TITLE: A few questions about Kontsevich formality
QUESTION [21 upvotes]: [K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I".
Background
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.
Questions
(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.

REPLY [5 votes]: Here are answers to questions 2, 3 and 1 (in this order). 
2) Let $X = \operatorname{Spec} A$ be a smooth affine variety over a field $K$ of characteristic $0$. Then there is a canonical bijection between Poisson deformations of $A$ and associative deformations of $A$ (all up to gauge equivalence). This was first proved (in slightly greater generality, yet assuming $K$ contains the real numbers) in my paper

Deformation Quantization in Algebraic Geometry, Amnon Yekutieli, 
  Advances in Mathematics 198 (2005), 383-432. Erratum: Advances in Mathematics  217 (2008), 2897-2906.

The condition about real numbers is not essential for the proof; all that is needed is a Formality Theorem for the power series ring 
$K[[t_1, \ldots, t_n]]$ satifying the invariance properties of Kontsevich. 
It should be noted that the proof above is of a global nature. A local argument can only be made on an affine scheme $X$ that admits etale coordinates (namely an etale map to $\mathbf{A}^n_K$). Bigger affine schemes (that do not admit such coordinates) require a gluing argument, based on formal geometry, and the vanishing of some cohomological obstructions.  
There are other later proofs of this same result (listed in another answer). I am only familiar with the proof by Van den Bergh (which is a variation on my proof).   
3) If $X$ is an arbitrary smooth scheme (over a field $K$ of characteristic $0$), then quantization is possible, but only in a stacky sense. This was suggested by Kontsevich in his paper 

M. Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), no. 3, 271-294.

The precise statement, and its proof, are in the paper (still not published!) 

[TDQ] Amnon Yekutieli, Twisted Deformation Quantization of Algebraic Varieties, http://arxiv.org/abs/0905.0488 .

In this more general situation one has to consider twisted (i.e. stacky) Poisson deformations of the sheaf $\mathcal{O}_X$, and twisted associative deformations of the sheaf $\mathcal{O}_X$. 
A twisted associative deformation $\mathcal{A}$ is a refined sort of stack of algebroids. The category of coherent left $\mathcal{A}$-modules is a deformation (as a stack of abelian categories, in the sense studied by Lowen and Van den Bergh) of the category of coherent $\mathcal{O}_X$-modules. This also goes in reverse: $\mathcal{A}$ can be recovered from its module category (basically by Morita theory). 
A twisted Poisson deformation is a new kind of algebro-geometric object. There is no Morita theory connected to it. 
Theorem 0.10.1 of [TDQ] states that there is a canonical bijection, called the twisted quantization map, between twisted Poisson deformations of $\mathcal{O}_X$  and twisted associative deformations of $\mathcal{O}_X$ (up to twisted gauge equivalence). 
There are lecture notes on this work: 
http://www.math.bgu.ac.il/~amyekut/lectures/twisted-defs-2013/notes.pdf 
and a survey article: 
http://www.math.bgu.ac.il/~amyekut/publications/tw-defs-surv/tw-defs-surv.html
The main paper [TDQ] relies on several auxiliary papers, including:

Central Extensions of Gerbes, Amnon Yekutieli, 
  Advances in Mathematics 225, Issue 1 (2010), 445-486.
MC Elements in Pronilpotent DG Lie Algebras,
  Amnon Yekutieli, J. Pure Appl. Algebra 216 (2012), 2338–2360
Deformations of Affine Varieties and the Deligne Crossed Groupoid,
  Amnon Yekutieli, Journal of Algebra 382 (2013), 115–143.
Combinatorial Descent Data for Gerbes,
  Amnon Yekutieli, to appear in J. Noncommutative Geometry. Eprint arXiv:1109.1919 at http://arxiv.org. 

There are several intriguing open questions in this subject. For instance Question 0.10.2 of [TDQ] regarding twisted deformation quantization of Calabi-Yau surfaces. 
1) The compatibility of the Kontsevich formality morphism with the associative DG algebra structures on polyvector fields and polydifferential operators is made precise and proved in the paper 

Hochschild cohomology and Atiyah classes,
  Damien Calaque and Michel Van den Bergh,
  Advances in Mathematics 224 (2010) 1839–1889. 

As I recall, they prove that the universal formality morphism is (in additions to being L_infty) an A_2 morphism (namely A_infty truncated at level 2). This is enough to deduce the formula for twisting the HKR morphism (by the square root of the Todd class) in order to respect the multiplications.