# 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]}

## See also

## References

- ↑ Hazewinkel, Michiel; 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 - ↑ Martens, Johan; Thaddeus, Michael,
*Variations on a theme of Grothendieck*, arXiv:1210.8161

- Birkhoff, George David (1909), "Singular points of ordinary linear differential equations",
*Transactions of the American Mathematical Society*,**10**(4): 436–470, doi:10.2307/1988594, ISSN 0002-9947, JFM 40.0352.02, JSTOR 1988594 - Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère 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.