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 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).[2]

Statement

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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

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The same result holds in algebraic geometry for algebraic vector bundle over   for any field  .[3] It also holds for   with one or two orbifold points, and for chains of projective lines meeting along nodes. [4]

Applications

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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   where n is the degree of the fat point at  . Since the only subvarieties are points, we have a complete classification of coherent sheaves.

See also

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References

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  1. ^ Grothendieck, Alexander (1957). "Sur la classification des fibrés holomorphes sur la sphère de Riemann". American Journal of Mathematics. 79 (1): 121–138. doi:10.2307/2372388. JSTOR 2372388. S2CID 120532002.
  2. ^ Birkhoff, George David (1909). "Singular points of ordinary linear differential equations". Transactions of the American Mathematical Society. 10 (4): 436–470. doi:10.2307/1988594. ISSN 0002-9947. JFM 40.0352.02. JSTOR 1988594.
  3. ^ 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.
  4. ^ Martens, Johan; Thaddeus, Michael (2016). "Variations on a theme of Grothendieck". Compositio Mathematica. 152: 62–98. arXiv:1210.8161. Bibcode:2012arXiv1210.8161M. doi:10.1112/S0010437X15007484. S2CID 119716554.

Further reading

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