# Algebraic Subellipticity and Dominability of Blow-Ups of Affine Spaces

### Finnur Lárusson

School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia### Tuyen Trung Truong

School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia

## 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 $\Cal A$ if it is the complement of an algebraic subvariety of codimension at least 2 in an algebraic manifold that is Zariski-locally isomorphic to $\Bbb C^{n}$. A manifold of Class $\Cal A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\Cal A$ blown up at finitely many points is of Class $\Cal A$. Our main result is that a manifold of Class $\Cal A$ 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 $\Bbb C^{2}$.

## Cite this article

Finnur Lárusson, Tuyen Trung Truong, Algebraic Subellipticity and Dominability of Blow-Ups of Affine Spaces. Doc. Math. 22 (2017), pp. 151–163

DOI 10.4171/DM/562