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

Abstract: 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 $(\wedge^kT^*)(M)$ on $M$. For an open set $U$ of $M$, $\omega$ is said to be an $A$-holomorphic differential $k$-form on $U$, if $\omega$ is an $A$-holomorphic section of $(\wedge^kT^*)(M)$ on $U$. So, if the set of all $A$-holomorphic differential $k$-forms on $U$ is denoted by $\Omega_{M}^k(U)$, then ${\Omega_{M}^k(U)}_{U}$ is a sheaf of modules on the structure sheaf $O_M$ of the $A$-manifold $M$ and the cohomology group $H^l(M,\Omega_{M}^k)$ with the coefficient sheaf ${\Omega_{M}^k(U)}_{U}$ 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 $H^l(M,\Omega_{M}^k)$ as an $A$-module. Furthermore, we try to define the structure sheaf of a manifold that is locally a continuous family of $\mathbb C$-manifolds (and also the one of an analytic family). Directing attention to a finite family of $\mathbb C$-manifolds, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of $\mathbb C$-manifolds. Also, we state a few related problems. One of them is the following. Let $n\in \mathbb N$. Then, does there exist a $\mathbb C^n$-manifold $N$ such that for any $\mathbb C$-manifolds $M_1, M_2, \cdots, M_{n-1}$ and $M_n$, $N$ can not be embedded in the direct product $M_1\times M_2 \times \cdots \times M_{n-1} \times M_n$ as a $\mathbb C^n$-manifold ? So, we propose something that is likely to be a candidate for such a $\mathbb C^2$-manifold $N$.