A theorem of completeness for families of compact analytic spaces

Abstract

A sufficient condition is given for a family of compact analytic spaces to be complete. This condition generalizes to analytic spaces the Theorem of Completeness of Kodaira and Spencer [6]. It contains, as a special case, the rigidity theorem proved by Schuster in [11]. Introduction. If X is a family of deformations of a compact complex manifold, XO, parametrized by a germ of complex manifold, (S, 0), Kodaira and Spencer have proved in [6] that X contains all small deformations of XO (i.e. is (locally) complete) provided that the Kodaira-Spencer map po: To(S) -? H'(Xo, ?o) is surjective. This condition is equivalent to the statement that X contains all deformations of XO parametrized by the analytic space, D, whose underlying topological space is a point, and whose structure sheaf is the ring of dual numbers. The paper [6] appeared before Kuranishi's Completeness Theorem [7], [8]. A very simple proof can be given of the Kodaira-Spencer Theorem if we use Kuranishi's result. We give this proof in ?1. If XO has singularities, it is quite plausible that an analog of the Kuranishi space exists; but, at the moment, no proof exists. A formal analog of the Kuranishi space can be shown to exist, however. In this paper we will prove a generalization of some results of Artin [1] which will allow us to prove the completeness theorem using this formal analog. The Theorem of Completeness is useful in answvering questions of whether a certain property of XO is shared by all small deformations of XO-for it allows one to recognize that a family of spaces having the special property is complete. We have used the theorem in this way in our paper [13]. Some applications are given in ?4. I am indebted to Michael Artin for a sketch of the proof of Theorem 2.2. 1. Quasi-representable functors. By a family of compact complex analytic spaces, we mean a pair of analytic spaces X, S together with a flat, proper morphism T: X-* S. If XO is a given compact analytic space, a family of deformations of XO is a family, (X, r, S), of compact spaces together with a distinguished point so E S Received by the editors December 8, 1970. AMS 1969 subject classifications. Primary 3247; Secondary 1810.