DOCUMENTA MATHEMATICA, Vol. 22 (2017), 151-163

Finnur Lárusson and Tuyen Trung Truong

Algebraic Subellipticity and Dominability of Blow-Ups of Affine Spaces

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 $\A$ if it is the complement of an algebraic subvariety of codimension at least 2 in an algebraic manifold that is Zariski-locally isomorphic to $\Cn$. A manifold of Class $\A$ is algebraically subelliptic and hence Oka, and a manifold of Class $\A$ blown up at finitely many points is of Class $\A$. Our main result is that a manifold of Class $\A$ 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 $\C2$.

2010 Mathematics Subject Classification: Primary 14R10. Secondary 14E15, 14M20, 32S45, 32Q99

Keywords and Phrases: Blow-up, affine space, subelliptic, spray, dominable, strongly dominable, Oka manifold.

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