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
- [math]\displaystyle{ E \otimes \mathbb{C} ; }[/math]
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.
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:
- [math]\displaystyle{ J: E \to E }[/math]
such that J acts as the square root i of −1 on fibers: if [math]\displaystyle{ J_x: E_x \to E_x }[/math] is the map on fiber-level, then [math]\displaystyle{ J_x^2 = -1 }[/math] as a linear map. If E is a complex vector bundle, then the complex structure J can be defined by setting [math]\displaystyle{ J_x }[/math] to be the scalar multiplication by [math]\displaystyle{ i }[/math]. 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,
- [math]\displaystyle{ (a + ib) v = a v + J(b v). }[/math]
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 [math]\displaystyle{ \overline{E} }[/math] 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: [math]\displaystyle{ E_{\mathbb{R}} \to \overline{E}_\mathbb{R} = E_{\mathbb{R}} }[/math] is conjugate-linear, and E and its conjugate E are isomorphic as real vector bundles.
The k-th Chern class of [math]\displaystyle{ \overline{E} }[/math] is given by
- [math]\displaystyle{ c_k(\overline{E}) = (-1)^k c_k(E) }[/math].
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 [math]\displaystyle{ E^* = \operatorname{Hom}(E, \mathcal{O}) }[/math] through the metric, where we wrote [math]\displaystyle{ \mathcal{O} }[/math] 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:
- [math]\displaystyle{ (E \otimes \mathbb{C})_{\mathbb{R}} = E \oplus E }[/math]
(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.
See also
References
- Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9
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