Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars.
Conversely, any real vector bundle
can be promoted to a complex vector bundle, the complexification whose fibers are
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.
is a complex manifold and if the local trivializations are biholomorphic.
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
is a real vector bundle with a complex structure
can be turned into a complex vector bundle by setting: for any real numbers
, Example: A complex structure on the tangent bundle of a real manifold
A theorem of Newlander and Nirenberg says that an almost complex structure
is "integrable" in the sense it is induced by a structure of a complex manifold if and only if a certain tensor involving
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:
is conjugate-linear, and E and its conjugate E are isomorphic as real vector bundles.
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: (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.