Line Bundles on Complex Tori

Abstract

In this chapter we describe the structure of the group Pic(X) of holomorphic line bundles on a complex torus X = V/Λ. The main result is the Appell-Humbert Theorem, which says that Pic(X) is an extension of the Néron-Severi group NS(X) by the group Hom(Λ, ℂ₁) of characters of Λ with values in the circle group ℂ₁. The group NS(X) turns out to be the group of hermitian forms H on V satisfying Im H (Λ, Λ) ⊆ ℤ. The theorem was proven for dimension 2 by Humbert [1] applying a result of Appell [1] and by Lefschetz [1] in general. The present formulation appears in Weil [3] and Mumford [2].