Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps

Abstract

Abstract. We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain Ω ⋐ C n and any connected complex manifold Y , the space O(Ω, Y ) contains a dense holomorphic disc. Our second result states that Y is an Oka manifold if and only if for any Stein space X there exists a dense entire curve in every path component of O(X, Y ).In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain Ω ⋐ C n , any fixed-point-free automorphism of Ω and any connected complex manifold Y , there exists a universal map Ω → Y . We also characterize Oka manifolds by the existence of universal maps.

Elliptic characterization and localization of Oka manifolds

“…Remark 2.8. (1) In the previous paper [12], we proved that several holomorphic flexibility properties such as strong C-connectedness characterize Oka manifolds if we generalize these properties to spaces of holomorphic maps. Since we may consider Condition Ell 1 as strong dominability for spaces of holomorphic maps (cf.…”

“…We also use the technique in the previous paper [12] where we proved other characterizations of Oka manifolds. We may reduce the approximation problem in the definition of Oka manifolds (Definition 1.1) to the following lemma (cf.…”

“…By Theorem 3.2, we obtain a holomorphic map H : Ω × C → Y such that H(•, j) ∈ U j , j = 0, 1. This implies that Y is an Oka manifold by [12,Theorem 3.2].…”