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

TL;DR

This paper develops the theory of holomorphic differential forms on complex manifolds over commutative Banach algebras, explores their cohomology, and discusses the structure of manifolds formed by continuous families of complex manifolds, proposing related open problems.

Contribution

It introduces a framework for defining and studying $A$-holomorphic differential forms and their cohomology on $A$-manifolds, extending classical complex geometry concepts to Banach algebra settings.

Findings

Defined $A$-holomorphic differential forms and their sheaves on $A$-manifolds.

Established that the cohomology groups are $A$-modules, preserving algebraic structure.

Proposed the possibility of Dolbeault theorem for continuous sums of complex manifolds and posed related embedding 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 $N$.