Local Euler Obstruction Research Articles

Microlocal study of topological Radon transforms and real projective duality

Various topological properties of projective duality between real projective varieties and their duals are obtained by making use of the microlocal theory of (subanalytically) constructible sheaves developed by Kashiwara [M. Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985) 193–209] and Kashiwara–Schapira [M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren Math. Wiss., vol. 292, Springer, Berlin–Heidelberg–New York, 1990]. In particular, we prove in the real setting some results similar to the ones proved by Ernström in the complex case [L. Ernström, Topological Radon transforms and the local Euler obstruction, Duke Math. J. 76 (1994) 1–21]. For this purpose, we describe the characteristic cycles of topological Radon transforms of constructible functions in terms of curvatures of strata in real projective spaces.

EULER OBSTRUCTION, POLAR MULTIPLICITIES AND EQUISINGULARITY OF MAP GERMS IN $\mathcal{O}(n,p), n < p$

There are two main goals in this article, one of them is to minimize the number of invariants needed to obtain Whitney equisingular one parameter families of finitely determined holomorphic map germs ft:(ℂn,0) → (ℂp,0), with n < p. The other is to show how to compute the local Euler obstruction of the stable types which appear in a finitely determined map germ in these dimensions. The polar multiplicities of all stable types and the 0-stable singularities are the invariants that guarantee the Whitney equisingularity of such families and the polar multiplicities are the numbers that also allow us to compute the local Euler obstruction. Therefore our first step is to describe all stable types which appear when n < p and show the relationship between the polar multiplicities in each stable type. Using the fact that these polar multiplicities are upper semi-continuous we minimize the number of invariants that guarantee Whitney equisingularity of such a family. We also apply the relationship between the polar multiplicities in each stable type and a result of Lê and Teissier to show how to compute the local Euler obstruction of the stable types which appear in these dimensions.

Invariants, equisingularity and Euler obstruction of map germs from ℂnto ℂn

We study how to minimize the number of invariants that is sufficient for the Whitney equisingularity of a one parameter deformation of corank one finitely determined holomorphic germ ƒ : (ℂn, 0) → (ℂn, 0). According to a result of Gaffney, these are the 0-stable invariants and all polar multiplicities which appear in the stable types of a stable deformation of the germ. First we describe all stable types, then we show how the invariants in the source and the target are related and reduce the number using these relations. We also investigate the relationship between the local Euler obstruction and the polar multiplicities of the stable types. We show an algebraic formula for the local Euler obstruction in terms of the polar multiplicities and show that the Euler obstruction is an invariant for the Whitney equisingularity.

Milnor numbers and Euler obstruction*

Using a geometric approach, we determine the relations between the local Euler obstruction Eu f of a holomorphic function f and several generalizations of the Milnor number for functions on singular spaces.

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

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.

EULER OBSTRUCTION AND DEFECTS OF FUNCTIONS ON SINGULAR VARIETIES

Several authors have proved Lefschetz type formulas for the local Euler obstruction. In particular, a result of this type has been proved that turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of the paper is to determine what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ‘defect’) of such functions. An interpretation of this defect is given in terms of vanishing cycles, which allows it to be calculated algebraically. When the function has an isolated singularity, the invariant can be defined geometrically, via obstruction theory. This invariant unifies the usual concepts of the Milnor number of a function and the local Euler obstruction of an analytic set.

