Birkhoff–Grothendieck theorem

Short description: Classifies holomorphic vector bundles over the complex projective line

In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^1$ is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck (1957, Theorem 2.1), It also holds for ${\mathbb{\mathbb{P}}}^1$ with one or two orbifold points, and for chains of projective lines meeting along nodes.

Applications

One application of this theorem is it gives a classification of all coherent sheaves on ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^1$. We have two cases, vector bundles and coherent sheaves supported along a subvariety, so ${𝒪}(k),{{𝒪}}_{nx}$ where n is the degree of the fat point at $x\in {\mathbb{ℂ}\mathbb{\mathbb{P}}}^1$. Since the only subvarieties are points, we have a complete classification of coherent sheaves.

See also

• Algebraic geometry of projective spaces
• Euler sequence
• Splitting principle
• K-theory
• Jumping line

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.