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Sur l’existence des fibrés vectoriels holomorphes sur les surfaces non- algébriques. (On the existence of holomorphic vector bundles on non- algebraic surfaces). (French) Zbl 0624.32017

Let X be a compact complex manifold of dimension n and E be a topological vector bundle of rang r over X. A general problem is to decide whether E carries the structure of a holomorphic vector bundle. If \(r=1\), this is true, if and only if the first Chern class \(c_ 1(E)\) is an element of the Neron-Severi group NS(X). In this paper the authors consider the case where X is a complex surface, i.e. \(\dim_{{\mathbb{C}}}X=2\) and \(r\geq 2\). They solve the problem stated above for filtrable holomorphic vector bundles and give a necessary condition for a topological vector bundle to carry a holomorphic structure in terms of the first two Chern classes of E.
Reviewer: K.Oeljeklaus

MSC:

32J15 Compact complex surfaces
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
55R25 Sphere bundles and vector bundles in algebraic topology
57R20 Characteristic classes and numbers in differential topology
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