非交换del Pezzo曲面与Calabi-Yau代数
Noncommutative del Pezzo surfaces and Calabi-Yau algebras
摘要：
三维向量空间中具有E_r型孤立拟齐次椭圆奇点的超曲面，r= 6，7，8，具有自然Poisson结构.我们证明了相应类型E_r的del Pezzo曲面族提供了该Poisson结构的半泛Poisson变形。我们还构造了这样的del Pezzo曲面的坐标环的变形量子化。为此，我们首先将多项式代数C[x，y，z]变形为具有生成元x，y，z和以下3个关系的非交换代数（其中[u，v]_t = uv-t.vu）：[x，y]_t=F_1（z），[y，z]_t=F_2（x），[z，x]_t=F_3（y）.这就给出了一族由复数t和三元组F=（F_1，F_2，F_3）参数化的Calabi-Yau代数A（F），其中三元组F是一个特定次数的单变量多项式。我们的del Pezzo曲面的坐标环的量子化是由形式为A（F）/（g）的非交换代数提供的，其中（g）代表由中心元素g生成的A（F）的理想，如果F足够一般，则该理想生成代数A（F）的中心。
英文摘要：
The hypersurface in a 3-dimensional vector space with an isolated quasi-homogeneous elliptic singularity of type E_r,r=6,7,8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type E_r 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,y,z] to a noncommutative algebra with generators x,y,z and the following 3 relations (where [u,v]_t = uv- t.vu): [x,y]_t=F_1(z), [y,z]_t=F_2(x), [z,x]_t=F_3(y). This gives a family of Calabi-Yau algebras A(F) parametrized by a complex number t and a triple F=(F_1,F_2,F_3), of polynomials in one variable of specifically chosen degrees. Our quantization of the coordinate ring of a del Pezzo surface is provided by noncommutative algebras of the form A(F)/(g) where (g) stands for the ideal of A(F) generated by a central element g, which generates the center of the algebra A(F) if F is generic enough.
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