Let $D$ be a bounded strictly pseudoconvex domain in ${ℂ}^2$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$. This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in ${ℂ}^2$.

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. ...

Let $D_j\subset {ℂ}^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, j = 1,...,N. Put $X:={\bigcup }_{j=1}^{N}A₁\times ...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset {ℂ}^{k₁+...+k_N}$. Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f̂|}_{X\setminus M}=f$. The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

Let $\Omega$ an open set in ${\mathbf{C}}^4$ near $z_0\in \partial \Omega$, $\lambda$ a suitable holomorphic function near $z_0$. If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\bar{\partial }u=\lambda f$, ($f$ is a $(0,1)$ form, $\bar{\partial }$ closed in $U\left(z_0\right)$ in $U\left(z_0\right)$ with supp$\left(u\right)\subset \bar{\Omega }\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U\left(z_0\right)$, as holomorphic fonctions in $\Omega \cap V\left(z_0\right)$.

The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: $|{\sum }_{n=0}^{\infty }aₙzⁿ|<1$, |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $|{\sum }_{n=0}^{k}aₙzⁿ|<1$, |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are...

Let $D\subset \subset {ℂ}^n,n\ge 2$, be a domain with $C^2$-boundary and $K\subset \partial D$ be a compact set such that $\partial D\setminus K$ is connected. We study univalent analytic extension of CR-functions from $\partial D\setminus K$ to parts of $D$. Call $K$ CR-convex if its $A\left(D\right)$-convex hull, $A\left(D\right)-\mathrm{hull}\left(K\right)$, satisfies $K=\partial D\cap A\left(D\right)-\mathrm{hull}\left(K\right)$ ($A\left(D\right)$ denoting the space of functions, which are holomorphic on $D$ and continuous up to $\partial D$). The main theorem of the paper gives analytic extension to $\partial D\setminus A\left(D\right)-\mathrm{hull}\left(K\right)$, if $K$ is CR- convex.

Let $V$ be an algebraic variety in $C^n$ and when $k\ge 0$ is an integer then $\mathrm{Pol}(V,k)$ denotes all holomorphic functions $f\left(z\right)$ on $V$ satisfying $\left|f\left(z\right)\right|\le C_f(1+|z|{)}^k$ for all $z\in V$ and some constant $C_f$. We estimate the least integer $ϵ(V,k)$ such that every $f\in \mathrm{Pol}(V,k)$ admits an extension from $V$ into $C^n$ by a polynomial $P(z_1,...,z_n)$, of degree $k+ϵ(V,k)$ at most. In particular ${lim}_{k\&gt;\infty }ϵ(V,k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$. The existence of the finite integer $ϵ(V,k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem. ...