A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces

Abstract

Abstract. This paper presents the proof of the relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces using conic neighbourhoods of holomorphic sections over 1-convex spaces. A proof of a version of Cartan's Theorem A for 1-convex spaces is also given.

Approximation of Holomorphic Mappings on 1-Convex Domains

“…The next result is a direct corollary of Theorem 8.1 and the main result in [24]. However, it also follows from our main Theorem 3.3 (independently of [24]) by a simple induction argument, which we sketch in the proof. Corollary 8.2.…”

“…, l}) with Oka fiber such that the submersion π : Z| D → D satisfies Condition E. IfD is O(D ′ )-convex, then for every continuous πsection f 0 :D ′ → Z which is of class A k onD there exists a homotopy f t :D ′ → Z of continuous π-sections of class A k onD such that f t is close to f 0 in C k (D) and agrees with f 0 on E for every t ∈ [0, 1], and the section f 1 is of class A k onD ′ . Theorem 3.3 still holds if we replaceD by an arbitrary compact O(D ′ )convex set K ⊂ D ′ and assume that the initial section f 0 is holomorphic in a neighborhood of K. By using Theorem 3.3 inductively, we immediately obtain the corresponding Oka principle in the open case, that is, for sections of a holomorphic fiber bundle Z → X with Oka fiber over a 1-convex base X (see Corollary 8.2 and compare with the main result in [24]).…”

“…An attempt in this direction was made in [24, §4] (see also [25,Corollary 2.6]). However, the proof in [24] is incomplete (see Sec. 7 below or the Appendix in [11]).…”

“…7. The only thing that is incorrect or incomplete in [24] is the proof of the existence of a holomorphic spray of π-sections, which has prescribed core and is dominating on the complement of the exceptional set (see Sec. 7 below for further explanation).…”

The Oka Principle for Sections of Stratified Fiber Bundles

“…The case of Theorem 1.4 when X is a Stein manifold (without singularities) and π : Z → X is a holomorphic fiber bundle whose fiber Y admits a dominating holomorphic spray is due to Gromov [31]. For exposition and extensions of Gromov's work see the papers [10,11,21,22,23,45]; for a homotopy theoretic point of view see Lárusson [38,39,40].…”