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].
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© 2004 Springer-Verlag Berlin Heidelberg
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Birkenhake, C., Lange, H. (2004). Line Bundles on Complex Tori. In: Complex Abelian Varieties. Grundlehren der mathematischen Wissenschaften, vol 302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06307-1_4
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DOI: https://doi.org/10.1007/978-3-662-06307-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05807-3
Online ISBN: 978-3-662-06307-1
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