2018/10/13 by Hiroki Yagisita, Yagisita, Hiroki
Mathematics · #Holomorphic and Operator Theory #Algebraic and Geometric Analysis #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1810.05829
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 (\wedgekT^*)(M) on M. For an open set U of M, ω is said to be an A-holomorphic differential k-form on U, if ω is an A-holomorphic section of (\wedgekT^*)(M) on U. So, if the set of all A-holomorphic differential k-forms on U is denoted by ΩMk(U), then \ΩMk(U)\U is a sheaf of modules on the structure sheaf OM of the A-manifold M and the cohomology group Hl(M,ΩMk) with the coefficient sheaf \ΩMk(U)\U is an OM(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 Hl(M,ΩMk) 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∈ \mathbb N. Then, does there exist a \mathbb Cn-manifold N such that for any \mathbb C-manifolds M1, M2, ⋯, Mn-1 and Mn, N can not be embedded in the direct product M1× M2 × ⋯ × Mn-1 × Mn as a \mathbb Cn-manifold ? So, we propose something that is likely to be a candidate for such a \mathbb C2-manifold N.