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Complex vector bundle

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In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.

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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 Ex ⊗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.

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 → E

such that J acts as the square root i of -1 on fibers: if J x : E x → 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 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 Ex,

( a + i b ) 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 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 → E ¯ R = E R is conjugate-linear, and E and its conjugate E are isomorphic as real vector bundles.

The k-th Chern class of E ¯ is given by

c k ( E ¯ ) = ( − 1 ) k c k ( E ) .

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

If E has a hermitian metric, then the conjugate bundle E is isomorphic to the dual bundle E ∗ = Hom ⁡ ( E , O ) through the metric, where we wrote 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.

References

Complex vector bundle Wikipedia
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