Invariants of Singularities with Group Actions

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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.