The Poincaré Index on Singular Varieties

In this paper, we discuss a few simple methods for computing the local topological index and its various analogs vector fields differential forms given on complex varieties with singularities of different types. They are based properties regular meromorphic logarithmic forms, dualizing (canonical) module related constructions. particular, show how to compute Cohen–Macaulay, Gorenstein monomial curves, normal non-normal surfaces some others. contrast known traditional approaches, use neither computers, nor integration, perturbations, deformations, resolution singularities, spectral sequences or other standard tools pure mathematics.