A class of non-algebraic threefolds (Q1112216)

A holomorphic vector bundle E of rank r on a complex manifold X is called irreducible if it has no coherent subsheaf F of rank r', with \(0<r'<r.\) This notion which is specific to non-algebraic manifolds was brought forward by G. Elencwajg and O. Forster and further studied by C. Banica and J. Le Potier in connection with the problem of existence of holomorphic structures on a given topological vector bundle. In this paper a holomorphic vector bundle is called strongly irreducible if it remains irreducible after any base-change X'\(\to X\). For X a compact surface without curves and \(r=2\), this happens exactly when the threefold \({\mathbb{P}}(E)\) has no divisors. The existence of strongly irreducible bundles on some classes of compact surfaces is proved.