Compact quotients of large domains in complex projective space

Lárusson, Finnur

Annales de l'Institut Fourier, Tome 48 (1998), p. 223-246 / Harvested from Numdam

Résumé

Nous étudions les variétés complexes compactes qui possèdent un revêtement qui soit un domaine dans l’espace projectif de dimension $n$ dont le complémentaire $E$ est un ensemble non vide de mesure de Hausdorff de dimension $(2n-2)$ égale à zéro. De telles variétés n’existent que si $n\ge 3$. Elles n’appartiennent pas à la classe ${\cal C}$, et par conséquent elles ne sont ni Kähler ni Moishezon, leur dimension de Kodaira est $-\infty $, leurs groupes fondamentaux sont des groupes de Klein généralisés et elles sont connexes par chaînes rationnelles. Nous considérons aussi les deux classes principales d’exemples connus en dimension 3 : les variétés de Blanchard, pour lesquels $E$ est une droite, et les revêtements généralisés de Schottky construits par Nori. Nous déterminons leur corps de fonctions méromorphes et décrivons les surfaces qu’elles contiennent.

Abstract

We study compact complex manifolds covered by a domain in $n$-dimensional projective space whose complement $E$ is non-empty with $(2n-2)$-dimensional Hausdorff measure zero. Such manifolds only exist for $n\ge 3$. They do not belong to the class ${\cal C}$, so they are neither Kähler nor Moishezon, their Kodaira dimension is $-\infty $, their fundamental groups are generalized Kleinian groups, and they are rationally chain connected. We also consider the two main classes of known 3-dimensional examples: Blanchard manifolds, for which $E$ is a line, and the generalized Schottky coverings constructed by Nori. We determine their function fields and describe the surfaces they contain.

Détail

Publié le : 1998-01-01

MR 99d:32035 | Zbl 0912.32020

DOI : https://doi.org/10.5802/aif.1616

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