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# Birkhoff-Grothendieck theorem

In mathematics, the Birkhoff-Grothendieck theorem concerns properties of vector bundles over complex projective space $mathbb\left\{CP\right\}^1$. It reduces every vector bundle over $mathbb\left\{CP\right\}^1$ 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.

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

$mathcal\left\{E\right\}congmathcal\left\{O\right\}\left(a_1\right)oplus cdots oplus mathcal\left\{O\right\}\left(a_n\right).$

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