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 ${\displaystyle \mathbb {CP} ^{1}}$ 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

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

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

${\displaystyle {\mathcal {E}}\cong {\mathcal {O}}(a_{1})\oplus \cdots \oplus {\mathcal {O}}(a_{n}).}$

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

The same result holds in algebraic geometry for algebraic vector bundle over ${\displaystyle \mathbb {P} _{k}^{1}}$ for any field ${\displaystyle k}$.[1] It also holds for ${\displaystyle \mathbb {P} ^{1}}$ with one or two orbifold points, and for chains of projective lines meeting along nodes. [2]