The Index of Differential Forms on Complete Intersections

The article is devoted to the development of a homological approach to the problem of calculating the local topological index of holomorphic differential $1$-forms given on complex space. In the study of complete intersections our method is based on the construction of Lebelt and Cousin resolutions, as well as on the simplest properties of the generalized and usual Koszul complexes, regular meromorphic differential forms, and the residue map. In particular, we show that the index of a differential $1$-form with an isolated singularity is equal to the dimension of the local analytical algebra of a zero-dimensional germ which is determined by the ideal generated by the interior product of the form and all Hamiltonian vector fields of the complete intersection. Moreover, in the quasihomogeneous case, the index can be expressed explicitly in terms of values of classical symmetric functions. We also discuss some other methods for computing the homological index of $1$-forms given on analytic spaces with singularities of various types.