New Proof for the Existence of Locally Complete Families of Complex Structures

The purpose of the present paper is to give a simpler new proof and some improvement of the theory the writer developed in [5]. We start with an explanation of the problem. Take a compact C ∞ manifold M and a complex analytic structure M on M. We ask to what extent we can deform the structure M. By “Reform the structure M” we mean that we have a parameter space T with reference point t 0 and an assignment of a complex analytic structure M t for each t in T with M t 0 = M in such a way that M t depends nicely on t. Now, assume that we have such a family {M t : t ∊ T}, a space S with reference point s 0, and a nice mapping τ: S → T with τ (s 0) = t 0. Then the assignment s → M τ(s) is a family of deformations of M, which is called the family induced by τ(s) from the family {M t : t ∊ T}. To answer the question posed above, we would like to construct a universal family, i. e. a family {M t : t ∊ T} such that any family of deformations of M is homeomorphic to a family induced from {M t : t ∊ T}. Among such universal families, we also like to have one which is, in a sense, the most economical one. We are here interested in the local aspect of the theory, i. e. in the germs of families of deformations of M at the reference points.