Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Abstract

The hypersurface in ${\mathbb{C}}^3$ with an isolated quasi-homogeneous elliptic singularity of type ${\tilde{E}}_r$, $r=6,7,8$, 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 $\mathcal{C}[x_1,x_2,x_3]$ 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 ${\mathfrak{A}}^t(\Phi )$ 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 ${\mathfrak{A}}^t(\Phi )/《\Psi 》$, where $《\Psi 》\subset {\mathfrak{A}}^t(\Phi )$ stands for the ideal generated by a central element Ψ which generates the center of the algebra ${\mathfrak{A}}^t(\Phi )$ if $\Phi$ is generic enough.