  An n-dimensional complex manifold is a manifold by biholomorphic mappings
between open sets of the finite direct product of the complex number field. On
the other hand, when A is a commutative Banach algebra, Lorch gave a definition
that an A-valued function on an open set of A is holomorphic. The definition of
a holomorphic function by Lorch can be straightforwardly generalized to an
A-valued function on an open set of the finite direct product of A. Therefore,
a manifold modeled on the finite direct product of A (an n-dimensional
A-manifold) is easily defined. However, in my opinion, it seems that so many
nontrivial examples were not known (including the case of n=1, that is, Riemann
surfaces).
  By the way, if X is a compact Hausdorff space, then the algebra C(X) of all
complex valued continuous functions on X is the most basic example of a
commutative Banach algebra (furthermore, a commutative C*-algebra). In this
note, we see that if the set of all continuous cross sections of a continuous
family M of compact complex manifolds (a topological deformation M of compact
complex analytic structures) on X is denoted by G(M), then the structure of a
C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of
complex vector bundles on X is introduced into G(M). Therefore, especially, if
X is contractible, then G(M) is a finite-dimensional C(X)-manifold.
