Complex vector bundle

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{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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