Birkhoff-Grothendieck theorem

Birkhoff-Grothendieck theorem

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.

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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