Line Bundles on Complex Tori
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  applying a result of Appell  and by Lefschetz  in general. The present formulation appears in Weil  and Mumford .
KeywordsExact Sequence Line Bundle Analytic Representation Chern Class Hermitian Form
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