In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over
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Statement
More precisely, the statement of the theorem is as the following.
Every holomorphic vector bundle
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