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Line Bundles on Complex Tori

  • Herbert Lange
  • Christina Birkenhake
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 302)

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].

Keywords

Exact Sequence Line Bundle Analytic Representation Chern Class Hermitian Form 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Herbert Lange
    • 1
  • Christina Birkenhake
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

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