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 ${\displaystyle \mathbb {CP} ^{1}}$ 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]

More precisely, the statement of the theorem is as the following.

Every holomorphic vector bundle ${\displaystyle {\mathcal {E}}}$ on ${\displaystyle \mathbb {CP} ^{1}}$ is holomorphically isomorphic to a direct sum of line bundles:

${\displaystyle {\mathcal {E}}\cong {\mathcal {O}}(a_{1})\oplus \cdots \oplus {\mathcal {O}}(a_{n}).}$

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.

The same result holds in algebraic geometry for algebraic vector bundle over ${\displaystyle \mathbb {P} _{k}^{1}}$ for any field ${\displaystyle k}$.^[3] It also holds for ${\displaystyle \mathbb {P} ^{1}}$ with one or two orbifold points, and for chains of projective lines meeting along nodes. ^[4]

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

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

• 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.

• 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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