Let $M$ be a two-dimensional complex manifold and $f:M\to M$ a holomorphic map. Let $S\subset M$ be a curve made of fixed points of $f$, i.e. $\mathrm{Fix}\left(f\right)=S$. We study the dynamics near $S$ in case $f$ acts as the identity on the normal bundle of the regular part of $S$. Besides results of local nature, we prove that if $S$ is a globally and locally irreducible compact curve such that $S·S\<0$ then there exists a point $p\in S$ and a holomorphic $f$-invariant curve with $p$ on the boundary which is attracted by $p$ under the action of $f$. These results...

Let $X$ be a reduced $n$-dimensional complex space, for which the set of singularities consists of finitely many points. If $X^{\text{'}}\subseteq X$ denotes the set of smooth points, the author considers a holomorphic vector bundle $E\to X^{\text{'}}\setminus A$, equipped with a Hermitian metric $h$, where $A$ represents a closed analytic subset of complex codimension at least two. The results, surveyed in this paper, provide criteria for holomorphic extension of $E$ across $A$, or across the singular points of $X$ if $A=⌀$. The approach taken here is...

We prove a uniqueness result for Coleff-Herrera currents which in particular means that if $f=(f_1,...,f_m)$ defines a complete intersection, then the classical Coleff-Herrera product associated to $f$ is the unique Coleff-Herrera current that is cohomologous to $1$ with respect to the operator ${\delta }_f-\overline{\partial }$, where ${\delta }_f$ is interior multiplication with $f$. From the uniqueness result we deduce that any Coleff-Herrera current on a variety $Z$ is a finite sum of products of residue currents with support on $Z$ and holomorphic...

Soit $U_1$ un ouvert dans ${\mathbf{C}}^m$. Soit ${\pi }_1:\mathbf{S}\to U_1$ une famille holomorphe de structures complexes sur une surface de Riemann non-compacte $M$, avec ${\mathbf{S}}_{t_0}={\pi }_1^{-1}(t_0)=M$. ($\mathbf{S}=\mathbf{S}(M\times U_1)$ est une structure complexe sur le produit différentiable $M\times U_1$). Soit $M_1$ un domaine relativement compact dans $M$. On démontre : pour tout voisinage de Stein $U$ de $t_0$, assez petit, la famille ${\pi }_1:\mathbf{S}(M_1\times U)\to U$ est isomorphe à la famille $\pi :\Omega \to \pi \left(\Omega \right)$, où $\Omega$ est un de la variété produit $M\times {\mathbf{C}}^m$, $\pi$ étant la projection $M\times {\mathbf{C}}^m\to {\mathbf{C}}^m$. On donne aussi un résultat analogue pour le cas des variations différentiables. ...