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 $\Omega \cap V\left(z_0\right)$.

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 ${lim}_{k\>\infty }ϵ(V,k)$ 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. ...

We give an elementary proof of the Briançon-Skoda theorem. The theorem gives a criterionfor when a function $\phi$ belongs to an ideal $I$ of the ring of germs of analytic functions at $0\in {ℂ}^n$; more precisely, the ideal membership is obtained if a function associated with $\phi$ and $I$ is locally square integrable. If $I$ can be generated by $m$ elements,it follows in particular that $\overline{I^{min(m,n)}}\subset I$, where $\overline{J}$ denotes the integral closure of an ideal $J$.

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]. ...

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. ...

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f:W:=(A\times Y)\cup (X\times B)\to Z$ extends to a holomorphic mapping $\widehat{f}$ on $\widehat{W}:=\left\{(z,w)\in X\times Y:\tilde{\omega }(z,A,X)+\tilde{\omega }(w,B,Y)\<1\right\}$ such that $\widehat{f}=f$ on $W\cap \widehat{W},$ where $\tilde{\omega }(·,A,X)$ (resp. $\tilde{\omega }(·,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$). Generalizations...