Holomorphic Bundles Trivializable by Proper Surjective Holomorphic Map

Given a compact complex manifold M, we investigate the holomorphic vector bundles E on M such that phi*E is holomorphically trivial for some surjective holomorphic map phi, to M, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a f lat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal G-bundles, where G is a connected reductive complex affine algebraic group.