Birkhoff-Grothendieck theorem

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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.

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

\mathcal{E}\cong\mathcal{O}(a_1)\oplus \cdots \oplus \mathcal{O}(a_n).

[edit] References

  • Okonek, C.; Schneider, M. & Spindler, H. (1980), Vector bundles on complex projective spaces, Progress in Mathematics, Birkhäuser .


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