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
:$E \otimes \mathbb ;$
whose fibers are $E_x\otimes_\R \C$.

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 $\mathrm i$ 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 $\mathrm i$. 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 $E_x$,
:$(a + \mathrm ib) v = a v + J(b v).$

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 $\overline$ 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: $E_ \to \overline_\mathbb = E_$ is conjugate-linear, and ''E'' and its conjugate are isomorphic as real vector bundles.

The ''k''-th Chern class of $\overline$ is given by
:$c_k(\overline) = (-1)^k c_k(E)$.
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 $E^* = \operatorname(E, \mathcal)$ through the metric, where we wrote $\mathcal$ 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'':
:$(E \otimes \mathbb)_ = E \oplus E$
(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.

See also

*Holomorphic vector bundle
*K-theory

References

* {{Citation , last1=Milnor , first1=John Willard , author1-link=John Milnor , last2=Stasheff , first2=James D. , author2-link=Jim Stasheff, title=Characteristic classes , publisher=Princeton University Press; University of Tokyo Press , series=Annals of Mathematics Studies , isbn=978-0-691-08122-9 , year=1974 , volume=76

Vector bundles