Invariants of Singularities with Group Actions

矢量场的指数，特别是关于奇异(实数或复数)变种的指数，一直受到深入的研究。申请人和S.M.Gusein-Zade也对这项研究做出了贡献。特别是，他们建议还考虑1-形式的索引和1-形式或奇异变种上的矢量场的集合。申请人和Gusein-Zade还研究了其他不变量，如复奇点的Poincaré级数和单调Zeta函数。在申请者与A.Takahashi以及后来与Gusein-Zade的合作中，事实证明，研究奇点及其对称群对于更好地理解对偶现象是很重要的。因此，这个项目的主要目标是在有限群对簇的作用下探索这些不变量。这在某种程度上意味着开创性的工作。它计划研究概念和陈述的等变和奥比福尔德类比。其中一个目的也是将结果应用到Orbillold Landau-Ginzburg模型中。

Indices of vector fields, in particular on singular (real or complex) varieties, have been subject to intensive investigations. The applicant and S.M.Gusein-Zade also contributed to this research. In particular, they suggested considering also indices of 1-forms and of collections of 1-forms or vector fields on singular varieties. The applicant and Gusein-Zade have also studied other invariants like Poincaré series and monodromy zeta functions of complex singularities. In joint work of the applicant with A.Takahashi and later with Gusein-Zade, it turned out that it is important to study singularities together with their groups of symmetries to get among other things a better understanding of duality phenomena. Therefore the main objective of this project is to explore these invariants in the presence of the action of a finite group on the variety. This means to some extent pioneering work. It is planned to study both equivariant and orbifold analogues of the notions and statements. One of the aims is also to apply the results to orbifold Landau-Ginzburg models.