Birkhoff–Grothendieck theorem

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In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over  \mathbb{CP}^1 is a direct sum of holomorphic line bundles. The theorem was proved by Grothendieck (1957, Theorem 2.1), and is more or less equivalent to Birkhoff factorization introduced by Birkhoff (1909).

Statement[edit]

More precisely, the statement of the theorem is as the following.

Every holomorphic vector bundle  \mathcal{E} on  \mathbb{CP}^1 is holomorphically isomorphic to a direct sum of line bundles:

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

The notation implies each summand is a Serre twist some number of times of the trivial bundle. The representation is unique up to permuting factors.

Generalization[edit]

The same result holds in algebraic geometry for algebraic vector bundle over \mathbb{P}^1_k for any field k.[1]

See also[edit]

References[edit]

  1. ^ Hazewinkel, Michael; Martin, Clyde F. (1982), "A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line", Journal of Pure and Applied Algebra 25 (2): 207–211, doi:10.1016/0022-4049(82)90037-8