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A complex vector bundle is a vector bundle whose fibers are complex vector spaces.
A complex vector bundle with complex 1-dimensional fibers is a complex line bundle.
More precisely, a complex vector bundle is a real vector bundle $E\to M$ together with a lifting of its $\mathbf{R}$-module structure $\mathbf{R}\to End(E)$ to a homomorphism of $\mathbf{R}$-algebras $\mathbf{C}\to End(E)$.
The latter lifting is also known as a complex structure on a real vector bundle.
In terms of cocycles, complex vector bundles can be described using cocycle data where the transition maps are complex-linear maps.
Any holomorphic vector bundle over a complex manifold has an underlying complex vector bundle.
Conversely, given a complex vector bundle over a complex manifold, if its transition maps are holomorphic, then it is a holomorphic vector bundle.
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