The Poincare index and the χy-characteristic of Hirzebruch

In the paper, we discuss several methods for computing the homology of contravariant and covariant versions of the classical De Rham complex on analytic spaces. Our approach is based on the theory of holomorphic and regular meromorphic differential forms, and is applicable in different settings depending on concrete types of varieties. Among other things, we describe how to compute by elementary calculations the homological index of vector fields and differential forms given on Cohen-Macaulay curves, graded normal surfaces, complete intersections and some others. In these situations, making use of ideas of X. Gomez-Mont, we derive explicit expressions for the local topological index of Poincare and its generalizations. Furthermore, applying similar methods to the study of certain other complexes, we investigate some challenges, relating to the computation of classical topological-analytical invariants, such as the Euler characteristic of the Milnor fibre of an isolated singularity, the multiplicity of the discriminant of the versal deformation, the dimension of torsion and cotorsion modules, and so on.