Encyclopedia

Complex vector bundle

Wikipedia contributorsAdapted for Fathomly · View authors

2 min read
Source updated July 22, 2026

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.

01Complex 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.

02Conjugate 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.

Watch videos about Complex vector bundleExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Complex vector bundle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Complex vector bundle

01Complex structure

02Conjugate bundle

Sources and credits

Related topics

Holomorphic vector bundle

K-theory