Algebraic Deformations of Polarized Varieties
Let V be a projectively embeddable complete non-singular variety of dimension n > 1. Let f be a projective embedding of V, U a non-singular variety, W a non-singular variety and φ a morphism of W onto U such that φ-1(u0) = f(V) for some point u0 of U. Denote by ∑(V) the set of all those complete non-singular fibres φ-1(u), u ∈ U, as we consider all possible (f, U, W). Suppose that we call members of ∑(V) (algebraic) deformations of V and propose to study ∑(V) from the stand point of algebraic geometry, as a generalization of the case of curves. This has been taken up at least locally by Kodaira, Spencer, Kuranishi and others in the case of characteristic 0 from a little more general point of view of complex manifolds (cf. [9] and references given in [16]).
REDUCTION OF ALGEBRAIC VARIETIES WITH RESPECT TO A DISCRETE VALUATION OF THE BASIC FIELD.
In the arithmetic theory of algebraic varieties it is useful to have a general theory on "reduction modulo 4" of varieties. Here "reduction" means, roughly spoken, the following process: let V be an algebraic variety defined over a field k with a discrete valuation, and denote by p the maximal ideal of that valuation; then we have an algebraic variety VP over the residue field K of t defined by the equations for v modulo p; such a variety V may be called the variety obtained from the variety V by the reduction modulo p. When one tries to construct the general theory of reduction in this sense, the first thing to do will be to investigate how the algebraico-geometric properties of V behave in the reduction process. The object of this paper is concerned with that problem. We shall give a precise definition of the reduction of varieties and the reduction of cycles on a variety. In the case when the ambient varieties v and V are both absolutely irreducible, we shall show that the reduction of cycles on V defines a linear mapping from the group of rational 1-cycles over Ic into the group of rational V-cycles over K of the same dimension; and shall show that the operations on cycles (the intersection-product, the direct product and the algebraic projection) are all preserved by that mapping (Section 4). If we consider reduction as specialization, this result may be regarded as a specialization-theory of cycles on algebraic varieties, in a certain sense. In fact, from this reduction-theory of cycles we can easily obtain a more general specialization-theory of cycles (Section 5). The latter will be a generalization of the specialization-theory of cycles which has been given by T. Matsusaka [4] and P. Samuel [6]. Our theory may be also called a "local" theory of varieties at a divisor p, whereas 4 is not a usual divisor on varieties. We shall also show the following fact: let K be a field with a set of infinitely many valuations which satisfies a certain condition (condition (I) in Section 6); if v is an absolutely irreducible variety defined over K, then for almost all those valuations, it remains absolutely irreducible by the reduction process.
Foundations of Algebraic Geometry
Algebraic preliminaries Algebraic theory of specializations Analytic theory of specializations The geometric language Intersection-multiplicities (special case) General intersection-theory The geometry on abstract varieties The calculus of cycles Divisors and linear systems Comments and discussions Appendix I. Normal varieties and normalization Appendix II. Characterization of the $i$-symbol by its properties Appendix III. Varieties over topological fields Index of definitions.