ZERO-DIMENSIONAL SYMPLECTIC ISOLATED COMPLETE INTERSECTION SINGULARITIES
ZERO-DIMENSIONAL SYMPLECTIC ISOLATED COMPLETE INTERSECTION SINGULARITIES
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零维辛孤立完全相交奇点
DOI：
10.5427/jsing.2012.6c
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
W. Domitrz
中科院分区：
文献类型：
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作者：
W. Domitrz
我们研究 0 维孤立完全交集奇点的局部辛代数。我们使用代数限制的方法对这些辛奇点进行分类。我们证明该分类中存在非平凡的辛不变量。 V. I. Arnold 在 (A1) 中介绍了奇异簇的辛分类问题。 Arnold 表明，平面曲线的 A2k 奇点（相对于参数化曲线的标准 A 等价的轨道）恰好分裂为 2k + 1 个辛奇点（相对于参数化曲线的辛等价的轨道）。他通过参数化曲线与最接近的平滑拉格朗日子流形的不同阶次切线来区分不同的辛奇异性。阿诺德提出了用局部代数与辛结构的相互作用来表达这些新的辛不变量的问题，他建议将这种相互作用称为局部辛代数。许多作者主要在奇异曲线的情况下研究了这个问题。在 (IJ1) 中，G. Ishikawa 和 S. Janeczko 对二维辛空间中的曲线辛奇点进行了分类。二维流形上的辛形式是光滑流形上体积形式的特例。 (IJ1) 中的结果推广到奇异品种的体积保持分类和任意维度的地图在 (DR) 中获得。所有微分同胚细菌的作用轨道与满足准同质性的特殊弱形式的细菌的 C 解析类别中的体积保持轨道一致。变体的弱准同质性是非负权重 si ≥ 0 且 P ii > 0 的准同质性。 P. A. Kolgushkin 将参数化曲线的稳定简单辛奇点分类在 C 解析类别 ((K)) 中。在 (DJZ2) 中，辛空间的奇异拟齐次子集的局部辛代数是通过辛形式对这些子集的代数限制来解释的。将达布-吉文塔尔定理 ((AG)) 推广到 (DJZ2) 中获得的辛空间的任意子集的萌芽，将拟齐次子集的萌芽的辛分类问题简化为辛形式的代数限制的分类问题
We study the local symplectic algebra of the 0-dimensional iso- lated complete intersection singularities. We use the method of algebraic re- strictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification. The problem of symplectic classification of singular varieties was introduced by V. I. Arnold in (A1). Arnold showed that the A2k singularity of a planar curve (the orbit with respect to the standard A-equivalence of parameterized curves) split into exactly 2k + 1 symplectic singularities (orbits with respect to the symplectic equivalence of parameterized curves). He distinguished different symplectic singu- larities by different orders of tangency of the parameterized curve to the nearest smooth Lagrangian submanifold. Arnold posed a problem of expressing these new symplectic invariants in terms of the local algebra's interaction with the symplectic structure and he proposed to call this interaction the local symplectic algebra. This problem was studied by many authors mainly in the case of singular curves. In (IJ1) G. Ishikawa and S. Janeczko classified symplectic singularities of curves in the 2-dimensional symplectic space. A symplectic form on a 2-dimensional man- ifold is a special case of a volume form on a smooth manifold. The generaliza- tion of results in (IJ1) to volume-preserving classification of singular varieties and maps in arbitrary dimensions was obtained in (DR). The orbit of the action of all diffeomorphism-germs agrees with the volume-preserving orbit in the C-analytic category for germs which satisfy a special weak form of quasi-homogeneity e.g. the weak quasi-homogeneity of varieties is a quasi-homogeneity with non-negative weightsi ≥ 0 and P ii > 0. P. A. Kolgushkin classified stably simple symplectic singularities of parameter- ized curves in the C-analytic category ((K)). In (DJZ2) the local symplectic algebra of singular quasi-homogeneous subsets of a symplectic space was explained by the algebraic restrictions of the symplectic form to these subsets. The generalization of the Darboux-Givental theorem ((AG)) to germs of arbitrary subsets of the symplectic space obtained in (DJZ2) reduces the problem of symplectic classification of germs of quasi-homogeneous subsets to the problem of classification of algebraic restrictions of symplectic forms to these