Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 Alexander Grothendieck, 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 G. D. Birkhoff on the Riemann–Hilbert problem. Michael Atiyah gave the classification of vector bundles on elliptic curves.

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