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

TL;DR

This paper explores finite-dimensional complex manifolds over commutative Banach algebras, particularly C(X), and introduces a structure on the space of continuous sections of a family of compact complex manifolds, showing it becomes a finite-dimensional C(X)-manifold when X is contractible.

Contribution

It generalizes the concept of complex manifolds to the setting of commutative Banach algebras and introduces a new structure on the space of continuous sections of complex manifold families.

Findings

G(M) has a C(X)-manifold structure.

When X is contractible, G(M) is finite-dimensional.

The framework includes classical complex manifolds as special cases.

Abstract

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.