Abelian Varieties

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.