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In mathematics, the Birkhoff-Grothendieck theorem concerns properties of vector bundles over complex projective space $mathbb\{CP\}^1$. It reduces every vector bundle over $mathbb\{CP\}^1$ into direct sum of tautological line bundles, which enables one to deal with the bundle in a practical way. More precisely, the statement of the theorem is as the following.## References

Every holomorphic vector bundle $mathcal\{E\}$ on $mathbb\{CP\}^1$ can be written as a direct sum of line bundles:

- $mathcal\{E\}congmathcal\{O\}(a\_1)oplus\; cdots\; oplus\; mathcal\{O\}(a\_n).$

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Last updated on Saturday April 12, 2008 at 16:14:28 PDT (GMT -0700)

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Last updated on Saturday April 12, 2008 at 16:14:28 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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