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 E can be promoted to a complex vector bundle, the complexification

whose fibers are E_x ⊗_RC.

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 can be thought of as a real vector bundle with an additional structure, the complex structure. By definition, a complex structure is a bundle map between a real vector bundle E and itself:

such that J acts as the square root i of -1 on fibers: if ${\displaystyle J_{x}:E_{x}\to E_{x}}$ is the map on fiber-level, then ${\displaystyle J_{x}^{2}=-1}$ as a linear map. If E is a complex vector bundle, then the complex structure J can be defined by setting ${\displaystyle J_{x}}$ to be the scalar multiplication by ${\displaystyle i}$. Conversely, if E is a real vector bundle with a complex structure J, then E can be turned into a complex vector bundle by setting: for any real numbers a, b and a real vector v in a fiber E_x,

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