Editing Complex vector bundle

In mathematics, a '''complex vector bundle''' is a [[vector bundle]] whose fibers are [[complex vector space]]s.

Any complex vector bundle can be viewed as a [[real vector bundle]] through the [[restriction of scalars]]. Conversely, any real vector bundle <math>E</math> can be promoted to a complex vector bundle, the [[complexification]]
:<math>E \otimes \mathbb{C} ;</math>
whose fibers are <math>E_x\otimes_\R \C</math>.

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 vector bundle|oriented]]; in particular, one can take its [[Euler class]].

A complex vector bundle is a [[holomorphic vector bundle]] if <math>X</math> is a [[complex manifold]] and if the local trivializations are [[biholomorphic]].

== Complex structure ==
{{see also|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 <math>E</math> and itself:
:<math>J: E \to E</math>
such that <math>J</math> acts as the square root <math>\mathrm i</math> of <math>-1</math> on fibers: if <math>J_x: E_x \to E_x</math> is the map on fiber-level, then <math>J_x^2 = -1</math> as a linear map. If <math>E</math> is a complex vector bundle, then the complex structure <math>J</math> can be defined by setting <math>J_x</math> to be the scalar multiplication by <math>\mathrm i</math>. Conversely, if <math>E</math> is a real vector bundle with a complex structure <math>J</math>, then <math>E</math> can be turned into a complex vector bundle by setting: for any real numbers <math>a</math>, <math>b</math> and a real vector <math>v</math> in a fiber <math>E_x</math>,
:<math>(a + \mathrm ib) v = a v + J(b v).</math>

'''Example''': A complex structure on the tangent bundle of a real manifold <math>M</math> is usually called an [[almost complex structure]]. A [[theorem of Newlander and Nirenberg]] says that an almost complex structure <math>J</math> is "integrable" in the sense it is induced by a structure of a complex manifold if and only if a certain tensor involving <math>J</math> vanishes.

== Conjugate bundle ==
{{see also|Complex conjugate vector space}}

If ''E'' is a complex vector bundle, then the '''conjugate bundle''' <math>\overline{E}</math> 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: <math>E_{\mathbb{R}} \to \overline{E}_\mathbb{R} = E_{\mathbb{R}}</math> is conjugate-linear, and ''E'' and its conjugate {{overline|''E''}} are isomorphic as real vector bundles.

The ''k''-th [[Chern class]] of <math>\overline{E}</math> is given by
:<math>c_k(\overline{E}) = (-1)^k c_k(E)</math>.
In particular, ''E'' and {{overline|''E''}} are not isomorphic in general.

If ''E'' has a [[hermitian metric]], then the conjugate bundle {{overline|''E''}} is isomorphic to the [[dual bundle]] <math>E^* = \operatorname{Hom}(E, \mathcal{O})</math> through the metric, where we wrote <math>\mathcal{O}</math> 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'':
:<math>(E \otimes \mathbb{C})_{\mathbb{R}} =  E \oplus E</math>
(since ''V''⊗<sub>'''R'''</sub>'''C''' = ''V''⊕''i''{{zwnj}}''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 ==
*[[Holomorphic vector bundle]]
*[[K-theory]]

== References ==
* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | last2=Stasheff | first2=James D. |author2-link=Jim Stasheff| title=Characteristic classes | publisher=Princeton University Press; University of Tokyo Press | series=Annals of Mathematics Studies | isbn=978-0-691-08122-9 | year=1974 | volume=76}}






[[Category:Vector bundles]]