Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1204.14004
Etingof, Pavel; Ginzburg, Victor
Noncommutative del Pezzo surfaces and Calabi-Yau algebras. (English)
[J] J. Eur. Math. Soc. (JEMS) 12, No. 6, 1371-1416 (2010). ISSN 1435-9855; ISSN 1435-9863/e

Let $A={\Bbb C}[x_1,x_2,x_3]$ be the polynomial ${\Bbb C}$-algebra in 3 variables, $t$ a non-zero complex number and choose a polynomial $\Phi_k\in {\Bbb C}[x_k]$ for each $1\leq k\leq 3$. Then the noncommutative ${\Bbb C}$-algebras ${\cal U}^t(\Phi)$ generated by $x_1,x_2,x_3$ with the relations: $x_1x_2-tx_2x_1=\Phi_3(x_3)$, $x_2x_3-tx_2x_1=\Phi_1(x_1)$, $x_3x_1-tx_1x_3=\Phi_2(x_2)$ are noncommutative deformations of $A$ and form a family of Calabi-Yau algebras. Here it constructs a deformation-quantization of the coordinate ring of a del Pezzo surface of type $E_r$, $6\leq r\leq 8$ considering noncommutative algebras of the form ${\cal U}^t(\Phi)/\langle\langle\Psi\rangle\rangle$, where $\langle\langle\Psi\rangle\rangle$ is the ideal generated by a central element $\Psi$, which generates the center of ${\cal U}^t(\Phi)$ if $\Phi$ is generic enough. Also it shows that the family of del Pezzo surfaces of type $E_r$ provides a semiuniversal Poisson deformation of the Poisson structure inherited by hypersurfaces in ${\Bbb C}^3$ with an isolated quasi-homogeneous elliptic singularity of type $E_r$.
[Dorin-Mihail Popescu (Bucureşti)]

MSC 2000:: *14B07 Deformations of singularities (local theory)
14H52 Elliptic curves
14J32 Calabi-Yau manifolds
13C14 Cohen-Macaulay modules

Keywords: del Pezzo surfaces; Poisson structures; Calabi-Yau deformations; Hochschild cohomology

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]