Oka-1 manifolds

Antonio Alarcon, Franc Forstneric

Published 2023-03-28Version 1

We introduce a new class of complex manifolds: Oka-1 manifolds. They are characterized by the property that holomorphic maps from any open Riemann surface satisfy the Runge approximation and the Weierstrass interpolation condition. We prove that every complex manifold which is dominable by tubes of complex lines is an Oka-1 manifold. In particular, a manifold dominable by $\mathbb C^n$ at most points is an Oka-1 manifold. This provides many examples of Oka-1 manifolds among compact algebraic surfaces, including all Kummer and all elliptic K3 surfaces. We also show that every compact rationally connected manifold is an Oka-1 manifold. The class of Oka-1 manifolds is invariant under Oka maps inducing a surjective homomorphism of fundamental groups; this includes holomorphic fibre bundles with connected Oka fibres. In another direction, we prove that every bordered Riemann surface admits a holomorphic map with dense image in any connected complex manifold.