# 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 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 on is holomorphically isomorphic to a direct sum of line bundles:

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 for any field .[1] It also holds for with one or two orbifold points, and for chains of projective lines meeting along nodes. [2]