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

Abstract

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 $di{m}_{\mathbb{C}}{H}^p(X,\mathcal{F})<+\infty$ 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 $2\le q\le n-1$ 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 $x$.

Submitted to Complex Variables and Elliptic Equations