The Picard functor for curves and their Jacobians

Abstract

In the last chapter of volume I we constructed the Jacobian of a compact Riemann surface S. The Jacobian was defined as the group of isomorphism classes of holomorphic line bundles on S. Our main result asserted that the Jacobian had the structure of a complex torus, and assuming theorems of Lefschetz and Chow we proved that this torus is a projective algebraic variety. We heavily relied on transcendental methods.