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Vector bundles with trivial determinant and second Chern class one on some non-Kähler surfaces. (English) Zbl 1037.32020
From the introduction: We investigate holomorphic rank-2 vector bundles with trivial determinant and second Chern class one on some non-Kähler surfaces. The main difficulty one encounters when dealing with holomorphic vector bundles over nonprojective manifolds is the presence of nonfiltrable such bundles (that is, bundles with no filtration by torsion-free coherent subsheaves) or even of irreducible ones, that is, bundles with no proper coherent subsheaf [see for instance the exhaustive monograph: V. Brînzănescu, “Holomorphic vector bundles over compact complex surfaces”, Lect. Notes Math. 1624 (1996; Zbl 0848.32024)]. However, as we show, on a certain class of non-Kähler surfaces such vector bundles are filtrable, hence one may classify them via the classical techniques of classification of extensions as it is furthermore shown.
##### MSC:
 32L05 Holomorphic bundles and generalizations
##### Keywords:
holomorphic vector bundle; coherent sheaf
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