In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces. which is the classical approach, or as locally free sheaves on algebraic curves C in a more general, algebraic setting (which can for example admit singular points).
Some foundational results on classification were known in the 1950s. The result of Grothendieck (1957), that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of Birkhoff (1909) on the Riemann–Hilbert problem.
Atiyah (1957) gave the classification of vector bundles on elliptic curves.
The Riemann–Roch theorem for vector bundles was proved by Weil (1938), before the 'vector bundle' concept had really any official status. Although, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. He was seeking a generalization of the Jacobian variety, by passing from holomorphic line bundles to higher rank. This idea would prove fruitful, in terms of moduli spaces of vector bundles. following on the work in the 1960s on geometric invariant theory.
See also
Hitchin system
References
Atiyah, M. (1957). "Vector bundles over an elliptic curve". Proc. London Math. Soc. VII: 414–452. doi:10.1112/plms/s3-7.1.414. Also in Collected Works vol. I
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.
Grothendieck, A. (1957). "Sur la classification des fibrés holomorphes sur la sphère de Riemann". Amer. J. Math. 79 (1): 121–138. doi:10.2307/2372388. JSTOR 2372388.
Weil, André (1938). "Zur algebraischen Theorie der algebraischen Funktionen". Journal für die reine und angewandte Mathematik. 179: 129–133. doi:10.1515/crll.1938.179.129.
Topics in algebraic curves
Rational curves
Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic
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
De Franchis theorem Faltings's theorem Hurwitz's automorphisms theorem Hurwitz surface Hyperelliptic curve
Plane curves
AF+BG theorem Bézout's theorem Bitangent Cayley–Bacharach theorem Conic section Cramer's paradox Cubic plane curve Fermat curve Genus–degree formula Hilbert's sixteenth problem Nagata's conjecture on curves Plücker formula Quartic plane curve Real plane curve
Riemann surfaces
Belyi's theorem Bring's curve Bolza surface Compact Riemann surface Dessin d'enfant Differential of the first kind Klein quartic Riemann's existence theorem Riemann–Roch theorem Teichmüller space Torelli theorem
Constructions
Dual curve Polar curve Smooth completion
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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