This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7, 8, 9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be...

We consider the representation theory for a class of log-canonical surface singularities in the sense of reflexive (or equivalently maximal Cohen-Macaulay) modules and in the sense of finite dimensional representations of the local fundamental group. A detailed classification and enumeration of the indecomposable reflexive modules is given, and we prove that any reflexive module admits an integrable connection and hence is induced from a finite dimensional representation of the local fundamental...

An effective formula for the Łojasiewicz exponent of a polynomial mapping of ℂ² into ℂ² at an isolated zero in terms of the resultant of its components is given.

Nous considérons un germe $\omega$ de 1-forme analytique dans ${ℝ}^2,0$ dont le 1-jet est $ydy$. Nous montrons que si l’équation $\omega =0$ définit un centre (i.e toutes les courbes solutions sont des cycles) il existe une involution analytique de ${ℝ}^2,0$ préservant le portrait de phase du système. Géométriquement ceci signifie que les centres analytiques nilpotents sont obtenus par image réciproque par des applications pli. Un théorème de conjugaison équivariante permet d’obtenir une classification complète de ces centres.

Let $X_1$ and $X_2$ be two compact strongly pseudoconvex CR manifolds of dimension $2n-1\ge 5$ which bound complex varieties $V_1$ and $V_2$ with only isolated normal singularities in ${ℂ}^{N1}$ and ${ℂ}^{N2}$ respectively. Let $S_1$ and $S_2$ be the singular sets of $V_1$ and $V_2$ respectively and $S_2$ is nonempty. If $2n-N_2-1\ge 1$ and the cardinality of $S_1$ is less than 2 times the cardinality of $S_2$, then we prove that any non-constant CR morphism from $X_1$ to $X_2$ is necessarily a CR biholomorphism. On the other hand, let $X$ be a compact strongly pseudoconvex CR manifold of...

We present the notion of scattering monodromy for a two degree of freedom hyperbolic oscillator and apply this idea to determine the Picard-Lefschetz monodromy of the isolated singular point of a quadratic function of two complex variables.

We investigate the quadratic homogeneous holomorphic vector fields on ${\mathbf{C}}^n$ that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove...

Let $𝒰$ be an open neighborhood of the origin in ${ℂ}^{n+1}$ and let $f:(𝒰,\mathbf{0})\to (ℂ,0)$ be complex analytic. Let $z_0$ be a generic linear form on ${ℂ}^{n+1}$. If the relative polar curve ${\Gamma }_{f,z_0}^1$ at the origin is irreducible and the intersection number $({\Gamma }_{f,z_0}^1{·V\left(f\right))}_{\mathbf{0}}$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $({\Gamma }_{f,z_0}^1{·V\left(f\right))}_{\mathbf{0}}$ is not prime.