# 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 ℂ3 with an isolated quasi-homogeneous elliptic singularity of type *Ē__r*, *r* = 6, 7, 8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type *Er* 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):

*xi* *xj* − *t*·*xi* *xj* = Φ_k_ (*xk*), Φ_k_ ∈ ℂ[*xk*].

This gives a family of Calabi-Yau algebras **A***t*(Φ) parametrized by a complex number *t* ∈ ℂ× and a triple Φ = (Φ1, Φ2, Φ3) 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