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).
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:
- 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 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.
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