**Edoardo Ballico**

## Holomorphic Vector Bundles on Holomorphically
Convex Complex Manifolds

**Abstract:**

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 $\partial (U)$.

**Keywords:**

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

**MSC 2000:** 32L05, 32E05, 32F10