Orbifold concepts in equivariant singularity theory

等变奇点理论中的轨道概念

• 批准号: 346364045
• 负责人: Professor Dr. Wolfgang Ebeling
• 金额: --
• 依托单位: Institut für Algebraische Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/346364045?language=en
• 关键词: Orbifold; concepts; equivariant; singularity; theory

Invariants of topological spaces and complex analytic varieties with actions of groups (say, finite ones) play an important role, in particular in topology, algebraic geometry and mathematical physics. The applicant and his coauthors S. M. Gusein-Zade and A. Takahashi also contributed to this research. In particular, they have described a number of symmetries between so called invertible polynomials with finite group symmetries (orbifold Landau-Ginzburg models). They have defined and studied some indices of invariant or equivariant vector fields and 1-forms and of their collections. The main objective of the project is a further development of the theory of invariants in the presence of a finite group action. The special focus will be on the orbifold setting and on orbifold type invariants. More precisely, it is planned to continue the search for symmetries between invariants of Berglund-Hübsch dual invertible polynomials (also with actions of possibly non-abelian dual groups), to study algebraic formulae for equivariant indices of 1-forms and vector fields, and to investigate orbifold analogues of the Milnor lattice. The proposed research also includes a study of generalizations of the McKay correspondence, of Chern characteristic numbers of orbifolds, and of the Orlik-Randell conjecture.

拓扑空间的不变量和具有群作用的复解析簇（例如有限群）起着重要的作用，特别是在拓扑学、代数几何和数学物理中。申请人及其共同作者S. M. Gusein-Zade和A. Takahashi也为这项研究做出了贡献。特别是，他们已经描述了一些所谓的可逆多项式与有限群对称性（轨道朗道-金兹伯格模型）之间的对称性。他们定义并研究了不变或等变向量场和1-形式及其集合的指数。该项目的主要目标是在有限群作用下进一步发展不变量理论。特别的重点将放在orbifold设置和orbifold类型不变量。更确切地说，计划继续寻找伯格伦-许布施对偶可逆多项式（也可能与非阿贝尔对偶群的作用）的不变量之间的对称性，研究1-形式和向量场的等变指数的代数公式，并研究米尔诺格的轨道类似物。拟议的研究还包括研究的McKay对应，陈省身特征数的orbifolds，和Orlik-Randell猜想的推广。