Let $M$ be a complex manifold (possibly non-compact). Let $E\to M$ be a flat globally generated holomorphic vector bundle. Is the bundle necessarily holomorphically trivial?
Note that flatness implies that all Chern classes vanish. I know the description of Chern classes in terms of the degenaracy locus of generic sections but the references I know (e.g. "Principles of Algebraic Geometry" - Griffiths and Harris) assume compactness. Therefore I am not sure whether or not I can apply this description in this case. If the description still holds, then $c_1(E)=0 =c_{\text{top}}(E)$ plus the bundle being globally generated should imply holomorphic triviality. Because $c_1(E)=0$ means $r$ generic sections are lineraly independent and $c_{\text{top}}(E)=0$ means a generic section is non vanishing.