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 ${\displaystyle E}${\displaystyle E} can be promoted to a complex vector bundle, the complexification

${\displaystyle E\otimes \mathbb {C} ;}${\displaystyle E\otimes \mathbb {C} ;}

whose fibers are ${\displaystyle E_{x}\otimes _{\mathbb {R} }\mathbb {C} }${\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 ${\displaystyle X}${\displaystyle X} is a complex manifold and if the local trivializations are biholomorphic.

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 ${\displaystyle E}${\displaystyle E} and itself:

${\displaystyle J:E\to E}${\displaystyle J:E\to E}

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

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

Example: A complex structure on the tangent bundle of a real manifold ${\displaystyle M}${\displaystyle M} is usually called an almost complex structure. A theorem of Newlander and Nirenberg says that an almost complex structure ${\displaystyle J}${\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 ${\displaystyle J}${\displaystyle J} vanishes.

If E is a complex vector bundle, then the conjugate bundle ${\displaystyle {\overline {E}}}${\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: ${\displaystyle E_{\mathbb {R} }\to {\overline {E}}_{\mathbb {R} }=E_{\mathbb {R} }}${\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 ${\displaystyle {\overline {E}}}${\displaystyle {\overline {E}}} is given by

${\displaystyle c_{k}({\overline {E}})=(-1)^{k}c_{k}(E)}${\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 ${\displaystyle E^{*}=\operatorname {Hom} (E,{\mathcal {O}})}${\displaystyle E^{*}=\operatorname {Hom} (E,{\mathcal {O}})} through the metric, where we wrote ${\displaystyle {\mathcal {O}}}${\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:

${\displaystyle (E\otimes \mathbb {C} )_{\mathbb {R} }=E\oplus E}${\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.

• Holomorphic vector bundle
• K-theory

• 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

• Vector bundles

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