We show that the converse of the aproximation theorem of Andreotti and Grauert does not hold. More precisely we construct a $4$-complete open subset $D\subset {ℂ}^6$ (which is an analytic complement in the unit ball) such that the restriction map $H^3({ℂ}^6,ℱ)\to H^3(D,ℱ)$ has a dense image for every $ℱ\in Coh\left({ℂ}^6\right)$ but the pair $(D,{ℂ}^6)$ is not a $4$-Runge pair.

We characterize intrinsically two classes of manifolds that can be properly embedded into spaces of the form ${\mathbb{P}}^N\setminus {\mathbb{P}}^{N-q}$. The first theorem is a compactification theorem for pseudoconcave manifolds that can be realized as $\overline{X}\setminus (\overline{X}\cap {\mathbb{P}}^{N-q})$ where $\overline{X}\subset {\mathbb{P}}^N$ is a projective variety. The second theorem is an embedding theorem for holomorphically convex manifolds into ${\mathbb{P}}^1\times {ℂ}^N$.

In this paper we show that a 1-convex (i.e., strongly pseudoconvex) manifold $X$, with 1- dimensional exceptional set $S$ and finitely generated second homology group $H_2(X,\mathbb{Z})$, is embeddable in ${ℂ}^m\times ℂ{\mathbb{P}}_n$ if and only if $X$ is Kähler, and this case occurs only when $S$ does not contain any effective curve which is a boundary.