Euler Obstruction and Defects of Functions on Singular Varieties

Abstract

Abstract. Several authors have proved Lefschetz type formulae for the local Euler obstruction. In particular, a result of this type is proved in [BLS]. The formula proved in that paper 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 this work is to understand what prevents the local Euler obstruction of 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 that we define below. We give an interpretation of this defect in terms of vanishing cycles, which allows us to calculate it algebraically, using [Ma2,Ma3]. When the function has an isolated singularity, our invariant can be defined geometrically, via obstruction theory. We notice that this invariant unifies the usual concepts of the Milnor number of a function and of the local Euler obstruction of an analytic set. INTRODUCTION.This work is a natural continuation of [BLS], where it is proved a Lefschetztype formula for the local Euler obstruction. More precisely, let (X, 0) be an equidimensional complex analytic singularity germ in an open set U ⊂ C N . We endow (X, 0) with a Whitney stratification {V i } and consider a general complex linear form l : C N → C. The formula of [BLS] says that the local Euler obstruction of X at 0 satisfies:where B ε is a small ball around 0 in C N , t 0 ∈ C \ {0} is sufficiently near {0} and Eu X (V i ) is the Euler obstruction of X at any point of the stratum V i .

Milnor numbers and Euler obstruction*

“…The number α q may be interpreted as the intersection number within T * C N between dF and the conormal T * Xreg . Therefore our Proposition 2.3 may be compared to [BMPS,Corollary 5.4], which is proved by using different methods. J. Schürmann informed us that such a result can also be obtained using the techniques of [Sch].…”

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

“…Let ω = df for the germ f of a holomorphic function on (C N , 0). Then Eu X,0 df differs from the Euler obstruction Eu f,X (0) defined in [3] by the sign (−1) n . The reason is that for the germ of a holomorphic function with an isolated critical point on (C n , 0) one has Eu f,X (0) = (−1) n µ f (see [3,Remark 3.4]), whence Eu X,0 df = µ f (µ f is the Milnor number of the germ f ).…”

“…In [3], there is introduced the notion of the local Euler obstruction Eu f,X (0) of a holomorphic function f : (X, 0) → (C, 0) with an isolated critical point on the germ of a complex analytic variety (X, 0). It is defined through an appropriately constructed gradient vector field tangent to the variety X.…”

“…if its differential df has an isolated singular point there) we describe a connection between the radial index of the differential df and the Euler characteristic of the Milnor fibre of the germ f . In the described situation there is also an appropriately adapted notion of the local Euler obstruction of a 1-form on the germ of a complex analytic variety [3]. We describe a connection between the radial index and the local Euler obstruction.…”

Global Euler obstruction and polar invariants

“…In the local setting, various authors have proved "Lefschetz type" formulas for the local Euler obstruction, see for instance [Du1,LT,BLS,Sch1,BMPS]; we refer to the bibliography for background on this topic.…”