Holomorphic vector bundle on Hopf manifolds with Abelian fundamental groups (Q705094)

Let \(X\) be a Hopf manifold with Abelian fundamental group and \(E\) a rank \(r\) holomorphic vector bundle on \(X\) with trivial pull-back to the universal covering \(W:= C^n\backslash \{0\}\) of \(X\). Here the authors prove the existence of a line bundle \(L\) on \(X\) such that \(E\otimes L\) has a nowhere vanishing section and study the vector bundles filtrations of \(E\). \textit{D. Mall} in [Math. Ann. 294, No. 4, 719--740 (1992; Zbl 0756.32018)] did the case in which \(X\) is primary, i.e \(\pi _1(X) \cong Z\).