Georgian Mathematical Journal 13 (2006), No. 1, 007--010
Copyright Heldermann Verlag 2006
Holomorphic Vector Bundles on Holomorphically Convex Complex Manifolds
Edoardo Ballico
Dept. of Mathematics, University of Trento, 38050 Povo (TN), Italy
ballico@science.unitn.it
[Abstract-pdf]
Let $X$ be a holomorphically convex complex manifold and ${\rm Exc} (X) \subseteq X$ the union of all positive dimensional compact analytic subsets of $X$. We assume that ${\rm Exc} (X) \ne X$ 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 ${\rm Exc}(X)$ 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: 32L05, 32E05, 32F10
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