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Register a new userFinite-dimensional complex manifolds on commutative Banach algebras and continuous families of compact complex manifolds
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Hiroki Yagisita
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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.
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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 $nin mathbb N$. 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 $M_1times M_2 times cdots times M_{n-1} times M_n$ as a $mathbb C^n$-manifold ? So, we propose something that is likely to be a candidate for such a $mathbb C^2$-manifold $N$.
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