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Youssef Alaoui

Publications and source records attributed to Youssef Alaoui.

10 recordsLinked to original sources
arXiv2026

The local Steiness problem with singularities

In this article, we prove that if $Π: X\rightarrow Ω$ is an unbranched Riemann domain with $Ω$ Stein of dimension $n$ and $Π$ a locally $q$-complete morphism, then $X$ is cohomologically $q$-complete if $n\geq 3$ and $1\leq q\leq n-2$ or if $Ω$ has dimension $2$ and $1\leq q\leq 2$. This generalizes a well-known result which is obtained in ~\cite{ref3} for $q=1$ when $X$ and $Ω$ have isolated singularities and, gives in particular a positive answer to the local Steiness problem, namely if $X$ is a Stein space and $Ω$ a locally Stein open subset of $X$, then $Ω$ is Stein.

math.CV↗
arXiv2026

A generalization of Hartog's extension of line bundles

In this article, we prove that if $X$ is a complex manifold of dimension $n\geq 4$ such that there exists a $q$-convex with corners function $f\in F_{q}(X)$, then every holomorphic line bundle over $\{f>c\}$ extends uniquely to $X$ if $1\leq q\leq n-3$. This generalizes a well-known result obtained in \cite{ref5} for $q$-complete with corners complex manifolds with a corresponding exhaustion function $f \in F_{q}(X)$, when $n \geq 3q$.

math.CV↗
arXiv2025

Increasing unions of Stein spaces with singularities

We show that if $X$ is a Stein space and, if $Ω\subset X$ is exhaustable by a sequence $Ω_1 \subset Ω_2 \subset \ldots \subset Ω_n \subset \ldots$ of open Stein subsets of $X$, then $Ω$ is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for $X=\mathbb{C}^n$ and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension 2, we prove that the same result follows if we assume only that $Ω\subset \subset X$ is a domain of holomorphy in a Stein normal space. It is known, however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets $X_1 \subset X_2 \subset \cdots \subset X_n \subset \cdots$, it does not follow in general that $X$ is holomorphically-convex or holomorphically-separate (even if $X$ has no singularities). One can even obtain 2-dimensional complex manifolds on which all holomorphic functions are constant.

math.CV↗
arXiv2025

On $q$-complete and $q$-concave with corners complex manifolds

It is proved that if there exists a positive and continuous function $f$ on an $n$-dimensional complex manifold $X$, $q$-convex with corners outside a compact set $K\subset X$ and which exhausts $X$ from below, then $dim_{\mathbb{C}}H^{p}(X,{\mathcal{F}})<+\infty$ for any coherent analytic sheaf ${\mathcal{F}}$ on $X$ if $p<n-q$. It is known from the theory of Andreotti and Grauert that if a complex space $X$ is $q$-complete, then $X$ is cohomoloogically $q$-complete. Until now it is not known in general if these two conditions are equivalent. The aim of section $4$ of this article is to provide a counterexample to the conjecture posed by Andreotti and Grauert ~\cite{ref2} to show that a cohomologically $q$-complete space is not necessarily $q$-complete. In section $5$ of this article, we will prove that there exist for each pair of integers $(n,q)$ with $2\leq q\leq n-1$ a $q$-complete with corners open subset $D$ of $\mathbb{P}^{n}$ and $\mathcal{F}\in coh(\mathbb{P}^{n})$ such that $D$ is not cohomologically $\hat{q}$-complete with respect to ${\mathcal{F}}$. Here $\hat{q}=n-[\frac{n-1}{q}]$, where $[x]$ denotes the integral part of $x$.

math.CV↗
arXiv2011

On the integral homology and counterexamples to the Andreotti-Grauert conjecture

In this paper, we prove by means of a counterexample that there exist pair of integers (n,p) with $n\geq 3$, $2\leq p\leq n-1$, and open sets $D$ in $C^{n}$ which are cohomologically $p$-complete with respect to the structure sheaf of $D$ such that the cohomology group $H_{n+p}(D,Z)$ does not vanish. In particular $D$ is not $p$-complete.

math.CV↗
arXiv2008

On q-Runge domains

In $[2]$, Coltoiu gave an example of a domain $D\subset\complexes^{6}$ which is 4-complete such that for every ${\mathcal{F}}\in Coh(\complexes^{6})$ the restriction map $H^{3}(\complexes^{6},{\mathcal{F}})\to H^{3}(D,{\mathcal{F}})$ has a dense image but $D$ is not 4-Runge in $\complexes^{6}$. Here, we prove that for every integers $n\geq 4$ and $1\leq q\leq n$ there exists a domain $D\subset \complexes^{n}$ which is not ($\tilde{q}-1$)-Runge in $\complexes^{n}$ but such that for any coherent analytic sheaf ${\mathcal{F}}$ on $\complexes^{n}$ the restriction map $H^{p}(\complexes^{n},{\mathcal{F}})\to H^{3}(D,{\mathcal{F}})$ has a dense image for all $p\geq \tilde{q}-2$ if $q$ does not divide $n$, where $\tilde{q}=n-[\frac{n}{q}]+1$ and $[\frac{n}{q}]$ denotes the integral part of $\frac{n}{q}$.

math.CV↗
arXiv2008

Unbranched Riemann domains over Stein spaces and Cartier divisors

It is proved that an unbranched Riemann domain $\Pi : X\rightarrow Y$ over an arbitrary Stein complex space of dimension $n\geq 2$ is Stein if and only if $X$ is cohomologically $2$-complete with respect to the structure sheaf ${\mathcal{O}}_{X}$ and every topologically trivial holomorphic line bundle over $X$ is associated to a Cartier divisor.

math.CV↗
arXiv2007

Holomorphic fiber bundle with Stein base and Stein fibers

In this article, we prove that if $Π: X\to Ω$ is a surjective holomorphic map, with $Ω$ a Stein space and $X$ a complex manifold of dimension $n\geq 3,$ and if, for every $x\in Ω$ there exists an open neighborhood $U$ such that $Π^{-1}(U)$ is Stein, then $X$ is Stein

math.CV↗
arXiv2001

The Runge approximation theorem for generalized polynomial hulls

It is known from the Runge approximation theorem that every function which is holomorphic in a neighborhood of a compact polynomially convex set $K\subset \complexes^{n}$ can be approximated uniformly on $K$ by analytic polynomials. We shall here prove the same result when the role of the polynomially convex hull $\hat {K}$ is played by the generalized polynomial hull $h_{q}(K)$ introduced by Basener and which can be defined, for each integer $q\in {0,...,n-1}$, by $h_{q}(K)=\displaystyle\bigcup_{P\in \complexes [z_{1},...,z_{n}]} A_{P}$ where $A_{P}=\{z\in \complexes^{n}: |P(z)|\leq δ_{K}(P,z)\}$, and where $δ_{K}(P,z)$ denotes the lowest value of $||P||_{K\cap f^{-1}(0)}$ when $f$ ranges in the set of holomorphic polynomial maps $\complexes^{n}\to \complexes^{q}$ vanishing at $z$.

math.CV↗