# Noncommutative del Pezzo surfaces and Calabi-Yau algebras

### Pavel Etingof

Massachusetts Institute of Technology, Cambridge, United States### Victor Ginzburg

University of Chicago, USA

## Abstract

The hypersurface in $C_{3}$ with an isolated quasi-homogeneous elliptic singularity of type $E~_{r}$, $r=6,7,8$, 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 $C[x_{1},x_{2},x_{3}]$ 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 $(1,2,3)$:

This gives a family of Calabi-Yau algebras $A_{t}(Φ)$ parametrized by a complex number $t∈C_{×}$ and a triple $Φ=(Φ_{1},Φ_{2},Φ_{3})$ 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 $A_{t}(Φ)/《Ψ》$, where $《Ψ》⊂A_{t}(Φ)$ stands for the ideal generated by a central element Ψ which generates the center of the algebra $A_{t}(Φ)$ if $Φ$ is generic enough.

## Cite this article

Pavel Etingof, Victor Ginzburg, Noncommutative del Pezzo surfaces and Calabi-Yau algebras. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1371–1416

DOI 10.4171/JEMS/235