Complex Tori and Abelian Varieties

This book takes the classical theory of complex tori and complex abelian varieties as a pretext to go through more modern aspects of complex algebraic and analytic geometry. Starting with complex elliptic curves, it moves on to the higher-dimensional case, giving characterizations from different points of view of those complex tori which are abelian varieties, i.e., those that can be holomorphically embedded in a projective space. This allows the author, on the one hand, to illuminate the computations of nineteenth century mathematicians, and on the other to familiarize the reader with more recent theories. Complex tori are ideal in this respect: one can perform ''hands-on'' computations, without the theory being totally trivial. Standard theorems about abelian varieties are proved, and moduli spaces are discussed. The last chapter includes recent results on the geometry and topology of some subvarieties of a complex torus. Each chapter is followed by a varying number of exercises.