nLab complex vector bundle





Complex geometry



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 EME\to M together with a lifting of its R\mathbf{R}-module structure REnd(E)\mathbf{R}\to End(E) to a homomorphism of R\mathbf{R}-algebras CEnd(E)\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.


Oka-Grauert principle

The Oka-Grauert principle states that for any Stein manifold XX the holomorphic and the topological classification of complex vector bundles on XX coincide. The original reference is (Grauert 58).

Relation to holomorphic vector bundles

See at Koszul-Malgrange theorem.


  • Emery Thomas, Complex structures on real vector bundles (JSTOR)

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.