A counter-example to the Andreotti–Grauert conjecture

In $1962$, Andreotti and Grauert showed that every $q$-complete complex space $X$ is cohomologically $q$-complete, that is for every coherent analytic sheaf ${\mathcal{F}}$ on $X$, the cohomology group $H^{p}(X, {\mathcal{F}})$ vanishes if $p\geq q$. Since then the question whether the reciprocal statements of these theorems are true have been subject to extensive studies, where more specific assumptions have been added. Until now it is not known if these two conditions are equivalent. Using test cohomology classes, it was shown however that if $X$ is a Stein manifold and, if $D\subset X$ is an open subset which has $C^{2}$ boundary such that $H^{p}(D, {\mathcal{O}}_{D})=0$ for all $p\geq q$, then $D$ is $q$-complete. The aim of the present article is to give a counterexample to the conjecture posed in $1962$ by Andreotti and Grauert [ref1] to show that a cohomologically $q$-complete space is not necessarily $q$-complete. More precisely, we show that there exist for each $n\geq 3$ open subsets $\Omega\subset\mathbb{C}^{n}$ such that for every ${\mathcal{F}}\in coh(\Omega)$, the cohomology groups $H^{p}(\Omega, {\mathcal{F}})$ vanish for all $p\geq n-1$ but $\Omega$ is not $(n-1)$-complete.