On the separation of cohomology groups of increasing unions of $(1, 1)$ convex-concave manifolds

Abstract

We construct a complex manifold $X$, $dimX \geq 3$, which is an increasing union of (1, 1) convex-concave open subsets having the same fixed convex boundary, and a holomorphic line bundle $L$ on $X$, such that the cohomology group $H^{1}(X,L)$ is not separated.The manifold $X$ is constructed as a proper modification of the (1, 1) convex-concave manifold $\mathbb{C}^{k} \backslash \{0\}$ at a discrete subset. It is also remarked that an increasing union of 1-concave manifolds has always separated cohomology (for locally free sheaves).