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

HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES