# Birkhoff–Grothendieck theorem

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

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

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

## References

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