Edoardo Ballico

Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds

Let $X$ be a holomorphically convex complex manifold and $Exc(X) \subseteq X$ the union of all positive dimensional compact analytic subsets of $X$. We assume that $Exc(X) \ne X$ and $X$ is not a Stein manifold. Here we prove the existence of a holomorphic vector bundle $E$ on $X$ such that $(E\vert U)\oplus \mathcal {O}_U^m$ is not holomorphically trivial for every open neighborhood $U$ of $Exc(X)$ and every integer $m \ge 0$. Furthermore, we study the existence of holomorphic vector bundles on such a neighborhood $U$, which are not extendable across a $2$-concave point of $\partial (U)$.

Holomorphic vector bundle, holomorphically convex complex manifold, Stein space, $q$-concave complex space.

MSC 2000: 32L05, 32E05, 32F10