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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
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Just as a slightly off topic comment, is that one can think about this problem in characteristic p, as there is an isomorphism of chain complexes between the tangent complex and the Hochschild cochain complex. One expects that it fails, because of the failing of Duflo theory but there is no example that I know of... –  Daniel Pomerleano Jul 30 '10 at 19:38

6 Answers 6

up vote 5 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

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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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.

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Great answer :-) –  Mariano Suárez-Alvarez Jul 20 at 2:10

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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