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).

Contents

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 vector bundles over \mathbb{P}^1_k for k any field.

See also

References