Vector bundles on algebraic curves can be studied as on , which is a classical approach, or as locally free sheaves on algebraic curves C in a more general algebraic environment (which, for example, allows special dots ).
Some fundamental classification results were known in the 1950s. The result of Grothendieck [1] that holomorphic vector bundles on the Riemann sphere are sums of 1-dimensional bundles is often called the Birkhoff – Grothendieck theorem , since it follows from Birkhoff’s earlier work [2] .
Atiyah [3] gave a classification of vector bundles on elliptic curves .
The Riemann – Roch theorem for vector bundles was proved by Weil [4] even before the concept of vector bundle got a real and official status, although the corresponding ruled surfaces were classical objects. See . Weil considered the possibility of generalizing the by moving from to higher ranks. This idea turned out to be fruitful, which was expressed in studies of moduli spaces of vector bundles, starting with work in the 1960s on the .
Notes
- ↑ Grothendieck, 1957 .
- ↑ Birkhoff, 1909 .
- ↑ Atiyah, 1957 .
- ↑ Weil, 1938 .
Literature
- Atiyah M. Vector bundles over an elliptic curve // Proc. London Math. Soc .. - 1957. - T. VII . - S. 414–452 . - DOI : 10.1112 / plms / s3-7.1.414 .
- George David Birkhoff . Singular points of ordinary linear differential equations // Transactions of the American Mathematical Society . - 1909. - T. 10 , no. 4 . - S. 436-470 . - ISSN 0002-9947 . - DOI : 10.2307 / 1988594 .
- Grothendieck A. Sur la classification des fibrés holomorphes sur la sphère de Riemann // Amer. J. Math. . - 1957.- T. 79 . - S. 121–138 . - DOI : 10.2307 / 2372388 .
- André Weil . Zur algebraischen Theorie der algebraischen Funktionen // Journal für die reine und angewandte Mathematik . - 1938.- T. 179 . - S. 129–133 . - DOI : 10.1515 / crll.1938.179.129 .