Birkhoff–Grothendieck theorem

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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[edit]

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[edit]

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[edit]

References[edit]

  1. ^ 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 
  2. ^ Martens, Johan; Thaddeus, Michael, Variations on a theme of Grothendieck, arXiv:1210.8161Freely accessible