Narrow Results

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 $\mathrm{\Omega }⋐{ℂ}^n$ and any connected complex manifold $Y$, the space ${𝒪}(\mathrm{\Omega },Y)$ 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 $\mathrm{\Omega }⋐{ℂ}^n$, any fixed-point-free automorphism of $\mathrm{\Omega }$ and any connected complex manifold $Y$, there exists a universal map $\mathrm{\Omega }\to Y$. We also characterize Oka manifolds by the existence of universal maps.

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition ${\mathrm{\text{Ell}}}_1$, and the equivariant localization principle is also given.