(see also Chern-Weil theory, parameterized homotopy theory)
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex vector bundle is a vector bundle with respect complex vector spaces.
A complex vector bundle with complex 1-dimensional fibers is a complex line bundle.
The Oka-Grauert principle states that for any Stein manifold $X$ the holomorphic and the topological classification of complex vector bundles on $X$ coincide. The original reference is (Grauert 58).
complex vector bundle
In the context of GAGA: