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]

Statement

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.

Generalization

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]

Applications

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 ${\displaystyle {\mathcal {O}}(k),{\mathcal {O}}_{np}}$ where n is the degree of the fat point at ${\displaystyle x}$. 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

References

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.
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

• Okonek, C.; Schneider, M.; Spindler, H. (1980). Vector bundles on complex projective spaces. Progress in Mathematics. Birkhäuser.