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An n-dimensional complex manifold is a manifold by biholomorphic mappingsbetween open sets of the finite direct product of the complex number field. Onthe other hand, when A is a commutative Banach algebra, Lorch gave a definitionthat an A-valued function on an open set of A is holomorphic. The definition ofa holomorphic function by Lorch can be straightforwardly generalized to anA-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-dimensionalA-manifold) is easily defined. However, in my opinion, it seems that so manynontrivial examples were not known (including the case of n=1, that is, Riemannsurfaces).  By the way, if X is a compact Hausdorff space, then the algebra C(X) of allcomplex valued continuous functions on X is the most basic example of acommutative Banach algebra (furthermore, a commutative C*-algebra). In thisnote, we see that if the set of all continuous cross sections of a continuousfamily M of compact complex manifolds (a topological deformation M of compactcomplex analytic structures) on X is denoted by G(M), then the structure of aC(X)-manifold modeled on the C(X)-modules of all continuous cross sections ofcomplex vector bundles on X is introduced into G(M). Therefore, especially, ifX is contractible, then G(M) is a finite-dimensional C(X)-manifold.

Finite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds