Noncommutative del Pezzo surfaces and Calabi-Yau algebras

  • Pavel Etingof

    Massachusetts Institute of Technology, Cambridge, United States
  • Victor Ginzburg

    University of Chicago, USA


The hypersurface in with an isolated quasi-homogeneous elliptic singularity of type , , has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type 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 to a noncommutative algebra with generators and the following 3 relations labelled by cyclic parmutations of :

This gives a family of Calabi-Yau algebras parametrized by a complex number and a triple 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 , where stands for the ideal generated by a central element Ψ which generates the center of the algebra if is generic enough.

Cite this article

Pavel Etingof, Victor Ginzburg, Noncommutative del Pezzo surfaces and Calabi-Yau algebras. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1371–1416

DOI 10.4171/JEMS/235