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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2019, Issue 753

# Schottky groups acting on homogeneous rational manifolds

Christian Miebach
• Univ. Littoral Côte d’Opale, EA 2797 – LMPA – Laboratoire de mathématiques pures et appliquées Joseph Liouville, F-62228 Calais, France
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/ Karl Oeljeklaus
• Aix-Marseille Univ, CNRS, Centrale Marseille, I2M, UMR 7373, CMI, 39, rue F. Joliot-Curie, 13453 Marseille Cedex 13, France
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Published Online: 2016-12-14 | DOI: https://doi.org/10.1515/crelle-2016-0065

## Abstract

We systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of $\mathrm{SL}\left(2,ℂ\right)/\mathrm{\Gamma }$ for Γ a discrete free loxodromic subgroup of $\mathrm{SL}\left(2,ℂ\right)$, previously obtained by A. Guillot.

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Revised: 2016-10-25

Published Online: 2016-12-14

Published in Print: 2019-08-01

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR-10-BLAN-0118

The authors would like to thank A. T. Huckleberry and P. Heinzner for invitations to the Ruhr-Universität Bochum, Germany, where part of the work was done. The first author is grateful for the hospitality of the Institut de Mathématiques de Marseille (I2M) as well as for an invitation to the Fakultät für Mathematik of the Universität Duisburg-Essen by D. Greb. The second author is partially supported by the ANR project MNGNK, decision #ANR-10-BLAN-0118.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 753, Pages 23–56, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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