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 formula_1 is a direct sum of holomorphic line bundles. The theorem was proved by , and is more or less equivalent to Birkhoff factorization introduced by . Statement. More precisely, the statement of the theorem is as the following. Every holomorphic vector bundle formula_2 on formula_1 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 formula_5 for any field formula_6. It also holds for formula_7 with one or two orbifold points, and for chains of projective lines meeting along nodes. Applications. One application of this theorem is it gives a classification of all coherent sheaves on formula_8. We have two cases, vector bundles and coherent sheaves supported along a subvariety, so formula_9 where n is the degree of the fat point at formula_10. Since the only subvarieties are points, we have a complete classification of coherent sheaves. Birkhoff