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 scalers. Conversely, any real vector bundle E can be promoted to a complex vector bundle, the "complexification" E ⊗ C of E: its fibers are

$E_{x}\otimes _{\mathbb {R} }\mathbb {C}$.

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

The basic invariant of a complex vector bundle is a Chern class.

Complex structure

A complex vector bundle can be thought of 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:

$J:E\to E$

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

$(a+ib)v=av+(Jb)v.$

See also: Almost complex manifold

Conjugate bundle

If E is a complex vector bundle, then the conjugate bundle ${\overline {E}}$ of E is obtained by having complex numbers acting through the complex conjugates of the numbers. Thus, the identity map of the underlying real vector bundles: $E\to {\overline {E}}$ is conjugate-linear and E and its conjugate ${\overline {E}}$ are isomorphic as real vector bundle.

The k-th Chern class of ${\overline {E}}$ is given by

$c_{k}({\overline {E}})=(-1)^{k}c_{k}(E)$.

If E has a hermitian metric, then ${\overline {E}}$ is isomorphic to the dual bundle $E^{*}=\operatorname {Hom} (E,{\mathcal {O}})$ through the metric, where we wrote ${\mathcal {O}}$ for the trivial complex line bundle.

See also

• holomorphic vector bundle
• K-theory

References

• Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9