In mathematics, a complex vector bundle is a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
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
can be promoted to a complex vector bundle, the
complexification
:
whose fibers are
.
Any complex vector bundle over a
paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
admits a
hermitian metric.
The basic invariant of a complex vector bundle is a
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
. A complex vector bundle is canonically
oriented; in particular, one can take its
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle o ...
.
A complex vector bundle is a
holomorphic vector bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
if
is a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
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
and itself:
:
such that
acts as the square root
of
on fibers: if
is the map on fiber-level, then
as a linear map. If
is a complex vector bundle, then the complex structure
can be defined by setting
to be the scalar multiplication by
. Conversely, if
is a real vector bundle with a complex structure
, then
can be turned into a complex vector bundle by setting: for any real numbers
,
and a real vector
in a fiber
,
:
Example: A complex structure on the tangent bundle of a real manifold
is usually called an
almost complex structure. 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
vanishes.
Conjugate bundle
If ''E'' is a complex vector bundle, then the conjugate bundle
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 are isomorphic as real vector bundles.
The ''k''-th
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...
of
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 through the metric, where we wrote
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''⊕''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
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
*
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
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