Abstract
AbstractLet Γ(M) be the set of all global continuous cross sections of a continuous family M of compact complex manifolds on a compact Hausdorff space X. In this paper, we introduce a C(X)-manifold structure on Γ(M). Especially, if X is contractible, then Γ(M) is a finite-dimensional C(X)-manifold. Here, C(X) denotes the Banach algebra of all complex-valued continuous functions on X.
Highlights
An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the nite direct product Cn of the complex number eld
The de nition of a holomorphic function by Lorch can be straightforwardly generalized to a Fréchet di erentiable mapping from an open set of a Banach A-module to a Banach A-module. (The de nition of a Banach A-module is given in De nition 2.4.) That is, it is said to be A-holomorphic, if the Fréchet derivatives are A-linear
We introduce a concept of compact continuous families of complex manifolds (De nition 1.3)
Summary
Precise de nitions will be given . we believe that there is no di culty in reading this section if the reader knows the very basic concepts of complex analytic families and Banach algebras. If X is a compact Hausdor space, the algebra C(X) of all complex valued continuous functions on X is the most basic example of a commutative Banach algebra (a commutative C*-algebra). The set of all global continuous cross sections of an Hermitian vector bundle on X is a Banach C(X)-module (Example 2.7). We see that if the set of all global continuous cross sections of a compact continuous family M on the base space X is denoted by Γ(M), the structure of a C(X)-manifold modeled on the C(X)-modules of all global continuous cross sections of some Hermitian vector bundles on X is introduced into Γ(M) (Corollary 3.9). The spray corresponding to LeviCivita connection (or the canonical connection) of an Hermitian manifold is only di erentiable, and generally not holomorphic Overcoming this di culty is the technically major part in our proof.
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