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 ${\displaystyle \mathbb {CP} ^{1}}$ is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck (1957, Theorem 2.1), and is more or less equivalent to Birkhoff factorization introduced by George David Birkhoff (1909).