Periods of Integrals on Algebraic Manifolds, I. (Construction and Properties of the Modular Varieties)

I. 0. Introduction. (a) The general problem we have in mind is to investigate the periods of integrals on an algebraic variety V defined over a function field 5. In practice, this will mean that we are given an algebraic family of algebraic varieties VttEB where the general member V = Vt of this family is an ordinary polarized, non-singular algebraic manifold, and we wish to study the behavior of the period matrix U (t) of Vt as a function of t. In order to discuss Q (t), we should think of the periods as a (not everywhere defined) mapping '1: B -> M where M, the modular variety associated to V, repersents the totality of inequivalent period matrices satisfying the bilinear relations imposed by the topological manifold underlying V. In this paper (Part I) we shall study the variety 111. Many of he classical results, which arise when V is a curve and M is the Siegel upperhalf-space factored by the modular group, will go through. However, there are some striking differences which turn up, and which seem to be best explained by the presence of higher order period relations. In Part II we shall study the local properties of the period mapping 41 and, in Part TTT, we shall look into the global behavior of '1. Some of these results have been announced in the Proceedings of the National Academy of Sciences (U. S. A. ), Vol. 55 (5), 1303-1309; (6), 13921395, and Vol. 56 (2), 413-416. It is my pleasure to express gratitude to several colleagues who, through conversation and correspondence, have been of immense help in studying this question on periods of integrals.