On the complex structure of a manifold of sections (Q923386)

Let E be a fibre bundle over a compact \(C^{\infty}\)-manifold M. Assume further that each fibre of E is a complex manifold. In this paper, structures on the space of all sections \(C^{\infty}(E)\) are discussed in ILH-category, developed by \textit{H. Omori} [Infinite dimensional Lie transformation groups, Lect. Notes Math. 427 (1974; Zbl 0328.58005)]. The main theorems are as follows. Theorem. \(C^{\infty}(E)\) is an (ILH-) complex manifold. As a special case, the space of all \(C^{\infty}\)-maps from M to a complex manifold N, denoted by \(C^{\infty}(M,N)\), is an (ILH-) complex manifold. Theorem. Suppose that M is endowed with an almost complex structure. Then the set of all holomorphic maps from M to N is a complex analytic subset of \(C^{\infty}(M,N)\), in the sense that it is the inverse image of a point by an (ILH-) holomorphic map. The above complex structure of the set of all holomorphic maps coincides with one introduced in a complex- analytical way developed by \textit{A. Douady} [Ann. Inst. Fourier 16, No.1, 1-95 (1966; Zbl 0146.311)] or \textit{M. Namba} [Families of meromorphic functions on compact Riemann surfaces, Lect. Notes Math. 767 (1979; Zbl 0417.32008)] (at least as reduced complex spaces). The arguments of this paper are purely differential-geometric, i.e. without \({\bar \partial}\)- theory.