Lesson plan
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.
Article summary
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.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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