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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 can be described as yields a "corrected" corrected HKR map, where the correction " 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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[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 $D$, D$(see section 8.2 of [K]), and it can be described as a "corrected" HKR map, where the correction comes from the square root of the$\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$is a smooth (compact?) real manifold, I think that all of the statements above are proved in Kontsevich's deformation quantization paper. [K]. When$X$is a smooth affine variety, I think that the statements should all still be true. Where can I find proofs? (2) 3) Moreover, the last section of Kontsevich's deformation quantization paper [K] suggests that the statements are all still true when$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$. Is this correct? If so, where can I find proofs? In the second-to-last sentence of the deformation quantization paper, [K], Kontsevich seems to claim that the statements for varieties are corollaries of the statements for real manifolds, but I don't see how this can possibly be true. In the last sentence of the paper, he says that he will prove these statements "in the next paper", but I'm not sure which paper "the next paper" is, nor am I even sure that it exists, since "Deformation quantization of Poisson manifolds, II" doesn't exist. P.S. I am not sure how to tag this question. Feel free to tag it as you wish. 6 deleted 68 characters in body ## Background Let$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$is given by the Hochschild cup product)product of Hochschild cochains). I believe "Kontsevich formality" refers to the statement that, while the HKR map is not a quasi-isomorphism --- or even a morphism --- of dg Lie algebras, there is an$L_\infty$quasi-isomorphism$U$from$T$to$D$, and therefore$D$is in fact formal as a dg Lie algebra. Moreover, this quasi-isomorphism$U$is compatible with the dg algebra structures on$T$and$D$, and upon passing to cohomology, it becomes an isomorphism of graded algebras which can be described as a "corrected" HKR mapon cohomology, where the correction comes from the square root of the$\hat{A}$class of$X$. See this previous MO question. ## Questions (0) Are all of my statements above correct? (1) When$X$is a smooth (compact?) real manifold, I think that all of the statements above are proved in Kontsevich's deformation quantization paper. When$X$is a smooth affine variety, I think that the statements should all still be true. Where can I find proofs? (2) Moreover, the last section of Kontsevich's deformation quantization paper suggests that the statements are all still true when$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 the deformation quantization paper, 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.