Suppose that (E,π,M) is a complex vector bundle. I’ve seen it suggested in a few places that if E and M are complex manifolds, and π is a holomorphic map, then (E,π,M) is in fact a holomorphic vector bundle. In other words, there exist local trivializations which are biholomorphisms. Is this true? Edit: As explained by Mike Miller in the comments, this cannot be true as stated, because the complex vector bundle might have an ‘anti-holomorphic’ structure to being with. A better question would be whether there is some holomorphic vector bundle structure (E,π′,M), which is isomorphic as a complex vector bundles to (E,π,M). Alternatively, one could drop the requirement that (E,π,M) be a complex vector bundle, and instead ask what milder conditions (e.g. π is a smooth constant rank submersion) would be sufficient to guarantee that (E,π,M) is holomorphic (assuming still that E and M are complex manifolds and π is holomorphic). Source.