  Given a compact complex manifold $M$, we investigate the holomorphic vector
bundles $E$ on $M$ such that $\varphi^* E$ is trivial for some surjective
holomorphic map $\varphi$, to $M$, from some compact complex manifold. We prove
that these are exactly those holomorphic vector bundles that admit a flat
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.
