Compact quotients of large domains in complex projective space
type: Item
instance of: scholarly article
title: Compact quotients of large domains in complex projective space
zbMATH Open document ID: 0912.32020
author: Finnur Lárusson
publication date: 1998-09-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75277
review text: 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.
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 32J18
Mathematics Subject Classification ID: 32M99
zbMATH DE Number: 1202026
zbMATH Keywords: compact quotients
zbMATH Keywords: covering space
zbMATH Keywords: Schottky coverings
zbMATH Keywords: Blanchard manifolds
zbMATH Keywords: Kato manifolds
reviewed by: Gian Mario Besana
MaRDI profile type: Publication
OpenAlex ID: W1983964458
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DOI: 10.5802/AIF.1616
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1202026