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[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.

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Questions (2) and (3) seem to have been settled. The bounty is for question (1) --- an explanation of how the dgLa/$L_\infty$ quasi-isomorphism and the dga/$A_\infty$ quasi-isomorphism are related. –  Kevin H. Lin Jul 30 '10 at 1:41
    
How about Section 3 of math.jussieu.fr/~keller/publ/emalca.pdf as a reference? –  Timo Schürg Jul 30 '10 at 6:23
    
I meant section 3 of chapter 1 up there. –  Timo Schürg Jul 30 '10 at 6:33
    
@Timo: Thanks, that looks good. I'll take a look at it soon. In the meantime, can you post that link in an actual answer? –  Kevin H. Lin Jul 30 '10 at 7:50
    
Have you looked at the papers of Ramadoss math/0512104 & math/0603127, and by Calaque and van den Bergh, 0708.2725? –  Aaron Bergman Jul 30 '10 at 10:37
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5 Answers

up vote 4 down vote accepted

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

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I can explain why this proof is simpler than it might look if there's demand. It revolves around the proper interpretation of their intermediate complex Xi. –  James Griffin Jul 31 '10 at 10:58
    
@James: Thank you. If you can elaborate on this, it would be much appreciated! –  Kevin H. Lin Aug 5 '10 at 5:09
    
Hi Kevin, sorry to have not gotten back sooner. I'd very much like to elaborate on this, I spent a lot of effort on that paper and would very much like to share. Also I don't want to forget what I've learnt so the chance to explain it is a welcome one! However it wont be for a few days as I am currently finishing up a paper, but after that. –  James Griffin Aug 12 '10 at 8:08
    
The only drawback of the very nice linked paper is that it works over a field of characteristic zero, while these results are true over any commutative ground ring. –  Fernando Muro Nov 26 '11 at 9:30
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@Fernando Muro: what makes you believe that these results are true over any commutative ring? –  DamienC Dec 13 '11 at 14:46
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Hi Kevin, even if the question is answered I would like to add a few remarks.

(0) the claim that

this quasi-isomorphism $U$ is compatible with the dg algebra structures on $T$ and $D$

is not exactly true. It is compatible only on tangent cohomology.

(1) I agree with Daniel and James that there exists a $G_\infty$-quasi-isomorphism between $T$ and $D$ (this implies compatibility at the level of tangent cohomology, and is strictly stronger). But until recently this was not known if Kontsevich's $L_\infty$-quasi-isomorphism can be upgraded to a $G_\infty$-one. A recent paper of Willwacher solves (in a positive way) this question (EDIT: Willwacher makes the comparison with Tamarkin's $G_\infty$-quasi-isomorphism in Section 10).

(2) the proof for smooth affine varieties is essentially the same as the one for smooth differentiable manifolds. Both rely

  • either on the exitence of a connection in the tangent bundle (see the papers of Dolgushev, e.g. this one).

  • or (equivalently) on acyclicity of sheaves of sections of bundles (see e.g. my paper with Michel Van den Bergh).

(3) references are

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?

One could do this, but one instead works on the sheaf level. Consider $T$ and $D$ as sheaves and prove that they are $L_\infty$- (or $G_\infty$-)quasi-isomorphic as sheaves of DG Lie (or $G_\infty$)-algebras.

(3+$\epsilon$) "Deformation quantization of Poisson manifolds, II" does not exist, but there is "Deformation quantization of algebraic varieties" (quite sketchy). You might also be interested by the very inspiring paper "Operads and motives in deformation quantization".

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Its only a reference, and I don't feel competent to give a summary. I think the answer to question 1 can be found in http://www.math.jussieu.fr/~keller/publ/emalca.pdf . At least it mentions the analogy to the Duflo isomorphism, which is similiar to what you are asking about. It also takes a map of vector spaces, does some magic and it ends up being a map of algebras.

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I would guess that the the proper statement to understand (1) which mixes the Lie structure and the algebra structure is that this map is some sort of map of homotopy Gerstenhaber algebras. I don't really know this stuff(or anything else), but my impression based upon work of Fred Cohen is that the precise statement is that this should be a map of modules over the homology of the little disc operad E2, which I guess acts on the Hochschild cochains by the proof of the Deligne Conjecture.

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Theorem 11 in Section 4.4 of this survey by Dolgushev, Tamarkin, and Tsygan answers my question (2).

Theorem 12 kind of addresses my question (3), but the approach there seems to be different from the approach that I am imagining. I am more interested in the Hochschild complex of the derived category. However, I would not be surprised if the Hochschild complex of the derived category of the variety is related to the "sheaf of Hochschild complexes" on the variety, probably the former is the global sections of the latter?

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WHat do you mean by "Hochschild complex of the derived category", exactly? –  Mariano Suárez-Alvarez Jul 23 '10 at 21:35
    
Something like this mathoverflow.net/questions/189/… –  Kevin H. Lin Jul 23 '10 at 21:48
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Try arxiv.org/abs/math/0111094 for relations between various sheafy definitions of Hochschild cohomology. I think all you need is smoothness to relate the $\mathcal{H}om_{X\times X}(\mathcal{O}_X,\mathcal{O}_X)$ definition there to the definition in terms of endomorphisms of the identity functor of the derived category. It should be in Toen's math/0408337. –  Aaron Bergman Jul 24 '10 at 2:02
    
Thanks, Aaron! I think that does it. –  Kevin H. Lin Jul 24 '10 at 4:19
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