Approximation of holomorphic mappings on strongly pseudoconvex domains

Abstract

Abstract. Let D be a relatively compact strongly pseudoconvex domain in a Stein manifold S. We prove that for every complex manifold We also establish the Oka property for sections of continuous or smooth fiber bundles overD which are holomorphic over D and whose fiber enjoys the Convex approximation property (Theorem 1.7).

Manifolds of holomorphic mappings from strongly pseudoconvex domains

“…It is easily verified that the transition map between any such pair of charts is holomorphic (the argument given in [11] for A(D, Y ) applies in all cases; for the Sobolev classes see [33,Theorem 9.10]). The collection of all such charts defines a holomorphic Banach (resp.…”

“…The method in [11] can be used to obtain the same result in the more general case when h : X →D is a smooth submersion ontoD which is holomorphic over D; an example is a smooth fiber bundle overD which is holomorphic over D. However, for holomorphic submersions which extend holomorphically to a neighborhood ofD in S, the above construction is simpler than the one in [11]. Indeed, we do not need a new splitting lemma for each of the function spaces -we only need it for the space A h (D, E) where h : E →D is a complex vector bundle which is holomorphic over D.…”

“…By Proposition 4.1 in [11] there exists a map F :D×rB N → X of class A(D×rB N ) for some r > 0 and N ∈ N such that for all z ∈D we have…”

“…We choose to present a different proof which is based on the method of holomorphic sprays developed in [10,11]. This method explicitly produces a holomorphic vector bundle π : E → U as in Theorem 1.1 whose total space E contains a biholomorphic copy of a Stein neighborhood of f (D).…”

“…In [11] the authors constructed a dominating spray with a given central section, holomorphic over D and continuous overD, by successively gluing sprays over small subsets ofD. The key ingredient of this construction is a Cartan type splitting lemma with control up to the boundary [10, Theorem 3.2]; its proof uses a sup-norm bounded linear solution operator to the∂-equation for (0, 1)-forms on strongly pseudoconvex domains.…”