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
[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 .[3] It also holds for with one or two orbifold points, and for chains of projective lines meeting along nodes. [4]
Applications
[edit]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
[edit]References
[edit]- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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
[edit]- Okonek, Christian; Schneider, Michael; Spindler, Heinz (1980). Vector Bundles on Complex Projective Spaces. Modern Birkhäuser Classics. Birkhäuser Basel. doi:10.1007/978-3-0348-0151-5. ISBN 978-3-0348-0150-8.
- Salamon, S. M.; Burstall, F. E. (1987). "Tournaments, Flags, and Harmonic Maps". Mathematische Annalen. 277 (2): 249–266. doi:10.1007/BF01457363. S2CID 120270501.
External links
[edit]- Roman Bezrukavnikov. 18.725 Algebraic Geometry (LEC # 24 Birkhoff–Grothendieck, Riemann-Roch, Serre Duality) Fall 2015. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons BY-NC-SA.
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