A complex vector bundle is a vector bundle 
whose fiber 
is a complex
vector space (Cohen 1998, p. 3). It is not necessarily a complex
manifold, even if its base manifold 
is a complex manifold.
A complex vector bundle is a holomorphic
vector bundle when 
and 
are complex manifolds
and its local trivializations
are holomorphic maps that restrict on each fiber to isomorphisms of complex vector spaces. These conditions imply
that the projection map 
is a holomorphic map (Khesin
and Wendt 2009, pp. 155-156).
Complex Vector Bundle
See also
Bundle, Complex Vector Space, Holomorphic Vector Bundle, Manifold, Topological K-Theory, Vector SpacePortions of this entry contributed by Todd Rowland
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References
Cohen, R. L. The Topology of Fiber Bundles. Lecture notes, p. 3, 1998. https://math.stanford.edu/~ralph/fiber.pdf.Khesin, B. and Wendt, R. The Geometry of Infinite-Dimensional Groups. Berlin, Germany: Springer-Verlag, pp. 155-156, 2009. https://doi.org/10.1007/978-3-540-77263-7.Referenced on Wolfram|Alpha
Complex Vector BundleCite this as:
Weisstein, Eric W., with contributions by Todd Rowland. "Complex Vector Bundle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexVectorBundle.html