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

Abstract

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.

This work has been partially supported by the National Science Foundation under Grant NSF GP-1904.

Received May 27, 1964.