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 $\C^{n}$.
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 $\C^{2}$.

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.

Full text: dvi.gz 26 k, dvi 58 k, ps.gz 222 k, pdf 145 k.

Home Page of DOCUMENTA MATHEMATICA