Algebraic geometry and complex analysis (i. e. the theory of complex manifolds) have made remarkable progress in the last several decades. Let us briefly review a part of their development as an introduction to various problems treated in the articles in this volume.

First, in the 1940's, a big reform began in algebraic geometry. One of the main aims was to establish solid foundations for this fascinating branch of mathematics, whose underpinnings had become the subject of criticism because of the highly intuitive, rather than rigorous, treatment of its early pioneers, the Italian algebraic geometers. From the algebraic (or rather “abstract”) viewpoint, this was carried out by Zariski, Weil and others (cf. Zariski (3, vol. I), Weil (1, 2)). Their methods are based on abstract algebra, making no use of topological or transcendental methods, and thus are applicable not only to algebraic varieties over the field of complex numbers, but also to those over an arbitrary ground field. In particular, Weil's method, including the abstract theory of abelian varieties, opened a new way for the application of algebraic geometry to number theory ; we mention here only Weil's proof (2) of the Riemann hypothesis for curves over a finite field. The reader is referred to the 1950 Congress talks of Weil (3) and Zariski (2) for their basic ideas.