Wikipedia / Complex vector bundle
Wikipedia / 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 can be promoted to a complex vector bundle, the complexification
.
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 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 and itself:
,
Example is usually called an almost complex structure. A theorem of Newlander and Nirenberg vanishes.
Conjugate bundle
If E is a complex vector bundle, then the conjugate bundle of E is conjugate-linear, and E and its conjugate are isomorphic as real vector bundles.
The k-th Chern class is given by
.
In particular, E and are not isomorphic in general.
If E has a hermitian metric, then the conjugate bundle is isomorphic to the dual bundle 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.