Orbifold concepts in equivariant singularity theory

拓扑空间的不变量和群作用下的复解析簇(比如有限群)在拓扑学、代数几何和数学物理中起着重要的作用。申请人和他的合著者S.M.Gusein-Zade和A.Takahashi也对这项研究做出了贡献。特别地，他们描述了具有有限群对称性的所谓的可逆多项式之间的一些对称性(奥比博尔德Landau-Ginzburg模型)。他们定义和研究了不变或等变向量场和1-形式及其集合的一些指标。该项目的主要目标是进一步发展有限群作用下的不变量理论。特别的焦点将放在orb orbold设置和orb orbold类型不变量上。更确切地说，计划继续寻找Berglund-Hübsch对偶可逆多项式(也有可能是非交换对偶群)的不变量之间的对称性，研究1-形式和向量场的同变指数的代数公式，并研究Milnor格的orbilold类似物。所提出的研究还包括对McKay对应的推广的研究，对Orbilold的陈特征数的推广的研究，以及对Orlik-Randell猜想的推广的研究。

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.