Complex vector bundle explained

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 ⊗ 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.

A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations are biholomorphic.

Complex structure

See also: Linear 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\toE

such that J acts as the square root i of −1 on fibers: if

J_x:E_x\toE_x

is the map on fiber-level, then

	2
J
	x

=-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

. 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=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_R\to\overline{E}_R=E_R

is conjugate-linear, and E and its conjugate are isomorphic as real vector bundles.

The k-th Chern class of

\overline{E}

is given by

c_{k(\overline{E})}=(-1)^kc_k(E)

.In particular, E and are not isomorphic in general.

E^*=\operatorname{Hom}(E,l{O})

through the metric, where we wrote

l{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 ⊗ C)_R=E ⊕ 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.