A note on non-filtrable holomorphic bundles. (Q1408190)

On a non-Kähler elliptic surface \(X\) there exist non-filtrable holomorphic rank 2 vector bundles [\textit{C. Banica} and \textit{J. Le Potier}, J. Reine Angew. Math. 378, 1--31 (1987; Zbl 0624.32017) and \textit{M. Toma}, Holomorphic vector bundles on non-algebraic surfaces, Dissertation, Bayreuth (1992; Zbl 0773.32021)]. By definition a holomorphic rank 2 vector bundle \(E\) is non-filtrable if it does not admit coherent subsheaves of rank 1. The authors prove in this paper that any such bundle is obtained as an elementary modification of a direct image of a line bundle on a double covering of the surface \(X\).