Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Etingof, 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 -