algebra* structures on $T$ and $D$ (see section 8.2 of [K]), and it yields a "corrected HKR map" which is a dg algebra quasi-isomorphism. The "correction" comes from the square root of the $\hat{A}$ class of $X$. See [this previous MO question](https://mathoverflow.net/questions/14861/is-there-a-refinement-of-the-hochschild-kostant-rosenberg-theorem-for-cohomology/).


Questions
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(0) Are all of my statements above correct?


(1) In what way is the $L\_\infty$ morphism $U$ compatible with the dg *algebra* structures? I don't understand what this means.


(2) When $X$ is a smooth (compact?) real manifold, I think that all of the statements above are proved in [K]. When $X$ is a smooth affine variety, I think that the statements should all still be true. Where can I find proofs?


(3) Moreover, the last section of [K] suggests that the statements are all still true when $X$ is a smooth *possibly non-affine* variety. For a general smooth variety, though, instead of taking the Hochschild cochain complex of $A = \Gamma(X;\mathcal{O}\_X)$, presumably we should take the Hochschild cochain complex of the (dg?) derived category of $X$. Is this correct? If so, where can I find proofs?


In the second-to-last sentence of [K], Kontsevich seems to claim that the statements for varieties are corollaries of the statements for real manifolds, but I don't see how this can possibly be true. In the last sentence of the paper, he says that he will prove these statements "in the next paper", but I'm not sure which paper "the next paper" is, nor am I even sure that it exists, since "Deformation quantization of Poisson manifolds, II" doesn't exist.


P.S. I am not sure how to tag this question. Feel free to tag it as you wish.



A few questions about Kontsevich formality
To (1): Daniel is right, there is a map of homotopy Gerstenhaber algebras between the two algebras. However the full story is quite complicated and to show that the hochschild cochains form a homotopy Gerstenhaber algebra is hard, it's known as the Deligne conjecture. I don't know the details of the proof.


Recall that a Poisson algebra is a commutative algebra with a Lie bracket and these two products satisfy a Leibniz identity. A Gerstenhaber algebra is a bit like a Poisson algebra, except the Lie bracket is of degree 1 not 0. The bracket satisfies a graded Leibniz identity wrt to the commutative algebra structure.


The formality morphism as homotopy Gerstenhaber algebras restricts to a formality morphism as homotopy Lie algebras and to a formality morphism as homotopy commutative algebras.


In my view the simplest proof of the formality of the Hochschild cochains of a nice enough algebra as a homotopy Gerstenhaber algebra is contained in


<http://arxiv.org/abs/math.KT/0605141>


