The Local Euler Obstruction

The local Euler obstruction was first introduced by R. MacPherson in [117] as an ingredient for the construction of characteristic classes of singular complex algebraic varieties. An equivalent definition was given by J.-P. Brasselet and M.-H. Schwartz in [33] using vector fields. Their viewpoint brings the local Euler obstruction into the framework of "indices of vector fields on singular varieties," though the definition only considers radial vector fields. This approach is most convenient for our study which is based on [29, 32] and shows relations with other indices. There are various other definitions and interpretations, in particular due to Gonzalez-Sprinberg [72], Verdier, Le-Teissier and others. The survey [27] provides an overview on the subject. Section 1 below is devoted to the definition of the local Euler obstruction and some of its main properties. The behavior of the local Euler obstruction relatively to hyperplane sections is described in Sect. 2, following [29]. In Sect. 3 and the thereafter we study a generalization of the local Euler obstruction introduced in [32] and called the Euler obstruction of the function, or also the "Euler defect"; this is an invariant associated to map-germs on singular varieties. MacPherson's local Euler obstruction corresponds to the square of the function distance to the given point. It is shown in [150], and explained in the last section of this chapter, that this invariant can be expressed in terms of the number of critical points in the regular part of a Morsification of the function.