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{C} ; 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.

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{C} ; 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. 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. If E is a complex vector bundle, then the conjugate bundle \overline{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_{\mathbb{R}} \to \overline{E}_\mathbb{R} = E_{\mathbb{R}} is conjugate-linear, and E and its conjugate are isomorphic as real vector bundles. The k-th Chern class of \overline{E} is given by c_k(\overline{E}) = (-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{Hom}(E, \mathcal{O}) through the metric, where we wrote \mathcal{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 \otimes \mathbb{C})_{\mathbb{R}} = E \oplus E (since V⊗ R C = 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.

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