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Yes, for $\mathbb{C}^n$ itself, since vector bundles are (holomorphically) trivial. See Griffiths and Adams " Topics in Algebraic and Analytic Geometry" p 209. I would need to think about the case of submanifolds, before giving an answer. But definitely NO for nontrivial projective varieties: an ample line bundle won't be a summand of a trivial vector bundle. Proof: If it were, then its dual would be generated by global sections, and this is impossible.