Introduction to compact Riemann surfaces, Jacobians, and abelian varieties. (English) Zbl 0815.14018

Waldschmidt, Michel (ed.) et al., From number theory to physics. Lectures of a meeting on number theory and physics held at the Centre de Physique, Les Houches (France), March 7-16, 1989. Berlin: Springer-Verlag. 64-211 (1992).
The theory of compact Riemann surfaces which arose in the last century turned after efforts of the most preeminant mathematicians into a very big field, in which it is rather hard to obtain systematic education. But nowadays the different constructions connected with Riemann surfaces were involved into different fields of applied mathematics and theoretical physics. In the same time the researcher who met the necessity to use abelian functions has a big problem with the literature. Therefore it is extremely valuable to overlook the theory of Riemann surfaces speaking on the modern language and keeping in mind current applications. – The paper under review excellent accomplishes this conception.
The author starts with a detailed discussion of the different definitions of Riemann surfaces and then gives a short survey of the theory of compact Riemann surfaces almost without proofs but with bibliographical comments. The chapter is supplied by important appendices devoted to topology of surfaces and holomorphic bundles on Riemann surfaces. The proofs given there seem to be simpler than that given in the usual analysis on Riemann surfaces.
The second chapter deals with Jacobians. The Jacobi inversion problem is stated, and it is valuable, that the nonhyperelliptic case is also discussed. A special section in this chapter is devoted to the classification of Jacobians written for the “field theoretical physicists”. The last sections contain Abel’s theorem and “historical digression”.
The last section discusses Riemann bilinear relations, Riemann conditions, theta functions, Riemann’s theorem and abelian varieties from the algebro-geometrical point of view.
For the entire collection see [Zbl 0784.00021].


14H55 Riemann surfaces; Weierstrass points; gap sequences
14H40 Jacobians, Prym varieties
14K30 Picard schemes, higher Jacobians