José Seade

Euler obstruction and indices of vector fields

Abstract In this paper we give a formula to calculate the Euler obstruction of a complex analytic singularity in the spirit of the Lefschetz Theorem on hyperplane sections. Namely, we show that the Euler obstruction of a singularity is obtained from the Euler obstructions of the singularities of a general hyperplane section passing near the singularity.

On Real Singularities with a Milnor Fibration

In this article we study the singularities defined by real analytic maps

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

\left( {\mathbb{R}^m ,0} \right) \to \left( {\mathbb{R}^2 ,0} \right)

A note on the eta function for quotients of PSL2(R) by co-compact Fuchsian groups

with an isolated critical point at the origin, having a Milnor fibration. It is known [14] that if such a map has rank 2 on a punctured neighbourhood of the origin, then one has a fibre bundle φ : S m−1 − → S 1, where K is the link. In this case we say that f satisfies the Milnor condition at 0 ∈ ℝ m . However, the map φ may not be the obvious map \( \frac{f} {{\parallel f\parallel }} \) as in the complex case [14, 9]. If f satisfies the Milnor condition at 0 ∈ ℝ m and for every sufficiently small sphere around the origin the map \( \frac{f} {{\parallel f\parallel }} \) defines a fibre bundle, then we say that f satisfies the strong Milnor condition at 0 ∈ ℝ m . In this article we first use well known results of various authors to translate “the Milnor condition” into a problem of finite determinacy of map germs, and we study the stability of these singularities under perturbations by higher order terms. We then complete the classification, started in [20, 21] of certain families of singularities that satisfy the (strong) Milnor condition. The simplest of these are the singularities in ℝ2 n ≅ ℂ n of the form \(\{ \sum _{i = 1}^nz_i^{{a_i}}z_i^{ - {b_i}} = 0, {a_i} > {b_i} \geqslant 1\}\) We prove that these are topologically equivalent (but not analytically equivalent!) to Brieskorn-Pham singularities.

Complex Schottky Groups

The Case of Manifolds.- The Schwartz Index.- The GSV Index.- Indices of Vector Fields on Real Analytic Varieties.- The Virtual Index.- The Case of Holomorphic Vector Fields.- The Homological Index and Algebraic Formulas.- The Local Euler Obstruction.- Indices for 1-Forms.- The Schwartz Classes.- The Virtual Classes.- Milnor Number and Milnor Classes.- Characteristic Classes of Coherent Sheaves on Singular Varieties.

Global Euler obstruction and polar invariants

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gomez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

Fibred links and a construction of real singularities via complex geometry

In this chapter we introduce some fundamental concepts in the theory of complex Kleinian groups that we study in the sequel. We begin with an example in \(\mathbb{P}^{2}_\mathbb{C}\) that illustrates the diversity of possibilities one has when defining the notion of “limit set”. In this example we see that there are several nonequivalent such notions, each having its own interest.

Morse Theory and the topology of holomorphic foliations near an isolated singularity

proportionality factor [7] provided F has no elliptic elements.) Our calculation uses little of the geometry of r\ PSLZ (R) but requires substantial information about the representations of SL2 (R). The group PSL2 (R) acts transitively and freely on r, .X, the space of unit tangent vectors to the upper half-plane .&‘, and may be identified with the orbit of (i, 1). If we give .X the standard Poincare’ metric (dx’ + dy2)/y2 and give T1 2 the induced metric; this metric is invariant under PSL,(R) and the basis vectors