L’objectif dans ce travail est de présenter une généralisation pour l’obstruction d’Euler locale d’une fonction holomorphe singulière à l’origine dans le cas d’une application holomorphe $f:(V,0)\to ({ℂ}^k,0)$, où $(V,0)$ est un germe de variété analytique complexe, équidimensionnel de dimension $n\ge k$. Le résultat principal (Théorème 6.1) exprime l’obstruction d’Euler locale, définie pour un $k$-repère par Brasselet, Seade, Suwa, en fonction de l’obstruction d’Euler relative à $f$.

Let $Y$ be a divisor on a smooth algebraic variety $X$. We investigate the geometry of the Jacobian scheme of $Y$, homological invariants derived from logarithmic differential forms along $Y$, and their relationship with the property that $Y$ be a free divisor. We consider arrangements of hyperplanes as a source of examples and counterexamples. In particular, we make a complete calculation of the local cohomology of logarithmic forms of generic hyperplane arrangements.

This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.

Lines on hypersurfaces with isolated singularities are classified. New normal forms of simple singularities with respect to lines are obtained. Several invariants are introduced.

In this paper we study a notion of local volume for Cartier divisors on arbitrary blow-ups of normal complex algebraic varieties of dimension greater than one, with a distinguished point. We apply this to study an invariant for normal isolated singularities, generalizing a volume defined by J. Wahl for surfaces. We also compare this generalization to a different one arising in recent work of T. de Fernex, S. Boucksom, and C. Favre.

There is a well known relation between simple algebraic groups and simple singularities, cf. [5], [28]. The simple singularities appear as the generic singularity in codimension two of the unipotent variety of simple algebraic groups. Furthermore, the semi-universal deformation and the simultaneous resolution of the singularity can be constructed in terms of the algebraic group. The aim of these notes is to extend this kind of relation to loop groups and simple elliptic singularities. It is the...

A mapping $F:ℝⁿ\to {ℝ}^m$ is called overdetermined if m > n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping $F:ℝⁿ\to {ℝ}^m$ can be reduced to the case m = n.

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber $f^{-1}\left(t₀\right)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}\left(t₀\right)$. We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula...

For every polynomial F in two complex variables we define the Łojasiewicz exponents ${ℒ}_{p,t}\left(F\right)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents ${ℒ}_{p,t}\left(F\right)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.

Electro-muscular disruption (EMD) devices such as TASER M26 and X26 have been used as a less-than-lethal weapon. Such EMD devices shoot a pair of darts toward an intended target to generate an incapacitating electrical shock. In the use of the EMD device, there have been controversial questions about its safety and effectiveness. To address these questions, we need to investigate the distribution of the current density J inside the target produced by the EMD device. One approach is to develop a computational...

Nous nous donnons, dans l’anneau des germes de fonctions holomorphes à l’origine de ${ℂ}^2$, une fonction $f$ définissant une singularité isolée et nous nous intéressons à l’équation $u{f}_x^{\prime }+v{f}_y^{\prime }=wf$, lorsque la fonction $w$ est donnée. Nous introduisons les multiplicités d’intersection relatives de $w$ et $f_y^{\prime }$ le long des branches de $f$ et nous étudions les solutions à l’aide de ces valuations. Grâce aux résultats ainsi démontrés, nous construisons explicitement une équation fonctionnelle vérifiée par $f$.