Compactifications of \({\mathbb{C}}^ 3\). III (Q809234)

[For parts I and II see the author and \textit{M. Schneider}, Math. Ann. 280, No.1, 129-146 (1988; Zbl 0651.14025) and the author, Math. Ann. 283, No.1, 121-137 (1989; Zbl 0671.14020).] Let X be a smooth compact complex threefold with \(b_ 2(X)=1\) which contains \({\mathbb{C}}^ 3:=X/Y\) as the complement of an (irreducible) divisor Y. It is shown that X admits a non-constant meromorphic function f, which implies that X is Moishezon, according to the first paper of this series. (If no such f would exist, Y were not Moishezon. A contradiction is then derived from Kodaira's classification of surfaces, together with more recent results of Enoki to deal with the class VII with curves (because Y cannot be normal either).