Strictly pseudoconvex domains and algebraic varieties
The paper presents a few well-known facts about compact complex surfaces \(X\) which contain a smooth compact complex curve \(S\) with positive self-intersection. For example: \(X\) is algebraic (Theorem 4.2); there exist only finitely many irreducible compact complex curves in \(X\) which do not intersect \(S\) and their intersection matrix is negative definite (Theorem 4.1). \(X\) is rational if \(S\) is rational (Theorem 5.1). The considerations are based on the theorem of Grauert, that the holomorphic reduction of a strictly pseudoconvex domain is a Stein space and the holomorphic reduction map is proper with at most finitely many nontrivial fibers. The author claims to give a modified version of the Kodaira-embedding-theorem in higher dimensions (Theorem 4.3), which is not covered by the original proof. But for \(\dim X\geq 3\) the assumptions of Theorem 4.3 obviously imply the existence of an ample holomorphic line bundle on \(X\), in fact they are even stronger.
scientific article; zbMATH DE number 1158908