Interludium: the geometry of Abelian varieties

Abstract

For the convenience of the reader who wishes to go on reading the rest of the text we include here a chapter about Abelian varieties. There is nothing new in this chapter, our intention was to give a compact and coherent presentation of the theory of Abelian varieties in a form suitable for applications to integrable systems. Our exposition is partly algebraic partly analytic, we think that both approaches highlight different aspects of the theory of Abelian varieties, see for example the theorems of Abel, Jacobi and Riemann in Paragraph 4.3. The main references for the theory of Abelian varieties are [LB], [Kern] and [Mum2] and [Mum3]. However the relevant chapters in [ACGH], [GH] and [Mum4] are also highly recommended to learn this subject.