On the Locally Complete Families of Complex Analytic Structures

Abstract

The main purpose of this paper is to prove that there is an effective and locally complete holomorphic family of deformations for any compact complex analytic manifold. As is known by examples (cf. [1, ? 16]), the parameter spaces of locally complete holomorphic families of complex analytic structures might have singular points. So we formulate in ? 1 the definition of holomorphic families of complex analytic structures over analytic sets in a form which is suitable to our use. The rest of ? 1 is devoted to establishing several results which we use in the proof of our main theorem. In ? 2, we construct a locally complete holomorphic family of complex analytic structures for any compact complex analytic manifold. Construction is by means of vector-valued differential forms associated with complex analytic structures. In ? 3, we prove that the effective family we constructed is also locally complete at points in the parameter space sufficiently near the reference point. The writer would like to express his thanks to Professors K. Kodaira and D. C. Spencer for their useful advice during the preparation of this paper.