Noncommutative del Pezzo surfaces and Calabi-Yau algebras

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 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.