Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Abstract

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 three relations labeled by cyclic parmutations (i, j, k) of (1, 2, 3): xixj − t · xjxi = 8k(xk), 8k ∈ C[xk]. This gives a family of Calabi–Yau algebras At(8) parametrized by a complex number t ∈ C and a triple 8 = (81,82,83) 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(8)/〈〈9〉〉, where 〈〈9〉〉 ⊂ At(8) stands for the ideal generated by a central element 9 which generates the center of the algebra At(8) if 8 is generic enough.