Algebraic Subellipticity and Dominability of Blow-Ups of Affine Spaces

Abstract

Little is known about the behaviour of the Oka property of a complex manifold with respect to blowing up a submanifold. A manifold is of Class $\mathcal{A}$ if it is the complement of an algebraic subvariety of codimension at least 2 in an algebraic manifold that is Zariski-locally isomorphic to ${\mathbb{C}}^n$. A manifold of Class $\mathcal{A}$ is algebraically subelliptic and hence Oka, and a manifold of Class $\mathcal{A}$ blown up at finitely many points is of Class $\mathcal{A}$. Our main result is that a manifold of Class $\mathcal{A}$ blown up along an arbitrary algebraic submanifold (not necessarily connected) is algebraically subelliptic. For algebraic manifolds in general, we prove that strong algebraic dominability, a weakening of algebraic subellipticity, is preserved by an arbitrary blow-up with a smooth centre. We use the main result to confirm a prediction of Forster's famous conjecture that every open Riemann surface may be properly holomorphically embedded into ${\mathbb{C}}^2$.