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Complex vector bundle

Vector bundle whose fibres carry complex structure

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 {\displaystyle E} can be promoted to a complex vector bundle, the 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 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 is a holomorphic vector bundle if X {\displaystyle X} is a complex manifold and if the local trivializations are biholomorphic.

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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 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. A 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

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 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, then the conjugate bundle E is isomorphic to the 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⊗RC = V⊕i‌V 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.