In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if $(X,0)\subset (\mathbb {C}^{n},0)$, $(Y,0)\subset (\mathbb {C}^{m},0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: if $X\subset \mathbb {C}^{n}$, $Y\subset \mathbb {C}^{m}$ are algebraic sets and $\phi \colon X\to Y$ is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb {P}^{n+m}(\mathbb {C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$.

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$. We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$. As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

Le diagramme d’Eisenbud et Neumann d’un germe est un arbre qui représente ce germe et permet d’en calculer les invariants. On donne une démonstration algébrique d’un résultat caractérisant l’ensemble des quotients jacobiens d’un germe d’application $(f,\,g)$ à partir du diagramme d’Eisenbud et Neumann de $fg$.

We prove the existence of complexified real arrangements with the same combinatorics but different embeddings in ${\mathbb P}^2$. Such a pair of arrangements has an additional property: they admit conjugated equations on the ring of polynomials over ${\mathbb Q}(\sqrt{5})$.

an index of a collection of 1-forms on a complex isolated complete intersection singularity corresponding to a chern number is defined and – in the case when the 1-forms are complex analytic – expressed as the dimension of a certain algebra.

Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.

This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\mathbb{Q}$-Gorenstein variety $X$ we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective $\mathbb{Q}$-Cartier divisors on $X$ whose support contains the singular locus of $X$.