Remarks on open surfaces (Q1199253)

(i) Let \(X\) be a compactifiable Stein surface. It is proved that \(X\) has an algebraic compactification. (ii) Let \(X\) be a compactifiable strongly pseudoconvex surface: for some algebraic complex variety \(M\) and its closed analytic subvariety \(D\), \(M\) is biholomorphically equivalent to \(M\backslash D\). Suppose that the Kodaira dimension of \(M\) is nonnegative. It is proved that \(M\) is determined up to birational equivalence. (iii) Suppose now that \(M_ j\) is an algebraic compactification of \(X_ j\) with boundary \(D_ j=M_ j\backslash X_ j\), that \(X_ j\) is strongly pseudoconvex, and that \(X\) is biholomorphically equivalent to \(X_ j\) for \(j=1,2\). After blowing up one point of \(D_ j\) if necessary, we can assume that \(M_ j\) is equipped with morphisms \(\pi_ j:M_ j\to B_ j\), where \(B_ j\) is a smooth complete curve of genus \(g_ j\) and the general fibre \(F_ j\) is a \(P^ 1\). We set \(\alpha_ j=D_ j\cdot f_ j\) and let \(\beta_ j\) be the number of fibers of \(\pi_ j\) contained in \(D_ j\). Suppose that \(\theta:X_ 1\to X_ 2\) is biholomorphic, but \(M_ 1\) and \(M_ 2\) are not biholomorphically equivalent. Then it is proved that \(g_ j\leq 1\) and, if \(g_ j=1\) then \(\beta_ j=0\) and \(\alpha_ j=1\) or 2. (iv) Suppose that \(X\) is strongly pseudoconvex but not Stein and that there exists an algebraic compactification \(M\). Then \(M\) is proved to be unique to bimeromorphic equivalence except, possibly, for some cases, which are listed in the paper.