A holomorphic vector bundle is a complex vector bundle 
over a complex manifold whose total
space 
is a complex manifold, whose projection 
and local trivializations are holomorphic
maps, and whose transition functions are
holomorphic maps. The fibers 
are copies of 
,
and the transition functions act linearly
on them. The simplest example is a holomorphic
line bundle, where each fiber is a copy of 
.
Holomorphic Vector Bundle
See also
Complex Manifold, Hermitian Metric, Holomorphic Function, Holomorphic Line Bundle, Holomorphic Tangent Bundle, Vector BundlePortions of this entry contributed by Todd Rowland
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Cite this as:
Weisstein, Eric W., with contributions by Todd Rowland. "Holomorphic Vector Bundle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HolomorphicVectorBundle.html