Bănică, Constantin; Le Potier, Joseph 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 J. Reine Angew. Math. 378, 1-31 (1987). 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 Cited in 9 ReviewsCited in 14 Documents 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 Keywords:compact complex surface; Chern class; filtrable holomorphic vector bundles; topological vector bundle PDFBibTeX XMLCite \textit{C. Bănică} and \textit{J. Le Potier}, J. Reine Angew. Math. 378, 1--31 (1987; Zbl 0624.32017) Full Text: DOI EuDML