# Complex vector bundle

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In mathematics, a **complex vector bundle** is a [vector bundle](/source/Vector_bundle) whose [fibers](/source/Fiber_(mathematics)) are [complex vector spaces](/source/Complex_vector_space).

Any complex vector bundle can be viewed as a [real vector bundle](/source/Real_vector_bundle) through the [restriction of scalars](/source/Restriction_of_scalars). Conversely, any real vector bundle E {\displaystyle E} can be promoted to a complex vector bundle, the [complexification](/source/Complexification)

- E ⊗ C ; {\displaystyle E\otimes \mathbb {C} ;}

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

Any complex vector bundle over a [paracompact space](/source/Paracompact_space) admits a [hermitian metric](/source/Hermitian_metric).

The basic invariant of a complex vector bundle is a [Chern class](/source/Chern_class). A complex vector bundle is canonically [oriented](/source/Oriented_vector_bundle); in particular, one can take its [Euler class](/source/Euler_class).

A complex vector bundle is a [holomorphic vector bundle](/source/Holomorphic_vector_bundle) if X {\displaystyle X} is a [complex manifold](/source/Complex_manifold) and if the local trivializations are [biholomorphic](/source/Biholomorphic).

## Complex structure

See also: [Linear complex structure](/source/Linear_complex_structure)

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 {\displaystyle E} and itself:

- J : E → E {\displaystyle J:E\to E}

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

- ( a + i b ) v = a v + J ( b v ) . {\displaystyle (a+\mathrm {i} b)v=av+J(bv).}

**Example**: A complex structure on the tangent bundle of a real manifold M {\displaystyle M} is usually called an [almost complex structure](/source/Almost_complex_structure). A [theorem of Newlander and Nirenberg](/source/Theorem_of_Newlander_and_Nirenberg) says that an almost complex structure J {\displaystyle J} is "integrable" in the sense it is induced by a structure of a complex manifold if and only if a certain tensor involving J {\displaystyle J} vanishes.

## Conjugate bundle

See also: [Complex conjugate vector space](/source/Complex_conjugate_vector_space)

If *E* is a complex vector bundle, then the **conjugate bundle** E ¯ {\displaystyle {\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 R → E ¯ R = E R {\displaystyle E_{\mathbb {R} }\to {\overline {E}}_{\mathbb {R} }=E_{\mathbb {R} }} is conjugate-linear, and *E* and its conjugate *E* are isomorphic as real vector bundles.

The *k*-th [Chern class](/source/Chern_class) of E ¯ {\displaystyle {\overline {E}}} is given by

- c k ( E ¯ ) = ( − 1 ) k c k ( E ) {\displaystyle c_{k}({\overline {E}})=(-1)^{k}c_{k}(E)} .

In particular, *E* and *E* are not isomorphic in general.

If *E* has a [hermitian metric](/source/Hermitian_metric), then the conjugate bundle *E* is isomorphic to the [dual bundle](/source/Dual_bundle) E ∗ = Hom ⁡ ( E , O ) {\displaystyle E^{*}=\operatorname {Hom} (E,{\mathcal {O}})} through the metric, where we wrote O {\displaystyle {\mathcal {O}}} for the trivial complex line bundle.

If *E* is a real vector bundle, then the underlying real vector bundle of the complexification of *E* is a direct sum of two copies of *E*:

- ( E ⊗ C ) R = E ⊕ E {\displaystyle (E\otimes \mathbb {C} )_{\mathbb {R} }=E\oplus E}

(since *V*⊗**R****C** = *V*⊕*i*‌*V* for any real vector space *V*.) If a complex vector bundle *E* is the complexification of a real vector bundle *E'*, then *E'* is called a [real form](/source/Real_form) of *E* (there may be more than one real form) and *E* is said to be defined over the real numbers. If *E* has a real form, then *E* is isomorphic to its conjugate (since they are both sum of two copies of a real form), and consequently the odd Chern classes of *E* have order 2.

## See also

- [Holomorphic vector bundle](/source/Holomorphic_vector_bundle)

- [K-theory](/source/K-theory)

## References

- [Milnor, John Willard](/source/John_Milnor); [Stasheff, James D.](/source/Jim_Stasheff) (1974), *Characteristic classes*, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press, [ISBN](/source/ISBN_(identifier)) [978-0-691-08122-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-08122-9)

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Adapted from the Wikipedia article [Complex vector bundle](https://en.wikipedia.org/wiki/Complex_vector_bundle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Complex_vector_bundle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.