Remarks on open surfaces
type: Item
instance of: scholarly article
title: Remarks on open surfaces
zbMATH Open document ID: 0783.32011
published in: Osaka Journal of Mathematics
publication date: 1993-01-16T00:00:00Z
review text: (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.
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 93860
zbMATH Keywords: pseudoconvex surface
zbMATH Keywords: Stein surface
zbMATH Keywords: biholomorphic mapping
author: G. K. Sankaran
reviewed by: Julian Ławrynowicz
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 93860