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.

Related concepts

• complex vector space

  • complex line

• complex vector bundle

  • universal complex vector bundle

  • stable vector bundle

  • complex line bundle

  • holomorphic vector bundle

  • complex oriented cohomology theory

• real vector bundle

• quaternionic vector bundle

References

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

In the context of GAGA:

• Jean-Pierre Demailly, section 3 of Analytic methods in algebraic geometry, lecture notes 2009 (pdf)