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| Format: | Preprint |
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2007
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| Online Access: | https://arxiv.org/abs/0710.3358 |
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| _version_ | 1866918156472156160 |
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| author | Alaoui, Youssef |
| author_facet | Alaoui, Youssef |
| contents | 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 $x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0710_3358 |
| institution | arXiv |
| publishDate | 2007 |
| record_format | arxiv |
| spellingShingle | On $q$-complete and $q$-concave with corners complex manifolds Alaoui, Youssef Complex Variables 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 $x$. |
| title | On $q$-complete and $q$-concave with corners complex manifolds |
| topic | Complex Variables |
| url | https://arxiv.org/abs/0710.3358 |