Noncommutative del Pezzo surfaces and Calabi-Yau algebras

Abstract

The hypersurface in ℂ 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): x ixj - t · xjxi = Φk(xk), φk ∈ ℂ[x k]. This gives a family of Calabi-Yau algebras zt(Φ) parametrized by a complex number t ∈ Cdx 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 z t(Φ)/〈〈ψ〈〈, where 〈〈Ψ〈 〈 ⊂ z[t(Φ) stands for the ideal generated by a central element Ψ which generates the center of the algebra 2[t(Φ) if Φ is generic enough.