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Abelian Varieties

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

Abstract

An abelian variety is by definition a complex torus admitting a positive definite line bundle. The Riemann Relations are necessary and sufficient conditions for a complex torus to be an abelian variety. They were given by Riemann in the special case of the Jacobian variety of a curve (see Chapter 11). For the general statement we refer to Poincaré-Picard [1] and Frobenius [2], although it was apparently known to Riemann and Weierstraß. Another characterization of abelian varieties is due to Lefschetz [1] p. 367: a complex torus is an abelian variety if and only if it admits the structure of an algebraic variety. Lefschetz showed that if L is a positive definite line bundle on a complex torus X, then L n is very ample for any n ≥ 3, i.e. the map φL n : X → ℙ N associated to the line bundle L n is an embedding.

Keywords

Line Bundle Algebraic Variety Theta Function Abelian Variety Hermitian Form 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

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

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