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 ${\mathbb {CP}}^{1}$ is a direct sum of holomorphic line bundles. The theorem was proved by Grothendieck (1957, Theorem 2.1), and is more or less equivalent to Birkhoff factorization introduced by Birkhoff (1909).

Statement

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

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

{\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 ${\mathbb {P}}_{k}^{1}$ for any field $k$ .^[1] It also holds for $\mathbb {P} ^{1}$ with one or two orbifold points, and for chains of projective lines meeting along nodes. ^[2]

References

↑ 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, Variations on a theme of Grothendieck, arXiv:1210.8161

Birkhoff, George David (1909), "Singular points of ordinary linear differential equations", Transactions of the American Mathematical Society, 10 (4): 436–470, ISSN 0002-9947, JFM 40.0352.02, JSTOR 1988594, doi:10.2307/1988594
Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère de Riemann", American Journal of Mathematics, 79: 121•138, doi:10.2307/2372388 .
Okonek, C.; Schneider, M.; Spindler, H. (1980), Vector bundles on complex projective spaces, Progress in Mathematics, Birkhäuser .

Topics in algebraic curves

Rational curves

Elliptic curves

Analytic theory	Elliptic function Elliptic integral Fundamental pair of periods Modular form
Arithmetic theory	Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell–Weil theorem Nagell–Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof–Elkies–Atkin algorithm
Applications	Elliptic curve cryptography Elliptic curve primality

Higher genus

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Constructions

Structure of curves

Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
Moduli	ELSV formula Gromov–Witten invariant Hodge bundle Moduli of algebraic curves Stable curve
Morphisms	Hasse–Witt matrix Riemann–Hurwitz formula Prym variety Weber's theorem
Singularities	Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

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Birkhoff–Grothendieck theorem

Statement

Generalization

See also

References