On a naturality of Chern-Mather classes

Introduction. The Chern-Schwartz-MacPher- son class (or more precisely natural transformation) is the unique natural transformation from the co- variant constructible function functor to the covari- ant integral homology functor satisfying the normal- ization that the value of the characteristic function of a nonsingular compact complex analytic variety is equal to the Poincare dual of the total Chern cohomology class of the tangent bundle. The ex- istence of such a transformation was conjectured by Deligne and Grothendieck and was proved by MacPherson (10). The novelity of MacPherson's proof is introducing the notion of local Euler obstruc- tion (which was independently introduced by Kashi- wara (7) also) and assigning the Chern-Mather class to this local Euler obstruction, not to the character- istic function. Although the Chern-Mather class is a very geometrically simple homology class, the assign- ment of the Chern-Mather class to the characteristic function does not give such a natural transformation. It is often said that few functorial proper- ties are know for the Chern-Mather class (e.g., see (6, Note, page 377, right after Example 19.1.7)), al- though the assignment of the Chern-Mather class to the local Euler obstruction is perfectly natural, which is the main part of MacPherson's proof. In this paper, using this fine naturality of this assingment, we interpret the Chern-Mather class in the same way as the Chern-Schwartz-MacPherson class. Further- more, by introducing the notion of a q-deformed local Euler obstruction which unifies the charac- teristic function and the local Euler obstruction, we give a q-deformed Chern-Schwartz-MacPherson class natural transformation, which specializes to the Chern-Mather class natural transformation for q = 0 and the Chern-Schwartz-MacPherson class natural transformation for q = 1.

Topological Radon transforms and the local Euler obstruction

Topological Radon transforms and the local Euler obstruction

MacPherson's and Fulton's Chern classes of hypersurfaces

In this note we compare two notions of Chern class of an algebraic scheme X (over C) specializing to the Chern class of the tangent bundle c(TX) ∩ [X] when X is nonsingular. The first of such notions is MacPherson’s Chern class, defined by means of Mather-Chern classes and local Euler obstructions [5]. MacPherson’s Chern class is functorial with respect to a push-forward defined via topological Euler characteristics of fibers; in particular, mapping to a point shows that the degree of the zero-dimensional component of MacPherson’s Chern class of a complete variety X equals the Euler characteristic χ(X) of X. We denote MacPherson’s Chern class of X by cMP (X). The second notion is Fulton’s intrinsic class of schemes X ′ that can be embedded in a nonsingular variety M : Fulton shows ([3], Example 4.2.6) that the class cF (X ′) = c(TM) ∩ s(X ′,M) is independent of the choice of embedding of X ′. This class has the advantage of being defined over arbitrary fields and in a completely algebraic fashion, but does not satisfy at first sight nice functorial properties: cf. [3], p. 377. (MacPherson’s class can also be defined algebraically over any field of characteristic 0: this is done in [4].) To state our result we need to remind the reader that if W is a scheme supported on a Cartier divisor X of a nonsingular variety M , then the Segre class of W in M can be written in terms of the Segre class of X and the Segre class of the residual scheme J to X in W : for a precise statement of this fact, see [3], Proposition 9.2, or section 2 below. By modifying this expression, we can make sense of the “Segre class” in M of an object “X \J” in which J is intuitively speaking “removed” from X. Since this object has a Segre class, we can define its Fulton–Chern class as above. Here is our result:

A formula for Segre classes of singular projective varieties

The Segre class of a singular projective variety X is that of the normal cone of the diagonal in the product X x X. This class was introduced by K. W. Johnson and W. Fulton to study immersions and embeddings. In our previous work we related the Segre classes and the Chern-Mather classes for hypersurfaces with codimension one singularities and Xn c P2 with isolated singularities. In this paper we generalize these results to the case of Xn c FN with singularities of codimension N — n (N < In). The notion of Segre classes (of cones) has become of increasing importance as a key ingredient for constructing or analyzing various invariants, e.g., in intersection theory and group representation theory, etc. The Segre class treated in this note is the Segre class of a singular projective variety, which was introduced by K. W. Johnson (and W. Fulton) to study immersions and embeddings of singular projective varieties [4]. This is the Segre class of the normal cone C^{X x X) of the diagonal Δ in the product X x X. We call this class Johnson's Segre class, denoted by S*(X). Another well-studied characteristic class of a singular variety is MacPherson's Chern class, the existence of which was conjectured by Deligne and Grothendieck. R. MacPherson [7] constructed this Chern class, using Chern-Mather classes and introducing the notion of local Euler obstruction. A. Dubson [2] gave a more concrete description for MacPherson's Chern class C*(X) : Let S?χ be a (in fact, any) Whitney stratification of X with the smooth part of X as the top-dimensional stratum and let C^(X) denote the Chern-Mather class of X. Then ms •