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

E\otimes \mathbb {C} ;

whose fibers are $E_{x}\otimes _{\mathbb {R} }\mathbb {C}$ .

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 $X$ is a complex manifold and if the local trivializations are biholomorphic.

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:

J:E\to E

such that $J$ acts as the square root $\mathrm {i}$ of $-1$ on fibers: if $J_{x}:E_{x}\to E_{x}$ is the map on fiber-level, then $J_{x}^{2}=-1$ as a linear map. If $E$ is a complex vector bundle, then the complex structure $J$ can be defined by setting $J_{x}$ to be the scalar multiplication by $\mathrm {i}$ . 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 $E_{x}$ ,

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

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 ${\overline {E}}$ 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: $E_{\mathbb {R} }\to {\overline {E}}_{\mathbb {R} }=E_{\mathbb {R} }$ is conjugate-linear, and E and its conjugate E are isomorphic as real vector bundles.

The k-th Chern class of ${\overline {E}}$ is given by

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 $E^{*}=\operatorname {Hom} (E,{\mathcal {O}})$ through the metric, where we wrote ${\mathcal {O}}$ 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\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.

References

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

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