Noncommutative del Pezzo surfaces and Calabi-Yau algebras

The hypersurface in C3 with an isolated quasi-homogeneous elliptic singularity of type Ẽr, 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[x1, x2, x3] to a noncommutative algebra with generators x1, x2, x3 and the following 3 relations labelled by cyclic parmutations (i, j, k) of (1, 2, 3): xixj − t·xjxi = Φk(xk), Φk ∈ C[xk]. This gives a family of Calabi-Yau algebras At(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 At(Φ)/〈〈Ψ〉〉, where 〈〈Ψ〉〉 ⊂ At(Φ) stands for the ideal generated by a central element Ψ which generates the center of the algebra At(Φ) if Φ is generic enough.