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 if it is the complement of an algebraic subvariety of codimension at least 2 in an algebraic manifold that is Zariski-locally isomorphic to . A manifold of Class is algebraically subelliptic and hence Oka, and a manifold of Class blown up at finitely many points is of Class . Our main result is that a manifold of Class 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 .
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Finnur Lárusson, Tuyen Trung Truong, Algebraic Subellipticity and Dominability of Blow-Ups of Affine Spaces. Doc. Math. 22 (2017), pp. 151–163DOI 10.4171/DM/562