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 [math]\displaystyle{ \mathbb{CP}^1 }[/math] is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck (1957, Theorem 2.1),Cite error: Closing </ref> missing for <ref> tag
It also holds for [math]\displaystyle{ \mathbb{P}^1 }[/math] with one or two orbifold points, and for chains of projective lines meeting along nodes.
[1]
Applications
One application of this theorem is it gives a classification of all coherent sheaves on [math]\displaystyle{ \mathbb{CP}^1 }[/math]. We have two cases, vector bundles and coherent sheaves supported along a subvariety, so [math]\displaystyle{ \mathcal{O}(k), \mathcal{O}_{nx} }[/math] where n is the degree of the fat point at [math]\displaystyle{ x \in \mathbb{CP}^1 }[/math]. Since the only subvarieties are points, we have a complete classification of coherent sheaves.
See also
References
- ↑ Martens, Johan; Thaddeus, Michael (2016), "Variations on a theme of Grothendieck", Compositio Mathematica 152: 62–98, doi:10.1112/S0010437X15007484, Bibcode: 2012arXiv1210.8161M
Further reading
- 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. http://eudml.org/doc/164249.
External links
- 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.
 |  |
