Vector Bundles with Trivial Determinant and Second Chern Class One on Some Nonkähler Surfaces
نویسندگان
چکیده
In this paper we investigate holomorhic rank-2 vector bundles with trivial determinant and second Chern class one on some nonKähler surfaces. The main dificulty 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 [Br1]). However, as we shall show, on a certain class of nonKähler surfaces which will be described below -, such vector bundles are filtrable, hence one may classify them via the classical technique of classification of extensions as it is furthermore shown. The class of surfaces we are working on is the class of nonKähler elliptic principal bundles, (that is surfaces X which admit a bundle map π : X → B over a smooth projective curve B with fiber and structure group a fixed elliptic smooth curve F ) satisfying the extra conditions: NS(X) = 0 and B is nonhypereliptic. For nonKä hler elliptic principal bundles, it was proven by Br̂ınzănescu in [Br2] (see also [Br3]) that
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