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).

1 Statement
2 Generalization
3 See also
4 References

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.

References

Birkhoff, George David (1909), "Singular points of ordinary linear differential equations", Transactions of the American Mathematical Society 10 (4): 436–470, ISSN 0002-9947, JFM 40.0352.02, JSTOR 1988594
Grothendieck, Alexander (1957), "Sur la classification des fibres holomorphes sur la sphere de Riemann", American Journal of Mathematics 79: 121•138, doi:10.2307/2372388 .

Okonek, C.; Schneider, M.; Spindler, H. (1980), Vector bundles on complex projective spaces, Progress in Mathematics, Birkhäuser .

Birkhoff–Grothendieck theorem

Contents

Statement

Generalization

See also

References