Moduli Theory and Classification Theory of Algebraic Varieties

The rough classification of algebraic varieties is obtained by dividing the smooth, projective, algebraic varieties into classes up to birational equivalence according to the structure of the m-canonical maps and the Albanese maps. This classification leads to the Enriques' classification for surfaces. For higher dimensional varieties the main known results of this theory are described in Ueno's Lecture Motes and in Lectures 1 and 11. The classification of 3-dimensional varieties seems to be possible. (For curves, the classification obtained is almost trivial and divides the curves into classes according to the genus.)
The fine classification is the explicit study of the varieties of the various classes obtained by the rough classification by fibre space methods or by moduli theory. But the rough classification and the fine classification cannot be separated. Certain results from the rough classification point out for which types of algebraic varieties a moduli theory should exist and what the properties of this moduli theory should be. Conversely, to do the hard part of the rough classification, for example the proof of Conjecture C from page 10 for fibre spaces f : V —*-W, the fine classification and a good moduli theory for certain types of algebraic varieties of dimension é n-1 must be available.
It is the purpose of this note, to provide a systematic treatment of moduli theory and to describe the interplay between the rough classification and the fine classification of algebraic varieties, as far as it is known.
In this way, many of the recent developments in moduli theory such as the theory of period maps, the projectivity of moduli spaces, the theory of fine moduli spaces and the compactification of moduli spaces, appear natural and their interaction becomes clear.
These notes, consisting of eleven lectures and an appendix on classical invariant theory, are based on lectures which I gave in the fall of 1975 at the University of Montreal and in the spring of 1976 at Tokyo University.