Homological index for 1-forms and a Milnor number for isolated singularities

Details Homological index for 1-forms and a Milnor number for isolated singularities Homological index for 1-forms and a Milnor number for isolated singularities

2003-07-17

arxiv.org/abs/math/0307239v1

Mathematics

Algebraic Geometry

Scientific paper

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by X. G\'omez-Mont and G.-M. Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by R.-O. Buchweitz and G.-M. Greuel for such singularities.

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