Compact quotients of large domains in complex projective space
Let \(\Omega\) be a domain in \(\mathbb{C}^n\) with a large complement. For example assume \(\Omega\) bounded. It is known that if \(\Omega\) is the Galois cover of a compact complex manifold \(M\) then \(M\) is projective algebraic. The paper under review is a contribution toward the study of \(n\)-dimensional compact complex manifolds \(M\) covered by domains \(\Omega\) in \(n\)-dimensional complex projective space with a small complement (more precisely such that the \(n-2\) Hausdorff measure of the complement vanishes). General properties of this class of manifolds are presented. Their dimension must be at least three. Their Kodaira dimension is always negative. They are neither Kähler nor Moishezon, indeed they are not even bimeromorphically equivalent to compact Kähler manifolds. Their fundamental group is a generalized Kleinian group. They are rationally chain connected and they contain many rational curves (one for every point \(P\) outside a countable union of proper submanifolds and for every tangent direction in a dense subset of the tangent space at \(P\)). The complement of the covering domain contains at least a line and it can be described in terms of rational curves contained in \(M\). Two classes of three-dimensional examples (Schottky and Blanchard manifolds) are studied in detail. The relationship of three-dimensional \(M\) with Kato manifolds is also described.
scientific article; zbMATH DE number 1202026
scientific article