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 Ex ⊗R 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 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 $$i$$. 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 Ex,
 * $$(a + ib) v = a v + J(b v).$$

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 $\overline{E}$ 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 $\overline{E}$ are not isomorphic in general.

If E has a hermitian metric, then the conjugate bundle $\overline{E}$ is isomorphic to the dual bundle $$E^* = \operatorname{Hom}(E, \mathcal{O})$$ 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⊕i'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.