Birkhoff–Grothendieck theorem

In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line.

is a direct sum of holomorphic line bundles.

The theorem was proved by Alexander Grothendieck (1957, Theorem 2.1),[1] and is more or less equivalent to Birkhoff factorization introduced by George David Birkhoff (1909).

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.

with one or two orbifold points, and for chains of projective lines meeting along nodes.

[4] One application of this theorem is it gives a classification of all coherent sheaves on

We have two cases, vector bundles and coherent sheaves supported along a subvariety, so

Since the only subvarieties are points, we have a complete classification of coherent sheaves.