# Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Pavel Etingof; Victor Ginzburg

Journal of the European Mathematical Society (2010)

- Volume: 012, Issue: 6, page 1371-1416
- ISSN: 1435-9855

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topEtingof, Pavel, and Ginzburg, Victor. "Noncommutative del Pezzo surfaces and Calabi-Yau algebras." Journal of the European Mathematical Society 012.6 (2010): 1371-1416. <http://eudml.org/doc/277421>.

@article{Etingof2010,

abstract = {The hypersurface in $\mathbb \{C\}^3$ with an isolated quasi-homogeneous elliptic singularity of type $\widetilde\{E\}_r,r=6,7,8$, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_r$ provides a semiuniversal Poisson deformation of that Poisson structure.
We also construct a deformation-quantization of the coordinate ring of such a del Pezzo
surface. To this end, we first deform the polynomial algebra $\mathbb \{C\}[x_1,x_2,x_3]$ to a noncommutative algebra with generators $x_1,x_2,x_3$ and the following 3 relations labelled by cyclic parmutations $(i,j,k)$ of $(1,2,3)$: $x_ix_j-t\cdot x_jx_i=\Phi _k(x_k)$, $\Phi _k\in \mathbb \{C\}[x_k]$. This gives a family of Calabi-Yau algebras $\mathfrak \{A\}^t(\Phi )$ parametrized by a complex number $t\in \mathbb \{C\}^\times $ and a triple
$\Phi =(\Phi _1,\Phi _2,\Phi _3)$, of polynomials of specifically chosen degrees.
Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form $\mathfrak \{A\}^t(\Phi )/\langle \langle \Psi \rangle \rangle $,
where $\langle \langle \Psi \rangle \rangle \subset \mathfrak \{A\}^t(\Phi )$ stands for the ideal generated by a central element $\Psi $ which generates the center of the algebra $\mathfrak \{A\}^t(\Phi )$ if $\Phi $ is generic enough.},

author = {Etingof, Pavel, Ginzburg, Victor},

journal = {Journal of the European Mathematical Society},

keywords = {del Pezzo surfaces; Poisson structures; Calabi-Yau deformations; Hochschild cohomology; del Pezzo surfaces; Poisson structures; Calabi-Yau deformations; Hochschild cohomology},

language = {eng},

number = {6},

pages = {1371-1416},

publisher = {European Mathematical Society Publishing House},

title = {Noncommutative del Pezzo surfaces and Calabi-Yau algebras},

url = {http://eudml.org/doc/277421},

volume = {012},

year = {2010},

}

TY - JOUR

AU - Etingof, Pavel

AU - Ginzburg, Victor

TI - Noncommutative del Pezzo surfaces and Calabi-Yau algebras

JO - Journal of the European Mathematical Society

PY - 2010

PB - European Mathematical Society Publishing House

VL - 012

IS - 6

SP - 1371

EP - 1416

AB - The hypersurface in $\mathbb {C}^3$ with an isolated quasi-homogeneous elliptic singularity of type $\widetilde{E}_r,r=6,7,8$, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type $E_r$ provides a semiuniversal Poisson deformation of that Poisson structure.
We also construct a deformation-quantization of the coordinate ring of such a del Pezzo
surface. To this end, we first deform the polynomial algebra $\mathbb {C}[x_1,x_2,x_3]$ to a noncommutative algebra with generators $x_1,x_2,x_3$ and the following 3 relations labelled by cyclic parmutations $(i,j,k)$ of $(1,2,3)$: $x_ix_j-t\cdot x_jx_i=\Phi _k(x_k)$, $\Phi _k\in \mathbb {C}[x_k]$. This gives a family of Calabi-Yau algebras $\mathfrak {A}^t(\Phi )$ parametrized by a complex number $t\in \mathbb {C}^\times $ and a triple
$\Phi =(\Phi _1,\Phi _2,\Phi _3)$, of polynomials of specifically chosen degrees.
Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form $\mathfrak {A}^t(\Phi )/\langle \langle \Psi \rangle \rangle $,
where $\langle \langle \Psi \rangle \rangle \subset \mathfrak {A}^t(\Phi )$ stands for the ideal generated by a central element $\Psi $ which generates the center of the algebra $\mathfrak {A}^t(\Phi )$ if $\Phi $ is generic enough.

LA - eng

KW - del Pezzo surfaces; Poisson structures; Calabi-Yau deformations; Hochschild cohomology; del Pezzo surfaces; Poisson structures; Calabi-Yau deformations; Hochschild cohomology

UR - http://eudml.org/doc/277421

ER -

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