vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
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 together with a lifting of its -module structure to a homomorphism of -algebras .
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.
The Oka-Grauert principle states that for any Stein manifold the holomorphic and the topological classification of complex vector bundles on coincide. The original reference is (Grauert 58).
See at Koszul-Malgrange theorem.
complex vector bundle
In the context of GAGA:
Last revised on June 10, 2023 at 13:11:53. See the history of this page for a list of all contributions to it.