Radial Index and Euler Obstruction of a 1-Form on a Singular Variety

Abstract

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A connection between the radial index and the local Euler obstruction of a 1-form is described. This gives an expression for the local Euler obstruction of the differential of a function in terms of Euler characteristics of some Milnor fibres.

On indices of 1-forms on determinantal singularities

“…dim e X i reflects the difference between the radial index of a complex 1-form and of its real part, see, e.g., [2]). Therefore the sum of the indices of the zeros of the 1-form ω on the entire stratum X i \ X i−1 is equal to…”

“…Applying one of the three possible resolutions of the variety X over a point of X i \ X i−1 , we glue in a standard fibre G In the formula expressing the radial index of a 1-form in terms of the Euler obstructions of the 1-form on the different strata [2], there participate certain integer coefficients n i . Up to sign they are equal to the reduced Euler characteristics of generic hyperplane sections of normal slices of the variety at points of different strata of a Whitney stratification.…”

Indices of Vector Fields on Singular Varieties: An Overview

“…But the systematic study of these indices was begun by W. Ebeling and S. Gusein-Zade in a series of articles (see for instance [16,18,19,20]). This has been, to some extent, a study parallel to the one for vector fields, outlined in this article.…”

Variedades determinantais e singularidades de matrizes

“…Para variedades determinantais com singularidade isolada, a relação entre PH-índice e ó ındice radial, (ver [14]),é dada por…”