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 ⊗_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.

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 E_x,

$(a + ib) v = a v + J(b v).$

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 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.

See also

• holomorphic vector bundle
• K-theory

References

• Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9