An implicit function theorem for sprays and applications to Oka theory

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 [Formula: see text], and the equivariant localization principle is also given.

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

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 [Formula: see text] and any connected complex manifold [Formula: see text], the space [Formula: see text] contains a dense holomorphic disc. Our second result states that [Formula: see text] is an Oka manifold if and only if for any Stein space [Formula: see text] there exists a dense entire curve in every path component of [Formula: see text]. 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 [Formula: see text], any fixed-point-free automorphism of [Formula: see text] and any connected complex manifold [Formula: see text], there exists a universal map [Formula: see text]. We also characterize Oka manifolds by the existence of universal maps.

STRATIFIED TRANSVERSALITY OF HOLOMORPHIC MAPS

We discuss genericity and stability of transversality of holomorphic maps to complex analytic stratifications. We prove that the set of maps between Stein manifolds and Oka manifolds transverse to a countable collection of submanifolds in the target is dense in the space of holomorphic maps with the weak topology. This greatly generalizes earlier results on the genericity of transverse maps by Forstnerič and by Kaliman and Zaidenberg. As an application we show that the Whitney (a)-regularity of a complex analytic stratification is necessary and sufficient for the stability of transverse holomorphic maps between a Stein manifold and an Oka manifold. This gives an analogue of a theorem in the real case due to Trotman.