Complex vector bundle

In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.

Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ${\displaystyle E}$ can be promoted to a complex vector bundle, the complexification

${\displaystyle E\otimes \mathbb {C} ;}$

whose fibers are ${\displaystyle E_{x}\otimes _{\mathbb {R} }\mathbb {C} }$.

Any complex vector bundle over a paracompact space admits a hermitian metric.

The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class.

A complex vector bundle is a holomorphic vector bundle if ${\displaystyle X}$ is a complex manifold and if the local trivializations are biholomorphic.