nLab
complex vector bundle
Contents
Context
Bundles
bundles

covering space

retractive space

fiber bundle , fiber ∞-bundle

numerable bundle

principal bundle , principal ∞-bundle

associated bundle , associated ∞-bundle

vector bundle , 2-vector bundle , (∞,1)-vector bundle

real , complex /holomorphic , quaternionic

topological , differentiable , algebraic

with connection

bundle of spectra

natural bundle

equivariant bundle

Complex geometry
Contents
Definition
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.

Connections to holomorphic vector bundles
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 .

Properties
Oka-Grauert principle
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 ).

Relation to holomorphic vector bundles
See at Koszul-Malgrange theorem .

References
Emery Thomas, Complex structures on real vector bundles (JSTOR )
In the context of GAGA :

Last revised on June 10, 2023 at 13:11:53.
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