Birkhoff–Grothendieck theorem

(Redirected from Birkhoff-Grothendieck theorem) 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 $\mathbb {CP} ^{1}$ is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck (1957, Theorem 2.1), and is more or less equivalent to Birkhoff factorization introduced by George David 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 $k$ . It also holds for $\mathbb {P} ^{1}$ 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 $\mathbb {CP} ^{1}$ . We have two cases, vector bundles and coherent sheaves supported along a subvariety, so ${\mathcal {O}}(k),{\mathcal {O}}_{nx}$ where n is the degree of the fat point at $x\in \mathbb {CP} ^{1}$ . Since the only subvarieties are points, we have a complete classification of coherent sheaves.

References

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. S2CID 120532002.
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.
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 (2016). "Variations on a theme of Grothendieck". Compositio Mathematica. 152: 62–98. arXiv:1210.8161. Bibcode:2012arXiv1210.8161M. doi:10.1112/S0010437X15007484. S2CID 119716554.

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.

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

Plane curves

Riemann surfaces

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 (Algebraic curves)
Singularities	A_k singularity Acnode Crunode Cusp Delta invariant Tacnode
Vector bundles	Birkhoff–Grothendieck theorem Stable vector bundle Vector bundles on algebraic curves

This topology-related article is a stub. You can help Misplaced Pages by expanding it.

Categories:

Misplaced Pages