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# Another section

$\twoheadrightarrow$

#### Deformation of Algebraic Varieties

Let $X$ be a smooth algebraic variety over a field $\mathbb{k}$ of characteristic $0$. The analogue of the HKR Theorem here is this:

###### Theorem

(Swan, Yekutieli). There is a canonical isomorphism

(1)$\operatorname{Ext}^i_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \bigoplus_q \operatorname{H}^{i - q}(X, \bigwedge^q (\mathcal{T}_X)) .$

Here $\mathcal{T}_X$ is the tangent sheaf of $X$, and $X$ is embedded diagonally in $X^2$.

This is a consequence of the following result. Let $\mathcal{C}_{cd, X}$ be the sheaf of continuous Hochschild cochains of $X$. It is a bounded below complex of quasi-coherent $\mathcal{O}_X$-modules.

###### Theorem

(Yekutieli).

1. There is a canonical isomorphism

(2)$\operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \mathcal{C}_{cd, X}$

in the derived category of $\mathcal{O}_X$-modules.

2. There is a canonical quasi-isomorphism of complexes of sheaves

(3)$\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \to \mathcal{C}_{cd, X} .$
3. Therefore there is a canonical isomorphism

(4)$\operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$

in the derived category of $\mathcal{O}_X$-modules.

The relation to deformation quantization is this: $\mathcal{C}_{cd, X}$ is a shift by $1$ of the sheaf of $\mathcal{D}_{poly, X}$ of polydifferential operators (viewed only as a complex of quasi-coherent $\mathcal{O}_X$-modules). Similarly, $\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q]$ is the shift by $1$ of the sheaf $\mathcal{T}_{poly, X}$ of polyvector fields. Thus item 2 in the theorem above says that there is a canonical $\mathcal{O}_X$-linear quasi-isomorphism

(5)$\mathcal{T}_{poly, X} \to \mathcal{D}_{poly, X} .$

Trying to replicate the global formality theorem of Kontsevich, one would like to upgrade this to an $\mathrm{L}_{\infty}$ quasi-isomorphism. However, it seems that in general this cannot be done directly, but only after a suitable resolution.

Here is our result. (See also Van den Bergh.) Any quasi-coherent sheaf $\mathcal{M}$ on $X$ admits a canonical flasque resolution called the mixed resolution:

(6)$\mathcal{M} \to \operatorname{Mix}(\mathcal{M}) .$

This βmixesβ the jet resolution with the Cech resolution (corresponding to an affine open covering of $X$ that we supress). In particular we get quasi-isomorphisms of sheaves of DG algebras

(7)$\mathcal{T}_{poly, X} \to \operatorname{Mix}(\mathcal{T}_{poly, X})$

and

(8)$\mathcal{D}_{poly, X} \to \operatorname{Mix}(\mathcal{D}_{poly, X}) .$
###### Theorem

(Yekutieli). There is an $\mathrm{L}_{\infty}$ quasi-isomorphism

(9)$\Psi : \operatorname{Mix}(\mathcal{T}_{poly, X}) \to \operatorname{Mix}(\mathcal{D}_{poly, X})$

whose $1$-st order term commutes with the HKR quasi-isomorphism above. It is independent on choices up to homotopy.

A Poisson deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of Poisson $\mathbb{k}[[\hbar]]$-algebras on $X$, with an isomorphism $\mathbb{k} \otimes_{\mathbb{k}[[\hbar]]} \mathcal{A} \cong \mathcal{O}_X$ called an augmentation. Likewise an associative deformation of $\mathcal{O}_X$ is a sheaf $\mathcal{A}$ of associative unital (but noncommutative) $\mathbb{k}[[\hbar]]$-algebras on $X$, with an augmentation to $\mathcal{O}_X$.

Theorem 3 implies:

###### Theorem

(Yekutieli). Assume that the cohomology groups $\operatorname{H}^{1}(X, \mathcal{O}_X)$ and $\operatorname{H}^{2}(X, \mathcal{O}_X)$ vanish. Then there is a canonical bijection

(10)$\mathrm{quant} : \quad \frac{ \{ \text{ Poisson deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}} \quad \xrightarrow{\, \simeq \,} \quad \frac{ \{ \text{ associative deformations of} \; \mathcal{O}_X \} }{\text{isomorphism}} .$

A proof of this theorem when $X$ is affine is here. For the full statement see this paper.

For twisted (or stacky) deformations there is a corresponding (but much more difficult to state and prove). See the paper and the survey.

 hello $x_2^3$ *hello* $x_2^3$ hello $x_2^3$ *hello* $x_2^3$
 hello $x_2^3$ *hello* $x_2^3$ hello $x_2^3$ *hello* $x_2^3$

$\begin{array}{c} \cdots &\to& C_3 &\stackrel{\delta}{\to} & C_2 & \stackrel{\delta}{\to} & C_1 & \stackrel{\overset{\delta_t}{\to}} {\underset{\delta_s}{\to}} & C_0 \\ & & \downarrow & & \downarrow & & \downarrow^{\rlap{\delta_s}} & & \downarrow^{\rlap{=}} \\ \cdots &\to& C_0 &\stackrel{=}{\to}& C_0 &\stackrel{=}{\to}& C_0 &\stackrel{=}{\to}& C_0 \end{array}$

$\underset{\circlearrowleft}{\bullet} \underset{}{\leftrightarrow} \underset{\circlearrowleft}{\bullet}$

?? Grothendieck inequality $<>$ ***

1. $[\mathcal{I},\mathcal{A}]$ is really just the underlying category with hom-collections given by $A_0(A,B)=V_0(I,\mathcal{A}(A,B))$.
2. $\mathcal{A}(-,-)$ is the fully faithful two-variable hom-functor from $A_0^{op}\times A_0\to V_0$, with $\mathcal{A}(f,g)$ defined as the composite $\mathcal{A}(B,C)\stackrel{l^{-1}r^{-1}}{\to}I\otimes\mathcal{A}(B,C)\otimes I\stackrel{f\otimes id\otimes g}{\to}\mathcal{A}(C,D)\otimes\mathcal{A}(B,C)\otimes\mathcal{A}(A,B)\stackrel{(\circ^{\mathcal{A}})^2}\mathcal{A}(A,D)$ in $V_0$
3. $[\mathcal{I},F]$ is the functor from $A_0$ to $B_0$ underlying the enriched functor $F$. This is defined by letting $Ff$ be the composite $I\stackrel{f}{\to}\mathcal{A}(A,B)\stackrel{F_{A,B}}\mathcal{B}(FA,FB)$ where $F_{A,B}$ is the family of morphisms in $V_0$ defining the enriched functor $F$.
4. The natural transformation $\bar F\colon\cat A(-,-)\to\cat B(F-,F-)$ has for its components exactly the maps $F_{A,B}$ above: i.e. $\bar F_{A,B}=F_{A,B}$. * *
category: meta

Revised on April 13, 2014 23:27:02 by Adeel Khan (132.252.63.38)