Abel’s Theorem

Abstract

The line bundles over a given complex curve are in a one-to-one correspondence with the linear equivalence classes of divisors on this curve. Such a class has an integer-valued characteristic, the degree. Since divisors consist of points of the curve, it is natural to expect that the set of classes of divisors of the same degree is endowed with additional structures. It must be a topological space and, moreover, a complex variety. Abel’s theorem identifies the space of classes of divisors of zero degree on a curve of genus g with a g-dimensional complex torus, the Jacobian of the curve.