nLab
Sandbox

Every wiki needs a sandbox! Just test between the horizontal rules below (*** in the source) and don't worry about messing things up.

Sandbox

Section

Another section


β† \twoheadrightarrow

Deformation of Algebraic Varieties

Let XX be a smooth algebraic variety over a field π•œ\mathbb{k} of characteristic 00. The analogue of the HKR Theorem here is this:

Theorem

(Swan, Yekutieli). There is a canonical isomorphism

(1)Ext X 2 i(π’ͺ X,π’ͺ X)≅⨁ qH iβˆ’q(X,β‹€ q(𝒯 X)). \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 𝒯 X\mathcal{T}_X is the tangent sheaf of XX, and XX is embedded diagonally in X 2X^2.

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

Theorem

(Yekutieli).

  1. There is a canonical isomorphism

    (2)Rℋℴ𝓂 X 2(π’ͺ X,π’ͺ X)β‰…π’ž cd,X \operatorname{R} \mathcal{Hom}_{X^2}(\mathcal{O}_X, \mathcal{O}_X) \cong \mathcal{C}_{cd, X}

    in the derived category of π’ͺ X\mathcal{O}_X-modules.

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

    (3)⨁ qβ‹€ q(𝒯 X)[βˆ’q]β†’π’ž cd,X. \bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] \to \mathcal{C}_{cd, X} .
  3. Therefore there is a canonical isomorphism

    (4)Rℋℴ𝓂 X 2(π’ͺ X,π’ͺ X)≅⨁ qβ‹€ q(𝒯 X)[βˆ’q] \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 π’ͺ X\mathcal{O}_X-modules.

The relation to deformation quantization is this: π’ž cd,X\mathcal{C}_{cd, X} is a shift by 11 of the sheaf of π’Ÿ poly,X\mathcal{D}_{poly, X} of polydifferential operators (viewed only as a complex of quasi-coherent π’ͺ X\mathcal{O}_X-modules). Similarly, ⨁ qβ‹€ q(𝒯 X)[βˆ’q]\bigoplus_q \bigwedge^q (\mathcal{T}_X)[-q] is the shift by 11 of the sheaf 𝒯 poly,X\mathcal{T}_{poly, X} of polyvector fields. Thus item 2 in the theorem above says that there is a canonical π’ͺ X\mathcal{O}_X-linear quasi-isomorphism

(5)𝒯 poly,Xβ†’π’Ÿ poly,X. \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 L ∞\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 β„³\mathcal{M} on XX admits a canonical flasque resolution called the mixed resolution:

(6)β„³β†’Mix(β„³). \mathcal{M} \to \operatorname{Mix}(\mathcal{M}) .

This β€œmixes” the jet resolution with the Cech resolution (corresponding to an affine open covering of XX that we supress). In particular we get quasi-isomorphisms of sheaves of DG algebras

(7)𝒯 poly,Xβ†’Mix(𝒯 poly,X) \mathcal{T}_{poly, X} \to \operatorname{Mix}(\mathcal{T}_{poly, X})

and

(8)π’Ÿ poly,Xβ†’Mix(π’Ÿ poly,X). \mathcal{D}_{poly, X} \to \operatorname{Mix}(\mathcal{D}_{poly, X}) .
Theorem

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

(9)Ξ¨:Mix(𝒯 poly,X)β†’Mix(π’Ÿ poly,X) \Psi : \operatorname{Mix}(\mathcal{T}_{poly, X}) \to \operatorname{Mix}(\mathcal{D}_{poly, X})

whose 11-st order term commutes with the HKR quasi-isomorphism above. It is independent on choices up to homotopy.

A Poisson deformation of π’ͺ X\mathcal{O}_X is a sheaf π’œ\mathcal{A} of Poisson π•œ[[ℏ]]\mathbb{k}[[\hbar]]-algebras on XX, with an isomorphism π•œβŠ— π•œ[[ℏ]]π’œβ‰…π’ͺ X\mathbb{k} \otimes_{\mathbb{k}[[\hbar]]} \mathcal{A} \cong \mathcal{O}_X called an augmentation. Likewise an associative deformation of π’ͺ X\mathcal{O}_X is a sheaf π’œ\mathcal{A} of associative unital (but noncommutative) π•œ[[ℏ]]\mathbb{k}[[\hbar]]-algebras on XX, with an augmentation to π’ͺ X\mathcal{O}_X.

Theorem 3 implies:

Theorem

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

(10)quant:{ Poisson deformations ofπ’ͺ X}isomorphism→≃{ associative deformations ofπ’ͺ X}isomorphism. \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 XX 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 x_2^3 *hello* $x_2^3$
hello x 2 3 x_2^3 *hello* $x_2^3$
hello x 2 3 x_2^3 *hello* $x_2^3$
hello x 2 3 x_2^3 *hello* $x_2^3$

β‹― β†’ C 3 β†’Ξ΄ C 2 β†’Ξ΄ C 1 β†’Ξ΄ sβ†’Ξ΄ t C 0 ↓ ↓ ↓ Ξ΄ s ↓ = β‹― β†’ C 0 β†’= C 0 β†’= C 0 β†’= C 0 \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}
type theorycategory theory
syntaxsemantics
natural deductionuniversal construction
dependent sum typedependent sum
type theorycategory theory
syntaxsemantics
natural deductionuniversal construction
dependent sum typedependent sum

?? Grothendieck inequality <>&lt;&gt; ***

  1. [ℐ,π’œ][\mathcal{I},\mathcal{A}] is really just the underlying category with hom-collections given by A 0(A,B)=V 0(I,π’œ(A,B))A_0(A,B)=V_0(I,\mathcal{A}(A,B)).
  2. π’œ(βˆ’,βˆ’)\mathcal{A}(-,-) is the fully faithful two-variable hom-functor from A 0 opΓ—A 0β†’V 0A_0^{op}\times A_0\to V_0, with π’œ(f,g)\mathcal{A}(f,g) defined as the composite π’œ(B,C)β†’l βˆ’1r βˆ’1IβŠ—π’œ(B,C)βŠ—Iβ†’fβŠ—idβŠ—gπ’œ(C,D)βŠ—π’œ(B,C)βŠ—π’œ(A,B)π’œ(∘ π’œ) 2(A,D)\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 0V_0
  3. [ℐ,F][\mathcal{I},F] is the functor from A 0A_0 to B 0B_0 underlying the enriched functor FF. This is defined by letting FfFf be the composite Iβ†’fπ’œ(A,B)ℬF A,B(FA,FB)I\stackrel{f}{\to}\mathcal{A}(A,B)\stackrel{F_{A,B}}\mathcal{B}(FA,FB) where F A,BF_{A,B} is the family of morphisms in V 0V_0 defining the enriched functor FF.
  4. The natural transformation FΒ―:catA(βˆ’,βˆ’)β†’catB(Fβˆ’,Fβˆ’)\bar F\colon\cat A(-,-)\to\cat B(F-,F-) has for its components exactly the maps F A,BF_{A,B} above: i.e. FΒ― A,B=F A,B\bar F_{A,B}=F_{A,B}. * *
    category: meta

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