# 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} ⊗_{R} **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.

## 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* and itself:

such that *J* acts as the square root *i* of -1 on fibers: if is the map on fiber-level, then as a linear map. If *E* is a complex vector bundle, then the complex structure *J* can be defined by setting to be the scalar multiplication by . 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},

**Example**: A complex structure on the tangent bundle of a real manifold *M* is usually called an almost complex structure. A theorem of Newlander and Nirenberg says that an almost complex structure *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* vanishes.

## Conjugate bundle

If *E* is a complex vector bundle, then the **conjugate bundle** 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: is conjugate-linear, and *E* and its conjugate *E* are isomorphic as real vector bundles.

The *k*-th Chern class of is given by

- .

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

If *E* has a hermitian metric, then the conjugate bundle *E* is isomorphic to the dual bundle through the metric, where we wrote 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*:

(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 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

## References

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