Holomorphic differential forms of complex manifolds on commutative Banach algebras and a few related problems

Let A be a commutative Banach algebra. Let M be a complex manifold on A (an A-manifold). Then, we define an A-holomorphic vector bundle (∧ k T *)(M) on M. For an open set U of M , ω is said to be an A-holomorphic differential k-form on U , if ω is an A-holomorphic section of (∧ k T *)(M) on U. So, if the set of all A-holomorphic differential k-forms on U is denoted by Ω k M (U), then {Ω k M (U)} U is a sheaf of modules on the structure sheaf O M of the A-manifold M and the cohomology group H l (M, Ω k M) with the coefficient sheaf {Ω k M (U)} U is an O M (M)-module and therefore, in particular, an A-module. There is no new thing in our definition of a holomorphic differential form. However, this is necessary to get the cohomology group H l (M, Ω k M) as an A-module. Furthermore, we try to define the structure sheaf of a manifold that is locally a continuous family of C-manifolds (and also the one of an analytic family). Directing attention to a finite family of C-manifolds, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of C-manifolds. Also, we state a few related problems. One of them is the following. Let n ∈ N. Then, does there exist a C n-manifold N such that (1) there exists k ∈ N such that N can be embedded in the k-dimensional Eu-clidean R n-space (R k) n as an R n-submanifold but (2) for any C-manifolds M 1 , M 2 , · · · , M n−1 and M n , N can not be embedded in the direct product M 1 × M 2 × · · · × M n−1 × M n as a C n-submanifold ? So, we propose something that is likely to be a candidate for such a C 2-manifold N .